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5,600 | 6,068 | Learning feed-forward one-shot learners
Luca Bertinetto?
University of Oxford
luca@robots.ox.ac.uk
Jo?o F. Henriques?
University of Oxford
joao@robots.ox.ac.uk
Philip H. S. Torr
University of Oxford
philip.torr@eng.ox.ac.uk
Jack Valmadre?
University of Oxford
jvlmdr@robots.ox.ac.uk
Andrea Vedaldi
University of Oxford
vedaldi@robots.ox.ac.uk
Abstract
One-shot learning is usually tackled by using generative models or discriminative
embeddings. Discriminative methods based on deep learning, which are very
effective in other learning scenarios, are ill-suited for one-shot learning as they
need large amounts of training data. In this paper, we propose a method to learn the
parameters of a deep model in one shot. We construct the learner as a second deep
network, called a learnet, which predicts the parameters of a pupil network from
a single exemplar. In this manner we obtain an efficient feed-forward one-shot
learner, trained end-to-end by minimizing a one-shot classification objective in
a learning to learn formulation. In order to make the construction feasible, we
propose a number of factorizations of the parameters of the pupil network. We
demonstrate encouraging results by learning characters from single exemplars in
Omniglot, and by tracking visual objects from a single initial exemplar in the Visual
Object Tracking benchmark.
1
Introduction
Deep learning methods have taken by storm areas such as computer vision, natural language processing and speech recognition. One of their key strengths is the ability to leverage large quantities of
labelled data and extract meaningful and powerful representations from it. However, this capability is
also one of their most significant limitations since using large datasets to train deep neural network is
not just an option, but a necessity. It is well known, in fact, that these models are prone to overfitting.
Thus, deep networks seem less useful when the goal is to learn a new concept on the fly, from a few or
even a single example as in one shot learning. These problems are usually tackled by using generative
models [18, 13] or, in a discriminative setting, using ad-hoc solutions such as exemplar support vector
machines (SVMs) [14]. Perhaps the most common discriminative approach to one-shot learning is to
learn off-line a deep embedding function and then to define on-line simple classification rules such as
nearest neighbors in the embedding space [5, 16]. However, computing an embedding is a far cry
from learning a model of the new object.
In this paper, we take a very different approach and ask whether we can induce, from a single
supervised example, a full, deep discriminative model to recognize other instances of the same object
class. Furthermore, we do not want our solution to require a lengthy optimization process, but to be
computable on-the-fly, efficiently and in one go. We formulate this problem as the one of learning a
deep neural network, called a learnet, that, given a single exemplar of a new object class, predicts the
parameters of a second network that can recognize other objects of the same type.
?
The first three authors contributed equally, and are listed in alphabetical order.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Our model has several elements of interest. Firstly, if we consider learning to be any process that maps
a set of images to the parameters of a model, then it can be seen as a ?learning to learn? approach.
Clearly, learning from a single exemplar is only possible given sufficient prior knowledge on the
learning domain. This prior knowledge is incorporated in the learnet in an off-line phase by solving
millions of small one-shot learning tasks and back-propagating errors end-to-end. Secondly, our
learnet provides a feed-forward learning algorithm that extracts from the available exemplar the final
model parameters in one go. This is different from iterative approaches such as exemplar SVMs or
complex inference processes in generative modeling. It also demonstrates that deep neural networks
can learn at the ?meta-level? of predicting filter parameters for a second network, which we consider
to be an interesting result in its own right. Thirdly, our method provides a competitive, efficient, and
practical way of performing one-shot learning using discriminative methods.
1.1
Related work
Our work is related to several others in the literature. However, we believe to be the first to look at
methods that can learn the parameters of complex discriminative models in one shot.
One-shot learning has been widely studied in the context of generative modeling, which unlike our
work is often not focused on solving discriminative tasks. One very recent example is by Rezende et
al. [18], which uses a recurrent spatial attention model to generate images, and learns by optimizing a
measure of reconstruction error using variational inference [9]. They demonstrate results by sampling
images of novel classes from this generative model, not by solving discriminative tasks. Another
notable work is by Lake et al. [13], which instead uses a probabilistic program as a generative model.
This model constructs written characters as compositions of pen strokes, so although more general
programs can be envisioned, they demonstrate it only on Optical Character Recognition (OCR)
applications.
A different approach to one-shot learning is to learn an embedding space, which is typically done
with a siamese network [2]. Given an exemplar of a novel category, classification is performed in
the embedding space by a simple rule such as nearest-neighbor. Training is usually performed by
classifying pairs according to distance [5], or by enforcing a distance ranking with a triplet loss [16].
Our work departs from the paradigms of generative modeling and similarity learning, instead predicting the parameters of a neural network from a single exemplar image. It can be seen as a network
that effectively ?learns to learn?, generalizing across tasks defined by different exemplars.
The idea of parameter prediction was, to our knowledge, first explored by Schmidhuber [20] in a
recurrent architecture with one network that modifies the weights of another. Parameter prediction
has also been used for zero-shot learning (as opposed to one-shot learning), which is the related
problem of learning a new object class without a single example image, based solely on a description
such as binary attributes or text. Whereas it is usually framed as a modality transfer problem and
solved through transfer learning [21], Noh et al. [15] recently employed parameter prediction to
induce the weights of an image classifier from text for the problem of visual question answering.
Denil et al. [4] investigated the redundancy of neural network parameters, showing that it is possible
to linearly predict as many as 95% of the parameters in a layer given the remaining 5%. This is a
vastly different proposition from ours, which is to predict all of the parameters of a layer given an
external exemplar image, and to do so non-linearly.
2
One-shot learning as dynamic parameter prediction
Since we consider one-shot learning as a discriminative task, our starting point is standard discriminative learning. It generally consists of finding the parameters W that minimize the average loss L of
a predictor function ?(x; W ), computed over a dataset of n samples xi and corresponding labels `i :
n
1X
L(?(xi ; W ), `i ).
min
W n
i=1
(1)
Unless the model space is very small, generalization also requires constraining the choice of model,
usually via regularization. However, in the extreme case in which the goal is to learn W from a single
exemplar z of the class of interest, called one-shot learning, even regularization may be insufficient
and additional prior information must be injected into the learning process. The main challenge in
2
siamese
siamese learnet
learnet
Figure 1: Our proposed architectures predict the parameters of a network from a single example,
replacing static convolutions (green) with dynamic convolutions (red). The siamese learnet predicts
the parameters of an embedding function that is applied to both inputs, whereas the single-stream
learnet predicts the parameters of a function that is applied to the other input. Linear layers are
denoted by ? and nonlinear layers by ?. Dashed connections represent parameter sharing.
discriminative one-shot learning is to find a mechanism to incorporate domain-specific information in
the learner, i.e. learning to learn. Another challenge, which is of practical importance in applications
of one-shot learning, is to avoid a lengthy optimization process such as eq. (1).
We propose to address both challenges by learning the parameters W of the predictor from a single
exemplar z using a meta-prediction process, i.e. a non-iterative feed-forward function ? that maps
(z; W 0 ) to W . Since in practice this function will be implemented using a deep neural network, we
call it a learnet. The learnet depends on the exemplar z, which is a single representative of the class of
interest, and contains parameters W 0 of its own. Learning to learn can now be posed as the problem of
optimizing the learnet meta-parameters W 0 using an objective function defined below. Furthermore,
the feed-forward learnet evaluation is much faster than solving the optimization problem (1).
In order to train the learnet, we require the latter to produce good predictors given any possible
exemplar z, which is empirically evaluated as an average over n training samples zi :
n
min
0
W
1X
L(?(xi ; ?(zi ; W 0 )), `i ).
n i=1
(2)
In this expression, the performance of the predictor extracted by the learnet from the exemplar zi is
assessed on a single ?validation? pair (xi , `i ), comprising another exemplar and its label `i . Hence,
the training data consists of triplets (xi , zi , `i ). Notice that the meaning of the label `i is subtly
different from eq. (1) since the class of interest changes depending on the exemplar zi : `i is positive
when xi and zi belong to the same class and negative otherwise. Triplets are sampled uniformly with
respect to these two cases. Importantly, the parameters of the original predictor ? of eq. (1) now
change dynamically with each exemplar zi .
Note that the training data is reminiscent of that of siamese networks [2], which also learn from
labeled sample pairs. However, siamese networks apply the same model ?(x; W ) with shared
weights W to both xi and zi , and compute their inner-product to produce a similarity score:
n
1X
L(h?(xi ; W ), ?(zi ; W )i, `i ).
min
W n
i=1
(3)
There are two key differences with our model. First, we treat xi and zi asymmetrically, which results
in a different objective function. Second, and most importantly, the output of ?(z; W 0 ) is used to
parametrize linear layers that determine the intermediate representations in the network ?. This is
significantly different to computing a single inner product in the last layer (eq. (3)).
Eq. (2) specifies the optimization objective of one-shot learning as dynamic parameter prediction.
By application of the chain rule, backpropagating derivatives through the computational blocks
of ?(x; W ) and ?(z; W 0 ) is no more difficult than through any other standard deep network.
Nevertheless, when we dive into concrete implementations of such models we face a peculiar
challenge, discussed next.
2.1
The challenge of naive parameter prediction
In order to analyse the practical difficulties of implementing a learnet, we will begin with one-shot
prediction of a fully-connected layer, as it is simpler to analyse. This is given by
y = W x + b,
3
(4)
?(?)
?
?
?
??
Figure 2: Factorized convolutional layer (eq. (8)). The channels of the input x are projected to the
factorized space by M (a 1 ? 1 convolution), the resulting channels are convolved independently
with a corresponding filter prediction from w(z), and finally projected back using M 0 .
given an input x ? Rd , output y ? Rk , weights W ? Rk?d and biases b ? Rk .
We now replace the weights and biases with their functional counterparts, w(z) and b(z), representing
two outputs of the learnet ?(z; W 0 ) given the exemplar z ? Rm as input (to avoid clutter, we omit
the implicit dependence on W 0 ):
y = w(z)x + b(z).
(5)
While eq. (5) seems to be a drop-in replacement for linear layers, careful analysis reveals that it scales
extremely poorly. The main cause is the unusually large output space of the learnet w : Rm ? Rk?d .
For a comparable number of input and output units in a linear layer (d ' k), the output space of the
learnet grows quadratically with the number of units.
While this may seem to be a concern only for large networks, it is actually extremely difficult also for
networks with few units. Consider a simple linear learnet w(z) = W 0 z. Even for a very small fullyconnected layer of only 100 units (d = k = 100), and an exemplar z with 100 features (m = 100),
the learnet already contains 1M parameters that must be learned. Overfitting and space and time
costs make learning such a regressor infeasible. Furthermore, reducing the number of features in the
exemplar can only achieve a small constant-size reduction on the total number of parameters. The
bottleneck is the quadratic size of the output space dk, not the size of the input space m.
2.2
Factorized linear layers
A simple way to reduce the size of the output space is to consider a factorized set of weights, by
replacing eq. (5) with:
y = M 0 diag (w(z)) M x + b(z).
(6)
0
The product M diag (w(z)) M can be seen as a factorized representation of the weights, analogous
to the Singular Value Decomposition. The matrix M ? Rd?d projects x into a space where the
elements of w(z) represent disentangled factors of variation. The second projection M 0 ? Rk?d
maps the result back from this space.
Both M and M 0 contain additional parameters to be learned, but they are modest in size compared to
the case discussed in sect. 2.1. Importantly, the one-shot branch w(z) now only has to predict a set of
diagonal elements (see eq. (6)), so its output space grows linearly with the number of units in the
layer (i.e. w(z): Rm ? Rd ).
2.3
Factorized convolutional layers
The factorization of eq. (6) can be generalized to convolutional layers as follows. Given an input
tensor x ? Rr?c?d , weights W ? Rf ?f ?d?k (where f is the filter support size), and biases b ? Rk ,
0
0
the output y ? Rr ?c ?k of the convolutional layer is given by
y = W ? x + b,
(7)
where ? denotes convolution, and the biases b are applied to each of the k channels.
Projections analogous to M and M 0 in eq. (6) can be incorporated in the filter bank in different ways
and it is not obvious which one to pick. Here we take the view that M and M 0 should disentangle
the feature channels (i.e. third dimension of x) so that the predicted filters w(z) can operate on each
channel independently. As such, we consider the following factorization:
y = M 0 ? w(z) ?d M ? x + b(z),
4
(8)
???
???
???
???
???
z
x
Predicted filters w(z)
???
Activations
Figure 3: The predicted filters and the output of a dynamic convolutional layer in a single-stream
learnet trained for the OCR task. Different exemplars z define different filters w(z). Applying the
filters of each exemplar to the same input x yields different responses. Best viewed in colour.
???
???
???
???
???
???
z
x
Predicted filters w(z)
Activations
Figure 4: The predicted filters and the output of a dynamic convolutional layer in a siamese learnet
trained for the object tracking task. Best viewed in colour.
where M ? R1?1?d?d , M 0 ? R1?1?d?k , and w(z) ? Rf ?f ?d . Convolution with subscript d
denotes independent filtering of d channels, i.e. each channel of x ?d y is simply the convolution of
the corresponding channel in x and y. In practice, this can be achieved with filter tensors that are
diagonal in the third and fourth dimensions, or using d filter groups [12], each group containing a
single filter. An illustration is given in fig. 2. The predicted filters w(z) can be interpreted as a filter
basis, as described in the supplementary material (sec. A).
Notice that, under this factorization, the number of elements to be predicted by the one-shot branch
w(z) is only f 2 d (the filter size f is typically very small, e.g. 3 or 5 [5, 23]). Without the factorization,
it would be f 2 dk (the number of elements of W in eq. (7)). Similarly to the case of fully-connected
layers (sect. 2.2), when d ' k this keeps the number of predicted elements from growing quadratically
with the number of channels, allowing them to grow only linearly.
Examples of filters that are predicted by learnets are shown in figs. 3 and 4. The resulting activations
confirm that the networks induced by different exemplars do indeed possess different internal
representations of the same input.
3
Experiments
We evaluate learnets against baseline one-shot architectures (sect. 3.1) on two one-shot learning
problems in Optical Character Recognition (OCR; sect. 3.2) and visual object tracking (sect. 3.3).
All experiments were performed using MatConvNet [22].
3.1
Architectures
As noted in sect. 2, the closest competitors to our method in discriminative one-shot learning are
embedding learning using siamese architectures. Therefore, we structure the experiments to compare
against this baseline. In particular, we choose to implement learnets using similar network topologies
for a fairer comparison.
The baseline siamese architecture comprises two parallel streams ?(x; W ) and ?(z; W ) composed
of a number of layers, such as convolution, max-pooling, and ReLU, sharing parameters W (fig. 1.a).
The outputs of the two streams are compared by a layer ?(?(x; W ), ?(z; W )) computing a measure
of similarity or dissimilarity. We consider in particular: the dot product ha, bi between vectors a and
b, the Euclidean distance ka ? bk, and the weighted l1 -norm kw a ? w bk1 where w is a vector
of learnable weights and the Hadamard product).
The first modification to the siamese baseline is to use a learnet to predict some of the intermediate
shared stream parameters (fig. 1.b). In this case W = ?(z; W 0 ) and the siamese architecture writes
?(?(x; ?(z; W 0 )), ?(z; ?(z; W 0 ))). Note that the siamese parameters are still the same in the two
5
Table 1: Error rate for character recognition in foreign alphabets (chance is 95%).
Inner-product (%) Euclidean dist. (%) Weighted `1 dist. (%)
Siamese (shared)
48.5
37.3
41.8
Siamese (unshared)
47.0
41.0
34.6
Siamese (unshared, fact.)
48.4
?
33.6
Siamese learnet (shared)
51.0
39.8
31.4
Learnet
43.7
36.7
28.6
Modified Hausdorff distance
43.2
streams, whereas the learnet is an entirely new subnetwork whose purpose is to map the exemplar
image to the shared weights. We call this model the siamese learnet.
The second modification is a single-stream learnet configuration, using only one stream ? of the
siamese architecture and predicting its parameter using the learnet ?. In this case, the comparison
block ? is reinterpreted as the last layer of the stream ? (fig. 1.c). Note that: i) the single predicted
stream and learnet are asymmetric and with different parameters and ii) the learnet predicts both the
final comparison layer parameters ? as well as intermediate filter parameters.
The single-stream learnet architecture can be understood to predict a discriminant function from one
example, and the siamese learnet architecture to predict an embedding function for the comparison
of two images. These two variants demonstrate the versatility of the dynamic convolutional layer
from eq. (6).
Finally, in order to ensure that any difference in performance is not simply due to the asymmetry of
the learnet architecture or to the induced filter factorizations (sect. 2.2 and sect. 2.3), we also compare
unshared siamese nets, which use distinct parameters for each stream, and factorized siamese nets,
where convolutions are replaced by factorized convolutions as in learnet.
3.2
Character recognition in foreign alphabets
This section describes our experiments in one-shot learning on OCR. For this, we use the Omniglot
dataset [13], which contains images of handwritten characters from 50 different alphabets. These
alphabets are divided into 30 background and 20 evaluation alphabets. The associated one-shot
learning problem is to develop a method for determining whether, given any single exemplar of a
character in an evaluation alphabet, any other image in that alphabet represents the same character or
not. Importantly, all methods are trained using only background alphabets and tested on the evaluation
alphabets.
Dataset and evaluation protocol. Character images are resized to 28 ? 28 pixels in order to be able
to explore efficiently several variants of the proposed architectures. There are exactly 20 sample
images for each character, and an average of 32 characters per alphabet. The dataset contains a total
of 19,280 images in the background alphabets and 13,180 in the evaluation alphabets.
Algorithms are evaluated on a series of recognition problems. Each recognition problem involves
identifying the image in a set of 20 that shows the same character as an exemplar image (there is
always exactly one match). All of the characters in a single problem belong to the same alphabet.
At test time, given a collection of characters (x1 , . . . , xm ), the function is evaluated on each pair
(z, xi ) and the candidate with the highest score is declared the match. In the case of the learnet
architectures, this can be interpreted as obtaining the parameters W = ?(z; W 0 ) and then evaluating
a static network ?(xi ; W ) for each xi .
Architecture. The baseline stream ? for the siamese, siamese learnet, and single-stream learnet
architecture consists of 3 convolutional layers, with 2 ? 2 max-pooling layers of stride 2 between
them. The filter sizes are 5 ? 5 ? 1 ? 16, 5 ? 5 ? 16 ? 64 and 4 ? 4 ? 64 ? 512. For both the siamese
learnet and the single-stream learnet, ? consists of the same layers as ?, except the number of outputs
is 1600 ? one for each element of the 64 predicted filters (of size 5 ? 5). To keep the experiments
simple, we only predict the parameters of one convolutional layer. We conducted cross-validation to
choose the predicted layer and found that the second convolutional layer yields the best results for
both of the proposed variants.
Siamese nets have previously been applied to this problem by Koch et al. [10] using much deeper
networks applied to images of size 105 ? 105. However, we have restricted this investigation to
relatively shallow networks to enable a thorough exploration of the parameter space. A more powerful
6
algorithm for one-shot learning, Hierarchical Bayesian Program Learning [13], is able to achieve
human-level performance. However, this approach involves computationally expensive inference at
test time, and leverages extra information at training time that describes the strokes drawn by the
human author.
Learning. Learning involves minimizing the objective function specific to each method (e.g. eq. (2)
for learnet and eq. (3) for siamese architectures) and uses stochastic gradient descent (SGD) in all
cases. As noted in sect. 2, the objective is obtained by sampling triplets (zi , xi , `i ) where exemplars
zi and xi are congruous (`i = +1) or incongruous (`i = ?1) with 50% probability. We consider
100,000 random pairs for training per epoch, and train for 60 epochs. We conducted a random
search to find the best hyper-parameters for each algorithm (initial learning rate and geometric decay,
standard deviation of Gaussian parameter initialization, and weight decay).
Results and discussion. Tab. 1 shows the classification error obtained using variants of each
architecture. A dash indicates a failure to converge given a large range of hyper-parameters. The two
learnet architectures combined with the weighted `1 distance are able to achieve significantly better
results than other methods. The best architecture reduced the error from 37.3% for a siamese network
with shared parameters to 28.6% for a single-stream learnet.
While the Euclidean distance gave the best results for siamese networks with shared parameters,
better results were achieved by learnets (and siamese networks with unshared parameters) using a
weighted `1 distance. In fact, none of the alternative architectures are able to achieve lower error
under the Euclidean distance than the shared siamese net. The dot product was, in general, less
effective than the other two metrics.
The introduction of the factorization in the convolutional layer might be expected to improve the
quality of the estimated model by reducing the number of parameters, or to worsen it by diminishing
the capacity of the hypothesis space. For this relatively simple task of character recognition, the
factorization did not seem to have a large effect.
3.3
Object tracking
The task of single-target object tracking requires to locate an object of interest in a sequence of video
frames. A video frame can be seen as a collection F = {w1 , . . . , wK } of image windows; then, in a
one-shot setting, given an exemplar z ? F1 of the object in the first frame F1 , the goal is to identify
the same window in the other frames F2 , . . . , FM .
Datasets. The method is trained using the ImageNet Large Scale Visual Recognition Challenge
2015 [19], with 3,862 videos totalling more than one million annotated frames. Instances of objects
of thirty different classes (mostly vehicles and animals) are annotated throughout each video with
bounding boxes. For tracking, instance labels are retained but object class labels are ignored. We use
90% of the videos for training, while the other 10% are held-out to monitor validation error during
network training. Testing uses the VOT 2015 benchmark [11].
Architecture. We experiment with siamese and siamese learnet architectures (fig. 1) where the
learnet ? predicts the parameters of the second (dynamic) convolutional layer of the siamese streams.
Each siamese stream has five convolutional layers and we test three variants of those: variant (A) has
the same configuration as AlexNet [12] but with stride 2 in the first layer, and variants (B) and (C)
reduce to 50% the number of filters in the first two convolutional layers and, respectively, to 25% and
12.5% the number of filters in the last layer.
Training. In order to train the architecture efficiently from many windows, the data is prepared
as follows. Given an object bounding box sampled at random, a crop z double the size of that is
extracted from the corresponding frame, padding with the average image color when needed. The
border is included in order to incorporate some visual context around the exemplar object. Next,
` ? {+1, ?1} is sampled at random with 75% probability of being positive. If ` = ?1, an image x
is extracted by choosing at random a frame that does not contain the object. Otherwise, a second
frame containing the same object and within 50 temporal samples of the first is selected at random.
From that, a patch x centered around the object and four times bigger is extracted. In this way, x
contains both subwindows that do and do not match z. Images z and x are resized to 127 ? 127 and
255 ? 255 pixels, respectively, and the triplet (z, x, `) is formed. All 127 ? 127 subwindows in x are
considered to not match z except for the central 2 ? 2 ones when ` = +1.
7
Table 2: Tracking accuracy and number of tracking failures in the VOT 2015 Benchmark, as reported
by the toolkit [11]. Architectures are grouped by size of the main network (see text). For each group,
the best entry for each column is in bold. We also report the scores of 5 recent trackers.
Method
Accuracy Failures
Siamese (?=B)
0.465
105
Siamese (?=B; unshared)
0.447
131
Siamese (?=B; factorized)
0.444
138
Siamese learnet (?=B; ?=A) 0.500
87
Siamese learnet (?=B; ?=B)
0.497
93
DAT [17]
0.442
113
SO-DLT [23]
0.540
108
Method
Accuracy Failures
Siamese (?=C)
0.466
120
Siamese (?=C; factorized)
0.435
132
Siamese learnet (?=C; ?=A) 0.483
105
Siamese learnet (?=C; ?=C)
0.491
106
DSST [3]
0.483
163
MEEM [24]
0.458
107
MUSTer [7]
0.471
132
All networks are trained from scratch using SGD for 50 epoch of 50,000 sample triplets (zi , xi , `i ).
The multiple windows contained in x are compared to z efficiently by making the comparison layer
? convolutional (fig. 1), accumulating a logistic loss across spatial locations. The same hyperparameters (learning rate of 10?2 geometrically decaying to 10?5 , weight decay of 0.005, and small
mini-batches of size 8) are used for all experiments, which we found to work well for both the baseline
and proposed architectures. The weights are initialized using the improved Xavier [6] method, and
we use batch normalization [8] after all linear layers.
Testing. Adopting the initial crop as exemplar, the object is sought in a new frame within a radius of
the previous position, proceeding sequentially. This is done by evaluating the pupil net convolutionally,
as well as searching at five possible scales in order to track the object through scale space. The
approach is described in more detail in Bertinetto et al. [1].
Results and discussion. Tab. 2 compares the methods in terms of the official metrics (accuracy and
number of failures) for the VOT 2015 benchmark [11]. The ranking plot produced by the VOT toolkit
is provided in the supplementary material (fig. B.1). From tab. 2, it can be observed that factorizing
the filters in the siamese architecture significantly diminishes its performance, but using a learnet to
predict the filters in the factorization recovers this gap and in fact achieves better performance than
the original siamese net. The performance of the learnet architectures is not adversely affected by
using the slimmer prediction networks B and C (with less channels).
An elementary tracker based on learnet compares favourably against recent tracking systems, which
make use of different features and online model update strategies: DAT [17], DSST [3], MEEM [24],
MUSTer [7] and SO-DLT [23]. SO-DLT in particular is a good example of direct adaptation of
standard batch deep learning methodology to online learning, as it uses SGD during tracking to
fine-tune an ensemble of deep convolutional networks. However, the online adaptation of the model
comes at a big computational cost and affects the speed of the method, which runs at 5 frames-persecond (FPS) on a GPU. Due to the feed-forward nature of our one-shot learnets, they can track
objects in real-time at framerates in excess of 60 FPS, while achieving less tracking failures. We
consider, however, that our implementation serves mostly as a proof-of-concept, using tracking as an
interesting demonstration of one-shot-learning, and is orthogonal to many technical improvements
found in the tracking literature [11].
4
Conclusions
In this work, we have shown that it is possible to obtain the parameters of a deep neural network
using a single, feed-forward prediction from a second network. This approach is desirable when
iterative methods are too slow, and when large sets of annotated training samples are not available. We
have demonstrated the feasibility of feed-forward parameter prediction in two demanding one-shot
learning tasks in OCR and visual tracking. Our results hint at a promising avenue of research in
?learning to learn? by solving millions of small discriminative problems in an offline phase. Possible
extensions include domain adaptation and sharing a single learnet between different pupil networks.
Acknowledgements
This research was supported by Apical Ltd. and ERC grants ERC-2012-AdG 321162-HELIOS,
HELIOS-DFR00200 and ?Integrated and Detailed Image Understanding? (EP/L024683/1).
8
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9
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5,601 | 6,069 | Mixed vine copulas as joint models
of spike counts and local field potentials
Arno Onken
Istituto Italiano di Tecnologia
38068 Rovereto (TN), Italy
arno.onken@iit.it
Stefano Panzeri
Istituto Italiano di Tecnologia
38068 Rovereto (TN), Italy
stefano.panzeri@iit.it
Abstract
Concurrent measurements of neural activity at multiple scales, sometimes performed with multimodal techniques, become increasingly important for studying
brain function. However, statistical methods for their concurrent analysis are currently lacking. Here we introduce such techniques in a framework based on vine
copulas with mixed margins to construct multivariate stochastic models. These
models can describe detailed mixed interactions between discrete variables such
as neural spike counts, and continuous variables such as local field potentials.
We propose efficient methods for likelihood calculation, inference, sampling and
mutual information estimation within this framework. We test our methods on simulated data and demonstrate applicability on mixed data generated by a biologically
realistic neural network. Our methods hold the promise to considerably improve
statistical analysis of neural data recorded simultaneously at different scales.
1
Introduction
The functions of the brain likely rely on the concerted interaction of its microscopic, mesoscopic
and macroscopic systems. Concurrent recordings of signals at different scales, such as simultaneous
measurements of field potential and single-cell spiking activity or other multimodal measures such as
concurrent electrophysiological and fMRI measures, are leading to rapid advances in understanding
brain dynamics [16]. Analysis of these concurrent data is complicated by the great difference in
nature (e.g. discrete vs. continuous) and signal-to-noise ratio of each type of neural signal. To
take full advantage of these data, flexible statistical models that take into account many variables
with different statistics and their dependencies are needed. Recently, construction of multivariate
statistical models based on the concept of copulas has attracted a lot of attention [9]. Intuitively,
a copula represents a particular relationship between a set of random variables that, together with
separate margin models of the individual elements can be used to construct a joint statistical model.
This approach has become an indispensable tool in economics, finance and risk management in
both theoretical and practical applications [9, 13, 11]. Yet, despite their promise, application to
neuroscience has been limited [10, 14, 19]. The copula approach is general and, in principle,
applicable to model mixed discrete and continuous statistics. Specific cases of mixed discrete and
continuous copula-based models with parametric distributions were recently applied in clinical
applications [24, 7]. Racine [17] proposed nonparametric mixed copula distributions based on
kernel density estimators. In most studies, however, the elements of the copula-based multivariate
distributions are all continuous [9, 13, 11]. A reason for this is that in the general case, likelihood
calculation has exponential complexity in the number of discrete elements, limiting the usefulness
of the models. In particular, these methods are impractical for likelihood-based estimation of
information-theoretic quantities which requires many likelihood evaluations. Smith and Khaled [23]
recently proposed a copula-based framework with quadratic complexity, but limited to fully discrete
distributions. For valuable applications in neuroscience settings, however, we need a framework
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
that can overcome these limitations and cope with elements (i.e. number of neurons, activity sites)
that have different statistical properties - some continuous and others discrete - while still allowing
efficient likelihood calculation.
Here, we develop a framework to accomplish these goals by means of vine copulas with mixed
discrete and continuous margins. We describe methods to make numeric model selection, parameter
fitting and sampling scale efficiently with the number of elements and apply these methods to estimate
information-theoretic quantities. To demonstrate our framework, we draw samples from mixed
models and simulate mixed activity in a biologically realistic neural network. We then apply our
methods to these data and show that our methods outperform corresponding mixed independent and
fully continuous models.
2
Mixed vine copulas
Our goal is to construct multivariate distributions with arbitrary mixed margins and a wide range of
possible dependence structures. To accomplish this goal, we apply an approach that individuates the
margin part and the dependence part. The dependence is represented by a copula. Briefly, a copula
is defined as a multivariate distribution function with support on the unit hypercube and uniform
margins [13]. We will denote multivariate random variables by X with elements Xi . We denote
the cumulative distribution function (CDF) of X by FX with margin CDFs Fi . For consistency of
notation, we will denote probability density functions as well as probability mass functions by fX
with margins fi .
2.1
Mixed copula-based models
The great strength of copulas is their utility for constructing and decomposing multivariate distributions. Sklar?s Theorem [21, 13] lays out the theoretical foundations for this. According to this
theorem, every CDF FX can be decomposed into margins F1 , . . . , Fd and a copula C such that
FX (x1 , . . . , xd ) = C(F1 (x1 ), . . . , Fd (xd ))
(1)
and, conversely, margins F1 , . . . , Fd , a copula C and Eq. 1 can be used to construct a CDF FX . In
this decomposition, C is unique on the range of X. Sklar?s Theorem holds for mixed discrete and
continuous distributions and thus provides a method to construct multivariate mixed distributions
based on CDFs of copulas and margins. The important point here is that the approach yields a
cumulative distribution function FX of a multivariate random variable X, not its likelihood fX
which we need for inference and other tasks (c.f. Section 2.5). Thus, we need to calculate the
likelihood fX based on the cumulative distribution function FX .
W.l.o.g., let X1 , . . . , Xn be discrete and Xn+1 , . . . , Xd be continuous. By calculating the mixed
derivative of Eq. 1, we obtain the probability density function of the mixed distribution of X:
X
X
fX (x1 , . . . , xd ) =
???
(?1)m1 +???+mn
m1 =0,1
mn =0,1
d
Y
C
(F1 (x1 ? m1 ), . . . , Fn (xn ? mn ), Fn+1 (xn+1 ), . . . , Fd (xd ))
fi (xi ).
?un+1 . . . ?ud
i=n+1
?
d?n
(2)
Note that the number of terms in the sum grows exponentially with the number of discrete variables.
In general, the exponential number of terms prevents us from a direct evaluation of this equation.
Nevertheless, we will see in the next section that we need to calculate the probability density function
for likelihood-based estimation of differential entropy and mutual information. Therefore, we need an
efficient way to calculate the probability density function that is tractable for many discrete variables.
We will introduce methods to accomplish this in Section 2.5.
2.2
Information estimation with copulas and mixed margins
For continuous
R as well as mixed multivariate distributions, differential entropy h(X) is defined as
h(X) = ? fX (x) log2 fX (x)dx, where fX is a multivariate density which can also have mixed
margins like the one in Eq. 2 [6, 20]. With this, the mutual information I(X; Y ) between two
2
multivariate random variables X and Y with potentially mixed margins is given by I(X; Y ) =
h(X) + h(Y ) ? h(X, Y ), where h(X, Y ) is the joint differential entropy of the joint distribution
(X, Y ) with joint density fX,Y [6, 20]. For high dimensional distributions, evaluation of the integral
over the support of fX is unfeasible. However, we can estimate the differential entropy and thereby
the mutual information by means of classical Monte Carlo (MC) estimation [18]. We express the
entropy as an expectation over fX and approximate the expectation by the empirical average by
producing a large number of samples x1 , . . . , xk from X:
k
X
ck := ? 1
h(X) = EfX [? log2 fX (X)] ? h
log (fX (xj ))
(3)
k j=1 2
ck converges almost surely to h(X). Moreover, we
By the strong law of large numbers, h
h i
ck ?
c
ck : Var h
can assess the convergence of hk by estimating the sample variance of h
2
Pk
1
k ?h(X)
c . With this estimate, the term hc
q
is approximately stanj=1 ? log2 (fX (xj )) ? hk
k+1
c
Var[h
k]
dard normal distributed, allowing us to obtain confidence intervals for our differential entropy
estimate [18]. This shows that there are two requisites for the MC procedure to estimate entropy and
mutual information for a mixed distribution: 1) an efficient sampling procedure to produce samples
xj from X, and 2) a tractable method for calculating the density fX (xj ). We will introduce the
former in Section 2.4 and the latter in Section 2.5. In the next section we will describe a copula
decomposition that makes these efficient methods possible.
2.3
Pair copula constructions
The number of available high-dimensional copula families is quite limited while there are an abundant number of bivariate copula families. The pair copula construction provides a flexible way to
construct higher-dimensional copulas from bivariate copulas [1]. The idea of pair copula models is to
factorize the multivariate distribution into conditional distributions and to describe these conditional
distributions by means of bivariate copulas modeling dependence of two variables at a time. Special
pair copula constructions, called regular vine copula structures, assume conditional independence
between specific elements of the distribution, allowing us to circumvent the curse of dimensionality in
likelihood evaluation and sampling. More specifically, a vine can be represented as a hierarchical set
of trees where each node corresponds to a conditional distribution function and each edge corresponds
to a pair copula. The nodes of the lowest tree are the unconditional distribution margins with empty
conditioning sets. Each tree in the hierarchy incorporates additional variables into the conditioning
sets by means of its pair copulas. The results of these couplings then form the nodes of the next
tree in the hierarchy, thus extending the conditioning sets from tree to tree. Here we focus on the
canonical vine or C-vine in which each tree in the hierarchy has a unique node that is connected to all
other nodes [1].
In this section, F (xi |xj1 , . . . , xjk ) denotes the conditional cumulative distribution function of Xi
given Xj1 , . . . , Xjk . In the C-vine, the multivariate model likelihood is factorized as follows [1]:
fX (x1 , . . . , xd ) =
d
Y
k=1
f (xk )
d?1
Y d?j
Y
cj,i+j|1,...,j?1 (F (xj |x1 , . . . , xj?1 ), F (xi+j |x1 , . . . , xj?1 ))
j=1 i=1
(4)
The C-vine is a good choice if there are outstanding variables with important dependencies to many
other variables [2]. Such situations are commonly encountered in electrophysiology recordings where
the same electrode might record a local field potential (LFP, acting as the outstanding variable) and
statistically dependent spikes from nearby neurons.
2.4
Sampling from mixed canonical vines
For a vine with mixed margins, we sample from the corresponding continuous vine and apply the
inversion method with the inverse of the margin cumulative distribution function to obtain mixed
discrete and continuous samples.
?C
In the following, ?u
denotes the partial derivative of the copula C with respect to its first argument
1
?C
and ?u2 denotes the partial derivative with respect to the second argument. For mixed C-vine
3
sampling, we take the algorithm for sampling from a continuous C-vine copula with uniform
margins [1] and extend it by means of the inversion method to attach arbitrary continuous and discrete
margins. The algorithm requires (d ? 2)(d ? 1)/2 + d cumulative distribution function evaluations:
1. Sample w1 , . . . , wd i.i.d. uniform on [0, 1].
2. v1,1 = w1 .
3. x1 = F1?1 (v1,1 ).
4. For i = 2, . . . , d:
(a) vi,1 = wi .
?1
(b) For k = i ? 1, i ? 2, . . . , 1 : vi,1 ? Fi|1,...,k
(vi,1 , vk,k ),
where Fi|1,...,k =
?Ck,i|1,...,k?1
.
?u1
(c) xi = Fi?1 (vi,1 ).
(d) If i < d then for j = 1, . . . , i ? 1 : vi,j+1 ? Fi|1,...,j (vi,j , vj,j ),
where Fi|1,...,j =
?Cj,i|1,...,j?1
.
?u1
5. The result is x1 , . . . , xd .
The algorithm has quadratic complexity and is thus applicable to estimate information-theoretic
quantities following the scheme outlined in Section 2.2.
2.5
Tractable algorithm for calculating mixed canonical vine densities
Panagiotelis et al. [15] introduced an algorithm for calculating the likelihood of specific discrete
pair-copula decompositions. Notably, this algorithm has quadratic complexity in the number of
elements in the multivariate distribution. Here, we generalize this algorithm to the mixed margins
case and apply it to the C-vine. We apply a dynamic programming approach and build the likelihood
in a bottom up fashion from vine level T0 to level Td . The algorithm has quadratic complexity
and computes the density of a C-vine with mixed discrete and continuous margins. We abbreviate
+
?
c
Fi|A
:= Fi|A
:= P (Xi ? xi |XA = xA ) and Fi|A
:= P (Xi ? xi ? 1|XA = xA ). We write
fi|A := f (Xi = xi |XA = xA ) if Xi is continuous and fi|A := P (Xi = xi |XA = xA ) if Xi is
?C
ab
a
b
discrete. Moreover ?a, b ? {+, ?, c} : Ci,j|A
:= Ci,j|A (Fi|A
, Fj|A
). ?u
is the partial derivative of
1
?C
the copula C with respect to its first argument and ?u2 is the partial derivative with respect to C?s
second argument. Consequently, for w ? {u, v} we write
ab
?Ci,j|A
?w
:=
?Ci,j|A
a
b
?w (Fi|A , Fj|A ).
1. Level T0 : For i = 1, . . . , d: evaluate fi = Fi+ ? Fi? if Xi is discrete and fi = fi (xi ) if Xi
is continuous.
2. Levels T1 , T2 , . . . , Td?1 : For t = 1, . . . , d ? 1 and i = t + 1, . . . , d: Let It = {1, . . . , t}.
Let Lt = {1, . . . , t ? 1} if t > 1, and Lt = ? if t = 1.
(a)
?
ab
?
?a, b ? {+, ?} : Ct,i|L
if Xt and Xi discrete,
?
t
?
ac
?
?Ct,i|L
?
ac
t
??a ? {+, ?} : C
if Xt discrete and Xi continuous,
t,i|Lt and
?u2
cb
Evaluate
?C
t,i|Lt
cb
?
?b ? {+, ?} : Ct,i|L
and ?u
if Xt continuous and Xi discrete,
?
?
t
1
? ?C cc
c
2 c
?
? t,i|Lt , ?Ct,i|Lt and ? Ct,i|Lt
if Xt and Xi continuous.
?u1
?u2
?u1 ?u2
(5)
(b) Evaluate
? if Xi discrete:
+
?
fi|It = Fi|I
? Fi|I
,
(6)
t
t
where
? if Xt discrete:
+
Fi|I
=
t
++
?+
Ct,i|L
? Ct,i|L
t
t
ft|Lt
4
,
?
Fi|I
=
t
+?
??
Ct,i|L
? Ct,i|L
t
t
ft|Lt
.
(7)
? if Xt continuous:
+
Fi|I
t
=
c+
?Ct,i|L
t
?u2
,
?
Fi|I
t
=
c?
?Ct,i|L
t
?u2
.
(8)
? if Xt discrete and Xi continuous:
c
Fi|I
=
t
+c
?c
Ct,i|L
? Ct,i|L
t
t
ft|Lt
, fi|It =
c
?Fi|I
t
?xi
=
+c
?Ct,i|L
t
?u1
?
?c
?Ct,i|L
t
!
?u1
fi|Lt
, (9)
ft|Lt
? if Xt continuous and Xi continuous:
c
Fi|I
t
3. The result is f1,...,d = f1
=
cc
?Ct,i|L
t
Qd
i=2
?u2
,
fi|It =
c
?Fi|I
t
?xi
=
cc
? 2 Ct,i|L
t
?u1 ?u2
fi|Lt ,
(10)
fi|1,...,i?1 .
Like the sampling algorithm in Section 2.4, the likelihood algorithm has quadratic complexity and
is thus applicable to estimate information-theoretic quantities following the scheme outlined in
Section 2.2.
2.6
Inference
We can apply maximum likelihood methods to estimate model parameters, because we can directly
calculate the full likelihood of the model - even for high dimensions - following the procedure
Pk
outlined in Section 2.5. Let L(?, ?1 , . . . , ?d ) = j=1 log fX (xj ; ?, ?1 , . . . , ?d ) denote the log
likelihood of the joint probability density function, where ? denotes the parameters of the chosen
copula family. We can now apply the so-called inference for margins (IFM) method to estimate the
parameters [11]. The idea of this method is to break the joint optimization of all parameters up into
Pk
smaller optimization problems. For i = 1, . . . , d, let Li (?i ) = j=1 log fi (xi,j ; ?i ) denote the sum
of log likelihoods of the marginal distribution fi (xi,j ; ?i ), where ?1 , . . . , ?d are the parameters of the
chosen family of margins. The method proceeds in two steps. In the first step, the margin likelihoods
ci = argmax{Li (?i )}. In the second step, the full
are maximized separately: ?i = 1, . . . , d : ?
?i
b = argmax{L(?, ?
c1 , . . . , ?
cd )}.
likelihood is maximized given the estimated margin parameters as ?
?
Each of the individual optimization problems can be solved by means of a general multivariate
optimization algorithm such as the trust-region-reflective algorithm [4]. Joe and Xu [11] showed
that the IFM estimator is asymptotically efficient. The method is particularly attractive if the ratio
of margin parameters to copula parameters is big. If the number of copula parameters is too big to
be estimated in a single joint optimization, then the complexity of the copula model can be reduced
by truncating the vine tree of the C-vine (truncated vine [1]). This corresponds to an independence
assumption for higher vine levels and the validity of this simplification should be confirmed [22]. The
families of margin and copula distributions can be selected using the Akaike information criterion
(AIC) [3]: Each combination of family selections is scored by means of its AIC value and then the
best combination is chosen.
3
Validation on artificial data
We validated our framework by sampling from mixed vine-based models of different dimensionality
and by evaluating performance of various alternative models. Fig. 1 illustrates a 3-dimensional
example vine-based model with two continuous margins and one discrete margin. In the top row, we
show the probability density functions of the 2-dimensional margins obtained by integrating over one
margin each. One can appreciate the mixed distribution from the step-wise changes in probability
density in margin 2 and the smooth changes in margins 1 and 3. The bottom row shows scatter plots
of 3-dimensional samples projected onto each pair of margins. The distributions of samples nicely
reflect the corresponding densities.
We drew samples from this and other mixed vine distributions and fitted various models to these
samples. For model selection, we used normal and gamma distributions as options for continuous
5
5
25
20
15
10
5
Margin 3
Margin 3
Margin 2
10
25
20
15
10
5
0
2
10
5
0
?2
?2
0
Margin 1
2
0
2
25
20
15
10
5
0
Margin 3
0
Margin 3
Margin 2
?2
?2
0
Margin 1
2
5
10
25
20
15
10
5
0
5
Margin 2
10
Figure 1: Characteristics of a 3D mixed vine example. Margin 1 is standard normal distributed,
margin 2 is Poisson distributed with mean 5 and margin 3 is gamma distributed with shape 2 and
scale 4. The pairwise copulas are Gaussian with parameter 0.5, Student with correlation 0.5 and 2
degrees of freedom and Clayton with parameter 5 for margin pairs (1,2), (1,3) and (2,3) respectively.
Top row: Probability density functions of 2D margins. The lighter the color the higher is the density.
Bottom row: 2D margin scatter plots of 300 samples.
margins, Poisson, binomial and negative binomial distributions as options for discrete margins, and
Gaussian, Student, Clayton and rotated (90? , 180? , 270? ) Clayton copula families as options for
pair copula constructions. To quantify the gain of using a vine-based mixed model instead of a
mixed independent model, we drew samples from the vine-based mixed model and calculated the
cross-validated likelihood ratio (LR) statistic for nested models as D = 2(log(Lvine ) ? log(Lind )),
where Lvine denotes the likelihood of separate test-set samples under the vine-based model and Lind
denotes the likelihood of the samples under the corresponding independent model.
Figure 2: Model fit and entropy of simulated vine samples. Ground truth models are mixed vines
of different dimensionality (range 2 to 6 shown as dark brown to light brown lines) with margins
and copulas up to the respective dimension. Margins 1 to 3 and associated pairwise copulas are the
same as in Fig. 1. Margin 4 is binomial distributed with N = 6 and p = 0.4, margin 5 is negative
binomial distributed with N = 6 and p = 0.4 and margin 6 is standard normal distributed. The
pairwise copulas are Clayton survival, independent and Clayton rotated 90? for margin pairs (1,4),
(2,4) and (3,4) respectively, and Clayton rotated 270? , independent, Gaussian with parameter 0.5 and
independent for margin pairs (1,5), (2,5), (3,5) and (4,5) respectively and independent, independent,
Gaussian with parameter 0.5, independent and Student with parameters 0.5 and 2 for margin pairs
(1,6), (2,6), (3,6), (4,6) and (5,6) respectively and with parameter 5 for all Clayton based copulas. (AC) Cross-validated LR statistic between the ground truth model and the mixed vine-based model (A),
independent model (B) or mixed Gaussian model (C). (D,E) Normalized entropy difference between
the ground truth model and the independent model (D) or fully continuous vine-based model (E).
Lines denote averages over 30 repetitions as a function of the number of samples. Shaded areas
denote standard error.
Fig. 2A shows the LR statistic between the ground truth and the best-fitting mixed vine-based model
as a function of the number of samples for different dimensionality. The statistics were low in all cases
but increased with increasing dimensionality. The gain as quantified by the LR statistic of using the
full mixed vine-based model instead of the independent model, on the other hand, was moderate for
the bivariate model (D < 0.5) while being substantial for the 6-dimensional model (D ? 7). Wilks?
LR test on non-cross-validated data was highly significant whenever we used at least 32 samples
6
(p < 0.01). We also evaluated the fit of the multivariate Gaussian copula with mixed margins which
is nested in our mixed vine-based models and obtained by restricting all pairwise copula families to
be Gaussian. The LR statistics indicated substantially better fit than for the independent model but
the statistics were below those of the mixed vine-based model for most tested dimensions (Fig. 2C).
Unfortunately, a vine-based mixed model and the corresponding best-fitting fully continuous vinebased model are not directly comparable in this way due to the different weighting of discrete and
continuous elements (i.e. mass vs. density). Nevertheless, in an actual application it is easy to
determine which margins are discrete and which margins are continuous. Appropriate discrete or
continuous margins can therefore be selected easily. To extend our comparison to fully continuous
vine-based models, we estimated entropies of the mixed vine-based model, the corresponding independent model and of the best-fitting fully continuous model. We calculated the entropy differences
between these models and normalized with the entropy of the mixed vine-based model. Fig. 2D
shows the normalized entropy difference between the mixed vine-based model and the independent
model. The relative results are similar to those of the likelihood ratio statistic (Fig. 2B) suggesting
that in this case the entropy comparison is indicative of the performance gain. In Fig. 2E, we plot
the normalized entropy difference between the mixed vine-based model and the best-fitting fully
continuous model. Overall, the normalized differences of these models were smaller than for the
independent model. Similarly to the independent model, though, we found increasing differences for
increasing dimensionality of the models. All in all, our results suggest that our framework can yield
substantial advantages in terms of goodness of fit and in terms of estimated entropy in particular for
high-dimensional problems.
4
Application to simulated network activity
To evaluate our framework in a typical neuroscience setting, we applied our mixed vine-based model
to a biologically realistic neural network model. We simulated network activity with the Virtual
Electrode Recording Tool for EXtracellular potentials (VERTEX) [25] with network parameters
as in VERTEX tutorial 2. Briefly, the model contained a total of 5000 neurons with 85% of those
cell models representing layer 2/3 pyramidal neurons and 15% representing basket interneurons.
The spiking dynamics followed an adaptive exponential model. To simulate two different stimulus
conditions, we used random input currents with different means. We presented each stimulus condition
an equal number of times (corresponding to 1/2 probability of occurrence of either stimulus). The
network generated network oscillations in both conditions. To simulate a typical recording situation,
we recorded LFPs with two randomly placed electrodes and collected spike counts from the four
neurons closest to those electrodes. For each input condition, we ran the network 128 times and
collected one 6-dimensional mixed vector with the LFPs (continuous) and spike counts (discrete)
collected in a 100 ms interval from each network run. We then fitted the full mixed vine-based model,
the mixed independent model and the fully continuous vine-based model to these data. Importantly,
we fitted separate models for each stimulus condition and varied the number of samples per stimulus
condition between 8 and 128. This allowed us to estimate mutual information following the procedure
outlined in Section 2.2.
Similarly to Figs. 2B,C, Fig. 3A depicts the LR statistic between the best-fitting mixed vine-based
model and the corresponding independent model or mixed Gaussian model. We found relatively
small statistics for all sample sizes (D < 1). Nevertheless, Wilks? LR test indicated highly significant
improvement whenever we used at least 64 samples (p < 0.01). To evaluate the importance of the
mixed vine-based model when performing an information-theoretic analysis of the network activity,
we estimated mutual information between the modeled network activity (LFP and spike counts) and
the two stimulus conditions. Fig. 3B shows mutual information estimates that we obtained based on
the mixed independent, mixed Gaussian, continuous vine-based and mixed vine-based models. The
mixed Gaussian model yielded information estimates that were close to those of the mixed-vine based
models. Estimates based on the independent model and fully continuous model, on the other hand,
were both substantially different (overestimating and underestimating information, respectively) from
estimates that we obtained from the mixed vine-based model. The latter model is the most faithful one
with the most accurate information estimates. The overestimation of the independent model suggests
that spike counts and LFPs carry partly redundant information. The big differences in information
estimates further indicate that it can be important to take mixed margins and dependencies into
account for estimating mutual information, even if the LR statistic is low.
7
Figure 3: Analysis of simulated neural network activity obtained from the VERTEX tool [25].
Data samples are formed by the average LFP within 200 ? 300 ms after simulation onset from two
randomly chosen electrodes and spike counts from the four neurons in closest proximity to those
electrodes. One simulation run provided one sample only. The network was simulated with two
different input conditions: Input currents following an Ornstein-Uhlenbeck process had a mean value
of 330 pA for the excitatory population and 190 pA for the inhibitory population in condition 1,
and 300 pA for the excitatory population and 40 pA for the inhibitory population in condition 2.
In both conditions, standard deviation was 90 pA for the excitatory population and 50 pA for the
inhibitory population. (A) LR statistic between the best-fitting mixed vine-based model and the
best-fitting mixed independent model (blue) or mixed Gaussian model (red) as a function of the
number of samples (i.e. number of simulations in each condition) averaged over stimulus conditions.
(B) Mutual information between the neural activity and the two input conditions estimated from the
mixed independent model (blue), mixed Gaussian model (red), continuous vine-based model (green)
or mixed vine-based model (black) as a function of the number of samples. Lines denote averages
over 30 repetitions. Shaded areas denote standard error.
5
Discussion
We developed a complete framework based on vine copulas for modeling multivariate data that are
partly discrete and partly continuous. Our framework includes methods for sampling, likelihood
calculation and inference. We combined these procedures to estimate entropy and mutual information
by means of MC integration. In particular, our methods provide the possibility to construct joint
statistical models of LFPs and spike counts. In a biologically realistic network simulation we demonstrated that our mixed vine-based model provides a fit that is better than that of the corresponding
independent model and showed that mutual information estimates of fully continuous and mixed
independent models can strongly differ even if the likelihood ratio statistic suggests otherwise. For
LFP and spike count data, a mixed model with detailed dependence structures can make full use of
all available statistical data. This also makes it possible to construct optimal Bayesian decoders for
inferring the presented stimulus from both LFPs and spike counts. Moreover, our model provides
the possibility to investigate the statistical dependencies between LFPs and spike counts. Contrary
to other analysis methods for analyzing mixed LFPs and spiking [12, 5] our framework follows a
purely data-driven approach. Even high-dimensional distributions can be fitted, because all inference
operations have quadratic complexity. However, entropy and MI estimation can be problematic,
because MC integration can become unfeasible for very high-dimensional problems. One possible
remedy is to use our models for maximum likelihood decoding and then estimate information based
on decoding performance [8]. We note that our models are based on pair-constructions and thus
cannot model arbitrary higher-order dependencies. We stress, however, that higher-order correlations
do occur in the vine tree and depend on both the vine-tree selection and the copula families. Thus,
selecting the right vine-tree and copula families can - to a limited extent - account for higher-order
correlations. In general, however, limited sample numbers make it difficult to reliably estimate higherorder correlations in real neuroscience applications. The parametric nature of our model framework
also makes it possible to introduce dependencies on external variables. Directions for future research
include applications to experimentally recorded data and detailed evaluation of observed dependency
structures.
Acknowledgments. This work was supported by the European Commission?s Horizon 2020 Programme (H2020-MSCA-IF-2014) under grant agreement number 659227 (?STOMMAC?).
8
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9
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5,602 | 607 | Recognition-based Segmentation
of On-line Hand-printed Words
M. Schenkel*, H. Weissman, I. Guyon, C. Nohl, D. Henderson
AT&T Bell Laboratories, Holmdel, NJ 07733
* Swiss Federal Institute of Technology, CH-8092 Zurich
Abstract
This paper reports on the performance of two methods for
recognition-based segmentation of strings of on-line hand-printed
capital Latin characters. The input strings consist of a timeordered sequence of X-Y coordinates, punctuated by pen-lifts. The
methods were designed to work in "run-on mode" where there is no
constraint on the spacing between characters. While both methods
use a neural network recognition engine and a graph-algorithmic
post-processor, their approaches to segmentation are quite different. The first method, which we call IN SEC (for input segmentation), uses a combination of heuristics to identify particular penlifts as tentative segmentation points. The second method, which
we call OUTSEC (for output segmentation), relies on the empirically trained recognition engine for both recognizing characters and
identifying relevant segmentation points.
1
INTRODUCTION
We address the problem of writer independent recognition of hand-printed words
from an 80,OOO-word English dictionary. Several levels of difficulty in the recognition
of hand-printed words are illustrated in figure 1. The examples were extracted from
our databases (table 1). Except in the cases of boxed or clearly spaced characters,
segmenting characters independently of the recognition process yields poor recognition performance. This has motivated us to explore recognition-based segmentation
techniques.
723
724
Schenkel, Weissman, Guyon, Nohl, and Henderson
Table 1: Databases used for training and testing. DB2 contains words one
to five letters long, but only four and five letter words are constrained to be legal
English words. DB3 contains legal English words of any length from an 80,000 word
dictionary.
uppercase
database
DBl
DB2
DBS
data
nature
boxed letters
short words
English words
pad
used
AT&T
Grid
Wacom
ffiJ[1g~
r~
2 ty.i
F!f.!r
L (:)rr:::rp
training
set size
9000
8000
-
test set
size
1500
1000
600
approx. #
of donors
250
400
25
(a) boxed
(b) spaced
(c) pen-lifts
(d) connected
Figure 1: Examples of styles that can be found in our databases:
(a) DB1; (b) DB2; (c), (d) DB2 and DB3 . The line thickness or darkness is
alternated at each pen-lift.
The basic principle of recognition-based segmentation is to present to the recognizer
many "tentative characters". The recognition scores ultimately determine the string
segmentation. We have investigated two different recognition-based segmentation
methods which differ in their definition of the tentative characters, but have very
similar recognition engines.
The data collection device provides pen trajectory information as a sequence of
(x, y) coordinates at regular time intervals (10-15 ms). We use a preprocessing
technique which preserves this information by keeping a finely sampled sequence of
feature vectors along the pen trajectory (Guyon et al. 1991, Weissman et al. 1992).
The recognizer is a Time Delay Neural Network (T DN N) (Lang and Hinton 1988,
Waibel et al. 1989, Guyon et al. 1991). There is one output per class, in this case
26 outputs, providing a score for all the capital letters of the Latin alphabet.
The critical step in the segmentation process is the postprocessing which disentangles various word hypotheses using the character recognition scores provided by the
TDN N. For this purpose, we use conventional dynamic programming algorithms.
In addition we use a dictionary that checks the solution and returns a list of similar legal words. The best word hypotheses, subject to this list, is again chosen by
dynamic programming algorithms.
Recognition-based segmentation relies on the recognizer to give low confidence
Recognition-based Segmentation of On-line Hand-printed Words
scores for wrong tentative characters corresponding to a segmentation mistake. Recognizers trained only on valid characters usually perform poorly on such a task.
We use "segmentation-driven training" techniques which allow the training of wrong
tentative characters, produced by the segmentation engine itself, as negative examples. This additional training has reduced our error rates by more than a factor of
two.
In section 2 we describe the INSEG method which uses tentative characters delineated by heuristic segmentation points. It is expected to be most appropriate for
hand-printed capital letters since nearly all writers separate these letters by pen-lifts.
This method was inspired by a similar technique used for Optical Character Recognition (OCR) (Burges et al. 1992). In section 3 we present an alternative method,
OUTSEG, which expects the recognition engine to learn empirically (learning by
examples) both to recognize characters and to identify relevant segmentation points.
This second method bears similarities with the OCR methods proposed by Matan
et al. (1991) or Keeler et al. (1991). In section 4 we compare the two methods and
present experimental results.
2
SEGMENTATION IN INPUT SPACE
Figure 2 shows the different steps of the IN SEG process. Module 1 is used to define
"tentative characters" delineated by "tentative cuts" (spaces or pen-lifts). The
tentative characters are then handed to module 2 which performs the preprocessing
and the scoring of the characters with a T DN N. The recognition results are then
gathered into an interpretation graph . In module 3 the best path through that
graph is found with the Viterbi algorithm .
stroke
detector
& grouper
pen input
~~~
qff
1-2 ,:::.(
"\.
0-2
2-3
1-3
tentative characters
preprocessor
&TDNN
u-,
R
0-2
Z
'-2
E
,-3
E
2-3
E
3-4
graph
best path
search
J
2-4
J
3-S
F
.!;!
ItR E EFIt
Figure 2: Processing steps of the IN SEG method.
725
726
Schenkel, Weissman, Guyon, Nohl, and Henderson
In figure 3 we show a simplified representation of an interpretation graph built by our
system. Each tentative character (denoted {i, j}) has a double index: the tentative
cut i at the character starting point and the tentative cut j at the character end
point. We denote by X {i, j} the node associated to the score of letter X for the
tentative character {i,j}. A path through the graph starts at a node X{O,.} and
ends at a node Y {., m}, where is the word starting point and m the last pen-lift.
In between, only transitions of the kind X{.,i} -+ Y{i,.} are allowed to prevent
character overlapping.
?
To avoid searching through too complex a graph, we need to perform some pruning.
The spatial relationship between strokes is used to discard unlikely tentative cuts.
For instance, strokes with a large horizontal overlap are bundled. The remaining
tentative characters are then grouped in different ways to form alternative tentative
characters. Tentative characters separated by a large horizontal spatial interval are
never considered for grouping.
Figure 3: Graph obtained with the input segmentation method.
The grey shading in each box indicates the recognition scores (the darker, the
stronger the recognition score and the higher the recognition confidence).
In table 2 we present the results obtained with the T DN N recognizer used by
Guyon et al. (1991), with 4 convolutional layers and 6,252 weights. Characters
are preprocessed individually, which provides the network with a fixed dimension
input.
3
SEGMENTATION IN OUTPUT SPACE
In contrast with IN SEC, the OUTSEC method does not rely on human designed
segmentation hints: the neural network learns both recognition and segmentation
features from examples.
Recognition-based Segmentation of On-line Hand-printed Words
Tentative characters are produced simply in that a window is swept over the input
sequence in small steps. At each step the content of the window is taken to be a
tentative character. Successive characters usually overlap considerably.
L
(X)
time (i)
1111111111111111111111111111111111111
012...
m
~
Figure 4: T DN N outputs of the OUTSEG system.
The grey curve indicates the best path through the graph, using duration modeling.
The word "LOOP" was correctly recognized in spite of the ligatures which prevent
segmentation on the basis of pen-lifts.
In figure 4, we show the outputs of our TDN N recognizer when the word "LOOP"
is processed. The main matrix is a simplified representation of our interpretation graph. Tentative character numbers i (i E {I, 2, ... , m}), run along the
time direction. Each column contains the scores of all possible interpretations
X (X E {A, B, C, ... , Z, nil}) of a given tentative character. The bottom line is the
nil interpretation score which approximates the probability that the present input
is not a character (meaningless character): P(nil{i}linput) 1- (P(A{i}linput) +
P(B{i}linput) + ... + P(Z{i} Iinput?
=
The connections between nodes reflect a model of character durations. A simple
way of enforcing duration is to allow only the following transitions:
X {i} ~ X {i + I},
nil{i} ~ nil{i+l},
X {i} ~ nil{i + I},
nil{i} ~ X {i + I},
where X stands for a certain letter. A character interpretation can be followed by
727
728
Schenkel, Weissman, Guyon, Nohl, and Henderson
the same interpretation but cannot be followed immediately by another character
interpretation: they must be separated by nil. This permits distinguishing between
letter duration and letter repetition (such as the double "0" in our example). The
best path in the graph is found by the Viterbi algorithm.
In fact, this simple pattern of connections corresponds to a Markov model of duration, with exponential decay. We implemented a slightly fancier model which
allows the generation of any duration distribution (Weissman et al. 1992) to help
prevent character omission or insertion. In our experiments, we selected two Poisson
distributions to model character and the nil-class duration respectively.
We use a T D N N recognizer with 3 layers and 10, 817 weights. The sequence of
recognition scores is obtained by sweeping the neural network over the input. Because of the convolutional structure of the T DN N, there are many identical computations between two successive calls of the recognizer and only about one sixth
of the network connections have to be reevaluated for each new tentative character.
As a consequence, although the OUTSEG system processes many more tentative
characters than the IN SEG system does, the overall computation time is about
the same.
4
COMPARISON OF RESULTS AND CONCLUSIONS
Table 2: Comparison of the performance of the two segmentation methods
using a TDN N recognizer.
II
on DB2
INSEG
OUTSEG
on DB3
INSEG
OUTSEG
Error without dictionary
% char.
% word
9
18
21
10
% char.
% word
8
33
11
48
II
Error with dictionary
% char.
% word
8.5
15
17
8
% char.
% word
13
5
7
21
I
We summarize in table 2 the results obtained with our two segmentation methods.
To complement the results obtained with database DB2, we used (without retraining) database DB3 as a control, containing words of any length from the English
dictionary. In our current versions, INSEG performs better than OUTSEG. The
OUTSEG method can handle connected letters (such as in the example of the word
"LOOP" in figure 4), while the INSEG method, which relies on pen lifts, cannot.
But, we discovered that very few people did not separate their characters by pen lifts
in the data we collected. On the other hand, an advantage of the IN SEG method
is that it can easily be used with recognizers other than the T DN N, whereas the
OUTSEG method relies heavily on the convolutional structure of the T DN N for
computational efficiency.
For comparison, we substituted two other neural network recognizers to the T DN N.
These networks use alternative input representations. The OCR - net was designed
for Optical Character Recognition (Le Cun et al. 1989) and uses pixel map inputs.
Recognition-based Segmentation of On-line Hand-printed Words
Its first layer performs local line orientation detection. The orientation - net has
an architecture similar to that of the OCR - net, but its first layer is removed
and local line orientation information, directly extracted from the pen trajectory,
is transmitted to the second layer (Weissbuch and Le Cun 1992). Without a dictionary, the OCR - net has an error rate more than twice that of the T DN N but the
orientation - net performs similarly. With dictionary the orientation - net has a
25% lower error rate than the T DN N. This improvement is attributed to better
second and third best recognition choices, which facilitates dictionary use.
Our best results to date (tables 3) were obtained with the INSEG method, using
two recognizers combined with a voting scheme: the T DN N and the orientationnet. For comparison purposes we mention the results obtained by a commercial
recognizer on the same data. One should notice that our dictionary is the same as
the one from which the data was drawn and is probably a larger dictionary than
the one used by the commercial system. Our results are substantially better than
those of the commercial system. On an absolute scale they are quite satisfactory if
we take into account that the test data was not cleaned at all and that more than
20% of the errors have been identified to be patterns written in cursive, misspelled
or totally illegible.
We expect the OUTSEG method to work best for cursive handwriting, which does
not exhibit trivial segmentation hints, but we do not have any direct evidence to
support this expectation as yet. Rumelhart (1992) had success with a version of
OUTSEG . Work is in progress to extend the capabilities of our systems to cursive
writing.
Table 3: Performance of our best system. For comparison, we mention in
parenthesis the performances obtained by a commercial recognizer on the same
data. The performance of the commercial system with dictionary (marked with a *)
are penalized because DB2 and DB3 include words not contained in its dictionary.
Method
DB2
DB3
Error without dictionary
% char.
% word
Error with dictionary
% char.
% word
7 (18)
6 (20)
7 (17*)
5 (18*)
13 (29)
23 (61)
10 (32*)
11 (49*)
Acknowledgments
We wish to thank the entire Neural Network group at Bell Labs Holmdel for their
supportive discussions. Helpful suggestions with the editing of this paper by L.
Jackel and B. Boser are gratefully acknowledged. We are grateful to Anne Weissbuch, Yann Le Cun and Jan Ben for giving us their Neural Networks to tryon
our IN SEG method. We are indebted to Howard Page for providing comparison
figures with the commercial recognizer. The experiments were performed with the
neural network simulators of B. Boser, Y. Le Cun and L. Bottou who we thank for
their help and advice.
729
730
Schenkel, Weissman, Guyon, Nohl, and Henderson
References
I. Guyon, P. Albrecht, Y. Le Cun, J . Denker and W. Hubbard. Design of a neural
network character recognizer for a touch terminal. Pattern Recognition, 24(2), 1991.
H. Weissman, M. Schenkel, I. Guyon, C. Nohl and D. Henderson. Recognitionbased Segmentation of On-line Run-on Handprinted Words: Input vs. Output
Segmentation. Submitted to Pattern Recognition, October 1992.
K. J. Lang and G . E. Hinton. A time delay neural network architecture for speech
recognition. Technical Report CMU-cs-88-152, Carnegie-Mellon University, Pittsburgh PA, 1988.
A. Waibel, T. Hanazawa, G. Hinton, K. Shikano and K. Lang. Phoneme recognition
using time-delay neural networks. IEEE Transactions on Acoustics, Speech and
Signal Processing, 37:328-339, March 1989.
C. J. C. Burges, O. Matan, Y. Le Cun, D. Denker, L. D. Jackel, C . E. Stenard,
C. R. Nohl and J. I. Ben. Shortest path segmentation: A method for training neural
networks to recognize character strings. In IJCNN'92, volume 3, Baltimore, 1992.
IEEE.
O. Matan, C. J . C. Burges, Y. Le Cun and J. Denker. Multi-digit recognition using
a Space Dispacement Neural Network. In J. E. Moody et al., editor, Advances in
Neural Information Processing Systems 4, Denver, 1992. Morgan Kaufmann.
J . Keeler, D. E. Rumelhart and W-K. Leow. Integrated segmentation and recognition of hand-printed numerals. In R. Lippmann et aI., editor, Advances in Neural
Information Processing Systems 3, pages 557-563, Denver, 1991. Morgan Kaufmann.
Y. Le Cun, L.D. Jackel, B. Boser, J .S. Denker, H.P. Graf, I. Guyon, D. Henderson,
R.E. Howard and W. Hubbard. Handwritten digit recognition: Application of
neural network chips and automatic learning. IEEE Communications Magazine,
pages 41-46, November 1989.
A. Weissbuch and Y. Le Cun. Private communication. 1992.
D. Rumelhart et al. Integrated segmentation and recognition of cursive handwriting.
In Third NEC symposium Computational Learning and Cognition, Princeton, New
Jersey, 1992 (to appear).
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5,603 | 6,070 | Asynchronous Parallel Greedy Coordinate Descent
Yang You ?, + XiangRu Lian?, + Ji Liu ? Hsiang-Fu Yu ?
Inderjit S. Dhillon ? James Demmel ? Cho-Jui Hsieh ?
+
?
?
equally contributed
University of California, Davis
University of Rochester
?
?
University of Texas, Austin
University of California, Berkeley
youyang@cs.berkeley.edu, xiangru@yandex.com, jliu@cs.rochester.edu
{rofuyu,inderjit}@cs.utexas.edu, demmel@eecs.berkeley.edu
chohsieh@cs.ucdavis.edu
Abstract
In this paper, we propose and study an Asynchronous parallel Greedy Coordinate
Descent (Asy-GCD) algorithm for minimizing a smooth function with bounded
constraints. At each iteration, workers asynchronously conduct greedy coordinate
descent updates on a block of variables. In the first part of the paper, we analyze the
theoretical behavior of Asy-GCD and prove a linear convergence rate. In the second
part, we develop an efficient kernel SVM solver based on Asy-GCD in the shared
memory multi-core setting. Since our algorithm is fully asynchronous?each core
does not need to idle and wait for the other cores?the resulting algorithm enjoys
good speedup and outperforms existing multi-core kernel SVM solvers including
asynchronous stochastic coordinate descent and multi-core LIBSVM.
1
Introduction
Asynchronous parallel optimization has recently become a popular way to speedup machine learning
algorithms using multiple processors. The key idea of asynchronous parallel optimization is to allow
machines work independently without waiting for the synchronization points. It has many successful
applications including linear SVM [13, 19], deep neural networks [7, 15], matrix completion [19, 31],
linear programming [26], and its theoretical behavior has been deeply studied in the past few
years [1, 9, 16].
The most widely used asynchronous optimization algorithms are stochastic gradient method (SG) [7,
9, 19] and coordinate descent (CD) [1, 13, 16], where the workers keep selecting a sample or a
variable randomly and conduct the corresponding update asynchronously. Although these stochastic
algorithms have been studied deeply, in some important machine learning problems a ?greedy?
approach can achieve much faster convergence speed. A very famous example is greedy coordinate
descent: instead of randomly choosing a variable, at each iteration the algorithm selects the most
important variable to update. If this selection step can be implemented efficiently, greedy coordinate
descent can often make bigger progress compared with stochastic coordinate descent, leading to a
faster convergence speed. For example, the decomposition method (a variant of greedy coordinate
descent) is widely known as best solver for kernel SVM [14, 21], which is implemented in LIBSVM
and SVMLight. Other successful applications can be found in [8, 11, 29].
In this paper, we study asynchronous greedy coordinate descent algorithm framework. The variable is
partitioned into subsets, and each worker asynchronously conducts greedy coordinate descent in one
of the blocks. To our knowledge, this is the first paper to present a theoretical analysis or practical
applications of this asynchronous parallel algorithm. In the first part of the paper, we formally define
the asynchronous greedy coordinate descent procedure, and prove a linear convergence rate under
mild assumption. In the second part of the paper, we discuss how to apply this algorithm to solve the
kernel SVM problem on multi-core machines. Our algorithm achieves linear speedup with number of
cores, and performs better than other multi-core SVM solvers.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
The rest of the paper is outlined as follows. The related work is discussed in Section 2. We propose
the asynchronous greedy coordinate descent algorithm in Section 3 and derive the convergence rate
in the same section. In Section 4 we show the details how to apply this algorithm for training kernel
SVM, and the experimental comparisons are presented in Section 5.
2
Related Work
Coordinate Descent. Coordinate descent (CD) has been extensively studied in the optimization
community [2], and has become widely used in machine learning. At each iteration, only one variable
is chosen and updated while all the other variables remain fixed. CD can be classified into stochastic
coordinate descent (SCD), cyclic coordinate descent (CCD) and greedy coordinate descent (GCD)
based on their variable selection scheme. In SCD, variables are chosen randomly based on some
distribution, and this simple approach has been successfully applied in solving many machine learning
problems [10, 25]. The theoretical analysis of SCD has been discussed in [18, 22]. Cyclic coordinate
descent updates variables in a cyclic order, and has also been applied to several applications [4, 30].
Greedy Coordinate Descent (GCD).The idea of GCD is to select a good, instead of random,
coordinate that can yield better reduction of objective function value. This can often be measured by
the magnitude of gradient, projected gradient (for constrained minimization) or proximal gradient
(for composite minimization). Since the variable is carefully selected, at each iteration GCD can
reduce objective function more than SCD or CCD, which leads to faster convergence in practice.
Unfortunately, selecting a variable with larger gradient is often time consuming, so one needs to
carefully organize the computation to avoid the overhead, and this is often problem dependent.
The most famous application of GCD is the decomposition method [14, 21] used in kernel SVM.
By exploiting the structure of quadratic programming, selecting the variable with largest gradient
magnitude can be done without any overhead; as a result GCD becomes the dominant technique in
solving kernel SVM, and is implemented in LIBSVM [5] and SVMLight [14]. There are also other
applications of GCD, such as non-negative matrix factorization [11], large-scale linear SVM [29],
and [8] proposed an approximate way to select variables in GCD. Recently, [20] proved an improved
convergence bound for greedy coordinate descent. We focus on parallelizing the GS-r rule in this
paper but our analysis can be potentially extended to the GS-q rule mentioned in that paper.
To the best of our knowledge, the only literature discussing how to parallelize GCD was in [23, 24].
A thread-greedy/block-greedy coordinate descent is a synchronized parallel GCD for L1 -regularized
empirical risk minimization. At an iteration, each thread randomly selects a block of coordinates
from a pre-partitioned block partition and proposes the best coordinate from this block along with its
increment (i.e., step size). Then all the threads are synchronized to perform the actual update to the
variables. However, the method can potentially diverge; indeed, this is mentioned in [23] about the
potential divergence when the number of threads is large. [24] establishes sub-linear convergence for
this algorithm.
Asynchronous Parallel Optimization Algorithms.In a synchronous algorithm each worker conducts local updates, and in the end of each round they have to stop and communicate to get the new
parameters. This is not efficient when scaling to large problem due to the curse of last reducer (all
the workers have to wait for the slowest one). In contrast, in asynchronous algorithms there is no
synchronization point, so the throughput will be much higher than a synchronized system. As a result,
many recent work focus on developing asynchronous parallel algorithms for machine learning as well
as providing theoretical guarantee for those algorithms [1, 7, 9, 13, 15, 16, 19, 28, 31].
In distributed systems, asynchronous algorithms are often implemented using the concept of parameter
servers [7, 15, 28]. In such setting, each machine asynchronously communicates with the server to
read or write the parameters. In our experiments, we focus on another multi-core shared memory
setting, where multiple cores in a single machine conduct updates independently and asynchronously,
and the communication is implicitly done by reading/writing to the parameters stored in the shared
memory space. This has been first discussed in [19] for the stochastic gradient method, and recently
proposed for parallelizing stochastic coordinate descent [13, 17].
This is the first work proposing an asynchronous greedy coordinate decent framework. The closest
work to ours is [17] for asynchronous stochastic coordinate descent (ASCD). In their algorithm, each
worker asynchronously conducts the following updates: (1) randomly select a variable (2) compute
the update and write to memory or server. In our AGCD algorithm, each worker will select the best
variable to update in a block, which leads to faster convergence speed. We also compare with ASCD
algorithm in the experimental results for solving the kernel SVM problem.
2
3
Asynchronous Greedy Coordinate Descent
We consider the following constrained minimization problem:
min
(1)
f (x),
x2?
where f is convex and smooth, ? ? RN is the constraint set, ? = ?1 ? ?2 ? ? ? ? ? ?N and each
?i , i = 1, 2, . . . , N is a closed subinterval of the real line.
Notation: We denote S to be the optimal solution set for (1) and PS (x), P? (x) to be the Euclidean
projection of x onto S, ?, respectively. We also denote f ? to be the optimal objective function value
for (1).
We propose the following Asynchronous parallel Greedy Coordinate Descent (Asy-GCD) for solving (1). Assume N coordinates are divided into n non-overlapping sets S1 [ . . . [ Sn . Let k be the
global counter of total number of updates. In Asy-GCD, each processor repeatedly runs the following
GCD updates:
? Randomly select a set Sk 2 {S1 , . . . , Sn } and pick the coordinate ik 2 Sk where the projected
gradient (defined in (2)) has largest absolute value.
? Update the parameter by
xk+1
P? (xk
rik f (xk )),
where is the step size.
Here the projected gradient defined by
r+
xk ) := xk
ik f (?
P? (xk
rik f (?
xk ))
(2)
is a measurement of optimality for each variable, where x
?k is current point stored in memory used
to calculate the update. The processors will run concurrently without synchronization. In order to
analyze Asy-GCD, we capture the system-wise global view in Algorithm 1.
Algorithm 1 Asynchronous Parallel Greedy Coordinate Descent (Asy-GCD)
Input: x0 2 ?, , K
Output: xK+1
1: Initialize k
0;
2: while k ? K do
3:
Choose Sk from {S1 , . . . , Sn } with equal probability;
4:
Pick ik = arg maxi2Sk kr+
x)k;
i f (?
5:
xk+1
P? (xk
rik f (?
xk ));
6:
k
k + 1;
7: end while
The update in the k th iteration is
xk+1
th
P? (xk
rik f (?
xk )),
where ik is the selected coordinate in k iteration, x
?k is the point used to calculate the gradient and
rik f (?
xk ) is a zero vector where the ik th coordinate is set to the corresponding coordinate of the
gradient of f at x
?k . Note that x
?k may not be equal to the current value of the optimization variable
xk due to asynchrony. Later in the theoretical analysis we will need to assume x
?k is close to xk using
the bounded delay assumption.
In the following we prove the convergence behavior of Asy-GCD. We first make some commonly
used assumptions:
Assumption 1.
1. (Bounded Delay) There is a set J(k) ? {k 1, . . . , k T } for each iteration k such that
X
x
?k := xk
(xj+1 xj ),
(3)
j2J(k)
where T is the upper bound of the staleness. In this ?inconsistent read? model, we assume
some of the latest T updates are not yet written back to memory. This is also used in some
previous papers [1, 17], and is more general than the ?consistent read? model that assumes
x
?k is equal to some previous iterate.
3
2. For simplicity, we assume each set Si , i 2 {1, . . . , n} has m coordinates.
3. (Lipschitzian Gradient) The gradient function of the objective rf (?) is Lipschitzian. That
is to say,
krf (x) rf (y)k ? Lkx yk 8x, 8y.
(4)
Under the Lipschitzian gradient assumption, we can define three more constants Lres , Ls and
Lmax . Define Lres to be the restricted Lipschitz constant satisfying the following inequality:
8i 2 {1, 2, ..., N } and t 2 R with x, x + tei 2 ?
(5)
Let ri be the operator calculating a zero vector where the ith coordinate is set to the ith
coordinate of the gradient. Define L(i) for i 2 {1, 2, . . . , N } as the minimum constant that
satisfies:
kri f (x) ri f (x + ?ei )k ? L(i) |?|.
(6)
Define Lmax := maxi2{1,...,N } L(i) . It can be seen that Lmax ? Lres ? L.
Let s be any positive integer bounded by N . Define Ls to be the minimal constant satisfying
the following inequality: 8x 2 ?, 8S ? {1, 2, ..., N } where |S| ? s:
P
P
rf (x) rf x + i2S ?i ei ? Ls
i2S ?i ei .
krf (x)
rf (x + ?ei )k ? Lres |?|,
4. (Global Error Bound) We assume that our objective f has the following property: when
= 3L1max , there exists a constant ? such that
PS (x)k 6 ?k?
x xk, 8x 2 ?.
(7)
?
?
Where x
? is defined by argminx0 2? hrf (x), x0 xi + 21 kx0 xk2 . This is satisfied by
strongly convex objectives and some weakly convex objectives. For example, it is proved
in [27] that the kernel SVM problem (9) satisfies the global error bound even when the
kernel is not strictly positive definite.
5. (Independence) All random variables in {Sk }k=0,1,??? ,K in Algorithm 1 are independent to
each other.
kx
We then have the following convergence result:
Theorem 2 (Convergence). Choose = 1/(3Lmax ) in Algorithm 1. Suppose n
upper bound for staleness T satisfies the following condition
p
nLmax
T (T + 1) 6
.
4eLres
Under Assumption 1, we have the following convergence rate for Algorithm 1:
?
?k
2Lmax b
?
E(f (xk ) f ) 6
1
(f (x0 ) f ? ).
L?2 n
where b is defined as
?
? 1
L2
p T
b=
+2
.
18 nLmax Lres
6 and that the
(8)
This theorem
p indicates a linear convergence rate under the global error bound and the condition
T 2 ? O( n). Since T is usually proportional to the total number cores involved in the computation,
this result suggests that one can have linear speedup as long as the total number of cores is smaller
than O(n1/4 ). Note that for n = N Algorithm 1 reduces to the standard asynchronous coordinate
descent algorithm (ASCD) and our result is essentially consistent with the one in [17], although they
use the optimally strong convexity assumption for f (?). The optimally strong convexity is a similar
condition to the global error bound assumption [32].
Here we briefly discuss the constants involved in the convergence rate. Using Gaussian kernel SVM on
covtype as a concrete sample, Lmax = 1 for Gaussian kernel, Lres is the maximum norm of columns
of kernel matrix (? 3.5), L is the 2-norm of Q (21.43 for covtype), and conditional number ? ? 1190.
As number of samples increased, the conditional number ? will become a dominant term, and this
also appears in the rate of serial greedy coordinate descent. In terms of speedup when increasing
L2T
number of threads (T ), although LT may grow, it only appears in b = ( 18pnLmax
+ 2) 1 , where
Lres
p
the first term inside b is usually small since there is a n in the demominator. Therefore, b ? 2 1 in
most cases, which means the convergence rate does not slow down too much when we increase T .
4
4
Application to Multi-core Kernel SVM
In this section, we demonstrate how to apply asynchronous parallel greedy coordinate descent to
solve kernel SVM [3, 6]. We follow the conventional notations for kernel SVM, where the variables
for the dual form are ? 2 Rn (instead of x in the previous section). Given training samples {ai }`i=1
with corresponding labels yi 2 {+1, 1}, kernel SVM solves the following quadratic minimization
problem:
?
1 T
minn
? Q? eT ? := f (?) s.t. 0 ? ? ? C,
(9)
?2R
2
where Q is an ` by ` symmetric matrix with Qij = yi yj K(ai , aj ) and K(ai , aj ) is the kernel
2
function. Gaussian kernel is a widely-used kernel function, where K(ai , aj ) = e kai aj k .
Greedy coordinate descent is the most popular way to solve kernel SVM. In the following, we first
introduce greedy coordinate descent for kernel SVM, and then discuss the detailed update rule and
implementation issues when applying our proposed Asy-GCD algorithm on multi-core machines.
4.1 Kernel SVM and greedy coordinate descent
When we apply coordinate descent to solve the dual form of kernel SVM (9), the one variable update
rule for any index i can be computed by:
?
i
= P[0, C] ?i
rfi (?)/Qii
(10)
?i
where P[0, C] is the projection to the interval [0, C] and the gradient is rfi (?) = (Q?)i 1. Note
that this update rule is slightly different from (2) by setting the step size to be = 1/Qii . For
quadratic functions this step size leads to faster convergence because i? obtained by (10) is the closed
form solution of
?
= arg min f (? + ei ),
and ei is the i-th indicator vector.
As in Algorithm 1, we choose the best coordinate based on the magnitude of projected gradient. In
this case, by definition
r+
P[0, C] ?i ri f (?) .
(11)
i f (?) = ?i
The success of GCD in solving kernel SVM is mainly due to the maintenance of the gradient
g := ri f (?) = (Q?)
1.
Consider the update rule (10): it requires O(`) time to compute (Q?)i , which is the cost for stochastic
coordinate descent or cyclic coordinate descent. However, in the following we show that GCD has
the same time complexity per update by using the trick of maintaining g during the whole procedure.
If g is available in memory, each element of the projected gradient (11) can be computed in O(1)
time, so selecting the variable with maximum magnitude of projected gradient only costs O(`) time.
The single variable update (10) can be computed in O(1) time. After the update ?i
?i + , the
g has to be updated by g
g + qi , where qi is the i-th column of Q. This also costs O(`) time.
Therefore, each GCD update only costs O(`) using this trick of maintaining g.
Therefore, for solving kernel SVM, GCD is faster than SCD and CCD since there is no additional
cost for selecting the best variable to update. Note that in the above discussion we assume Q can be
stored in memory. Unfortunately, this is not the case for large scale problems because Q is an ` by `
dense matrix, where ` can be millions. We will discuss how to deal with this issue in Section 4.3.
With the trick of maintaining g = Q? 1, the GCD for solving (9) can be summarized in Algorithm 2.
Algorithm 2 Greedy Coordinate Descent (GCD) for Dual Kernel SVM
1: Initial g = 1, ? = 0
2: For k = 1, 2, ? ? ?
3:
step 1: Pick i = arg maxi |r+
i f (?)| using g
4:
step 2: Compute i? by eq (10)
5:
step 3: g
g + ? qi
6:
step 4: ?i
?i + ?
5
(See eq (11))
4.2 Asynchronous greedy coordinate descent
When we have n threads in a multi-core shared memory machine, and the dual variables (or corresponding training samples) are partitioned into the same number of blocks:
S1 [ S2 [ ? ? ? [ Sn = {1, 2, ? ? ? , `} and Si \ Sj =
for all i, j.
Now we apply Asy-GCD algorithm to solve (9). For better memory allocation of kernel cache
(see Section 4.3), we bind each thread to a partition. The behavior of our algorithm still follows
Asy-GCD because the sequence of updates are asynchronously random. The algorithm is summarized
in Algorithm 3.
Algorithm 3 Asy-GCD for Dual Kernel SVM
1: Initial g = 1, ? = 0
2: Each thread t repeatedly performs the following updates in parallel:
3:
step 1: Pick i = arg maxi2St |r+
i f (?)| using g
4:
step 2: Compute i? by eq (10)
5:
step 3: For j = 1, 2, ? ? ? , `
6:
gj
gj + ? Qj,i using atomic update
7:
step 4: ?i
?i + ?
(See eq (11))
Note that each thread will read the `-dimensional vector g in step 2 and update g in step 3 in the
shared memory. For the read, we do not use any atomic operations. For the writes, we maintain the
correctness of g by atomic writes, otherwise some updates to g might be overwritten by others, and
the algorithm cannot converge to the optimal solution. Theorem 2, suggests a linear convergence rate
of our algorithm, and in the experimental results we will see the algorithm is much faster than the
widely used Asynchronous Stochastic Coordinate Descent (Asy-SCD) algorithm [17].
4.3 Implementation Issues
In addition to the main algorithm, there are some practical issues we need to handle in order to
make Asy-GCD algorithm scales to large-scale kernel SVM problems. Here we discuss these
implementation issues.
Kernel Caching.The main difficulty for scaling kernel SVM to large dataset is the memory requirement for storing the Q matrix, which takes O(`2 ) memory. In the GCD algorithm, step 2 (see eq (10))
requires a diagonal element of Q, which can be pre-computed and stored in memory. However, the
main difficulty is to conduct step 3, where a column of Q (denoted by qi )is needed. If qi is in the
memory, the algorithm only takes O(`) time; however, if qi is not in the memory, re-computing it
from scratch takes O(dn) time. As a result, how to maintain most important columns of Q in memory
is an important implementation issues in SVM software.
In LIBSVM, the user can specify the size of memory they want to use for storing columns of Q. The
columns of Q are stored in a linked-list in the memory, and when memory space is not enough the
Least Recent Used column will be kicked out (LRU technique).
In our implementation, instead of sharing the same LRU for all the cores, we create an individual
LRU for each core, and make the memory space used by a core in a contiguous memory space. As
a result, remote memory access will happen less in the NUMA system when there are more than 1
CPU in the same computer. Using this technique, our algorithm is able to scale up in a multi-socket
machine (see Figure 2).
Variable Partitioning.The theory of Asy-GCD allows any non-overlapping partition of the dual
variables. However, we observe a better partition that minimizes the between-cluster connections can
often lead to faster convergence. This idea has been used in a divide-and-conquer SVM algorithm [12],
and we use the same idea to obtain the partition. More specifically, we partition the data by running
kmeans algorithm on a subset of 20000 training samples to obtain cluster centers {cr }nr=1 , and then
assign each i to the nearest center: ?(i) = argminr kcr xi k. This steps can be easily parallelized,
and costs less than 3 seconds in all the datasets used in the experiments. Note that we include this
kmeans time in all our experimental comparisons.
5
Experimental Results
We conduct experiments to show that the proposed method Asy-GCD achieves good speedup in
parallelizing kernel SVM in multi-core systems. We consider three datasets: ijcnn1, covtype and
webspam (see Table 1 for detailed information). We follow the parameter settings in [12], where C
6
Table 1: Data statistics. ` is number of training samples, d is dimensionality, `t is number of testing
samples.
`
`t
d
C
ijcnn1
49,990
91,701
22 32 2
covtype
464,810 116,202 54 32 32
webspam 280,000 70,000 254 8 32
(a) ijcnn1 time vs obj
(b) webspam time vs obj
(c) covtype time vs obj
Figure 1: Comparison of Asy-GCD with 1?20 threads on ijcnn1, covtype and webspam datasets.
and are selected by cross validation. All the experiments are run on the same system with 20 CPUs
and 256GB memory, where the CPU has two sockets, each with 10 cores. We locate 64GB for kernel
caching for all the algorithms. In our algorithm, the 64GB will distribute to each core; for example,
for Asy-GCD with 20 cores, each core will have 3.2GB kernel cache.
We include the following algorithms/implementations into our comparison:
1. Asy-GCD: Our proposed method implemented by C++ with OpenMP. Note that the preprocessing time for computing the partition is included in all the timing results.
2. PSCD: We implement the asynchronous stochastic coordinate descent [17] approach for
solving kernel SVM. Instead of forming the whole kernel matrix in the beginning (which
cannot scale to all the dataset we are using), we use the same kernel caching technique as
Asy-GCD to scale up PSCD.
3. LIBSVM (OMP): In LIBSVM, there is an option to speedup the algorithm in multi-core environment using OpenMP (see http://www.csie.ntu.edu.tw/~cjlin/libsvm/
faq.html#f432). This approach uses multiple cores when computing a column of kernel
matrix (qi used in step 3 of Algorithm 2).
All the implementations are modified from LIBSVM (e.g., they share the similar LRU cache class),
so the comparison is very fair. We conduct the following two sets of experiments. Note that another
recent proposed DC-SVM solver [12] is currently not parallelizable; however, since it is a meta
algorithm and requires training a series of SVM problems, our algorithm can be naturally served as a
building block of DC-SVM.
5.1 Scaling with number of cores
In the first set of experiments, we test the speedup of our algorithm with varying number of cores.
The results are presented in Figure 1 and Figure 2. We have the following observations:
? Time vs obj (for 1, 2, 4, 10, 20 cores). From Fig. 1 (a)-(c), we observe that when we use
more CPU cores, the objective decreases faster.
? Cores vs speedup. From Fig. 2, we can observe that we got good strong scaling when we
increase the number of threads. Note that our computer has two sockets, each with 10 cores,
and our algorithm can often achieve 13-15 times speedup. This suggests our algorithm can
scale to multiple sockets in a Non-Uniform Memory Access (NUMA) system. Previous
asynchronous parallel algorithms such as HogWild [19] or PASSCoDe [13] often struggle
when scaling to multiple sockets.
5.2 Comparison with other methods
Now we compare the efficiency of our proposed algorithm with other multi-core parallel kernel SVM
solvers on real datasets in Figure 3. All the algorithms in this comparison are using 20 cores and
64GB memory space for kernel caching. Note that LIBSVM is solving the kernel SVM problem with
the bias term, so the objective function value is not showing in the figures.
We have the following observations:
7
(a) ijcnn1 cores vs speedup
(b) webspam cores vs speedup
(c) covtype cores vs speedup
Figure 2: The scalability of Asy-GCD with up to 20 threads.
(a) ijcnn1 time vs accuracy
(b) covtype time vs accuracy
(c) webspam time vs accuracy
(d) ijcnn1 time vs objective
(e) covtype time vs objective
(f) webspam time vs objective
Figure 3: Comparison among multi-core kernel SVM solvers. All the solvers use 20 cores and the
same amount of memory.
? Our algorithm achieves much faster convergence in terms of objective function value
compared with PSCD. This is not surprising because using the trick of maintaining g (see
details in Section 4) greedy approach can select the best variable to update, while stochastic
approach just chooses variables randomly. In terms of accuracy, PSCD is sometimes good in
the beginning, but converges very slowly to the best accuracy. For example, in covtype data
the accuracy of PSCD remains 93% after 4000 seconds, while our algorithm can achieve
95% accuracy after 1500 seconds.
? LIBSVM (OMP) is slower than our method. The main reason is that they only use multiple
cores when computing kernel values, so the computational power is wasted when the column
of kernel (qi ) needed is available in memory.
Conclusions In this paper, we propose an Asynchronous parallel Greedy Coordinate Descent (AsyGCD) algorithm, and prove a linear convergence rate under mild condition. We show our algorithm
is useful for parallelizing the greedy coordinate descent method for solving kernel SVM, and the
resulting algorithm is much faster than existing multi-core SVM solvers.
Acknowledgement XL and JL are supported by the NSF grant CNS-1548078. HFY and ISD
are supported by the NSF grants CCF-1320746, IIS-1546459 and CCF-1564000. YY and JD are
supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific
Computing Research, Applied Mathematics program under Award Number DE-SC0010200; by the
U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research
under Award Numbers DE-SC0008700 and AC02-05CH11231; by DARPA Award Number HR001112-2-0016, Intel, Google, HP, Huawei, LGE, Nokia, NVIDIA, Oracle and S Samsung, Mathworks
and Cray. CJH also thank the XSEDE and Nvidia support.
8
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5,604 | 6,071 | Generative Shape Models: Joint Text Recognition and
Segmentation with Very Little Training Data
Xinghua Lou, Ken Kansky, Wolfgang Lehrach, CC Laan
Vicarious FPC Inc., San Francisco, USA
xinghua,ken,wolfgang,cc@vicarious.com
Bhaskara Marthi, D. Scott Phoenix, Dileep George
Vicarious FPC Inc., San Francisco, USA
bhaskara,scott,dileep@vicarious.com
Abstract
Abstract: We demonstrate that a generative model for object shapes can achieve
state of the art results on challenging scene text recognition tasks, and with orders of magnitude fewer training images than required for competing discriminative methods. In addition to transcribing text from challenging images, our
method performs fine-grained instance segmentation of characters. We show that
our model is more robust to both affine transformations and non-affine deformations compared to previous approaches.
1
Introduction
Classic optical character recognition (OCR) tools focus on reading text from well-prepared scanned
documents. They perform poorly when used for reading text from images of real world scenes [1].
Scene text exhibits very strong variation in font, appearance, and deformation, and image quality
can be lowered by many factors, including noise, blur, illumination change and structured background. Fig. 1 shows some representative images from two major scene text datasets: International
Conference on Document Analysis and Recognition (ICDAR) 2013 and Street View Text (SVT).
Figure 1: Examples of text in real world scenes: ICDAR 2013 (left two columns) and SVT (right
two columns). Unlike classic OCR that handles well-prepared scanned documents, scene text recognition is difficult because of the strong variation in font, background, appearance, and distortion.
Despite these challenges, the machine learning and computer vision community have recently witnessed a surging interest in developing novel approaches for scene text recognition. This is driven
by numerous potential applications such as scene understanding for robotic control and augmented
reality, street sign reading for autonomous driving, and image feature extraction for large-scale image search. In this paper we present a novel approach for robust scene text recognition. Specifically,
we study the problem of text recognition in a cropped image that contains a single word, which is
usually the output from some text localization method (see [2] for a thorough review on this topic).
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Our core contribution is a novel generative shape model that shows strong generalization capabilities. Unlike many previous approaches that are based on discriminative models and trained on
millions of real world images, our generative model only requires hundreds of training images,
yet still effectively captures affine transformations and non-affine deformations. To cope with the
strong variation of fonts in real scenes, we also propose a greedy approach for selecting representative fonts from a large database of fonts. Finally, we introduce a word parsing model that is trained
using structured output learning.
We evaluated our approach on ICDAR 2013 and SVT and achieved state-of-the-art performance
despite using several orders of magnitude less of training data. Our results show that instead of
relying on a massive amount of supervision to train a discriminative model, a generative model
trained on uncluttered fonts with properly encoded invariance generalizes well to text in natural
images and is more interpretable.
2
Related Work
We only consider literature on recognizing scene text in English. There are two paradigms for
solving this problem: character detection followed by word parsing, and simultaneous character
detection and word parsing.
Character detection followed by word parsing is the more popular paradigm. Essentially, a character
detection method first finds candidate characters, and then a parsing model searches for the true
sequence of characters by optimizing some objective function. Many previous works following this
paradigm differ in the detection and parsing methods.
Character detection methods can be patch-based or bottom-up. Patch-based detection first finds
patches of (hopefully) single characters using over-segmentation [3] or the stroke width transformation [4], followed by running a character classifier on each patch. Bottom-up detection first creates
an image-level representation using engineered or learned features and then finds instances of characters by aggregating image-level evidence and searching for strong activations at every pixel. Many
different representations have been proposed such as Strokelets [5], convolutional neural networks
[6], region-based features [7], tree-structured deformable models [8] and simple shape template [9].
Both patch-based and bottom-up character detection have flawed localization because they cannot
provide accurate segmentation boundaries of the characters.
Unlike detection, word parsing methods in literature show strong similarity. They are generally
sequence models that utilize attributes of individual as well as adjacent candidate characters. They
differ in model order and inference techniques. For example, [10] considered the problem as a highorder Markov model in a Bayesian inference framework. A classic pairwise conditional random field
was also used by [8, 11, 4], and inference was carried out using message passing [11] and dynamic
programming [8, 4]. Acknowledging that a pairwise model cannot encode as useful features as a
high-order character n-gram, [3] proposed a patch-based sequence model that encodes up to 4thorder character n-grams and applied beam search to solve it.
A second paradigm is simultaneous character detection and word parsing, reading the text without an
explicit step for detecting the characters. For example, [12] proposed a graphical model that jointly
models the attributes, location, and class of characters as well as the language consistency of the
word they constitute. Inference was carried out using weighted finite-state transducers (WFSTs).
[13] took a drastically different approach: they used a lexicon of about 90k words to synthesize
about 8 million images of text, which were used to train a CNN that predicts a character at each
independent position. The main drawback of this ?all-in-one? approach is weak invariance and
insufficient robustness, since changes in any attribute such as spacing between characters may cause
the system to fail due to over-fitting to the training data.
3
Model
Our approach follows the first detection-parsing paradigm. First, candidate characters are detected
using a novel generative shape model trained on clean character images. Second, a parsing model is
used to infer the true word, and this parser is trained using max-margin structured output learning.
2
3.1
Generative Shape Model for Fonts
Unlike many vision problem such as distinguishing dogs from cats where many local discriminative
features can be informative, text in real scenes, printed or molded using some fonts, is not as easily
distinguishable from local features. For example, the curve ?^? at the bottom of ?O? also exists
in ?G?, ?U? and ?Q?. This special structure ?`? can be found in ?B?, ?E?, ?F?, ?H?, ?P? and ?R?.
Without a sense of the global structure, a naive accumulation of local features easily leads to false
detections in the presence of noise or when characters are printed tightly. We aim at building a model
that specifically accounts for the global structure, i.e. the entire shape of characters. Our model
is generative such that during testing time we obtain a segmentation together with classification,
making the final word parsing much easier due to better explaining-away.
Model Construction
During training we build a graph representation from rendered clean images of fonts, as shown in
Fig. 2. Since we primarily care about shape, the basic image-level feature representation relies only
on edges, making our model invariant to appearance such as color and texture. Specifically, given a
clean font image we use 16 oriented filters to detect edges, followed by local suppression that keeps
at most one edge orientation active per pixel (Fig. 2a). Then, ?landmark? features are generated by
selecting one edge at a time, suppressing any other edges within a fixed radius, and repeat (Fig. 2b).
We then create a pool variable centered around each landmark point such that it allows translation
pooling in a window around the landmark (Fig. 2b). To coordinate the the pool choices between
adjacent landmarks (thus the shape of the letter), we add ?lateral constraints? between neighboring
pairs of pools that lie on the same edge contour (blue dashed lines in Fig. 2c). All lateral constraints
are elastic, allowing for some degree of affine and non-affine deformation. This allows our model
to generalize to different variations observed in real images such as noise, aspect change, blur, etc.
In addition to contour laterals, we add lateral constraints between distant pairs of pixels (red dashed
lines in Fig. 2c) to further constrain the shapes this model can represent. These distant laterals are
greedily added one at a time, from shortest to longest, between pairs of features that with the current
constraints can deform more than ? times the deformation allowed by adding a direct constraint
between the features (typically ? ? 3).
Figure 2: Model construction process for our generative shape model for fonts. Given a clean
character images, we detect 16 oriented edge features at every pixel (a). We perform a sparsification
process that selects ?landmark? features from the dense edge map (b) and then add pooling window
(b). We then add lateral constraints to constrain the shape model (c). A factor graph representation
of our model is partially shown in (d). (best viewed in color)
Formally, our model can be viewed as a factor graph shown in Fig. 2d. Each pool variable centered
on a landmark feature is considered a random variable and is associated with unary factors corresponding to the translations of the landmark feature. Each lateral constraint is a pairwise factor. The
unary factors give positive scores when matching features are found in the test image. The pairwise
factor is parameterized with a single perturbation radius parameter, which is defined as the largest
allowed change in the relative position of features in the adjacent pools. This perturbation radius
forbids extreme deformation, giving ?? log probability if this lateral constraint is violated. During
testing, the state space of each random variable is the pooling window and lateral constraints are not
allowed to be violated.
During training, this model construction process is carried out independently for all letter images,
and each letter is rendered in multiple fonts.
3
Inference and Instance Detection The letter models can be considered to be tiling an input image
at all translations. Given a test image, finding all candidate character instances involves two steps: a
forward pass and backtracing.
The forward pass is a bottom-up procedure that accumulate evidence from the test image to compute the marginal distribution of the shape model at each pixel location, similar to an activation
heatmap. To speed-up the computation, we simplify our graph (Fig. 2c) into a minimum spanning
tree, computed with edge weights equal to the pixel distance between features. Moreover, we make
the pooling window as large as the entire image to avoid a scanning procedure. The marginals in
the tree can be computed exactly and quickly with a single iteration of belief propagation. After
non-maximum suppression, a few positions that have the strongest activation are selected for backtracing. This process is guaranteed to overestimate the true marginals in the original loopy graphical
model, so this forward pass admits some false positives. Such false positives occur more often when
the image has a tight character layout or a prominent texture or background.
Given the estimated positions of character instances, backtracing is performed in the original loopy
graph to further reduce false positives and to output a segmentation (by connecting the ?landmarks?)
of each instance in the test image. The backtracing procedure selects a single landmark feature,
constrains its position to one of the local maxima in its marginal from the forward pass, and then
performs MAP inference in the full loopy graph to estimate the positions of all other landmarks
in the model, which provides the segmentation. Because this inference is more accurate than the
forward pass, additional false positives can be pruned after backtracing.
In both the forward and backward pass, classic loopy belief propagation was sufficient.
Greedy Font Model Selection One challenge in scene text reading is covering the huge variation
of fonts in uncontrolled, real world images. It is not feasible to train on all fonts because it is
too computationally expensive and redundant. We resort to an automated greedy font selection
approach. Briefly, for some letter we render images for all fonts and then use the resulting images
to train shape models. These shape models are then tested on every other rendered image, yielding
a compatibility score (amount of matching ?landmark? features) between every pair of fonts of the
same letter. One font is considered representable by another if their compatibility score is greater
than a given threshold (=0.8). For each letter, we find and keep the fonts that can represent most
other fonts and remove it from the font candidate set together with all the fonts it represents. This
selection process is repeated until 90% of all fonts are represented. Usually the remaining 10% fonts
are non-typical and rare in real scenes.
3.2
Word Parsing using Structured Output Learning
Parsing Model Our generative shape models were trained independently on all font images. Therefore, no explaining-away is performed before parsing. The shape model shows high invariance and
sensitivity, yielding a rich list of candidate letters that contains many false positives. For example,
an image of letter ?E? may also trigger the following: ?F?, ?I?, ?L? and ?c?. Word parsing refers to
inferring the true word from this list of candidate letters.
Figure 3: Our parsing model represented as a high-order factor graph. Given a test image, the
shape model generates a list of candidate letters. A factor graph is created by adding edges between
hypothetical neighboring letters and considering these edges as random variables. Four types of
factors are defined: transition, smoothness, consistency, and singleton factors. The first two factors
characterize the likelihood of parsing path, while the latter two ensure valid parsing output.
Our parsing model can be represented as a high-order factor graph in Fig. 3. First, we build hypothetical edges between a candidate letter and every candidate letter on its right-hand side within
some distance. Two pseudo letters ?*? and ?#? are created, indicating start and end of the graph,
4
respectively. Edges are created from start to all possible head letters and similarly from end to
all possible tail letters. Each edge is considered as a binary random variable which, if activated,
indicates a pair of neighboring letters from the true word.
We define four types of factors. Transition factors (green, unary) describe the likelihood of a hypothetical pair of neighboring letters being true. Similarly, smoothness factors (blue, pairwise) describe
the likelihood of a triplet of consecutive letters. Two additional factors are added as constraints to
ensure valid output. Consistency factors (red, high-order) ensure that if any candidate letter has an
activated inward edge, it must have one activated outward edge. This is sometimes referred to as
?flow consistency?. Lastly, to satisfy the single word constraint, a singleton factor (purple, highorder) is added such that there must be a single activated edge from ?start?. Examples of these
factors are shown in Fig. 3.
Mathematically, assuming that potentials on the factors are provided, inferring the state of random
variables in the parsing factor graph is equivalent to solving the following optimization problem.
?
?
?
?
?
?
?X X
?
X X
T
T
S
S
? = arg maxz
z
?v (w )zv +
?u,v (w )zu zv
(1)
?
?
?
?
c?C u?In(c)
?c?C v?Out(c)
?
v?Out(c)
P
P
s.t.
?c ? C, u?In(c) zu = v?Out(c) zv ,
(2)
P
(3)
v?Out(?) zv = 1,
?c ? C, ?v ? Out(c), zv ? {0, 1}.
(4)
where, z = {zv } is the set of all binary random variables indexed by v; C is the set of all candidate
letters, and for candidate letter c in C, In(c) and Out(c) index the random variables that correspond
T
to the inward and outward edges of c, respectively; ?T
v (w ) is the potential of transition factor at v
S
T
S
(parameterized by weight vector w ) and ?u,v (w ) is the potential of smoothness factor from u to
v (parameterized by weight vector wS ); Constraints (2)?(4) ensure flow consistency, singleton, and
the binary nature of all random variables.
T
Parameter Learning Another issue is proper parameterization of the factor potentials, i.e. ?T
v (w )
S
S
and ?u,v (w ). Due to the complex nature of real world images, high dimensional parsing features
are required. For one example, consecutive letters of the true word are usually evenly spaced. For
another example, a character n-gram model can be used to resolve ambiguous letter detections and
improve parsing quality. We use Wikipedia as the source for building our character n-gram model.
S
T
S
Both ?T
v (w ) and ?u,v (w ) are linear models of some features and a weight vector. To learn the
best weight vector that directly maps the input-output dependency of the parsing factor graph, we
used the maximum-margin structured output learning paradigm [14].
Briefly, maximum-margin structured output learning attempts to learn a direct functional dependency between structured input and output by maximizing the margin between the compatibility
score of the ground truth solution and that of the second best solution. It is an extension to the classic support vector machine (SVM) paradigm. Usually, the compatibility score is a linear function of
some so-called joint feature vector (i.e. parsing features) and feature weights to be learned (i.e. wT
and wS here). We designed 18 parsing features, including the score of individual candidate letters,
color consistency between hypothetical neighboring pairs, alignment of hypothetical consecutive
triplets, and character n-grams up to third order.
Re-ranking Lastly, top scoring words from the second-order Viterbi algorithm are re-ranked using
statistical word frequencies from Wikipedia.
4
4.1
Experiments
Datasets
ICDAR ICDAR (?International Conference on Document Analysis and Recognition?) is a biannual
competition on text recognition. The ICDAR 2013 Robust Reading Competition was designed for
comparing scene text recognition approaches [1]. Unlike digital-born images like those used on the
5
web, real world image recognition is more challenging due to uncontrolled environmental and imaging conditions that result in strong variation in font, blur, noise, distortion, non-uniform appearance,
and background structure. We worked on two datasets: ICDAR 2013 Segmentation dataset and ICDAR 2013 Recognition dataset. In this experiment, we only consider letters, ignoring punctuations
and digits. All test images are cropped and each image contains only a single word, see examples in
Fig. 1.
SVT The Street View Text (SVT) dataset [15] was harvested from Google Street View. Image text in
this dataset exhibits high variability and often has low resolution. SVT provides a small lexicon and
was created for lexicon-driven word recognition. In our experiments, we did not restrict the setting
to a given small lexicon and instead used a general, large English lexicon. SVT does not contain
symbols other than letters.
4.2
Model Training
Training Generative Shape Model To ensure sufficient coverage of fonts, we obtained 492 fonts
from Google Fonts1 . Manual font selection is biased and inaccurate, and it is not feasible to train
on all fonts (492 fonts times 52 letters gives 25584 training images). After the proposed greedy
font selection process for all letters, we retained 776 unique training images in total (equivalent to
a compression rate of 3% if we would have trained on all fonts for all letters). Fig. 4 shows the
selected fonts for letter ?a? and ?A?, respectively.
Figure 4: Results of greedy font selection for letter ?a? and ?A?. Given a large font database of 492
fonts, this process leverages the representativeness of our generative shape model to significantly
reduce the number of training images required to cover all fonts.
Training Word Parsing Model Training the structured output prediction model is expensive in
terms of supervision because every training sample consists of many random variables, and the state
of every random variable has to be annotated (i.e. the entire parsing path). We prepared training data
for our parsing model automatically using the ICDAR 2013 Segmentation dataset [1] that provides
per-character segmentation of scene text. Briefly, we first detect characters and construct a parsing
graph for each image. We then find the true path in the parsing graph (i.e. a sequence of activated
random variables) by matching the detected characters to the ground truth segmentation. In total,
we used 630 images for training the parser using PyStruct2 .
Shape Model Invariance Study We studied the invariance of our model by testing on transformations of the training images. We considered scaling and rotation. For the former, our model
performs robust fitting when the scaling varies between 130% and 70%. For the later, the angle of
robust fitting is between -20 and +20 degrees.
4.3
Results and Comparison
Character Detection We first tested our shape model on the ICDAR 2013 Segmentation dataset.
Since this is pre-parsing and no explaining-away is performed, we specifically looked for high recall.
A detected letter and a true segmented letter is considered a match only when the letter classes match
and their segmentation masks strongly overlap with ? 0.8 IoU (intersection-over-union). Trained on
fonts selected from Google Fonts, we obtained a very high 95.0% recall, which is significantly better
than 68.7% by the best reported method on the dataset [1]. This attributes to the high invariance
encoded in our model from the lateral constraints. The generative nature of the model gives a
complete segmentation and classification instead of only letter classification (as most discriminative
models do). Fig. 5 shows some instances of letters detected by our model. They exhibit strong
variance in font and appearance. Note that two scales (?1, ?2) are used during testing.
1
2
https://www.google.com/fonts
https://pystruct.github.io
6
Figure 5: Examples of detected and segmented characters (?A?, ?E? and ?R?) from the ICDAR
2013 Segmentation dataset. Despite obvious differences in font, appearance, and imaging condition
and quality, our shape model shows high accuracy in localizing and segmenting them from the full
image. (best viewed in color and when zoomed in)
Word Parsing We compared our approach against top performing ones in the ICDAR 2013 Robust Reading Competition. Results are given in Table 4.3. Our model perform better than Google?s
PhotoOCR[3] with a margin of 2.3%. However, a more important message is that we achieved this
result using three orders of magnitude less training data: 1406 total images (776 letter font images
for training the shape models and 630 word images for training the parser) versus 5 million by
PhotoOCR. Two major factors attribute to our high efficiency. First, considering character detection, our model demonstrates strong generalization in practice. Data-intensive models like those in
PhotoOCR impose weaker structural priors and require significantly more supervision. Second, considering word parsing, our generative model solves recognition and segmentation together, allowing
the use of highly accurate parsing features. On the other hand, PhotoOCR?s neural-network based
character classifier is incapable of generating accurate character segmentation boundaries, making
the parsing quality bounded. Our observations on the SVT dataset are similar: using exactly the
same training data we achieved state-of-the-art 80.7% accuracy. Note that all reported results in our
experiments are case-sensitive. Fig. 6 demonstrates the robustness of our approach toward unusual
fonts, noise, blur, and distracting backgrounds.
Method
PicRead [1]
Deep Struc. Learn. [16]
PhotoOCR [3]
This paper
ICDAR
63.1%
81.8%
84.3%
86.2%
SVT
72.9%
71.7%
78.0%
80.7 %
Training Data Size
N/A
8,000,000 (synthetic)
7,900,000 (manually labeled + augmented)
1,406 (776 letter images + 630 word images)
Figure 6: Visualization of correctly parsed images from ICDAR (first two columns) and SVT (last
column) including per-character segmentation and parsing path. The numbers therein are local
potential values on the parsing factor graph. (best viewed in color and when zoomed in)
7
4.4
Further Analysis & Discussion
Failure Case Analysis Fig. 7 shows some typical failure cases for our system (left) and PhotoOCR
(right). Our system fails mostly when the image is severely corrupted by noise, blur, over-exposure,
or when the text is handwritten. PhotoOCR fails at some clean images where the text is easily
readable. This reflects the limited generalization of data-intensive models because of the diminishing
return of more training data. This comparison also shows that our approach is more interpretable:
we can quickly justify the failure reasons by viewing the letter segmentation boundaries overlaid on
the raw image. For example, over-exposure and blur cause edge features to drop out and thus fail
the shape model. On the contrary, it is not so straightforward to explain why a discriminative model
like PhotoOCR fails at some cases as shown in Fig. 7.
Figure 7: Examples of failure cases for our system and PhotoOCR. Typically our system fails when
the image is severely corrupted or contains handwriting. PhotoOCR is susceptible to failing at clean
images where the text is easily readable.
Language Model In our experiments, a language model plays two roles: in parsing as character ngram features and in re-ranking as word-level features. Ideally, a perfect perception system should
be able to recognize most text in ICDAR without the need for a language model. We turned off
the language model in our experiments and observed approximately a 15% performance drop. For
PhotoOCR in the same setting, the performance drop is more than 40%. This is due to the fact that
PhotoOCR?s core recognition model is a coarse understanding of the scene, and parsing is difficult
without the high quality character segmentation that our generative shape model provides.
Relation to Other Methods Here we discuss the connections and differences between our shape
model and two very popular vision models: deformable parts models (DPM) [17] and convolutional
neural networks (CNN) [18]. The first major distinction is that both DPM and CNN are discriminative while our model is generative. Only our model can generate segmentation boundaries without
any additional ad-hoc processing. Second, CNN does not model any global shape structure, depending solely on local discriminative features (usually in a hierarchical fashion) to perform classification. DPM accounts for some degree of global structure, as the relative positions of parts are
encoded in a star graph or tree structure. Our model imposes stronger global structure by using
short and long lateral constraints. Third, during model inference both CNN and DPM only perform
a forward pass, while ours also performs backtracing for accurate segmentation. Finally, regarding
invariance and generalization, we directly encode invariance into the model using the perturbation
radius in lateral constraints. This is proven very effective in capturing various deformations while
still maintaining the stability of the overall shape. Neither DPM nor CNN encode invariance directly
and instead rely on substantial data to learn model parameters.
5
Conclusion and Outlook
This paper presents a novel generative shape model for scene text recognition. Together with a
parser trained using structured output learning, the proposed approach achieved state-of-the-art performance on the ICDAR and SVT datasets, despite using orders of magnitude fewer training images
than many pure discriminative models require. This paper demonstrates that it is preferable to directly encode invariance and deformation priors in the form of lateral constraints. Following this
principle, even a non-hierarchical model like ours can outperform deep discriminative models.
In the future, we are interested in extending our model to a hierarchical version with reusable features. We are also interested in further improving the parsing model to account for missing edge
evidence due to blur and over-exposure.
8
References
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9
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5,605 | 6,072 | The Power of Adaptivity in Identifying Statistical
Alternatives
Kevin Jamieson, Daniel Haas, Ben Recht
University of California, Berkeley
Berkeley, CA 94720
{kjamieson,dhaas,brecht}@eecs.berkeley.edu
Abstract
This paper studies the trade-off between two different kinds of pure exploration:
breadth versus depth. We focus on the most biased coin problem, asking how
many total coin flips are required to identify a ?heavy? coin from an infinite bag
containing both ?heavy? coins with mean ?1 2 (0, 1), and ?light" coins with
mean ?0 2 (0, ?1 ), where heavy coins are drawn from the bag with proportion
? 2 (0, 1/2). When ?, ?0 , ?1 are unknown, the key difficulty of this problem lies in
distinguishing whether the two kinds of coins have very similar means, or whether
heavy coins are just extremely rare. While existing solutions to this problem require
some prior knowledge of the parameters ?0 , ?1 , ?, we propose an adaptive algorithm
that requires no such knowledge yet still obtains near-optimal sample complexity
guarantees. In contrast, we provide a lower bound showing that non-adaptive
strategies require at least quadratically more samples. In characterizing this gap
between adaptive and nonadaptive strategies, we make connections to anomaly
detection and prove lower bounds on the sample complexity of differentiating
between a single parametric distribution and a mixture of two such distributions.
1
Introduction
The trade-off between exploration and exploitation has been an ever-present trope in the online
learning literature. In contrast, this paper studies the trade-off between two different kinds of pure
exploration: breadth versus depth. Consider a bag that contains an infinite number of two kinds
of biased coins: ?heavy? coins with mean ?1 2 (0, 1) and ?light? coins with mean ?0 2 (0, ?1 ).
When a player picks a coin from the bag, with probability ? the coin is ?heavy? and with probability
(1 ?) the coin is ?light.? The player can flip any coin she picks from the bag as many times as she
wants, and the goal is to identify a heavy coin using as few total flips as possible. When ?, ?0 , ?1 are
unknown, the key difficulty of this problem lies in distinguishing whether the two kinds of coins have
very similar means, or whether heavy coins are just extremely rare. That is, how does one balance
flipping an individual coin many times to better estimate its mean against considering many new coins
to maximize the probability of observing a heavy one. Previous work has only proposed solutions
that rely on some or full knowledge ?, ?0 , ?1 , limiting their applicability. In this work we propose
the first algorithm that requires no knowledge of ?, ?0 , ?1 , is guaranteed to return a heavy coin with
probability at least 1
, and flips a total number of coins, in expectation, that nearly matches known
lower bounds. Moreover, our fully adaptive algorithm supports more general sub-Gaussian sources in
addition to just coins, and only ever has one ?coin? outside the bag at a given time, a constraint of
practical importance to some applications.
In addition, we connect the most biased coin problem to anomaly detection and prove novel lower
bounds on the difficulty of detecting the presence of a mixture versus just a single component of
a known family of distributions (e.g. X ? (1 ?)g?0 + ?g?1 versus X ? g? for some ?). We
show that in detecting the presence of a mixture distribution, there is a stark difference of difficulty
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
between when the underlying distribution parameters are known (e.g. ?, ?0 , ?1 ) and when they are
not. The most biased coin problem can be viewed as an online, adaptive mixture detection problem
where source distributions arrive one at a time that are either g?0 with probability (1 ?) or g?1
with probability ? (e.g. null or anomolous) and the player adaptively chooses how many samples
to take from each distribution (to increase the signal-to-noise ratio) with the goal of identifying an
anomolous distribution f?1 using as few total number of samples as possible. This work draws a
contrast between the power of an adaptive versus non-adaptive (e.g. taking the same number of
samples each time) approaches to this problem, specifically when ?, ?0 , ?1 are unknown.
1.1
Motivation and Related Work for the Most Biased Coin Problem
The most biased coin problem characterizes the inherent difficulty of real-world problems including
anomaly and intrusion detection and discovery of vacant frequencies in the radio spectrum. Our
interest in the problem stemmed from automated hiring of crowd workers: data labeling for machine
learning applications is often performed by humans, and recent work in the crowdsourcing literature
accelerates labeling by organizing workers into pools of labelers and paying them to wait for incoming
data [4, 12]. Workers hired on marketplaces such as Amazon?s Mechanical Turk [16] vary widely in
skill, and identifying high-quality workers as quickly as possible is an important challenge. We can
model each worker?s performance (e.g. accuracy or speed) as a random variable so that selecting
a good worker is equivalent to identifying a worker with a high mean. Since we do not observe a
worker?s expected performance directly, we must give them tasks from which we estimate it (like
repeatedly flipping a biased coin). Arlotto et al. [3] proposed a strategy with some guarantees for a
related problem but did not characterize the sample complexity of the problem, the focus of our work.
The most biased coin problem was first proposed by Chandrasekaran and Karp [8]. In that work,
it was shown that if ?, ?0 , ?1 were known then there exists an algorithm based on the sequential
probability ratio test (SPRT) that is optimal in that it minimizes the expected number of total flips to
find a ?heavy? coin whose posterior probability of being heavy is at least 1
, and the expected
sample complexity of this algorithm was upper-bounded by
?
?
??
16
1 ?
(1 ?)(1
)
+ log
.
(1)
(?1 ?0 )2
?
?
However, the practicality of the proposed algorithm is severely limited as it relies critically on
knowing ?, ?0 , and ?1 exactly. In addition, the algorithm returns to coins it has previously flipped
and thus requires more than one coin to be outside the bag at a time, ruling out some applications.
Malloy et al. [15] addressed some of the shortcomings of [9] (a preprint of [8]) by considering both
an alternative SPRT procedure and a sequential thresholding procedure. Both of these proposed
algorithms only ever have one coin out of the bag at a time. However, the former requires knowledge
of all relevant parameters ?, ?0 , ?1 , and the latter requires knowledge of ?, ?0 . Moreover, these results
are only presented for the asymptotic case where ! 0.
The most biased coin problem can be viewed through the lens of multi-armed bandits. In the
best-arm identification problem, the player has access to K distributions (arms) such that if arm
i 2 [K] is sampled (pulled), an iid random variable with mean ?i is observed; the objective is to
identify the arm associated with the highest mean with probability at least 1
using as few pulls
as possible (see [14] for a short survey). In the infinite armed bandit problem, the player is not
confined to K arms but an infinite reservoir of arms such that a draw from this reservoir results in
an arm with a mean ? drawn from some distribution; the objective is to identify the highest mean
possible after n total pulls for any n > 0 with probability 1
(see [7]). The most biased coin
problem is an instance of this latter game with the arm reservoir distribution of means ? defined as
P(? ?1 ?) = ?1? 0 + (1 ?)1? ?1 ?0 for all ?. Previous work has focused on an alternative
arm distribution reservoir that satisfies E? ? P(? ?? ?) ? E 0 ? for some ?? 2 [0, 1] where
E, E 0 are constants and is known [5, 21, 6, 7]. Because neither arm distribution reservoir can be
written in terms of the other, neither work subsumes the other. Note that one can always apply an
algorithm designed for the infinite armed bandit problem to any finite K-armed bandit problem by
defining the arm reservoir as placing a uniform distribution over the K arms. This is appealing when
K is very large and one wishes to guarantee nontrivial performance when the number of pulls is
much less than K 1 . The most biased problem is a special case of the K-armed reservoir distribution
1
where one arm has mean ?1 and K 1 arms have mean ?0 with ? = K
.
1
All algorithms for K-armed bandit problem known to these authors begins by sampling each arm once so
that until the number of pulls exceeds K, performance is no better than random selection.
2
Given that [8] and [15] are provably optimal algorithms for the most biased coin problem given
knowledge of ?, ?0 , ?1 , it is natural to consider a procedure that first estimates these unknown
parameters first and then uses these estimates in the algorithms of [8] or [15]. Indeed, in the parameterized arm reservoir setting discussed above, this is exactly what Carpentier and Valko [7]
propose to do, suggesting a particular estimator for given a lower bound b ? . They show that
this estimator is sufficient to obtain the same sample complexity result up to log factors as when
was known. Sadly, through upper and lower bounds we show that for the most biased coin problem
this estimate-then-explore approach requires quadratically more flips than our proposed algorithm
that adapts to these unknown parameters. Specifically, we show that when ?1 ?0 is sufficiently
small one cannot use a static estimation step to determine whether ? = 0 or ? > 0 unless a number
of samples quadratic in the optimal sample complexity are taken.
Our contributions to the most biased coin problem include a novel algorithm that never has more
than one coin outside the bag at a time, has no knowledge of the distribution parameters, supports
distributions on [0, 1] rather than just ?coins,? and comes within log factors of the known informationtheoretic lower bound and Equation 1 which is achieved by an algorithm that knows the parameters.
See Table 1 for an overview of the upper and lower bounds proved in this work for this problem.
We believe that our algorithm is the first solution to the most biased coin problem that does not
require prior knowledge of the problem parameters and that the same approach can be reworked to
solve more general instances of the infinite-armed bandit problem, including the -parameterized
and K-armed reservoir cases described of above. Finally, if an algorithm is desired for arbitrary arm
reservoir distributions, this work rules out an estimate-then-explore approach.
1.2
Problem Statement
Let ? 2 ? index a family of single-parameter probability density functions g? and fix ?0 , ?1 2 ?,
? 2 [0, 1/2]. For any ? 2 ? assume that g? is known to the procedure. Note that in the most biased
coin problem, g? =Bernoulli(?), but in general it is arbitrary (e.g. N (?, 1)). Consider a sequence of iid
Bernoulli random variables ?i 2 {0, 1} for i = 1, 2, . . . where each P(?i = 1) = 1 P(?i = 0) = ?.
Let Xi,j for j = 1, 2, . . . be a sequence of random variables drawn from g?1 if ?i = 1 and g?0
N
i
otherwise, and let {{Xi,j }M
j=1 }i=1 represent the sampling history generated by a procedure for some
N 2 N and (M1 , . . . , MN ) 2 NN . Any valid procedure behaves accordingly:
Algorithm 1 The most biased coin problem definition. Only the last distribution drawn may be
sampled or declared heavy, enforcing the rule that only one coin may be outside the bag at a time.
Initialize an empty history (N = 1, M = (0, 0, . . . )).
Repeat until heavy distribution declared:
Choose one of
1. draw a sample from distribution N , MN
MN + 1
2. draw a sample from the (N + 1)st distribution, MN +1 = 1, N
N +1
3. declare distribution N as heavy
Definition 1 We say a strategy for the most biased coin problem is -probably correct if for all
(?, ?0 , ?1 ) it identifies a ?heavy? g?1 distribution with probability at least 1
.
Definition 2 (Strategies for the most biased coin problem) An estimate-then-explore strategy
is a strategy that, for any fixed m 2 N, begins by sampling each successive coin exactly m times for
a number of coins that is at least the minimum necessary for any test to determine that ? 6= 0 with
probability at least 1
, then optionally continues sampling with an arbitrary strategy that declares
a heavy coin. An adaptive strategy is any strategy that is not an estimate-then-explore strategy.
We study the estimate-then-explore strategy because there exist optimal algorithms [8, 15] for the
most biased coin problem if ?, ?0 , ?1 are known, so it is natural to consider estimating these quantities
then using one of these algorithms. Note that the algorithm of [7] for the -parameterized infinite
armed bandit problem discussed above can be considered an estimate-then-explore strategy since it
first estimates by sampling a fixed number of samples from a set of arms, and then uses this estimate
to draw a fixed number of arms and applies a UCB-style algorithm to these arms. A contribution of
this work is showing that such a strategy is infeasible for the most biased coin problem.
3
For all strategies that are -probably correct and follow the interface of Algorithm 1, our goal is
PN
to provide lower and upper bounds on the quantity E[T ] := E[ i=1 Mi ] for any (?, ?0 , ?1 ) if N
denotes the final number of coins considered.
2
From Identifying Coins to Detecting Mixture Distributions
Addressing the most biased coin problem, [15] analyzes perhaps the most natural strategy: fix an
m 2 N and flip each successive coin exactly m times. The relevant questions are how large does m
have to be in order to guarantee correctness with probability 1
, and for a given m how long must
one wait to declare a ?heavy? coin? The authors partially answer these questions and we improve
upon them (see Section 3.2.1) which leads us to our study of the difficulty of detecting the presence of
a mixture distribution. As an example of the kind of lower bounds shown in this work, if we observe
a sequence of random variables X1 , . . . , Xn , consider the following hypothesis test:
2
H0 : 8i X1 , . . . , Xn ? N (?,
H1 : 8i X1 , . . . , Xn ? (1
)
for some ? 2 R,
?)N (?0 ,
2
) + ? N (?1 ,
2
(P1)
)
which will henceforth be referred to as Problem P1 or just (P1). We can show that if ?0 , ?1 , ? are
2
known and ? = ?0 , then it is sufficient to observe just max{1/?, ?2 (?1 ?0 )2 log(1/ )} samples to
determine the correct hypothesis with probability at least 1
. However, if ?0 , ?1 , ? are unknown
2
2
then it is necessary to observe at least max 1/?, ?(?1 ?0 )2 log(1/ ) samples in expectation
whenever
(?1 ?0 )2
2
? 1 and max{1/?, ?2 (?1
2
?0 ) 2
log(1/ )} otherwise (see Appendix C).
2
Recognizing (?1 2?0 ) as the KL divergence between two Gaussians of H1 , we observe startling
consequences for anomaly detection when the parameters of the underlying distributions are unknown:
if the anomalous distribution is well separated from the null distribution, then detecting an anomalous
component is only about as hard as observing just one anomalous sample (i.e. 1/?) multiplied by
the inverse KL divergence between the null and anomalous distributions. However, when the two
distributions are not well separated then the necessary sample complexity explodes to this latter
quantity squared. In Section 4 we will investigate adaptive methods for dramatically decreasing this
sample complexity.
Our lower bounds are based on the detection of the presence of a mixture of two distributions of an
exponential family versus just a single distribution of the same family. There has been extensive
work in the estimation of mixture distributions [13, 11] but this literature often assumes that the
mixture coefficient ? is bounded away from 0 and 1 to ensure a sufficient number of samples from
each distribution. In contrast, we highlight the regime when ? is arbitrarily small, as is the case in
statistical anomaly detection [10, 20, 2]. Property testing, e.g. unimodality, [1] is relevant but can
lack interpetability or strength in favor of generality. Considering the exponential family allowing us
to make interpretable statements about the relevant problem parameters in different regimes.
Preliminaries Let P and Q be two probability distributions with densities p and q, respectively. For
simplicity, assume p and
? q have
? the same support. Define the KL Divergence between P and Q
R
p(x)
as KL(P, Q) = log q(x) dp(x). Define the 2 Divergence between P and Q as 2 (P, Q) =
?2
R ? p(x)
R
q(x))2
1 dq(x) = (p(x)q(x)
dx. Note that by Jensen?s inequality
q(x)
?
?
? ?
KL(P, Q) = Ep log pq ? log Ep pq = log 2 (P, Q) + 1 ? 2 (P, Q).
(2)
Examples: If P = N (?1 ,
e
(?1
?0 ) 2
2
3
) and Q = N (?0 ,
2
) then KL(P, Q) =
(?1 ?0 )2
2 2
1. If P = Bernoulli(?1 ) and Q = Bernoulli(?0 ) then KL(P, Q) =
?1 ) log( 11 ??10 )
appendix.
2
?
(?1 ?0 )2 /2
?0 (1 ?0 ) [(?1 ?0 )(2?0 1)]+
and
2
(P, Q) =
(?1 ?0 )2
?0 (1 ?0 ) .
and
2
(P, Q) =
?1 log( ??10 )
+ (1
All proofs appear in the
Lower bounds
We present lower bounds on the sample complexity of -probably correct strategies for the most
biased coin problem that follow the interface of Algorithm 1. Lower bounds are stated for any
4
adaptive strategy in Section 3.1, non-adaptive strategies that may have knowledge of the parameters
but sample each distribution the same number of times in Section 3.2.1, and estimate-then-explore
strategies that do not have prior knowledge of the parameters in Section 3.2.2. Our lower bounds,
with the exception of the adaptive strategy, are based on the difficulty of detecting the presence of a
mixture distribution, and this reduction is explained in Section 3.2.
3.1
Adaptive strategies
The following theorem, reproduced from [15], describes the sample complexity of any -probably
correct algorithm for the most biased coin identification problem. Note that this lower bound holds
for any procedure even if it returns to previously seen distributions to draw additional samples and
even if it knows ?, ?0 , ?1 .
Theorem 1 [15, Theorem 2] Fix 2 (0, 1). Let T be the total number of samples taken of any
procedure that is -probably correct in identifying a heavy distribution. Then
?
1
(1
)
E[T ] c1 max
,
? ?KL(g?0 |g?1 )
whenever ? ? c2 where c1 , c2 2 (0, 1) are absolute constants.
The above theorem is directly applicable to the special case where g? is a Bernoulli distribution,
(1 ?0 ),?1 (1 ?1 )}
implying a lower bound of max 1 ? , 2 min{?0?(?
for the most biased coin problem.
2
1 ?0 )
The upper bounds of our proposed procedures for the most biased coin problem presented later will
be compared to this benchmark.
3.2
The detection of a mixture distribution and the most biased coin problem
First observe that identifying a specific distribution i ? N as heavy (i.e. ?i = 1) or determining that ?
is strictly greater than 0, is at least as hard as detecting that any of the distributions up to distribution N
is heavy. Thus, a lower bound on the total expected number of samples of all considered distributions
for this strictly easier detection problem is also a lower bound for the estimate-then-explore strategy
for the most biased coin identification problem.
The estimate-then-explore strategy fixes an m 2 N prior to starting the game and then samples each
distribution exactly m times, i.e. Mi = m for all i ? N for some N . To simplify notation let f?
denote the distribution of the sufficient statistics of these m samples. In general f? is a product
distribution, but when g? is a Bernoulli distribution, as in the biased coin problem, we can take f? to
be a Binomial distribution with parameters (m, ?). Now our problem is more succinctly described as:
e ? ?,
for some ? 2 ?
?
f ?0
H1 : 8i ?i ? Bernoulli(?), 8i Xi ?
f ?1
H0 : 8i Xi ? f?
if ?i = 0
if ?i = 1
(P2)
If ?0 and ?1 are close to each other, or if ? is very small, it can be very difficult to decide between H0
and H1 even if ?, ?0 , ?1 are known a priori. Note that when the parameters are known, one can take
e = {?0 }. However, when the parameters are unknown, one takes ?
e = ? to prove a lower bound on
?
the sample complexity of the estimate-then-explore algorithm, which is tasked with deciding whether
or not samples are coming from a mixture of distributions or just a single distribution within the
family. That is, lower bounds on the sample complexity when the parameters are known and unknown
follow by analyzing a simple binary and composite hypothesis test, respectively. In what follows, for
any event A, let Pi (A) and Ei [A] denote probability and expectation of A under hypothesis Hi for
i 2 {0, 1} (the specific value of ? in H0 will be clear from context). The next claim is instrumental
in our ability to prove lower bounds on the difficulty of the hypothesis tests.
Claim 1 Any procedure that is -probably correct also satisfies P0 (N < 1) ?
3.2.1
whenever ? = 0.
Sample complexity when parameters are known
e ?
Theorem 2 Fix 2 (0, 1). Consider the hypothesis test of Problem P2 for any fixed ? 2 ?
?. Let N be the random number of distributions considered before stopping and declaring a
5
N
hypothesis. If na procedure satisfies
P0 (N
1
, then
o
n < 1) ? oand P1 ([i=1 {?i = 1})
log(1/ )
log(1/ )
1
1
e
E1 [N ] max ? , KL(P1 |P0 )
max ? , 2 (P1 |P0 ) . In particular, if ? = {?0 } then
n1
log(1/ ) o
E1 [N ] max
, 2 2
.
? ?
(f?1 |f?0 )
The next corollary relates Theorem 2 to the most biased coin problem and is related to Malloy et al.
[15, Theorem 4] that considers the limit as ? ! 0 and assumes m is sufficiently large (specifically,
large enough for the Chernoff-Stein lemma to apply). In contrast, our result holds for all finite , ?, m.
Corollary 1 Fix 2 (0, 1). For any m 2 N consider a -probably correct strategy that flips each
coin exactly m times. If Nm is the number of coins considered before declaring a coin as heavy then
?
?
)
(1
) log log(1/
?
?0 (1 ?0 )
min E[mNm ]
.
m2N
?
(?1 ?0 )2
One can show the existence of such a strategy with a nearly matching upperbound when ?, ?0 , ?1 are
known (see Appendix B.1). Note that this is at least log(1/?) larger than the sample complexity of
(1) that can be achieved by an adaptive algorithm when the parameters are known.
3.2.2
Sample complexity when parameters are unknown
If ?, ?0 , and ?1 are unknown, we cannot test f?0 against the mixture (1 ?)f?0 + ?f?1 . Instead, we
have the general composite test of any individual distribution against any mixture, which is at least as
e = {?} for some particular worst-case setting of ?.
hard as the hypothesis test of Problem P2 with ?
Without any specific form of f? , it is difficult to pick a worst case ? that will produce a tight bound.
Consequently, in this section we consider single parameter exponential families (defined formally
below) to provide us with a class of distributions in which we can reason about different possible
values for ?. Since exponential families include Bernoulli, Gaussian, exponential, and many other
distributions, the following theorem is general enough to be useful in a wide variety of settings. The
constant C referred to in the next theorem is an absolute constant under certain conditions that we
outline in the following remark and corollary, its explicit form is given in the proof.
Theorem 3 Suppose f? for ? 2 ? ? R is a single parameter exponential family so that f? (x) =
e =
h(x) exp(?(?)x b(?(?))) for some scalar functions h, b, ? where ? is strictly increasing. If ?
{?? } where ?? = ? 1 (1 ?)?(?0 ) + ??(?1 ) and N is the stopping time of any procedure that
satisfies P0 (N < 1) ? and P1 ([N
1
, then
i=1 {?i = 1})
n
o
1
log( )
E1 [N ] max 1 ? , 1
.
2
C ( 2 ?(1 ?)(?(?1 ) ?(?0 ))2 )
where C is a constant that may depend on ?, ?0 , ?1 .
The following remark and corollary apply Theorem 3 to the special cases of Gaussian mixture model
detection and the most biased coin problem, respectively.
e in Problem P2 and
Remark 1 When ?, ?0 , ?1 are unknown, any procedure has no knowledge of ?
consequently it cannot rule out ? = ?? for H0 where ?? is defined in Theorem 3. If f? = N (?, 2 )
2
for known , then whenever (?1 2?0 ) ? 1 the constant C in Theorem 3 is an absolute constant and
2
2
consequently, E1 [N ] = ? ?(?1 ?0 )2 log(1/ ) . Conversely, when ?, ?0 , ?1 are known, then we
simply need to determine whether samples came? from N (?0 , 2 ) or (1? ?)N (?0 , 2 ) + ?N (?1 , 2 ),
2
and we show that it is sufficient to take just O
?2 (?1 ?0 )2
log(1/ ) samples (see Appendix C).
Corollary 2 Fix 2 [0, 1] and assume ?0 , ?1 are bounded sufficiently far from {0, 1} such that
2(?1 ?0 ) ? min{?0 (1 ?0 ), ?1 (1 ?1 )}. For any m let Nm be the number of coins a -probably
correct estimate-then-explore strategy that flips each coin m times in the exploration step. Then
1
c0 min{ m
, ?? (1 ?? )}
?? (1 ?? )
1
mE[Nm ]
.
?
?2 log( ) whenever m ?
2
(?1 ?0 )2
?(1 ?) ?(??1(1 ?0?)? )
where c0 is an absolute constant and ?? = ?
1
((1
6
?)?(?0 ) + ??(?1 )) 2 [?0 , ?1 ].
Remark 2 If ?, ?0 , ?1 are unknown, any estimate-then-explore strategy (or the strategy described in
Corollary 1) would be unable to choose an m that depended on these parameters, so we can treat it as
a constant. Thus, for the case when ?0 and ?1 are bounded away from {0, 1} (e.g. ?0 , ?1 2 [1/8, 7/8]),
the above corollary states that for any fixed m, whenever ?1 ?0 is sufficiently small the number
2
of samples necessary for these strategies to identify a heavy coin scales like ?(?1 1 ?0 )2 log(1/ ).
This is striking example of the difference when parameters are known versus when they are not and
effectively rules out an estimate-then-explore strategy for practical purposes.
Setting
Upper Bound
Fixed, known ?, ?0 , ?1
log(1/( ?))
,
??2
Adaptive, known ?, ?0 , ?1
1
?2
Est+Expl, unknown ?, ?0 , ?1
Unconsidered?
Adaptive, unknown ?, ?0 , ?1
c log(
1
?
Lower Bound
Thm. 7
[8, 15], Thm. 4
+ log( 1 )
1
??2
log(log(1/ )/?)
??2
) log(log(
??2
1
??2
)/ )
Thm. 5
[15]
1
??2
1 2
??2
Cor. 1
log( 1 )
1
??2
Cor. 2
[15]
Table 1: Upper and lower bounds on the expected sample complexity of different -probably correct
strategies. Fixed refers to the strategy of Corollary 1. For this table, we assume min{?0 (1
?0 ), ?1 (1 ?1 )} is lower bounded by a constant (e.g. ?0 , ?1 2 [1/8, 7/8]) and ? = ?1 ?0 is
sufficiently small. Also note that the upperbounds apply to distributions supported on [0, 1], not
just coins. All results without bracketed citations were unknown prior to this work. ? Due to our
discouraging lower bound for any estimate-then-explore strategy, it is inadvisable to propose an algorithm.
4
Near optimal adaptive algorithm
In this section we propose an algorithm that has no prior knowledge of the parameters ?, ?0 , ?1 yet
yields an upper bound that matches the lower bound of Theorem 1 up to logarithmic factors. We
assume that samples from heavy or light distributions are supported on [0, 1], and that drawn samples
are independent and unbiased estimators of the mean, i.e., E[Xi,j ] = ?i for ?i 2 {?0 , ?1 }. All
results can be easily extended to sub-Gaussian distributions. Consider Algorithm 2, an SPRT-like
procedure [18] for finding a heavy distribution given and lower bounds on ? and ? = ?1 ?0 . It
improves upon prior work by supporting arbitrary distributions on [0, 1] and requires only bounds
?, ?.
Algorithm 2 Adaptive strategy for heavy distribution identification with inputs ?0 , ?0 ,
Given 2 (0, 1/4), ?0 2 (0, 1/2), ?0 2 (0, 1).
Initialize n = d2 log(9)/?0 e, m = d64?0 2 log(14n/ )e, A = 8?0 1 log(21),
B = 8?0 1 log(14n/ ), k1 = 5, k2 = d8?0 2 log(2k1 / min{ /8, m 1 ?0 2 })e.
Draw k1 distributions and sample them each k2 times.
Estimate ?b0 = mini=1,...,k1 ?
bi,k2 , ? = ?b0 + ?0 /2.
Repeat for i = 1, . . . , n:
Draw distribution i.
Repeat for j = 1, . . . , m:
Sample distribution i and observe Xi,j .
Pj
If k=1 (Xi,k ? ) > B:
Declare distribution i to be heavy and Output distribution i.
Pj
Else if k=1 (Xi,k ? ) < A:
break.
Output null.
Theorem 4 If Algorithm 2 is run with 2 (0, 1/4), ?0 2 (0, 1/2), ?0 2 (0, 1), then the expected
number of total samples taken by the algorithm is no more than
c0 ? log(1/?0 ) + c00 log
?0 ?20
7
1
(3)
for some absolute constants c0 ,c00 , and all of the following hold: 1) with probability at least 1
,
a light distribution is not returned, 2) if ?0 ? ?1 ?0 and ?0 ? ?, then with probability 45 a heavy
0 ))
distribution is returned, and 3) the procedure takes no more than c log(1/(?
total samples.
? 0 ?2
0
The second claim of the theorem holds only with constant probability (versus with probability 1
)
since the probability of observing a heavy distribution among the n = d2 log(4)/?0 e distributions
only occurs with constant probability. One can show that if the outer loop of algorithm is allowed
to run indefinitely (with m and n defined as is), ?0 = ?1 ?0 , ?0 = ?, and ?b0 = ?0 , then a heavy
coin is returned with probability at least 1
and the expected number of samples is bounded by
(3). If a tight lower bound is known on either ? = ?1 ?0 or ?, there is only one parameter that is
unknown and the ?doubling trick?, along with Theorem 4, can be used to identify a heavy coin with
2
1
)/ )
)/ )
just log(log(?
and log(log(?
samples, respectively (see Appendix B.3).
??2
??2
Now consider Algorithm 3 that assumes no prior knowledge of ?, ?0 , ?1 , the first result for this setting
that we are aware of. We remark that while the placing of ?landmarks? (?k , ?k ) throughout the search
space as is done in Algorithm 3 appears elementary in hindsight, it is surprising that so few can cover
this two dimensional space since one has to balance the exploration of ? and ?. We believe similar a
similar approach may be generalized for more generic infinite armed bandit problems.
Algorithm 3 Adaptive strategy for heavy distribution identification with unknown parameters
Given > 0.
Initialize ` = 1, heavy distribution h = null.
Repeat until h is not null:
Set ` = 2` , ` = /(2`3 )
Repeat for k = 0, . . . , `:
q
k
Set ?k = 2 ` , ?k = 2?1k `
Run Algorithm 2 with ?0 = ?k , ?0 = ?k , = ` and Set h to its output.
If h is not null break
Set ` = ` + 1
Output h
Theorem 5 (Unknown ?, ?0 , ?1 ) Fix 2 (0, 1). If Algorithm 3 is run with then with probability
at least 1
a heavy distribution is returned and the expected number of total samples taken is
bounded by
log2 ( ??1 2 )
(? log2 ( ?12 ) + log(log2 ( ??1 2 )) + log(1/ ))
??2
for an absolute constant c.
c
5
Conclusion
While all prior works have required at least partial knowledge of ?, ?0 , ?1 to solve the most biased
coin problem, our algorithm requires no knowledge of these parameters yet obtain the near-optimal
sample complexity. In addition, we have proved lower bounds on the sample complexity of detecting
the presence of a mixture distribution when the parameters are known or unknown, with consequences
for any estimate-then-explore strategy, an approach previously proposed for an infinite armed bandit
problem. Extending our adaptive algorithm to arbitrary arm reservoir distributions is of significant
interest. We believe a successful algorithm in this vein could have a significant impact on how
researchers think about sequential decision processes in both finite and uncountable action spaces.
Acknowledgments Kevin Jamieson is generously supported by ONR awards N00014-15-1-2620, and N0001413-1-0129. This research is supported in part by NSF CISE Expeditions Award CCF-1139158, DOE Award
SN10040 DE-SC0012463, and DARPA XData Award FA8750-12-2-0331, and gifts from Amazon Web Services,
Google, IBM, SAP, The Thomas and Stacey Siebel Foundation, Apple Inc., Arimo, Blue Goji, Bosch, Cisco, Cray,
Cloudera, Ericsson, Facebook, Fujitsu, Guavus, HP, Huawei, Intel, Microsoft, Pivotal, Samsung, Schlumberger,
Splunk, State Farm and VMware.
8
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5,606 | 6,073 | Designing smoothing functions for improved
worst-case competitive ratio in online optimization
Reza Eghbali
Department of Electrical Engineering
University of Washington
Seattle, WA 98195
eghbali@uw.edu
Maryam Fazel
Department of Electrical Engineering
University of Washington
Seattle, WA 98195
mfazel@uw.edu
Abstract
Online optimization covers problems such as online resource allocation, online
bipartite matching, adwords (a central problem in e-commerce and advertising),
and adwords with separable concave returns. We analyze the worst case competitive ratio of two primal-dual algorithms for a class of online convex (conic)
optimization problems that contains the previous examples as special cases defined
on the positive orthant. We derive a sufficient condition on the objective function
that guarantees a constant worst case competitive ratio (greater than or equal to
1
2 ) for monotone objective functions. We provide new examples of online problems on the positive orthant that satisfy the sufficient condition. We show how
smoothing can improve the competitive ratio of these algorithms, and in particular
for separable functions, we show that the optimal smoothing can be derived by
solving a convex optimization problem. This result allows us to directly optimize
the competitive ratio bound over a class of smoothing functions, and hence design
effective smoothing customized for a given cost function.
1
Introduction
Given a proper convex cone K ? Rn , let ? : K 7? R be an upper semi-continuous concave function.
Consider the optimization problem
Pm
maximize
? ( t=1 At xt )
(1)
subject to
xt ? Ft ,
?t ? [m],
where for all t ? [m] := {1, 2, . . . , m}, xt ? Rl are the optimization variables and Ft are compact
convex constraint sets. We assume At ? Rn?l maps Ft to K; for example, when K = Rn+ and
Ft ? Rl+ , this assumption is satisfied if At has nonnegative entries. We consider problem (1) in the
online setting, where it can be viewed as a sequential game between a player (online algorithm) and
an adversary. At each step t, the adversary reveals At , Ft and the algorithm chooses x
?t ? Ft . The
performance of the algorithm is measured by its competitive ratio, i.e., the ratio of objective value
at x
?1 , . . . , x
?m to the offline optimum. Problem (1) covers (convex relaxations of) various online
combinatorial problems including online bipartite matching [14], the ?adwords? problem [16], and
the secretary problem [15]. More generally, it covers online linear programming (LP) [6], online
packing/covering with convex cost [3, 4, 7], and generalization of adwords [8]. In this paper, we
study the case where ??(u) ? K ? for all u ? K, i.e., ? is monotone with respect to the cone K.
The competitive performance of online algorithms has been studied mainly under the worst-case
model (e.g., in [16]) or stochastic models (e.g., in [15]). In the worst-case model one is interested
in lower bounds on the competitive ratio that hold for any (A1 , F1 ), . . . , (Am , Fm ). In stochastic models, adversary choses a probability distribution from a family of distributions to generate
(A1 , F1 ), . . . , (Am , Fm ), and the competitive ratio is calculated using the expected value of the
algorithm?s objective value. Online bipartite matching and its generalization, the ?adwords? problem,
are the two main problems that have been studied under the worst case model. The greedy algorithm
achieves a competitive ratio of 1/2 while the optimal algorithm achieves a competitive ratio of 1?1/e
(as bid to budget ratio goes to zero) [16, 5, 14, 13]. A more general version of Adwords in which
each agent (advertiser) has a concave cost has been studied in [8].
The majority of algorithms proposed for the problems mentioned above rely on a primal-dual
framework [5, 6, 3, 8, 4]. The differentiating point among the algorithms is the method of updating
the dual variable at each step, since once the dual variable is updated the primal variable can be
assigned using a simple complementarity condition. A simple and efficient method of updating
the dual variable is through a first order online learning step. For example, the algorithm stated in
[9] for online linear programming uses mirror descent with entropy regularization (multiplicative
weight updates algorithm) once written in the primal dual language. Recently, the work in [9] was
independently extended to random permutation model in [12, 2, 11]. In [2], the authors provide
competitive difference bound for online convex optimization under random permutation model as a
function of the regret bound for the online learning algorithm applied to the dual.
In this paper, we consider two versions of the greedy algorithm for problem (1), a sequential
update and a simultaneous update algorithm. The simultaneous update algorithm, Algorithm 2,
provides a direct saddle-point representation of what has been described informally in the literature
as ?continuous updates? of primal and dual variables. This saddle point representation allows us to
generalize this type of updates to non-smooth function. In section 2, we bound the competitive ratios
of the two algorithms. A sufficient condition on the objective function that guarantees a non-trivial
worst case competitive ratio is introduced. We show that the competitive ratio is at least 12 for a
monotone non-decreasing objective function. Examples that satisfy the sufficient condition (on
the positive orthant and the positive semidefinite cone) are given. In section 3, we derive optimal
algorithms, as variants of greedy algorithm applied to a smoothed version of ?. For example, Nesterov
smoothing provides optimal algorithm for the adwords problem. The main contribution of this paper
is to show how one can derive the optimal smoothing function (or from the dual point of view the
optimal regularization function) for separable ? on positive orthant by solving a convex optimization
problem. This gives an implementable algorithm that achieves the optimal competitive ratio derived
in [8]. We also show how this convex optimization can be modified for the design of smoothing
function specifically for the sequential algorithm. In contrast, [8] only considers continuous updates.
The algorithms considered in this paper and their general analysis are the same as those we considered
in [10]. In [10], the focus is on non-monotone functions and online problems on the positive
semidefinite cone. However, the focus of this paper is on monotone functions on the positive orthant.
In [10], we only considered Nesterov smoothing and only derived competitive ratio bounds for the
simultaneous algorithm.
Notation. Given a function ? : Rn 7? R, ? ? denotes the concave conjugate of ? defined as
? ? (y) = inf u hy, ui ? ?(u), for all y ? Rn . For a concave function ?, ??(u) denotes the set of
supergradients of ? at u, i.e., the set of all y ? Rn such that ?u0 ? Rn : ?(u0 ) ? hy, u0 ? ui+?(u).
The set ?? is related to the concave conjugate function ? ? as follows. For an upper semi-continuous
concave function ? we have ??(u) = argminy hy, ui ? ? ? (y). A differentiable function ? has
a Lipschitz continuous gradient with respect to k?k with continuity parameter 1/? > 0 if for all
?
?
u, u0 ? Rn , k??(u0 ) ? ??(u)k ? 1/? ku ? u0 k, where k?k is the dual norm to k?k.
The dual cone K ? of a cone K ? Rn is defined as K ? = {y | hy, ui ? 0 ?u ? K}. Two examples
of self-dual cones are the positive orthant Rn+ and the cone of n ? n positive semidefinite matrices
n
S+
. A proper cone (pointed convex cone with nonempty interior) K induces a partial ordering on Rn
which is denoted by ?K and is defined as x ?K y ? y ? x ? K.
1.1
Two primal-dual algorithms
The (Fenchel) dual problem for problem (1) is given by
Pm
T
?
minimize
t=1 ?t (At y) ? ? (y),
n
(2)
where the optimization variable is y ? R , and ?t denotes the support function for the set Ft defined
as ?t (z) = supx?Ft hx, zi. A pair (x? , y ? ) ? (F1 ? . . . ? Fm ) ? K ? is an optimal primal-dual pair
if and only if
2
x?t
?
argmaxhx, ATt y ? i,
x?Ft
m
X
y ? ??(
At x?t ),
?
?t ? [m].
t=1
Based on these optimality conditions, we consider two algorithms. Algorithm 1 updates the primal
and dual variables sequentially, by maintaining a dual variable y?t and using it to assign x
?t ?
argmaxx?Ft hx, ATt y?t i. The then algorithm updates the dual variable based on the second optimality
condition. By the assignment rule, we have At x
?t ? ??t (?
yt ), and the dual variable update can be
Pt
viewed as y?t+1 ? argminy h s=1 As x
?s , yi ? ? ? (y). Therefore, the dual update is the same as the
update in dual averaging [18] or Follow The Regularized Leader (FTRL) [20, 19, 1] algorithm with
regularization ?? ? (y).
Algorithm 1 Sequential Update
Initialize y?1 ? ??(0)
for t ? 1 to m do
Receive At , Ft
x
?t ? argmaxx?Ft hx, ATt y?t i
Pt
y?t+1 ? ??( s=1 As x
?s )
end for
Algorithm 2 updates the primal and dual variables simultaneously, ensuring that
t
X
x
?t ? argmaxhx, ATt y?t i,
y?t ? ??(
As x
?s ).
x?Ft
s=1
This algorithm is inherently more complicated than algorithm 1, since finding x
?t involves solving a
saddle-point problem. This can be solved by a first order method like mirror descent algorithm for
saddle point problems. In contrast, the primal and dual updates in algorithm 1 solve two separate
maximization and minimization problems 1 .
Algorithm 2 Simultaneous Update
for t ? 1 to m do
Receive At , Ft
Pt?1
(?
yt , x
?t ) ? arg miny maxx?Ft hy, At x + s=1 As x
?s i ? ? ? (y)
end for
2
Competitive ratio bounds and examples for ?
In this section, we derive bounds on the competitive ratios of Algorithms 1 and 2 by bounding their
respective duality gaps. We begin by stating a sufficient condition on ? that leads to non-trivial
competitive ratios, and we assume this condition holds in the rest of the paper. Roughly, one can
interpret this assumption as having ?diminishing returns? with respect to the ordering induced by a
cone. Examples of functions that satisfy this assumption will appear later in this section.
Assumption 1 Whenever u ?K v, there exists y ? ??(u) that satisfies y ?K ? z for all z ? ??(v).
When ? is differentiable, assumption 1 simplifies to u ?K v ? ??(u) ?K ? ??(v). That is, the
gradient, as a map from Rn (equipped with ?K ) to Rn (equipped with ?K ? ), is order-reversing.When
? is twice differentiable, assumption 1 is equivalent to hw, ?2 ?(u)vi ? 0, for all u, v, w ? K. For
example, this is equivalent to Hessian being element-wise non-positive when K = Rn+ .
Pm
Let define y?m+1 to be the minimum element in ??( t=1 At x
?t ) with respect
ordering ?K ? (such
Pto
m
an element
?t ) and Psim =
Pm exists in the superdifferential by Assumption (1)). Let Pseq = ? ( t=1 At x
? ( t=1 At x
?t ) denote the primal objective values for the primal solution produced by the algorithms
1
Also if the original problem is a convex relaxation of an integer program, meaning that each Ft = convFt
where Ft ? Zl , then x
?t can always be chosen to be integral while integrality may not hold for the solution of
the second algorithm.
3
Pm
Pm
1 and 2, and Dseq = t=1 ?t (ATt y?t ) ? ? ? (?
ym+1 ) and Dsim = t=1 ?t (ATt y?t ) ? ? ? (?
ym+1 ) denote
the corresponding dual objective values.
The next lemma provides a lower bound on the duality gaps of both algorithms.
Lemma 1 The duality gaps for the two algorithms can be lower bounded as
Psim ? Dsim ? ? ? (?
ym+1 ) + ?(0),
Pseq ? Dseq ? ? ? (?
ym+1 ) + ?(0) +
m
X
hAt x
?t , y?t+1 ? y?t i
t=1
Furthermore, if ? has a Lipschitz continuous gradient with parameter 1/? with respect to k?k,
Pm
2
1
Pseq ? Dseq ? ? ? (?
ym+1 ) + ?(0) ? 2?
?t k .
(3)
t=1 kAt x
Note that right hand side of (3) is exactly the regret bound of the FTRL algorithm (with a negative
sign) [19]. The proof is given in the appendix. To simplify the notation in the rest of the paper, we
assume ?(0) = 0 by replacing ?(u) with ?(u) ? ?(0). To quantify the competitive ratio of the
algorithms, we define ?? as
?? = sup {c | ? ? (y) ? c?(u), y ? ??(u), u ? K},
(4)
Since ? ? (y) + ?(u) = hy, ui for all y ? ??(u), ?? is equivalent to
?? = sup{c | hy, ui ? (c + 1)?(u), y ? ??(u) u ? K}.
(5)
Note that ?1 ? ?? ? 0, since for any u ? K and y ? ??(u), by concavity of ? and the fact
that y ? K ? , we have 0 ? hy, ui ? ?(u) ? ?(0). If ? is a linear function then ?? = 0, while if
0 ? ??(u) for some u ? K, then ?? = ?1.
The next theorem provides lower bounds on the competitive ratio of the two algorithms.
Theorem 1 If Assumption 1 holds, we have
Psim ?
1
D? ,
1 ? ??
Pseq ?
m
X
1
hAt x
?t , y?t+1 ? y?t i)
(D? +
1 ? ??
t=1
where D? is the dual optimal objective. If ? has a Lipschitz continuous gradient with parameter 1/?
with respect to k?k,
Pm
2
1
1
(D? ? 2?
?t k ).
Pseq ? 1??
(6)
t=1 kAt x
?
Pt
Pm
?s ?K s=1 As x
?s for all
Proof: Consider the simultaneous update algorithm. We have s=1 As x
Pt
t, since As Fs ?PK for all s. Since y?t ? ??( s=1 As x
?s ) and y?m+1 was picked to be the minimum
m
element in ??( s=1 As x
?s ) with respect to ?K ? , by Assumption 1, we have y?t ?K ? y?m+1 . Since
At x ? K for all x ? Ft , we get hAt x, y?t i ? hAt x, y?m+1 i; therefore, ?t (ATt y?t ) ? ?t (ATt y?m ). Thus
Dsim =
m
X
?t (ATt y?t ) ? ? ? (?
ym ) ?
m
X
t=1
?t (ATt y?m+1 ) ? ? ? (?
ym ) ? D? .
t=1
Now Lemma 1 and definition of ?? give the desired result. The proof for Algorithm 1 follows similar
steps.
We now consider examples of ? that satisfy Assumption 1 and derive lower bound on ?? for those
examples.
Examples on positive orthant. Let K = Rn+ and note that K ? = K. To simplify the notation we
use ? instead of ?Rn+ . Assumption 1 is satisfied for a twice differentiable function if and only if
Pn
the Hessian is element-wise non-positive over Rn+ . If ? is separable, i.e., ?(u) = i=1 ?i (ui ),
Assumption 1 is satisfied since by concavity for each ?i we have ??i (ui ) ? ??i (vi ) when ui ? vi .
In the basic adwords problem, for all t, Ft = {x ? Rl+ | 1T x ? 1}, At is a diagonal matrix with
non-negative entries, and
Pn
Pn
?(u) = i=1 ui ? i=1 (ui ? 1)+ ,
(7)
4
where (?)+ = max{?, 0}. In this problem, ? ? (y) = 1T (y ? 1). Since 0 ? ??(1) we have ?? = ?1
by (5); therefore, the competitive ratio of algorithm 2 is 12 . Let r = maxt,i,j At,i,j , then we have
Pm
?t , y?t+1 ? y?t i ? nr. Therefore, the competitive ratio of algorithm 1 goes to 12 as r (bid to
t=1 hAt x
budget ratio) goes to zero. In adwords with concave returns studied in [8], At is diagonal for all t and
? is separable 2 .
Pn
For any p ? 1 let Bp be the lp -norm ball. We can rewrite the penalty
function ? i=1 (ui ? 1)+ in
Pn
the adwords objective using the distance from B? : we have i=1 (ui ? 1)+ = d1 (u, B? ), where
d1 (?, C) is the l1 norm distance from set C. For p ? [1, ?) the function ?d1 (u, Bp ) although not
separable it satisfies Assumption 1. The proof is given in the supplementary materials.
n
Examples on the positive semidefinite cone. Let K = S+
and note that K ? = K. Two examples
that satisfy Assumption 1 are ?(U ) = log det(U + A0 ), and ?(U ) = trU p with p ? (0, 1). We
refer the reader to [10] for examples of online problems that entails log det in the objective function
and competitive ratio analysis of the simultanuous algorithm for these problems.
Smoothing of ? for improved competitive ratio
3
The technique of ?smoothing? an (potentially non-smooth) objective function, or equivalently adding
a strongly convex regularization term to its conjugate function, has been used in several areas. In
convex optimization, a general version of this is due to Nesterov [17], and has led to faster convergence
rates of first order methods for non-smooth problems. In this section, we study how replacing ?
with a appropriately smoothed function ?S helps improve the performance of the two algorithms
discussed in section 1.1, and show that it provides optimal competitive ratio for two of the problems
mentioned in section 2, adwords and online LP. We then show how to maximize the competitive
ratio of both algorithms for a separable ? and compute the optimal smoothing by solving a convex
optimization problem. This allows us to design the most effective smoothing customized for a given
?: we maximize the bound on the competitive ratio over the set of smooth functions.(see subsection
3.2 for details).
Let ?S denote an upper semi-continuous concave function (a smoothed version of ?), and suppose
?S satisfies Assumption 1. The algorithms we consider in this section are the same as Algorithms
1 and 2, but with ? replacing ?S . Note that the competitive ratio is computed with respect to the
original problem, that is the offline primal and dual optimal values are still the same P ? and D? as
before.
Pm
Pm
From LemmaP
1, we have that Dsim ? ?S ( t=1 At x
?t )?? ? (?
ym+1 ) and Dseq ? ?S ( t=1 At x
?t )?
m
? ? (?
ym+1 ) ? t=1 hAt x
?t , y?t+1 ? y?t i. To simplify the notation, assume ?S (0) = 0 as before. Define
??,?S = sup{c |? ? (y) ? ?S (u) + (c ? 1)?(u), y ? ??S (u), u ? K}.
Then the conclusion of Theorem 1 for Algorithms 1 and 2 applied to the smoothed function holds
with ?? replaced by ??,?S .
3.1
Nesterov Smoothing
We first consider Nesterov smoothing [17], and apply it to examples on non-negative orthant. Given a
proper upper semi-continuous concave function ? : Rn 7? R ? {??}, let
?S = (? ? + ?? )? .
Note that ?S is the supremal convolution of ? and ?. If ? and ? are separable, then ?S satisfies
Assumption 1 for K = Rn+ . Here we provide example of Nesterov smoothing for functions on
non-negative orthant.
Adwords: The optimal competitive
ratio for the Adwords problem is 1 ? e?1 . This is achieved by
Pm
e
smoothing ? with ?? (y) = i=1 (yi ? e?1
) log(e ? (e ? 1)yi ) ? 2yi , which gives
(
eui ?exp (ui )+1
ui ? [0, 1]
e?1
?S,i (ui ) ? ?S,i (0) =
1
ui > 1,
e?1
2
Note that in this case one can remove the assumption that ??i ? R+ since if y?t,i = 0 for some t and i,
then x
?s,i = 0 for all s ? t.
5
3.2
Computing optimal smoothing for separable functions on Rn+
We now tackle the problem of finding the optimal smoothing for separable functions on the positive
orthant, which as we show in an example at the endP
of this section is not necessarily
Pn given by Nesterov
n
smoothing. Given a separable monotone ?(u) = i=1 ?i (ui ) and ?S (u) = i=1 ?S,i (ui ) on Rn+
we have that ??,?S ? mini ??i ,?S,i .
To simplify the notation, drop the index i and consider ? : R+ 7? R. We formulate the problem
of finding ?S to maximize ??,?S as an optimization problem. In section 4 we discuss the relation
between
this optimization method and the optimal algorithm presented in [8]. We set ?S (u) =
Ru
y(s)ds
with y a continuous function (y ? C[0, ?)), and state the infinite dimensional convex
0
optimization problem with y as a variable,
minimize
?
Ru
(8)
subject to
y(s)ds ? ? ? (y(u)) ? ??(u),
?u ? [0, ?)
0
y ? C[0, ?),
where ? = 1 ? ??,?S (theorem 1 describes the dependence of the competitive ratios on this
parameter). Note that we have not imposed any condition on y to be non-increasing (i.e., the
corresponding ?S to be concave). The next lemma establishes that every feasible solution to the
problem (8) can be turned into a non-increasing solution.
Lemma 2 Let (y, ?) be a feasible solution for problem (8) and define y?(t) = inf s?t y(s). Then
(?
y , ?) is also a feasible solution for problem (8).
In particular if (y, ?) is an optimal solution, then so is (?
y , ?). The proof is given in the supplement.
Re-
visiting the adwords problem, we observe that the optimal solution is given by y(u) =
e?exp(u)
e?1
+
,
which is the derivative of the smooth function we derived using Nesterov smoothing in section 3.1.
The optimality of this y can be established by providing a dual certificate, a measure ? corresponding to the inequality constraint, that together with y satisfies the optimality condition. If we set
d? = exp (1 ? u)/(e ? 1) du, the optimality conditions are satisfied with ? = (1 ? 1/e)?1 . Also
note that if ? plateaus (e.g., as in the adwords objective), then one can replace problem (8) with a
problem over a finite horizon.
Theorem 2 Suppose ?(t) = c on [u0 , ?) (? plateaus). Then problem (8) is equivalent to
minimize
subject to
?
Ru
0
0
y(s)ds ? ? ? (y(u)) ? ??(u),
y(u ) = 0,
?u ? [0, u0 ]
(9)
0
y ? C[0, u ].
So for a function ? with a plateau, one can discretize problem (9) to get a finite dimensional problem,
minimize
?
subject to
h
Pt
s=1
y[s] ? ? ? (y[t]) ? ??(ht)
?t ? [d]
(10)
y[d] = 0,
where h = u /d is the discretization step. Figure 1a shows the optimal smoothing for the piecewise
linear function ?(u) = min(.75, u, .5u + .25) by solving problem (10). We point out that the
optimal smoothing for this function is not given by Nesterov?s smoothing (even though the optimal
smoothing can be derived by Nesterov?s smoothing for a piecewise linear function with only two
pieces, like the adwords cost function). Figure 1d shows the difference between the conjugate of the
optimal smoothing function and ? ? for the piecewise linear function, which we can see is not concave.
We simulated the performance of the simultaneous algorithm on a dataset with n = m, Ft simplex,
and At diagonal. We varied m in the range from 1 to 30 and for each m calculated the the smallest
competitive ratio achieved by the algorithm over (10m)2 random permutation of A1 , . . . , Am . Figure
1i depicts this quantity vs. m for the optimal smoothing
and the Nesterov smoothing. For the Nesterov
?
?
?
e
?
?
smoothing we used the function ? (y) = (y ? e?1 ) log( e ? ( e ? 1)y) ? 23 y.
Pm
In cases where a bound umax on t=1 At Ft is known, we can restrict t to [0, umax ] and discretize
problem (8) over this interval. However, the conclusion of Lemma 2 does not hold for a finite horizon
0
6
and we need to impose additional linear constraints y[t] ? y[t ? 1] to ensure the monotonicity of
y. We find the optimal smoothing
for two examples of this kind: ?(u) = log(1 + u) over [0, 100]
?
(Figure 1b), and ?(u) = u over [0, 100] (Figure 1c). Figure 1e shows the competitive ratio achieved
with the optimal smoothing of ?(u)
? = log(1 + u) over [0, umax ] as a function of umax . Figure 1f
depicts this quantity for ?(u) = u.
3.3
Competitive ratio bound for the sequential algorithm
In this section we provide a lower bound on the competitive ratio of the sequential algorithm
(Algorithm 1). Then we modify Problem (8) to find a smoothing function that optimizes this
competitive ratio bound for the sequential algorithm.
Theorem 3 Suppose ?S is differentiable on an open set containing K and satisfies Assumption 1. In
addition, suppose there exists c ? K such that At Ft ?K c for all t, then
1
D? ,
Pseq ?
1 ? ??,?S + ?c,?,?S
where ? is given by
?c,?,?S = inf{r | hc, ??S (0) ? ??S (u)i ? r?(u), u ? K}
Since ?S satisfies Assumption 1, we have y?t+1 ?K ? y?t . Therefore, we can write:
Pm
Pm
?t , y?t ? y?t+1 i ? t=1 hc, y?t ? y?t+1 i = hc, y?0 ? y?m+1 i
(11)
t=1 hAt x
Pm
Now by combining the duality gap
by Lemma 1 with 11, we get Dseq ? ?S ( t=1 At x
?t ) ?
Pgiven
m
? ? (?
ym+1 )+hc, ??S (0) ? ??S ( t=1 At x
?t )i. The conclusion follows from the definition of ??,?S ,
?c,?,?S and the fact that Dseq ? D? .
Proof:
Based on the result of the previous theorem we can modify the optimization problem set up in Section
3.2 for separable functions on Rn+ to maximize the lower bound on the competitive ratio of the
sequential algorithm. Note that when ? and ?S are separable, we have ?c,?,?S ? maxi ?ci ,?i ,?S i .
Therefore, similar to the previous section to simplify the notation we drop the index i and assume
? is a function of a scalar variable. The optimization problem for finding ?S that minimizes
?c,?,?S ? ??,?S is as follows:
minimize
subject to
?
Ru
0
y(s)ds + c(? 0 (0) ? y(u)) ? ? ? (y(u)) ? ??(u),
?u ? [0, ?)
(12)
y ? C[0, ?).
1
u?1
For adwords, the optimal solution is given by ? = 1?exp(?
and
y(u)
=
?
1
?
exp
,
1
1+c
c+1 )
+
?1
which gives a competitive ratio of 1 ? exp c+1
. In Figure 1h we have plotted the competitive
ratio achieved by solving problem 12 for ?(u) = log det(1 + u) with umax = 100 as a function
of c. Figure 1g shows the competitive ratio as a function of c for the piecewise linear function
?(u) = min(.75, u, .5u + .25).
4
Discussion and Related Work
We discuss results and papers from two communities, computer science theory and machine learning,
related to this work.
Online optimization. In [8], the authors proposed an optimal algorithm for adwords with differentiable concave returns (see examples in section 2). Here, ?optimal? means that they construct
an instance of the problem for which competitive ratio bound cannot be improved, hence showing
the bound is tight. The algorithm is stated and analyzed for a twice differentiable, separable ?(u).
The assignment rule for primal variables in their proposed algorithm is explained as a continuous
process. A closer look reveals that this algorithm falls in the framework of algorithm 2, with the only
difference being that at each step, (?
xt , y?t ) are chosen such that
x
?t ? argmaxhx, ATt y?t i
?i ? [n] : y?t,i = ??i (vi (ui )),
7
Pt
ui = ( t=1 As x
?s )i ,
0.8
0.6
5
25
4
20
3
15
2
10
0.4
0.2
?S
?
1
?S
?
0
0
0
0
0.5
1
1.5
2
0
20
40
60
u
0
100
40
0.9
0.2
0.1
0.4
0.6
0.8
1
0.85
0.8
0.9
0.85
0.8
0.75
0
1
500
0.7
1000
0
200
400
umax
y
0.4
1000
1
Optimal smoothing
Nesterov smoothing
0.95
competitive ratio
Competitive ratio
0.5
800
(f)
0.8
0.6
600
umax
(e)
(d)
0.7
100
0.95
0.7
-0.1
80
(c)
0.75
0
60
u
comp. ratio
0.3
0.2
20
(b)
0.95
comp. ratio
?S? (y) ? ?? (y)
(a)
Competitive ratio
80
u
0.4
0
?S
?
5
0.6
0.4
0.2
0.9
0.85
0.8
0.75
0.3
0
0
0.2
0.4
0.6
0.8
1
0
20
40
60
c
c
(g)
(h)
80
100
0.7
0
10
20
30
m
(i)
Figure 1: Optimal?smoothing for ?(u) = min(.75, u, .5u+.25) (a), ?(u) = log(1+u) over [0, 100]
(b), and ?(u) = u over [0, 100] (c). The competitive ratio
? achieved by the optimal smoothing as a
function of umax for ?(u) = log(1 + u) (e) and ?(u) = u (f). ?S? ? ? ? for the piecewise linear
function (d). The competitive ratio achieved by the optimal smoothing for the sequential algorithm as
a function of c for ?(u) = min(.75, u, .5u + .25) (g) and ?(u) = log(1 + u) with umax = 100 (h).
i, Competitive ratio of the simultaneous algorithm for ?(u) = min(.75, u, .5u + .25) as a function
of m with optimal smoothing and Nesterov smoothing (see text).
where vi : R+ 7? R+ is an increasing differentiable function given as a solution of a nonlinear
differential equation that involves ?i and may not necessarily have a closed form. The competitive
ratio is also given based on the differential equation. They prove that this gives the optimal competitive
ratio for the instances where ?1 = ?2 = . . . = ?m .
Note that this is equivalent of setting ?S,i (ui ) = ?(vi (ui ))). Since vi is nondecreasing ?S,i is a
concave function. On the other hand, given a concave function ?S,i (R+ ) ? ?i (R+ ), we can set
vi : R+ 7? R+ as vi (u) = inf{z | ?i (z) ? ?S,i (u)}. Our formulation in section 3.2 provides a
constructive way of finding the optimal smoothing. It also applies to non-smooth ?.
Online learning. As mentioned before, the dual update in Algorithm 1 is the same as in Follow-theRegularized-Leader (FTRL) algorithm with ?? ? as the regularization. This primal dual perspective
has been used in [20] for design and analysis of online learning algorithms. In the online learning
literature, the goal is to derive a bound on regret that optimally depends on the horizon, m. The goal
in the current paper is to provide competitive ratio for the algorithm that depends on the function ?.
Regret provides a bound on the duality gap, and in order to get a competitive ratio the regularization
function should be crafted based on ?. A general choice of regularization which yields an optimal
regret bound in terms of m is not enough for a competitive ratio argument, therefore existing results
in online learning do not address our aim.
8
References
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9
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5,607 | 6,074 | Proximal Deep Structured Models
Shenlong Wang
University of Toronto
slwang@cs.toronto.edu
Sanja Fidler
University of Toronto
fidler@cs.toronto.edu
Raquel Urtasun
University of Toronto
urtasun@cs.toronto.edu
Abstract
Many problems in real-world applications involve predicting continuous-valued
random variables that are statistically related. In this paper, we propose a powerful
deep structured model that is able to learn complex non-linear functions which
encode the dependencies between continuous output variables. We show that
inference in our model using proximal methods can be efficiently solved as a feedfoward pass of a special type of deep recurrent neural network. We demonstrate
the effectiveness of our approach in the tasks of image denoising, depth refinement
and optical flow estimation.
1
Introduction
Many problems in real-world applications involve predicting a collection of random variables that
are statistically related. Over the past two decades, graphical models have been widely exploited
to encode these interactions in domains such as computer vision, natural language processing and
computational biology. However, these models are shallow and only a log linear combination of
hand-crafted features is learned [34]. This limits the ability to learn complex patterns, which is
particularly important nowadays as large amounts of data are available, facilitating learning.
In contrast, deep learning approaches learn complex data abstractions by compositing simple nonlinear transformations. In recent years, they have produced state-of-the-art results in many applications such as speech recognition [17], object recognition [21], stereo estimation [38], and machine
translation [33]. In some tasks, they have been shown to outperform humans, e.g., fine grained
categorization [7] and object classification [15].
Deep neural networks are typically trained using simple loss functions. Cross entropy or hinge
loss are used when dealing with discrete outputs, and squared loss when the outputs are continuous.
Multi-task approaches are popular, where the hope is that dependencies of the output will be captured
by sharing intermediate layers among tasks [9].
Deep structured models attempt to learn complex features by taking into account the dependencies
between the output variables. A variety of methods have been developed in the context of predicting
discrete outputs [7, 3, 31, 39]. Several techniques unroll inference and show how the forward
and backward passes of these deep structured models can be expressed as a set of standard layers
[1, 14, 31, 39]. This allows for fast end-to-end training on GPUs.
However, little to no attention has been given to deep structured models with continuous valued
output variables. One of the main reasons is that inference (even in the shallow model) is much less
well studied, and very few solutions exist. An exception are Markov random fields (MRFs) with
Gaussian potentials, where exact inference is possible (via message-passing) if the precision matrix is
positive semi-definite and satisfies the spectral radius condition [36]. A family of popular approaches
convert the continuous inference problem into a discrete task using particle methods [18, 32]. Specific
solvers have also been designed for certain types of potentials, e.g. polynomials [35] and piecewise
convex functions [37].
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Proximal methods are a popular solution to perform inference in continuous MRFs when the potentials
are non-smooth and non-differentiable functions of the outputs [26]. In this paper, we show that
proximal methods are a special type of recurrent neural networks. This allows us to efficiently train a
wide family of deep structured models with continuous output variables end-to-end on the GPU. We
show that learning can simply be done via back-propagation for any differentiable loss function. We
demonstrate the effectiveness of our algorithm in the tasks of image denoising, depth refinement and
optical flow and show superior results over competing algorithms on these tasks.
2
Proximal Deep Structured Networks
In this section, we first introduce continuous-valued deep structured models and briefly review
proximal methods. We then propose proximal deep structured models and discuss how to do efficient
inference and learning in these models. Finally we discuss the relationship with previous work.
2.1
Continuous-valued Deep Structured Models
Given an input x ? X , let y = (y1 , ..., yN ) be the set of random variables that we are interested
QN
in predicting. The output space is a product space of all the elements: y ? Y = i=1 Yi , and
the domain of each individual variable yi is a closed subset in real-valued space, i.e. Yi ? R. Let
E(x, y; w) : X ? Y ? RK ? R be an energy function which encodes the problem that we are
interested in solving. Without loss of generality we assume that the energy decomposes into a sum of
functions, each depending on a subset of variables
X
X
E(x, y; w) =
fi (yi , x; wu ) +
f? (y? , x; w? )
(1)
?
i
where fi (yi ; x, w) : Yi ? X ? R is a function that depends on a single variable (i.e., unary term)
and f? (y? ) : Y? ? X ? R depends on a subset of variables y? = (yi )i?? defined on a domain
Y? ? Y. Note that, unlike standard MRF models, the functions fi and f? are non-linear functions of
the parameters.
The energy function is parameterized in terms of a set of weights w, and learning aims at finding the
value of these weights which minimizes a loss function. Given an input x, inference aims at finding
the best configuration by minimizing the energy function:
X
X
y? = arg min
fi (yi , x; wu ) +
f? (y? , x; w? )
(2)
y?Y
?
i
?
Finding the best scoring configuration y is equivalent to maximizing the posteriori distribution:
1
p(y|x; w) = Z(x;w)
exp(?E(x, y|w)), with Z(x; w) the partition function.
Standard multi-variate deep networks (e.g., FlowNet [11]) have potential functions which depend on
a single output variable. In this simple case, inference corresponds to a forward pass that predicts the
value of each variable independently. This can be interpreted as inference in a graphical model with
only unary potentials fi .
In the general case, performing inference in MRFs with continuous variables involves solving
a very challenging numerical optimization problem. Depending on the structure and properties
of the potential functions, various methods have been proposed. For instance, particle methods
perform approximate inference by performing message passing on a series of discrete MRFs [18, 32].
Exact inference is possible for a certain type of MRFs, i.e., Gaussian MRFs with positive semidefinite precision matrix. Efficient dedicated algorithms exist for a restricted family of functions,
e.g., polynomials [35]. If certain conditions are satisfied, inference is often tackled by a group of
algorithms called proximal methods [26]. In this section, we will focus on this family of inference
algorithms and show that they are a particular type of recurrent net. We will use this fact to efficiently
train deep structured models with continuous outputs.
2.2 A Review on Proximal Methods
Next, we briefly discuss proximal methods, and refer the reader to [26] for a thorough review.
Proximal algorithms are very generally applicable, but they are particularly successful at solving
non-smooth, non-differentiable, or constrained problems. Their base operation is evaluating the
proximal operator of a function, which involves solving a small convex optimization problem that
2
often admits a closed-form solution. In particular, the proximal operator proxf (x0 ) : R ? R of a
function is defined as
proxf (x0 ) = arg min(y ? x0 )2 + f (y)
y
If f is convex, the fixed points of the proximal operator of f are precisely the minimizers of f . In
other words, proxf (x? ) = x? iff x? minimizes f . This fix-point property motivates the simplest
proximal method called the proximal point algorithm which iterates x(n+1) = proxf (x(n) ). All the
proximal algorithms used here are based on this fix-point property. Note that even if the function
f (?) is not differentiable (e.g., `1 norm) there might exist a closed-form or easy to compute proximal
operator.
While the original proximal operator was designed for the purpose of obtaining the global optimum
in convex optimization, recent work has shown that proximal methods work well for non-convex
optimization as long as the proximal operator exists [20, 30, 5].
For multi-variate optimization problems the proximal operator might not be trivial to obtain (e.g.,
when having high-order potentials). In this case, a widely used solution is to decompose the highorder terms into small problems that can be solved through proximal operators. Examples of this
family of algorithms are half-quadratic splitting [13], alternating direction method of multipliers
[12] and primal-dual methods [2] . In this work, we focus on the non-convex multi-variate case.
2.3
Proximal Deep Structured Models
In order to apply proximal algorithms to tackle the inference problem defined in Eq. (2), we require
the energy functions fi and f? to satisfy the following conditions:
1. There exist functions hi and gi such that fi (yi , x; w) = gi (yi , hi (x, w)), where gi is a
distance function 1 ;
2. There exists a closed-form proximal operator for gi (yi , hi (x; w)) wrt yi .
3. There exist functions h? and g? such that f? (y? , x; w) can be re-written as f? (y? , x; w) =
T
h? (x; w)g? (w?
y? ).
4. There exists a proximal operator for either the dual or primal form of g? (?).
A fairly general family of deep structured models satisfies these conditions. Our experimental
evaluation will demonstrate the applicability in a wide variety of tasks including depth refinement,
image denoising as well as optical flow. If our potential functions satisfy the conditions above, we
can rewrite our objective function as follows:
X
X
T
E(x, y; w) =
gi (yi , hi (x; w)) +
h? (x; w)g? (w?
y? )
(3)
?
i
In this paper, we make the important observation that each iteration of most existing proximal solvers
contain five sub-steps: (i) compute the locally linear part; (ii) compute the proximal operator proxgi ;
(iii) deconvolve; (iv) compute the proximal operator proxg? ; (v) update the result through a gradient
descent step. Due to space restrictions, we show primal-dual solvers in this section, and refer the
reader to the supplementary material for ADMM, half-quadratic splitting and the proximal gradient
method.
The general idea of primal dual solvers is to introduce auxiliary variables z to decompose the highorder terms. We can then minimize z and y alternately through computing their proximal operator.
In particular, we can transform the primal problem in Eq. (3) into the following saddle point problem
X
X
X
T
min max
gi (yi , hi (x, wu )) ?
h? (x, w)g?? (z? ) +
h? (x, w)hw?
y ? , z? i
(4)
y?Y z?Z
i
?
?
where g?? (?) is the convex conjugate of g? (?): g?? (z? ) = sup{hz? , zi ? g? (z)|z ? Z} and the convex
conjugate of g?? is g? itself, if g? (?) is convex.
1
A function g : Y ? Y ? [0, ?) is called a distance function iff it satisfies the condition of non-negativity,
identity of indisernibles, symmetry and triangle inequality.
3
Figure 1: The whole architecture (top) and one iteration block (bottom) of our proximal deep
structured model.
The primal-dual method solves the problem in Eq. (4) by iterating the following steps: (i) fix y
and minimize the energy wrt z; (ii) fix z and minimize the energy wrt y; (iii) conduct a Nesterov
extrapolation gradient step. These iterative computation steps are:
? (t+1)
?
? z?
(t+1)
yi
?
? (t+1)
y?i
(t)
??
T ? (t)
h? (x;w) w? y? )
(t)
??
?T (t+1)
proxgi ,hi (x,w) (yi ? h? (x;w)
w?,i
z
)
(t+1)
(t+1)
(t)
yi
+ ?ex (yi
? yi )
= proxg?? (z? +
=
=
(5)
where y(t) is the solution at the t-th iteration, z(t) is an auxiliary variable and h(x, wu ) is the deep
unary network. Note that different functions gi and g? in (3) have different proximal operators.
It is not difficult to see that the inference process in Eq. (5) can be written as a feed-forward pass in a
recurrent neural network by stacking multiple computation blocks. In particular, the first step is a
convolution layer and the third step can be considered as a deconvolution layer sharing weights with
the first step. The proximal operators are non-linear activation layers and the gradient descent step
is a weighted sum. We also rewrite the scalar multiplication as a 1 ? 1 convolution. We refer the
reader to Fig. 1 for an illustration. The lower figure depicts one iteration of inference while the whole
inference process as a recurrent net is shown in the top figure.
Note that the whole inference process has two stages: first we compute the unaries h(x; wu ) with a
forward pass. Then we perform MAP inference through our recurrent network.
The first non-linearity for the primal dual method is the proximal operator of the dual function
of f? . This changes for other types of proximal methods. In the case of the alternating direction
method of multipliers (ADMM) the nonlinearity corresponds to the proximal operator of f? ; for
half-qudratic splitting it is the proximal operator of f? ?s primal form while the second non linearity is
a least-squares solver; if fi or f? is reduced to a quadratic function of y, the algorithm is simplified,
as the proximal operator of a quadratic function is a linear function [5]. We refer the reader to the
supplementary material for more details on other proximal methods.
2.4
Learning
gt N
Given training pairs composed of inputs {xn }N
n=1 and their corresponding output {yn }n=1 , learning
aims at finding parameters which minimizes a regularized loss function:
X
w? = arg min
`(yn? , yngt ) + ?r(w)
w
n
where `(?) is the loss, r(?) is a regularizer of the weight (we use `2 -norm in practice), yn? is the
minimizer of Eq. (3) for the n-th example and ? is a scalar. Given the conditions that both proxfi and
proxg? (or proxg?? ) are sub-differentiable wrt. w and y, back-propagation can be used to compute
the gradient efficiently. We refer the reader to Fig. 2 for an illustration of our learning algorithm.
Parameters such as the gradient steps ?? , ?? , ?ex in Eq. (5) are considered hyper-parameters in
proximal methods and are typically manually set. In contrast, we can learn them as they are 1 ? 1
convolution weights.
4
Algorithm: Learning Continuous-Valued Deep Structured Models
Repeat until stopping criteria
1.
2.
3.
4.
Forward pass to compute hi (x, w) and h? (x, w)
Compute y? i via forward pass in Eq. (5)
Compute the gradient via backward pass
Parameter update
Figure 2: Algorithm for learning proximal deep structured models.
Non-shared weights: The weights and gradient steps for high-order potentials are shared among
all the iteration blocks in the inference network, which guarantees the feed-forward pass to explicitly
minimize the energy function in Eq. (2). In practice we found that by removing the weight-sharing
and fixed gradient step constraints, we can give extra flexibility to our model, boosting the final
performance. This observation is consistent with the findings of shrinkage field [30] and inference
machines [27].
Multi-loss: Intermediate layer outputs y(t) should gradually converge towards the final output.
Motivated by this fact, we include a loss over the intermediate computations to accelerate convergence.
2.5
Discussion and Related Work
Our approach can be considered as a continuous-valued extension of deep structured models [3,
31, 39]. Unlike previous methods where the output lies in a discrete domain and inference is
conducted through a specially designed message passing layer, the output of the proposed method is
in continuous domain and inference is done by stacking convolution and non-linear activation layers.
Without deep unary potentials, our model is reduced to a generalized version of field-of-experts [28].
The idea of stacking shrinkage functions and convolutions as well as learning iteration-specific
weights was exploited in the learning iterative shrinkage algorithm (LISTA) [14]. LISTA can be
considered as a special case of our proposed model with sparse coding as the energy function and
proximal gradient as the inference algorithm. Our approach is also closely related to the recent
structured prediction energy networks (SPEN) [1], where our unary network is analogous to the
feature net in SPEN and the whole energy model is analogous to the energy net. Both SPEN and our
proposed method can be considered as a special case of optimization-based learning [8]. However,
SPEN utilizes plain gradient descent for inference while our network is proximal algorithm motivated.
Previous methods have tried to learn multi-variate regression networks for optical flow [11] and
stereo [24]. But none of these approaches model the interactions between output variables. Thus,
they can be considered a special case of our model, where only unary functions fi are present.
3
Experiments
We demonstrate the effectiveness of our approach in three different applications: image denoising,
depth refinement and optical flow estimation. We employ mxnet [4] with CUDNNv4 acceleration to
implement the networks, which we train end-to-end. Our experiments are conducted on a Xeon 3.2
Ghz machine with a Titan X GPU.
3.1
Image Denoising
We first evaluate our method for the task of image denoising (i.e., shallow unary) using the BSDS
image dataset [23]. We corrupt each image with Gaussian noise with standard deviation ? = 25. We
use the energy function typically employed for image denoising:
y? = arg min
y?Y
X
kyi ? xi k22 + ?
X
T
kwho,?
y? k1
(6)
?
i
According to the primal dual algorithm, the activation function for the first nonlinearity is the proximal
operator of the dual function of the `1 -norm: prox?? (z) = min(|z|, 1)?sign(z), which is the projection
onto an `? -norm ball. In practice we encode this function as prox?? (z) = max(min(z, 1), ?1). The
5
PSNR
Time (second)
BM3D [6] EPLL [40] LSSC [22] CSF [30] RTF [29] Ours Ours GPU
28.56
28.68
28.70
28.72
28.75
28.79
28.79
2.57
108.72
516.48
5.10
69.25
0.23
0.011
Table 1: Natural Image Denoising on BSDS dataset [23] with noise variance ? = 25.
16
32
64
3?3
28.43
28.48
28.49
5?5
28.57
28.64
28.68
7?7
28.68
28.76
28.79
Table 2: Performance of the proposed model with different hyper-parameters
Figure 3: Qualitative results for image denoising. Left to right: noisy input, ground-truth, our result.
second nonlinearity is the proximal operator of the primal function of the `2 -norm, which is a
weighted sum: prox`2 (y, ?) = x+?y
1+? .
For training, we select 244 images, following the configuration of [30]. We randomly cropped
128 ? 128 clean patches from the training images and obtained the noisy input by adding random
noise. We use mean square error as the loss function and set a weight decay strength of 0.0004 for all
settings. Note that for all the convolution and deconvolution layers, the bias is set to zero. MSRA
initialization [16] is used for the convolution parameters and the initial gradient step for each iteration
is set to be 0.02. We use adam [19] with a learning rate of t = 0.02 and hyper-parameters ?1 = 0.9
and ?2 = 0.999 as in Kingma et al. [19]. The learning rate is divided by 2 every 50 epoch, and we
use a mini-batch size of 32.
We compare against a number of recent state-of-the-art techniques [6, 40, 22, 30, 29]. 2 The Peak
Signal-to-Noise Ratio (PSNR) is used as a performance measure. As shown in Tab. 1 our proposed
method outperforms all methods in terms of accuracy and speed. The second best performing
method is RTF [29], while being two orders of magnitude slower than our approach. Our GPU
implementation achieves real-time performance with more than 90 frames/second. Note that a GPU
version of CSF is reported to run at 0.92s on a 512 ? 512 image on a GTX 480 [30]. However, since
GPU implementation is not available online, we cannot make proper comparisons.
Tab. 2 shows performance with different hyper-parameters (filter size, number of filters per each layer).
As we can see, larger receptive fields and more convolution filters slightly boost the performance.
Fig. 3 depicts the qualitative results of our model for the denoising task.
3.2
Depth Refinement
Due to specularities and intensity changes of structured light imaging, the sensor?s output depth is
often noisy. Thus, refining the depth to generate a cleaner, more accurate depth image is an important
task. We conduct the depth refinement experiment on the 7 Scenes dataset [25]. We follow the
configuration of [10], where the ground-truth depth was computed using KinectFusion [25]. The
noise [10] has a Poisson-like distribution and is depth-dependent, which is very different from the
image denoising experiment which contained Gaussian noise.
We use the same architecture as for the task of natural image denoising. The multi-stage mean square
error is used as loss function and the weight decay strength is set to be 0.0004. Adam (?1 = 0.9 and
2
We chose the model with the best performance for each competing algorithm. For the CSF method, we use
CSF57?7 ; for RTF we use RTF5 ; for our method, we pick 7 ? 7 ? 64 high-order structured network.
6
Figure 4: Qualitative results for depth refinement. Left to right: input, ground-truth, wiener filter,
bilateral filter, BM3D, Filter Forest, Ours.
PSNR
Wiener
32.29
Bilateral
30.95
LMS
24.37
BM3D [6]
35.46
FilterForest [10]
35.63
Ours
36.31
Table 3: Performance of depth refinement on dataset [10]
Figure 5: Optical flow: Left to right: first and second input, ground-truth, Flownet [11], ours.
?2 = 0.999) is used as the optimizer with a learning rate of 0.01. Data augmentation is used to avoid
overfitting, including random cropping, flipping and rotation. We used a mini-batch size of 16.
We train our model on 1000 frames of the Chess scene and test on the other scenes. PSNR is used to
evaluate the performance. As shown in Tab. 3, our approach outperforms all competing algorithms.
This shows that our deep structured network is able to handle non-additive non-Gaussian noise.
Qualitative results are shown in Fig. 4. Compared to the competing approaches, our method is able to
recover better depth estimates particularly along the depth discontinuities.
3.3
Optical Flow
We evaluate the task of optical flow estimation on the Flying Chairs dataset [11]. The size of training
images is 512 ? 384. We formulate the energy as follows:
X
X
T
y? = arg min
kyi ? fi (xl , xr ; wu )k1 + ?
kwho,?
y? k1
(7)
y?Y
?
i
where fi (xl , xr ; wu ) is a Flownet model [11], is a fully-convolutional encoder-decoder network
that predicts 2D optical flow per pixel. It has 11 encoding layers and 11 deconv layers with skip
connections. xl and xr are the left and right input images respectively and y is the desired optical
flow output. Note that we use the `1 -norm for both, the data and the regularization term. The first
nonlinearity activation function is the proximal operator of the `1 -norm?s dual function: prox?? (z) =
min(|z|, 1) ? sign(z), and the second non-linear activation function is the proximal operator of the
`1 -norm?s primal form: prox?,x (y, ?) = x ? min(|x ? y|, ?) ? sign(x), which is a soft shrinkage
function [26].
7
End-point-error
Flownet
4.98
Flownet + TV-l1
4.96
Our proposed
4.91
Table 4: Performance of optical flow on Flying chairs dataset [11]
We build a deep structured model with 5 iteration blocks. Each iteration block has 32 convolution
filters of size 7 ? 7 for both the convolution and deconvolution layers, which results in 10 convolution/deconv layers and 10 non-linearities. The multi-stage mean square error is used as the loss
function and the weight decay strength is set to be 0.0004.
Training is conducted on the training subset of the Flying Chairs dataset. Our unary model is
initialized with a pre-trained Flownet parameters. The high-order term is initialized with MSRA
random initialization [16]. The hyper-parameter ? in this experiment is pre-set to be 10. We use
random flipping, cropping and color-tuning for data augmentation, and employ the adam optimizer
with the same configuration as before (?1 = 0.9 and ?2 = 0.999) with a learning rate t = 0.005. The
learning rate is divided by 2 every 10 epoch and the mini-batch size is set to be 12.
We evaluate all approaches on the test set of the Flying chairs dataset. End-point error is used as a
measure of performance. The unary-only model (i.e. plain flownet) is used as baseline and we also
compare against a plain TV-l1 model with four pre-set gradient operators as post-processing. As
shown in Tab. 4 our method outperforms all the baselines. From Fig. 5 we can see that our method is
less noisy than Flownet?s output and better preserves the boundaries. Note that our current model is
isotropic. In order to further boost the performance, incorporating anisotropic filtering like bilateral
filtering is an interesting future direction.
4
Conclusion
We have proposed a deep structured model that learns non-linear functions encoding complex
dependencies between continuous output variables. We have showed that inference in our model
using proximal methods can be efficiently solved as a feed-foward pass on a special type of deep
recurrent neural network. We demonstrated our approach in the tasks of image denoising, depth
refinement and optical flow. In the future we plan to investigate other proximal methods and a wider
variety of applications.
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5,608 | 6,075 | Single Pass PCA of Matrix Products
Shanshan Wu
The University of Texas at Austin
shanshan@utexas.edu
Srinadh Bhojanapalli
Toyota Technological Institute at Chicago
srinadh@ttic.edu
Sujay Sanghavi
The University of Texas at Austin
sanghavi@mail.utexas.edu
Alexandros G. Dimakis
The University of Texas at Austin
dimakis@austin.utexas.edu
Abstract
In this paper we present a new algorithm for computing a low rank approximation
of the product AT B by taking only a single pass of the two matrices A and B. The
straightforward way to do this is to (a) first sketch A and B individually, and then
(b) find the top components using PCA on the sketch. Our algorithm in contrast
retains additional summary information about A, B (e.g. row and column norms
etc.) and uses this additional information to obtain an improved approximation from
the sketches. Our main analytical result establishes a comparable spectral norm
guarantee to existing two-pass methods; in addition we also provide results from
an Apache Spark implementation1 that shows better computational and statistical
performance on real-world and synthetic evaluation datasets.
1
Introduction
Given two large matrices A and B we study the problem of finding a low rank approximation of their
product AT B, using only one pass over the matrix elements. This problem has many applications in
machine learning and statistics. For example, if A = B, then this general problem reduces to Principal
Component Analysis (PCA). Another example is a low rank approximation of a co-occurrence matrix
from large logs, e.g., A may be a user-by-query matrix and B may be a user-by-ad matrix, so AT B
computes the joint counts for each query-ad pair. The matrices A and B can also be two large bag-ofword matrices. For this case, each entry of AT B is the number of times a pair of words co-occurred
together. As a fourth example, AT B can be a cross-covariance matrix between two sets of variables,
e.g., A and B may be genotype and phenotype data collected on the same set of observations. A low
rank approximation of the product matrix is useful for Canonical Correlation Analysis (CCA) [3].
For all these examples, AT B captures pairwise variable interactions and a low rank approximation is
a way to efficiently represent the significant pairwise interactions in sub-quadratic space.
Let A and B be matrices of size d ? n (d
n) assumed too large to fit in main memory. To obtain
a rank-r approximation of AT B, a naive way is to compute AT B first, and then perform truncated
singular value decomposition (SVD) of AT B. This algorithm needs O(n2 d) time and O(n2 ) memory
to compute the product, followed by an SVD of the n ? n matrix. An alternative option is to directly
run power method on AT B without explicitly computing the product. Such an algorithm will need to
access the data matrices A and B multiple times and the disk IO overhead for loading the matrices to
memory multiple times will be the major performance bottleneck.
For this reason, a number of recent papers introduce randomized algorithms that require only a few
passes over the data, approximately linear memory, and also provide spectral norm guarantees. The
1
The code can be found at https://github.com/wushanshan/MatrixProductPCA
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
key step in these algorithms is to compute a smaller representation of data. This can be achieved by
two different methods: (1) dimensionality reduction, i.e., matrix sketching [15, 5, 14, 6]; (2) random
sampling [7, 1]. The recent results of Cohen et al. [6] provide the strongest spectral norm guarantee
eT B
e to
of the former. They show that a sketch size of O(?
r/?2 ) suffices for the sketched matrices A
T e
e
achieve a spectral error of ?, where r? is the maximum stable rank of A and B. Note that A B is
not the desired rank-r approximation of AT B. On the other hand, [1] is a recent sampling method
with very good performance guarantees. The authors consider entrywise sampling based on column
norms, followed by a matrix completion step to compute low rank approximations. There is also a lot
of interesting work on streaming PCA, but none can be directly applied to the general case when A is
different from B (see Figure 4(c)). Please refer to Appendix D for more discussions on related work.
Despite the significant volume of prior work, there is no method that computes a rank-r approximation
of AT B when the entries of A and B are streaming in a single pass 2 . Bhojanapalli et al. [1] consider
a two-pass algorithm which computes column norms in the first pass and uses them to sample in
a second pass over the matrix elements. In this paper, we combine ideas from the sketching and
sampling literature to obtain the first algorithm that requires only a single pass over the data.
Contributions: We propose a one-pass algorithm SMP-PCA (which stands for Streaming Matrix
2 3
Product PCA) that computes a rank-r approximation of AT B in time O((nnz(A) + nnz(B)) ? ?r2 r? +
nr 6 ?4 r?3
).
?4
Here nnz(?) is the number of non-zero entries, ? is the condition number, r? is the maximum
stable rank, and ? measures the spectral norm error. Existing two-pass algorithms such as [1] typically
have longer runtime than our algorithm (see Figure 3(a)). We also compare our algorithm with the
simple idea that first sketches A and B separately and then performs SVD on the product of their
sketches. We show that our algorithm always achieves better accuracy and can perform arbitrarily
better if the column vectors of A and B come from a cone (see Figures 2, 4(b), 3(b)).
The central idea of our algorithm is a novel rescaled JL embedding that combines information from
matrix sketches and vector norms. This allows us to get better estimates of dot products of high
dimensional vectors compared to previous sketching approaches. We explain the benefit compared to
a naive JL embedding in Figure 2 and the related discussion; we believe it may be of more general
interest beyond low rank matrix approximations.
We prove that our algorithm recovers a low rank approximation of AT B up to an error that depends
on kAT B (AT B)r k and kAT Bk, decaying with increasing sketch size and number of samples
(Theorem 3.1). The first term is a consequence of low rank approximation and vanishes if AT B is
exactly rank-r. The second term results from matrix sketching and subsampling; the bounds have
similar dependencies as in [6].
We implement SMP-PCA in Apache Spark and perform several distributed experiments on synthetic
and real datasets. Our distributed implementation uses several design innovations described in
Section 4 and Appendix C.5 and it is the only Spark implementation that we are aware of that
can handle matrices that are large in both dimensions. Our experiments show that we improve by
approximately a factor of 2? in running time compared to the previous state of the art and scale
gracefully as the cluster size increases. The source code is available at [18].
In addition to better performance, our algorithm offers another advantage: It is possible to compute
low-rank approximations to AT B even when the entries of the two matrices arrive in some arbitrary
order (as would be the case in streaming logs). We can therefore discover significant correlations
even when the original datasets cannot be stored, for example due to storage or privacy limitations.
2
Problem setting and algorithms
Consider the following problem: given two matrices A 2 Rd?n1 and B 2 Rd?n2 that are stored in
disk, find a rank-r approximation of their product AT B. In particular, we are interested in the setting
where both A, B and AT B are too large to fit into memory. This is common for modern large scale
machine learning applications. For this setting, we develop a single-pass algorithm SMP-PCA that
computes the rank-r approximation without explicitly forming the entire matrix AT B.
2
One straightforward idea is to sketch each matrix individually and perform SVD on the product of the
sketches. We compare against that scheme and show that we can perform arbitrarily better using our rescaled JL
embedding.
2
Notations. Throughout the paper, we use A(i, j) or Aij to denote (i, j) entry for any matrix A. Let
Ai and Aj be the i-th column vector and j-th row vector. We use kAkF for Frobenius norm, and
kAk for spectral (or operator) norm. The optimal rank-r approximation of matrix A is Ar , which
can be found by SVD. For any positive integer n, let [n] denote the set {1, 2, ? ? ? , n}. Given a set
? ? [n1 ] ? [n2 ] and a matrix A 2 Rn1 ?n2 , we define P? (A) 2 Rn1 ?n2 as the projection of A on ?,
i.e., P? (A)(i, j) = A(i, j) if (i, j) 2 ? and 0 otherwise.
2.1
SMP-PCA
Our algorithm SMP-PCA (Streaming Matrix Product PCA) takes four parameters as input: the
desired rank r, number of samples m, sketch size k, and the number of iterations T . Performance
guarantee involving these parameters is provided in Theorem 3.1. As illustrated in Figure 1, our
algorithm has three main steps: 1) compute sketches and side information in one pass over A and B;
2) given partial information of A and B, estimate important entries of AT B; 3) compute low rank
approximation given estimates of a few entries of AT B. Now we explain each step in detail.
Figure 1: An overview of our algorithm. A single pass is performed over the data to produce the
e B
e and the column norms kAi k, kBj k, for i 2 [n1 ] and j 2 [n2 ]. We then
sketched matrices A,
f) through a biased sampling process, where P? (M
f) = M
f(i, j) if
compute the sampled matrix P? (M
f
(i, j) 2 ? and zero otherwise. Here ? represents the set of sampled entries. The (i, j)-th entry of M
f
is given in Eq. (2). Performing matrix completion on P? (M ) gives the desired rank-r approximation.
Algorithm 1 SMP-PCA: Streaming Matrix Product PCA
1: Input: A 2 Rd?n1 , B 2 Rd?n2 , desired rank: r, sketch size: k, number of samples: m, number
of iterations: T
2: Construct a random matrix ? 2 Rk?d , where ?(i, j) ? N (0, 1/k), 8(i, j) 2 [k] ? [d]. Perform
3:
4:
5:
6:
e = ?A, B
e = ?B, and kAi k, kBj k, for i 2 [n1 ] and
a single pass over A and B to obtain: A
j 2 [n2 ].
Sample each entry (i, j) 2 [n1 ] ? [n2 ] independently with probability q?ij = min{1, qij }, where
qij is defined in Eq.(1); maintain a set ? ? [n1 ] ? [n2 ] which stores all the sampled pairs (i, j).
f 2 Rn1 ?n2 , where M
f(i, j) is given in Eq. (2). Calculate P? (M
f) 2 Rn1 ?n2 , where
Define M
f) = M
f(i, j) if (i, j) 2 ? and zero otherwise.
P? ( M
f), ?, r, q?, T ), see Appendix A for more details.
Run WAltMin(P? (M
n1 ?r
b
Output: U 2 R
and Vb 2 Rn2 ?r .
Step 1: Compute sketches and side information in one pass over A and B. In this step we
e := ?A and B
e := ?B, where ? 2 Rk?d is a random matrix with entries being
compute sketches A
i.i.d. N (0, 1/k) random variables. It is known that ? satisfies an "oblivious Johnson-Lindenstrauss
(JL) guarantee" [15][17] and it helps preserving the top row spaces of A and B [5]. Note that any
sketching matrix ? that is an oblivious subspace embedding can be considered here, e.g., sparse JL
transform and randomized Hadamard transform (see [6] for more discussion).
e and B,
e we also compute the L2 norms for all column vectors, i.e., kAi k and kBj k, for
Besides A
i 2 [n1 ] and j 2 [n2 ]. We use this additional information to design better estimates of AT B in the
3
eT B
e to sample. Note that this is the only step
next step, and also to determine important entries of A
that needs one pass over data.
Step 2: Estimate important entries of AT B by rescaled JL embedding. In this step we use partial
eT B.
e We first
information obtained from the previous step to compute a few important entries of A
T
e B
e to sample, and then propose a novel rescaled JL embedding for
determine what entries of A
estimating those entries.
We sample entry (i, j) of AT B independently with probability q?ij = min{1, qij }, where
kAi k2
kBj k2
+
).
(1)
2n2 kAk2F
2n1 kBk2F
P
Let ? ? [n1 ] ? [n2 ] be the set of sampled entries (i, j). Since E( i,j qij ) = m, the expected number
of sampled entries is roughly m. The special form of qij ensures that we can draw m samples in
O(n1 + m log(n2 )) time; we show how to do this in Appendix C.5.
qij = m ? (
Note that qij intuitively captures important entries of AT B by giving higher weight to heavy rows
and columns. We show in Section 3 that this sampling actually generates good approximation to the
matrix AT B.
The biased sampling distribution of Eq. (1) is first proposed by Bhojanapalli et al. [1]. However, their
algorithm [1] needs a second pass to compute the sampled entries, while we propose a novel way of
estimating dot products, using information obtained in the first step.
f 2 Rn1 ?n2 as
Define M
f(i, j) = kAi k ? kBj k ?
M
eT B
ej
A
i
.
ei k ? kB
ej k
kA
(2)
f, instead, we only calculate M
f(i, j) for (i, j) 2 ?. This
Note that we will not compute and store M
f), where P? (M
f)(i, j) = M
f(i, j) if (i, j) 2 ? and 0 otherwise.
matrix is denoted as P? (M
2
Estimated dot product
JL embedding
Rescaled JL embedding
1
0
-1
-2
-1
-0.5
0
0.5
1
True dot product
(a)
(b)
Figure 2: (a) Rescaled JL embedding (red dots) captures the dot products with smaller variance
compared to JL embedding (blue triangles). Mean squared error: 0.053 versus 0.129. (b) Lower
figure illustrates how to construct unit-norm vectors from a cone with angle ?. Let x be a fixed
unit-norm vector, and let t be a random Gaussian vector with expected norm tan(?/2), we set y as
either x + t or (x + t) with probability half, and then normalize it. Upper figure plots the ratio of
T
eT Bk/kA
e
fk, when the column vectors of A and B are unit
spectral norm errors kAT B A
B M
f has better accuracy than A
eT B
e for all possible
vectors drawn from a cone with angle ?. Clearly, M
values of ?, especially when ? is small.
f is a better estimator than A
eT B.
e To estimate the
We now explain the intuition of Eq. (2), and why M
T
T e
e
e
e
e
(i, j) entry of A B, a straightforward way is to use Ai Bj = kAi k ? kBj k ? cos ?ij , where ?eij is the
4
ei and B
ej . Since we already know the actual column norms, a potentially
angle between vectors A
better estimator would be kAi k ? kBj k ? cos ?eij . This removes the uncertainty that comes from
distorted column norms3 .
eT B
ej (JL embedding) and M
f(i, j) (rescaled JL embedding)
Figure 2(a) compares the two estimators A
i
for dot products. We plot simulation results on pairs of unit-norm vectors with different angles. The
vectors have dimension 1,000 and the sketching matrix has dimension 10-by-1,000. Clearly rescaling
by the actual norms help reduce the estimation uncertainty. This phenomenon is more prominent
when the true dot products are close to ?1, which makes sense because cos ? has a small slope
when cos ? approaches ?1, and hence the uncertainty from angles may produce smaller distortion
compared to that from norms. In the extreme case when cos ? = ?1, rescaled JL embedding can
perfectly recover the true dot product4 .
In the lower part of Figure 2(b) we illustrate how to construct unit-norm vectors from a cone with
angle ?. Given a fixed unit-norm vector x, and a random Gaussian vector t with expected norm
tan(?/2), we construct new vector y by randomly picking one from the two possible choices x+t and
(x + t), and then renormalize it. Suppose the columns of A and B are unit vectors randomly drawn
T
eT Bk/kA
e
fk in
from a cone with angle ?, we plot the ratio of spectral norm errors kAT B A
B M
T e
f
e
Figure 2(b). We observe that M always outperforms A B and can be much better when ? approaches
zero, which agrees with the trend indicated in Figure 2(a).
Step 3: Compute low rank approximation given estimates of few entries of AT B. Finally we
compute the low rank approximation of AT B from the samples using alternating least squares:
X
f(i, j))2 ,
min
wij (eTi U V T ej M
(3)
U,V 2Rn?r
(i,j)2?
where wij = 1/?
qij denotes the weights, and ei , ej are standard base vectors. This is a popular
technique for low rank recovery and matrix completion (see [1] and the references therein). After T
f presented in the convenient factored form. This
iterations, we will get a rank-r approximation of M
subroutine is quite standard, so we defer the details to Appendix A.
3
Analysis
Now we present the main theoretical result. Theorem 3.1 characterizes the interaction between
the sketch size k, the sampling complexity m, the number of iterations T , and the spectral error
[
[
T B k, where A
T B is the output of SMP-PCA, and (AT B) is the optimal rank-r
k(AT B)r A
r
r
r
T
approximation of A B. Note that the following theorem assumes that A and B have the same size.
For the general case of n1 6= n2 , Theorem 3.1 is still valid by setting n = max{n1 , n2 }.
Theorem 3.1. Given matrices A 2 Rd?n and B 2 Rd?n , let (AT B)r be the optimal rank-r
?
kAk2 kBk2
approximation of AT B. Define r? = max{ kAkF2 , kBkF2 } as the maximum stable rank, and ? = 1? as
r
the condition number of (AT B)r , where i? is the i-th singular values of AT B.
[
T B be the output of Algorithm SMP-PCA. If the input parameters k, m, and T satisfy
Let A
r
C1 kAk2 kBk2 ?2 r3 max{?
r, 2 log(n)} + log (3/ )
?
,
(4)
kAT Bk2F
?2
?
?2
C2 r?2
kAk2F + kBk2F
nr3 ?2 log(n)T 2
m
?
?
,
(5)
T
kA BkF
?2
kAkF + kBkF
T log(
),
(6)
?
where C1 and C2 are some global constants independent of A and B. Then with probability at least
1
, we have
[
T B k ? ?kAT B
k(AT B)r A
(AT B)r kF + ? + ? r? .
(7)
r
k
3
We also tried using the cosine rule for computing the dot product, and another sketching method specifically
designed for preserving angles [2], but empirically those methods perform worse than our current estimator.
4
See http://wushanshan.github.io/files/RescaledJL_project.pdf for more results.
5
Remark 1. Compared to the two-pass algorithm proposed by [1], we notice that Eq. (7) contains
an additional error term ? r? . This extra term captures the cost incurred when we are approximating
entries of AT B by Eq. (2) instead of using the actual values. The exact tradeoff between ? and k is
given by Eq. (4). On one hand, we want to have a small k so that the sketched matrices can fit into
memory. On the other hand, the parameter k controls how much information is lost during sketching,
and a larger k gives a more accurate estimation of the inner products.
kAk2 +kBk2
Remark 2. The dependence on kAFT BkF F captures one difficult situation for our algorithm. If
kAT BkF is much smaller than kAkF or kBkF , which could happen, e.g., when many column
vectors of A are orthogonal to those of B, then SMP-PCA requires many samples to work. This is
reasonable. Imagine that AT B is close to an identity matrix, then it may be hard to tell it from an
all-zero matrix without enough samples. Nevertheless, removing this dependence is an interesting
direction for future research.
Remark 3. For the special case of A = B, SMP-PCA computes a rank-r approximation of matrix
AT A in a single pass. Theorem 3.1 provides an error bound in spectral norm for the residual matrix
[
T A . Most results in the online PCA literature use Frobenius norm as performance
(AT A)r A
r
measure. Recently, [10] provides an online PCA algorithm with spectral norm guarantee. They
?
achieves a spectral norm bound of ? 1? + r+1
, which is stronger than ours. However, their algorithm
requires a target dimension of O(r log n/?2 ), i.e., the output is a matrix of size n-by-O(r log n/?2 ),
while the output of SMP-PCA is simply n-by-r.
Remark 4. We defer our proofs to Appendix C. The proof proceeds in three steps. In Appendix C.2,
we show that the sampled matrix provides a good approximation of the actual matrix AT B. In
Appendix C.3, we show that there is a geometric decrease in the distance between the computed
b , Vb and the optimal ones U ? , V ? at each iteration of WAltMin algorithm. The spectral
subspaces U
norm bound in Theorem 3.1 is then proved using results from the previous two steps.
Computation Complexity. We now analyze the computation complexity of SMP-PCA. In Step
1, we compute the sketched matrices of A and B, which requires O(nnz(A)k + nnz(B)k) flops.
Here nnz(?) denotes the number of non-zero entries. The main job of Step 2 is to sample a set ?
and calculate the corresponding inner products, which takes O(m log(n) + mk) flops. Here we
define n as max{n1 , n2 } for simplicity. According to Eq. (4), we have log(n) = O(k), so Step 2
takes O(mk) flops. In Step 3, we run alternating least squares on the sampled matrix, which can
be completed in O((mr2 + nr3 )T ) flops. Since Eq. (5) indicates nr = O(m), the computation
complexity of Step 3 is O(mr2 T ). Therefore, SMP-PCA has a total computation complexity
O(nnz(A)k + nnz(B)k + mk + mr2 T ).
4
Numerical Experiments
Spark implementation. We implement our SMP-PCA in Apache Spark 1.6.2 [19]. For the purpose
of comparison, we also implement a two-pass algorithm LELA [1] in Spark5 . The matrices A
and B are stored as a resilient distributed dataset (RDD) in disk (by setting its StorageLevel
as DISK_ONLY). We implement the two passes of LELA using the treeAggregate method.
For SMP-PCA, we choose the subsampled randomized Hadamard transform (SRHT) [16] as the
sketching matrix. The biased sampling procedure is performed using binary search (see Appendix C.5
for how to sample m elements in O(m log n) time). After obtaining the sampled matrix, we run ALS
(alternating least squares) to get the desired low-rank matrices. More details can be found at [18].
Description of datasets. We test our algorithm on synthetic datasets and three real datasets:
SIFT10K [9], NIPS-BW [11], and URL-reputation [12]. For synthetic data, we generate matrices A and B as GD, where G has entries independently drawn from standard Gaussian distribution,
and D is a diagonal matrix with Dii = 1/i. SIFT10K is a dataset of 10,000 images. Each image is
represented by 128 features. We set A as the image-by-feature matrix. The task here is to compute
a low rank approximation of AT A, which is a standard PCA task. The NIPS-BW dataset contains
bag-of-words features extracted from 1,500 NIPS papers. We divide the papers into two subsets,
and let A and B be the corresponding word-by-paper matrices, so AT B computes the counts of
co-occurred words between two sets of papers. The original URL-reputation dataset has 2.4 million
5
To our best knowledge, this the first distributed implementation of LELA.
6
Runtime (sec) vs Cluster size
2000
1000
Spectral norm error
LELA
SMC-PCA
3000
!T B)
!
SVD(A
SMP-PCA
LELA
Optimal
0.2
0.15
0.2
0.15
0.1
0.05
0.05
1000
5
0.25
0.1
0
2
!T B)
!
SVD(A
SMP-PCA
LELA
Optimal
0.3
2000
1000
Sketch size (k)
10
(a)
2000
Sketch size (k)
(b)
Figure 3: (a) Spark-1.6.2 running time on a 150GB dataset. All nodes are m3.2xlarge EC2 instances.
See [18] for more details. (b) Spectral norm error achieved by three algorithms over two datasets:
eT B)
e by a factor of 1.8 for
SIFT10K (left) and NIPS-BW (right). SMP-PCA outperforms SVD(A
SIFT10K and 1.1 for NIPS-BW. The error of SMP-PCA keeps decreasing as the sketch size k grows.
URLs. Each URL is represented by 3.2 million features, and is indicated as malicious or benign. This
dataset has been used previously for CCA [13]. Here we extract two subsets of features, and let A and
B be the corresponding URL-by-feature matrices. The goal is to compute a low rank approximation
of AT B, the cross-covariance matrix between two subsets of features.
Sample complexity. In Figure 4(a) we present simulation results on a small synthetic data with
n = d = 5, 000 and r = 5. We observe that a phase transition occurs when the sample complexity
m = ?(nr log n). This agrees with the experimental results shown in previous papers, see, e.g., [4, 1].
For all rest experiments, unless otherwise specified, we set r = 5, T = 10, and m as 4nr log n.
Table 1: A comparison of spectral norm error over three datasets
Dataset
d
n
Algorithm
Sketch size k
Error
Synthetic
100,000
100,000
Optimal
LELA
SMP-PCA
2,000
0.0271
0.0274
0.0280
URLmalicious
792,145
10,000
Optimal
LELA
SMP-PCA
2,000
0.0163
0.0182
0.0188
URLbenign
1,603,985
10,000
Optimal
LELA
SMP-PCA
2,000
0.0103
0.0105
0.0117
Comparison of SMP-PCA and LELA. We now compare the statistical performance of SMP-PCA
and LELA [1] on three real datasets and one synthetic dataset. As shown in Figure 3(b) and Table 1,
LELA always achieves a smaller spectral norm error than SMP-PCA, which makes sense because
LELA takes two passes and hence has more chances exploring the data. Besides, we observe that as
the sketch size increases, the error of SMP-PCA keeps decreasing and gets closer to that of LELA.
In Figure 3(a) we compare the runtime of SMP-PCA and LELA using a 150GB synthetic dataset on
m3.2xlarge Amazon EC2 instances6 . The matrices A and B have dimension n = d = 100, 000. The
sketch dimension is set as k = 2, 000. We observe that the speedup achieved by SMP-PCA is more
prominent for small clusters (e.g., 56 mins versus 34 mins on a cluster of size two). This is possibly
due to the increasing spark overheads at larger clusters, see [8] for more related discussion.
eT B).
e In Figure 4(b) we repeat the experiment in Section 2
Comparison of SMP-PCA and SVD(A
eT B)
e refers to
by generating column vectors of A and B from a cone with angle ?. Here SVD(A
6
Each machine has 8 cores, 30GB memory, and 2?80GB SSD.
7
105
Ratio of errors vs theta
Spectral norm error
Spectral norm error
0.5
k = 400
0.4
k = 800
0.3
0.2
1
2
3
4
# Samples / nrlogn
(a)
100
0
? /4
? /2 3 ? /4
(b)
?
1
0.8
ATr Br
SMP-PCA
0.6
0.4
0.2
200 400 600 800 1000
Sketch size (k)
(c)
Figure 4: (a) A phase transition occurs when the sample complexity m = ?(nr log n). (b) This
eT B)
e over that of SMP-PCA. The columns of
figure plots the ratio of spectral norm error of SVD(A
A and B are unit vectors drawn from a cone with angle ?. We see that the ratio of errors scales to
infinity as the cone angle shrinks. (c) If the top r left singular vectors of A are orthogonal to those of
B, the product ATr Br is a very poor low rank approximation of AT B.
eT B)
e
computing SVD on the sketched matrices7 . We plot the ratio of the spectral norm error of SVD(A
over that of SMP-PCA, as a function of ?. Note that this is different from Figure 2(b), as now we
take the effect of random sampling and SVD into account. However, the trend in both figures are the
eT B)
e and can be arbitrarily better as ? goes to zero.
same: SMP-PCA always outperforms SVD(A
eT B)
e on two real datasets SIFK10K and NIPS-BW.
In Figure 3(b) we compare SMP-PCA and SVD(A
[
[
T B ||/||AT B||, where A
T B is
The y-axis represents spectral norm error, defined as ||AT B A
r
r
the rank-r approximation found by a specific algorithm. We observe that SMP-PCA outperforms
eT B)
e by a factor of 1.8 for SIFT10K and 1.1 for NIPS-BW.
SVD(A
eT B).
e The reasons are
Now we explain why SMP-PCA produces a more accurate result than SVD(A
f is a better estimator for AT B than A
eT B
e (Figure 2).
twofold. First, our rescaled JL embedding M
f, and
Second, the noise due to sampling is relatively small compared to the benefit obtained from M
T e
f
e
hence the final result computed using P? (M ) still outperforms SVD(A B).
Comparison of SMP-PCA and ATr Br . Let Ar and Br be the optimal rank-r approximation of A
and B, we show that even if one could use existing methods (e.g., algorithms for streaming PCA)
to estimate Ar and Br , their product ATr Br can be a very poor low rank approximation of AT B.
This is demonstrated in Figure 4(c), where we intentionally make the top r left singular vectors of A
orthogonal to those of B.
5
Conclusion
We develop a novel one-pass algorithm SMP-PCA that directly computes a low rank approximation
of matrix product, using ideas of matrix sketching and entrywise sampling. As a subroutine of our
algorithm, we propose rescaled JL for estimating entries of AT B, which has smaller error compared
? This we believe can be extended to other applications. Moreover,
to the standard estimator A?T B.
SMP-PCA allows the non-zero entries of A and B to be presented in any arbitrary order, and hence
can be used for steaming applications. We design a distributed implementation for SMP-PCA. Our
eT B),
e and is
experimental results show that SMP-PCA can perform arbitrarily better than SVD(A
significantly faster compared to algorithms that require two or more passes over the data.
Acknowledgements We thank the anonymous reviewers for their valuable comments. This research
has been supported by NSF Grants CCF 1344179, 1344364, 1407278, 1422549, 1302435, 1564000,
and ARO YIP W911NF-14-1-0258.
7
This can be done by standard power iteration based method, without explicitly forming the product matrix
T e
e
A B, whose size is too big to fit into memory according to our assumption.
8
References
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9
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5,609 | 6,076 | Learning values across many orders of magnitude
Hado van Hasselt
Arthur Guez
Matteo Hessel
Volodymyr Mnih
David Silver
Google DeepMind
Abstract
Most learning algorithms are not invariant to the scale of the signal that is being
approximated. We propose to adaptively normalize the targets used in the learning updates. This is important in value-based reinforcement learning, where the
magnitude of appropriate value approximations can change over time when we
update the policy of behavior. Our main motivation is prior work on learning to
play Atari games, where the rewards were clipped to a predetermined range. This
clipping facilitates learning across many different games with a single learning
algorithm, but a clipped reward function can result in qualitatively different behavior. Using adaptive normalization we can remove this domain-specific heuristic
without diminishing overall performance.
1
Introduction
Many machine-learning algorithms rely on a-priori access to data to properly tune relevant hyperparameters [Bergstra et al., 2011, Bergstra and Bengio, 2012, Snoek et al., 2012]. It is much harder
to learn efficiently from a stream of data when we do not know the magnitude of the function we
seek to approximate beforehand, or if these magnitudes can change over time, as is typically the case
in reinforcement learning when the policy of behavior improves over time.
Our main motivation is the work by Mnih et al. [2015], in which Q-learning [Watkins, 1989] is
combined with a deep convolutional neural network [cf. LeCun et al., 2015]. The resulting deep Q
network (DQN) algorithm learned to play a varied set of Atari 2600 games from the Arcade Learning
Environment (ALE) [Bellemare et al., 2013], which was proposed as an evaluation framework to test
general learning algorithms on solving many different interesting tasks. DQN was proposed as a
singular solution, using a single set of hyperparameters.
The magnitudes and frequencies of rewards vary wildly between different games. For instance, in
Pong the rewards are bounded by 1 and +1 while in Ms. Pac-Man eating a single ghost can yield
a reward of up to +1600. To overcome this hurdle, rewards and temporal-difference errors were
clipped to [ 1, 1], so that DQN would perceive any positive reward as +1, and any negative reward as
1. This is not a satisfying solution for two reasons. First, the clipping introduces domain knowledge.
Most games have sparse non-zero rewards. Clipping results in optimizing the frequency of rewards,
rather than their sum. This is a fairly reasonable heuristic in Atari, but it does not generalize to
many other domains. Second, and more importantly, the clipping changes the objective, sometimes
resulting in qualitatively different policies of behavior.
We propose a method to adaptively normalize the targets used in the learning updates. If these targets
are guaranteed to be normalized it is much easier to find suitable hyperparameters. The proposed
technique is not specific to DQN or to reinforcement learning and is more generally applicable in
supervised learning and reinforcement learning. There are several reasons such normalization can be
desirable. First, sometimes we desire a single system that is able to solve multiple different problems
with varying natural magnitudes, as in the Atari domain. Second, for multi-variate functions the
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
normalization can be used to disentangle the natural magnitude of each component from its relative
importance in the loss function. This is particularly useful when the components have different
units, such as when we predict signals from sensors with different modalities. Finally, adaptive
normalization can help deal with non-stationary. For instance, in reinforcement learning the policy of
behavior can change repeatedly during learning, thereby changing the distribution and magnitude of
the values.
1.1
Related work
Input normalization has long been recognized as important to efficiently learn non-linear approximations such as neural networks [LeCun et al., 1998], leading to research on how to achieve
scale-invariance on the inputs [e.g., Ross et al., 2013, Ioffe and Szegedy, 2015, Desjardins et al.,
2015]. Output or target normalization has not received as much attention, probably because in
supervised learning data sets are commonly available before learning commences, making it straightforward to determine appropriate normalizations or to tune hyper-parameters. However, this assumes
the data is available a priori, which is not true in online (potentially non-stationary) settings.
Natural gradients [Amari, 1998] are invariant to reparameterizations of the function approximation,
thereby avoiding many scaling issues, but these are computationally expensive for functions with
many parameters such as deep neural networks. This is why approximations are regularly proposed,
typically trading off accuracy to computation [Martens and Grosse, 2015], and sometimes focusing
on a certain aspect such as input normalization [Desjardins et al., 2015, Ioffe and Szegedy, 2015].
Most such algorithms are not fully invariant to the scale of the target function.
In the Atari domain several algorithmic variants and improvements for DQN have been proposed
[van Hasselt et al., 2016, Bellemare et al., 2016, Schaul et al., 2016, Wang et al., 2016], as well as
alternative solutions [Liang et al., 2016, Mnih et al., 2016]. However, none of these address the
clipping of the rewards or explicitly discuss the impacts of clipping on performance or behavior.
1.2
Preliminaries
n
Concretely, we consider learning from a stream of data {(Xt , Yt )}1
t=1 where the inputs Xt 2 R
k
and targets Yt 2 R are real-valued tensors. The aim is to update parameters ? of a function
f? : Rn ! Rk such that the output f? (Xt ) is (in expectation) close to the target Yt according to some
loss lt (f? ), for instance defined as a squared difference: lt (f? ) = 12 (f? (Xt ) Yt )> (f? (Xt ) Yt ).
A canonical update is stochastic gradient descent (SGD). For a sample (Xt , Yt ), the update is then
?t+1 = ?t ?r? lt (f? ), where ? 2 [0, 1] is a step size. The magnitude of this update depends on
both the step size and the loss, and it is hard to pick suitable step sizes when nothing is known about
the magnitude of the loss.
An important special case is when f? is a neural network [McCulloch and Pitts, 1943, Rosenblatt,
1962], which are often trained with a form of SGD [Rumelhart et al., 1986], with hyperparameters
that interact with the scale of the loss. Especially for deep neural networks [LeCun et al., 2015,
Schmidhuber, 2015] large updates may harm learning, because these networks are highly non-linear
and such updates may ?bump? the parameters to regions with high error.
2
Adaptive normalization with Pop-Art
We propose to normalize the targets Yt , where the normalization is learned separately from the
approximating function. We consider an affine transformation of the targets
Y?t = ?t 1 (Yt ?t ) ,
(1)
where ?t and ?t are scale and shift parameters that are learned from data. The scale matrix ?t
can be dense, diagonal, or defined by a scalar t as ?t = t I. Similarly, the shift vector ?t can
contain separate components, or be defined by a scalar ?t as ?t = ?t 1. We can then define a loss
on a normalized function g(Xt ) and the normalized target Y?t . The unnormalized approximation for
any input x is then given by f (x) = ?g(x) + ?, where g is the normalized function and f is the
unnormalized function.
At first glance it may seem we have made little progress. If we learn ? and ? using the same algorithm
as used for the parameters of the function g, then the problem has not become fundamentally different
2
or easier; we would have merely changed the structure of the parameterized function slightly.
Conversely, if we consider tuning the scale and shift as hyperparameters then tuning them is not
fundamentally easier than tuning other hyperparameters, such as the step size, directly.
Fortunately, there is an alternative. We propose to update ? and ? according to a separate objective
with the aim of normalizing the updates for g. Thereby, we decompose the problem of learning an
appropriate normalization from learning the specific shape of the function. The two properties that
we want to simultaneously achieve are
(ART) to update scale ? and shift ? such that ? 1 (Y ?) is appropriately normalized, and
(POP) to preserve the outputs of the unnormalized function when we change the scale and shift.
We discuss these properties separately below. We refer to algorithms that combine output-preserving
updates and adaptive rescaling, as Pop-Art algorithms, an acronym for ?Preserving Outputs Precisely,
while Adaptively Rescaling Targets?.
2.1
Preserving outputs precisely
Unless care is taken, repeated updates to the normalization might make learning harder rather than
easier because the normalized targets become non-stationary. More importantly, whenever we adapt
the normalization based on a certain target, this would simultaneously change the output of the
unnormalized function of all inputs. If there is little reason to believe that other unnormalized outputs
were incorrect, this is undesirable and may hurt performance in practice, as illustrated in Section 3.
We now first discuss how to prevent these issues, before we discuss how to update the scale and shift.
The only way to avoid changing all outputs of the unnormalized function whenever we update the
scale and shift is by changing the normalized function g itself simultaneously. The goal is to preserve
the outputs from before the change of normalization, for all inputs. This prevents the normalization
from affecting the approximation, which is appropriate because its objective is solely to make learning
easier, and to leave solving the approximation itself to the optimization algorithm.
Without loss of generality the unnormalized function can be written as
f?,?,?,W,b (x)
?
?g?,W,b (x) + ?
?
?(Wh? (x) + b) + ? ,
(2)
where h? is a parametrized (non-linear) function, and g?,W,b = Wh? (x) + b is the normalized
function. It is not uncommon for deep neural networks to end in a linear layer, and then h? can be the
output of the last (hidden) layer of non-linearities. Alternatively, we can always add a square linear
layer to any non-linear function h? to ensure this constraint, for instance initialized as W0 = I and
b0 = 0.
The following proposition shows that we can update the parameters W and b to fulfill the second
desideratum of preserving outputs precisely for any change in normalization.
Proposition 1. Consider a function f : Rn ! Rk defined as in (2) as
f?,?,?,W,b (x)
?
? (Wh? (x) + b) + ? ,
where h? : R ! R is any non-linear function of x 2 Rn , ? is a k ? k matrix, ? and b are
k-element vectors, and W is a k ? m matrix. Consider any change of the scale and shift parameters
from ? to ?new and from ? to ?new , where ?new is non-singular. If we then additionally change
the parameters W and b to Wnew and bnew , defined by
n
m
1
Wnew = ?new
?W
1
bnew = ?new
(?b + ?
and
?new ) ,
then the outputs of the unnormalized function f are preserved precisely in the sense that
f?,?,?,W,b (x) = f?,?new ,?new ,Wnew ,bnew (x) ,
8x .
This and later propositions are proven in the appendix. For the special case of scalar scale and
shift, with ? ? I and ? ? ?1, the updates to W and b become Wnew = ( / new )W and
bnew = ( b + ? ?new )/ new . After updating the scale and shift we can update the output of
the normalized function g?,W,b (Xt ) toward the normalized output Y?t , using any learning algorithm.
Importantly, the normalization can be updated first, thereby avoiding harmful large updates just before
they would otherwise occur. This observation is made more precise in Proposition 2 in Section 2.2.
3
Algorithm 1 SGD on squared loss with Pop-Art
For a given differentiable function h? , initialize ?.
Initialize W = I, b = 0, ? = I, and ? = 0.
while learning do
Observe input X and target Y
Use Y to compute new scale ?new and new shift ?new
1
1
W
?new
?W , b
?new
(?b + ? ?new )
?
?new ,
?
?new
h
h? (X)
J
(r? h?,1 (X), . . . , r? h?,m (X))
Wh + b ? 1 (Y ?)
?
? ?J W>
W
W ? h>
b
b ?
end while
(rescale W and b)
(update scale and shift)
(store output of h? )
(compute Jacobian of h? )
(compute normalized error)
(compute SGD update for ?)
(compute SGD update for W)
(compute SGD update for b)
Algorithm 1 is an example implementation of SGD with Pop-Art for a squared loss. It can be
generalized easily to any other loss by changing the definition of . Notice that W and b are updated
twice: first to adapt to the new scale and shift to preserve the outputs of the function, and then by
SGD. The order of these updates is important because it allows us to use the new normalization
immediately in the subsequent SGD update.
2.2
Adaptively rescaling targets
A natural choice is to normalize the targets to approximately have zero mean and unit variance. For
clarity and conciseness, we consider scalar normalizations. It is straightforward to extend to diagonal
or dense matrices. If we have data {(Xi , Yi )}ti=1 up to some time t, we then may desire
t
X
t
(Yi
?t )/
t
1X
(Yi
t i=1
and
=0
i=1
such that
?t =
t
t )?t 1
+
t Yt
and
2
t
= 1,
t
1X
Yi
t i=1
and
This can be generalized to incremental updates
?t = (1
?t ) 2 /
2
t
?2t ,
= ?t
t
=
1X 2
Y
t i=1 i
where ?t = (1
?2t .
t )?t 1
+
(3)
2
t Yt
. (4)
Here ?t estimates the second moment of the targets and t 2 [0, 1] is a step size. If ?t ?2t is positive
initially then it will always remain so, although to avoid issues with numerical precision it can be
useful to enforce a lower bound explicitly by requiring ?t ?2t ? with ? > 0. For full equivalence
to (3) we can use t = 1/t. If t = is constant we get exponential moving averages, placing more
weight on recent data points which is appropriate in non-stationary settings.
A constant has the additional benefit of never becoming negligibly small. Consider the first time a
target is observed that is much larger than all previously observed targets. If t is small, our statistics
would adapt only slightly, and the resulting update may be large enough to harm the learning. If t
is not too small, the normalization can adapt to the large target before updating, potentially making
learning more robust. In particular, the following proposition holds.
Proposition 2. When using updates (4) to adapt the normalization parameters
ized targets are bounded for all t by
p
p
(1
?t )/ t ? (1
t )/ t ? (Yt
t )/ t .
and ?, the normal-
For instance, if t = = 10 4 for all t, then the normalized target is guaranteed to be in ( 100, 100).
Note that Proposition 2 does not rely on any assumptions about the distribution of the targets. This is
an important result, because it implies we can bound the potential normalized errors before learning,
without any prior knowledge about the actual targets we may observe.
4
Algorithm 2 Normalized SGD
For a given differentiable function h? , initialize ?.
while learning do
Observe input X and target Y
Use Y to compute new scale ?
h
h? (X)
J
(rh?,1 (X), . . . , rh?,m (X))>
Wh + b Y
?
? ?J (? 1 W)> ? 1
W
W ? g>
b
b ?
end while
(store output of h? )
(compute Jacobian of h? )
(compute unnormalized error)
(update ? with scaled SGD)
(update W with SGD)
(update b with SGD)
It is an open question whether it is uniformly best to normalize by mean and variance. In the
appendix we discuss other normalization updates, based on percentiles and mini-batches, and derive
correspondences between all of these.
2.3
An equivalence for stochastic gradient descent
We now step back and analyze the effect of the magnitude of the errors on the gradients when using
regular SGD. This analysis suggests a different normalization algorithm, which has an interesting
correspondence to Pop-Art SGD.
We consider SGD updates for an unnormalized multi-layer function of form f?,W,b (X) =
Wh? (X) + b. The update for the weight matrix W is
Wt = Wt
1
+ ?t t h?t (Xt )> ,
where t = f?,W,b (X) Yt is gradient of the squared loss, which we here call the unnormalized error.
The magnitude of this update depends linearly on the magnitude of the error, which is appropriate
when the inputs are normalized, because then the ideal scale of the weights depends linearly on the
magnitude of the targets.1
Now consider the SGD update to the parameters of h? , ?t = ?t 1 ?Jt Wt> 1 t where Jt =
(rg?,1 (X), . . . , rg?,m (X))> is the Jacobian for h? . The magnitudes of both the weights W and
the errors depend linearly on the magnitude of the targets. This means that the magnitude of the
update for ? depends quadratically on the magnitude of the targets. There is no compelling reason
for these updates to depend at all on these magnitudes because the weights in the top layer already
ensure appropriate scaling. In other words, for each doubling of the magnitudes of the targets, the
updates to the lower layers quadruple for no clear reason.
This analysis suggests an algorithmic solution, which seems to be novel in and of itself, in which
we track the magnitudes of the targets in a separate parameter t , and then multiply the updates for
all lower layers with a factor t 2 . A more general version of this for matrix scalings is given in
Algorithm 2. We prove an interesting, and perhaps surprising, connection to the Pop-Art algorithm.
Proposition 3. Consider two functions defined by
and
f?,?,?,W,b (x) = ?(Wh? (x) + b) + ?
f?,W,b (x) = Wh? (x) + b ,
where h? is the same differentiable function in both cases, and the functions are initialized identically,
using ?0 = I and ? = 0, and the same initial ?0 , W0 and b0 . Consider updating the first function
using Algorithm 1 (Pop-Art SGD) and the second using Algorithm 2 (Normalized SGD). Then, for
1
any sequence of non-singular scales {?t }1
t=1 and shifts {?t }t=1 , the algorithms are equivalent in
1
the sense that 1) the sequences {?t }t=0 are identical, 2) the outputs of the functions are identical, for
any input.
The proposition shows a duality between normalizing the targets, as in Algorithm 1, and changing the
updates, as in Algorithm 2. This allows us to gain more intuition about the algorithm. In particular,
1
In general care should be taken that the inputs are well-behaved; this is exactly the point of recent work on
input normalization [Ioffe and Szegedy, 2015, Desjardins et al., 2015].
5
RMSE (log scale)
10000
1000
100
Pop-Art
Art
SGD
10
0
2500
5000
number of samples
Fig. 1a. Median RMSE on binary regression for SGD
without normalization (red), with normalization but
without preserving outputs (blue, labeled ?Art?), and
with Pop-Art (green). Shaded 10?90 percentiles.
Fig. 1b. `2 gradient norms for DQN during learning
on 57 Atari games with actual unclipped rewards (left,
red), clipped rewards (middle, blue), and using PopArt (right, green) instead of clipping. Shaded areas
correspond to 95%, 90% and 50% of games.
in Algorithm 2 the updates in top layer are not normalized, thereby allowing the last linear layer
to adapt to the scale of the targets. This is in contrast to other algorithms that have some flavor of
adaptive normalization, such as RMSprop [Tieleman and Hinton, 2012], AdaGrad [Duchi et al.,
2011], and Adam [Kingma and Adam, 2015] that each component in the gradient by a square root of
an empirical second moment of that component. That said, these methods are complementary, and it
is straightforward to combine Pop-Art with other optimization algorithms than SGD.
3
Binary regression experiments
We first analyze the effect of rare events in online learning, when infrequently a much larger target
is observed. Such events can for instance occur when learning from noisy sensors that sometimes
captures an actual signal, or when learning from sparse non-zero reinforcements. We empirically
compare three variants of SGD: without normalization, with normalization but without preserving
outputs precisely (i.e., with ?Art?, but without ?Pop?), and with Pop-Art.
The inputs are binary representations of integers drawn uniformly randomly between 0 and n =
210 1. The desired outputs are the corresponding integer values. Every 1000 samples, we present the
binary representation of 216 1 as input (i.e., all 16 inputs are 1) and as target 216 1 = 65, 535. The
approximating function is a fully connected neural network with 16 inputs, 3 hidden layers with 10
nodes per layer, and tanh internal activation functions. This simple setup allows extensive sweeps over
hyper-parameters, to avoid bias towards any algorithm by the way we tune these. The step sizes ? for
SGD and for the normalization are tuned by a grid search over {10 5 , 10 4.5 , . . . , 10 1 , 10 0.5 , 1}.
Figure 1a shows the root mean squared error (RMSE, log scale) for each of 5000 samples, before
updating the function (so this is a test error, not a train error). The solid line is the median of 50
repetitions, and shaded region covers the 10th to 90th percentiles. The plotted results correspond to
the best hyper-parameters according to the overall RMSE (i.e., area under the curve). The lines are
slightly smoothed by averaging over each 10 consecutive samples.
SGD favors a relatively small step size (? = 10 3.5 ) to avoid harmful large updates, but this slows
learning on the smaller updates; the error curve is almost flat in between spikes. SGD with adaptive
normalization (labeled ?Art?) can use a larger step size (? = 10 2.5 ) and therefore learns faster, but
has high error after the spikes because the changing normalization also changes the outputs of the
smaller inputs, increasing the errors on these. In comparison, Pop-Art performs much better. It
prefers the same step size as Art (? = 10 2.5 ), but Pop-Art can exploit a much faster rate for the
statistics (best performance with = 10 0.5 for Pop-Art and = 10 4 for Art). The faster tracking
of statistics protects Pop-Art from the large spikes, while the output preservation avoids invalidating
6
the outputs for smaller targets. We ran experiments with RMSprop but left these out of the figure as
the results were very similar to SGD.
4
Atari 2600 experiments
An important motivation for this work is reinforcement learning with non-linear function approximators such as neural networks (sometimes called deep reinforcement learning). The goal is to predict
and optimize action values defined as the expected sum of future rewards. These rewards can differ
arbitrarily from one domain to the next, and non-zero rewards can be sparse. As a result, the action
values can span a varied and wide range which is often unknown before learning commences.
Mnih et al. [2015] combined Q-learning with a deep neural network in an algorithm called DQN,
which impressively learned to play many games using a single set of hyper-parameters. However, as
discussed above, to handle the different reward magnitudes with a single system all rewards were
clipped to the interval [ 1, 1]. This is harmless in some games, such as Pong where no reward is ever
higher than 1 or lower than 1, but it is not satisfactory as this heuristic introduces specific domain
knowledge that optimizing reward frequencies is approximately is useful as optimizing the total score.
However, the clipping makes the DQN algorithm blind to differences between certain actions, such as
the difference in reward between eating a ghost (reward >= 100) and eating a pellet (reward = 25)
in Ms. Pac-Man. We hypothesize that 1) overall performance decreases when we turn off clipping,
because it is not possible to tune a step size that works on many games, 2) that we can regain much
of the lost performance by with Pop-Art. The goal is not to improve state-of-the-art performance,
but to remove the domain-dependent heuristic that is induced by the clipping of the rewards, thereby
uncovering the true rewards.
We ran the Double DQN algorithm [van Hasselt et al., 2016] in three versions: without changes,
without clipping both rewards and temporal difference errors, and without clipping but additionally
using Pop-Art. The targets are the cumulation of a reward and the discounted value at the next state:
Yt = Rt+1 + Q(St , argmax Q(St , a; ?); ? ) ,
(5)
a
where Q(s, a; ?) is the estimated action value of action a in state s according to current parameters ?,
and where ? is a more stable periodic copy of these parameters [cf. Mnih et al., 2015, van Hasselt
et al., 2016, for more details]. This is a form of Double Q-learning [van Hasselt, 2010]. We roughly
tuned the main step size and the step size for the normalization to 10 4 . It is not straightforward to
tune the unclipped version, for reasons that will become clear soon.
Figure 1b shows `2 norm of the gradient of Double DQN during learning as a function of number of
training steps. The left plot corresponds to no reward clipping, middle to clipping (as per original
DQN and Double DQN), and right to using Pop-Art instead of clipping. Each faint dashed lines
corresponds to the median norms (where the median is taken over time) on one game. The shaded
areas correspond to 50%, 90%, and 95% of games.
Without clipping the rewards, Pop-Art produces a much narrower band within which the gradients
fall. Across games, 95% of median norms range over less than two orders of magnitude (roughly
between 1 and 20), compared to almost four orders of magnitude for clipped Double DQN, and more
than six orders of magnitude for unclipped Double DQN without Pop-Art. The wide range for the
latter shows why it is impossible to find a suitable step size with neither clipping nor Pop-Art: the
updates are either far too small on some games or far too large on others.
After 200M frames, we evaluated the actual scores of the best performing agent in each game on 100
episodes of up to 30 minutes of play, and then normalized by human and random scores as described
by Mnih et al. [2015]. Figure 2 shows the differences in normalized scores between (clipped) Double
DQN and Double DQN with Pop-Art.
The main eye-catching result is that the distribution in performance drastically changed. On some
games (e.g., Gopher, Centipede) we observe dramatic improvements, while on other games (e.g.,
Video Pinball, Star Gunner) we see a substantial decrease. For instance, in Ms. Pac-Man the clipped
Double DQN agent does not care more about ghosts than pellets, but Double DQN with Pop-Art
learns to actively hunt ghosts, resulting in higher scores. Especially remarkable is the improved
performance on games like Centipede and Gopher, but also notable is a game like Frostbite which
went from below 50% to a near-human performance level. Raw scores can be found in the appendix.
7
1600%
Normalized differences
800%
400%
200%
100%
0%
-100%
-200%
-400%
-800%
Video Pinball
Star Gunner
James Bond
Double Dunk
Breakout
Time Pilot
Wizard of Wor
Defender
Phoenix
Chopper Command
Q*Bert
Battle Zone
Amidar
Skiing
Beam Rider
Tutankham
H.E.R.O.
River Raid
Seaquest
Ice Hockey
Robotank
Alien
Up and Down
Berzerk
Pong
Montezuma?s Revenge
Private Eye
Freeway
Pitfall
Gravitar
Surround
Space Invaders
Asteroids
Kangaroo
Crazy Climber
Bank Heist
Solaris
Yars Revenge
Asterix
Kung-Fu Master
Bowling
Ms. Pacman
Frostbite
Zaxxon
Road Runner
Fishing Derby
Boxing
Venture
Name This Game
Enduro
Krull
Tennis
Demon Attack
Centipede
Assault
Atlantis
Gopher
-1600%
Figure 2: Differences between normalized scores for Double DQN with and without Pop-Art on 57
Atari games.
Some games fare worse with unclipped rewards because it changes the nature of the problem. For
instance, in Time Pilot the Pop-Art agent learns to quickly shoot a mothership to advance to a next
level of the game, obtaining many points in the process. The clipped agent instead shoots at anything
that moves, ignoring the mothership. However, in the long run in this game more points are scored
with the safer and more homogeneous strategy of the clipped agent. One reason for the disconnect
between the seemingly qualitatively good behavior combined with lower scores is that the agents
are fairly myopic: both use a discount factor of = 0.99, and therefore only optimize rewards that
happen within a dozen or so seconds into the future.
On the whole, the results show that with Pop-Art we can successfully remove the clipping heuristic
that has been present in all prior DQN variants, while retaining overall performance levels. Double
DQN with Pop-Art performs slightly better than Double DQN with clipped rewards: on 32 out of 57
games performance is at least as good as clipped Double DQN and the median (+0.4%) and mean
(+34%) differences are positive.
5
Discussion
We have demonstrated that Pop-Art can be used to adapt to different and non-stationary target
magnitudes. This problem was perhaps not previously commonly appreciated, potentially because
in deep learning it is common to tune or normalize a priori, using an existing data set. This is not
as straightforward in reinforcement learning when the policy and the corresponding values may
repeatedly change over time. This makes Pop-Art a promising tool for deep reinforcement learning,
although it is not specific to this setting.
We saw that Pop-Art can successfully replace the clipping of rewards as done in DQN to handle
the various magnitudes of the targets used in the Q-learning update. Now that the true problem is
exposed to the learning algorithm we can hope to make further progress, for instance by improving
the exploration [Osband et al., 2016], which can now be informed about the true unclipped rewards.
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9
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5,610 | 6,077 | Online Bayesian Moment Matching for
Topic Modeling with Unknown Number of Topics
Wei-Shou Hsu and Pascal Poupart
David R. Cheriton School of Computer Science
University of Waterloo
Wateroo, ON N2L 3G1
{wwhsu,ppoupart}@uwaterloo.ca
Abstract
Latent Dirichlet Allocation (LDA) is a very popular model for topic modeling as
well as many other problems with latent groups. It is both simple and effective.
When the number of topics (or latent groups) is unknown, the Hierarchical Dirichlet Process (HDP) provides an elegant non-parametric extension; however, it is a
complex model and it is difficult to incorporate prior knowledge since the distribution over topics is implicit. We propose two new models that extend LDA in a
simple and intuitive fashion by directly expressing a distribution over the number
of topics. We also propose a new online Bayesian moment matching technique to
learn the parameters and the number of topics of those models based on streaming data. The approach achieves higher log-likelihood than batch and online HDP
with fixed hyperparameters on several corpora. The code is publicly available at
https://github.com/whsu/bmm.
1
Introduction
Latent Dirichlet Allocation (LDA) [3] recently emerged as the dominant framework for topic modeling as well as many other applications with latent groups. The Hierarchical Dirichlet Process
(HDP) [18] provides an elegant extension to LDA when the number of topics (latent groups) is
unknown. The non-parametric nature of HDPs is quite attractive since HDPs effectively allow an
unbounded number of topics to be inferred from the data. There is also a rich mathematical theory
underlying HDPs as well as attractive metaphors (e.g., stick breaking process, Chinese restaurant
franchise) to ease the understanding by those less comfortable with non-parametric statistics [18].
That being said, HDPs are not perfect. They do not expose an explicit distribution over the topics
that could allow practitioners to incorporate prior knowledge and to inspect the model?s posterior
confidence in different number of topics. Furthermore, the implicit distribution over the number of
topics is restricted to a regime where the number of topics grows logarithmically with the amount of
data in expectation [18]. For instance, this growth rate is insufficient for applications that exhibit a
power law distribution [6] ? a generalization of the HDP known as the hierarchical Pitman-Yor process [21] is often used instead. Existing inference algorithms for HDPs (e.g., Gibbs sampling [18],
variational inference [19, 24, 23, 4, 17]) are also fairly complex. As a result, practitioners often stick
with LDA and estimate the number of topics by repeatedly evaluating different number of topics by
cross-validation; however, this is an expensive procedure.
We propose two new models that extend LDA in a simple and intuitive fashion by directly expressing
a distribution over the number of topics under the assumption that an upper bound on the number of
topics is available. When the amount of data is finite, this assumption is perfectly fine since there
cannot be more topics than the amount of data. Otherwise, domain experts can often define a suitable
range for the number of topics and if they plan to inspect the resulting topics, they cannot inspect
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
an unbounded number of topics. We also propose a novel Bayesian moment matching algorithm
to compute a posterior distribution over the model parameters and the number of topics. Bayesian
learning naturally lends itself to online learning for streaming data since the posterior is updated
sequentially after each data point and there is no need to go over the data more than once. The main
issue is that the posterior becomes intractable. We approximate the posterior after each observed
word by a tractable distribution that matches some moments of the exact posterior (hence the name
Bayesian Moment Matching). The approach compares favorably to online HDP on several topic
modeling tasks.
2
Related work
Setting the number of topics to use can be treated as a model selection problem. One solution is to
train a topic model multiple times, each time with a different number of topics, and choose the number of topics that minimizes some cost function on a heldout test set. More recently nonparametric
Bayesian methods have been used to bypass the model selection problem. Hierarchical Dirichlet
process (HDP) [18] is the natural extension of LDA in this direction. With HDP, the number of topics is learned from data as part of the inference procedure. Gibbs sampling [7, 15] and Variational
Bayes [3, 20] are by far the most popular inference techniques for LDA. They have been extended
to HDP [18, 19, 17]. With the rise of streaming data, online variants of Variational Bayes have also
been developed for LDA [8] and HDP [24, 23, 4]. The first online variational technique [24] used
a truncation that effectively bounds the number of topics while subsequent techniques [23, 4] avoid
any fixed truncation to fully exploit the non-parametric nature of HDP. These online variational techniques perform stochastic gradient ascent on mini-batches, which reduces their data efficiency, but
improves computational efficiency.
We propose two new models that are simpler than HDP and express a distribution directly on the
number of topics. We extend online Bayesian moment matching (originally designed for LDA with
a fixed number of topics [14]) to learn the number of topics. This technique avoids mini-batches. It
approximates Bayesian learning by Assumed Density Filtering [13], which can be thought as a single
forward iteration of Expectation Propagation [12]. Note that Bayesian moment matching is different
from frequentist moment matching techniques such as spectral learning [1, 2, 9, 11]. In BMM, we
compute a posterior over the parameters of the model and approximate the posterior with a simpler
distribution that matches some moments of the exact posterior. In spectral learning, moments of the
empirical distribution of the data are used to find parameters that yield the same moments in the
model. This is usually achieved by a spectral (or tensor) decomposition of the empirical moments,
hence the name spectral learning. Although both BMM and spectral learning use the method of
moments, they match different moments in different distributions resulting in completely different
algorithms. While stochastic gradient descent can be used to compute tensor decompositions in an
online fashion [5, 10], no online variant of spectral learning has been developed to infer the number
of topics in LDA.
3
Models
We investigate the problem of online clustering of grouped discrete observations. Using terminology
from text processing, we will call each observation a word and each group a document. The observed
N
data set is then a corpus of N words, {wn }N
n=1 , along with the IDs, {dn }n=1 , of the documents to
which these words belong. We will let D denote the number of documents and V the number of
distinct words in the vocabulary. Figure 1 shows the generative models we are considering.
The basic model is LDA, in which the number of the topics T is fixed. We propose two extensions to
the basic model where the parameter T is unknown and inferred from data, with the assumption that
~ specifies
T ranges from 1 to K. Each ?~ specifies the topic distribution of a document, while each ?
the word distribution of a topic. In the rest of the paper, we will use ? to denote the collection of all
~ and ? the collection of all ??s
~ in the model.
??s
2
?~t
?
~d
~t
?
?~d
D
dn
tn
~?
?
~d
T
?~d
T
?~t
~t
?
D
wn
dn
tn
N
~?
?
~ k,d
?~t
T
?~k,d
~t
?
K
wn
DK
dn
tn
N
K
wn
N
Figure 1: Graphical representations of basic model with fixed number of topics (left), degenerate
Dirichlet model (middle), and triangular Dirichlet model (right)
3.1
Degenerate Dirichlet model
The generative process of the degenerate Dirichlet model (DDM), as shown in the middle in Figure 1,
~ K
~ D
works by first sampling the hyperparameters ~? , {~
?d }D
d=1 , and {?t }t=1 . The parameters T , {?d }d=1 ,
K
~
and {?t }t=1 are then sampled from the following conditional distributions:
P (T |~? ) = Discrete(T ; ~? )
P (?~d |~
?d , T ) = Dir(?~d ; ?
~ d, T )
~ t |?~t ) = Dir(?
~t; ?
~t )
P (?
where Dir(?~d ; ?
~ d , T ) denotes a degenerate Dirichlet distribution Dir(?~d ; ?
~ d0 ) with
?d,t for t ? T
0
?d,t
=
0
for t > T
and Discrete(T ; ~? ) is the general discrete distribution with probability P (T = k) = ?k for
k = 1, . . . , K. Finally, the N observations are generated by first sampling the topic indicators
tn according to the distribution P (tn |dn , ?) = ?dn ,tn . Note that since ?~dn is sampled from a degenerate Dirichlet, we have ?dn ,tn = 0 for tn > T . Given tn , the words are then sampled according
to the categorical distribution P (wn |tn , ?) = ?tn ,wn .
3.2
Triangular Dirichlet model
The triangular Dirichlet model (TDM), shown on the right in Figure 1, works in a similar way except
the document-topic distribution ? is represented by a three-dimensional array that is also indexed
by the number of topics T in addition to the document ID d and the topic ID t. Given T and d, the
topic t is drawn according to the probability P (t|d, ?, T ) = ?T,d,t for 1 ? t ? T . The array ?
therefore has a triangular shape in the first and third dimension. Again, we place a Dirichlet prior on
each ?~k,d : P (?~k,d |~
?k,d ) = Dir(?~k,d ; ?
~ k,d ). In this case, however, ?~k,d has no dependence on T .
4
Bayesian update by moment matching
Let Pn (?, ?, T ) denote the joint posterior probability of ?, ?, and T after seeing the first n observations. Then1
Pn (?, ?, T ) = P (?, ?, T |w1:n )
=
K
1 X
P (tn |?, T )P (wn |?, tn )Pn?1 (?, ?, T )
cn t =1
(1)
n
where cn = P (wn |w1:n?1 ).
From (1) we can see that after seeing each new observation wn , the number of terms in the posterior
is increased by a factor of K, resulting in an exponential complexity for exact Bayesian update.
Therefore, we will instead approximate Pn by a different distribution, whose parameters will be
estimated by moment matching.
In the derivations that follow, the dependence on the document IDs {dn }D
?,
n=1 and the hyperparameters ~
?, and ? is implicit and not shown.
1
3
4.1
Approximating distribution
To make the inference tractable, we approximate Pn using a factorized distribution: Pn (?, ?, T ) =
f? (?)f? (?)fT (T ).
For TDM, we choose the factorized distribution to have the exact same form as the prior distribution,
i.e.,
f? (?) =
K Y
D
Y
Dir(?~k,d ; ?
~ k,d )
(2)
k=1 d=1
K
Y
~ t ; ?~t )
Dir(?
(3)
fT (T ) = Discrete(T ; ~? )
(4)
f? (?) =
t=1
For DDM, we use the same f? and fT , but rather than choosing f? as degenerate Dirichlets again,
we instead approximate the posterior over ? using proper Dirichlet distributions to decouple ? from
T:
D
Y
f? (?) =
Dir(?~d ; ?
~ d)
(5)
d=1
4.2
Moment matching
Let x be a random variable with distribution p(x). The i-th moment of x about zero is defined as the
expectation of xi over p, and we denote it by Mxi (p):
Mxi (p) = Ep xi
(6)
For a K-dimensional Dirichlet distribution Dir(x1 , . . . , xK ; ?1 , . . . , ?K ), we can uniquely solve for
the parameters ?1 , . . . , ?K if we have K ? 1 first moments, Mx1 , . . . , MxK?1 , and one second
moment, Mx21 . Given the moments, we can determine the Dirichlet parameters as
?k = Mxk
Mx1 ? Mx21
(7)
Mx21 ? Mx21
for k = 1, . . . , K. Therefore, we can compute the parameters for f? and f? using (7): for ?d ,
replace ?k with ?d,k and xk with ?d,k ; and for ?t , replace ?k with ?t,k and xk with ?t,k .
The parameters for Discrete(T ; ~? ) are estimated directly as
?k = E[?T,k ]
(8)
where ? denotes the the Kronecker delta
?i,j =
4.3
1
0
if i = j
.
if i 6= j
(9)
Moment computation
From (7) and (8), we see that to approximate Pn by moment matching, we need to compute the first
and second moments of ? and ? as well as the expectation E[?T,k ] with respect to Pn . They can be
calculated using the Bayesian update equation (1).
To keep the notation uncluttered, let S~x,:m denote the sum of the first m elements in a vector ~x and
S~x the sum of all elements in ~x. We can then compute the moments of DDM as follows:
cn =
K
X
T =1
EPn [?T,k ] =
T
X
?dn ,tn ?tn ,wn
?T
S
~ dn ,:T S?
~t
t =1 ?
(10)
n
n
k
X
1
?dn ,tn ?tn ,wn
?k
cn
S
~ dn ,:k S?
~t
t =1 ?
n
4
n
(11)
M?d,t (Pn ) =
K
T
X
1 X
?dn ,tn ?tn ,wn ?d,t + ?d,dn ?t,tn
?T
cn
S~ dn ,:T S?~t
S?~ d ,:T + ?d,dn
t =1 ?
T =t
2
M?d,t
(Pn ) =
K
T
X
1 X
?dn ,tn ?tn ,wn ?d,t + ?d,dn ?t,tn ?d,t + 1 + ?d,dn ?t,tn
?T
cn
S~ dn ,:T S?~t
S?~ d ,:T + ?d,dn S?~ d ,:T + 1 + ?d,dn
t =1 ?
T =t
T =1
n
(14)
n
n
K
T
X
1 X
?dn ,tn ?tn ,wn ?t,w + ?t,tn ?w,wn ?t,w + 1 + ?t,tn ?w,wn
?T
cn
S
S?~t + ?t,tn
S?~t + 1 + ?t,tn
~ dn ,:T S?
~t
t =1 ?
T =1
(13)
n
n
K
T
X
1 X
?dn ,tn ?tn ,wn ?t,w + ?t,tn ?w,wn
M?t,w (Pn ) =
?T
cn
S
S?~t + ?t,tn
~ dn ,:T S?
~t
t =1 ?
M?2t,w (Pn ) =
(12)
n
n
(15)
n
For TDM, the moments are computed similarly except that T is used to index into ? rather than to
take partial sums. The equations are included in the supplement.
4.4
Parameter update
For TDM, the approximating distribution for the posterior has the exact same form as the prior;
therefore, the parameters we compute for Pn in the n-th update can be used directly as the parameters
for the prior in the (n + 1)-th update.
However, for DDM, the prior for ? consists of degenerate Dirichlet distributions conditionally dependent on T , whereas the approximating distribution for the posterior is a fully factorized distribution with proper Dirichlets. Therefore, we have to make a further approximation to match the
parameters of the two distributions.
When Pn is being used as the prior in the (n + 1)-th update, we use the same ? that was obtained
by moment matching during the n-th update, but it now has a different meaning. During the n-th
update, ? is computed as parameters of proper Dirichlet distributions, but in the next update, it is
used as parameters of a weighted sum of degenerate Dirichlet distributions. As a result, the DDM
has a natural bias towards smaller number of topics.
4.5
Algorithm summary
In summary, starting from a prior distribution, the algorithm successively updates the posterior by
first computing the exact moments according to the Bayesian update equation (1), and then updating the parameters by matching the moments with those of an approximating distribution. In the
case of TDM, the approximating distribution has the same form as the prior, whereas a simplified
distribution is used for DDM. Algorithm 1 summarizes the procedure for the two models.
Algorithm 1 Online Bayesian moment matching algorithm
1: Initialize ?, ?, and ~
?.
2: for n = 1, . . . , N do
3:
Read the n-th observation (dn , wn ).
4:
Compute moments according to (10)?(15) for DDM or equations in supplement for TDM.
5:
Update ?, ?, and ~? according to (7) and (8) with appropriate substitutions.
6: end for
5
Experiments
In this section, we discuss our experiments on a synthetic dataset and three real text corpora. The
TDM and DDM implementations are available at https://github.com/whsu/bmm. For both
models we initialized the hyperparameters to be ?d,t = 1 and ?t,w = ?1V for all d, t, and w. The
reason that ?t,w was not initialized to 1 was to encourage the algorithm to find topics with more
concentrated word distributions.
5
DDM
TDM
10
0
2
DDM
TDM
20
Predicted T
Predicted T
20
4
6
8
10
10
0
Actual T
2
4
6
8
10
Actual T
(a)
(b)
Figure 2: Number of topics discovered by the DDM and TDM on synthetic datasets using (a) uniform prior and (b) exponentially decreasing prior on T . The results are averaged over 100 randomly
generated datasets for each actual T . Error bars show plus/minus one standard deviation. Gray line
indicates the true number of topics that generated the datasets.
5.1
Synthetic data
We first ran some tests on synthetic data to see how well the models estimate the number of topics.
For this experiment, the actual number of topics T was varied from 1 to 10, and for each value of T ,
we generated 100 random datasets with D = 100, V = 200, and N =100,000. Each random dataset
was created by first sampling ? from Dir(~
?d |0.05) and ? from Dir(?~t |0.1). The observations were
then sampled from ? and ?.
1
We set K = 20 and used the uniform prior P (T ) = K
for T = 1, . . . , K. The estimated number
of topics is shown in Figure 2(a). Both models were able to discover more topics as the actual
number of topics increases. They tend to overestimate the number of topics because the initial value
?t,w = ?1V encourages topics with smaller number of words.
However, in both models, the modeler has direct control over the number of topics. If there is
reason to believe the data come from a smaller number of topics, the modeler can change the prior
distribution on T accordingly as is typical in a Bayesian framework.
For this example, we also tested on an exponentially decreasing prior P (T ) ? e?T for T =
1, . . . , K. The results are shown in Figure 2(b). In this case, TDM shows a slight decrease than
with a uniform prior, whereas DDM produces an estimate that is close to the true number of topics.
5.2
Text modeling
We compare the two proposed models by using them to model the distributions of three real text
corpora containing Reuters news articles, NIPS conference proceedings, and Yelp reviews. We also
include online HDP (oHDP) in the comparisons, as well as the basic moment matching (basic MM)
algorithm with different values of T . For online HDP, we used the gensim 0.10.3 [16] implementation with the default parameters except for the top-level truncation, which we set equal to the
maximum number of topics we used for DDM and TDM. Because DDM and TDM do not estimate
a global alpha as oHDP, for oHDP we include the results with both uniform alpha (oHDP unif) and
alpha that is learned (oHDP alpha).
We followed a similar experimental setup as in [22, 4]. Each dataset was divided into a training set
Dtrain and a test set Dtest based on document IDs. The words in the test set were further split into
two subsets W1 and W2 , where W1 contains the words in the first half of each document in the
test set, and W2 contains the second half. The evaluation metric used is the per-word log likelihood
2 |W1 ,Dtrain )
L = log p(W|W
where |W2 | denotes the total number of tokens in W2 .
2|
For each experiment we also report the number of topics inferred by DDM, TDM. We do not report
this number for online HDP because it is not returned by the implementation.
6
50
DDM
TDM
DDM
TDM
Basic MM
oHDP alpha
oHDP unif
-6.7
40
30
-6.9
T
Per-word log likelihood
-6.5
20
-7.1
10
-7.3
0
20
40
60
80
0
100
0
T
500
1000
(?1000)
n
(a)
(b)
Figure 3: Text modeling on Reuters-21578: (a) Per-word test log likelihood and (b) Number of
topics found as a function of number of observations.
5.2.1
Reuters-21578
The Reuters-21578 corpus contains 21,578 Reuters news articles in 1987. For this dataset,
we divided the data into training and test sets according to the LEWISSPLIT attribute that
is available as part of the distribution at http://www.daviddlewis.com/resources/
testcollections/reuters21578/. The text was passed through a stemmer, and stopwords
and words appearing in five or fewer documents were removed. This resulted in a total of 1,307,468
tokens and a vocabulary of 7,720 distinct words. We chose K to be 100 for both models with
1
uniform prior P (T ) = K
. Figure 3(a) shows the experimental results.
DDM discovered 39 topics while TDM found 36, and they both achieved similar per-word log
likelihood as the best models with fixed T showing that they were able to automatically determine
the number of topics necessary to model the data.
While both models found the a similar number of topics in the end, they progressed to the final values
in different ways. Fig. 3(b) shows the number of topics found by the two models as a function of
number of observations. DDM shows a logarithmically increasing trend as more words are observed,
whereas TDM follows a more irregular progression.
5.2.2
NIPS
We also tested the two models on 2,742 articles from the NIPS conference for the years 1988?2004.
We used the raw text versions available at http://cs.nyu.edu/?roweis/data.html
(1988?1999) and http://ai.stanford.edu/?gal/data.html (2000?2004). The first set
was used as the training set and the second as the test set. The corpus was again passed through a
stemmer, and stopwords and words appearing no more than 50 times were removed. After preprocessing we are left with 2,207,106 total words and a vocabulary of 4,383 unique words.
For this dataset we used K = 400 with the exponentially decreasing prior. DDM discovered 54
topics, and TDM found 89 topics. Figure 4(a) shows the per-word log likelihood on the test set.
In this experiment, both DDM and TDM obtained closed to the optimal likelihood compared to basic
MM.
5.2.3
Yelp
In our third experiment, we tested the models on a subset of the Yelp Academic Dataset (http:
//www.yelp.com/dataset_challenge). We took the 129,524 reviews in the dataset that
were given to businesses in the Food category. The reviews were randomly split so that 70% were
used for training and 30% for testing.
Similar preprocessing was performed. The corpus was passed through a stemmer, and stopwords
and words appearing no more than 50 times were removed. After preprocessing the corpus contains
a total of 5,317,041 words and a vocabulary of 5,640 distinct words.
7
-6.7
DDM
TDM
Basic MM
oHDP alpha
oHDP unif
-6.9
Per-word log likelihood
Per-word log likelihood
-6.7
-7.1
-7.3
-7.5
-7.7
DDM
TDM
Basic MM
oHDP alpha
oHDP unif
-6.9
-7.1
-7.3
-7.5
-7.7
0
100
200
300
400
0
20
40
T
60
80
100
T
(a)
(b)
Figure 4: Per-word test log likelihood of (a) NIPS and (b) Yelp.
For this dataset, we tested with K=100 using the exponentially decreasing prior on T . Figure 4(b)
shows the per-word log likelihood on the test set. DDM found the optimal number of topics while
both models achieved close to best likelihood on the test set compared to basic MM.
5.2.4
Comparison with online HDP
Because DDM and TDM do not estimate the global alpha, in the experiments we compute the test
likelihood using a uniform alpha. If we also use a uniform alpha for online HDP, DDM and TDM
achieve higher test likelihood. However, online HDP is able to learn the global alpha, which results
in higher likelihood. This is a shortcoming of our models, and we are exploring ways to estimate
the global alpha.
5.3
Additional experimental results
Additional experimental results may be found in the supplement, including running time of the experiments and samples of topics discovered in the Reuters and NIPS corpora, as well as experiments
on using the models as dimensionality reduction preprocessors in text classification.
6
Conclusions
In this paper we proposed two topic models that can be used when the number of topics is not known.
Unlike nonparametric Bayesian models, the proposed models provide explicit control over the prior
for the number of topics. We then presented an online learning algorithm based on Bayesian moment
matching, and experiments showed that reasonable topics could be recovered using the proposed
models. Additional experiments on text classification and visual inspection of the inferred topics
show that the clusters discovered were indeed semantically meaningful.
One unsolved problem is that the proposed models do not estimate the global alpha, resulting in
lower test likelihood compared to online HDP, which is able to estimate alpha. Developing a robust
way to estimate alpha will be the next step to improve the models.
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5,611 | 6,078 | On Mixtures of Markov Chains
Rishi Gupta?
Stanford University
Stanford, CA 94305
rishig@cs.stanford.edu
Ravi Kumar
Google Research
Mountain View, CA 94043
ravi.k53@gmail.com
Sergei Vassilvitskii
Google Research
New York, NY 10011
sergeiv@google.com
Abstract
We study the problem of reconstructing a mixture of Markov chains from the
trajectories generated by random walks through the state space. Under mild nondegeneracy conditions, we show that we can uniquely reconstruct the underlying
chains by only considering trajectories of length three, which represent triples of
states. Our algorithm is spectral in nature, and is easy to implement.
1
Introduction
Markov chains are a simple and incredibly rich tool for modeling, and act as a backbone in numerous
applications?from Pagerank for web search to language modeling for machine translation. While
the true nature of the underlying behavior is rarely Markovian [6], it is nevertheless often a good
mathematical assumption.
In this paper, we consider the case where we are given observations from a mixture of L Markov
chains, each on the same n states, with n ? 2L. Each observation is a series of states, and is
generated as follows: a Markov chain and starting state are selected from a distribution S, and then
the selected Markov chain is followed for some number of steps. The goal is to recover S and the
transition matrices of the L Markov chains from the observations.
When all of the observations follow from a single Markov chain (namely, when L = 1), recovering
the mixture parameters is easy. A simple calculation shows that the empirical starting distribution
and the empirical transition probabilities form the maximum likelihood Markov chain. So we are
largely interested in the case when L > 1.
As a motivating example, consider the usage of a standard maps app on a phone. There are a number
of different reasons one might use the app: to search for a nearby business, to get directions from
one point to another, or just to orient oneself. However, the users of the app never specify an explicit
intent, rather they swipe, type, zoom, etc., until they are satisfied. Each one of the latent intents can be
modeled by a Markov chain on a small state space of actions. If the assignment of each session to an
intent were explicit, recovering these Markov chains would simply reduce to several instances of the
L = 1 case. Here we are interested in the unsupervised setting of finding the underlying chains when
this assignment is unknown. This allows for a better understanding of usage patterns. For example:
? Common uses for the app that the designers had not expected, or had not expected to be
common. For instance, maybe a good fraction of users (or user sessions) simply use the app
to check the traffic.
? Whether different types of users use the app differently. For instance, experienced users
might use the app differently than first time users, either due to having different goals, or
due to accomplishing the same tasks more efficiently.
? Undiscoverable flows, with users ignoring a simple, but hidden menu setting, and instead
using a convoluted path to accomplish the same goal.
?
Part of this work was done while the author was visiting Google Research.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
The question of untangling mixture models has received a lot of attention in a variety of different
situations, particularly in the case of learning mixtures of Gaussians, see for example the seminal
work of [8], as well as later work by [5, 11, 15] and the references therein. This is, to the best of our
knowledge, the first work that looks at unraveling mixtures of Markov chains.
There are two immediate approaches to solving this problem. The first is to use the ExpectationMaximization (EM) algorithm [9]. The EM algorithm starts by guessing an initial set of parameters
for the mixture, and then performs local improvements that increase the likelihood of the proposed
solution. The EM algorithm is a useful benchmark and will converge to some local optimum, but it
may be slow to get there [12], and has are no guarantees on the quality of the final solution.
The second approach is to model the problem as a Hidden Markov Model (HMM), and employ the
machinery for learning HMMs, particularly the recent tensor decomposition methods [2, 3, 10]. As
in our case, this machinery relies on having more observed states than hidden states. Unfortunately,
directly modeling a Markov chain mixture as an HMM (or as a mixture of HMMs, as in [13]) requires
nL hidden states for n observed states. Given that, one could try adapting the tensor decomposition
arguments from [3] to our problem, which is done in Section 4.3 of [14]. However, as the authors note,
this requires accurate estimates for the distribution of trajectories (or trails) of length five, whereas our
results only require estimates for the distribution of trails of length three. This is a large difference in
the amount of data one might need to collect, as one would expect to need ?(nt ) samples to estimate
the distribution of trails of length t.
An entirely different approach is to assume a Dirichlet prior on the mixture, and model the problem
as learning a mixture of Dirichlet distributions [14]. Besides requiring the Dirichlet prior, this
method also requires very long trails. Finally, we would like to note a connection to the generic
identifiability results for HMMs and various mixture models in [1]. Their results are existential rather
than algorithmic, but dimension three also plays a central role.
Our contributions. We propose and study the problem of reconstructing a mixture of Markov chains
from a set of observations, or trajectories. Let a t-trail be a trajectory of length t: a starting state
chosen according to S along with t ? 1 steps along the appropriate Markov chain.
(i) We identify a weak non-degeneracy condition on mixtures of Markov chains and show that under
that non-degeneracy condition, 3-trails are sufficient for recovering the underlying mixture parameters.
We prove that for random instances, the non-degeneracy condition holds with probability 1.
(ii) Under the non-degeneracy condition, we give an efficient algorithm for uniquely recovering the
mixture parameters given the exact distribution of 3-trails.
(iii) We show that our algorithm outperforms the most natural EM algorithm for the problem in some
regimes, despite EM being orders of magnitude slower.
Organization. In Section 2 we present the necessary background material that will be used in the
rest of the paper. In Section 3 we state and motivate the non-degeneracy condition that is sufficient
for unique reconstruction. Using this assumption, in Section 4 we present our four-step algorithm
for reconstruction. In Section 5 we present our experimental results on synthetic and real data. In
Section 6 we show that random instances are non-degenerate with probability 1.
2
Preliminaries
Let [n] = {1, . . . , n} be a state space. We consider Markov chains defined on [n]. For a Markov
chain given by its n ? n transition matrix M , let M (i, j) denote the probability
of moving from state
P
i to state j. By definition, M is a stochastic matrix, M (i, j) ? 0 and j M (i, j) = 1. (In general
we use A(i, j) to denote the (i, j)th entry of a matrix A.)
For a matrix A, let A denote its transpose. Every n ? n matrix A of rank r admits a singular value
decomposition (SVD) of the form A = U ?V where U and V are n ? r orthogonal matrices and ?
is an r ? r diagonal matrix with non-negative entries. For an L ? n matrix B of full rank, its right
pseudoinverse B ?1 is an n ? L matrix of full rank such that BB ?1 = I; it is a standard fact that
pseudoinverses exist and can be computed efficiently when n ? L.
We now formally define a mixture of Markov chains (M, S). Let L ? 1 be an integer. Let
M = {M 1 , . . . , M L } be L transition matrices, all defined on [n]. Let S = {s1 , . . . , sL } be a
2
P
corresponding set of positive n-dimensional vectors of starting probabilities such that `,i s`i = 1.
Given M and S, a t-trail is generated as follows: first pick the chain ` and the starting state i with
probability s`i , and then perform a random walk according to the transition matrix M ` , starting from
i, for t ? 1 steps.
Throughout, we use i, j, k to denote states in [n] and ` to denote a particular chain. Let 1n be a
column vector of n 1?s.
Definition 1 (Reconstructing a Mixture of Markov Chains). Given a (large enough) set of trails
generated by a mixture of Markov chains and an L > 1, find the parameters M and S of the mixture.
Note that the number of parameters is O(n2 ? L). In this paper, we focus on a seemingly restricted
version of the reconstruction problem, where all of the given trails are of length three, i.e., every trail
is of the form i ? j ? k for some three states i, j, k ? [n]. Surprisingly, we show that 3-trails are
sufficient for perfect reconstruction.
By the definition of mixtures, the probability of generating a given 3-trail i ? j ? k is
X
s`i ? M ` (i, j) ? M ` (j, k),
(1)
`
which captures the stochastic process of choosing a particular chain ` using S and taking two steps in
M ` . Since we only observe the trails, the choice of the chain ` in the above process is latent. For
each j ? [n], let Oj be an n ? n matrix such that Oj (i, k) equals the value in (1). It is easy to see
that using O((n3 log n)/2 ) sample trails, every entry in Oj for every j is approximated to within
an additive ?. For the rest of the paper, we assume we know each Oj (i, k) exactly, rather than an
approximation of it from samples.
We now give a simple decomposition of Oj in terms of the transition matrices in M and the starting
probabilities in S. Let Pj be the L ? n matrix whose (`, i)th entry denotes the probability of
using chain `, starting in state i, and transitioning to state j, i.e., Pj (`, i) = s`i ? M ` (i, j). In a
similar manner, let Qj be the L ? n matrix whose (`, k)th entry denotes the probability of starting
in state j, and transitioning to state k under chain `, i.e., Qj (`, k) = s`j ? M ` (j, k). Finally, let
Sj = diag(s1j , . . . , sL
j ) be the L ? L diagonal matrix of starting probabilities in state j. Then,
Oj = Pj ? Sj?1 ? Qj .
This decomposition will form the key to our analysis.
3
(2)
Conditions for unique reconstruction
Before we delve into the details of the algorithm, we first identify a condition on the mixture (M, S)
such that there is a unique solution to the reconstruction problem when we consider trails of length
0
three. (To appreciate such a need, consider a mixture where two of the matrices M ` and M ` in
0
0
M are identical. Then for a fixed vector v, any s` and s` with s` + s` = v will give the same
observations, regardless of the length of the trails.) To motivate the condition we require, consider
again the sets of L ? n matrices P = {P1 , . . . , Pn } and Q = {Q1 , . . . , Qn } as defined in (2).
Together these matrices capture the n2 L ? 1 parameters of the problem, namely, n ? 1 for each of
the n rows of each of the L transition matrices M ` , and nL ? 1 parameters defining S. However,
together P and Q have 2n2 L entries, implying algebraic dependencies between them.
Definition 2 (Shuffle pairs). Two ordered sets X = {X1 , . . . , Xn } and Y = {Y1 , . . . , Yn } of L ? n
matrices are shuffle pairs if the jth column of Xi is identical to the ith column of Yj for all i, j ? [n].
Note that P and Q are shuffle pairs. We state an equivalent way of specifying this definition. Consider
a 2nL ? n2 matrix A(P, Q) that consists of a top and a bottom half. The top half is an nL ? n2 block
diagonal matrix with Pi as the ith block. The bottom half is a concatenation of n different nL ? n
block diagonal matrices; the ith block of the jth matrix is the jth column of ?Qi . A representation
of A is given in Figure 1. As intuition, note that in each column, the two blocks of L entries are the
same up to negation. Let F be the L ? 2nL matrix consisting of 2n L ? L identity matrices in a row.
It is straightforward to see that P and Q are shuffle pairs if and only if F ? A(P, Q) = 0.
Let the co-kernel of a matrix X be the vector space comprising the vectors v for which vX = 0. We
have the following definition.
3
Figure 1: A(P, Q) for L = 2, n = 4. When P and Q are shuffle pairs, each column has two copies
of the same L-dimensional vector (up to negation). M is well-distributed if there are no non-trivial
vectors v for which v ? A(P, Q) = 0.
Definition 3 (Well-distributed). The set of matrices M is well-distributed if the co-kernel of A(P, Q)
has rank L.
Equivalently, M is well-distributed if the co-kernel of A(P, Q) is spanned by the rows of F . Section 4
shows how to uniquely recover a mixture from the 3-trail probabilities Oj when M is well-distributed
and S has only non-zero entries. Section 6 shows that nearly all M are well-distributed, or more
formally, that the set of non well-distributed M has (Lebesgue) measure 0.
4
Reconstruction algorithm
We present an algorithm to recover a mixture from its induced distribution on 3-trails. We assume for
the rest of the section that M is well-distributed (see Definition 3) and S has only non-zero entries,
which also means Pj , Qj , and Oj have rank L for each j.
At a high level, the algorithm begins by performing an SVD of each Oj , thus recovering both Pj and
Qj , as in (2), up to unknown rotation and scaling. The key to undoing the rotation will be the fact
that the sets of matrices P and Q are shuffle pairs, and hence have algebraic dependencies.
More specifically, our algorithm consists of four high-level steps. We first list the steps and provide
an informal overview; later we will describe each step in full detail.
(i) Matrix decomposition: Using SVD, we compute a decomposition Oj = Uj ?j Vj and let Pj0 = Uj
and Q0j = ?j Vj . These are the initial guesses at (Pj , Qj ). We prove in Lemma 4 that there exist
L ? L matrices Yj and Zj so that Pj = Yj Pj0 and Qj = Zj Q0j for each j ? [n].
(ii) Co-kernel: Let P 0 = {P10 , . . . , Pn0 }, and Q0 = {Q01 , . . . , Q0n }. We compute the co-kernel of
matrix A(P 0 , Q0 ) as defined in Section 3, to obtain matrices Yj0 and Zj0 . We prove that there is a
single matrix R for which Yj = RYj0 and Zj = RZj0 for all j.
(iii) Diagonalization: Let R0 be the matrix of eigenvectors of (Z10 Y10 )?1 (Z20 Y20 ). We prove that there
is a permutation matrix ? and a diagonal matrix D such that R = D?R0 .
(iv) Two-trail matching: Given Oj it is easy to compute the probability distribution of the mixture
over 2-trails. We use these to solve for D, and using D, compute R, Yj , Pj , and Sj for each j.
4.1
Matrix decomposition
From the definition, both Pj0 and Q0j are L ? n matrices of full rank. The following lemma states that
the SVD of the product of two matrices A and B returns the original matrices up to a change of basis.
4
Lemma 4. Let A, B, C, D be L ? n matrices of full rank, such that AB = CD. Then there is an
L ? L matrix X of full rank such that C = X ?1 A and D = XB.
Proof. Note that A = ABB ?1 = CDB ?1 = CW for W = DB ?1 . Since A has full rank, W must
as well. We then get CD = AB = CW B, and since C has full column rank, D = W B. Setting
X = W completes the proof.
Since Oj = Pj (Sj?1 Qj ) and Oj = Pj0 Q0j , Lemma 4 implies that there exists an L ? L matrix Xj of
full rank such that Pj = Xj?1 Pj0 and Qj = Sj Xj Q0j . Let Yj = Xj?1 , and let Zj = Sj Xj . Note that
both Yj and Zj have full rank, for each j. Once we have Yj and Zj , we can easily compute both Pj
and Sj , so we have reduced our problem to finding Yj and Zj .
4.2
Co-kernel
Since (P, Q) is a shuffle pair, ((Yj Pj0 )j?[n] , (Zj Q0j )j?[n] ) is also a shuffle pair. We can write the
latter fact as B(Y, Z) A(P 0 , Q0 ) = 0, where B(Y, Z) is the L ? 2nL matrix comprising 2n matrices
concatenated together; first Yj for each j, and then Zj for each j. We know A(P 0 , Q0 ) from the
matrix decomposition step, and we are trying to find B(Y, Z). By well-distributedness, the co-kernel
of A(P, Q) has rank L. Let D be the 2nL ? 2nL block diagonal matrix with the diagonal entries
(Y1?1 , Y2?1 , . . . , Yn?1 , Z1?1 , Z2?1 , . . . , Zn?1 ). Then A(P 0 , Q0 ) = D A(P, Q). Since D has full rank,
the co-kernel of A(P 0 , Q0 ) has rank L as well.
We compute an arbitrary basis of the co-kernel of A(P 0 , Q0 )),2 and write it as an L ? 2nL matrix
as an initial guess B(Y 0 , Z 0 ) for B(Y, Z). Since B(Y, Z) lies in the co-kernel of A(P 0 , Q0 ), and has
exactly L rows, there exists an L ? L matrix R such that B(Y, Z) = R B(Y 0 , Z 0 ), or equivalently,
such that Yj = RYj0 and Zj = RZj0 for every j. Since Yj and Zj have full rank, so does R. Now our
problem is reduced to computing R.
4.3
Diagonalization
Recall from the matrix decomposition step that there exist matrices Xj such that Yj = Xj?1 and
Zj = Sj Xj . Hence Zj0 Yj0 = (R?1 Zj )(Yj R?1 ) = R?1 Sj R?1 . It seems difficult to compute R
directly from equations of the form R?1 Sj R?1 , but we can multiply any two of them together to get,
e.g., (Z10 Y10 )?1 (Z20 Y20 ) = RS1?1 S2 R?1 .
Since S1?1 S2 is a diagonal matrix, we can diagonalize RS1?1 S2 R?1 as a step towards computing
R. Let R0 be the matrix of eigenvectors of RS1?1 S2 R?1 . Now, R is determined up to a scaling and
ordering of the eigenvectors. In other words, there is a permutation matrix ? and diagonal matrix D
such that R = D?R0 .
4.4
Two-trail matching
First, Oj 1n = Pj Sj?1 Qj 1n = Pj 1L for each j, since each row of Sj?1 Qj is simply the set of
transition probabilities out of a particular Markov chain and state. Another way to see it is that both
Oj 1n and Pj 1L are vectors whose ith coordinate is the probability of the trail i ? j.
From the first three steps of the algorithm, we also have Pj = Yj Pj0 = RYj0 Pj0 = D?R0 Yj0 Pj0 .
Hence 1L D? = 1L P1 (R0 Y10 P10 )?1 = O1 1n (R0 Y10 P10 )?1 , where the inverse is a pseudoinverse.
We arbitrarily fix ?, from which we can compute D, R, Yj , and finally Pj for each j. From the
diagonalization step (Section 4.3), we can also compute Sj = R(Zj0 Yj0 )R for each j.
Note that the algorithm implicitly includes a proof of uniqueness, up to a setting of ?. Different
orderings of ? correspond to different orderings of M ` in M.
2
For instance, by taking the SVD of A(P 0 , Q0 ), and looking at the singular vectors.
5
5
Experiments
We have presented an algorithm for reconstructing a mixture of Markov chains from the observations,
assuming the observation matrices are known exactly. In this section we demonstrate that the
algorithm is efficient, and performs well even when we use empirical observations. In addition, we
also compare its performance against the most natural EM algorithm for the reconstruction problem.
Synthetic data. We begin by generating well distributed instances M and S. Let Dn be the
uniform distribution over the n-dimensional unit simplex, namely, the uniform distribution over
vectors in Rn whose coordinates are non-negative and sum to 1.
For a specific n and L, we generate an instance (M, S) as follows. For each state i and Markov chain
M ` , the set of transition probabilities leaving i is distributed as Dn . We draw each s` from Dn as
well, and then divide by L, so that the sum over all s` (i) is 1. In other words, each trail is equally
likely to come from any of the L Markov chains. This restriction has little effect on our algorithm,
but is needed to make EM tractable. For each instance, we generate T samples of 3-trails. The results
that we report are the medians of 100 different runs.
Metric for synthetic data. Our goal is exact recovery of the underlying instance M. Given two
n ? n matrices A and B, the error is the
P average total variation distance between the transition
probabilities: error(A, B) = 1/(2n) ? i,j |A(i, j) ? B(i, j)|. Given a pair of instances M =
{M 1 , . . . , M L } and N = {N 1 , . . . , N L } on the same state space [n], the recovery error is the
minimum average error over all matchings of chains in N to M. Let ? be a permutation on [L], then:
1X
recovery error(M, N ) = min
error(M ` , N ?(`) ).
?
L
`
`
p
Given all the pairwise errors error(M , N ), this minimum can be computed in time O(L3 ) by the
Hungarian algorithm. Note that the recovery error ranges from 0 to 1.
Real data. We use the last.fm 1K dataset3 , which contains the list of songs listened by heavy users
of Last.Fm. We use the top 25 artist genres4 as the states of the Markov chain. We consider the ten
heaviest users in the data set, and for each user, consider the first 3001 state transitions that change
their state. We break each sequence into 3000 3-trails. Each user naturally defines a Markov chain on
the genres, and the goal is to recover these individual chains from the observed mixture of 3-trails.
Metric for real data. Given a 3-trail from one of the users, our goal is to predict which user the
3-trail came from. Specifically, given a 3-trail t and a mixture of Markov chains (M, S), we assign t
to the Markov chain most likely to have generated it. A recovered mixture (M, S) thereby partitions
the observed 3-trails into L groups. The prediction error is the minimum over all matchings between
groups and users of the fraction of trails that are matched to the wrong user. The prediction error
ranges from 0 to 1 ? 1/L.
Handling approximations. Because the algorithm operates on real data, rather than perfect observation matrices, we make two minor modifications to make it more robust. First, in the diagonalization
0
step (Section 4.3), we sum (Zi0 Yi )?1 (Zi+1
Yi+1 )?1 over all i before diagonalizing to estimate R0 ,
instead of just using i = 1. Second, due to noise, the matrices M that we recover at the end need not
be stochastic. Following the work of [7] we normalize the values by first taking absolute values of all
entries, and then normalizing so that each of the columns sums to 1.
Baseline. We turn to EM as a practical baseline for this reconstruction problem. In our implementation, we continue running EM until the log likelihood changes by less than 10?7 in each iteration;
this corresponds to roughly 200-1000 iterations. Although EM continues to improve its solution past
this point, even at the 10?7 cutoff, it is already 10-50x slower than the algorithm we propose.
5.1
Recovery and prediction error
3
4
http://mtg.upf.edu/static/datasets/last.fm/lastfm-dataset-1K.tar.gz
http://static.echonest.com/Lastfm-ArtistTags2007.tar.gz
6
0.07
0.25
0.05
Median Recovery Error
Median Recovery Error
0.20
0.04
0.03
0.02
1.0
Alg
EM 100
EM 1000
Alg
EM 1000
0.8
Median Prediction Error
Alg
EM
0.06
0.15
0.10
0.05
0.6
0.4
0.2
0.01
0.00
0.0
0.2
0.4
0.6
Number of Samples
0.8
1.0
1e9
(a)
0.00
4
5
6
7
L
8
9
0.0
4
10
5
6
7
L
(b)
8
9
10
(c)
Figure 3: (a) Performance of EM and our algorithm vs number of samples (b) Performance of EM
and our algorithm vs L (synthetic data) (c) Performance of EM and our algorithm (real data)
For the synthetic data, we fix n = 6 and L = 3, and
for each of the 100 instances generate a progressively
larger set of samples. Recall that the number of unknown
parameters grows as ?(n2 L), so even this relatively simple setting corresponds to over 100 unknown parameters.
Figure 3(a) shows the median recovery error of both approaches. It is clear that the proposed method significantly
outperforms the EM approach, routinely achieving errors
of 10-90% lower. Furthermore, while we did not make
significant attempts to speed up EM, it is already over
10x slower than our algorithm at n = 6 and L = 3, and Figure 2: Performance of the algorithm
as a function of n and L for a fixed numbecomes even slower as n and L grow.
ber of samples.
In Figure 3(b) we study the error as a function of L. Our
approach is significantly faster, and easily outperforms
EM at 100 iterations. Running EM for 1000 iterations results with prediction error on par with our
algorithm, but takes orders of magnitude more time to complete.
Median Recovery Error
0.20
0.15
l=3
l=5
l=7
l=9
0.10
0.05
0.00
5
10
15
20
25
30
n
For the real data, there are n = 25 states, and we tried L = 4, . . . , 10 for the number of users. We run
EM for 500 iterations and show the results in Figure 3(c). While our algorithm slightly underperforms
EM, it is significantly faster in practice.
5.2
Dependence on n and L
To investigate the dependence of our approach on the size of the input, namely n and L, we fix the
number of samples to 108 but vary both the number of states from 6 to 30, as well as the number
of chains from 3 to 9. Recall that the number of parameters grows as n2 L, therefore, the largest
examples have almost 1000 parameters that we are trying to fit.
We plot the results in Figure 2. As expected, the error grows linearly with the number of chains. This
is expected ? since we are keeping the number of samples fixed, the relative error (from the true
observations) grows as well. It is therefore remarkable that the error grows only linearly with L.
We see more interesting behavior with respect to n. Recall that the proofs required n ? 2L.
Empirically we see that at n = 2L the approach is relatively brittle, and errors are relatively high.
However, as n increases past that, we see the recovery error stabilizes. Explaining this behavior
formally is an interesting open question.
6
Analysis
We now show that nearly all M are well-distributed (see Definition 3), or more formally, that the set
of non well-distributed M has (Lebesgue) measure 0 for every L > 1 and n ? 2L.
We first introduce some notation. All arrays and indices are 1-indexed. In previous sections, we
have interpreted i, j, k, and ` as states or as indices of a mixture; in this section we drop these
interpretations and just use them as generic indices.
7
For vectors v1 , . . . , vn ? RL , let v[n] denote (v1 , . . . , vn ), and let ?(v1 , . . . , vn ) denote the vi ?s
concatenated together to form a vector in RnL . Let vi [j] denote the jth coordinate of vector vi .
We first show that there exists at least one well-distributed P for each n and L.
Lemma 5 (Existence of a well-distributed P). For every n and L with n ? 2L, there exists a P for
which the co-kernel of A(P, Q) has rank L.
Proof. It is sufficient to show it for n = 2L, since for larger n we can pad with zeros. Also, recall
that F ? A(P, Q) = 0 for any P, where F is the L ? 2nL matrix consisting of 2n identity matrices
concatenated together. So the co-kernel of any A(P, Q) has rank at least L, and we just need to show
that there exists a P where the co-kernel of A(P, Q) has rank at most L.
Now, let e` be the `th basis vector in RL . Let P ? = (P1? , . . . , Pn? ), and let p?ij denote the jth column
of Pi? . We set p?ij to the (i, j)th entry of
?
e1
? eL
? .
? .
? .
?
? e2
?e
? 1
?e
? L
? .
? ..
e2
e2
e1
???
???
eL
e1
eL
..
.
e2
eL
e2
e1
???
???
eL
?
eL?1
eL?1 ?
..
.. ?
?
.
. ?
?
e3 ? ? ?
e1
e3 ? ? ?
e1 ?
.
e2 ? ? ?
eL
e1 ? ? ? eL?1 ?
?
?
e1 ? ? ? eL?1 eL?1 eL ? ? ? eL?2 ?
..
..
.. ?
.
.
. ?
e3 ? ? ?
e1
e1
e2 ? ? ?
eL
e
if i ? L or j ? L
?
Formally, pij = j?i+1
, where subscripts are taken mod L. Note that we can
ej?i ,
if i, j > L
E E
split the above matrix into four L ? L blocks
where E 0 is a horizontal ?rotation? of E.
E E0
Now, let a[n] , b[n] be any vectors in RL such that v = ?(a1 , . . . , an , b1 , . . . , bn ) ? R2nL is in the
co-kernel of A(P ? , Q? ). Recall this means v ? A(P ? , Q? ) = 0. Writing out the matrix A, it is not
too hard to see that this holds if and only if hai , p?ij i = hbj , p?ij i for each i and j.
Consider the i and j where p?ij = e1 . For each k ? [L], we have ak [1] = bk [1] from the upper
left quadrant, ak [1] = bL+k [1] from the upper right quadrant, aL+k [1] = bk [1] from the lower left
quadrant, and aL+k [1] = bL+(k+1 (mod L)) [1] from the lower right quadrant. It is easy to see that
these combine to imply that ai [1] = bj [1] for all i, j ? [n].
A similar argument for each l ? [L] shows that ai [l] = bj [l] for all i, j and l. Equivalently, ai = bj
for each i and j, which means that v lives in a subspace of dimension L, as desired.
We now bootstrap from our one example to show that almost all P are well-distributed.
Theorem 6 (Almost all P are well-distributed). The set of non-well-distributed P has Lebesgue
measure 0 for every n and L with n ? 2L.
Proof. Let A0 (P, Q) be all but the last L rows of A(P, Q). For any P, let h(P) =
det |A0 (P, Q) A0 (P, Q)|. Note that h(P) is non-zero if and only if P is well-distributed. Let P ? be
the P ? from Lemma 5. Since A0 (P ? , Q? ) has full row rank, h(P ? ) 6= 0. Since h is a polynomial
function of the entries of P, and h is non-zero somewhere, h is non-zero almost everywhere [4].
7
Conclusions
In this paper we considered the problem of reconstructing Markov chain mixtures from given
observation trails. We showed that unique reconstruction is algorithmically possible under a mild
technical condition on the ?well-separatedness? of the chains. While our condition is sufficient, we
conjecture it is also necessary; proving this is an interesting research direction. Extending our analysis
to work for the noisy case is also a plausible research direction, though we believe the corresponding
analysis could be quite challenging.
8
References
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9
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5,612 | 6,079 | High Dimensional Structured Superposition Models
Arindam Banerjee
Dept of Computer Science & Engineering
University of Minnesota, Twin Cities
banerjee@cs.umn.edu
Qilong Gu
Dept of Computer Science & Engineering
University of Minnesota, Twin Cities
guxxx396@cs.umn.edu
Abstract
High dimensional superposition models characterize observations using parameters
which can be written as a sum of multiple component parameters, each with its
own structure, e.g., sum of low rank and sparse matrices, sum of sparse and rotated
sparse vectors, etc. In this paper, we consider general superposition models which
allow sum of any number of component parameters, and each component structure
can be characterized by any norm. We present a simple estimator for such models,
give a geometric condition under which the components can be accurately estimated,
characterize sample complexity of the estimator, and give high probability nonasymptotic bounds on the componentwise estimation error. We use tools from
empirical processes and generic chaining for the statistical analysis, and our results,
which substantially generalize prior work on superposition models, are in terms of
Gaussian widths of suitable sets.
1
Introduction
For high-dimensional structured estimation problems [3, 15], considerable advances have been made
in accurately estimating a sparse or structured parameter ? ? Rp even when the sample size n is
far smaller than the ambient dimensionality of ?, i.e., n p. Instead of a single structure, such as
sparsity or low rank, recent years have seen interest in parameter estimation when the parameter ? is
Pk
a superposition or sum of multiple different structures, i.e., ? = i=1 ?i , where ?1 may be sparse, ?2
may be low rank, and so on [1, 6, 7, 9, 11, 12, 13, 23, 24].
In this paper, we substantially generalize the non-asymptotic estimation error analysis for such
superposition models such that (i) the parameter ? can be the superposition of any number of
component parameters ?i , and (ii) the structure in each ?i can be captured by any suitable norm
Ri (?i ). We will analyze the following linear measurement based superposition model
y=X
k
X
?i + ? ,
(1)
i=1
where X ? Rn?p is a random sub-Gaussian design or compressive matrix, k is the number of
components, ?i is one component of the unknown parameters, y ? Rn is the response vector, and
? ? Rn is random noise independent of X. The structure in each component ?i is captured by any
suitable norm Ri (?), such that Ri (?i ) has a small value, e.g., sparsity captured by k?i k1 , low-rank
(for matrix ?i ) captured by the nuclear norm k?i k? , etc. Popular models such as Morphological
Component Analysis (MCA) [10] and Robust PCA [6, 9] can be viewed as a special cases of this
framework (see Section D).
The superposition estimation problem can be posed as follows: Given (y, X) generated following (1),
estimate component parameters {??i } such that all the component-wise estimation errors ?i = ??i ??i? ,
where ?i? is the population mean, are small. Ideally, we want to obtain high-probability non-asymptotic
Pk
bounds on the total componentwise error measured as i=1 k??i ? ?i? k2 , with the bound improving
(getting smaller) with increase in the number n of samples.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
We propose the following estimator for the superposition model in (1):
2
k
X
s.t. Ri (?i ) ? ?i , i = 1, . . . , k ,
?i
min
y ? X
{?1 ,...,?k }
i=1
(2)
2
?
?
where ?i are suitable constants. In this paper, we focus on
?i =
? the case where
? Ri (?i ), e.g., if ?i is
?
?
s-sparse with k?i k2 = 1 and Ri (?) = k ? k1 , then ?i = s so that Ri (?i ) ? s, noting that recent
advances [16] can be used to extend our results to more general settings.
The superposition estimator in (2) succeeds if a certain geometric condition, which we call structural
coherence (SC), is satisfied by certain sets (cones) associated with the component norms Ri (?). Since
the estimate ??i = ?i? + ?i is in the feasible set of the optimization problem (2), the error vector
?i satisfies the constraint Ri (?i? + ?i ) ? ?i where ?i = Ri (?i? ). The SC condition is a geometric
relationship between the corresponding error cones Ci = cone{?i |Ri (?i? + ?i ) ? Ri (?i? )}.
If SC is satisfied, then we can show that the sum of componentwise estimation error can be bounded
with high probability, and the bound takes the form:
?
k
X
maxi w(Ci ? Bp ) + log k
?
?
?
k?i ? ?i k2 ? c
,
(3)
n
i=1
where n is the sample size, k is the number of components, and w(Ci ? Bp ) is the Gaussian width
p
[3, 8, 22] of the intersection of the error cone C?
i with the unit Euclidean ball Bp ? R . Interestingly,
the estimation error decreases at the rate of 1/ n, similar to the case of single parameter estimators
[15, 3], and depends only logarithmically on the number of components k. Further, while dependency
of the error on Gaussian width of the error cone has been established in recent results involving
a single parameter [3, 22], the bound in (3) depends on the maximum of the Gaussian width of
individual error cones, not their sum. The analysis thus gives a general way to construct estimators
for superposition problems along with high-probability non-asymptotic upper bounds on the sum of
componentwise errors. To show the generality of our work, we review and compare related work in
Appendix B.
Notation: In this paper, we use k.k to denote vector norm, and |||.||| to denote operator norm. For
example, k.k2 is the Euclidean norm for a vector or matrix, and |||.|||? is the nuclear norm of a matrix.
We denote cone{E} as the smallest closed cone that contains a given set E. We denote h., .i as the
inner product.
The rest of this paper is organized as follows: We start with a deterministic estimation error bound
in Section 2, while laying down the key geometric and statistical quantities involved in the analysis.
In Section 3, we discuss the geometry of the structural coherence (SC) condition, and in Section
4 show that the geometric SC condition implies statistical restricted eigenvalue (RE) condition. In
Section 5, we develop the main error bound on the sum of componentwise errors which hold with high
probability for sub-Gaussian designs and noise. We apply our error bound to practical problems in
Section 6, and present experimental results in Section 7. We conclude in Section 8. In the Appendix,
we compare an estimator using ?infimal convolution?[18] of norms with our estimator (2) for the
noiseless case, and provide some addition examples and experiments. The proofs of all technical
results are also in the Appendix.
2
Error Structure and Recovery Guarantees
In this section, we start with some basic results and, under suitable assumptions, provide a deterministic bound for the componentwise estimation error in superposition models. Subsequently, we
will show that the assumptions made here hold with high probability as long as a purely geometric
non-probabilistic condition characterized by structural coherence (SC) is satisfied.
Let {??i } be a solution to the superposition estimation problem in (2), {?i? } be the optimal (population)
parameters involved in the true data generation process. Let ?i = ??i ? ?i? be the error vector for
component i of the superposition. Our goal is to provide a preliminary understanding of the structure
of error sets where ?i live, identify conditions under which a bound on the total componentwise
Pk
error i=1 k??i ? ?i? k2 will hold, and provide a preliminary version of such a bound, which will be
subsequently refined to the form in (3) in Section 5. Since ??i = ?i? + ?i lies in the feasible set of (2),
2
as discussed in Section 1, the error vectors ?i will lie in the error sets Ei = {?i ? Rp |Ri (?i? +?i ) ?
Ri (?i? )} respectively. For the analysis, we will be focusing on the cone of such error sets, given by
Ci = cone{?i ? Rp |Ri (?i? + ?i ) ? Ri (?i? )} .
(4)
P
P
P
k
k
k
Let ?? = i=1 ?i? , ?? = i=1 ??i , and ? = i=1 ?i , so that ? = ?? ? ?? . From the optimality of ??
as a solution to (2), we have
? 2 ? ky ? X?? k2 ? kX?k2 ? 2? T X? ,
ky ? X ?k
(5)
using ?? = ?? + ? and y = X?? + ?. In order to establish recovery guarantees, under suitable
assumptions we construct a lower bound to kX?k2 , the left hand side of (5). The lower bound is a
generalized form of the restricted eigenvalue (RE) condition studied in the literature [4, 5, 17]. We
also construct an upper bound to ? T X?, the right hand side of (5), which needs to carefully analyze
the noise-design (ND) interaction, i.e., between the noise ? and the design X.
We start by assuming that a generalized form of RE condition is satisfied by the superposition of
errors: there exists a constant ? > 0 such that for all ?i ? Ci , i = 1, 2, . . . , k:
k
k
X
1
X
?
X
?i
? ?
k?i k2 .
(6)
(RE)
n
i=1
i=1
2
The above RE condition considers the following set:
nP
o
Pk
k
H=
?
:
?
?
C
,
k?
k
=
1
.
i
i
i 2
i=1 i
i=1
(7)
which involves all the k error cones, and the lower bound is over the sum of norms of the component
wise errors. If k = 1, the RE condition in (6) above simplifies to the widely studied RE condition
in the current literature on Lasso-type and Dantzig-type estimators [4, 17, 3] where only one error
cone is involved. If we set all components but ?i to zero, then (6) becomes the RE condition only for
component i. We also note that the general RE condition as explicitly stated in (6) has been implicitly
? defined as
used in [1] and [24]. For subsequent analysis, we introduce the set H
nP
o
Pk
k
?=
H
?
:
?
?
C
,
k?
k
?
1
.
(8)
i
i
i 2
i=1 i
i=1
?
noting that H ? H.
The general RE condition in (6) depends on the random design matrix X, and is hence an inequality
which will hold with certain probability depending on X and the set H. For superposition problems,
the probabilistic RE condition as in (6) is intimately related to the following deterministic structural
coherence (SC) condition on the interaction of the different component cones Ci , without any
explicit reference to the random design matrix X: there is a constant ? > 0 such that for all
?i ? Ci , i = 1, . . . , k,
k
k
X
X
(SC)
?i
? ?
k?i k2 .
(9)
i=1
i=1
2
If k = 1, the SC condition is trivially satisfied with ? = 1. Since most existing literature on highdimensional structured models focus on the k = 1 setting [4, 17, 3], there was no reason to study the
SC condition carefully. For k > 1, the SC condition (9) implies a non-trivial relationship among the
Pk
component cones. In particular, if the SC condition is true, then the sum i=1 ?i being zero implies
that each component ?i must also be zero. As presented in (9), the SC condition comes across as
an algebraic condition. In Section 3, we present a geometric characterization of the SC condition
[13], and illustrate that the condition is both necessary and sufficient for accurate recovery of each
component. In Section 4, we show that for sub-Gaussian design matrices X, the SC condition in
(9) in fact implies that the RE condition in (6) will hold with high probability, after the number of
samples crosses a certain sample complexity, which depends on the Gaussian width of the component
cones. For now, we assume the RE condition in (6) to hold, and proceed with the error bound analysis.
To establish recovery guarantee, following (5), we need an upper bound on the interaction between
noise ? and design X [3, 14]. In particular, we consider the noise-design (ND) interaction
?
1
(ND)
sn (?) = inf s : sup ? ? T Xu ? ?s2 n ,
(10)
s>0
n
u?sH
3
Figure 1: Geometry of SC condition
when k = 2. The error sets E1 and
E2 are respectively shown as blue an
green squares, and the corresponding error cones are C1 and C2 respectively. ?C1
is the reflection of error cone C1 . If ?C1
and C2 do not share a ray, i.e., the angle
? between the cones is larger than 0, then
?0 < 1, and the SC condition will hold.
where ? > 0 is a constant, and sH is the scaled version of H where the scaling factor is s > 0. Here,
sn (?) denotes the minimal scaling needed on H such that one obtains a uniform bound over ? ? sH
of the form: n1 ? T X? ? ?s2n (?). Then, from the basic inequality in (5), with the bounds implied by
the RE condition and the ND interaction, we have
k
X
1
1 ? T
?
? kX?k2 ? ?
? X? ? ?
(11)
k?i k2 ? ?sn (?) ,
n
n
i=1
which implies a bound on the component-wise error. The main deterministic bound below states the
result formally:
Theorem 1 (Deterministic bound) Assume that the RE condition in (6) is satisfied in H with paPk
rameter ?. Then, if ?2 > ?, we have i=1 k?i k2 ? 2sn (?).
The above bound is deterministic and holds only when the RE condition in (6) is satisfied with
constant ? such that ?2 > ?. In the sequel, we first give a geometric characterization of the SC
condition in Section 3, and show that the SC condition implies the RE condition with high probability
in Section 4. Further, we give a high probability characterization of sn (?) based on the noise ? and
design X in terms of the Gaussian widths of the component cones, and also illustrate how one can
choose ? in Section 5. With these characterizations, we will obtain the desired component-wise error
bound of the form (3).
3
Geometry of Structural Coherence
In this section, we give a geometric characterize the structural coherence (SC) condition in (9). We
start with the simplest case of two vectors x, y. If they are not reflections of each other, i.e., x 6= ?y,
then the following relationship holds:
Proposition 2 If there exists a ? < 1 such that ?hx, yi ? ?kxk2 kyk2 , then
q
kx + yk2 ? 1??
2 (kxk2 + kyk2 ) .
(12)
Next, we generalize the condition of Proposition 2 to vectors in two different cones C1 and C2 . Given
the cones, define
?0 =
sup
? hx, yi .
(13)
x?C1 ?S p?1 ,y?C2 ?S p?1
By construction, ?hx, yi ? ?0 kxk2 kyk2 for all
p x ? C1 and y ? C2 . If ?0 < 1, then (12) continues to
hold for all x ? C1 and y ? C2 with
constant
(1 ? ?0 )/2 > 0. Note that this corresponds to the SC
p
condition with k = 2 and ? = (1 ? ?0 )/2. We can interpret this geometrically as follows: first
reflect cone C1 to get ?C1 , then ? is the cosine of the minimum angle between ?C1 and C2 . If ?0 = 1,
then ?C1 and C2 share a ray, and structural coherence does not hold. Otherwise, ?0 < 1, implying
?C1 ? C2 = {0}, i.e., the two cones intersect only at the origin, and structural coherence holds.
For the general case involving k cones, denote
?i =
supP
hu, vi .
(14)
u??Ci ?Sp?1 ,v?
j6=i
Cj ?Sp?1
P
In recent work, [13] concluded that if ?i < 1 for each i = 1, . . . , k then ?Ci and j6=i Cj does not
share a ray, and the original signal can be recovered in noiseless case. We show that the condition
above in fact implies ? > 0 for the SC condition in (9), which is sufficient for accurate recovery even
in the noisy case. In particular, with ? := maxi ?i , we have the following result:
4
Theorem 3 (Structural Coherence (SC) Condition) Let ? := maxi ?i with ?i as defined in (14).
If ? < 1, then there exists a ? > 0 such that for any ?i ? Ci , i = 1, . . . , k, the SC condition in (9)
holds, i.e.,
P
Pk
k
(15)
i=1 ?i
? ? i=1 k?i k2 .
2
Thus, the SC condition is satisfied in the general case as long
Pas the reflection ?Ci of any cone Ci
does not intersect, i.e., share a ray, with the Minkowski sum j6=i Cj of the other cones.
4
Restricted Eigenvalue Condition for Superposition Models
Assuming that the SC condition is satisfied by the error cones {Ci }, i = 1, . . . , k, in this section we
show that the general RE condition in (6) will be satisfied with high probability when the number of
samples n in the sub-Gaussian design matrix X ? Rn?p crosses the sample complexity n0 . We give
a precise characterization of the sample complexity n0 in terms of the Gaussian width of the set H.
Our analysis is based on the results and techniques in [20, 14], and we note that [3] has related results
using mildly different techniques. We start with a restricted eigenvalue condition on C. For a random
vector Z ? Rp , we define marginal tail function for an arbitrary set E as
Q? (E; Z) = inf u?E P (|hZ, ui| ? ?) ,
(16)
noting that it is deterministic given the set E ? Rp . Let i , i = 1, . . . , n, be independent Rademacher
random variables, i.e., random variable with probability 12 of being either +1 or ?1, and let Xi , i =
1, . . . , n, be independent copies of Z. We define empirical width of E as
Pn
Wn (E; Z) = supu?E hh, ui, where h = ?1n i=1 i Xi .
(17)
With this notation, we recall the following result from [20]:
Lemma 1 Let X ? Rn?p be a random design matrix with each row the independent copy of
sub-Gaussian random vector Z. Then for any ?, ?, t > 0, we have
?
(18)
inf kXuk2 ? ?? nQ2?? (H; Z) ? 2Wn (H; Z) ? ??t
u?H
t2
with probability at least 1 ? e? 2 .
In order to obtain lower bound of ? in RE condition (6), we need to lower bound Q2?? (H; Z) and
upper bound Wn (H; Z). To lower bound Q2?? (H; Z), we consider the spherical cap
Pk
A = ( i=1 Ci ) ? S p?1 .
(19)
From [20, 14], one can obtain a lower bound to Q? (A; Z) based on the Paley-Zygmund inequality.
The Paley-Zygmund inequality lower bound the tail distribution of a random variable by its second
momentum. Let u be an arbitrary vector, we use the following version of the inequality.
P (|hZ, ui| ? 2?) ?
[E|hZ,ui|?2?]2+
E|hZ,ui|2
(20)
In the current context, the following result is a direct consequence of SC condition, which shows that
Q2?? (H; Z) is lower bounded by Q? (A; Z), which in turn is strictly bounded away from 0 . The
proof of Lemma 2 is given in Appendix H.1.
Lemma 2 Let sets H and A be as defined in (7) and (19) respectively. If the SC condition in (9)
holds, then the marginal tail functions of the two sets have the following relationship:
Q?? (H; Z) ? Q? (A; Z).
(21)
Next we discuss how to upper bound the empirical width Wn (H; Z). Let set E be arbitrary, and
random vector g ? N (0, Ip ) be a standard Gaussian random vector in Rp . The Gaussian width [3] of
E is defined as
w(E) = E suphg, ui.
(22)
u?E
Empirical width Wn (H; Z) can be seen as the supremum of a stochastic process. One way to upper
bound the supremum of a stochastic process is by generic chaining [19, 3, 20], and by using generic
5
chaining we can upper bound the stochastic process by a Gaussian process, which is the Gaussian
width.
As we can bound Q2?? (H; Z) and Wn (H; Z), we come to the conclusion on RE condition. Let X ?
Rn?p be a random matrix where each row is an independent copy of the sub-Gaussian random vector
Z ? Rp , and where Z has sub-Gaussian norm |||Z|||?2 ? ?x [21]. Let ? = inf u?S p?1 E[|hZ, ui|] so
that ? > 0 [14, 20]. We have the following lower bound of the RE condition. The proof of Theorem
4 is based on the proof of [20, Theorem 6.3], and we give it in appendix H.2.
Theorem 4 (Restricted Eigenvalue Condition) Let X be the sub-Gaussian design matrix that satisfies the assumptions above. If the SC condition (9) holds with a ? > 0, then with probability at least
1 ? exp(?t2 /2), we have
?
inf kXuk2 ? c1 ? n ? c2 w(H) ? c3 ?t
(23)
u?H
where c1 , c2 and c3 are positive constants determined by ?x , ?? and ?.
?
To get a ? > 0 in (6), one can simply choose t = (c1 ? n ? c2 w(H))/2c3 ?. Then as long as
n > c4 w2 (H)/?2 for c4 = c22 /c21 , we have
?
? = inf u?H ?1n kXuk2 ? 21 c1 ? ? c2 w(H)
> 0,
n
with high probability.
From the discussion above, if SC condition holds and the sample size n is large enough, then we can
find a matrix X such that RE condition holds. On the other hand, once there is a matrix X such that
RE condition holds, then we can show that SC must also be true. Its proof is give in Appendix H.3.
Proposition 5 If X is a matrix such that the RE condition (6) holds for ?i ? Ci , then the SC
condition (9) holds.
Proposition 5 demonstrates that SC condition is a necessary condition for the possibility of RE.
If SC condition does not hold, then there is {?i } such that ?i 6= 0 for some i = 1, . . . , k, but
Pk
Pk
Pk
k i=1 ?i k2 = 0 which implies i=1 ?i = 0. Then for every matrix X, we have X i=1 ?i = 0,
and RE condition is not possible.
5
General Error Bound
Recall that the error bound in Theorem 1 is given in terms of the noise-design (ND) interaction
n
? o
sn (?) = inf s>0 s : supu?sC ?1n ? T Xu ? ?s2 n .
(24)
In this section, we give a characterization of the ND interaction, which yields the final bound on the
componentwise error as long as n ? n0 , i.e., the sample complexity is satisfied.
Let ? be a centered sub-Gaussian random vector, and its sub-Gaussian norm |||?|||?2 ? ?? . Let X be
a row-wise i.i.d. sub-Gaussian random matrix, for each row Z, its sub-Gaussian norm |||Z|||?2 ? ?x .
The ND interaction can be bounded by the following conclusion, and the proof of lemma 3 is given
in appendix I.1.
Lemma 3 Let design X ? Rn?p be a row-wise i.i.d. sub-Gaussian random matrix, and noise
?
H)
? . for some constant c > 0
? ? Rn be a centered sub-Gaussian random vector. Then sn (?) ? c w(
? n
? ? c3 exp(?c4 n). Constant c depends on ?x and ?? .
with probability at least 1 ? c1 exp(?c2 w2 (H))
? and H respectively. From definition,
In lemma 3 and theorem 6, we need the Gaussian width of H
?
? and
both H and H is related to the union of different cones; therefore bounding the width of H
?
H may be difficult. We have the following bound of w(H) and w(H) in terms of the width of the
component spherical caps. The proof of Lemma 4 is given in Appendix I.2.
? be as defined in (7) and (8) respectively. Then, we
Lemma 4 (Gaussian width bound) Let ?
H and H
?
? = O maxi w(Ci ? Bp ) + log k .
have w(H) = O maxi w(Ci ? Sp?1 ) + log k and w(H)
By applying lemma 4, we can derive the error bound using the Gaussian width of individual error
cone. From our conclusion on deterministic bound in theorem 1, we can choose an appropriate ?
such that ?2 > ?. Then, by combining the result of theorem 1, theorem 4, lemma 3 and lemma 4, we
have the final form of the bound, as originally discussed in (3):
6
Theorem 6 For estimator (2), let Ci = cone{? : Ri (?i? + ?) ? Ri (?i? )}, design X be a random matrix with each row an independent copy of sub-Gaussian random vector Z, noise ? be
a centered sub-Gaussian random vector, and Bp ? Rp be the centered unit euclidean ball. If
sample size n > c(maxi w2 (Ci ? Sp?1 ) + log k)/?2 , then we have with probability at least
1 ? ?k1 exp(??2 maxi w2 (Ci ? Sp?1 )) ? ?3 exp(??4 n),
?
Pk
)+ log k
?i ? ?? k2 ? C maxi w(Ci ?B
?p
k
?
,
(25)
2
i
i=1
? n
for constants c, C > 0 that depend on sub-Gaussian norms |||Z|||?2 and |||?|||?2 .
Thus, assuming the SC condition in (9) is satisfied, the sample complexity and error bound of the
estimator depends on the largest Gaussian width, rather than the sum of Gaussian widths. The result
can be viewed as a direct generalization of existing results for k = 1, when the SC condition is always
satisfied, and the sample complexity and error is given by w2 (C1 ? Sp?1 ) and w(C1 ? Bp ) [3, 8].
6
Application of General Bound
In this section, we instantiate the general error bounds on Morphological Component Analysis
(MCA), and low-rank and sparse matrix decomposition. The comprehensive results are provided in
Appendix D.
6.1 Morphological Component Analysis
In Morphological Component Analysis [10], we consider the following linear model
y = X(?1? + ?2? ) + ?
(26)
where vector ?1? is sparse and ?2? is sparse under a rotation Q. Consider the following estimator
min ky ? X(?1 + ?2 )k22
?1 ,?2
s.t. k?1 k1 ? k?1? k1 , kQ?2 k1 ? kQ?2? k1 ,
(27)
where vector y ? Rn is the observation, vectors ?1 , ?2 ? Rp are the parameters we want to estimate,
matrix X ? Rn?p is a sub-Gaussian random design, matrix Q ? Rp?p is orthogonal. We assume ?1?
and Q?2? are s1 -sparse and s2 -sparse vectors respectively. Function kQ.k1 is still a norm. In general,
we can derive the following error bound from Theorem 6:
q
q
s1 log p
s2 log p
?
?
k?1 ? ?1 k2 + k?2 ? ?2 k2 = O max
,
.
n
n
6.2 Low-rank and Sparse Matrix Decomposition
To recover a sparse matrix and low-rank matrix from their sum [6, 9], one can use L1 norm to induce
sparsity and nuclear norm to induce low-rank. These two kinds of norm ensure that the sparsity and
the rank of the estimated matrices are small. Suppose we have a rank-r matrix L? and a sparse matrix
S ? with s nonzero entries, S ? , L? ? Rd1 ?d2 . Our observation Y comes from the following problem
Yi = hXi , L? + S ? i + Ei , i = 1, . . . , n,
where each Xi ? Rd1 ?d2 is a sub-Gaussian random design matrix. Ei is the noise matrix. The
estimator takes the form:
n
X
min
(Yi ? hXi , L + Si)2 s.t. |||L|||? ? |||L? |||? , kSk1 ? kS ? k1 .
(28)
L,S
i=1
By using Theorem 6, and existing results on Gaussian widths, the error bound is given by
q
q
s log(d1 d2 )
r(d1 +d2 ?r)
,
.
kL ? L? k2 + kS ? S ? k2 = O max
n
n
7
Experimental Results
In this section, we confirm the theoretical results in this paper with some simple experiments. We
show our experimental results under different settings. In our experiments we focus on MCA when
k = 2. The design matrix X are generated from Gaussian distribution such that every entry of X
7
1
2.5
Fraction of Successful Recovery
? ? 0.026
?
? = 1/ 2
?=0
k?1 ? ?1? k2 + k?2 ? ?2? k2
2
1.5
1
0.5
0
100
200
300
400
500
600
700
800
900
0.6
0.4
0.2
0
0
1000
Samples
p=20
p=40
p=80
p=160
0.8
0
5
10
15
20
25
30
35
40
Samples
(a)
(b)
Figure 2: (a)?
Effect of parameter ? on estimation error when noise ? 6= 0. We choose the parameter
? to be 0, 1/ 2, and a random sample. (b) Effect of dimension p on fraction of successful recovery
in noiseless case. Dimension p varies in {20, 40, 50, 150}
subjects to N (0, 1). The noise ? is generated from Gaussian distribution such that every entry of ?
subjects to N (0, 1). We implement our algorithm 1 in MATLAB. We use synthetic data in all our
experiments, and let the true signal
?1 = (1, . . . , 1, 0 . . . , 0), Q?2 = (1, . . . , 1, 0 . . . , 0)
| {z }
| {z }
s1
s2
We generate our data in different ways for our three experiments.
7.1 Recovery From Noisy Observation
In our first experiment, we test the impact of ? on the estimation error. We choose three different
matrices Q, and ? is determined the choice of Q. The first Q is given by random sampling: we sample
a random orthogonal matrix Q such that Qij > 0, and ? is lower bounded?by (42). The second and
third Q is given by identity matrix I and its negative ?I; therefore ? = 1/ 2 and ? = 0 respectively.
We choose dimension p = 1000, and let s1 = s2 = 1. The number of samples n varied between 1
and 1000. Observation y is given by y = X(?1? + ?2? ) + ?. In this experiment, given Q, for each
n, we generate 100 pairs of X and w. For each (X, w) pair, we get a solution ??1 and ??2 . We take
the average over all k??1 ? ?1? k2 + k??2 ? ?2? k2 . Figure 2(a) shows the plot of number of samples vs
the average error. From figure 2(a), we can see that the error curve given by random Q lies between
curves given by two extreme cases, and larger ? gives lower curve. In Appendix E, we provide an
additional experiment using k-support norm [2].
7.2 Recovery From Noiseless Observation
In our second experiment, we test how the dimension p affects the successful recovery of true value.
In this experiment, we choose different dimension p with p = 20, p = 40, p = 80, and p = 160. We
let s1 = s2 = 1. To avoid the impact of ?, for each sample size n, we sample 100 random orthogonal
matrices Q. Observation y is given by y = X(?1? + ?2? ). For each solution ??1 and ??2 of (41), we
calculate the proportion of Q such that k??1 ? ?1? k2 + k??2 ? ?2? k2 ? 10?4 . We increase n from 1
to 40, and the plot we get is figure 2(b). From figure 2(b) we can find that the sample complexity
required to recover ?1? and ?2? increases with dimension p.
8
Conclusions
We present a simple estimator for general superposition models and give a purely geometric characterization, based on structural coherence, of when accurate estimation of each component is possible.
Further, we establish sample complexity of the estimator and upper bounds on componentwise
estimation error and show that both, interestingly, depend on the largest Gaussian width among the
spherical caps induced by the error cones corresponding to the component norms. Going forward, it
will be interesting to investigate specific component structures which satisfy structural coherence,
and also extend our results to allow more general measurement models.
Acknowledgements: The research was also supported by NSF grants IIS-1563950, IIS-1447566,
IIS-1447574, IIS-1422557, CCF-1451986, CNS- 1314560, IIS-0953274, IIS-1029711, NASA grant
NNX12AQ39A, and gifts from Adobe, IBM, and Yahoo.
8
References
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9
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5,613 | 608 | Summed Weight Neuron Perturbation: An O(N)
Improvement over Weight Perturbation.
Barry Flower and Marwan Jabri
SEDAL
Department of Electrical Engineering
University of Sydney
NSW 2006 Australia
Abstract
The algorithm presented performs gradient descent on the weight space
of an Artificial Neural Network (ANN), using a finite difference to
approximate the gradient The method is novel in that it achieves a computational complexity similar to that of Node Perturbation, O(N3), but
does not require access to the activity of hidden or internal neurons.
This is possible due to a stochastic relation between perturbations at the
weights and the neurons of an ANN. The algorithm is also similar to
Weight Perturbation in that it is optimal in terms of hardware requirements when used for the training ofVLSI implementations of ANN's.
1 INTRODUCTION
Optimization of the weights of an ANN may be performed by, the application of a gradient descent teclmique. The gradient may be calculated directly as in Backpropagation, or it
may be approximated by a Finite Difference Method which is what we concern ourselves
with in this paper. These methods lend themselves to the task of training hardware implementations of ANNs where real estate is at a premium and synaptic density is of great
importance. Neuron Perturbation (NP), as described by the Madaline Rule ill (MRllI)
(Widrow and Lehr, 1990), is a teclmique that approximates the gradient of the Mean
Square Error (MSE) with respect to the change at a given neuron by applying a small perturbation to the input of the neuron and measuring the change in the MSE. The weight
dE
I1w ij = -tl'-:l-'Xr
onet.I
(1)
update is then calculated from the product of this gradient measure and the activation of
212
Summed Weight Neuron Perturbation: An (O)N Improvement over Weight Perturbation
the neuron from which the weight is fed, as described by (1).
Weight Perturbation (WP), as described by Jabri and Flower (Jabri and Flower, 1992) is a
neural network training techniques based on gradient descent using a Finite Difference
method to approximate the gradient. The gradient of the MSE with respect to a weight is
approximated by applying a small pertubation to the weight and measuring the change in
the MSE. This gradient is then used to calculated the weight update such that:
aW r'J
iJE
= -11? :l-..
uw ..
(2)
I)
The advantages of WP over NP are that it performs better when limited precision weights
are used, as shown by Xie and Jabri (Xie and Jabri, 1992), and is optimal with respect to
hardware requirements when used to train VLSI implementations of ANNs. However, WP
has O(~) computational complexity whilst NP has O(N3) computational complexity.
Summed Weight Neuron Perturbation (SWNP) is similar to NP in that it has a computational complexity of O(N3) but it has the added advantage that the activation of internal
neurons does not need to be known. The cost of this reduced computational complexity is
that SWNP needs to save the perturbation vector used.
In the following sections a description of the SWNP algorithm is provided and, finally,
some experimental results are presented.
2 THE SUMMED WEIGHT NEURON PERTURBATION
ALGORITHM
A subsection of a feedforward ANN containing N neurons is shown in Figure 1. on which
nomenclature the following derivation is based.
FIGURE 1: Description Of Indices Used To Describe The Neurons Weights And
Perturbations In An ANN.
In a feedforward network of size N neurons the activation of a given neuron is determined
by:
Xi (P) = Ii (net i (p? ,
and
net i (p) = ~WilXl (p) ,
(3)
and Ii (y) is the ith neuron transfer function, Xi (p) is the activation of the ith neuron for
the pth pattern, and W iI is the weight connecting the Ith neuron's output to the ith neuron's
input. The error function, (MSE), is defmed as in (4), where T is the set of output neurons
and dt (p) is the expected value of the output on the kth neuron. The change in E (p)
213
214
Flower and Jabri
with respect to a given weight may then be expressed as (5).
1
2:
E(p) =
E
(dt(p) -xt (p?2.
(4)
tE T
oE (p)
dw ij
oE (p)
= anet i (p)
.Xj (p) .
(5)
The first term of on the right-hand side of (5) can be determined using a Finite Difference,
which in this case is a Forward Difference, so that:
AEr . (p)
oE(p)
=-------c- =
~.
onet i (p)
f
+0
(6)
(ri ),
where,
and
AEr(p)
= Er, (p) -E(p),
,
(7)
r i is the perturbation applied to the ith neuron, E r., (p)
is the error for the pth pattern
with a perturbation applied to the ith neuron and E (p) is the error for the pth pattern
without a perturbation applied to any neurons. The error introduced by the approximation
is represented by the last term on the right-hand side in (6).
The perturbation of one or more of the weights that are inputs to the qth neuron can be
thought of as being equal to some perturbation applied directly to that neuron. Hence:
rq
= ~Yqrl(P)'
(8)
where Yql is the perturbation applied to weight w ql . As will be shown, perturbing the qth
neuron by perturbing all the weights feeding into it, enables the sign of the gradient
o~~)
to be determined without performing the product on the right-hand side of (5).
fJ
Further more, the activation of hidden neurons, (i.e. Xj (p) in (5? need not be known. The
contribution of the perturbation of weight w ij to the perturbation of the ith neuron is
(9)
Yi/j(p) .
Let us take the degenerate case where there is only one weight for the ith neuron. Then the
gradient of the MSE with respect to weight w ij is:
AEr (p) Xj (p)
=
'()
y.x). p
fJ
+0
(r.)
f
ll.Er . (p)
=
'
Yij
+0 (r.) ,
f
(10)
Summed Weight Neuron Perturbation: An (O)N Improvement over Weight Perturbation
noting that
Xj (p)
has been eliminated. In the general case where the ith neuron has more
than one weight the gradient with respect to weight w ij is shown in (11).
oE(p)
Ow fJ..
Mr_ (p) Xj (p)
=
r.
I
+O(ri )
f
=
Mr. (p)
+ O(r.)
I
':P ij
f
(11)
where,
r.
f
':P ij
= x. (p) .
(12)
J
The form of (10) and (11) are the same and it will be shown that y.. can be substituted for
fJ
':P fJ.. in (1) due to a stochastic relationship between them.
Let us represent the sign of yfJ.. and ':P lJ.. as either +1 or -1 such that:
and
v fJ..
I':PiJ~
= \TI'
T
..
fJ
(13)
The set of all possible states for the system represented by the vector (1.1"J
v ..) , assuming
fJ f}
y .. and ':P .. are never zero, is:
lJ
fJ
{(-1, -1), (-1, 1), (1, -1), (1, 1)} .
and it can be seen that when 1.1 fJ..
= v fJ..
(14)
then the sign of the gradient of the MSE with
respect to weight w ij given by (0) is the same as that given by (11). If the sign of Yij is
chosen randomly then the probability of 1.1fJ..
= v fJ.. being true is 0.5, from (4), and so (0)
will generate a gradient that is in the correct direction 50% of the time. This in itself is not
sufficient to allow the network to be trained as it will take as many steps in the incorrect
direction as the correct direction if the steps themselves are of the same size, (i.e. the magnitude of r.f is the same for a step in the correct direction as a step in the incorrect direction).
Fortunately it can be shown that the size of the steps in the correct direction are greater
than those in the incorrect direction. Let us take the case where a particular yf}.. is chosen
such that
1.1fJ.. = v IJ...
Now by substituting (8), (2) and (13) into (5) we get:
(15)
215
216
Flower and Jabri
~ Yitxt (p)
-
x?J
(16)
-----'---
~YitXt (p)
Yij
x?J
rearranging to give,
(17)
~YitXt (p)
Yi/j
which implies that the contribution to r. made by the pertUIbation y.. is of the same sign
v
I
r I.. Let us designate this neuron pertUIbation as r.I (A) . Now we take the other possible
as
case where,
JI ij
*
(18)
V ij'
assuming every other parameter is the same, and only the sign of y .. is changed. The
IJ
equality in (17) is now untrue and the contribution to r.I made by the perturbation yIJ.. is of
the opposite sign as
r I.. Let us designate this neuron perturbation as r I.(B) . From (8) we
can determine that,
(19)
Equation (19) shows the relationship between the two possible states of the system where
r. (A) represents the summed neuron perturbation for a selected weight perturbation y ..
I
that generates a step in the corrected direction and
r i (B)
v
is similar but for a step in the
incorrect direction. Clearly the correct step is always calculated from an approximated
gradient that is larger than that for an incorrect step as the neuron perturbation is larger.
The weight update rule then becomes:
Mr. (p)
~Wij = -Tl.
I
(20)
Yij
The algorithm for SWNP is shown as pseudo code in Figure 2.
2.1 HARDWARE COMPATmILITY OF SWNP
This optimisation technique is ideally suited to the training of hardware implementations
of ANN's whether they consist of discrete components or are VLSI technology. The speed
up over WP of 0 (N) achieved is at the cost of an 0 (N) storage requirement but this
sto~ge can be achieved with a single bit per neuron. SWNP is the same order of complexity as NP but does not require access to the activation of internal neurons and therefore can
treat a network as a "black box" into which an input vector and weight matrix is fed and an
Summed Weight Neuron Perturbation: An (O)N Improvement over Weight Perturbation
output vector is received.
While (total error> error threshold) {
For (all patterns in training set) {
Select next pattern and training vector,
Forward Prop.;Measure, (calculate) and save error;
Accumulate total error;
For (all non-input neurons) {
For (all weights of current neuron) {
} Apply & Save perturbation of random polarity;
Forward Prop.;Measure, (calculate) and save &!rror;
For (all weights of current neuron) {
Restore value of weight;
Calculate weight delta using saved perturbation value;
} If (Online Mode) Update current weight;
If (Online Mode)
}
Forward Prop.; Measure, (calculate) and save new error;
If (Batch Mode) {
For (all weights)
Update current weight;
} }
}
FIGURE 2: Algorithm in Pseudo Code for Summed Weight Neuron Perturbation.
3
TEST RESULTS USING SWNP
The results for a series of tests are shown in the next three tables and are summarised in
Figure 4. The headings are, N the number of neurons in the network, P the number of patterns in the training set, FF-SWNP the number of feedforward passes for the SWNP
Algorithm, FF-WP the number of feedforward passes for the WP Algorithm, and RA
the ratio between the number of feedforward passes for WP against SWNP. The feedforward passes are recorded to 1 significant figure.
no
The results for a series of simulations comparing the performance of SWNP against WP
are shown in Table 1. The simulations utilised floating point synaptic and neuron precis1ons.
The results for a series of simulations comparing the performance of SWNP against WP
are shown in Table 2. The simulations utilised limited synaptic precision, (i.e. 6 bits) and
floating point neuron precisions
The results for a series of experiments comparing the performance of SWNP against WP
are shown in Table 2. Note: the training algorithm are the variations of WP and SWNP
that are combined with the Random Search Algorithm (RSA). The results reported are
averaged over 10 trials.
An example of the training error trajectories of WP and SWNP for the Monk 2 problem
are shown in Figure 3.
217
218
Flower and Jabri
Table 1: Performance Of SWMP Versus WP, Comparing Feedforward Operations To
Convergence. (Simulations With Floating Point Precision)
PROBLEM
N
P
FF-SWNP
FF-WP
ERROR
RATIO
XOR
3
4
1.6xlQ3
1.9xlcP
0.0125
1. 22
4 Encoder
5
4
0.9xlcP
1.8xlcY
0.0125
1.84
8 Encoder
11
8
1.5xlOS
4.5xlOS
0.0125
2.88
ICEG
15
119
3.7xlOS
7.9xlrf'
0.0125
21.34
Table 2: Performance Of SWNP Versus WP, Comparing Feedforward Operations To
Convergence. (Simulations With Limited Precision)
PROBLEM
N
P
SWNP
WP
ERROR
RATIO
Monk 1
4
129
1.OxlOS
1.9x106
0.001
19.38
MonIa
17
169
3.6xlOS
6.8x106
0.0005
18.71
Monk3
17
122
1.2xl06
7.1xl06
0.022
5.87
IECG 55
5
8
1.6xl04
7.2xl04
0.0001
4.2
Table 3: Performance Of SWNP Versus WP, Comparing Feedforward Operations To
Convergence. (Hardware Implementation)
PROBLEM
N
P
SWNP
WP
ERROR
RATIO
ECG 55
ECG045
5
5
8
8
3.1xlQ3
1.1xl04
3.6xlQ3
2.Oxl04
0.00001
0.001
1.13
1.78
--
MONX 2 PROBLEM
YBKa lIT'
..,CO
lIWRP
",",co
_co
>:"'co
:IOOCO
1I0CO
ItDCO
,OOCO
1:"'CO
'OOCO
lOCO
?lCO
ooCO
:lOCO
000
OCO
2).(1)
?),(1)
?).IX)
tom
10000
12).00
leO)
J!IIIOC2B
FIGURE 3: Comparison ofWP and SWNP For Monk 2 Problem
Summed Weight Neuron Perturbation: An (O)N Improvement over Weight Perturbation
FIGURE 4: Comparison of the number of Feedforward passes performed to achieve
convergence on a range of problems using SWNP and WP.
20
18
16
14
12
10
8
6
4
2
mwsm SWNP Feedforward Passes
_
WP Feedforward Passes
XORx~02
4ENCODERxI
8ENCODERxI
ICEGxl
MONK. 1xl
MONK.2xl
MONlOxl
lCEG 55xl
ECG 55xl
ECG045xl
4 CONCLUSION
The algorithm presented, SWNP, performs gradient descent on the weight space of an
ANN, using a fInite difference to approximate the gradient. The method is novel in that it
achieves 0 (N3 ) computational complexity similar to that of Node Perturbation but does
not require access to the activity of hidden or internal neurons. The algorithm is also similar to Weight Perturbation in that it is optimal in terms of hardware requirements when
used for the training of VLSI implementations of ANN's. Results are presented that show
the algorithm in operation on floating point simulations, limited precision simulations and
an actual hardware implementation of an ANN.
References
labri, M. and Flower, B. (1992). Weight perturbation: An optimal architecture and learning
technique for analog vlsi feedforward and recurrent multilayer networks. IEEE
Transactions on Neural Networks, 3(1):154-157.
Widrow, B. and Lehr, M. A. (1990). 30 years of adaptive neural networks: Perceptron,
madaline, and backpropagation. Proceedings of the IEEE, 78(9):1415-1442.
Xie, Y. and lahri, M. (1992). Analysis of the effects of quantization in multilayer neural
networks using a statistical model. IEEE Transactions on Neural Networks,
3(2):334-338.
219
| 608 |@word sydney:1 trial:1 effect:1 true:1 implies:1 hence:1 direction:9 equality:1 added:1 correct:5 saved:1 wp:20 simulation:8 stochastic:2 australia:1 ll:1 nsw:1 defmed:1 gradient:18 kth:1 ow:1 require:3 feeding:1 series:4 designate:2 yij:5 qth:2 performs:3 fj:14 current:4 comparing:6 assuming:2 code:2 index:1 activation:6 novel:2 great:1 relationship:2 polarity:1 ratio:4 madaline:2 ql:1 substituting:1 ji:1 achieves:2 enables:1 perturbing:2 update:5 analog:1 implementation:7 approximates:1 selected:1 sedal:1 accumulate:1 significant:1 monk:5 neuron:48 ith:9 finite:5 descent:4 lco:1 clearly:1 node:2 always:1 perturbation:39 access:3 introduced:1 incorrect:5 improvement:5 expected:1 ra:1 themselves:2 yi:2 seen:1 flower:7 fortunately:1 lj:2 greater:1 mr:2 pattern:6 actual:1 hidden:3 relation:1 vlsi:4 wij:1 becomes:1 provided:1 determine:1 barry:1 ii:3 lend:1 ill:1 what:1 restore:1 summed:9 whilst:1 equal:1 never:1 untrue:1 eliminated:1 technology:1 pseudo:2 represents:1 every:1 lit:1 ti:1 oco:1 multilayer:2 optimisation:1 np:5 represent:1 achieved:2 randomly:1 engineering:1 treat:1 floating:4 ourselves:1 versus:3 pass:7 black:1 pij:1 ecg:2 sufficient:1 co:5 lehr:2 estate:1 limited:4 noting:1 feedforward:13 range:1 averaged:1 changed:1 xj:5 last:1 architecture:1 heading:1 rsa:1 loco:2 opposite:1 backpropagation:2 side:3 xr:1 allow:1 perceptron:1 whether:1 thought:1 calculated:4 forward:4 made:2 adaptive:1 get:1 nomenclature:1 measuring:2 anet:1 pth:3 storage:1 transaction:2 applying:2 cost:2 approximate:3 hardware:8 reported:1 reduced:1 generate:1 aw:1 marwan:1 xi:2 combined:1 rule:2 search:1 density:1 sign:7 yf:1 delta:1 per:1 table:7 dw:1 transfer:1 summarised:1 discrete:1 variation:1 rearranging:1 connecting:1 mse:7 threshold:1 recorded:1 containing:1 ije:1 jabri:8 substituted:1 approximated:3 uw:1 year:1 de:1 electrical:1 tl:2 calculate:4 ff:4 sto:1 oe:4 precision:6 performed:2 utilised:2 rq:1 xl:4 bit:2 complexity:7 onet:2 ideally:1 ix:1 trained:1 contribution:3 activity:2 square:1 aer:3 xt:1 xor:1 er:2 n3:4 ri:2 concern:1 consist:1 generates:1 quantization:1 speed:1 represented:2 importance:1 performing:1 derivation:1 train:1 leo:1 teclmique:2 trajectory:1 describe:1 department:1 magnitude:1 te:1 artificial:1 labri:1 anns:2 suited:1 synaptic:3 larger:2 against:4 expressed:1 encoder:2 rror:1 pertubation:1 x106:2 itself:1 online:2 advantage:2 equation:1 subsection:1 net:2 prop:3 ann:10 product:2 ge:1 fed:2 iceg:1 xl04:3 change:4 dt:2 xie:3 operation:4 tom:1 degenerate:1 achieve:1 apply:1 determined:3 corrected:1 description:2 box:1 total:2 experimental:1 premium:1 save:5 batch:1 convergence:4 hand:3 requirement:4 select:1 internal:4 widrow:2 recurrent:1 mode:3 ij:12 received:1 |
5,614 | 6,080 | Truncated Variance Reduction: A Unified Approach
to Bayesian Optimization and Level-Set Estimation
Ilija Bogunovic1 , Jonathan Scarlett1 , Andreas Krause2 , Volkan Cevher1
1
Laboratory for Information and Inference Systems (LIONS), EPFL
2
Learning and Adaptive Systems Group, ETH Z?urich
{ilija.bogunovic,jonathan.scarlett,volkan.cevher}@epfl.ch, krausea@ethz.ch
Abstract
We present a new algorithm, truncated variance reduction (T RU VA R), that treats
Bayesian optimization (BO) and level-set estimation (LSE) with Gaussian processes in a unified fashion. The algorithm greedily shrinks a sum of truncated
variances within a set of potential maximizers (BO) or unclassified points (LSE),
which is updated based on confidence bounds. T RU VA R is effective in several
important settings that are typically non-trivial to incorporate into myopic algorithms, including pointwise costs and heteroscedastic noise. We provide a general
theoretical guarantee for T RU VA R covering these aspects, and use it to recover
and strengthen existing results on BO and LSE. Moreover, we provide a new result
for a setting where one can select from a number of noise levels having associated
costs. We demonstrate the effectiveness of the algorithm on both synthetic and
real-world data sets.
1
Introduction
Bayesian optimization (BO) [1] provides a powerful framework for automating design problems,
and finds applications in robotics, environmental monitoring, and automated machine learning, just
to name a few. One seeks to find the maximum of an unknown reward function that is expensive
to evaluate, based on a sequence of suitably-chosen points and noisy observations. Numerous BO
algorithms have been presented previously; see Section 1.1 for an overview.
Level-set estimation (LSE) [2] is closely related to BO, with the added twist that instead of seeking
a maximizer, one seeks to classify the domain into points that lie above or below a certain threshold. This is of considerable interest in applications such as environmental monitoring and sensor
networks, allowing one to find all ?sufficiently good? points rather than the best point alone.
While BO and LSE are closely related, they are typically studied in isolation. In this paper, we provide a unified treatment of the two via a new algorithm, Truncated Variance Reduction (T RU VA R),
which enjoys theoretical guarantees, good computational complexity, and the versatility to handle
important settings such as pointwise costs, non-constant noise, and multi-task scenarios. The main
result of this paper applies to the former two settings, and even the fixed-noise and unit-cost case,
we refine existing bounds via a significantly improved dependence on the noise level.
1.1 Previous Work
Three popular myopic techniques for Bayesian optimization are expected improvement (EI), probability of improvement (PI), and Gaussian process upper confidence bound (GP-UCB) [1, 3], each
of which chooses the point maximizing an acquisition function depending directly on the current
posterior mean and variance. In [4], the GP-UCB-PE algorithm was presented for BO, choosing
the highest-variance point within a set of potential maximizers that is updated based on confidence
bounds. Another relevant BO algorithm is BaMSOO [5], which also keeps track of potential maximizers, but instead chooses points based on a global optimization technique called simultaneous
1
online optimization (SOO). An algorithm for level-set estimation with GPs is given in [2], which
keeps track of a set of unclassified points. These algorithms are computationally efficient and have
various theoretical guarantees, but it is unclear how best to incorporate aspects such as pointwise
costs and heteroscedastic noise [6]. The same is true for the Straddle heuristic for LSE [7].
Entropy search (ES) [8] and its predictive version [9] choose points to reduce the uncertainty of
the location of the maximum, doing so via a one-step lookahead of the posterior rather than only
the current posterior. While this is more computationally expensive, it also permits versatility with
respect to costs [6], heteroscedastic noise [10], and multi-task scenarios [6]. A recent approach
called minimum regret search (MRS) [11] also performs a look-ahead, but instead chooses points to
minimize the regret. To our knowledge, no theoretical guarantees have been provided for these.
The multi-armed bandit (MAB) [12] literature has developed alongside the BO literature, with the
two often bearing similar concepts. The MAB literature is far too extensive to cover here, but we
briefly mention some variants relevant to this paper. Extensive attention has been paid to the bestarm identification problem [13], and cost constraints have been incorporated in a variety of forms
[14]. Moreover, the concept of ?zooming in? to the optimal point has been explored [15]. In general,
the assumptions and analysis techniques in the MAB and BO literature are quite different.
1.2
Contributions
We present a unified analysis of Bayesian optimization and level-set estimation via a new algorithm
Truncated Variance Reduction (T RU VA R). The algorithm works by keeping track of a set of potential maximizers (BO) or unclassified points (LSE), selecting points that shrink the uncertainty within
that set up to a truncation threshold, and updating the set using confidence bounds. Similarly to ES
and MRS, the algorithm performs a one-step lookahead that is highly beneficial in terms of versatility. However, unlike these previous works, our lookahead avoids the computationally expensive
task of averaging over the posterior distribution and the observations.
Also in contrast with ES and MRS, we provide theoretical bounds for T RU VA R characterizing the
cost required to achieve a certain accuracy in finding a near-optimal point (BO) or in classifying
each point in the domain (LSE). By applying this to the standard BO setting, we not only recover
existing results [2, 4], but we also strengthen them via a significantly improved dependence on the
noise level, with better asymptotics in the small noise limit. Moreover, we provide a novel result for
a setting in which the algorithm can choose the noise level, each coming with an associated cost.
Finally, we compare our algorithm to previous works on several synthetic and real-world data sets,
observing it to perform favorably in a variety of settings.
2
Problem Setup and Proposed Algorithm
Setup: We seek to sequentially optimize an unknown reward function f (x) over a finite domain
D.1 At time t, we query a single point xt 2 D and observe a noisy sample yt = f (xt ) + zt , where
zt ? N (0, 2 (xt )) for some known noise function 2 (?) : D ! R+ . Thus, in general, some points
may be noisier than others, in which case we have heteroscedastic noise [10]. We associate with
each point a cost according to some known cost function c : D ! R+ . If both 2 (?) and c(?) are
set to be constant, then we recover the standard homoscedastic and unit-cost setting.
We model f (x) as a Gaussian process (GP) [16] having mean zero and kernel function k(x, x0 ),
normalized so that k(x, x) = 1 for all x 2 D. The posterior distribution of f given the points and
observations up to time t is again a GP, with the posterior mean and variance given by [10]
?t (x) = kt (x)T Kt + ?t
1
(1)
yt
1
= k(x, x) kt (x)T Kt + ?t
kt (x),
(2)
?
?t
?
?
where kt (x) = k(xi , x) i=1 , Kt = k(xt , xt0 ) t,t0 , and ?t = diag( 2 (x1 ), . . . , 2 (xt )). We also
let t2 1|x (x) denote the posterior variance of x upon observing x along with x1 , ? ? ? , xt 1 .
t (x)
1
2
Extensions to continuous domains are discussed in the supplementary material.
2
Confidence
Selected point
Target
Max. lower bound
Potential maximizers
(a) t = 6
(b) t = 7
(c) t = 8
(d) t = 9
Figure 1: An illustration of the T RU VA R algorithm. In (a), (b), and (c), three points within the set
of potential maximizers Mt are selected in order to bring the confidence bounds to within the target
range, and Mt shrinks during this process. In (d), the target confidence width shrinks as a result of
the last selected point bringing the confidence within Mt to within the previous target.
We consider both Bayesian optimization, which consists of finding a point whose function value is
as high as possible, and level-set estimation, which consists of classifying the domain according into
points that lie above or below a given threshold h. The precise performance criteria for these settings
are given in Definition 3.1 below. Essentially, after spending a certain cost we report a point (BO)
or a classification (LSE), but there is no preference on the values of f (xt ) for the points xt chosen
before coming to such a decision (in contrast with other notions such as cumulative regret).
T RU VA R algorithm: Our algorithm is described in Algorithm 1, making use of the updates
described in Algorithm 2. The algorithm keeps track of a sequence of unclassified points Mt ,
representing potential maximizers for BO or points close to h for LSE. This set is updated
based on the confidence bounds depending on constants (i) . The algorithm proceeds in epochs,
1/2
where in the i-th epoch it seeks to bring the confidence (i) t (x) of points within Mt below a target value ?(i) . It does this by greedily minimizing the sum of truncated variances
P
2
x2Mt 1 max{ (i) t 1|x (x), ?(i) } arising from choosing the point x, along with a normalization
and division by c(x) to favor low-cost points. The truncation by ?(i) in this decision rule means that
once the confidence of a point is below the current target value, there is no preference in making it
any lower (until the target is decreased). Once the confidence of every point in Mt is less than a
factor 1 + above the target value, the target confidence is reduced according to a multiplication by
r 2 (0, 1). An illustration of the process is given in Figure 1, with details in the caption.
For level-set estimation, we also keep track of the sets Ht and Lt , containing points believed to have
function values above and below h, respectively. The constraint x 2 Mt 1 in (5)?(7) ensures that
{Mt } is non-increasing with respect to inclusion, and Ht and Lt are non-decreasing.
Algorithm 1 Truncated Variance Reduction (T RU VA R)
Input: Domain D, GP prior (?0 , 0 , k), confidence bound parameters > 0, r 2 (0, 1), { (i) }i 1 ,
?(1) > 0, and for LSE, level-set threshold h
1: Initialize the epoch number i = 1 and potential maximizers M(0) = D.
2: for t = 1, 2, . . . do
3:
Choose
P
P
2
2
2
2
x2Mt 1 max{ (i) t 1 (x), ?(i) }
x2Mt 1 max{ (i) t 1|x (x), ?(i) }
xt = arg max
. (3)
c(x)
x2D
4:
5:
6:
Observe the noisy function sample yt , and update according to Algorithm 2 to obtain Mt ,
?t , t , lt and ut , as well as Ht and Lt in the case of LSE
1/2
while maxx2Mt (i) t (x) ? (1 + )?(i) do
Increment i, set ?(i) = r ? ?(i 1) .
The choices of (i) , , and r are discussed in Section 4. As with previous works, the kernel is
assumed known in our theoretical results, whereas in practice it is typically learned from training
data [3]. Characterizing the effect of model mismatch or online hyperparameter updates is beyond
the scope of this paper, but is an interesting direction for future work.
3
Algorithm 2 Parameter Updates for T RU VA R
Input: Selected points and observations {xt0 }tt0 =1 ; {yt0 }tt0 =1 , previous sets Mt 1 , Ht 1 , Lt
1/2
parameter (i) , and for LSE, level-set threshold h.
1: Update ?t and t according to (1)?(2), and form the upper and lower confidence bounds
1/2
2: For BO, set
or for LSE, set
H t = Ht
Mt = x 2 Mt
1
1/2
ut (x) = ?t (x) + (i) t (x), `t (x) = ?t (x)
(i)
?
Mt = x 2 Mt 1 : ut (x)
max `t (x) ,
x2Mt 1
[ x 2 Mt
1
1
: ut (x)
: `t (x) > h ,
t (x).
1
(4)
(5)
h and `t (x) ? h
Lt = L t
1,
(6)
[ x 2 Mt
1
: ut (x) < h .
(7)
Some variants of our algorithm and theory are discussed in the supplementary material due to lack
of space, including pure variance reduction, non-Bayesian settings [3], continuous domains [3], the
batch setting [4], and implicit thresholds for level-set estimation [2].
3
Theoretical Bounds
In order to state our results for BO and LSE in a unified fashion, we define a notion of ?-accuracy
for the two settings. That is, we define this term differently in the two scenarios, but then we provide
theorems that simultaneously apply to both. All proofs are given in the supplementary material.
Definition 3.1. After time step t of T RU VA R, we use the following terminology:
? For BO, the set Mt is ?-accurate if it contains all true maxima x? 2 arg maxx f (x), and all of
its points satisfy f (x? ) f (x) ? ?.
? For LSE, the triplet (Mt , Ht , Lt ) is ?-accurate if all points in Ht satisfy f (x) > h, all points in
Lt satisfy f (x) < h, and all points in Mt satisfy |f (x) h| ? 2? .
Pt
In both cases, the cumulative cost after time t is defined as Ct = t0 =1 c(xt0 ).
We use 2? in the LSE setting instead of ? since this creates a region of size ? where the function value
lies, which is consistent with the BO setting. Our performance criterion for level-set estimation is
slightly different from that of [2], but the two are closely related.
3.1
General Result
Preliminary definitions: Suppose that the { (i) } are chosen to ensure valid confidence bounds,
i.e., lt (x) ? f (x) ? ut (x) with high probability; see Theorem 3.1 and its proof below for such
choices. In this case, we have after the i-th epoch that all points are either already discarded (BO)
or classified (LSE), or are known up to the confidence level (1 + )?(i) . For the points with such
confidence, we have ut (x) lt (x) ? 2(1 + )?(i) , and hence
ut (x) ? lt (x) + 2(1 + )?(i) ? f (x) + 2(1 + )?(i) ,
(8)
and similarly lt (x) f (x) 2(1 + )?(i) . This means that all points other than those within a gap
of width 4(1 + )?(i) must have been discarded or classified:
Mt ? x : f (x)
Mt ? x : |f (x)
f (x? )
4(1 + )?(i) =: M (i)
h| ? 2(1 + )?(i) =: M (i)
(LSE)
Since no points are discarded or classified initially, we define M (0) = D.
4
(BO)
(9)
(10)
For a collection of points S = (x01 , . . . , x0|S| ), possibly containing duplicates, we write the total cost
P|S|
as c(S) = i=1 c(x0i ). Moreover, we denote the posterior variance upon observing the points up to
time t 1 and the additional points in S by t 1|S (x). Therefore, c(x) = c({x}) and t 1|x (x) =
t 1|{x} (x). The minimum cost (respectively, maximum cost) is denoted by cmin = minx2D c(x)
(respectively, cmax = maxx2D c(x)).
Finally, we introduce the quantity
n
o
C ? (?, M ) = min c(S) : max 0|S (x) ? ? ,
(11)
S
x2M
representing the minimum cost to achieve a posterior standard deviation of at most ? within M .
Main result: In all of our results, we make the following assumption.
Assumption 3.1. The kernel k(x, x0 ) is such that the variance reduction function
t,x (S)
2
t (x)
=
2
t|S (x)
(12)
is submodular [17] for any time t, and any selected points (x1 , . . . , xt ) and query point x.
This assumption has been used in several previous works based on Gaussian processes, and sufficient
conditions for its validity can be found in [18, Sec. 8]. We now state the following general guarantee.
Theorem 3.1. Fix ? > 0 and 2 (0, 1), and suppose there exist values {C(i) } and { (i) } such that
?
?
?(i)
|M (i 1) | (i)
C(i) C ?
,
M
+ cmax ,
(13)
(i 1) log
2 2
1/2
?(i)
(i)
and
(i)
2 log
Then if T RU VA R is run with these choices of
C? =
P
|D|
i0 ?i C(i0 )
6 c2min
?
(14)
.
until the cumulative cost reaches
X
C(i) ,
(i)
i : 4(1+ )?(i
then with probability at least 1
2 2
(15)
1) >?
, we have ?-accuracy.
While this theorem is somewhat abstract, it captures the fact that the algorithm improves when points
having a lower cost and/or lower noise are available, since both of these lead to a smaller value of
C ? (?, M ); the former by directly incurring a smaller cost, and the latter by shrinking the variance
more rapidly. Below, we apply this result to some important cases.
3.2
Results for Specific Settings
Homoscedastic and unit-cost setting: Define the maximum mutual information [3]
1
log det IT + 2 KT ,
(16)
T = max
x1 ,...,xT 2
and consider the case that 2 (x) = 2 and c(x) = 1. In the supplementary material, we provide a
1
theorem with a condition for ?-accuracy of the form T
?? C1 ?T2 T + 1 with C1 = log(1+
2) ,
thus matching [2, 4] up to logarithmic factors. In the following, we present a refined version that has
a significantly better dependence on the noise level, thus exemplifying that a more careful analysis
of (13) can provide improvements over the standard bounding techniques.
2
2
?
Corollary 3.1. Fix ? > 0 and 2 (0, 1), define T = 2 log |D|T
, and set ?(1) = 1 and r = 12 .
6
There exist choices of (i) (not depending on the time horizon T ) such that we have ?-accuracy with
probability at least 1
once the following condition holds:
?
?
l
96(1 + )2
6(1 + )2
32(1 + )2 m
16(1 + )2 |D| T
2
T
2
+
C
+
2
log
log
,
T T
1
T
T
2
2
2
?2
?
?2
(17)
where C1 =
1
log(1+
2)
. This condition is of the form T
5
??
2
T
?2
T
+
C1
T
2
T
+1 .
The choices ?(1) = 1 and r = 12 are made for mathematical convenience, and a similar result follows
for any other choices ?(1) > 0 and r 2 (0, 1), possibly with different constant factors.
As 2 ! 1 (i.e., high noise), both of the above-mentioned bounds have noise dependence O? ( 2 ),
since log(1 + ? 1 ) = O(? 1 ) as ? ! 1. On the other hand, as 2 ! 0 (i.e., low noise), C1 is
logarithmic, and Corollary 3.1 is significantly better provided that ? ? .
Choosing the noise and cost: Here we consider the setting that there is a domain of points D0
that the reward function depends on, and alongside each point we can choose a noise variance 2 (k)
(k = 1, . . . , K). Hence, D = D0 ?{1, ? ? ? , K}. Lower noise variances incur a higher cost according
to a cost function c(k).
Corollary 3.2. For each k = 1, ? ? ? , K, let T ? (k) denote the smallest value of T such that (17) holds
|D|T 2 c2 ? 2
with 2 (k) in place of 2 , and with T = 2 log 6 c2max . Then, under the preceding setting, there
min
exist choices of (i) (not depending on T ) such that we have ?-accuracy with probability at least 1
once the cumulative cost reaches mink c(k)T ? (k).
This result roughly states that we obtain a bound as good as that obtained by sticking to any fixed
choice of noise level. In other words, every choice of noise (and corresponding cost) corresponds to a
different version of a BO or LSE algorithm (e.g., [2, 4]), and our algorithm has a similar performance
guarantee to the best among all of those. This is potentially useful in avoiding the need for running
an algorithm once per noise level and then choosing the best-performing one. Moreover, we found
numerically that beyond matching the best fixed noise strategy, we can strictly improve over it by
mixing the noise levels; see Section 4.
4
Experimental Results
We evaluate our algorithm in both the level-set estimation and Bayesian optimization settings.
Parameter choices: As with previous GP-based algorithms that use confidence bounds, our theoretical choice of (i) in T RU VA R is typically overly conservative. Therefore, instead of using (14)
directly, we use a more aggressive variant with similar dependence on the domain size and time:
2
(i) = a log(|D|t(i) ), where t(i) is the time at which the epoch starts, and a is a constant. Instead
of the choice a = 2 dictated by (14), we set a = 0.5 for BO to avoid over-exploration. We found
exploration to be slightly more beneficial for LSE, and hence set a = 1 for this setting. We found
T RU VA R to be quite robust with respect to the choices of the remaining parameters, and simply set
?(1) = 1, r = 0.1, and = 0 in all experiments; while our theory assumes > 0, in practice there
is negligible difference between choosing zero and a small positive value.
Level-set estimation: For the LSE experiments, we use a common classification rule in all al? t = {x : ?t (x)
gorithms, classifying the points according to the posterior mean as H
h} and
?
Lt = {x : ?t (x) < h}. The classification accuracy is measured by the F1 -score (i.e., the harmonic
mean of precision and recall) with respect to the true super- and sub-level sets.
We compare T RU VA R against the GP-based LSE algorithm [2], which we name via the authors?
surnames as GCHK, as well as the state-of-the-art straddle (STR) heuristic [7] and the maximum
variance rule (VAR) [2]. Descriptions can be found in the supplementary material. GCHK includes
1/2
an exploration constant t , and we follow the recommendation in [2] of setting t = 3.
Lake data (unit cost): We begin with a data set from the domain of environmental monitoring of
inland waters, consisting of 2024 in situ measurements of chlorophyll concentration within a vertical
transect plane, collected by an autonomous surface vessel in Lake Z?urich [19]. As in [2], our goal
is to detect regions of high concentration. We evaluate each algorithm on a 50 ? 50 grid of points,
with the corresponding values coming from the GP posterior that was derived using the original data
(see Figure 2d). We use the Mat?ern-5/2 ARD kernel, setting its hyperparameters by maximizing the
likelihood on the second (smaller) available dataset. The level-set threshold h is set to 1.5.
In Figure 2a, we show the performance of the algorithms averaged over 100 different runs; here
the randomness is only with respect to the starting point, as we are in the noiseless setting. We observe that in this unit-cost case, T RU VA R performs similarly to GCHK and STR. All three methods
outperform VAR, which is good for global exploration but less suited to level-set estimation.
6
1
0.8
0.8
0.9
0.6
0.4
TruVaR
GCHK
STR
VAR
0.2
F1 score
1
F1 score
F1 score
1
0.6
0.4
0.2
0
20
40
60
Time
80
100
TruVaR
GCHK high noise
GCHK medium noise
GCHK small noise
0.5
0
120
0.7
0.6
TruVaR
GCHK
0
0
0.8
0.5
1
Cost (?104 )
1.5
2
0.1
0.15
?104
0.2 0.25 0.3
Cost (?104 )
0.35
0.4
(a) Lake data, unit-cost
(b) Lake data, varying cost
(c) Synthetic data, varying noise
(d) Inferred concentration function
(e) Points chosen by GCHK
(f) Points chosen by T RU VA R
Figure 2: Experimental results for level-set estimation.
0.27
Averaged Regret
Median Regret
10-2
10-4
TruVaR
EI
GP-UCB
TruVaR
EI
GP-UCB
ES
MRS
100
Validation Error
TruVaR
EI
GP-UCB
ES
MRS
100
10-2
10-4
0.26
0.25
10-6
10-6
0
20
40
60
80
100
120
0.24
0
20
40
60
80
100
120
0
20
Time
Time
(a) Synthetic, median
(b) Synthetic, outlier-adjusted mean
40
60
80
100
Time
(c) SVM data
Figure 3: Experimental results for Bayesian optimization.
Lake data (varying cost): Next, we modify the above setting by introducing pointwise costs that
are a function of the previous sampled point x0 , namely, cx0 (x) = 0.25|x1 x01 | + 4(|x2 | + 1),
where x1 is the vessel position and x2 is the depth. Although we did not permit such a dependence
on x0 in our original setup, the algorithm itself remains unchanged. Our choice of cost penalizes the
distance traveled |x1 x01 |, as well as the depth of the measurement |x2 |. Since incorporating costs
into existing algorithms is non-trivial, we only compare against the original version of GCHK that
ignores costs.
In Figure 2b, we see that TruVaR significantly outperforms GCHK, achieving a higher F1 score
for a significantly smaller cost. The intuition behind this can be seen in Figures 2e and 2f, where
we show the points sampled by TruVaR and GCHK in one experiment run, connecting all pairs of
consecutive points. GCHK is designed to pick few points, but since it ignores costs, the distance
traveled is large. In contrast, by incorporating costs, T RU VA R tends to travel small distances, often
even staying in the same x1 location to take measurements at multiple depths x2 .
Synthetic data with multiple noise levels: In this experiment, we demonstrate Corollary 3.2 by
considering the setting in which the algorithm can choose the sampling noise variance and incur the
associated cost. We use a synthetic function sampled from a GP on a 50 ? 50 grid with an isotropic
squared exponential kernel having length scale l = 0.1 and unit variance, and set h = 2.25. We use
three different noise levels, 2 2 {10 6 , 10 3 , 0.05}, with corresponding costs {15, 10, 2}.
We run GCHK separately for each of the three noise levels, while running T RU VA R as normal
and allowing it to mix between the noise levels. The resulting F1 -scores are shown in Figure 2c.
The best-performing version of GCHK changes throughout the time horizon, while T RU VA R is
consistently better than all three. A discussion on how T RU VA R mixes between the noise levels can
be found in the supplementary material.
7
Bayesian optimization. We now provide the results of two experiments for the BO setting.
Synthetic data: We first conduct a similar experiment as that in [8, 11], generating 200 different
test functions defined on [0, 1]2 . To generate a single test function, 200 points are chosen uniformly
at random from [0, 1]2 , their function values are generated from a GP using an isotropic squared exponential kernel with length scale l = 0.1 and unit variance, and the resulting posterior mean forms
the function on the whole domain [0, 1]2 . We subsequently assume that samples of this function are
corrupted by Gaussian noise with 2 = 10 6 . The extension of T RU VA R to continuous domains is
straightforward, and is explained in the supplementary material. For all algorithms considered, we
evaluate the performance according to the regret of a single reported point, namely, the one having
the highest posterior mean.
We compare the performance of T RU VA R against expected improvement (EI), GP-upper confidence
bound (GP-UCB), entropy search (ES) and minimum regret search (MRS), whose acquisition functions are outlined in the supplementary material. We use publicly available code for ES and MRS
[20]. The exploration parameter t in GP-UCB is set according to the recommendation in [3] of
dividing the theoretical value by five, and the parameters for ES and MRS are set according to the
recommendations given in [11, Section 5.1].
Figure 3a plots the median of the regret, and Figure 3b plots the mean after removing outliers (i.e.,
the best and worst 5% of the runs). In the earlier rounds, ES and MRS provide the best performance,
while T RU VA R improves slowly due to exploration. However, the regret of T RU VA R subsequently
drops rapidly, giving the best performance in the later rounds after ?zooming in? towards the maximum. GP-UCB generally performs well with the aggressive choice of t , despite previous works?
experiments revealing it to perform poorly with the theoretical value.
Hyperparameter tuning data: In this experiment, we use the SVM on grid dataset, previously used
in [21]. A 25 ? 14 ? 4 grid of hyperparameter configurations resulting in 1400 data points was preevaluated, forming the search space. The goal is to find a configuration with small validation error.
We use a Mat?ern-5/2 ARD kernel, and re-learn its hyperparameters by maximizing the likelihood
after sampling every 3 points. Since the hyperparameters are not fixed in advance, we replace Mt 1
by D in (5) to avoid incorrectly ruling points out early on, allowing some removed points to be
added again in later steps. Once the estimated hyperparameters stop to vary significantly, the size
of the set of potential maximizers decreases almost monotonically. Since we consider the noiseless
setting here, we measure performance using the simple regret, i.e., the best point found so far.
We again average over 100 random starting points, and plot the resulting validation error in Figure 3c. Even in this noiseless and unit-cost setting that EI and GP-UCB are suited to, we find that
T RU VA R performs slightly better, giving a better validation error with smaller error bars.
5
Conclusion
We highlight the following aspects in which T RU VA R is versatile:
? Unified optimization and level-set estimation: These are typically treated separately, whereas
T RU VA R and its theoretical guarantees are essentially identical in both cases
? Actions with costs: T RU VA R naturally favors cost-effective points, as this is directly incorporated into the acquisition function.
? Heteroscedastic noise: T RU VA R chooses points that effectively shrink the variance of other
points, thus directly taking advantage of situations in which some points are noisier than others.
? Choosing the noise level: We provided novel theoretical guarantees for the case that the algorithm can choose both a point and a noise level, cf., Corollary 3.2.
Hence, T RU VA R directly handles several important aspects that are non-trivial to incorporate into
myopic algorithms. Moreover, compared to other BO algorithms that perform a lookahead (e.g.,
ES and MRS), T RU VA R avoids the computationally expensive task of averaging over the posterior
and/or measurements, and comes with rigorous theoretical guarantees.
Acknowledgment: This work was supported in part by the European Commission under Grant
ERC Future Proof, SNF Sinergia project CRSII2-147633, SNF 200021-146750, and EPFL Fellows
Horizon2020 grant 665667.
8
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9
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5,615 | 6,081 | Convolutional Neural Networks on Graphs
with Fast Localized Spectral Filtering
Micha?l Defferrard
Xavier Bresson
Pierre Vandergheynst
EPFL, Lausanne, Switzerland
{michael.defferrard,xavier.bresson,pierre.vandergheynst}@epfl.ch
Abstract
In this work, we are interested in generalizing convolutional neural networks
(CNNs) from low-dimensional regular grids, where image, video and speech are
represented, to high-dimensional irregular domains, such as social networks, brain
connectomes or words? embedding, represented by graphs. We present a formulation of CNNs in the context of spectral graph theory, which provides the necessary mathematical background and efficient numerical schemes to design fast
localized convolutional filters on graphs. Importantly, the proposed technique offers the same linear computational complexity and constant learning complexity
as classical CNNs, while being universal to any graph structure. Experiments on
MNIST and 20NEWS demonstrate the ability of this novel deep learning system
to learn local, stationary, and compositional features on graphs.
1
Introduction
Convolutional neural networks [19] offer an efficient architecture to extract highly meaningful statistical patterns in large-scale and high-dimensional datasets. The ability of CNNs to learn local
stationary structures and compose them to form multi-scale hierarchical patterns has led to breakthroughs in image, video, and sound recognition tasks [18]. Precisely, CNNs extract the local stationarity property of the input data or signals by revealing local features that are shared across
the data domain. These similar features are identified with localized convolutional filters or kernels,
which are learned from the data. Convolutional filters are shift- or translation-invariant filters, meaning they are able to recognize identical features independently of their spatial locations. Localized
kernels or compactly supported filters refer to filters that extract local features independently of the
input data size, with a support size that can be much smaller than the input size.
User data on social networks, gene data on biological regulatory networks, log data on telecommunication networks, or text documents on word embeddings are important examples of data lying on
irregular or non-Euclidean domains that can be structured with graphs, which are universal representations of heterogeneous pairwise relationships. Graphs can encode complex geometric structures
and can be studied with strong mathematical tools such as spectral graph theory [6].
A generalization of CNNs to graphs is not straightforward as the convolution and pooling operators
are only defined for regular grids. This makes this extension challenging, both theoretically and
implementation-wise. The major bottleneck of generalizing CNNs to graphs, and one of the primary
goals of this work, is the definition of localized graph filters which are efficient to evaluate and learn.
Precisely, the main contributions of this work are summarized below.
1. Spectral formulation. A spectral graph theoretical formulation of CNNs on graphs built
on established tools in graph signal processing (GSP). [31].
2. Strictly localized filters. Enhancing [4], the proposed spectral filters are provable to be
strictly localized in a ball of radius K, i.e. K hops from the central vertex.
3. Low computational complexity. The evaluation complexity of our filters is linear w.r.t. the
filters support?s size K and the number of edges |E|. Importantly, as most real-world graphs
are highly sparse, we have |E| n2 and |E| = kn for the widespread k-nearest neighbor
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Input graph signals
Feature extraction
e.g. bags of words
Classification
Convolutional layers
Fully connected layers
Output signals
e.g. labels
Graph coarsening
Graph signal filtering
3. Sub-sampling
4. Pooling
1. Convolution
2. Non-linear activation
Figure 1: Architecture of a CNN on graphs and the four ingredients of a (graph) convolutional layer.
(NN) graphs, leading to a linear complexity w.r.t the input data size n. Moreover, this
method avoids the Fourier basis altogether, thus the expensive eigenvalue decomposition
(EVD) necessary to compute it as well as the need to store the basis, a matrix of size n2 .
That is especially relevant when working with limited GPU memory. Besides the data, our
method only requires to store the Laplacian, a sparse matrix of |E| non-zero values.
4. Efficient pooling. We propose an efficient pooling strategy on graphs which, after a rearrangement of the vertices as a binary tree structure, is analog to pooling of 1D signals.
5. Experimental results. We present multiple experiments that ultimately show that our formulation is (i) a useful model, (ii) computationally efficient and (iii) superior both in accuracy and complexity to the pioneer spectral graph CNN introduced in [4]. We also show
that our graph formulation performs similarly to a classical CNNs on MNIST and study the
impact of various graph constructions on performance. The TensorFlow [1] code to reproduce our results and apply the model to other data is available as an open-source software.1
2
Proposed Technique
Generalizing CNNs to graphs requires three fundamental steps: (i) the design of localized convolutional filters on graphs, (ii) a graph coarsening procedure that groups together similar vertices and
(iii) a graph pooling operation that trades spatial resolution for higher filter resolution.
2.1
Learning Fast Localized Spectral Filters
There are two strategies to define convolutional filters; either from a spatial approach or from a
spectral approach. By construction, spatial approaches provide filter localization via the finite size
of the kernel. However, although graph convolution in the spatial domain is conceivable, it faces
the challenge of matching local neighborhoods, as pointed out in [4]. Consequently, there is no
unique mathematical definition of translation on graphs from a spatial perspective. On the other
side, a spectral approach provides a well-defined localization operator on graphs via convolutions
with a Kronecker delta implemented in the spectral domain [31]. The convolution theorem [22]
defines convolutions as linear operators that diagonalize in the Fourier basis (represented by the
eigenvectors of the Laplacian operator). However, a filter defined in the spectral domain is not
naturally localized and translations are costly due to the O(n2 ) multiplication with the graph Fourier
basis. Both limitations can however be overcome with a special choice of filter parametrization.
Graph Fourier Transform. We are interested in processing signals defined on undirected and
connected graphs G = (V, E, W ), where V is a finite set of |V| = n vertices, E is a set of edges and
W ? Rn?n is a weighted adjacency matrix encoding the connection weight between two vertices.
A signal x : V ? R defined on the nodes of the graph may be regarded as a vector x ? Rn
where xi is the value of x at the ith node. An essential operator in spectral graph analysis is the
graph Laplacian [6], which combinatorial definition is L = D ?W ? Rn?n where D ? Rn?n is the
1
https://github.com/mdeff/cnn_graph
2
P
diagonal degree matrix with Dii = j Wij , and normalized definition is L = In ? D?1/2 W D?1/2
where In is the identity matrix. As L is a real symmetric positive semidefinite matrix, it has a
n
complete set of orthonormal eigenvectors {ul }n?1
l=0 ? R , known as the graph Fourier modes, and
their associated ordered real nonnegative eigenvalues {?l }n?1
l=0 , identified as the frequencies of the
graph. The Laplacian is indeed diagonalized by the Fourier basis U = [u0 , . . . , un?1 ] ? Rn?n
such that L = U ?U T where ? = diag([?0 , . . . , ?n?1 ]) ? Rn?n . The graph Fourier transform of a
signal x ? Rn is then defined as x
? = U T x ? Rn , and its inverse as x = U x
? [31]. As on Euclidean
spaces, that transform enables the formulation of fundamental operations such as filtering.
Spectral filtering of graph signals. As we cannot express a meaningful translation operator in
the vertex domain, the convolution operator on graph ?G is defined in the Fourier domain such that
x ?G y = U ((U T x) (U T y)), where is the element-wise Hadamard product. It follows that a
signal x is filtered by g? as
y = g? (L)x = g? (U ?U T )x = U g? (?)U T x.
A non-parametric filter, i.e. a filter whose parameters are all free, would be defined as
g? (?) = diag(?),
where the parameter ? ? Rn is a vector of Fourier coefficients.
(1)
(2)
Polynomial parametrization for localized filters. There are however two limitations with nonparametric filters: (i) they are not localized in space and (ii) their learning complexity is in O(n),
the dimensionality of the data. These issues can be overcome with the use of a polynomial filter
g? (?) =
K?1
X
?k ? k ,
(3)
k=0
where the parameter ? ? RK is a vector of polynomial coefficients.
P The value at vertex j of the
filter g? centered at vertex i is given by (g? (L)?i )j = (g? (L))i,j = k ?k (Lk )i,j , where the kernel
is localized via a convolution with a Kronecker delta function ?i ? Rn . By [12, Lemma 5.2],
dG (i, j) > K implies (LK )i,j = 0, where dG is the shortest path distance, i.e. the minimum number
of edges connecting two vertices on the graph. Consequently, spectral filters represented by K th order polynomials of the Laplacian are exactly K-localized. Besides, their learning complexity is
O(K), the support size of the filter, and thus the same complexity as classical CNNs.
Recursive formulation for fast filtering. While we have shown how to learn localized filters
with K parameters, the cost to filter a signal x as y = U g? (?)U T x is still high with O(n2 ) operations because of the multiplication with the Fourier basis U . A solution to this problem is to
parametrize g? (L) as a polynomial function that can be computed recursively from L, as K multiplications by a sparse L costs O(K|E|) O(n2 ). One such polynomial, traditionally used
in GSP to approximate kernels (like wavelets), is the Chebyshev expansion [12]. Another option, the Lanczos algorithm [33], which constructs an orthonormal basis of the Krylov subspace
KK (L, x) = span{x, Lx, . . . , LK?1 x}, seems attractive because of the coefficients? independence.
It is however more convoluted and thus left as a future work.
Recall that the Chebyshev polynomial Tk (x) of order k may be computed by the stable recurrence
relation Tk (x) = 2xTk?1 (x) ? Tk?2
p(x) with T0 = 1 and T1 = x. These polynomials form an
orthogonal basis for L2 ([?1,
1],
dy/
1 ? y 2 ), the Hilbert space of square integrable functions with
p
2
respect to the measure dy/ 1 ? y . A filter can thus be parametrized as the truncated expansion
g? (?) =
K?1
X
?
?k Tk (?),
(4)
k=0
? ?
of order K ? 1, where the parameter ? ? RK is a vector of Chebyshev coefficients and Tk (?)
n?n
? = 2?/?max ? In , a diagonal matrix
R
is the Chebyshev polynomial of order k evaluated at ?
of scaled eigenvalues that lie in [?1, 1]. The filtering operation can then be written as y = g? (L)x =
PK?1
n?n
?
?
is the Chebyshev polynomial of order k evaluated at the
k=0 ?k Tk (L)x, where Tk (L) ? R
?
? ? Rn , we can use the recurrence
scaled Laplacian L = 2L/?max ? In . Denoting x
?k = Tk (L)x
?
? The entire filtering operation
relation to compute x
?k = 2L?
xk?1 ? x
?k?2 with x
?0 = x and x
?1 = Lx.
y = g? (L)x = [?
x0 , . . . , x
?K?1 ]? then costs O(K|E|) operations.
3
Learning filters. The j th output feature map of the sample s is given by
ys,j =
Fin
X
g?i,j (L)xs,i ? Rn ,
(5)
i=1
where the xs,i are the input feature maps and the Fin ? Fout vectors of Chebyshev coefficients
?i,j ? RK are the layer?s trainable parameters. When training multiple convolutional layers with
the backpropagation algorithm, one needs the two gradients
S
X
?E
?E
=
[?
xs,i,0 , . . . , x
?s,i,K?1 ]T
??i,j
?ys,j
s=1
and
F
out
X
?E
?E
=
g?i,j (L)
,
?xs,i
?ys,j
j=1
(6)
where E is the loss energy over a mini-batch of S samples. Each of the above three computations
boils down to K sparse matrix-vector multiplications and one dense matrix-vector multiplication for
a cost of O(K|E|Fin Fout S) operations. These can be efficiently computed on parallel architectures
by leveraging tensor operations. Eventually, [?
xs,i,0 , . . . , x
?s,i,K?1 ] only needs to be computed once.
2.2
Graph Coarsening
The pooling operation requires meaningful neighborhoods on graphs, where similar vertices are
clustered together. Doing this for multiple layers is equivalent to a multi-scale clustering of the graph
that preserves local geometric structures. It is however known that graph clustering is NP-hard [5]
and that approximations must be used. While there exist many clustering techniques, e.g. the popular spectral clustering [21], we are most interested in multilevel clustering algorithms where each
level produces a coarser graph which corresponds to the data domain seen at a different resolution.
Moreover, clustering techniques that reduce the size of the graph by a factor two at each level offers
a precise control on the coarsening and pooling size. In this work, we make use of the coarsening
phase of the Graclus multilevel clustering algorithm [9], which has been shown to be extremely efficient at clustering a large variety of graphs. Algebraic multigrid techniques on graphs [28] and the
Kron reduction [32] are two methods worth exploring in future works.
Graclus [9], built on Metis [16], uses a greedy algorithm to compute successive coarser versions of
a given graph and is able to minimize several popular spectral clustering objectives, from which we
chose the normalized cut [30]. Graclus? greedy rule consists, at each coarsening level, in picking an
unmarked vertex i and matching it with one of its unmarked neighbors j that maximizes the local
normalized cut Wij (1/di + 1/dj ). The two matched vertices are then marked and the coarsened
weights are set as the sum of their weights. The matching is repeated until all nodes have been explored. This is an very fast coarsening scheme which divides the number of nodes by approximately
two (there may exist a few singletons, non-matched nodes) from one level to the next coarser level.
2.3
Fast Pooling of Graph Signals
Pooling operations are carried out many times and must be efficient. After coarsening, the vertices
of the input graph and its coarsened versions are not arranged in any meaningful way. Hence, a
direct application of the pooling operation would need a table to store all matched vertices. That
would result in a memory inefficient, slow, and hardly parallelizable implementation. It is however
possible to arrange the vertices such that a graph pooling operation becomes as efficient as a 1D
pooling. We proceed in two steps: (i) create a balanced binary tree and (ii) rearrange the vertices.
After coarsening, each node has either two children, if it was matched at the finer level, or one, if it
was not, i.e the node was a singleton. From the coarsest to finest level, fake nodes, i.e. disconnected
nodes, are added to pair with the singletons such that each node has two children. This structure is
a balanced binary tree: (i) regular nodes (and singletons) either have two regular nodes (e.g. level
1 vertex 0 in Figure 2) or (ii) one singleton and a fake node as children (e.g. level 2 vertex 0), and
(iii) fake nodes always have two fake nodes as children (e.g. level 1 vertex 1). Input signals are
initialized with a neutral value at the fake nodes, e.g. 0 when using a ReLU activation with max
pooling. Because these nodes are disconnected, filtering does not impact the initial neutral value.
While those fake nodes do artificially increase the dimensionality thus the computational cost, we
found that, in practice, the number of singletons left by Graclus is quite low. Arbitrarily ordering the
nodes at the coarsest level, then propagating this ordering to the finest levels, i.e. node k has nodes
2k and 2k + 1 as children, produces a regular ordering in the finest level. Regular in the sense that
adjacent nodes are hierarchically merged at coarser levels. Pooling such a rearranged graph signal is
4
5
0 1 2 3 4 5 6 7 8 9 10 11
6
0
1
4
8
2 3
7 11
3
2
0
1
1
0
4
10
0 1 2 3 4 5
2
5
9
0 1 2
Figure 2: Example of Graph Coarsening and Pooling. Let us carry out a max pooling of size 4
(or two poolings of size 2) on a signal x ? R8 living on G0 , the finest graph given as input. Note
that it originally possesses n0 = |V0 | = 8 vertices, arbitrarily ordered. For a pooling of size 4,
two coarsenings of size 2 are needed: let Graclus gives G1 of size n1 = |V1 | = 5, then G2 of size
n2 = |V2 | = 3, the coarsest graph. Sizes are thus set to n2 = 3, n1 = 6, n0 = 12 and fake nodes
(in blue) are added to V1 (1 node) and V0 (4 nodes) to pair with the singeltons (in orange), such that
each node has exactly two children. Nodes in V2 are then arbitrarily ordered and nodes in V1 and
V0 are ordered consequently. At that point the arrangement of vertices in V0 permits a regular 1D
pooling on x ? R12 such that z = [max(x0 , x1 ), max(x4 , x5 , x6 ), max(x8 , x9 , x10 )] ? R3 , where
the signal components x2 , x3 , x7 , x11 are set to a neutral value.
analog to pooling a regular 1D signal. Figure 2 shows an example of the whole process. This regular
arrangement makes the operation very efficient and satisfies parallel architectures such as GPUs as
memory accesses are local, i.e. matched nodes do not have to be fetched.
3
3.1
Related Works
Graph Signal Processing
The emerging field of GSP aims at bridging the gap between signal processing and spectral graph
theory [6, 3, 21], a blend between graph theory and harmonic analysis. A goal is to generalize
fundamental analysis operations for signals from regular grids to irregular structures embodied by
graphs. We refer the reader to [31] for an introduction of the field. Standard operations on grids
such as convolution, translation, filtering, dilatation, modulation or downsampling do not extend
directly to graphs and thus require new mathematical definitions while keeping the original intuitive
concepts. In this context, the authors of [12, 8, 10] revisited the construction of wavelet operators
on graphs and techniques to perform mutli-scale pyramid transforms on graphs were proposed in
[32, 27]. The works of [34, 25, 26] redefined uncertainty principles on graphs and showed that
while intuitive concepts may be lost, enhanced localization principles can be derived.
3.2
CNNs on Non-Euclidean Domains
The Graph Neural Network framework [29], simplified in [20], was designed to embed each node in
an Euclidean space with a RNN and use those embeddings as features for classification or regression
of nodes or graphs. By setting their transition function f as a simple diffusion instead of a neural
net with a recursive relation, their state vector becomes s = f (x) = W x. Their point-wise output
function g? can further be set as x
? = g? (s, x) = ?(s ? Dx) + x = ?Lx + x instead of another
neural net. The Chebyshev polynomials of degree K can then be obtained with a K-layer GNN, to
be followed by a non-linear layer and a graph pooling operation. Our model can thus be interpreted
as multiple layers of diffusions and node-local operations.
The works of [11, 7] introduced the concept of constructing a local receptive field to reduce the
number of learned parameters. The idea is to group together features based upon a measure of
similarity such as to select a limited number of connections between two successive layers. While
this model reduces the number of parameters by exploiting the locality assumption, it did not attempt
to exploit any stationarity property, i.e. no weight-sharing strategy. The authors of [4] used this
idea for their spatial formulation of graph CNNs. They use a weighted graph to define the local
neighborhood and compute a multiscale clustering of the graph for the pooling operation. Inducing
weight sharing in a spatial construction is however challenging, as it requires to select and order the
neighborhoods when a problem-specific ordering (spatial, temporal, or otherwise) is missing.
A spatial generalization of CNNs to 3D-meshes, a class of smooth low-dimensional non-Euclidean
spaces, was proposed in [23]. The authors used geodesic polar coordinates to define the convolu5
Model
Architecture
Classical CNN
Proposed graph CNN
C32-P4-C64-P4-FC512
GC32-P4-GC64-P4-FC512
Accuracy
99.33
99.14
Table 1: Classification accuracies of the proposed graph CNN and a classical CNN on MNIST.
tion on mesh patches, and formulated a deep learning architecture which allows comparison across
different manifolds. They obtained state-of-the-art results for 3D shape recognition.
The first spectral formulation of a graph CNN, proposed in [4], defines a filter as
g? (?) = B?,
(7)
n?K
K
where B ? R
is the cubic B-spline basis and the parameter ? ? R is a vector of control points.
They later proposed a strategy to learn the graph structure from the data and applied the model to
image recognition, text categorization and bioinformatics [13]. This approach does however not
scale up due to the necessary multiplications by the graph Fourier basis U . Despite the cost of
computing this matrix, which requires an EVD on the graph Laplacian, the dominant cost is the need
to multiply the data by this matrix twice (forward and inverse Fourier transforms) at a cost of O(n2 )
operations per forward and backward pass, a computational bottleneck already identified by the
authors. Besides, as they rely on smoothness in the Fourier domain, via the spline parametrization,
to bring localization in the vertex domain, their model does not provide a precise control over the
local support of their kernels, which is essential to learn localized filters. Our technique leverages
on this work, and we showed how to overcome these limitations and beyond.
4
Numerical Experiments
In the sequel, we refer to the non-parametric and non-localized filters (2) as Non-Param, the filters
(7) proposed in [4] as Spline and the proposed filters (4) as Chebyshev. We always use the Graclus
coarsening algorithm introduced in Section 2.2 rather than the simple agglomerative method of [4].
Our motivation is to compare the learned filters, not the coarsening algorithms.
We use the following notation when describing network architectures: FCk denotes a fully connected layer with k hidden units, Pk denotes a (graph or classical) pooling layer of size and stride
k, GCk and Ck denote a (graph) convolutional layer with k feature maps. All FCk, Ck and GCk
layers are followed by a ReLU activation max(x, 0). The final layer is always a softmax regression
and the loss energy E is the cross-entropy with an `2 regularization on the weights of all FCk layers.
Mini-batches are of size S = 100.
4.1
Revisiting Classical CNNs on MNIST
To validate our model, we applied it to the Euclidean case on the benchmark MNIST classification
problem [19], a dataset of 70,000 digits represented on a 2D grid of size 28 ? 28. For our graph
model, we construct an 8-NN graph of the 2D grid which produces a graph of n = |V| = 976 nodes
(282 = 784 pixels and 192 fake nodes as explained in Section 2.3) and |E| = 3198 edges. Following
standard practice, the weights of a k-NN similarity graph (between features) are computed as
kzi ? zj k22
Wij = exp ?
,
(8)
?2
where zi is the 2D coordinate of pixel i.
This is an important sanity check for our model, which must be able to extract features on any graph,
including the regular 2D grid. Table 1 shows the ability of our model to achieve a performance very
close to a classical CNN with the same architecture. The gap in performance may be explained
by the isotropic nature of the spectral filters, i.e. the fact that edges in a general graph do not
possess an orientation (like up, down, right and left for pixels on a 2D grid). Whether this is a
limitation or an advantage depends on the problem and should be verified, as for any invariance.
Moreover, rotational invariance has been sought: (i) many data augmentation schemes have used
rotated versions of images and (ii) models have been developed to learn this invariance, like the
Spatial Transformer Networks [14]. Other explanations are the lack of experience on architecture
design and the need to investigate better suited optimization or initialization strategies.
The LeNet-5-like network architecture and the following hyper-parameters are borrowed from the
TensorFlow MNIST tutorial2 : dropout probability of 0.5, regularization weight of 5 ? 10?4 , initial
2
https://www.tensorflow.org/versions/r0.8/tutorials/mnist/pros
6
1400
Accuracy
Linear SVM
Multinomial Naive Bayes
Softmax
65.90
68.51
66.28
FC2500
FC2500-FC500
64.64
65.76
GC32
68.26
Table 2: Accuracies of the proposed graph
CNN and other methods on 20NEWS.
Chebyshev
Non-Param / Spline
1200
time (ms)
Model
1000
800
600
400
200
0
2000
4000
6000
8000
10000
number of features (words)
12000
Figure 3: Time to process a mini-batch of S = 100
20NEWS documents w.r.t. the number of words n.
Accuracy
Dataset
Architecture
MNIST
MNIST
GC10
GC32-P4-GC64-P4-FC512
Non-Param (2)
Spline (7) [4]
Chebyshev (4)
95.75
96.28
97.26
97.15
97.48
99.14
Table 3: Classification accuracies for different types of spectral filters (K = 25).
Model
Architecture
Time (ms)
CPU GPU
Classical CNN
Proposed graph CNN
C32-P4-C64-P4-FC512
GC32-P4-GC64-P4-FC512
210
1600
31
200
Speedup
6.77x
8.00x
Table 4: Time to process a mini-batch of S = 100 MNIST images.
learning rate of 0.03, learning rate decay of 0.95, momentum of 0.9. Filters are of size 5 ? 5 and
graph filters have the same support of K = 25. All models were trained for 20 epochs.
4.2
Text Categorization on 20NEWS
To demonstrate the versatility of our model to work with graphs generated from unstructured data,
we applied our technique to the text categorization problem on the 20NEWS dataset which consists
of 18,846 (11,314 for training and 7,532 for testing) text documents associated with 20 classes [15].
We extracted the 10,000 most common words from the 93,953 unique words in this corpus. Each
document x is represented using the bag-of-words model, normalized across words. To test our
model, we constructed a 16-NN graph with (8) where zi is the word2vec embedding [24] of word
i, which produced a graph of n = |V| = 10, 000 nodes and |E| = 132, 834 edges. All models
were trained for 20 epochs by the Adam optimizer [17] with an initial learning rate of 0.001. The
architecture is GC32 with support K = 5. Table 2 shows decent performances: while the proposed
model does not outperform the multinomial naive Bayes classifier on this small dataset, it does
defeat fully connected networks, which require much more parameters.
4.3
Comparison between Spectral Filters and Computational Efficiency
Table 3 reports that the proposed parametrization (4) outperforms (7) from [4] as well as nonparametric filters (2) which are not localized and require O(n) parameters. Moreover, Figure 4
gives a sense of how the validation accuracy and the loss E converges w.r.t. the filter definitions.
Figure 3 validates the low computational complexity of our model which scales as O(n) while [4]
scales as O(n2 ). The measured runtime is the total training time divided by the number of gradient
steps. Table 4 shows a similar speedup as classical CNNs when moving to GPUs. This exemplifies
the parallelization opportunity offered by our model, who relies solely on matrix multiplications.
Those are efficiently implemented by cuBLAS, the linear algebra routines provided by NVIDIA.
4.4
Influence of Graph Quality
For any graph CNN to be successful, the statistical assumptions of locality, stationarity, and compositionality regarding the data must be fulfilled on the graph where the data resides. Therefore, the
learned filters? quality and thus the classification performance critically depends on the quality of
7
80
training loss
validation accuracy
100
60
Chebyshev
Non-Param
Spline
40
20
0
500
1000
1500
2000
6.5
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
Chebyshev
Non-Param
Spline
500
1000
1500
2000
Figure 4: Plots of validation accuracy and training loss for the first 2000 iterations on MNIST.
Architecture
8-NN on 2D Euclidean grid
random
97.40
99.14
96.88
95.39
GC32
GC32-P4-GC64-P4-FC512
Table 5: Classification accuracies with different graph constructions on MNIST.
bag-of-words
67.50
word2vec
pre-learned learned
66.98
68.26
approximate
random
67.86
67.75
Table 6: Classification accuracies of GC32 with different graph constructions on 20NEWS.
the graph. For data lying on Euclidean space, experiments in Section 4.1 show that a simple k-NN
graph of the grid is good enough to recover almost exactly the performance of standard CNNs. We
also noticed that the value of k does not have a strong influence on the results. We can witness the
importance of a graph satisfying the data assumptions by comparing its performance with a random
graph. Table 5 reports a large drop of accuracy when using a random graph, that is when the data
structure is lost and the convolutional layers are not useful anymore to extract meaningful features.
While images can be structured by a grid graph, a feature graph has to be built for text documents
represented as bag-of-words. We investigate here three ways to represent a word z: the simplest option is to represent each word as its corresponding column in the bag-of-words matrix while, another
approach is to learn an embedding for each word with word2vec [24] or to use the pre-learned embeddings provided by the authors. For larger datasets, an approximate nearest neighbors algorithm
may be required, which is the reason we tried LSHForest [2] on the learned word2vec embeddings.
Table 6 reports classification results which highlight the importance of a well constructed graph.
5
Conclusion and Future Work
In this paper, we have introduced the mathematical and computational foundations of an efficient
generalization of CNNs to graphs using tools from GSP. Experiments have shown the ability of the
model to extract local and stationary features through graph convolutional layers. Compared with
the first work on spectral graph CNNs introduced in [4], our model provides a strict control over the
local support of filters, is computationally more efficient by avoiding an explicit use of the Graph
Fourier basis, and experimentally shows a better test accuracy. Besides, we addressed the three
concerns raised by [13]: (i) we introduced a model whose computational complexity is linear with
the dimensionality of the data, (ii) we confirmed that the quality of the input graph is of paramount
importance, (iii) we showed that the statistical assumptions of local stationarity and compositionality
made by the model are verified for text documents as long as the graph is well constructed.
Future works will investigate two directions. On one hand, we will enhance the proposed framework
with newly developed tools in GSP. On the other hand, we will explore applications of this generic
model to important fields where the data naturally lies on graphs, which may then incorporate external information about the structure of the data rather than artificially created graphs which quality
may vary as seen in the experiments. Another natural and future approach, pioneered in [13], would
be to alternate the learning of the CNN parameters and the graph.
8
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9
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5,616 | 6,082 | Sampling for Bayesian Program Learning
Kevin Ellis
Brain and Cognitive Sciences
MIT
ellisk@mit.edu
Armando Solar-Lezama
CSAIL
MIT
asolar@csail.mit.edu
Joshua B. Tenenbaum
Brain and Cognitive Sciences
MIT
jbt@mit.edu
Abstract
Towards learning programs from data, we introduce the problem of sampling
programs from posterior distributions conditioned on that data. Within this setting,
we propose an algorithm that uses a symbolic solver to efficiently sample programs.
The proposal combines constraint-based program synthesis with sampling via
random parity constraints. We give theoretical guarantees on how well the samples
approximate the true posterior, and have empirical results showing the algorithm is
efficient in practice, evaluating our approach on 22 program learning problems in
the domains of text editing and computer-aided programming.
1
Introduction
Learning programs from examples is a central problem in artificial intelligence, and many recent
approaches draw on techniques from machine learning. Connectionist approaches, like the Neural
Turing Machine [1, 2] and symbolic approaches, like Hierarchical Bayesian Program Learning [3,
4, 5], couple a probabilistic learning framework with either gradient- or sampling-based search
procedures. In this work, we consider the problem of Bayesian inference over program spaces. We
combine solver-based program synthesis [6] and sampling via random projections [7], showing how
to sample from posterior distributions over programs where the samples come from a distribution
provably arbitrarily close to the true posterior. The new approach is implemented in a system called
P ROGRAM S AMPLE and evaluated on a set of program induction problems that include list and string
manipulation routines.
1.1
Motivation and problem statement
Consider the problem of learning string edit programs, a well studied domain for programming
by example. Often end users provide these examples and are unwilling to give more than one
instance, which leaves the target program highly
ambiguous. We model this ambiguity by sampling string edit programs, allowing us to learn
from very few examples (Figure 1) and offer
different plausible solutions. Our sampler also
incorporates a description-length prior to bias
us towards simpler programs.
Input
?1/21/2001?
Output
?01?
substr(pos(?0?,-1),-1)
const(?01?)
substr(-2,-1)
?last 0 til end?
?output 01?
?take last two?
Figure 1: Learning string manipulation programs
by example (top input/output pair). Our system receives data like that shown above and then sampled
the programs shown below.
Another program learning domain comes from computer-aided programming, where the goal is
to synthesize algorithms from either examples or formal specifications. This problem can be ill
posed because many programs may satisfy the specification or examples. When this ambiguity
arises, P ROGRAM S AMPLE proposes multiple implementations with a bias towards shorter or simpler
ones. The samples can also be used to efficiently approximate the posterior predictive distribution,
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
effectively integrating out the program. We show P ROGRAM S AMPLE learning routines for counting
and recursively sorting/reversing lists while modeling the uncertainty over the correct algorithm.
Because any model can be represented as a (probabilistic or deterministic) program, we need to
carefully delimit the scope of this work. The programs we learn are a subset of those handled
by constraint-based program synthesis tools. This means that the program is finite (bounded size,
bounded runtime, bounded memory consumption), can be modeled in a constraint solver (like a SAT
or SMT solver), and that the program?s high-level structure is already given as a sketch [6], which
can take the form of a recursive grammar over expressions. The sketch defines the search space and
imparts prior knowledge. For example, we use one sketch when learning string edit programs and a
different sketch when learning recursive list manipulation programs.
More formally, our sketch specifies a finite set of programs, S, as well as a measure of the programs?
description length, which we write as |x| for x ? S. This defines a prior (? 2?|x| ). For each
program learning problem, we have a specification (such as consistency with input/output examples)
and want to sample from S conditioned upon the specification holding, giving the posterior over
programs ? 2?|x| 1[specification holds for x]. Throughout the rest of the paper, we write p(?) to
mean this posterior distribution, and write X to mean the set of all programs in S consistent with the
P
?|x|
specification. So the problem is to sample from p(x) = 2 Z where Z = x?X 2?|x| .
We can invoke a solver, which enumerates members of X, possibly subject to extra constraints, but
without any guarantees on the order of enumeration. Throughout this work we use a SAT solver, and
encode x ? X in the values of n Boolean decision variables. With a slight abuse of notation we will
use x to refer to both a member of X and an assignment to those n decision variables. An assignment
to j th variable we write as xj for 1 ? j ? n. Section 1.2 briskly summarizes the constraint-solving
program synthesis approach.
1.2
Program synthesis by constraint solving
The constraint solving approach to program synthesis, pioneered in [6, 8], synthesizes programs by
(1) modeling the space of programs as assignments to Boolean decision variables in a constraint
satisfaction problem; (2) adding constraints to enforce consistency with a specification; (3) asking the
solver to find any solution to the constraints; and (4) reinterpreting that solution as a program.
Figure 2 illustrates this approach for the toy problem of synthesizing programs in a language consisting
of single-bit operators. Each program has one input (i in Figure 2) which it transforms using nand
gates. The grammar in Figure 2a is the sketch. If we inline the grammar, we can diagram the space of
all programs as an AND/OR graph (Figure 2b), where xj are Boolean decision variables that control
the program?s structure. For each of the input/output examples (Figure 2d) we have constraints that
model the program execution (Figure 2c) and enforce the desired output (P1 taking value 1). After
solving for a satisfying assignment to the xj ?s, we can read these off as a program (Figure 2e). In this
work we measure the description length of a program x as the number of bits required to specify its
structure (so |x| is a natural number).1 P ROGRAM S AMPLE further constrains unused bits to take
a canonical form, such as all being zero. This causes the mapping between programs x ? X and
variable assignments {xj }N
j=1 to be one-to-one.
1.3
Algorithmic contribution
In the past decade different groups of researchers have concurrently developed solver-based techniques
for (1) sampling of combinatorial spaces [9, 7, 10, 11] and (2) program synthesis [6, 8]. This work
merges these two lines of research to attack the problem of program learning in a probabilistic
setting. We use program synthesis tools to convert a program learning problem into a SAT formula.
Then, rather than search for one program (formula solution), we augment the formula with random
constraints that cause it to (approximately) sample the space of programs, effectively ?upgrading?
our SAT solver from a program synthesizer to a program sampler.
The groundbreaking algorithms in [9] gave the first scheme (XORSample) for sampling discrete
spaces by adding random constraints to a constraint satisfaction problem. While one could use a tool
like Sketch to reduce a program learning problem to SAT and then use an algorithm like XORSample,
1
This is equivalent to the assumption that x is drawn from a probabilistic grammar specified by the sketch
2
Program ::= i
| nand(Program,Program)
(a) Sketch
i ? 0 ? P1 ? 1
x1 ? (P1 ? i)
P1
x1
x1
i
nand
Program(i = 0) = 1
P2
(d) Specification
x1 = 0, x2 = 1, x3 = 1
Program = nand(i, i)
(e) A constraint solution; |x| = 3 bits
P3
......
x2
x2
i
nand
......
x1 ? (P1 ? P2 ? P3 )
x2 ? (P2 ? i)
x2 ? (P2 ? P4 ? P5 )
x3 ? (P3 ? i)
??????
(c) Constraints for SAT solver
(b) Program space
Figure 2: Synthesizing a program via sketching and constraint solving. Typewriter font refers
to pieces of programs or sketches, while math font refers to pieces of a constraint satisfaction problem.
The variable i is the program input.
PAWS, or WeightGen [9, 7, 10] to sample programs from a description length prior, doing so can
be surprisingly inefficient2 . The efficiency of these sampling algorithms depends critically on a
x p(x)
quantity called the distribution?s tilt, introduced in [10] as max
minx p(x) . When there are a few very likely
(short) programs and many extremely unlikely (long) programs, the posterior over programs becomes
extremely tilted. Recent work has relied on upper bounding the tilt, often to around 20 [10]. For
program sampling problems, we usually face very high tilt upwards of 250 . Our main algorithmic
contribution is a new approach that extends these techniques to distributions with high tilt, such as
those encountered in program induction.
2
The sampling algorithm
Given the distribution p(?) on the program space X, it is always possible to define a higher dimensional
space E (an embedding) and a mapping F : E ? X such that sampling uniformly from E and
applying F will give us approximately p-distributed samples [7]. But, when the tilt of p(?) becomes
large, we found that such an approach is no longer practical.3
Our approach instead is to define an F 0 : E ? X such that uniform samples on E map to a
distribution q(?) that is guaranteed to have low tilt, but whose KL divergence from p(?) is low. The
discrepancy between the distributions p(?) and q(?) can be corrected through rejection sampling.
Sampling uniformly from E is itself not trivial, but a variety of techniques exist to approximate
uniform sampling by adding random XOR constraints (random projections mod 2) to the set E,
which is extensively studied in [9, 12, 10, 13, 11]. These techniques introduce approximation error
that can be made arbitrarily small at the expense of lower efficiency. Figure 3 illustrates this process.
2.1
Getting high-quality samples
Low-tilt approximation. We introduce a parameter into the sampling algorithm, d, that parameterizes q(?). The parameter d acts as a threshold, or cut-off, for the description length of a program;
the distribution q(?) acts as though any program with description length exceeding d can be encoded
using d bits. Concretely,
?|x|
2
, if |x| ? d
q(x) ?
(1)
2?d ,
otherwise
If we could sample exactly from q(?), we could reject a sample x with probability 1 ? A(x) where A
is
1,
if |x| ? d
A(x) ?
(2)
2?|x|+d , otherwise
2
In many cases, slower than rejection sampling or enumerating all of the programs
[10] take a qualitatively different approach from [7] not based on an embedding, but which still becomes
prohibitively expensive in the high-tilt regime.
3
3
and get exact samples from p(?), where the acceptance rate would approach 1 exponentially quickly
in d. We have the following result; see supplement for proofs.
Proposition 1. Let x ? X be a sample from q(?). The probability of accepting x is at least
1
where x? = arg minx |x|.
1+|X|2|x? |?d
The distribution q(?) is useful because we can
guarantee that it has tilt bounded by 2d?|x? | . Introducing the proposal q(?) effectively reifies the
tilt, making it a parameter of the sampling algorithm, not the distribution over programs. We
now show how to approximately sample from
q(?) using a variant of the Embed and Project
framework [7].
The embedding. The idea is to define a new set
of programs, which we call E, such that short
programs are included in the set much more often than long programs. Each program x will be
represented in E by an amount proportional to
2? min(|x|,d) , thus proportional to q(x), such that
sampling elements uniformly from E samples
according to q(?).
Figure 3: P ROGRAM S AMPLE twice distorts the
posterior distribution p(?). First it introduces a
parameter d that bounds the tilt; we correct for
this by accepting samples w.p. A(x). Second it
samples from q(?) by drawing instead from r(?),
where KL(q||r) can be made arbitrarily small by
appropriately setting another parameter, K. The
distribution of samples is A(x)r(x).
We embed X within the larger set E by introducing d auxiliary variables, written (y1 , ? ? ? , yd ), such that every element of E is a tuple of an
element of x = (x1 , ? ? ? , xn ) and an assignment to y = (y1 , ? ? ? , yd ):
^
E = {(x, y) : x ? X,
|x| ? j ? yj = 1}
(3)
1?j?d
Suppose we sample (x, y) uniformly from E. Then the probability of getting a particular x ? X
is proportional to |{(x0 , y) ? E : x0 = x}| = |{y : |x| ? j ? yj = 1}| = 2min(0,d?|x|) which is
proportional to q(x). Notice that |E| grows exponentially with d, and thus with the tilt of the q(?).
This is the crux of the inefficiency of sampling from high-tilt distributions in these frameworks: these
auxiliary variables combine with the random constraints to entangle otherwise independent Boolean
decision variables, while also increasing the number of variables and clauses.
The random projections. We could sample exactly from E by invoking the solver |E| + 1 times to
get every element of E, but in general it will have O(|X|2d ) elements, which could be very large.
Instead, we ask the solver for all the elements of E consistent with K random constraints such that
(1) few elements of E are likely to satisfy (?survive?) the constraints, and (2) any element of E is
approximately equally likely to satisfy the constraints. We can then sample a survivor uniformly to
get an approximate sample from E, an idea introduced in the XORSample0 algorithm [9]. Although
simple compared to recent approaches [10, 14, 15], it suffices for our theoretical and empirical results.
Our random constraints take the form of XOR, or parity constraints, which are random projections
mod 2. Each constraint fixes the parity of a random subset of SAT variables in x to either 1 or 0;
thus any x survives a constraint with probability 12 . A useful feature of random parity constraints
is that whether an assignment to the SAT variables survives is independent of whether another,
different assignment survives, which has been exploited to create a variety of approximate sampling
algorithms [9, 12, 10, 13, 11].
2
Then the K constraints are of the form h ( xy ) ? b where h is a K ? (d + n) binary matrix and b is a
K-dimensional binary vector. If no solutions satisfy the K constraints then the sampling attempt is
rejected. These samples are close to uniform in the following sense:
1
Proposition 2. The probability of sampling (x, y) is at least |E|
? 1+2K1 /|E| and the probability of
getting any sample at all is at least 1 ? 2K /|E|.
So we get approximate samples from E as long asP
|E|2?K is not small. In reference to Figure 3,
we call the distribution of these samples r(x) = y r(x, y). Schemes more sophisticated than
XORSample0 , like [7], also guarantee upper bounds on sampling probability, but we found that these
4
Algorithm 1 P ROGRAM S AMPLE
Input: Program space X, number of samples N , failure probability ?, parameters ? > 0, ? > 0
Output: N samples
Set |x? | = minx?X |x|
Set BX = ApproximateUpperBoundModelCount(X,?/2)
Set d = d? + log BX + |x? |e V
Define E = {(x, y) : x ? X, 1?j?d |x| ? j =? yj = 1}
Set BE = ApproximateLowerBoundModelCount(E,?/2)
Set K = blog BE ? ?c
Initialize samples = [ ]
repeat
Sample h uniformly from {0, 1}(d+n)?K
Sample b uniformly from {0, 1}K
Enumerate S = {(x, y) where h(x, y) = b ? x ? X}
if |S| > 0 then
Sample (x, y) uniformly from S
if Uniform(0, 1) < 2d?|x| then
samples = samples + [x]
end if
end if
until |samples| = N
return samples
were unnecessary for our main result, which is that the KL between p(?) and A(x)r(x) goes to zero
exponentially quickly in a new quantity we call ?:
Proposition
3. Write
Ar(x) to mean the distribution proportional to A(x)r(x). Then D(p||Ar) <
1+2??
log 1 + 1+2? where ? = log |E| ? K and ? = d ? log |X| ? |x? |.
So we can approximate the true distribution p(?) arbitrarily well, but at the expense of either more calls
to the solver (increasing ?) or a larger embedding (increasing ?; our main algorithmic contribution).
See supplement for theoretical and empirical analyses of this accuracy/runtime trade-off.
Proposition 3 requires knowing minx |x| to set K and d. We compute minx |x| using the iterative
minimization routine in [16]; in practice this is very efficient for finite program spaces. We also need
to calculate |X| and |E|, which are model counts that are in general difficult to compute exactly.
However, many approximate model counting schemes exist, which provide upper and lower bounds
that hold with arbitrarily high probability. We use Hybrid-MBound [13] to upper bound |X| and
lower bound |E| that each individually hold with probability at least 1 ? ?/2, thus giving lower
bounds on both the ? and ? parameters of Proposition 3 with probability at least 1 ? ? and thus an
upper bound on the KL divergence. Algorithm 1 puts these ideas together.
3
Experimental results
We evaluated P ROGRAM S AMPLE on program learning problems in a text editing domain and a
list manipulation domain. For each domain, we wrote down a sketch and produced SAT formulas
using the tool in [6], specifying a large but finite set of possible programs. This implicitly defined a
description-length prior, where |x| is the number of bits required to specify x in the SAT encoding.
We used CryptoMiniSAT [17], which can efficiently handle parity constraints.
3.1
Learning Text Edit Scripts
We applied our program sampling algorithm to a suite of programming by demonstration problems
within a text editing domain. Here, the challenge is to learn a small text editing program from very
few examples and apply that program to held out inputs. This problem is timely, given the widespread
use of the FlashFill program synthesis tool, which now ships by default in Microsoft Excel [18]
and can learn sophisticated edit operations in real time from examples. We modeled a subset of
5
the FlashFill language; our goal here is not to compete with FlashFill, which is cleverly engineered
for its specific domain, but to study the behavior of our more general-purpose program learner in a
real-world task. To impart domain knowledge, we used a sketch equivalent to Figure 4.
Because FlashFill?s training set is not yet
public, we drew text editing problems
from [19] and adapted them to our subset of FlashFill, giving 19 problems, each
with 5 training examples. The supplement
contains these text edit problems.
Program ::= Term | Program + Term
Term
::= String | substr(Pos,Pos)
Pos
::= Number
| pos(String,String,Number)
Number ::= 0 | 1 | 2 | ...
| -1 | -2 | ...
String ::= Character
| Character + String
Character ::= a | b | c | ...
We are interested both in the ability of
the learner to generalize and in P RO GRAM S AMPLE ?s ability to generate samples quickly. Table 1 shows the average time per sampling attempt using P RO - Figure 4: The sketch (program space) for learning text
GRAM S AMPLE , which is on the order of edit scripts
a minute. These text edit problems come
from distributions with extremely high tilt:
often the smallest program is only tens of bits long, but the program space contains (implausible)
solutions with over 100 bits. By putting d to |x? | ? n we eliminate the tilt correction and recover a
variant of the approaches in [7]. This baseline does not produce any samples for any of our text edit
problems in under an hour.4 Other baselines also failed to produce samples in a reasonable amount
of time (see supplement). For example, pure rejection sampling (drawing from the prior) is also
infeasible, with consistent programs having prior probability ? 2?50 in some cases.
The learner generalizes to unseen examples, as Figure 5 shows. We evaluated the performance of the
learner on held out test examples while varying training set size, and compare with baselines that
either (1) enumerate programs in the arbitrary order provided by the underlying solver, or (2) takes the
most likely program under p(x) (MDL learner). The posterior is sharply peaked, with most samples
being from the MAP solution, and so our learner does about as well as the MDL learner. However,
sampling offers an (approximate) predictive posterior over predictions on the held out examples; in a
real world scenario, one would offer the top C predictions to the user and let them choose, much like
how spelling correction works. This procedure allows us to offer the correct predictions more often
than the MDL learner (Figure 6), because we correctly handle ambiguous problems like in Figure 1.
We see this as a primary strength of the sampling approach to Bayesian program learning: when
learning from one or a few examples, a point estimate of the posterior can often miss the mark.
Figure 5: Generalization when learning text
edit operations by example. Results averaged
across 19 problems. Solid: 100 samples from
P ROGRAM S AMPLE . Dashed: enumerating
100 programs. Dotted: MDL learner. Test
cases past 1 (respectively 2,3) examples are
held out when trained on 1 (respectively 2,3)
examples.
Figure 6: Comparing the MDL learner
(dashed black line) to program sampling when
doing one-shot learning. We count a problem
as ?solved? if the correct joint prediction to
the test cases is in the top C most frequent
samples.
4
Approximate model counting of E was also intractable in this regime, so we used the lower bound
|E| ? 2d?|x? | + |X| ? 1
6
Table 1: Average solver time to generate a sample measured in seconds. See Figure 9 and 5 for
training set sizes. n ? 180, 65 for text edit, list
manipulation domains, respectively. w/o tilt correction, sampling text edit & count takes > 1 hour.
text edit
sort
reverse
count
3.2
Large set
49?3
1549?155
326?42
?1
Medium set
21 ?1
905 ?58
141 ?18
?1
Small set
84 ?3
463 ?65
39 ?3
?1
Figure 7: Sampling frequency vs.
ground truth probability on a counting
task with ? = 3 and ? = 4.
Learning list manipulation algorithms
One goal of program synthesis is computer-aided programming [6], which is the automatic generation
of executable code from either declarative specifications or examples of desired behavior. Systems
with this goal have been successfully applied to, for example, synthesizing intricate bitvector routines
from specifications [18]. However, when learning from examples, there is often uncertainty over
the correct program. While past approaches have handled this uncertainty within an optimization
framework (see [20, 21, 16]), we show that P ROGRAM S AMPLE can sample algorithms.
We take as our goal to learn recursive routines
for sorting, reversing, and counting list elements
from input/output examples, particularly in the
ambiguous, unconstrained regime of few examples. We used a sketch with a set of basis primitives capable of representing a range of list manipulation routines equivalent to Figure 8.
Program ::=
(if Bool List
(append RecursiveList
RecursiveList
RecursiveList))
Bool ::= (<= Int) | (>= Int)
Int ::= 0 | (1+ Int) | (1- Int)
| (length List) | (head List)
List ::= nil | (filter Bool List)
| X | (tail List) | (list Int)
RecursiveList ::= List
| (recurse List)
A description-length prior that penalizes longer
programs allowed learning of recursive list manipulation routines (from production Program)
and a non-recursive count routine (from production Int); see Figure 9, which shows average
accuracy on held out test data when trained on Figure 8: The sketch (program space) for learning
variable numbers of short randomly generated list manipulation routines; X is program input
lists. With the large training set (5?11 examples) P ROGRAM S AMPLE recovers a correct implementation, and with less data it recovers a
distribution over programs that functions as a
probabilistic algorithm despite being composed
of only deterministic programs.
For some of these tasks the number of consistent
programs is small enough that we can enumerate
all of them, allowing us to compare our sampler
with ground-truth probabilities. Figure 7 shows
this comparison for a counting problem with 80
consistent programs, showing empirically that
the tilt correction and random constraints do not
significantly perturb the distribution.
Table 1 shows the average solver time per sample. Generating recursive routines like sorting and reversing is much more costly than
generating the nonrecursive counting routine.
The constraint-based approach propositionalizes higher-order constructs like recursion, and Figure 9: Learning to manipulate lists. Trained on
so reasoning about them is much more costly. lists of length ? 3; tested on lists of length ? 14.
Yet counting problems are highly tilted due to
count?s short implementation, which makes them intractable without our tilt correction.
7
4
4.1
Discussion
Related work
There is a vast literature on program learning in the AI and machine learning communities. Many
employ a (possibly stochastic) heuristic search over structures using genetic programming [22] or
MCMC [23]. These approaches often find good programs and can discover more high-level structure
than our approach. However, they are prone to getting trapped in local minima and, when used as a
sampler, lack theoretical guarantees. Other work has addressed learning priors over programs in a
multitask setting [4, 5]. We see our work as particularly complementary to these methods: while they
focus on learning the structure of the hypothesis space, we focus on efficiently sampling an already
given hypothesis space (the sketch). Several recent proposals for recurrent deep networks can learn
algorithms [2, 1]. We see our system working in a different regime, where we want to quickly learn
an algorithm from a small number of examples or an ambiguous specification.
The program synthesis community has several recently proposed learners that work in an optimization
framework [20, 21, 16]. By computing a posterior over programs, we can more effectively represent
uncertainty, particularly in the small data limit, but at the cost of more computation.
P ROGRAM S AMPLE borrows heavily from a line of work started in [9, 13] on sampling of combinatorial spaces using random XOR constraints. An exciting new approach is to use sparse XOR
constraints [14, 15] , which might sample more efficiently from our embedding of the program space.
4.2
Limitations of the approach
Constraint-based synthesis methods tend to excel in domains where the program structure is restricted
by a sketch [6] and where much of the program?s description length can be easily computed from
the program text. For example, P ROGRAM S AMPLE can synthesize text editing programs that are
almost 60 bits long in a couple seconds, but spends 10 minutes synthesizing a recursive sorting
routine that is shorter but where the program structure is less restricted. Constraint-based methods
also require the entire problem to be represented symbolically, so they have trouble when the function
to be synthesized involves difficult to analyze building blocks such as numerical routines. For such
problems, stochastic search methods [23, 22] can be more effective because they only need to run the
functions under consideration. Finally, past work shows empirically that these methods scale poorly
with data set size, although this can be mitigated by considering data incrementally [21, 20].
The requirement of producing representative samples imposes additional overhead on our approach,
so scalability can more limited than for standard symbolic techniques on some problems. For
example, our method requires 1 MAP inference query, and 2 queries to an approximate model
counter. These serve to ?calibrate? the sampler, and its cost can be amortized because they only has
to be invoked once in order to generate an arbitrary number of iid samples. Approximate model
counters like MBound [13] have complexity comparable with that of generating a sample, but the
complexity can depend on the number of solutions. Thus, for good performance, P ROGRAM S AMPLE
requires that there not be too many programs consistent with the data?the largest spaces considered
in our experiments had ? 107 programs. This limitation, together with the general performance
characteristics of symbolic techniques, means that the approach will work best for ?needle in a
haystack? problems, where the space of possible programs is large but restricted in its structure, and
where only a small fraction of the programs satisfy the constraints.
4.3
Future work
This work could naturally extend to other domains that involve inducing latent symbolic structure
from small amounts of data, such as semantic parsing to logical forms [24], synthesizing motor
programs [3], or learning relational theories [25]. These applications have some component of
transfer learning, and building efficient program learners that can transfer inductive biases across
tasks is a prime target for future research.
Acknowledgments
We are grateful for feedback from Adam Smith, Kuldeep Meel, and our anonymous reviewers. Work
supported by NSF-1161775 and AFOSR award FA9550-16-1-0012.
8
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9
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5,617 | 6,083 | Poisson?Gamma Dynamical Systems
Aaron Schein
College of Information and Computer Sciences
University of Massachusetts Amherst
Amherst, MA 01003
aschein@cs.umass.edu
Mingyuan Zhou
McCombs School of Business
The University of Texas at Austin
Austin, TX 78712
mingyuan.zhou@mccombs.utexas.edu
Hanna Wallach
Microsoft Research New York
641 Avenue of the Americas
New York, NY 10011
hanna@dirichlet.net
Abstract
We introduce a new dynamical system for sequentially observed multivariate count
data. This model is based on the gamma?Poisson construction?a natural choice for
count data?and relies on a novel Bayesian nonparametric prior that ties and shrinks
the model parameters, thus avoiding overfitting. We present an efficient MCMC inference algorithm that advances recent work on augmentation schemes for inference
in negative binomial models. Finally, we demonstrate the model?s inductive bias
using a variety of real-world data sets, showing that it exhibits superior predictive
performance over other models and infers highly interpretable latent structure.
1
Introduction
Sequentially observed count vectors y (1) , . . . , y (T ) are the main object of study in many real-world
applications, including text analysis, social network analysis, and recommender systems. Count data
pose unique statistical and computational challenges when they are high-dimensional, sparse, and
overdispersed, as is often the case in real-world applications. For example, when tracking counts of
user interactions in a social network, only a tiny fraction of possible edges are ever active, exhibiting
bursty periods of activity when they are. Models of such data should exploit this sparsity in order to
scale to high dimensions and be robust to overdispersed temporal patterns. In addition to these characteristics, sequentially observed multivariate count data often exhibit complex dependencies within and
across time steps. For example, scientific papers about one topic may encourage researchers to write
papers about another related topic in the following year. Models of such data should therefore capture
the topic structure of individual documents as well as the excitatory relationships between topics.
The linear dynamical system (LDS) is a widely used model for sequentially observed data, with
many well-developed inference techniques based on the Kalman filter [1, 2]. The LDS assumes
that each sequentially observed V -dimensional vector r (t) is real valued and Gaussian distributed:
r (t) ? N (? ? (t) , ?), where ? (t) ? RK is a latent state, with K components, that is linked to the
observed space via ? ? RV ?K . The LDS derives its expressive power from the way it assumes
that the latent states evolve: ? (t) ? N (? ? (t?1) , ?), where ? ? RK?K is a transition matrix that
captures between-component dependencies across time steps. Although the LDS can be linked to
non-real observations via the extended Kalman filter [3], it cannot efficiently model real-world count
data because inference is O((K + V )3 ) and thus scales poorly with the dimensionality of the data [2].
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Many previous approaches to modeling sequentially observed count data rely on the generalized
linear modeling framework [4] to link the observations to a latent Gaussian space?e.g., via the
Poisson?lognormal link [5]. Researchers have used this construction to factorize sequentially observed count matrices under a Poisson likelihood, while modeling the temporal structure using
well-studied Gaussian techniques [6, 7]. Most of these previous approaches assume a simple Gaussian state-space model?i.e., ? (t) ? N (? (t?1) , ?)?that lacks the expressive transition structure
of the LDS; one notable exception is the Poisson linear dynamical system [8]. In practice, these
approaches exhibit prohibitive computational complexity in high dimensions, and the Gaussian
assumption may fail to accommodate the burstiness often inherent to real-world count data [9].
We present the Poisson?gamma dynamical system
(PGDS)?a new dynamical system, based on the
gamma?Poisson construction, that supports the expressive transition structure of the LDS. This model naturally handles overdispersed data. We introduce a new
Bayesian nonparametric prior to automatically infer
the model?s rank. We develop an elegant and efficient
algorithm for inferring the parameters of the transition
structure that advances recent work on augmentation
schemes for inference in negative binomial models [10]
and scales with the number of non-zero counts, thus
exploiting the sparsity inherent to real-world count data.
We examine the way in which the dynamical gamma?
Poisson construction propagates information and derive
the model?s steady state, which involves the Lambert
W function [11]. Finally, we use the PGDS to analyze
a diverse range of real-world data sets, showing that it
exhibits excellent predictive performance on smoothing and forecasting tasks and infers interpretable latent
structure, an example of which is depicted in figure 1.
2
25
20
15
10
5
1988
1991
1994
1997
2000
Figure 1: The time-step factors for three
components inferred by the PGDS from a
corpus of NIPS papers. Each component
is associated with a feature factor for each
word type in the corpus; we list the words
with the largest factors. The inferred structure tells a familiar story about the rise and
fall of certain subfields of machine learning.
Poisson?Gamma Dynamical Systems
We can represent a data set of V -dimensional sequentially observed count vectors y (1) , . . . , y (T ) as a
(t)
V ? T count matrix Y . The PGDS models a single count yv ? {0, 1, . . .} in this matrix as follows:
P
PK
(t)
(t)
(t?1)
K
yv(t) ? Pois(? (t) k=1 ?vk ?k ) and ?k ? Gam(?0 k2 =1 ?kk2 ?k2 , ?0 ),
(1)
(t)
where the latent factors ?vk and ?k are both positive, and represent the strength of feature v in
component k and the strength of component k at time step t, respectively. The scaling factor ? (t)
captures the scale of the counts at time step t, and therefore obviates the need to rescale the data as a
preprocessing step. We refer to the PGDS as stationary if ? (t) = ? for t = 1, . . . , T . We can view the
feature factors as a V ?K matrix ? and the time-step factors as a T ?K matrix ?. Because we can also
collectively view the scaling factors and time-step factors as a T ? K matrix ?, where element ?tk =
(t)
? (t) ?k , the PGDS is a form of Poisson matrix factorization: Y ? Pois(? ?T ) [12, 13, 14, 15].
The PGDS is characterized by its expressive transition structure, which assumes that each time-step
(t)
factor ?k is drawn from a gamma distribution, whose shape parameter is a linear combination of the
K factors at the previous time step. The latent transition weights ?11 , . . . , ?k1 k2 , . . . , ?KK , which we
can view as a K ? K transition matrix ?, capture the excitatory relationships between components.
(t)
(t)
The vector ? (t) = (?1 , . . . , ?K ) has an expected value of E[? (t) | ? (t?1) , ?] = ? ? (t?1) and is therefore analogous to a latent state in the the LDS. The concentration parameter ?0 determines the variance
of ? (t) ?specifically, Var (? (t) | ? (t?1) , ?) = (? ? (t?1) ) ?0?1 ?without affecting its expected value.
To model the strength of each component, we introduce K component weights ? = (?1 , . . . , ?K )
and place a shrinkage prior over them. We assume that the time-step factors and transition weights
for component k are tied to its component weight ?k . Specifically, we define the following structure:
(1)
?k ? Gam(?0 ?k , ?0 ) and ? k ? Dir(?1 ?k , . . . , ??k . . . , ?K ?k ) and ?k ? Gam( ?K0 , ?),
2
(2)
PK
where ? k = (?1k , . . . , ?Kk ) is the k th column of ?. Because k1 =1 ?k1 k = 1, we can interpret
?k1 k as the probability of transitioning from component k to component k1 . (We note that interpreting
? as a stochastic transition matrix relates the PGDS to the discrete hidden Markov model.) For a fixed
value of ?0 , increasing K will encourage many of the component weights to be small. A small value of
(1)
?k will shrink ?k , as well as the transition weights in the k th row of ?. Small values of the transition
weights in the k th row of ? therefore prevent component k from being excited by the other components
(t)
and by itself. Specifically, because the shape parameter for the gamma prior over ?k involves a
(t?1)
linear combination of ?
and the transition weights in the k th row of ?, small transition weights
(t)
will result in a small shape parameter, shrinking ?k . Thus, the component weights play a critical
role in the PGDS by enabling it to automatically turn off any unneeded capacity and avoid overfitting.
Finally, we place Dirichlet priors over the feature factors and draw the other parameters from a noninformative gamma prior: ?k = (?1k , . . . , ?V k ) ? Dir(?0 , . . . , ?0 ) and ? (t) , ?, ? ? Gam(0 , 0 ).
The PGDS therefore has four positive hyperparameters to be set by the user: ?0 , ?0 , ?0 , and 0 .
Bayesian nonparametric interpretation: As KP? ?, the component weights and their correspond?
ing feature factor vectors constitute a draw G = k=1 ?k 1?k from a gamma process GamP (G0 , ?),
where ? is a scale parameter and G0 is a finite and continuous base measure over a complete separable
metric space ? [16]. Models based on the gamma process have an inherent shrinkage mechanism
because the number of atoms with
weights greater than ? > 0 follows a Poisson distribution with a fiR?
nite mean?specifically, Pois(?0 ? d? ? ?1 exp (?? ?)), where ?0 = G0 (?) is the total mass under
the base measure. This interpretation enables us to view the priors over ? and ? as novel stochastic
processes, which we call the column-normalized relational gamma process and the recurrent gamma
process, respectively. We provide the definitions of these processes in the supplementary material.
Non-count observations: The PGDS can also model non-count data by linking the observed vectors
(t)
(t)
to latent counts. A binary observation bv can be linked to a latent Poisson count yv via the Bernoulli?
PK
(t)
(t)
(t)
(t)
(t)
Poisson distribution: bv = 1(yv ? 1) and yv ? Pois(?
k=1 ?vk ?k ) [17]. Similarly, a
(t)
(t)
real-valued observation rv can be linked to a latent Poisson count yv via the Poisson randomized
gamma distribution [18]. Finally, Basbug and Engelhardt [19] recently showed that many types of
non-count matrices can be linked to a latent count matrix via the compound Poisson distribution [20].
3
MCMC Inference
MCMC inference for the PGDS consists of drawing samples of the model parameters from their joint
posterior distribution given an observed count matrix Y and the model hyperparameters ?0 , ?0 , ?0 ,
0 . In this section, we present a Gibbs sampling algorithm for drawing these samples. At a high level,
our approach is similar to that used to develop Gibbs sampling algorithms for several other related
models [10, 21, 22, 17]; however, we extend this approach to handle the unique properties of the
PGDS. The main technical challenge is sampling ? from its conditional posterior, which does not
have a closed form. We address this challenge by introducing a set of auxiliary variables. Under this
augmented version of the model, marginalizing over ? becomes tractable and its conditional posterior
has a closed form. Moreover, by introducing these auxiliary variables and marginalizing over ?,
we obtain an alternative model specification that we can subsequently exploit to obtain closed-form
conditional posteriors for ?, ?, and ?. We marginalize over ? by performing a ?backward filtering?
pass, starting with ? (T ) . We repeatedly exploit the following three definitions in order to do this.
PN
Definition 1: If y? = n=1 yn , where yn ? Pois(?n ) are independent Poisson-distributed random variPN
ables, then (y1 , . . . , yN ) ? Mult(y? , ( PN?1 ? , . . . , PN?N ? )) and y? ? Pois( n=1 ?n ) [23, 24].
n=1
n
n=1
n
c
Definition 2: If y ? Pois(c ?), where c is a constant, and ? ? Gam(a, b), then y ? NB(a, b+c
)
is a negative binomial?distributed random variable. We can equivalently parameterize it as
y ? NB(a, g(?)), where g(z) = 1 ? exp (?z) is the Bernoulli?Poisson link [17] and ? = ln (1 + cb ).
Definition 3: If y ? NB(a, g(?)) and l ? CRT(y, a) is a Chinese restaurant table?distributed
random variable, then y and l are equivalently jointly distributed as y ? SumLog(l, g(?)) and
l ? Pois(a ?) [21]. The sum logarithmic distribution is further defined as the sum of l independent
Pl
and identically logarithmic-distributed random variables?i.e., y = i=1 xi and xi ? Log(g(?)).
3
Marginalizing over ?: We first note that we can re-express the Poisson likelihood in equation 1
PK
(t)
(t)
(t)
(t)
(t)
in terms of latent subcounts [13]: yv = yv? = k=1 yvk and yvk ? Pois(? (t) ?vk ?k ). We then
P
P
(t)
(t)
(t)
(t)
V
V
define y?k = v=1 yvk . Via definition 1, we obtain y?k ? Pois(? (t) ?k ) because v=1 ?vk = 1.
(T )
We start with ?k because none of the other time-step factors depend on it in their priors. Via
(T )
definition 2, we can immediately marginalize over ?k to obtain the following equation:
PK
(T )
(T )
(T ?1)
, g(? (T ) )), where ? (T ) = ln (1 + ??0 ).
y?k ? NB(?0 k2 =1 ?kk2 ?k2
(3)
(T ?1)
(T )
Next, we marginalize over ?k
. To do this, we introduce an auxiliary variable: lk
?
PK
(T )
(T ?1)
(T )
(T )
CRT(y?k , ?0 k2 =1 ?kk2 ?k2
). We can then re-express the joint distribution over y?k and lk as
(T )
(T )
(T )
y?k ? SumLog(lk , g(? (T ) ) and lk
? Pois(? (T ) ?0
PK
k2 =1 ?kk2
(T ?1)
?k2
).
(4)
(T ?1)
We are still unable to marginalize over ?k
because it appears in a sum in the parameter of the
(T )
Poisson distribution over lk ; however, via definition 1, we can re-express this distribution as
PK
(T )
(T )
(T )
(T )
(T ?1)
lk = lk? = k2 =1 lkk2 and lkk2 ? Pois(? (T ) ?0 ?kk2 ?k2
).
(5)
(T )
(T )
k1 =1 lk1 k . Again via definition 1, we can express the distribution
(T )
(T )
(T ?1)
over l?k as l?k ? Pois(? (T ) ?0 ?k
). We note that this expression does not depend on
PK
the transition weights because k1 =1 ?k1 k = 1. We also note that definition 1 implies that
(T )
(T )
(T )
(T ?1)
(T ?1)
(T )
(l1k , . . . , lKk ) ? Mult(l?k , (?1 , . . . , ?K )). Next, we introduce mk
= y?k
+ l?k , which
(T ?1)
(T )
summarizes all of the information about the data at time steps T ? 1 and T via y?k
and l?k ,
(T ?1)
(T )
respectively. Because y?k
and l?k are both Poisson distributed, we can use definition 1 to obtain
We then define l?k
=
PK
(T ?1)
mk
(T ?1)
? Pois(?k
(? (T ?1) + ? (T ) ?0 )).
(6)
(T ?1)
Combining this likelihood with the gamma prior in equation 1, we can marginalize over ?k
:
PK
(T ?1)
(T ?1)
(T ?2)
mk
? NB(?0 k2 =1 ?kk2 ?k2
, g(? (T ?1) )), where ? (T ?1) = ln (1 + ? ?0 + ? (T ) ). (7)
(T ?1)
We then introduce lk
(T ?1)
lk
(T ?1)
? CRT(mk
, ?0
(T ?2)
PK
k2 =1
?kk2 ?k2
) and re-express the joint distribu-
(T ?1)
mk
tion over
and
as the product of a Poisson and a sum logarithmic distribution, similar to
(T ?2)
equation 4. This then allows us to marginalize over ?k
to obtain a negative binomial distribution.
(1)
(1)
We can repeat the same process all the way back to t = 1, where marginalizing over ?k yields mk ?
(t)
NB(?0 ?k , g(? (1) )). We note that just as mk summarizes all of the information about the data at time
(t)
steps t, . . . , T , ? (t) = ln (1 + ??0 + ? (t+1) ) summarizes all of the information about ? (t) , . . . , ? (T ) .
As we mentioned previously, introducing
these auxiliary variables and marginalizing
over ? also enables us to define an alternative model specification that we can exploit
to obtain closed-form conditional posteriors for ?, ?, and ?. We provide part of its
generative process in figure 2. We define
(T )
(T )
(T +1)
(T +1)
mk = y?k + l?k
, where l?k
= 0,
(T +1)
and ?
= 0 so that we can present the
alternative model specification concisely.
(1)
lk? ? Pois(? (1) ?0 ?k )
(t)
(t)
(t)
(l1k , . . . , lKk ) ? Mult(l?k , (?1k , . . . , ?Kk )) for t > 1
P
(t)
(t)
lk? = K
k2 =1 lkk2 for t > 1
(t)
(t)
mk ? SumLog(lk? , g(? (t) ))
(t)
(t+1)
(y?k , l?k
(t)
(t)
(t)
) ? Bin(mk , ( ?(t) +?? (t+1) ? ,
0
(t)
? (t+1) ?0
))
? (t) +? (t+1) ?0
(t)
(y1k , . . . , yV k ) ? Mult(y?k , (?1k , . . . , ?V k ))
Figure 2: Alternative model specification.
(t)
Steady state: We draw particular attention to the backward pass ? (t) = ln (1 + ??0 + ? (t+1) ) that
propagates information about ? (t) , . . . , ? (T ) as we marginalize over ?. In the case of the stationary
PGDS?i.e., ? (t) = ??the backward pass has a fixed point that we define in the following proposition.
4
Proposition 1: The backward pass has a fixed point of ? ? = ?W?1 (? exp (?1 ?
?
?0 ))
?1?
?
?0 .
The function W?1 (?) is the lower real part of the Lambert W function [11]. We prove this proposition
in the supplementary material. During inference, we perform the O(T ) backward pass repeatedly.
The existence of a fixed point means that we can assume the stationary PGDS is in its steady state and
replace the backward pass with an O(1) computation1 of the fixed point ? ? . To make this assumption,
(T +1)
(T )
(T +1)
we must also assume that l?k
? Pois(? ? ?0 ?k ) instead of l?k
= 0. We note that an analogous
steady-state approximation exists for the LDS and is routinely exploited to reduce computation [25].
Gibbs sampling algorithm: Given Y and the hyperparameters, Gibbs sampling involves resampling
each auxiliary variable or model parameter from its conditional posterior. Our algorithm involves a
?backward filtering? pass and a ?forward sampling? pass, which together form a ?backward filtering?
forward sampling? algorithm. We use ? \ ?(?t) to denote everything excluding ? (t) , . . . , ? (T ) .
Sampling the auxiliary variables: This step is the ?backward filtering? pass. For the stationary PGDS
(T +1)
(T )
in its steady state, we first compute ? ? and draw (l?k
| ?) ? Pois(? ? ?0 ?k ). For the other vari(T +1)
ants of the model, we set l?k
= ? (T +1) = 0. Then, working backward from t = T, . . . , 2, we draw
PK
(t)
(t)
(t+1)
(t?1)
(lk? | ? \ ?(?t) ) ? CRT(y?k + l?k , ?0 k2 =1 ?kk2 ?k2 ) and
(8)
(t)
(t)
(t)
(lk1 , . . . , lkK | ? \ ?(?t) ) ? Mult(lk? , ( PK
(t?1)
(t?1)
?kK ?K
?k1 ?1
k2
(t?1)
=1 ?kk2 ?k
, . . . , PK
(t)
After using equations 8 and 9 for all k = 1, . . . , K, we then set l?k =
state variants, we also set ?
(t)
= ln (1 +
? (t)
?0
+?
(t+1)
k2
2
(t)
k1 =1 lk1 k .
PK
(t?1)
=1 ?kk2 ?k
)).
(9)
2
For the non-steady-
); for the steady-state variant, we set ? (t) = ? ? .
Sampling ?: We sample ? from its conditional posterior by performing a ?forward sampling? pass,
(2)
(T +1)
starting with ? (1) . Conditioned on the values of l?k , . . . , l?k
and ? (2) , . . . , ? (T +1) obtained via
the ?backward filtering? pass, we sample forward from t = 1, . . . , T , using the following equations:
(1)
(1)
(2)
(?k | ? \ ?) ? Gam(y?k + l?k + ?0 ?k , ?0 + ? (1) + ? (2) ?0 ) and
PK
(t)
(t)
(t+1)
(t?1)
(?k | ? \ ?(?t) ) ? Gam(y?k + l?k
+ ?0 k2 =1 ?kk2 ?k2 , ?0 + ? (t) + ? (t+1) ?0 ).
(10)
(11)
Sampling ?: The alternative model specification, with ? marginalized out, assumes that
(t)
(t)
(t)
(l1k , . . . , lKk ) ? Mult(l?k , (?1k , . . . , ?Kk )). Therefore, via Dirichlet?multinomial conjugacy,
PT (t)
PT (t)
PT (t)
(? k | ? \ ?) ? Dir(?1 ?k + t=1 l1k , . . . , ??k + t=1 lkk , . . . , ?K ?k + t=1 lKk ).
(12)
Sampling ? and ?: We use the alternative model specification to obtain closed-form conditional
posteriors for ?k and ?. First, we marginalize over ? k to obtain a Dirichlet?multinomial distribution.
When augmented with a beta-distributed auxiliary variable, the Dirichlet?multinomial distribution is
proportional to the negative binomial distribution [26]. We draw such an auxiliary variable, which we
use, along with negative binomial augmentation schemes, to derive closed-form conditional posteriors
for ?k and ?. We provide these posteriors, along with their derivations, in the supplementary material.
We also provide the conditional posteriors for the remaining model parameters??, ? (1) , . . . , ? (T ) ,
and ??which we obtain via Dirichlet?multinomial, gamma?Poisson, and gamma?gamma conjugacy.
4
Experiments
In this section, we compare the predictive performance of the PGDS to that of the LDS and that
of gamma process dynamic Poisson factor analysis (GP-DPFA) [22]. GP-DPFA models a single
PK
(t)
(t)
count in Y as yv ? Pois( k=1 ?k ?vk ?k ), where each component?s time-step factors evolve
as a simple gamma Markov chain, independently of those belonging to the other components:
(t)
(t?1) (t)
?k ? Gam(?k
, c ). We consider the stationary variants of all three models.2 We used five data
(t)
sets, and tested each model on two time-series prediction tasks: smoothing?i.e., predicting yv given
1
2
Several software packages contain fast implementations of the Lambert W function.
We used the pykalman Python library for the LDS and implemented GP-DPFA ourselves.
5
(1)
(t?1)
(t+1)
(T +s)
(T )
(1)
(T )
yv , . . . , y v
given yv , . . . , yv for
, yv
, . . . , yv ?and forecasting?i.e., predicting yv
some s ? {1, 2, . . .} [27]. We provide brief descriptions of the data sets below before reporting results.
Global Database of Events, Language, and Tone (GDELT): GDELT is an international relations data
set consisting of country-to-country interaction events of the form ?country i took action a toward
country j at time t,? extracted from news corpora. We created five count matrices, one for each year
from 2001 through 2005. We treated directed pairs of countries i?j as features and counted the
number of events for each pair during each day. We discarded all pairs with fewer than twenty-five
total events, leaving T = 365, around V ? 9, 000, and three to six million events for each matrix.
Integrated Crisis Early Warning System (ICEWS): ICEWS is another international relations event data
set extracted from news corpora. It is more highly curated than GDELT and contains fewer events.
We therefore treated undirected pairs of countries i?j as features. We created three count matrices,
one for 2001?2003, one for 2004?2006, and one for 2007?2009. We counted the number of events for
each pair during each three-day time step, and again discarded all pairs with fewer than twenty-five
total events, leaving T = 365, around V ? 3, 000, and 1.3 to 1.5 million events for each matrix.
State-of-the-Union transcripts (SOTU): The SOTU corpus contains the text of the annual SOTU
speech transcripts from 1790 through 2014. We created a single count matrix with one column per
year. After discarding stopwords, we were left with T = 225, V = 7, 518, and 656,949 tokens.
DBLP conference abstracts (DBLP): DBLP is a database of computer science research papers. We
used the subset of this corpus that Acharya et al. used to evaluate GP-DPFA [22]. This subset corresponds to a count matrix with T = 14 columns, V = 1, 771 unique word types, and 13,431 tokens.
NIPS corpus (NIPS): The NIPS corpus contains the text of every NIPS conference paper from 1987
to 2003. We created a single count matrix with one column per year. We treated unique word types
as features and discarded all stopwords, leaving T = 17, V = 9, 836, and 3.1 million tokens.
2500
900
800
700
600
500
400
300
200
100
0
2000
1500
1000
500
0
1988
1992
1995
1998
2002
Israel?Palestine
Russia?USA
China?USA
Iraq?USA
Mar 2009
Jun 2009
Aug 2009
Oct 2009
Dec 2009
(t)
Figure 3: yv over time for the top four features in the NIPS (left) and ICEWS (right) data sets.
Experimental design: For each matrix, we created four masks indicating some randomly selected
subset of columns to treat as held-out data. For the event count matrices, we held out six (noncontiguous) time steps between t = 2 and t = T ? 3 to test the models? smoothing performance, as
well as the last two time steps to test their forecasting performance. The other matrices have fewer
time steps. For the SOTU matrix, we therefore held out five time steps between t = 2 and t = T ? 2,
as well as t = T . For the NIPS and DBLP matrices, which contain substantially fewer time steps
than the SOTU matrix, we held out three time steps between t = 2 and t = T ? 2, as well as t = T .
For each matrix, mask, and model combination, we ran inference four times.3 For the PGDS and
GP-DPFA, we performed 6,000 Gibbs sampling iterations, imputing the missing counts from the
?smoothing? columns at the same time as sampling the model parameters. We then discarded the
first 4,000 samples and retained every hundredth sample thereafter. We used each of these samples to
predict the missing counts from the ?forecasting? columns. We then averaged the predictions over the
samples. For the LDS, we ran EM to learn the model parameters. Then, given these parameter values,
we used the Kalman filter and smoother [1] to predict the held-out data. In practice, for all five data
sets, V was too large for us to run inference for the LDS, which is O((K + V )3 ) [2], using all V
features. We therefore report results from two independent sets of experiments: one comparing all
three models using only the top V = 1, 000 features for each data set, and one comparing the PGDS
to just GP-DPFA using all the features. The first set of experiments is generous to the LDS because
the Poisson distribution is well approximated by the Gaussian distribution when its mean is large.
3
For the PGDS and GP-DPFA we used K = 100. For the PGDS, we set ?0 = 1, ?0 = 50, ?0 = 0 = 0.1.
We set the hyperparameters of GP-DPFA to the values used by Acharya et al. [22]. For the LDS, we used the
default hyperparameters for pykalman, and report results for the best-performing value of K ? {5, 10, 25, 50}.
6
Table 1: Results for the smoothing (?S?) and forecasting (?F?) tasks. For both error measures, lower
? of each data set.
values are better. We also report the number of time steps T and the burstiness B
Mean Relative Error (MRE)
T
? Task
B
PGDS
GP-DPFA
Mean Absolute Error (MAE)
LDS
PGDS
GP-DPFA
LDS
GDELT 365 1.27 S 2.335 ?0.19 2.951 ?0.32 3.493 ?0.53 9.366 ?2.19 9.278 ?2.01
F 2.173 ?0.41 2.207 ?0.42 2.397 ?0.29 7.002 ?1.43 7.095 ?1.67
10.098 ?2.39
7.047 ?1.25
ICEWS 365 1.10 S 0.808 ?0.11 0.877 ?0.12 1.023 ?0.15 2.867 ?0.56 2.872 ?0.56
F 0.743 ?0.17 0.792 ?0.17 0.937 ?0.31 1.788 ?0.47 1.894 ?0.50
3.104 ?0.60
1.973 ?0.62
SOTU 225 1.45 S 0.233 ?0.01 0.238 ?0.01 0.260 ?0.01 0.408 ?0.01 0.414 ?0.01
F 0.171 ?0.00 0.173 ?0.00 0.225 ?0.01 0.323 ?0.00 0.314 ?0.00
0.448 ?0.00
0.370 ?0.00
DBLP 14 1.64 S 0.417 ?0.03 0.422 ?0.05 0.405 ?0.05 0.771 ?0.03 0.782 ?0.06
F 0.322 ?0.00 0.323 ?0.00 0.369 ?0.06 0.747 ?0.01 0.715 ?0.00
0.831 ?0.01
0.943 ?0.07
NIPS 17 0.33 S
F
0.415 ?0.07 0.392 ?0.07 1.609 ?0.43 29.940 ?2.95 28.138 ?3.08 108.378 ?15.44
0.343 ?0.01 0.312 ?0.00 0.642 ?0.14 62.839 ?0.37 52.963 ?0.52 95.495 ?10.52
Results: We used two error measures?mean relative error (MRE) and mean absolute error (MAE)?
to compute the models? smoothing and forecasting scores for each matrix and mask combination. We
then averaged these scores over the masks. For the data sets with multiple matrices, we also averaged
the scores over the matrices. The two error measures differ as follows: MRE accommodates
the
|y (t) ??
y (t) |
scale of the data, while MAE does not. This is because relative error?which we define as v (t)v ,
1+yv
(t)
(t)
where yv is the true count and y?v is the prediction?divides the absolute error by the true count and
thus penalizes overpredictions more harshly than underpredictions. MRE is therefore an especially
natural choice for data sets that are bursty?i.e., data sets that exhibit short periods of activity that far
exceed their mean. Models that are robust to these kinds of overdispersed temporal patterns are less
likely to make overpredictions following a burst, and are therefore rewarded accordingly by MRE.
In table 1, we report the MRE and MAE scores for the experiments using the top V = 1, 000 features.
We also report the average burstiness of each data set. We define the burstiness of feature v in matrix
(t+1)
(t)
PT
(t)
?v = 1 PT ?1 |yv ?yv | , where ?
Y to be B
?v = 1
yv . For each data set, we calculated
T ?1
t=1
?
?v
T
t=1
the burstiness of each feature in each matrix, and then averaged these values to obtain an average
? The PGDS outperformed the LDS and GP-DPFA on seven of the ten prediction
burstiness score B.
tasks when we used MRE to measure the models? performance; when we used MAE, the PGDS
outperformed the other models on five of the tasks. In the supplementary material, we also report the
results for the experiments comparing the PGDS to GP-DPFA using all the features. The superiority
of the PGDS over GP-DPFA is even more pronounced in these results. We hypothesize that the
difference between these models is related to the burstiness of the data. For both error measures, the
only data set for which GP-DPFA outperformed the PGDS on both tasks was the NIPS data set. This
data set has a substantially lower average burstiness score than the other data sets. We provide visual
(t)
evidence in figure 3, where we display yv over time for the top four features in the NIPS and ICEWS
data sets. For the former, the features evolve smoothly; for the latter, they exhibit bursts of activity.
Exploratory analysis: We also explored the latent structure inferred by the PGDS. Because its
parameters are positive, they are easy to interpret. In figure 1, we depict three components inferred
from the NIPS data set. By examining the time-step factors and feature factors for these components,
we see that they capture the decline of research on neural networks between 1987 and 2003, as well
as the rise of Bayesian methods in machine learning. These patterns match our prior knowledge.
In figure 4, we depict the three components with the largest component weights inferred by the PGDS
from the 2003 GDELT matrix. The top component is in blue, the second is in green, and the third
is in red. For each component, we also list the sixteen features (directed pairs of countries) with
the largest feature factors. The top component (blue) is most active in March and April, 2003. Its
features involve USA, Iraq (IRQ), Great Britain (GBR), Turkey (TUR), and Iran (IRN), among others.
This component corresponds to the 2003 invasion of Iraq. The second component (green) exhibits a
noticeable increase in activity immediately after April, 2003. Its top features involve Israel (ISR),
Palestine (PSE), USA, and Afghanistan (AFG). The third component exhibits a large burst of activity
7
6
5
4
3
2
1
Jan 2003
Mar 2003
May 2003
Aug 2003
Oct 2003
Dec 2003
Figure 4: The time-step factors for the top three components inferred by the PGDS from the 2003
GDELT matrix. The top component is in blue, the second is in green, and the third is in red. For each
component, we also list the features (directed pairs of countries) with the largest feature factors.
in August, 2003, but is otherwise inactive. Its top features involve North Korea (PRK), South Korea
(KOR), Japan (JPN), China (CHN), Russia (RUS), and USA. This component corresponds to the
six-party talks?a series of negotiations between these six countries for the purpose of dismantling
North Korea?s nuclear program. The first round of talks occurred during August 27?29, 2003.
0
1
2
3
4
5
6
7
8
9
In figure 5, we also show the component weights for the top ten com- 6
ponents, along with the corresponding subset of the transition matrix 543
?. There are two components with weights greater than one: the com- 21
0
ponents that are depicted in blue and green in figure 4. The transition
weights in the corresponding rows of ? are also large, meaning that
other components are likely to transition to them. As we mentioned
previously, the GDELT data set was extracted from news corpora.
Therefore, patterns in the data primarily reflect patterns in media coverage of international affairs. We therefore interpret the latent structure
inferred by the PGDS in the following way: in 2003, the media briefly
covered various major events, including the six-party talks, before
quickly returning to a backdrop of the ongoing Iraq war and Israeli?
Palestinian relations. By inferring the kind of transition structure
depicted in figure 5, the PGDS is able to model persistent, long-term
0 1 2 3 4 5 6 7 8 9
temporal patterns while accommodating the burstiness often inherent
to real-world count data. This ability is what enables the PGDS to Figure 5: The latent tranachieve superior predictive performance over the LDS and GP-DPFA. sition structure inferred by
the PGDS from the 2003
GDELT matrix. Top: The
5 Summary
component weights for the
top ten components, in deWe introduced the Poisson?gamma dynamical system (PGDS)?a new creasing order from left to
Bayesian nonparametric model for sequentially observed multivariate right; two of the weights are
count data. This model supports the expressive transition structure greater than one. Bottom:
of the linear dynamical system, and naturally handles overdispersed The transition weights in the
data. We presented a novel MCMC inference algorithm that remains corresponding subset of the
efficient for high-dimensional data sets, advancing recent work on aug- transition matrix. This strucmentation schemes for inference in negative binomial models. Finally, ture means that all compowe used the PGDS to analyze five real-world data sets, demonstrating nents are likely to transition
that it exhibits superior smoothing and forecasting performance over to the top two components.
two baseline models and infers highly interpretable latent structure.
Acknowledgments
We thank David Belanger, Roy Adams, Kostis Gourgoulias, Ben Marlin, Dan Sheldon, and Tim
Vieira for many helpful conversations. This work was supported in part by the UMass Amherst CIIR
and in part by NSF grants SBE-0965436 and IIS-1320219. Any opinions, findings, conclusions, or
recommendations are those of the authors and do not necessarily reflect those of the sponsors.
8
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9
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5,618 | 6,084 | Fast -free Inference of Simulation Models with
Bayesian Conditional Density Estimation
George Papamakarios
School of Informatics
University of Edinburgh
g.papamakarios@ed.ac.uk
Iain Murray
School of Informatics
University of Edinburgh
i.murray@ed.ac.uk
Abstract
Many statistical models can be simulated forwards but have intractable likelihoods.
Approximate Bayesian Computation (ABC) methods are used to infer properties
of these models from data. Traditionally these methods approximate the posterior
over parameters by conditioning on data being inside an -ball around the observed
data, which is only correct in the limit ? 0. Monte Carlo methods can then draw
samples from the approximate posterior to approximate predictions or error bars on
parameters. These algorithms critically slow down as ? 0, and in practice draw
samples from a broader distribution than the posterior. We propose a new approach
to likelihood-free inference based on Bayesian conditional density estimation.
Preliminary inferences based on limited simulation data are used to guide later
simulations. In some cases, learning an accurate parametric representation of the
entire true posterior distribution requires fewer model simulations than Monte Carlo
ABC methods need to produce a single sample from an approximate posterior.
1
Introduction
A simulator-based model is a data-generating process described by a computer program, usually
with some free parameters we need to learn from data. Simulator-based modelling lends itself
naturally to scientific domains such as evolutionary biology [1], ecology [24], disease epidemics [10],
economics [8] and cosmology [23], where observations are best understood as products of underlying
physical processes. Inference in these models amounts to discovering plausible parameter settings
that could have generated our observed data. The application domains mentioned can require properly
calibrated distributions that express uncertainty over plausible parameters, rather than just point
estimates, in order to reach scientific conclusions or make decisions.
As an analytical expression for the likelihood of parameters given observations is typically not available for simulator-based models, conventional likelihood-based Bayesian inference is not applicable.
An alternative family of algorithms for likelihood-free inference has been developed, referred to as
Approximate Bayesian Computation (ABC). These algorithms simulate the model repeatedly and
only accept parameter settings which generate synthetic data similar to the observed data, typically
gathered in a real-world experiment.
Rejection ABC [21], the most basic ABC algorithm, simulates the model for each setting of proposed
parameters, and rejects parameters if the generated data is not within a certain distance from the
observations. The accepted parameters form a set of independent samples from an approximate
posterior. Markov Chain Monte Carlo ABC (MCMC-ABC) [13] is an improvement over rejection
ABC which, instead of independently proposing parameters, explores the parameter space by perturbing the most recently accepted parameters. Sequential Monte Carlo ABC (SMC-ABC) [2, 5] uses
importance sampling to simulate a sequence of slowly-changing distributions, the last of which is an
approximation to the parameter posterior.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Conventional ABC algorithms such as the above suffer from three drawbacks. First, they only
represent the parameter posterior as a set of (possibly weighted or correlated) samples. A samplebased representation easily gives estimates and error bars of individual parameters, and model
predictions. However these computations are noisy, and it is not obvious how to perform some other
computations using samples, such as combining posteriors from two separate analyses. Second, the
parameter samples do not come from the correct Bayesian posterior, but from an approximation
based on assuming a pseudo-observation that the data is within an -ball centred on the data actually
observed. Third, as the -tolerance is reduced, it can become impractical to simulate the model
enough times to match the observed data even once. When simulations are expensive to perform,
good quality inference becomes impractical.
We propose a parametric approach to likelihood-free inference, which unlike conventional ABC does
not suffer from the above three issues. Instead of returning samples from an -approximation to
the posterior, our approach learns a parametric approximation to the exact posterior, which can be
made as accurate as required. Preliminary fits to the posterior are used to guide future simulations,
which can reduce the number of simulations required to learn an accurate approximation by orders
of magnitude. Our approach uses conditional density estimation with Bayesian neural networks,
and draws upon advances in parametric density estimation, stochastic variational inference, and
recognition networks, as discussed in the related work section.
2
2.1
Bayesian conditional density estimation for likelihood-free inference
Simulator-based models and ABC
Let ? be a vector of parameters controlling a simulator-based model, and let x be a data vector
generated by the model. The model may be provided as a probabilistic program that can be easily
simulated, and implicitly defines a likelihood p(x | ?), which we assume we cannot evaluate. Let
p(?) encode our prior beliefs about the parameters. Given an observation xo , we are interested in the
parameter posterior p(? | x = xo ) ? p(x = xo | ?) p(?).
As the likelihood p(x = xo | ?) is unavailable, conventional Bayesian inference cannot be carried out.
The principle behind ABC is to approximate p(x = xo | ?) by p(kx ? xo k < | ?) for a sufficiently
small value of , and then estimate the latter?e.g. by Monte Carlo?using simulations from the
model. Hence, ABC approximates the posterior by p(? | kx ? xo k < ), which is typically broader
and more uncertain. ABC can trade off computation for accuracy by decreasing , which improves the
approximation to the posterior but requires more simulations from the model. However, the approximation becomes exact only when ? 0, in which case simulations never match the observations,
p(kx ? xo k < | ?) ? 0, and existing methods break down. In this paper, we refer to p(? | x = xo )
as the exact posterior, as it corresponds to setting = 0 in p(? | kx ? xo k < ).
In most practical applications of ABC, x is taken to be a fixed-length vector of summary statistics
that is calculated from data generated by the simulator, rather than the raw data itself. Extracting
statistics is often necessary in practice, to reduce the dimensionality of the data and maintain
p(kx ? xo k < | ?) to practically acceptable levels. For the purposes of this paper, we will make no
distinction between raw data and summary statistics, and we will regard the calculation of summary
statistics as part of the data generating process.
2.2
Learning the posterior
Rather than using simulations from the model in order to estimate an approximate likelihood,
p(kx ? xo k < | ?), we will use the simulations to directly estimate p(? | x = xo ). We will run
simulations for parameters drawn from a distribution, p?(?), which we shall refer to as the proposal
prior. The proposition below indicates how we can then form a consistent estimate of the exact
posterior, using a flexible family of conditional densities, q? (? | x), parameterized by a vector ?.
Proposition 1. We assume that each of a set of N pairs (?n , xn ) was independently generated by
?n ? p?(?) and xn ? p(x | ?n ).
(1)
Q
In the limit N ? ?, the probability of the parameter vectors n q? (?n | xn ) is maximized w.r.t. ?
if and only if
p?(?)
q? (? | x) ?
p(? | x),
(2)
p(?)
2
provided a setting of ? that makes q? (? | x) proportional to
p(?)
?
p(?)
p(? | x) exists.
Intuition: if we simulated enough parameters from the prior, the density estimator q? would learn a
conditional of the joint prior model over parameters and data, which is the posterior p(? | x). If we
simulate parameters drawn from another distribution, we need to ?importance reweight? the result. A
more detailed proof can be found in Section A of the supplementary material.
The proposition above suggests the following procedure for learning the posterior: (a) propose a
set of parameter vectors {?n } from the proposal prior; (b) for each ?n run the simulator to obtain a
corresponding data vector xn ; (c) train q? with maximum likelihood on {?n , xn }; and (d) estimate
the posterior by
p(?)
p?(? | x = xo ) ?
q? (? | xo ).
(3)
p?(?)
This procedure is summarized in Algorithm 2.
2.3
Choice of conditional density estimator and proposal prior
In choosing the types of density estimator q? (? | x) and proposal prior p?(?), we need to meet the
following criteria: (a) q? should be flexible enough to represent the posterior but easy to train with
maximum likelihood; (b) p?(?) should be easy to evaluate and sample from; and (c) the right-hand
side expression in Equation (3) should be easily evaluated and normalized.
We draw upon work on conditional neural density estimation and take q? to be a Mixture Density
Network (MDN) [3] with fully parameterized covariance
matrices. That is, q? takes the form of a
P
mixture of K Gaussian components q? (? | x) = k ?k N (? | mk , Sk ), whose mixing coefficients
{?k }, means {mk } and covariance matrices {Sk } are computed by a feedforward neural network
parameterized by ?, taking x as input. Such an architecture is capable of representing any conditional
distribution arbitrarily accurately?provided the number of components K and number of hidden
units in the neural network are sufficiently large?while remaining trainable by backpropagation.
The parameterization of the MDN is detailed in Section B of the supplementary material.
We take the proposal prior to be a single Gaussian p?(?) = N (? | m0 , S0 ), with mean m0 and full
covariance matrix S0 . Assuming the prior p(?) is a simple distribution (uniform or Gaussian, as is
typically the case in practice), then this choice allows us to calculate p?(? | x = xo ) in Equation (3)
analytically. That is, p?(? | x = xo ) will be a mixture of K Gaussians, whose parameters will be a
function of {?k , mk , Sk } evaluated at xo (as detailed in Section C of the supplementary material).
2.4
Learning the proposal prior
Simple rejection ABC is inefficient because the posterior p(? | x = xo ) is typically much narrower
than the prior p(?). A parameter vector ? sampled from p(?) will rarely be plausible under
p(? | x = xo ) and will most likely be rejected. Practical ABC algorithms attempt to reduce the
number of rejections by modifying the way they propose parameters; for instance, MCMC-ABC and
SMC-ABC propose new parameters by perturbing parameters they already consider plausible, in the
hope that nearby parameters remain plausible.
In our framework, the key to efficient use of simulations lies in the choice of proposal prior. If we
take p?(?) to be the actual prior, then q? (? | x) will learn the posterior for all x, as can be seen from
Equation (2). Such a strategy however is grossly inefficient if we are only interested in the posterior
for x = xo . Conversely, if p?(?) closely matches p(? | x = xo ), then most simulations will produce
samples that are highly informative in learning q? (? | x) for x = xo . In other words, if we already
knew the true posterior, we could use it to construct an efficient proposal prior for learning it.
We exploit this idea to set up a fixed-point system. Our strategy becomes to learn an efficient proposal
prior that closely approximates the posterior as follows: (a) initially take p?(?) to be the prior p(?);
(b) propose N samples {?n } from p?(?) and corresponding samples {xn } from the simulator, and
train q? (? | x) on them; (c) approximate the posterior using Equation (3) and set p?(?) to it; (d) repeat
until p?(?) has converged. This procedure is summarized in Algorithm 1.
In the procedure above, as long as q? (? | x) has only one Gaussian component (K = 1) then p?(?)
remains a single Gaussian throughout. Moreover, in each iteration we initialize q? with the density
3
Algorithm 2: Training of posterior
initialize q? (? | x) with K components
// if q? available by Algorithm 1
// initialize by replicating its
// one component K times
for n = 1..N do
sample ?n ? p?(?)
sample xn ? p(x | ?n )
end
train q? (? | x) on {?n , xn }
p(?)
q? (? | xo )
p?(? | x = xo ) ? p(?)
?
Algorithm 1: Training of proposal prior
initialize q? (? | x) with one component
p?(?) ? p(?)
repeat
for n = 1..N do
sample ?n ? p?(?)
sample xn ? p(x | ?n )
end
retrain q? (? | x) on {?n , xn }
p?(?) ? p(?)
q? (? | xo )
p(?)
?
until p?(?) has converged;
estimator learnt in the iteration before, thus we keep training q? throughout. This initialization allows
us to use a small sample size N in each iteration, thus making efficient use of simulations.
As we shall demonstrate in Section 3, the procedure above learns Gaussian approximations to the true
posterior fast: in our experiments typically 4?6 iterations of 200?500 samples each were sufficient.
This Gaussian approximation can be used as a rough but cheap approximation to the true posterior,
or it can serve as a good proposal prior in Algorithm 2 for efficiently fine-tuning a non-Gaussian
multi-component posterior. If the second strategy is adopted, then we can reuse the single-component
neural density estimator learnt in Algorithm 1 to initialize q? in Algorithm 2. The weights in the final
layer of the MDN are replicated K times, with small random perturbations to break symmetry.
2.5
Use of Bayesian neural density estimators
To make Algorithm 1 as efficient as possible, the number of simulations per iteration N should
be kept small, while at the same time it should provide a sufficient training signal for q? . With
a conventional MDN, if N is made too small, there is a danger of overfitting, especially in early
iterations, leading to over-confident proposal priors and an unstable procedure. Early stopping could
be used to avoid overfitting; however a significant fraction of the N samples would have to be used
as a validation set, leading to inefficient use of simulations.
As a better alternative, we developed a Bayesian version of the MDN using Stochastic Variational
Inference (SVI) for neural networks [12]. We shall refer to this Bayesian version of the MDN as
MDN-SVI. An MDN-SVI has two sets of adjustable parameters of the same size, the means ?m and
the log variances ?s . The means correspond to the parameters ? of a conventional MDN. During
training, Gaussian noise of variance exp ?s is added to the means independently for each training
example (?n , xn ). The Bayesian interpretation of this procedure is that it optimizes a variational
Gaussian posterior with a diagonal covariance matrix over parameters ?. At prediction time, the noise
is switched off and the MDN-SVI behaves like a conventional MDN with ? = ?m . Section D of the
supplementary material details the implementation and training of MDN-SVI. We found that using
an MDN-SVI instead of an MDN improves the robustness and efficiency of Algorithm 1 because
(a) MDN-SVI is resistant to overfitting, allowing us to use a smaller number of simulations N ; (b) no
validation set is needed, so all samples can be used for training; and (c) since overfitting is not an
issue, no careful tuning of training time is necessary.
3
Experiments
We showcase three versions of our approach: (a) learning the posterior with Algorithm 2 where q?
is a conventional MDN and the proposal prior p?(?) is taken to be the actual prior p(?), which we
refer to as MDN with prior; (b) training a proposal prior with Algorithm 1 where q? is an MDN-SVI,
which we refer to as proposal prior; and (c) learning the posterior with Algorithm 2 where q? is an
MDN-SVI and the proposal prior p?(?) is taken to be the one learnt in (b), which we refer to as MDN
with proposal. All MDNs were trained using Adam [11] with its default parameters.
We compare to three ABC baselines: (a) rejection ABC [21], where parameters are proposed from the
prior and are accepted if kx ? xo k < ; (b) MCMC-ABC [13] with a spherical Gaussian proposal,
whose variance we manually tuned separately in each case for best performance; and (c) SMC4
Proposal prior
MDN with proposal
10
?
5
Mean
75% of mass
99% of mass
xo = 0
10
5
0
?
True posterior
MDN with prior
?5
?2
?1
0
?
1
2
3
?15
0
?5
?10
?3
Mean
75% of mass
99% of mass
xo = 0
?10
?10
?5
0
x
5
10
15
?15
?10
?5
0
x
5
10
15
Figure 1: Results on mixture of two Gaussians. Left: approximate posteriors learnt by each strategy
for xo = 0. Middle: full conditional density q? (?|x) leant by the MDN trained with prior. Right:
full conditional density q? (?|x) learnt by the MDN-SVI trained with proposal prior. Vertical dashed
lines show the location of the observation xo = 0.
ABC [2], where the sequence of ?s was exponentially decayed, with a decay rate manually tuned
separately in each case for best performance. MCMC-ABC was given the unrealistic advantage of
being initialized with a sample from rejection ABC, removing the need for an otherwise necessary
burn-in period. Code for reproducing the experiments is provided in the supplementary material and
at https://github.com/gpapamak/epsilon_free_inference.
3.1
Mixture of two Gaussians
The first experiment is a toy problem where the goal is to infer the common mean ? of a mixture of
two 1D Gaussians, given a single datapoint xo . The setup is
p(?) = U(? | ?? , ?? ) and p(x | ?) = ? N x | ?, ?12 + (1 ? ?) N x | ?, ?22 ,
(4)
where ?? = ?10, ?? = 10, ? = 0.5, ?1 = 1, ?2 = 0.1 and xo = 0. The posterior can be calculated
analytically, and is proportional to an equal mixture of two Gaussians centred at xo with variances ?12
and ?22 , restricted to [?? , ?? ]. This problem is often used in the SMC-ABC literature to illustrate the
difficulty of MCMC-ABC in representing long tails. Here we use it to demonstrate the correctness of
our approach and its ability to accurately represent non-Gaussian long-tailed posteriors.
Figure 1 shows the results of neural density estimation using each strategy. All MDNs have one
hidden layer with 20 tanh units and 2 Gaussian components, except for the proposal prior MDN
which has a single component. Both MDN with prior and MDN with proposal learn good parametric
approximations to the true posterior, and the proposal prior is a good Gaussian approximation to it.
We used 10K simulations to train the MDN with prior, whereas the prior proposal took 4 iterations
of 200 simulations each to train, and the MDN with proposal took 1000 simulations on top of the
previous 800. The MDN with prior learns the posterior distributions for a large range of possible
observations x (middle plot of Figure 1), whereas the MDN with proposal gives accurate posterior
probabilities only near the value actually observed (right plot of Figure 1).
3.2
Bayesian linear regression
In Bayesian linear regression, the goal is to infer the parameters ? of a linear map from noisy
observations of outputs at known inputs. The setup is
Q
p(?) = N (? | m, S) and p(x | ?) = i N xi | ? T ui , ? 2 ,
(5)
where we took m = 0, S = I, ? = 0.1, randomly generated inputs {ui } from a standard Gaussian,
and randomly generated observations xo from the model. In our setup, ? and x have 6 and 10
dimensions respectively. The posterior is analytically tractable, and is a single Gaussian.
All MDNs have one hidden layer of 50 tanh units and one Gaussian component. ABC methods were
run for a sequence of decreasing ?s, up to their failing points. To measure the approximation quality
to the posterior, we analytically calculated the KL divergence from the true posterior to the learnt
posterior (which for ABC was taken to be a Gaussian fit to the set of returned posterior samples).
The left of Figure 2 shows the approximation quality vs ; MDN methods are shown as horizontal
5
KL divergence
10
2
MDN with prior
Proposal prior
MDN with prop.
101
100
10?1
10?2 ?1
10
10
0
10
1
# simulations (per effective sample for ABC)
Rej. ABC
MCMC-ABC
SMC-ABC
103
107
True posterior
MDN with prior
Proposal prior
Rej. ABC
MCMC-ABC
SMC-ABC
MDN with prior
Proposal prior
MDN with prop.
106
105
104
MDN with prop.
MCMC-ABC
SMC-ABC
103
102
101
100 ?2
10
10?1
100
101
102
KL divergence
103
104
?0.20 ?0.15 ?0.10 ?0.05
?1
0.00
0.05
Figure 2: Results on Bayesian linear regression. Left: KL divergence from true posterior to
approximation vs ; lower is better. Middle: number of simulations vs KL divergence; lower left is
better. Note that number of simulations is total for MDNs, and per effective sample for ABC. Right:
Posterior marginals for ?1 as computed by each method. ABC posteriors (represented as histograms)
correspond to the setting of that minimizes the KL in the left plot.
lines. As is decreased, ABC methods sample from an increasingly better approximation to the
true posterior, however they eventually reach their failing point, or take prohibitively long. The best
approximations are achieved by MDN with proposal and a very long run of SMC-ABC.
The middle of Figure 2 shows the increase in number of simulations needed to improve approximation
quality (as decreases). We quote the total number of simulations for MDN training, and the number
of simulations per effective sample for ABC. Section E of the supplementary material describes how
the number of effective samples is calculated. The number of simulations per effective sample should
be multiplied by the number of effective samples needed in practice. Moreover, SMC-ABC will not
work well with only one particle, so many times the quoted cost will always be needed. Here, MDNs
make more efficient use of simulations than Monte Carlo ABC methods. Sequentially fitting a prior
proposal was more than ten times cheaper than training with prior samples, and more accurate.
3.3
Lotka?Volterra predator-prey population model
The Lotka?Volterra model is a stochastic Markov jump process that describes the continuous time
evolution of a population of predators interacting with a population of prey. There are four possible
reactions: (a) a predator being born, (b) a predator dying, (c) a prey being born, and (d) a prey being
eaten by a predator. Positive parameters ? = (?1 , ?2 , ?3 , ?4 ) control the rate of each reaction. Given
a set of statistics xo calculated from an observed population time series, the objective is to infer ?.
We used a flat prior over log ?, and calculated a set of 9 statistics x. The full setup is detailed in
Section F of the supplementary material. The Lotka?Volterra model is commonly used in the ABC
literature as a realistic model which can be simulated, but whose likelihood is intractable. One of
the properties of Lotka?Volterra is that typical nature-like observations only occur for very specific
parameter settings, resulting in narrow, Gaussian-like posteriors that are hard to recover.
The MDN trained with prior has two hidden layers of 50 tanh units each, whereas the MDN-SVI
used to train the proposal prior and the MDN-SVI trained with proposal have one hidden layer of 50
tanh units. All three have one Gaussian component. We found that using more than one components
made no difference to the results; in all cases the MDNs chose to use only one component and switch
the rest off, which is consistent with our observation about the near-Gaussianity of the posterior.
We measure how well each method retrieves the true parameter values that were used to generate
xo by calculating their log probability under each learnt posterior; for ABC a Gaussian fit to the
posterior samples was used. The left panel of Figure 3 shows how this log probability varies with ,
demonstrating the superiority of MDN methods over ABC. In the middle panel we can see that MDN
training with proposal makes efficient use of simulations compared to training with prior and ABC;
note that for ABC the number of simulations is only for one effective sample. In the right panel, we
can see that the estimates returned by MDN methods are more confident around the true parameters
compared to ABC, because the MDNs learn the exact posterior rather than an inflated version of it
like ABC does (plots for the other three parameters look similar).
We found that when training an MDN with a well-tuned proposal that focuses on the plausible region,
an MDN with fewer parameters is needed compared to training with the prior. This is because the
6
10
MDN with prior
Proposal prior
MDN with prop.
5
0
?5
10
?2
10
?1
10
0
10
1
106
Rej. ABC
MCMC-ABC
SMC-ABC
MDN with prior
Proposal prior
MDN with prop.
105
104
103
?3.8
?4.2
?4.4
2
?4.6
101
?4.8
10
100
?5
0
5
10
15
Neg. log probability of true parameters
True value
?4.0
log ?1
Rej. ABC
MCMC-ABC
SMC-ABC
# simulations (per effective sample for ABC)
Neg. log probability of true parameters
15
?5.0
Rej. MCMC SMC
ABC ABC ABC
MDN
prior
Prop.
prior
MDN
prop.
Figure 3: Results on Lotka?Volterra. Left: negative log probability of true parameters vs ; lower
is better. Middle: number of simulations vs negative log probability; lower left is better. Note that
number of simulations is total for MDNs, but per effective sample for ABC. Right: Estimates of log ?1
with 2 standard deviations. ABC estimates used many more simulations with the smallest feasible .
MDN trained with proposal needs to learn only the local relationship between x and ? near xo , as
opposed to in the entire domain of the prior. Hence, not only are savings achieved in number of
simulations, but also training the MDN itself becomes more efficient.
3.4
M/G/1 queue model
The M/G/1 queue model describes the processing of a queue of continuously arriving jobs by a
single server. In this model, the time the server takes to process each job is independently and
uniformly distributed in the interval [?1 , ?2 ]. The time interval between arrival of two consecutive
jobs is independently and exponentially distributed with rate ?3 . The server observes only the time
intervals between departure of two consecutive jobs. Given a set of equally-spaced percentiles xo of
inter-departure times, the task is to infer parameters ? = (?1 , ?2 , ?3 ). This model is easy to simulate
but its likelihood is intractable, and it has often been used as an ABC benchmark [4, 16]. Unlike
Lotka?Volterra, data x is weakly informative about ?, and hence the posterior over ? tends to be
broad and non-Gaussian. In our setup, we placed flat independent priors over ?1 , ?2 ? ?1 and ?3 , and
we took x to be 5 equally spaced percentiles, as detailed in Section G of the supplementary material.
The MDN trained with prior has two hidden layers of 50 tanh units each, whereas the MDN-SVI
used to train the proposal prior and the one trained with proposal have one hidden layer of 50 tanh
units. As observed in the Lotka?Volterra demo, less capacity is required when training with proposal,
as the relationship to be learned is local and hence simpler, which saves compute time and gives a
more accurate final posterior. All MDNs have 8 Gaussian components (except the MDN-SVI used
to train the proposal prior, which always has one), which, after experimentation, we determined are
enough for the MDNs to represent the non-Gaussian nature of the posterior.
Figure 4 reports the log probability of the true parameters under each posterior learnt?for ABC,
the log probability was calculated by fitting a mixture of 8 Gaussians to posterior samples using
Expectation-Maximization?and the number of simulations needed to achieve it. As before, MDN
methods are more confident compared to ABC around the true parameters, which is due to ABC computing a broader posterior than the true one. MDN methods make more efficient use of simulations,
since they use all of them for training and, unlike ABC, do not throw a proportion of them away.
4
Related work
Regression adjustment. An early parametric approach to ABC is regression adjustment, where a
parametric regressor is trained on simulation data in order to learn a mapping from x to ?. The
learnt mapping is then used to correct for using a large , by adjusting the location of posterior
samples gathered by e.g. rejection ABC. Beaumont et al. [1] used linear regressors, and later Blum
and Fran?ois [4] used neural networks with one hidden layer that separately predicted the mean
and variance of ?. Both can be viewed as rudimentary density estimators and as such they are a
predecessor to our work. However, they were not flexible enough to accurately estimate the posterior,
and they were only used within some other ABC method to allow for a larger . In our work, we
make conditional density estimation flexible enough to approximate the posterior accurately.
7
2
MDN with prior
Proposal prior
MDN with prop.
1
0
?1
?2
?3
10
?4
10
?3
10
?2
10
?1
108
106
105
10
12
Rej. ABC
MCMC-ABC
SMC-ABC
MDN with prior
Proposal prior
MDN with prop.
107
4
True value
10
8
?2
Rej. ABC
MCMC-ABC
SMC-ABC
3
# simulations (per effective sample for ABC)
Neg. log probability of true parameters
4
103
6
4
102
10
2
1
100
?3
?2
?1
0
1
2
Neg. log probability of true parameters
3
0
Rej. MCMC SMC
ABC ABC ABC
MDN
prior
Prop.
prior
MDN
prop.
Figure 4: Results on M/G/1. Left: negative log probability of true parameters vs ; lower is better.
Middle: number of simulations vs negative log probability; lower left is better. Note that number
of simulations is total for MDNs, and per effective sample for ABC. Right: Estimates of ?2 with 2
standard deviations; ABC estimates correspond to the lowest setting of used.
Synthetic likelihood. Another parametric approach is synthetic likelihood, where parametric models
are used to estimate the likelihood p(x | ?). Wood [24] used a single Gaussian, and later Fan et al.
[7] used a mixture Gaussian model. Both of them learnt a separate density model of x for each ? by
repeatedly simulating the model for fixed ?. More recently, Meeds and Welling [14] used a Gaussian
process model to interpolate Gaussian likelihood approximations between different ??s. Compared
to learning the posterior, synthetic likelihood has the advantage of not depending on the choice of
proposal prior. Its main disadvantage is the need of further approximate inference on top of it in order
to obtain the posterior. In our work we directly learn the posterior, eliminating the need for further
inference, and we address the problem of correcting for the proposal prior.
Efficient Monte Carlo ABC. Recent work on ABC has focused on reducing the simulation cost of
sample-based ABC methods. Hamiltonian ABC [15] improves upon MCMC-ABC by using stochastically estimated gradients in order to explore the parameter space more efficiently. Optimization Monte
Carlo ABC [16] explicitly optimizes the location of ABC samples, which greatly reduces rejection
rate. Bayesian optimization ABC [10] models p(kx ? xo k | ?) as a Gaussian process and then uses
Bayesian optimization to guide simulations towards the region of small distances kx ? xo k. In our
work we show how a significant reduction in simulation cost can also be achieved with parametric
methods, which target the posterior directly.
Recognition networks. Our use of neural density estimators for learning posteriors is reminiscent of
recognition networks in machine learning. A recognition network is a neural network that is trained
to invert a generative model. The Helmholtz machine [6], the variational auto-encoder [12] and
stochastic backpropagation [22] are examples where a recognition network is trained jointly with the
generative network it is designed to invert. Feedforward neural networks have been used to invert
black-box generative models [18] and binary-valued Bayesian networks [17], and convolutional
neural networks have been used to invert a physics engine [25]. Our work illustrates the potential of
recognition networks in the field of likelihood-free inference, where the generative model is fixed,
and inference of its parameters is the goal.
Learning proposals. Neural density estimators have been employed in learning proposal distributions
for importance sampling [20] and Sequential Monte Carlo [9, 19]. Although not our focus here, our
fit to the posterior could also be used within Monte Carlo inference methods. In this work we see
how far we can get purely by fitting a series of conditional density estimators.
5
Conclusions
Bayesian conditional density estimation improves likelihood-free inference in three main ways: (a) it
represents the posterior parametrically, as opposed to as a set of samples, allowing for probabilistic
evaluations later on in the pipeline; (b) it targets the exact posterior, rather than an -approximation
of it; and (c) it makes efficient use of simulations by not rejecting samples, by interpolating between
samples, and by gradually focusing on the plausible parameter region. Our belief is that neural density
estimation is a tool with great potential in likelihood-free inference, and our hope is that this work
helps in establishing its usefulness in the field.
8
Acknowledgments
We thank Amos Storkey for useful comments. George Papamakarios is supported by the Centre for
Doctoral Training in Data Science, funded by EPSRC (grant EP/L016427/1) and the University of
Edinburgh, and by Microsoft Research through its PhD Scholarship Programme.
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9
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5,619 | 6,085 | Bi-Objective Online Matching and Submodular
Allocations
Hossein Esfandiari
University of Maryland
College Park, MD 20740
hossein@cs.umd.edu
Nitish Korula
Google Research
New York, NY 10011
nitish@google.com
Vahab Mirrokni
Google Research
New York, NY 10011
mirrokni@google.com
Abstract
Online allocation problems have been widely studied due to their numerous practical applications (particularly to Internet advertising), as well as considerable
theoretical interest. The main challenge in such problems is making assignment
decisions in the face of uncertainty about future input; effective algorithms need to
predict which constraints are most likely to bind, and learn the balance between
short-term gain and the value of long-term resource availability.
In many important applications, the algorithm designer is faced with multiple
objectives to optimize. In particular, in online advertising it is fairly common to
optimize multiple metrics, such as clicks, conversions, and impressions, as well
as other metrics which may be largely uncorrelated such as ?share of voice?, and
?buyer surplus?. While there has been considerable work on multi-objective offline
optimization (when the entire input is known in advance), very little is known
about the online case, particularly in the case of adversarial input. In this paper,
we give the first results for bi-objective online submodular optimization, providing
almost matching upper and lower bounds for allocating items to agents with two
submodular value functions. We also study practically relevant special cases of
this problem related to Internet advertising, and obtain improved results. All our
algorithms are nearly best possible, as well as being efficient and easy to implement
in practice.
1
Introduction
As a central optimization problem with a wide variety of applications, online resource allocation
problems have attracted a large body of research in networking, distributed computing, and electronic
commerce. Here, items arrive one at a time (i.e. online), and when each item arrives, the algorithm
must irrevocably assign it to an agent; each agent has a limited resource budget / capacity for items
assigned to him. A big challenge in developing good algorithms for these problems is to predict future
binding constraints or learn future capacity availability, and allocate items one by one to agents who
are unlikely to hit their capacity in the future. Various stochastic and adversarial models have been
proposed to study such online allocation problems, and many techniques have been developed for
these problems. For stochastic input, a natural approach is to build a predicted instance (for instance,
via sampling, or using historical data), and some of these techniques solve a dual linear program
to learn dual variables that are used by the online algorithm moving forward [6, 10, 2, 23, 16, 18].
However, stochastic approaches may provide poor results on some input (for example, when there
are unexpected spikes in supply / demand), and hence such problems have been extensively studied
in adversarial models as well. Here, the algorithm typically maintains a careful balance between
greedily exploiting the current item by assigning it to agents with high value for it, and assigning the
item to a lower-value agent for whom the value is further from the distribution of ?typical? items they
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
have received. Again, primal-dual techniques have been applied to learn the dual variables used by
the algorithm in an online manner [17, 3, 9].
A central practical application of such online algorithms is the online allocation of impressions or
page-views to ads on the Internet [9, 2, 23, 5, 7]. Such problems are present both in the context of
sponsored search advertising where advertisers have global budget constraints [17, 6, 3], or in display
advertising where each ad campaign has a desired goal or a delivery constraint [9, 10, 2, 23, 5, 7].
Many of these online optimization techniques apply to general optimization problems including the
online submodular welfare maximization problem (SWM) [20, 13].
For many real-world optimization problems, the goal is to optimize multiple objective functions [14,
1]. For instance, in Internet advertising, such objectives might include revenue, clicks, or conversions.
A variety of techniques have been developed for multi-objective optimization problems; however, in
most cases, these techniques are only applicable for offline multi-objective optimization problems [21,
26], and they do not apply to online settings, especially for online competitive algorithms that work
against an adversarial input [17, 9] or in the presence of traffic spikes [18, 8] or hard-to-predict traffic
patterns [5, 4, 22].
Our contributions. Motivated by the above applications and the increasing need to satisfy multiple
objectives, we study a wide class of multi-objective online optimization problems, and present both
hardness results and (almost tight) bi-objective approximation algorithms for them. In particular, we
study resource allocation problems in which a sequence of items (also referred to as impressions)
i from an unknown set I arrive one by one, and we have to allocate each item to one agent (for
example, one advertiser) a in a given set of agents A. Each agent a has two monotone submodular
set functions fa , ga : 2I ? R associated with it. Let Sa be the set of items assigned to bin a as
a result of online allocation decisions. The goal of the online
allocation algorithm
is to maximize
P
P
two social welfare functions based on fa ?s and ga , i.e., a?A fa (Sa ) and a?A ga (Sa ). We first
present almost tight online approximation algorithms for the general online bi-objective submodular
welfare maximization problem (see Theorems 2.3 and 2.5, and Fig. 1). We show that a simple random
selection rule along with the greedy algorithm (when each item arrives, randomly pick one objective
to greedily optimize) results in almost optimal algorithms. Our allocation rule is thus both very
fast to run and trivially easy to implement. The main technical result of this part is the hardness
result showing that the achieved approximation factor is almost tight unless P=NP. Furthermore, we
consider special cases of this problem motivated by online ad allocation. In particular, for the special
cases of online budgeted allocation and online weighted matching, motivated by sponsored search
and display advertising (respectively), we present improved primal-dual-based algorithms along with
improved hardness results for these problems (see, for example, the tight Theorem 3.1).
Related Work. It is known that the greedy algorithm leads to a 1/2-approximation for the submodular
social welfare maximization problem (SWM) [11], and this problem admits a 1 ? 1/e-approximation
in the offline setting [24], which is tight [19]. However, for the online setting, the problem does
not admit a better than 1/2-approximation algorithm unless P= NP [12]. Bi-objective online
allocation problems have been studied in two previous papers [14, 1]. The first paper presents [14]
an online bi-objective algorithm for the problem of maximizing a general weight function and the
cardinality function, and the second paper [1] presents results for the combined budgeted allocation
and cardinality constraints. Our results in this paper improve and generalize those results for
more general settings. Submodular partitioning problems have also been studied based on mixed
robust/average-case objectives [25].
Our work is related to online ad allocation problems, including the Display Ads Allocation (DA)
problem [9, 10, 2, 23], and the Budgeted Allocation (AdWords) problem [17, 6]. In both of these
problems, the publisher must assign online impressions to an inventory of ads, optimizing efficiency
or revenue of the allocation while respecting pre-specified contracts. The Display Ad (DA) problem is
the online matching problem described above with a single weight objective [9, 7]. In the Budgeted
Allocation problem, the publisher allocates impressions resulting from search queries. Advertiser
a has a budget B(a) on the total spend, instead of a bound n(a) on the number of impressions.
Assigning impression i to advertiser a consumes wia units of a?s budget instead of 1 of the n(a)
slots, as in the DA problem. For both of these problems, 1 ? 1e -approximation algorithms have been
designed under the assumption of large capacities [17, 3, 9]. None of the above papers for adversarial
models studies multiple objectives at the same time.
2
2
2.1
Bi-Objective Online Submodular Welfare Maximization
Model and Overview
For any allocation S, let Sa denote the set of items assigned to agent a ? A by this allocation. In the
classic Submodular Welfare Maximization problem (SWM) for which there is a single monotone
submodular objective, each agent a ? A is associated withP
a submodular function fa defined on
the set of items I. The welfare of allocation S is defined as a fa (Sa ), and the goal of SWM is to
maximize this welfare. In the classic SWM, the natural greedy algorithm is to assign each item (when
it arrives) to the agent whose gain increases the most. This greedy algorithm (note that it is an online
algorithm) is (1/2 + 1/n)-competitive, and this is the best possible [15].
In this section, we consider the extension of online SWM to two monotone submodular functions.
Formally, each agent a ? A is associated with two submodular functions
a - defined on I.
P - fa and gP
The goal is to find an allocation S that does well on both objectives a fa (Sa ) and a ga (Sa ). We
measure the performance of the
for each objective:
Palgorithm by comparison to the offline optimum
P
Let S ?f = arg maxallocations S a fa (Sa ) and S ?g = arg maxallocations S a ga (Sa ). P
An algorithm
AP
is (?, ?)-competitive
if,
for
every
input,
it
produces
an
allocation
S
such
that
a fa (Sa ) ?
P
P
?f
?g
? a fa (Sa ) and a ga (Sa ) ? ? a ga (Sa ).
A (1, 1)-competitive algorithm would be one that finds an allocation which is simultaneously optimal
in both objectives, but since the objectives are distinct, no single allocation may maximize both,
even ignoring computational difficulties or lack of knowledge of the future. One could attempt to
maximize a linear combination of the two submodular objectives, but since the linear combination is
itself submodular, this is no harder than the classic online SWM. Instead, we provide algorithms with
the stronger guarantee that they are simultaneously competitive with the optimal solution for each
objective separately. Further, our algorithms are parametrized, so the user can balance the importance
of the two objectives.
Similar to previous approaches for bi-objective online allocation problems [14], we run two simultaneous greedy algorithms, each based on one of the objective functions. Upon arrival of each online
item, with probability p we pass the item to the greedy algorithm based on the objective function f ,
and with probability 1 ? p we pass the item to the greedy algorithm based on g.
First, as a warmup, we provide a charging argument to show that the greedy algorithm for (singleobjective) SWM is 1/2-competitive. This charging argument is similar to the usual primal-dual
analysis for allocation problems. However, since the objective functions are not linear, it may not be
possible to interpret the proof using a primal-dual technique. Later, we modify our charging argument
and show that if we run the greedy algorithm for SWM but only consider items for allocation with
p
probability p, the competitive ratio is 1+p
. (Note that a naive analysis would yield a competitive ratio
of p/2, since we lose a factor of p in the sampling and a factor of 1/2 due to the greedy algorithm.)
Since our algorithm for bi-objective online SWM passes items to the ?first? greedy algorithm with
probability p and passes items to the second greedy algorithm with probability 1 ? p, the modified
p
charging argument immediately implies that our algorithm is ( 1+p
, 1?p
2?p ) competitive, as we state in
Theorem 2.3 below. Also, using a factor-revealing framework, assuming N P 6= RP , we provide an
almost tight hardness result, which holds even if the objective functions have the simpler ?coverage?
structure. Both our competitive ratio and the associated hardness result are presented in Figure 1.
2.2
Algorithm for Bi-Objective online SWM
We define some notation and ideas that we use to bound the competitive ratio of our algorithm. Let Gr
be the greedy algorithm and let Opt be a fixed optimum allocation. For an agent j, and an algorithm
Alg, let Algj be the set of online items allocated to the agent j by Alg; Optj denotes the set of online
items allocated to j in Opt. Trivially, for any two agents j and k, we have Algj ? Algk = ?.
For each online item i we define a variable ?i , and for each agent j we define a variable ?j . In order
to bound the competitive ratio of the algorithm
c, it suffices
P Alg byP
to set the values of ?i s and ?j s
n
m
such that 1) the value of Alg is at least c
i=1 ?i +
j=1 ?j and 2) the value of Opt is at most
Pn
Pm
i=1 ?i +
j=1 ?j .
3
Figure 1: The lower (blue) curve is the competitive ratio of our algorithm, and the red curve is the
upper bound on the competitive ratio of any algorithm.
Theorem 2.1. (Warmup) The greedy algorithm is 0.5-competitive for online SWM.
Proof. For each online itemPi, let ?i be the marginal gain by Gr from allocating item i upon its
n
arrival. It is easy to see that i=1 ?i is equal to the value of Gr. For each agent j, P
let ?j be the total
m
value of the allocation to j at the end of the algorithm. By definition, we know that j=1 ?j is equal
P
Pm
n
to the value of Gr. Thus, the value of Gr is clearly 0.5
i=1 ?i +
j=1 ?j .
Recall that fj (.) denotes
the valuation function of agent j. Below, we show that fj (Optj ) is upperP
bounded by ?j + i?Optj ?i . Note that for distinct agents j and k, Optj and Optk are disjoint. Thus,
Pn
Pm
by summing over all agents, we can upper-bound the value of Opt by i=1 ?i + j=1 ?j . This
means that the competitive ratio of Gr is 0.5.
P
Now, we just need to show that for any agent j we have fj (Optj ) ? ?j + i?Optj ?i . Note that for
any item i ? Optj , the value of ?i is at least the marginal gain that would have been obtained from
assigning i to j when it arrives. Applying submodularity of fj , we have ?i ? fj (Grj ? i) ? fj (Grj ).
Moreover, by definition we have ?j = fj (Grj ). Thus, we have:
X
X
?j +
?i ? fj (Grj ) +
(fj (Grj ? i) ? fj (Grj ))
i?Optj
i?Optj
? fj (Grj ) + fj (Grj ? Optj ) ? fj (Grj )
= fj (Grj ? Optj ) ? fj (Optj ),
where the second inequality follows by submodularity, and the last inequality by monotonicity. This
completes the proof.
Lemma 2.2. Let Grp be an algorithm that with probability p passes each online item to Gr for
p
-competitive for online SWM.
allocation, and leaves it unmatched otherwise. Grp is 1+p
Proof. The proof here is fairly similar to Theorem 2.1. For each online item i, set ?i to be the
marginal gain that would have been achieved from allocating item i upon its arrival (assuming i is
passed to Gr), given the current allocation of items. Note that ?i is a random variable (depending on
the outcome of previous decisions to pass items to Gr or not), but it is independent of the coin toss
that determines whether it is passed to Gr, and so the expected marginal gain of allocating item i,
(given P
all previous allocations) is pE[?i ]. Thus, by linearity of expectation, the expected value of Grp
n
is pE[ i=1 ?i ]. On the other hand, for each agent
Pmj, set ?j to be the value of the actual allocations
to j at the end of the algorithm. Again, we have j=1 ?j equal to the value of Grp . Combining these
P
Pm
n
1
E[?
]
+
E[?
]
two, we conclude that the expected value of Grp is equal to 1+1/p
i
j .
i=1
j=1
4
As before, we show that fj (Optj ) is upper-bounded by ?j +
that the competitive ratio of Grp is
1
1+1/p
=
P
i?Optj
?i . Therefore, we can conclude
p
1+p .
P
It remains only to show that for any agent j, we have fj (Optj ) ? ?j + i?Optj ?i . This is exactly
the same as our proof for Theorem 2.1: By submodularity of fj we have, ?i ? fj (Grp (j) ? i) ?
fj (Grp (j)), and by definition we have ?j = fj (Grp (j)). We provide the complete proof in the full
version.
The main theorem of this section follows immediately.
p
Theorem 2.3. For any 0 < p < 1, there is a ( 1+p
, 1?p
2?p )-competitive algorithm for bi-objective
online SWM.
2.3
Hardness of Bi-Objective online SWM
We now prove that Theorem 2.3 is almost tight, by describing a hard instance for bi-objective online
SWM. To describe this instance, we define notions of super nodes and super edges, which capture
the hardness of maximizing a submodular function even in the offline setting. Using the properties of
super nodes and edges, we construct and analyze a hard example for bi-objective online SWM.
Our construction generalizes that of Kapralov et al. [12], who prove the upper bound corresponding to
the two points (0.5, 0) and (0, 0.5) in the curve shown in Figure 1. They use the following result: For
any fixed c0 and 0 it is NP-hard to distinguish between the following two cases for offline SWM with
n agents and m = kn items. This holds even for submodular functions with ?coverage? valuations.
? There is an allocation with value n.
? For any l ? c0 , no allocation allocates kl items and gets a value more than 1 ? e?l + 0 .
Intuitively, in the former case, we can assign k items to each agent and obtain value 1 per agent. In the
latter case, even if we assign 2k items (however they are split across agents), we can obtain total value
at most 0.865. It also follows that there exist ?hard? instances such that there is an optimal solution
of value n, but for any l < 1, any assigment of ml edges obtains value at most (1 ? e?l + 0 )n.
We now define a super edge to be a hard instance of offline SWM as defined above. We refer to the
set of agents in a super edge as the agent super node, and the set of items in the super edge as the
item super node. If two super edges share a super node, it means that they share the agents / items
corresponding to that super node in the same order. If (in expectation) we allocate ml items of a
super edge, we say the load of that super edge is l. Similarly, if (in expectation) we allocate ml items
to an agent super node, we say the load of that super node is l. Using the definition of super edge and
super node, the hardness result of Kapralov et al. [12] gives us the following lemma:
Lemma 2.4. Assume RP 6= N P and let be an arbitrary small constant. If the (expected) load of a
randomized polynomial algorithm on an agent super node is l, the expected welfare of all agents is at
most (1 ? e?l + )n.
Now with Lemma 2.4 in hand, we are ready to present an upper bound for bi-objective online SWM.
Theorem 2.5. Assume RP 6= N P . The competitive ratio (?, ?) of any algorithm for bi-objective
online SWM is upper bounded by the red curve in Figure 1. More precisely (assuming w.l.o.g. that
2
/6
? ? ?), for any ? ? [0, 1], there is no algorithm with ? > 0.5+?
and ? > ??.
1+? 2
3
Bi-Objective Online Weighted Matching
In this section, we consider two special cases of bi-objective online SWM, each of which generalizes
the (single objective) online weighted matching problem (with free disposal). Here, each item i has
f
g
two weights wij
and wij
for agent j, and each agent j has (large) capacity Cj . The weights of item i
are revealed when it arrives, and the algorithm must allocate it to some agent immediately.
In the first model, after the algorithm terminates, and each agent j has received items Sj , it chooses a
P P
f
subset Sj0 ? Sj of at most Cj items. The total value in the first objective is then j i?S 0 wij
, and
j
P P
g
in the second objective j i?S 0 wij . Intuitively, each agent must pick a subset of its items, and it
j
5
Exponential Weight Algorithm.
Set ?j to 0 for each agent j.
Upon arrival of each item i:
1. If there is agent j with wij ? ?j > 0
(a) Let j be the agent that maximizes wij ? ?j
(b) Assign i to j, and set ?i to wij ? ?j .
(c) Let w1 , w2 , . . . , wCj be the weights of the Cj highest weight items, matched to j in
a non-increasing order.
PCj
(d) Set ?j to
j=1
j?1
wj 1+ C1
j
Cj ((1+1/Cj )Cj ?1)
.
2. Else: Leave i unassigned.
Figure 2: Exponential weight algorithm for online matching with free disposal.
gets paid its (additive) value for these items. In the (single-objective) case where each agent can only
be allocated Cj items, this is the online weighted b-matching problem, where vertices are arriving
online, and we have edge weights in the bipartite (item, agent) graph. This problem is completely
intractable in the online setting, while the free disposal variant [9] in which additional items can be
assigned, but at most Cj items count towards the objective, is of theoretical and practical interest.
In the second model, after the algorithm terminates and agent j has received items Sj , it chooses two
(not necessarily disjoint) subsets Sj0f and Sj0g ; items in Sj0f are counted towards the first objective,
and those in Sj0g are counted towards the second objective.
Theorem 3.1. For any (?, ?) such that ? + ? ? 1 ? 1e , there is an (?, ?)-competitive algorithm for
the first model of the bi-objective online weighted matching. For any constant > 0, there is no such
algorithm when ? + ? > 1 ? 1e + .
To obtain the positive result, with probability p, run the exponential weight algorithm (see Figure 2)
for the first objective (for all items), and with probability 1 ? p run the exponential weight algorithm
for the second objective for all items; this combination is (p(1 ? 1e ), (1 ? p)(1 ? 1e ))-competitive.
We deffer the proof of this and the matching hardness results to the full version.
Having given matching upper and lower bounds for the first model, we now consider the second model,
where if we assign a set Sj of items / edges to an agent j we can select two subsets Sj0f , Sj0g ? Sj
and use them for the first and second objective functions respectively.
1
1
Theorem 3.2. There is a (p(1 ? e1/p
), (1 ? p)(1 ? e1/(1?p)
))-competitive algorithm for the biobjective online weighted matching problem in the second model as minj {Cj } tends to infinity.
Theorem 3.3. The competitive ratio of any algorithm for bi-objective online weighted matching in
the second model is upper bounded by the curve in figure 3.
4
Bi-Objective Online Budgeted Allocation
In this section, we consider the bi-objective online allocation problem where one of the objectives is
a budgeted allocation problem and the other objective function is weighted matching. Here, each
item i has a weight wij and a bid bij for agent j. Each agent j has a capacity Cj and a budget Bj . If
an agent is allocated items SjP
, for the first objective (weighted matching), it choosesPa subset Sj0 of
at most Cj items; its score is i?S 0 wij . For the second objective, its score is min{ i?Sj bij , Bj }.
j
Note that in the second objective, the agent does not need to choose a subset; it obtains the sum of the
bids of all items assigned to it, capped at its budget Bj .
Clearly, if we set all bids bij to 1, the goal of the budgeted allocation part will be maximizing the
cardinality. Thus, this is a clear generalization of the bi-objective online allocation to maximize
weight and cardinality, and the same hardness results hold here.
6
Figure 3: The blue curve is the competitive ratio of our algorithm in the second model, while the red
line and the green curves are the upper bounds on the competitive ratio of any algorithm.
As is standard, throughout this section we assume that the bid of each agent for each item is
vanishingly small compared to the budget of each bidder. Interestingly, again here, we provide a
1
1
(p(1 ? e1/p
), (1 ? p)(1 ? e1/(1?p)
))-competitive algorithm, which is almost tight. At the end, as a
1
1
corollary of our results, we provide a a (p(1 ? e1/p
), (1 ? p)(1 ? e1/(1?p)
))-competitive algorithm,
for the case that both objectives are budgeted allocation problems (with separate budgets).
Our algorithm here is roughly the same as for two weight objectives. For each item, with probability
1 ? p, we pass it to the Exponential Weight algorithm for matching, and allocate it based on its weight.
With the remaining probability p, we assign the algorithm based on its bids and count it towards the
Budgeted Allocation objective. However, the algorithm we use for Budgeted Allocation is slightly
different: We virtually run the Balance algorithm of Mehta et al. [17] for Budgeted Allocation (Fig. 4),
as though we were assigning all items (not just those passed to this algorithm), but with each item?s
bids scaled down by a factor of p. For those p fraction of items to be assigned by the Budgeted
Allocation algorithm, assign them according to the recommendation of the virtual Balance algorithm.
1
Theorem 3.2 from the previous section shows that our algorithm is (1 ? p)(1 ? e1/(1?p)
)-competitive
against the optimum weighted matching objective. Thus, in the rest of this section, we only need
1
to show that this algorithm is p(1 ? e1/p
)-competitive against the optimum Budgeted Allocation
solution. First, using a primal dual approach, we show that the outcome of the virtual Balance
1
algorithm (that runs on p fraction of the value of each item) is p(1 ? e1/p
) against the optimum with
the actual weights. Then, using the Hoeffding inequality, we show that the expected value of our
allocation for the budgeted allocation objective is fairly close to the virtual algorithm?s value, i.e. the
difference between the competitive ratio of our allocation and the virtual allocation is o(1).
max
b
i,j ij
Lemma 4.1. When
? 0, the total allocation of the virtual balance algorithm that runs on
Bj
1
p fraction of the value of each bid is at least p(1 ? e1/p
) times that of the optimum with the actual
values.
The proof of this lemma is similar to the analysis of Buchbinder et al. [3] for the basic Budgeted
Allocation problem. We provide this proof in the full version.
max
b
i,j ij
Lemma 4.2. For any constant p, assuming
? 0, the budgeted allocation value of our
Bj
algorithm tends to the value of Balance with p fraction of each bid, with high probability.
In the virtual Balance algorithm, we allocate p fraction of each item, while in our real algorithm, we
allocate every item according to the virtual Balance algorithm with probability p. Since each item?s
bids are small compared to the budgets, the lemma follows from a straightforward concentration
argument. We present the complete proof in the full version.
The following lemma is an immediate result of combining Lemma 4.1 and Lemma 4.2.
7
Virtual Balance algorithm on p fraction of values.
Set ?j and yj to 0 for each agent j.
Upon arrival of each item i:
1. If i has a neighbor with bij (1 ? ?j ) > 0
(a) Let j be the agent that maximizes bij (1 ? ?j )
(b) Assign i to j i.e. set xij to 1.
(c) Set ?i to bij (1 ? ?j ).
(d) Increase yj by
bij
Bj
(e) Increase ?j by
eyj ?1/p bij
1?e?1/p Bj
2. Else: Leave i unassigned.
Figure 4: Maintaining solution to primal and dual LPs.
max
b
i,j ij
? 0, our algorithm is p(1 ?
Lemma 4.3. For any constant p, assuming
Bj
against the optimum budgeted allocation solution.
1
)-competitive
e1/p
Lemma 4.3 immediately gives us the following theorem.
max
b
i,j ij
1
? 0, there is a (p(1 ? e1/p
Theorem 4.4. For any constant p, assuming
), (1 ? p)(1 ?
Bj
1
))-competitive algorithm for the bi-objective online allocation with two budgeted allocation
e1/(1?p)
objectives.
Moreover, if we pass each item to the exponential weight algorithm with probability p, the expected
1
size of the output matching is at least p(1 ? e1/p
) that of the optimum [14]. Together with Lemma
4.3, this gives us the following theorem.
max
b
i,j ij
1
Theorem 4.5. For any constant p, assuming
? 0, there is a (p(1 ? e1/p
), (1 ? p)(1 ?
Bj
1
))-competitive algorithm for the bi-objective online allocation with a budgeted allocation
e1/(1?p)
objective and a weighted matching objective.
5
Conclusions
In this paper, we gave the first algorithms for several bi-objective online allocation problems. Though
these are nearly tight, it would be interesting to consider other models for bi-objective online
allocation, special cases where one may be able to go beyond our hardness results, and other
objectives such as fairness to agents.
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9
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5,620 | 6,086 | Learning HMMs with Nonparametric Emissions via
Spectral Decompositions of Continuous Matrices
Kirthevasan Kandasamy?
Carnegie Mellon University
Pittsburgh, PA 15213
kandasamy@cs.cmu.edu
Maruan Al-Shedivat?
Carnegie Mellon University
Pittsburgh, PA 15213
alshedivat@cs.cmu.edu
Eric P. Xing
Carnegie Mellon University
Pittsburgh, PA 15213
epxing@cs.cmu.edu
Abstract
Recently, there has been a surge of interest in using spectral methods for estimating
latent variable models. However, it is usually assumed that the distribution of the
observations conditioned on the latent variables is either discrete or belongs to
a parametric family. In this paper, we study the estimation of an m-state hidden
Markov model (HMM) with only smoothness assumptions, such as H?lderian
conditions, on the emission densities. By leveraging some recent advances in
continuous linear algebra and numerical analysis, we develop a computationally
efficient spectral algorithm for learning nonparametric HMMs. Our technique is
based on computing an SVD on nonparametric estimates of density functions by
viewing them as continuous matrices. We derive sample complexity bounds via
concentration results for nonparametric density estimation and novel perturbation
theory results for continuous matrices. We implement our method using Chebyshev
polynomial approximations. Our method is competitive with other baselines on
synthetic and real problems and is also very computationally efficient.
1
Introduction
Hidden Markov models (HMMs) [1] are one of the most popular statistical models for analyzing time
series data in various application domains such as speech recognition, medicine, and meteorology. In
an HMM, a discrete hidden state undergoes Markovian transitions from one of m possible states to
another at each time step. If the hidden state at time t is ht , we observe a random variable xt 2 X
drawn from an emission distribution, Oj = P(xt |ht = j). In its most basic form X is a discrete set
and Oj are discrete distributions. When dealing with continuous observations, it is conventional to
assume that the emissions Oj belong to a parametric class of distributions, such as Gaussian.
Recently, spectral methods for estimating parametric latent variable models have gained immense
popularity as a viable alternative to the Expectation Maximisation (EM) procedure [2?4]. At a high
level, these methods estimate higher order moments from the data and recover the parameters via
a series of matrix operations such as singular value decompositions, matrix multiplications and
pseudo-inverses of the moments. In the case of discrete HMMs [2], these moments correspond
exactly to the joint probabilities of the observations in the sequence.
Assuming parametric forms for the emission densities is often too restrictive since real world
distributions can be arbitrary. Parametric models may introduce incongruous biases that cannot be
reduced even with large datasets. To address this problem, we study nonparametric HMMs only
assuming some mild smoothness conditions on the emission densities. We design a spectral algorithm
for this setting. Our methods leverage some recent advances in continuous linear algebra [5, 6]
which views two-dimensional functions as continuous analogues of matrices. Chebyshev polynomial
approximations enable efficient computation of algebraic operations on these continuous objects [7,
8]. Using these ideas, we extend existing spectral methods for discrete HMMs to the continuous
nonparametric setting. Our main contributions are:
?
Joint lead authors.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
1. We derive a spectral learning algorithm for HMMs with nonparametric emission densities. While
the algorithm is similar to previous spectral methods for estimating models with a finite number
of parameters, many of the ideas used to generalise it to the nonparametric setting are novel, and,
to the best of our knowledge, have not been used before in the machine learning literature.
2. We establish sample complexity bounds for our method. For this, we derive concentration results
for nonparametric density estimation and novel perturbation theory results for the aforementioned
continuous matrices. The perturbation results are new and might be of independent interest.
3. We implement our algorithm by approximating the density estimates via Chebyshev polynomials
which enables efficient computation of many of the continuous matrix operations. Our method outperforms natural competitors in this setting on synthetic and real data and is computationally more
efficient than most of them. Our Matlab code is available at github.com/alshedivat/nphmm.
While we focus on HMMs in this exposition, we believe that the ideas presented in this paper can be
easily generalised to estimating other latent variable models and predictive state representations [9]
with nonparametric observations using approaches developed by Anandkumar et al. [3].
Related Work: Parametric HMMs are usually estimated using maximum likelihood principle via EM
techniques [10] such as the Baum-Welch procedure [11]. However, EM is a local search technique,
and optimization of the likelihood may be difficult. Hence, recent work on spectral methods has
gained appeal. Our work builds on Hsu et al. [2] who showed that discrete HMMs can be learned
efficiently, under certain conditions. The key idea is that any HMM can be completely characterised
in terms of quantities that depend entirely on the observations, called the observable representation,
which can be estimated from data. Siddiqi et al. [4] show that the same algorithm works under slightly
more general assumptions. Anandkumar et al. [3] proposed a spectral algorithm for estimating more
general latent variable models with parametric observations via a moment matching technique.
That said, we are aware of little work on estimating latent variable models, including HMMs, when
the observations are nonparametric. A commonly used heuristic is the nonparametric EM [12], which
lacks theoretical underpinnings. This should not be surprising because EM is degenerate for most
nonparametric problems as a maximum likelihood procedure [13]. Only recently, De Castro et al.
[14] have provided a minimax-type of result for the nonparametric setting. In their work, Siddiqi et al.
[4] proposed a heuristic based on kernel smoothing, to modify the discrete algorithm for continuous
observations. Further, their procedure cannot be used to recover the joint or conditional probabilities
of a sequence, which would be needed to compute probabilities of events and other inference tasks.
Song et al. [15, 16] developed an RKHS-based procedure for estimating the Hilbert space embedding
of an HMM. While they provide theoretical guarantees, their bounds are in terms of the RKHS
distance of the true and estimated embeddings. This metric depends on the choice of the kernel and it
is not clear how it translates to a suitable distance measure on the observation space such as an L1 or
L2 distance. While their method can be used for prediction and pairwise testing, it cannot recover the
joint and conditional densities. On the contrary, our model provides guarantees in terms of the more
interpretable total variation distance and is able to recover the joint and conditional probabilities.
2
A Pint-sized Review of Continuous Linear Algebra
We begin with a pint-sized review on continuous linear algebra which treats functions as continuous
analogues of matrices. Appendix A contains a quart-sized review. Both sections are based on [5, 6].
While these objects can be viewed as operators on Hilbert spaces which have been studied extensively
in the years, the above line of work simplified and specialised the ideas to functions.
A matrix F 2 Rm?n is an m ? n array of numbers where F (i, j) denotes the entry in row i, column j.
m or n could be (countably) infinite. A column qmatrix (quasi-matrix) Q 2 R[a,b]?m is a collection
of m functions defined on [a, b] where the row index is continuous and column index is discrete.
Writing Q = [q1 , . . . , qm ] where qj : [a, b] ! R is the j th function, Q(y, j) = qj (y) denotes the value
of the j th function at y 2 [a, b]. Q> 2 Rm?[a,b] denotes a row qmatrix with Q> (j, y) = Q(y, j).
A cmatrix (continuous-matrix) C 2 R[a,b]?[c,d] is a two dimensional function where both row and
column indices are continuous and C(y, x) is the value of the function at (y, x) 2 [a, b] ? [c, d].
C > 2 R[c,d]?[a,b] denotes its transpose with C > (x, y) = C(y, x). Qmatrices and cmatrices permit
all matrix multiplications with suitably defined inner products. For example, if R 2 R[c,d]?m and
Rd
C 2 R[a,b]?[c,d] , then CR = T 2 R[a,b]?m where T (y, j) = c C(y, s)R(s, j)ds.
2
A cmatrix has a singular P
value decomposition (SVD). If C 2 R[a,b]?[c,d] , it decomposes as an
1
2
infinite sum, C(y, x) =
???
0
2
j=1 j uj (y)vj (x), that converges in L . Here 1
are the singular values of C. {uj }j 1 and {vj }j 1 are functions that form orthonormal bases for
L2 ([a, b]) and L2 ([c, d]), respectively. We can write the SVD as C = U ?V > by writing the singular
vectors as infinite qmatrices U = [u1 , u2 . . . ], V = [v1 , v2 . . . ], and ? = diag( 1 , 2 . . . ). If only
m < 1 first singular values are nonzero, we say that C is of rank m. The SVD of a qmatrix
Q 2 R[a,b]?m is, Q = U ?V > where U 2 R[a,b]?m and V 2 Rm?m have orthonormal columns
and ? = diag( 1 , . . . , m ) with 1 ? ? ?
0. The rank of a column qmatrix is the number
m
of linearly independent columns (i.e. functions) and is equal to the number of nonzero singular
values. Finally, as for the finite matrices, the pseudo inverse of the cmatrix C is C ? = V ? 1 U >
with ? 1 = diag(1/ 1 , 1/ 2 , . . . ). The pseudo inverse of a qmatrix is defined similarly.
3
Nonparametric HMMs and the Observable Representation
Notation: Throughout this manuscript, we will use P to denote probabilities of events while p will
denote probability density functions (pdf). An HMM characterises a probability distribution over a
sequence of hidden states {ht }t 0 and observations {xt }t 0 . At a given time step, the HMM can
be in one of m hidden states, i.e. ht 2 [m] = {1, . . . , m}, and the observation is in some bounded
continuous domain X . Without loss of generality, we take2 X = [0, 1]. The nonparametric HMM
will be completely characterised by the initial state distribution ? 2 Rm , the state transition matrix
T 2 Rm?m and the emission densities Oj : X ! R, j 2 [m]. ?i = P(h1 = i) is the probability
that the HMM would be in state i at the first time step. The element T (i, j) = P(ht+1 = i|ht = j)
of T gives the probability that a hidden state transitions from state j to state i. The emission
function, Oj : X ! R+ , describes the pdf of the observation conditioned
on the hidden state j, i.e.
R
Oj (s) = p(xt = s|ht = j). Note that we have Oj (x) > 0, 8x and Oj (?) = 1 for all j 2 [m]. In
[0,1]?m
this exposition, we denote the emission densities by the qmatrix, O = [O1 , . . . , Om ] 2 R+
.
e
e
In addition, let O(x)
= diag(O1 (x), . . . , Om (x)), and A(x) = T O(x).
Let x1:t = {x1 , . . . , xt }
be an ordered sequence and xt:1 = {xt , . . . , x1 } denote its reverse. For brevity, we will overload
notation for A for sequences and write A(xt:1 ) = A(xt )A(xt 1 ) . . . A(x1 ). It is well known [2, 17]
that the joint probability density of the sequence x1:t can be computed via p(x1:t ) = 1>
m A(xt:1 )?.
Key structural assumption: Previous work on estimating HMMs with continuous observations
typically assumed that the emissions, Oj , take a parametric form, e.g. Gaussian. Unlike them, we
only make mild nonparametric smoothness assumptions on Oj . As we will see, to estimate the HMM
well in this problem we will need to estimate entire pdfs well. For this reason, the nonparametric
setting is significantly more difficult than its parametric counterpart as the latter requires estimating
only a finite number of parameters. When compared to the previous literature, this is the crucial
distinction and the main challenge in this work.
Observable Representation: The observable representation is a description of an HMM in terms of
quantities that depend on the observations [17]. This representation is useful for two reasons: (i) it
depends only on the observations and can be directly estimated from the data; (ii) it can be used to
compute joint and conditional probabilities of sequences even without the knowledge of T and O and
therefore can be used for inference and prediction. First, we define the joint densities, P1 , P21 , P321 :
P1 (t) = p(x1 = t),
P21 (s, t) = p(x2 = s, x1 = t),
P321 (r, s, t) = p(x3 = r, x2 = s, x1 = t),
where xi , i = 1, 2, 3 denotes the observation at time i. Denote P3x1 (r, t) = P321 (r, x, t) for all x. We
will find it useful to view both P21 , P3x1 2 R[0,1]?[0,1] as cmatrices. We will also need an additional
qmatrix U 2 R[0,1]?m such that U > O 2 Rm?m is invertible. Given one such U , the observable
representation of an HMM is described by the parameters b1 , b1 2 Rm and B : [0, 1] ! Rm?m ,
b1 = U > P1 ,
>
b1 = (P21
U ) ? P1 ,
B(x) = (U > P3x1 )(U > P21 )?
(1)
As before, for a sequence, xt:1 = {xt , . . . , x1 }, we define B(xt:1 ) = B(xt )B(xt 1 ) . . . B(x1 ). The
following lemma shows that the first m left singular vectors of P21 are a natural choice for U .
Lemma 1. Let ? > 0, T and O be of rank m and U be the qmatrix composed of the first m left
singular vectors of P21 . Then U > O is invertible.
2
We discuss the case of higher dimensions in Section 7.
3
To compute the joint and conditional probabilities using the observable representation, we maintain
an internal state, bt , which is updated as we see more observations. The internal state at time t is
bt =
B(xt 1:1 )b1
.
>
b1 B(xt 1:1 )b1
(2)
This definition of bt is consistent with b1 . The following lemma establishes the relationship between
the observable representation and the internal states to the HMM parameters and probabilities.
Lemma 2 (Properties of the Observable Representation). Let rank(T ) = rank(O) = m and U > O
be invertible. Let p(x1:t ) denote the joint density of a sequence x1:t and p(xt+1:t+t0 |x1:t ) denote the
conditional density of xt+1:t+t0 given x1:t in a sequence x1:t+t0 . Then the following are true.
1. b1 = U > O?
>
2. b1 = 1>
m (U O)
4. bt+1 = B(xt )bt /(b>
1 B(xt )bt ).
1
3. B(x) = (U > O)A(x)(U > O)
1
8 x 2 [0, 1].
5. p(x1:t ) = b>
1 B(xt:1 )b1 .
6. p(xt+t0 :t+1 |x1:t ) = b>
1 B(xt+t0 :t+1 )bt .
The last two claims of the Lemma 2 show that we can use the observable representation for computing
the joint and conditional densities. The proofs of Lemmas 1 and 2 are similar to the discrete case and
mimic Lemmas 2, 3 & 4 of Hsu et al. [2].
4
Spectral Learning of HMMs with Nonparametric Emissions
The high level idea of our algorithm, NP-HMM-SPEC, is as follows. First we will obtain density
estimates for P1 , P21 , P321 which will then be used to recover the observable representation b1 , b1 , B
by plugging in the expressions in (1). Lemma 2 then gives us a way to estimate the joint and
conditional probability densities. For now, we will assume that we have N i.i.d sequences of triples
(j)
(j)
(j)
(j)
{X (j) }N
= (X1 , X2 , X3 ) are the observations at the first three time steps. We
j=1 where X
describe learning from longer sequences in Section 4.3.
4.1 Kernel Density Estimation
The first step is the estimation of the joint probabilities which requires a nonparametric density
estimate. While there are several techniques [18], we use kernel density estimation (KDE) since it is
easy to analyse and works well in practice. The KDE for P1 , P21 , and P321 take the form:
!
!
!
N
N
(j)
(j)
(j)
X
X
1
1
t
X
1
1
s
X
t
X
1
2
1
Pb1 (t) =
K
,
Pb21 (s, t) =
K
K
,
N j=1 h1
h1
N j=1 h221
h21
h21
!
!
!
N
(j)
(j)
(j)
X
1
1
r
X
s
X
t
X
3
2
1
Pb321 (r, s, t) =
K
K
K
.
(3)
N j=1 h3321
h321
h321
h321
Here K : [0, 1] ! R is a symmetric function called a smoothing kernel and satisfies (at the very
R1
R1
least) 0 K(s)ds = 1, 0 sK(s)ds = 0. The parameters h1 , h21 , h321 are the bandwidths, and are
typically decreasing with N . In practice they are usually chosen via cross-validation.
4.2
The Spectral Algorithm
Algorithm 1 NP-HMM-SPEC
(j)
(j)
(j)
Input: Data {X (j) = (X1 , X2 , X3 )}N
j=1 , number of states m.
? Obtain estimates Pb1 , Pb21 , Pb321 for P1 , P21 , P321 via kernel density estimation (3).
b 2 R[0,1]?m be the first m left singular vectors of Pb21 .
? Compute the cmatrix SVD of Pb21 . Let U
b is a Rm?m valued function.
? Compute the parameters observable representation. Note that B
bb1 = U
b > Pb1 ,
bb1 = (P > U
b ?b
21 ) P1 ,
4
b
b > Pb3x1 )(U
b > Pb21 )?
B(x)
= (U
The algorithm, given above in Algorithm 1, follows the roadmap set out at the beginning of this
section. While the last two steps are similar to the discrete HMM algorithm of Hsu et al. [2], the
SVD, pseudoinverses and multiplications are with q/c-matrices. Once we have the estimates bb1 , bb1 ,
b
and B(x)
the joint and predictive (conditional) densities can be estimated via (see Lemma 2):
b
b
pb(x1:t ) = bb>
1 B(xt:1 )b1 ,
b
b
pb(xt+t0 :t+1 |x1:t ) = bb>
1 B(xt+t0 :t+1 )bt .
(4)
b in (2). Theoretically, these
Here bbt is the estimated internal state obtained by plugging in bb1 , bb1 , B
estimates can be negative in which case they can be truncated to 0 without affecting the theoretical
results in Section 5. However, in our experiments these estimates were never negative.
4.3
Implementation Details
C/Q-Matrix operations using Chebyshev polynomials: While our algorithm and analysis are
conceptually well founded, the important practical challenge lies in the efficient computation of the
many aforementioned operations on c/q-matrices. Fortunately, some very recent advances in the
numerical analysis literature, specifically on computing with Chebyshev polynomials, have rendered
the above algorithm practical [6, Ch.3-4]. Due to the space constraints, we provide only a summary.
Chebyshev polynomials is a family of orthogonal polynomials on compact intervals, known to be
an excellent approximator of one-dimensional functions [19, 20]. A recent line of work [5, 8] has
extended the Chebyshev technology to two dimensional functions enabling the mentioned operations
and factorisations such as QR, LU and SVD [6, Sections 4.6-4.8] of continuous matrices to be carried
efficiently. The density estimates Pb1 , Pb21 , Pb321 are approximated by Chebyshev polynomials to
within machine precision. Our implementation makes use of the Chebfun library [7] which provides
an efficient implementation for the operations on continuous and quasi matrices.
Computation time: Representing the KDE estimates Pb1 , Pb21 , Pb321 using Chebfun was roughly
linear in N and is the brunt of the computational effort. The bandwidths for the three KDE estimates
are chosen via cross validation which takes O(N 2 ) effort. However, in practice the cost was
dominated by the Chebyshev polynomial approximation. In our experiments we found that NPHMM-SPEC runs in linear time in practice and was more efficient than most alternatives.
Training with longer sequences: When training with longer sequences we can use a sliding window
of length 3 across the sequence to create the triples of observations needed for the algorithm. That
is, given N samples each of length `(j) , j = 1, . . . , N , we create an augmented dataset of triples
(j)
(j)
(j)
(j)
{ {(Xt , Xt+1 , Xt+2 )}`t=1 2 }N
j=1 and run NP-HMM-SPEC with the augmented data. As is with
conventional EM procedures, this requires the additional assumption that the initial state is the
stationary distribution of the transition matrix T .
5
Analysis
We now state our assumptions and main theoretical results. Following [2, 4, 15] we assume i.i.d
sequences of triples are used for training. With longer sequences, the analysis should only be modified
to account for the mixing of the latent state Markov chain, which is inessential for the main intuitions.
We begin with the following regularity condition on the HMM.
Assumption 3. ? > 0 element-wise. T 2 Rm?m and O 2 R[0,1]?m are of rank m.
The rank condition on O means that emission pdfs are linearly independent. If either T or O are
rank deficient, then the learner may confuse state outputs, which makes learning difficult3 . Next,
while we make no parametric assumptions on the emissions, some smoothness conditions are used
to make density estimation tractable. We use the H?lder class, H1 ( , L), which is standard in the
nonparametrics literature. For = 1, this assumption reduces to L-Lipschitz continuity.
Assumption 4. All emission densities belong to the H?lder class, H1 ( , L). That is, they satisfy,
d? Oj (s)
ds?
for all ? ? b c, j 2 [m], s, t 2 [0, 1]
d? Oj (t)
? L|s
dt?
t|
|?|
.
Here b c is the largest integer strictly less than .
3
Siddiqi et al. [4] show that the discrete spectral algorithm works under a slightly more general setting.
Similar results hold for the nonparametric case too but will restrict ourselves to the full rank setting for simplicity.
5
Under the above assumptions we bound the total variation distance between the true and the estimated
densities of a sequence, x1:t . Let ?(O) = 1 (O)/ m (O) denote the condition number of the
observation qmatrix. The following theorem states our main result.
Theorem 5. Pick any sufficiently small ? > 0 and a failure probability 2 (0, 1). Let t 1. Assume
that the HMM satisfies Assumptions 3 and 4 and the number of samples N satisfies,
? ?2+ 3
? ?1+ 23
3
N
?(O)2+
t
1
1+ 23
Cm
log
.
4
4+
log(N )
?
m (P21 )
Then,
with probability at least 1
, the estimated joint density for a t-length sequence satisfies
R
|p(x1:t ) pb(x1:t )|dx1:t ? ?. Here, C is a constant depending on and L and pb is from (4).
Synopsis: Observe that the sample complexity depends critically on the conditioning of O and P21 .
The closer they are to being singular, the more samples is needed to distinguish different states and
learn the HMM. It is instructive to compare the results above with the discrete case result of Hsu et al.
2
t2
1
[2], whose sample complexity bound4 is N & m m?(O)
(P21 )4 ?2 log . Our bound is different in two
regards. First, the exponents are worsened by additional ? 1 terms. This characterizes the difficulty
of the problem in the nonparametric setting. While we do not have any lower bounds, given the
current understanding of the difficulty of various nonparametric tasks [21?23], we think our bound
might be unimprovable. As the smoothness of the densities increases ! 1, we approach the
parametric sample complexity. The second difference is the additional log(N ) term on the left hand
side. This is due to the fact that we want the KDE to concentrate around its expectation in L2 over
[0, 1], instead of just point-wise. It is not clear to us whether the log can be avoided.
To prove Theorem 5, first we will derive some perturbation theory results for c/q-matrices; we will
need them to bound the deviation of the singular values and vectors when we use Pb21 instead of
P21 . Some of these perturbation theory results for continuous linear algebra are new and might be of
independent interest. Next, we establish a concentration result for the kernel density estimator.
5.1
Some Perturbation Theory Results for C/Q-matrices
The first result is an analog of Weyl?s theorem which bounds the difference in the singular values
in terms of the operator norm of the perturbation. Weyl?s theorem has been studied for general
operators [24] and cmatrices [6]. We have given one version in Lemma 21 of Appendix B. In addition
to this, we will also need to bound the difference in the singular vectors and the pseudo-inverses
of the truth and the estimate. To our knowledge, these results are not yet known. To that end, we
establish the following results. Here k (A) denotes the k th singular value of a c/q-matrix A.
? E 2 R[0,1]?[0,1] where
Lemma 6 (Simplified Wedin?s Sine Theorem for Cmatrices). Let A, A,
[a,b]?m
?
?
?
A = A + E and rank(A) = m. Let U, U 2 R
be
qthe first m left singular vectors of A and A
? > U xk2 kxk2 1 2kEk2 2 / m (A)
? 2.
respectively. Then, for all x 2 Rm , kU
L
? E 2 R[a,b]?m and A? = A + E. Then,
Lemma 7 (Pseudo-inverse Theorem for Qmatrices). Let A, A,
?
A?? ) ? 3 max{ 1 (A? )2 , 1 (A? )2 } 1 (E).
1 (A
5.2
Concentration Bound for the Kernel Density Estimator
Next, we bound the error for kernel density estimation. To obtain the best rates under H?lderian
assumptions on O, the kernels used in KDE need to be of order . A order kernel satisfies,
Z 1
Z 1
Z 1
K(s)ds = 1,
s? K(s)ds = 0, for all ? ? b c,
s K(s)ds ? 1.
(5)
0
0
0
Such kernels can be constructed using Legendre polynomials [18]. Given N i.i.d samples from a d
dimensional density f , where d 2 {1, 2, 3} and f 2 {P1 , P21 , P321 }, for appropriate choices of the
bandwidths h1 , h21 , h321 , the KDE f? 2 {Pb1 , Pb21 , Pb321 } concentrates around f . Informally, we show
?
?
?
?
2
d
P kf? f kL2 > " . exp
log(N ) 2 +d N 2 +d "2 .
(6)
4
Hsu et al. [2] provide a more refined bound but we use this form to simplify the comparison.
6
0.25
0.1
0.05
0.39
0.388
0.386
104
3
105
10
10
103
10
2
101
5
0.25
0.1
True
MG-HMM
NP-HMM-BIN
NP-HMM-EM
NP-HMM-HSE
NP-HMM-SPEC
0.37
Prediction absolute error
MG-HMM
NP-HMM-BIN
NP-HMM-EM
NP-HMM-SPEC
0.366
MG-HMM
NP-HMM-EM
NP-HMM-HSE
NP-HMM-SPEC
104
0.362
104
10
3
102
MG-HMM
NP-HMM-EM
NP-HMM-HSE
NP-HMM-SPEC
0.358
0.05
5
10
50
Number of training sequences # 103
100
5
10
50
Number of training sequences # 103
105
Number of training sequences
Number of training sequences
Number of training sequences
0.5
4
10
Training time, sec
103
4
100
103
0.384
0.025
Predictive L1 error
10
True
MG-HMM
NP-HMM-BIN
NP-HMM-EM
NP-HMM-HSE
NP-HMM-SPEC
Training time, sec
MG-HMM
NP-HMM-BIN
NP-HMM-EM
NP-HMM-SPEC
Prediction absolute error
Predictive L1 error
0.5
100
101
5
10
50
100
Number of training sequences
Figure 1: The upper and lower panels correspond to m = 4 m = 8 respectively. All figures are in log-log
scale and the x-axis is the number of triples used for training. Left: L1 error between true conditional density
p(x6 |x1:5 ), and the estimate for each method. Middle: The absolute error between the true observation and a
one-step-ahead prediction. The error of the true model is denoted by a black dashed line. Right: Training time.
d
for all sufficiently small " and N/ log N & " 2+ . Here ., & denote inequalities ignoring constants.
See Appendix C for a formal statement. Note that when the observations are either discrete or
parametric, it is possible to estimate the distribution using O(1/"2 ) samples to achieve " error in a
suitable metric, say, using the maximum likelihood estimate. However, the nonparametric setting is
inherently more difficult and therefore the rate of convergence is slower. This slow convergence is
also observed in similar concentration bounds for the KDE [25, 26].
A note on the Proofs: For Lemmas 6, 7 we follow the matrix proof in Stewart and Sun [27] and
derive several intermediate results for c/q-matrices in the process. The main challenge is that several
properties for matrices, e.g. the CS and Schur decompositions, are not known for c/q-matrices. In
addition, dealing with various notions of convergences with these infinite objects can be finicky. The
main challenge with the KDE concentration result is that we want an L2 bound ? so usual techniques
(such as McDiarmid?s [13, 18]) do not apply. We use a technical lemma from Gin? and Guillou [26]
which allows us to bound the L2 error in terms of the VC characteristics of the class of functions
induced by an i.i.d sum of the kernel. The proof of theorem 5 just mimics the discrete case analysis
of Hsu et al. [2]. While, some care is needed (e.g. kxkL2 ? kxkL1 does not hold for functional
norms) the key ideas carry through once we apply Lemmas 21, 6, 7 and (6). A more refined bound on
N that is tighter in polylog(N ) terms is possible ? see Corollary 25 and equation 13 in the appendix.
6
Experiments
We compare NP-HMM-SPEC to the following. MG-HMM: An HMM trained using EM with the
emissions modeled as a mixture of Gaussians. We tried 2, 4 and 8 mixtures and report the best result.
NP-HMM-BIN: A naive baseline where we bin the space into n intervals and use the discrete spectral
algorithm [2] with n states. We tried several values for n and report the best. NP-HMM-EM: The
Nonparametric EM heuristic of [12]. NP-HMM-HSE: The Hilbert space embedding method of [15].
Synthetic Datasets: We first performed a series of experiments on synthetic data where the true
distribution is known. The goal is to evaluate the estimated models against the true model. We
generated triples from two HMMs with m = 4 and m = 8 states and nonparametric emissions. The
details of the set up are given in Appendix E. Figure 1 presents the results.
First we compare the methods on estimating the one step ahead conditional density p(x6 |x1:5 ). We
report the L1 error between the true and estimated models. In Figure 2 we visualise the estimated one
step ahead conditional densities. NP-HMM-SPEC outperforms all methods on this metric. Next, we
compare the methods on the prediction performance. That is, we sample sequences of length 6 and test
how well a learned model can predict x6 conditioned on x1:5 . When comparing on squared error, the
best predictor is the mean of the distribution. For all methods we use the mean of pb(x6 |x1:5 ) except
7
0.4
0.2
0
Predictive density
0.6
1
Truth
NP-HMM-HKZ-BIN
0.8
0.6
0.4
0.2
0
-1
-0.5
0
0.5
1
1
Truth
NP-HMM-EM
0.8
Predictive density
1
Truth
MG-HMM
Predictive density
Predictive density
1
0.8
0.6
0.4
0.2
0
-1
-0.5
X
0
0.5
1
Truth
NP-HMM-SPEC
0.8
0.6
0.4
0.2
0
-1
X
-0.5
0
0.5
1
X
-1
-0.5
0
0.5
1
X
Figure 2: True and estimated one step ahead densities p(x4 |x1:3 ) for each model. Here m = 4 and N = 104 .
Dataset
Internet Traffic
Laser Gen
Patient Sleep
MG-HMM
NP-HMM-BIN
NP-HMM-HSE
NP-HMM-SPEC
0.143 ? 0.001
0.33 ? 0.018
0.330 ? 0.002
0.188 ? 0.004
0.31 ? 0.017
0.38 ? 0.011
0.0282 ? 0.0003
0.19 ? 0.012
0.197 ? 0.001
0.016 ? 0.0002
0.15 ? 0.018
0.225 ? 0.001
Table 1: The mean prediction error and the standard error on the 3 real datasets.
for NP-HMM-HSE for which we used the mode since the mean cannot be computed. No method can
do better than the true model (shown via the dotted line) in expectation. NP-HMM-SPEC achieves
the performance of the true model with large datasets. Finally, we compare the training times of all
methods. NP-HMM-SPEC is orders of magnitude faster than NP-HMM-HSE and NP-HMM-EM.
Note that the error of MG-HMM?a parametric model?stops decreasing even with large data. This
is due to the bias introduced by the parametric assumption. We do not train NP-HMM-EM for longer
sequences because it is too slow. A limitation of the NP-HMM-HSE method is that it cannot recover
conditional probabilities, so we exclude it from that experiment. We could not include the method
of [4] in our comparisons since their code was not available and their method is not straightforward
to implement. Further, their method cannot compute joint/predictive probabilities.
Real Datasets: We compare all the above methods (except NP-HMM-EM which was too slow) on
prediction error on 3 real datasets: internet traffic [28], laser generation [29] and sleep data [30]. The
details on these datasets are in Appendix E. For all methods we used the mode of the conditional
distribution p(xt+1 |x1:t ) as the prediction as it performed better. For NP-HMM-SPEC, NP-HMMHSE,NP-HMM-BIN we follow the procedure outlined in Section 4.3 to create triples and train with
the triples. In Table 1 we report the mean prediction error and the standard error. NP-HMM-HSE
and NP-HMM-SPEC perform better than the other two methods. However, NP-HMM-SPEC was
faster to train (and has other attractive properties) when compared to NP-HMM-HSE.
7
Conclusion
We proposed and studied a method for estimating the observable representation of a Hidden Markov
Model whose emission probabilities are smooth nonparametric densities. We derive a bound on the
sample complexity for our method. While our algorithm is similar to existing methods for discrete
models, many of the ideas that generalise it to the nonparametric setting are new. In comparison
to other methods, the proposed approach has some desirable characteristics: we can recover the
joint/conditional densities, our theoretical results are in terms of more interpretable metrics, the
method outperforms baselines and is orders of magnitude faster to train.
In this exposition only focused on one dimensional observations. The multidimensional case is
handled by extending the above ideas and technology to multivariate functions. Our algorithm and the
analysis carry through to the d-dimensional setting, mutatis mutandis. The concern however, is more
practical. While we have the technology to perform various c/q-matrix operations for d = 1 using
Chebyshev polynomials, this is not yet the case for d > 1. Developing efficient procedures for these
operations in the high dimensional settings is a challenge for the numerical analysis community and is
beyond the scope of this paper. That said, some recent advances in this direction are promising [8, 31].
While our method has focused on HMMs, the ideas in this paper apply for a much broader class
of problems. Recent advances in spectral methods for estimating parametric predictive state representations [32], mixture models [3] and other latent variable models [33] can be generalised to the
nonparamatric setting using our ideas. Going forward, we wish to focus on such models.
Acknowledgements
The authors would like to thank Alex Townsend, Arthur Gretton, Ahmed Hefny, Yaoliang Yu, and
Renato Negrinho for the helpful discussions. This work was supported by NIH R01GM114311,
AFRL/DARPA FA87501220324.
8
References
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9
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5,621 | 6,087 | Dimension-Free Iteration Complexity of Finite Sum
Optimization Problems
Yossi Arjevani
Weizmann Institute of Science
Rehovot 7610001, Israel
yossi.arjevani@weizmann.ac.il
Ohad Shamir
Weizmann Institute of Science
Rehovot 7610001, Israel
ohad.shamir@weizmann.ac.il
Abstract
Many canonical machine learning problems boil down to a convex optimization
problem with a finite sum structure. However, whereas much progress has been
made in developing faster algorithms for this setting, the inherent limitations of
these problems are not satisfactorily addressed by existing lower bounds. Indeed,
current bounds focus on first-order optimization algorithms, and only apply in
the often unrealistic regime where the number of iterations is less than O(d/n)
(where d is the dimension and n is the number of samples). In this work, we
extend the framework of Arjevani et al. [3, 5] to provide new lower bounds, which
are dimension-free, and go beyond the assumptions of current bounds, thereby
covering standard finite sum optimization methods, e.g., SAG, SAGA, SVRG,
SDCA without duality, as well as stochastic coordinate-descent methods, such as
SDCA and accelerated proximal SDCA.
1
Introduction
Many machine learning tasks reduce to Finite Sum Minimization (FSM) problems of the form
n
min F (w) :=
w?Rd
1X
fi (w),
n i=1
(1)
where fi are L-smooth and ?-strongly convex. In recent years, a major breakthrough was made
when a linear convergence rate was established for this setting (SAG [16] and SDCA [18]), and since
then, many methods have been developed to achieve better convergence rate. However, whereas a
large body of literature is devoted for upper bounds, the optimal convergence rate with respect to the
problem parameters is not quite settled.
Let us discuss existing lower bounds for this setting, along with their shortcomings, in detail. One
approach to obtain lower bounds for this setting is to consider the average of carefully handcrafted
functions defined on n disjoint sets of variables. This approach was taken by Agarwal and Bottou [1]
who derived a lower bound for FSM under the first-order oracle model (see Nemirovsky and Yudin
[12]). In this model, optimization algorithms are assumed to access a given function by issuing queries
to an external first-order oracle procedure. Upon receiving a query point in the problem domain, the
oracle reports the corresponding function value and gradient. The construction used by Agarwal and
Bottou consisted of n different quadratic functions which are adversarially determined based on the
first-order queries being issued during the optimization process. The resulting bound in this case does
not apply to stochastic algorithms, rendering it invalid for current state-of-the-art methods. Another
instantiation of this approach was made by Lan [10] who considered n disjoint copies of a quadratic
function proposed by Nesterov in [13, Section 2.1.2]. This technique is based on the assumption that
any iterate generated by the optimization algorithm lies in the span of previously acquired gradients.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
This assumption is rather permissive and is satisfied by many first-order algorithms, e.g., SAG and
SAGA [6]. However, the lower bound stated in the paper faces limitations in a few aspects. First, the
validity of the derived bound is restricted to d/n iterations. In many datasets, even if d, n are very
large, d/n is quite small. Accordingly, the admissible regime of the lower bound is often not very
interesting. Secondly, it is not clear how the proposed construction can be expressed as a Regularized
Loss Minimization (RLM) problem with linear predictors (see Section 4). This suggests that methods
specialized in dual RLM problems, such as SDCA and accelerated proximal SDCA [19], can not be
addressed by this bound. Thirdly, at least the formal theorem requires assumptions (such as querying
in the span of previous gradients, or sampling from a fixed distribution over the individual functions),
which are not met by some state-of-the-art methods, such as coordinate descent methods, SVRG [9]
and without-replacements sampling algorithms [15].
Another relevant approach in this setting is to model the functional form of the update rules. This
approach was taken by Arjevani et al. [3] where new iterates are assumed to be generated by a
recurrent application of some fixed linear transformation. Although this method applies to SDCA
?
and produces a tight lower bound of ?((n
+ 1/?) ln(1/)), its scope is rather limited. In recent
work, Arjevani and Shamir [5] considerably generalized parts of this framework by introducing
the class of first-order oblivious optimization algorithms, whose step sizes are scheduled regardless
of the function under consideration, and deriving tight lower bounds for general smooth convex
minimization problems (note that obliviousness rules out, e.g., quasi-Newton methods where gradients
obtained at each iteration are multiplied by matrices which strictly depend on the function at hand,
see Definition 2 below).
In this work, building upon the framework of oblivious algorithms, we take a somewhat more
abstract point of view which allows us to easily incorporate coordinate-descent methods, as well
as stochastic algorithms. Our framework subsumes the vast majority of optimization methods for
machine learning problems, in particular, it applies to SDCA, accelerated proximal SDCA, SDCA
without duality [17], SAG, SAGA, SVRG and acceleration schemes [7, 11]), as well as to a large
number of methods for smooth convex optimization (i.e., FSM with n = 1), e.g., (stochastic) Gradient
descent (GD), Accelerated Gradient Descent (AGD, [13]), the Heavy-Ball method (HB, [14]) and
stochastic coordinate descent.
Under this structural assumption, we derive lower bounds for FSM (1), according to which the
iteration complexity, i.e., the number of iterations required to obtain an -optimal solution in terms of
function value, is at least1
p
? + n(? ? 1) ln(1/)),
?(n
(2)
where ? denotes the condition number of F (w) (that is, the smoothness parameter over the strong
convexity parameter). To the best of our knowledge, this is the first tight lower bound to address all
the algorithms mentioned above. Moreover, our bound is dimension-free and thus applies to settings
in machine learning which are not covered in the current literature (e.g., when n is ?(d)). We also
derive a dimension-free nearly-optimal lower bound for smooth convex optimization of
1/?
? (L(? ? 2)/)
,
for any ? ? (2, 4), which holds for any oblivious stochastic first-order algorithm. It should be noted
that our lower bounds remain valid under any source of randomness which may be introduced into
the optimization process (by the oracle or by the optimization algorithm). In particular, our bounds
hold in cases where the variance of the iterates produced by the algorithm converges to zero, a highly
desirable property of optimization algorithms in this setting.
Two implications can be readily derived from this lower bound. First, obliviousness forms a real
barrier for optimization algorithms, and whereas non-oblivious algorithms may achieve a super-linear
convergence rate at later stages of the optimization process (e.g., quasi-newton), or practically zero
error after ?(d) iterations (e.g. Center of Gravity method, MCG), oblivious algorithms are bound
to linear convergence indefinitely, as demonstrated by Figure 1. We believe that this indicates that
a major progress can be made in solving machine learning problems by employing non-oblivious
methods for settings where d n. It should be further noted that another major advantage of
1
Following standard conventions, here tilde notation hides logarithmic factors in the parameters of a given
class of optimization problems, e.g., smoothness parameter and number of components.
2
non-oblivious algorithms is their ability to obtain optimal convergence rates without an explicit
specification of the problem parameters (e.g., [5, Section 4.1]).
104
GD
AGD
HB
L-BFGS
Lower Bound
102
Error
100
10-2
10-4
10-6
10-8
0
500
1000
1500
2000
2500
Number of Iterations
Figure 1: Comparison of first-order methods based on the function used by Nesterov in [13, Section
2.1.2] over R500 . Whereas L-BFGS (with a memory size of 100) achieves a super-linear convergence
rate after ?(d) iterations, the convergence rate of GD, AGD and HB remains linear as predicted by
our bound.
Secondly, many practitioners have noticed that oftentimes sampling the individual functions without
replacement at each iteration performs better than sampling with replacement (e.g., [18, 15], see also
[8, 20]). The fact that our lower bound holds regardless of how the individual functions are sampled
and is attained using with-replacement sampling (e.g., accelerated proximal SDCA), implies that,
in terms of iteration complexity, one should expect to gain no more than log factors in the problem
parameters when using one method over the other (it is noteworthy that when comparing with and
without replacement samplings, apart from iteration complexity, other computational resources, such
as limited communication in distributed settings [4], may significantly affect the overall runtime).
2
2.1
Framework
Motivation
Due to difficulties which arise when studying the complexity of general optimization problems under
discrete computational models, it is common to analyze the computational hardness of optimization
algorithms by modeling the way a given algorithm interacts with the problem instances (without
limiting its computational resources). In the seminal work of Nemirovsky and Yudin [12], it is shown
that algorithms which access the function at hand exclusively by querying a first-order oracle require
at least
?
? min d, ? ln(1/) ,
?
?>0
(3)
p
?
?(min{d ln(1/), L/}),
?=0
oracle calls to obtain an -optimal solution, where L and ? are the smoothness and the strong
convexity parameter, respectively (note that, here and throughout this section we refer to FSM
problems with n = 1). This lower bound is tight and its dimension-free part is attained by Nesterov?s
well-known accelerated gradient descent, and by MCG otherwise. The fact that this approach is
based on information considerations alone is very appealing and renders it valid for any first-order
algorithm. However, discarding the resources needed for executing a given algorithm, in particular
the per-iteration cost (in time and space), the complexity boundaries drawn by this approach are
too crude from a computational point of view. Indeed, the per-iteration cost of MCG, the only
method known with oracle complexity of O(d ln(1/)), is excessively high, rendering it prohibitive
for high-dimensional problems.
We are thus led into the question of how well can a given optimization algorithm perform assuming
that its per-iteration cost is constrained? Arjevani et al. [3, 5] adopted a more structural approach
3
where instead of modeling how information regarding the function at hand is being collected, one
models the update rules according to which iterates are being generated. Concretely, they proposed
the framework of p-CLI optimization algorithms where, roughly speaking, new iterates are assumed
to form linear combinations of the previous p iterates and gradients, and the coefficients of these
linear combinations are assumed to be either stationary (i.e., remain fixed throughout the optimization
process) or oblivious. Based on this structural assumption, they showed that the iteration complexity
? ?? ln(1/)). The fact that this lower
of minimizing smooth and strongly convex functions is ?(
bound is stronger than (3), in the sense that it does not depend on the dimension, confirms that
controlling the functional form of the update rules allows one to derive tighter lower bounds. The
framework of p-CLIs forms the nucleus of our formulation below.
2.2
Definitions
When considering lower bounds one must be very precise as to the scope of optimization algorithms
to which they apply. Below, we give formal definitions for oblivious stochastic CLI optimization algorithms and iteration complexity (which serves as a crude proxy for their computational complexity).
Definition 1 (Class of Optimization Problems). A class of optimization problems is an ordered triple
(F, I, Of ), where F is a family of functions defined over some domain designated by domF, I is
the side-information given prior to the optimization process and Of is a suitable oracle which upon
receiving x ? domF and ? in the parameter set ?, returns Of (x, ?) ? dom(F) for a given f ? F
(we shall omit the subscript in Of when f is clear from the context).
For example, in FSM, F contains functions as defined in (1), the side-information contains the
smoothness parameter L, the strong convexity parameter ? and the number of components n (although
it carries a crucial effect on the iteration complexity, e.g., [5], in this work, we shall ignore the sideinformation and assume that all the parameters of the class are given). We shall assume that both
first-order and coordinate-descent oracles (see 10,11 below) are allowed to be used during the
optimization process. Formally, this is done by introducing an additional parameter which indicates
which oracle is being addressed. This added degree of freedom does not violate our lower bounds.
We now turn to rigorously define CLI optimization algorithms. Note that, compared with the definition
of first-order p-CLIs provided in [5], here, in order to handle coordinate-descent and first-order oracles
in a unified manner, we base our formulation on general oracle procedures.
Definition 2 (CLI). An optimization algorithm is called a Canonical Linear Iterative (CLI) optimization algorithm over a class of optimization problems (F, I, Of ), if given an instance f ? F
(0)
and initialization points {wi }i?J ? dom(F), where J is some index set, it operates by iteratively
generating points such that for any i ? J ,
X
(k+1)
(k) (k)
wi
?
Of wj ; ?ij , k = 0, 1, . . .
(4)
j?J
(k)
holds, where ?ij ? ? are parameters chosen, stochastically or deterministically, by the algorithm,
possibly depending on the side-information. If the parameters do not depend on previously acquired
oracle answers, we say that the given algorithm is oblivious. Lastly, algorithms with |J | ? p, for
some p ? N, are denoted by p-CLI.
(k)
Note that assigning different weights to different terms in (4) can be done through ?ij ? ?
(e.g., oracle 10 below). This allows a succinct definition for obliviousness. Lastly, we define
iteration complexity.
Definition 3 (Iteration Complexity). The iteration complexity of a given CLI w.r.t. a given problem
class (F, I, Of ) is defined to be the minimal number of iterations K such that
(k)
E[f (w1 ) ? min f (w)] < ,
w?domF
?f ? F, k ? K
where the expectation is taken over all the randomness introduced into the optimization process
(k)
(choosing w1 merely serves as a convention and is not necessary for our bounds to hold).
4
2.3
Proof Technique - Deriving Lower Bounds via Approximation Theory
Consider the following parametrized class of L-smooth and ?-strongly convex optimization problems,
min f? (w) :=
w?R
?w2
? w,
2
? ? [?, L].
(5)
Clearly, the minimizer of f? are w? (?) := 1/?, with norm bounded by 1/?. For simplicity, we will
consider a special case, namely, vanilla gradient descent (GD) with step size 1/L, which produces
new iterates as follows
1
? (k)
1
w(k+1) (?) = w(k) (?) ? f?0 (w(k) (?)) = 1 ?
w (?) + .
L
L
L
Setting the initialization point to be w(0) (?) = 0, we derive an explicit expression for w(k) (?):
k?1
1X
k
w(k) (?) =
(?1)i
(?/L)i .
(6)
L i=0
i+1
Approximating polynomials
1
GD, w(1)(2)
GD, w(2)(2)
GD, w(3)(2)
0.9
GD, w(4)(2)
AGD, w(1)(2)
AGD, w(2)(2)
0.8
AGD, w(3)(2)
AGD, w(4)(2)
1/2
0.7
0.6
0.5
0.4
0.3
0.2
1
1.5
2
2.5
3
3.5
4
Figure 2: The first four iterates of GD and AGD, which form polynomials in ?, the parameter of
problem (5), are compared to 1/? over [1, 4].
It turns out that each w(k) (?) forms a univariate polynomial whose degree is at most k. Furthermore,
since f? (w) are L-smooth ?-strongly convex for any ? ? [?, L], standard convergence analysis for
k
GD (e.g., [13], Theorem 2.1.14) guarantees that |w(k) (?) ? w? (?)| ? (1 ? 2/(1 + ?)) 2 |w? (?)|,
where ? denotes the condition number. Substituting Equation (6) for w(k) (?) yields
k2
1 k?1
k
2
1
X
i
i
(?1)
(?/L) ? 1/? ?
1?
.
max
?
i+1
1+?
??[?,L] L
i=0
Thus, we see that the faster the convergence rate of a given optimization algorithm is, the better
the induced sequence of polynomials (w(k) (?))k?0 approximate 1/? w.r.t. the maximum norm
k ? kL? ([?,L]) over [?, L]. In Fig. 2, we compare the first 4 polynomials induced by GD and AGD.
Not surprisingly, AGD polynomials approximates 1/? better than those of GD.
Now, one may ask, assuming that iterates of a given optimization algorithm A for (5) can be
expressed as polynomials sk (?) whose degree does not exceed the iteration number, just how fast
can these iterates converge to the minimizer? Since the convergence rate is bounded from below by
ksk (?) ? 1/?kL? ([?,L]) , we may address the following question instead:
min ks(?) ? 1/?kL? ([?,L]) ,
s(?)?Pk
(7)
where Pk denotes the set of univariate polynomials whose degree does not exceed k. Problem (7) and
other related settings are main topics of study in approximation theory. Accordingly, our technique
5
for proving lower bounds makes an extensive use of tools borrowed from this area. Specifically, in a
paper from 1899 [21] Chebyshev showed that
?
1
(c ? c2 ? 1)k
min s(?) ?
?
, c > 1,
(8)
? ? c
L? ([?1,1])
c2 ? 1
s(?)?Pk
by which we derive the following theorem (see Appendix A.1 for a detailed proof).
? ?? ln(1/)).
Theorem 1. The number of iterations required by A to get an -optimal solution is ?(
In the following sections, we apply oblivious CLI on various parameterized optimization problems
so that the resulting iterates are polynomials in the problem parameters. We then apply arguments
similar to the above
A similar reduction, from optimization problems to approximation problems, was used before in a
few contexts to analyze the iteration complexity of deterministic CLIs (e.g., [5, Section 3], see also
Conjugate Gradient convergence analysis [14]). But, what if we allow random algorithms? should we
expect the same iteration complexity? To answer this, we use Yao?s minimax principle according to
which the performance of a given stochastic optimization algorithm w.r.t. its worst input are bounded
from below by the performance of the best deterministic algorithm w.r.t. distributions over the input
space. Thus, following a similar reduction one can show that the convergence rate of stochastic
algorithms is bounded from below by
Z L
1
min
|s(?) ? 1/?|
d?.
(9)
L??
s(?)?Pk ?
That is, a lower bound for the stochastic case can be attained by considering an approximation
problem w.r.t. weighted L1 with the uniform distribution over [?, L]. Other approximation problems
considered in this work involve L2 -norm and different distributions. We provide a schematic
description of our proof technique in Scheme 2.1.
S CHEME
G IVEN
2.1
F ROM O PTIMIZATION P ROBLEMS TO A PPROXIMATION P ROBLEMS
F , A SUITABLE ORACLE O
Sk OVER SOME PARAMETERS SET H .
k
A SUBSET OF FUNCTIONS {f? ? F |? ? H}, S . T. w (?) ? Sk .
?
THE MINIMIZER w (?) FOR ANY f?
?
FROM BELOW THE BEST APPROXIMATION FOR w (?) W. R . T. Sk
?
AND A NORM k ? k, I . E ., min{ks(?) ? w (?)k | s(?) ? Sk }
A CLASS OF FUNCTIONS
AND A SEQUENCE OF SETS OF FUNCTION
C HOOSE
C OMPUTE
B OUND
3
Lower Bound for Finite Sums Minimization Methods
Having described our analytic approach, we now turn to present some concrete applications, starting
with iteration complexity lower bounds in the context of FSM problems (1). In what follows, we
derive a lower bound on the iteration complexity of oblivious (possibly stochastic) CLI algorithms
equipped with first-order and coordinate-descent oracles for FSM. Strictly speaking, we focus on
optimization algorithms equipped with both generalized first order oracle,
O(w; A, B, c, j) = A?fj (w) + Bw + c,
A, B ? Rd?d , c ? Rd , j ? [n],
(10)
and steepest coordinate-descent oracle
O(w; i, j)
= w + t? ei ,
t? ? argmin fj (w1 , . . . , wi?1 , wi + t, wi+1 , . . . , wd ), j ? [n], (11)
t?R
where ei denotes the i?th unit vector. We remark that coordinate-descent steps w.r.t. partial gradients
can be implemented using (10) by setting A to be some principal minor of the unit matrix. It should
be further noted that our results below hold for scenarios where the optimization algorithm is free to
call a different oracle at different iterations.
First, we sketch the proof of the lower bound for deterministic oblivious CLIs. Following Scheme
2.1, we restrict our attention to a parameterized subset of problems. We assume2 d > 1 and denote by
2
Clearly, in order to derive a lower bound for coordinate-descent algorithms, we must assume d > 1. If only
a first-order oracle is allowed, then the same lower bound as in Theorem 2 can be derived for d = 1.
6
HFSM the set of all (?1 , . . . , ?n ) ? Rn such that all the entries equal ?(L ? ?)/2, except for some
j ? [n], for which ?j ? [?(L ? ?)/2, (L ? ?)/2]. Now, given ? := (?1 , . . . , ?n ) ? HFSM we define
n
1X 1 >
F? (w) :=
w Q?i w ? q> w , where
(12)
n i=1 2
? R? ?
?
? L+?
?
?i
2
2
L+?
? R?
?
? ?
?
? ?
i
?
?
?
2
? 2 ?
?
?
?
Q?i := ?
? , q := ? 0 ? .
? . ?
?
?
..
? . ?
?
?
.
.
?
0
It is easy to verify that the minimizers of (12) are
?
?>
R?
R?
, ?
, 0, . . . , 0? .
w? (?) = ? ?
Pn
Pn
L+?
1
1
?
?
2 L+?
+
2
+
i=1 i
i=1 i
2
n
2
n
(13)
We would like to show that the coordinates of the iterates of deterministic oblivious CLIs, which
minimize F? using first-order and coordinate-descent oracles, form multivariate polynomials in ?
of total degrees (the maximal sum of powers over all the terms) which does not exceed the iteration
(k)
number. Indeed, if the coordinates of wi (?) are multivariate polynomial in ? of total degree at
most k, then the coordinates of the vectors returned by both oracles
(k)
(k)
First-order oracle: O(wj ; A, B, c, j) = A(Q?j wi
Coordinate-descent oracle:
(k)
O(wj ; i, j)
(k)
? q) + Bwi
+ c,
(14)
(k)
= I ? (1/(Q?j )ii )ei (Q?j )i,? wi ? qi /(Q?j )ii ei ,
are multivariate polynomials of total degree of at most k + 1, as all the parameters (A, B, C, i and j)
do not depend on ? (due to obliviousness) and the rest of the terms (Q?j , q, I, 1/(Q?j )ii , (Q?j )i,? , ei
and qi ) are either linear in ?j or constants. Now, since the next iterates are generated simply by
summing up all the oracle answers, they also form multivariate polynomials of total degree of at most
(k)
k + 1. Thus, denoting the first coordinate of w1 (?) by s(?) and using Inequality (8), we get the
following bound
R?
(k)
?
max kw1 (?) ? w (?)k ?
s(?) ? ?
(15)
P
n
??HFSM
1
+
?
2 L+?
i
i=1
2
n
L? ([?,L])
?q
? ?(1) ? q
??1
n
+1?1
??1
n
+1+1
?k/n
?
,
(16)
where ?(1) designates a constant which does not depend on k (but may depend on the problem
parameters). Lastly, this implies that for any deterministic oblivious CLI and any iteration number,
there exists some ? ? HFSM such that the convergence rate of the algorithm, when applied on F? ,
is bounded from below by Inequality (16). We note that, as opposed to other related lower bounds,
e.g., [10], our proof is non-constructive. As discussed in subsection 2.3, this type of analysis can be
extended to stochastic algorithms by considering (15) w.r.t. other norms such as weighted L1 -norm.
We now arrive at the following theorem whose proof, including the corresponding logarithmic factors
and constants, can be found in Appendix A.2.
Theorem 2. The iteration complexity of oblivious (possibly stochastic) CLIs for FSM (1) equipped
with first-order (10) and coordinate-descent oracles (11), is bounded from below by
p
? + n(? ? 1) ln(1/)).
?(n
The lower bound stated in Theorem 2 is tight and is attained by, e.g., SAG combined with an
acceleration scheme (e.g., [11]). Moreover, as mentioned earlier, our lower bound does not depend
on the problem dimension (or equivalently, holds for any number of iterations, regardless of d and
7
n), and covers coordinate descent methods with stochastic or deterministic coordinate schedule
(in the special case where n = 1, this gives a lower bound for minimizing smooth and strongly
convex functions by performing steepest coordinate descent steps). Also, our bound implies that
using mini-batches for tackling FSM does not reduce the overall iteration complexity. Lastly, it is
noteworthy that the n term in the lower bound above holds for any algorithm accompanied with an
incremental oracle, which grants access to at most one individual function each time.
We also derive a nearly-optimal lower bound for smooth non-strongly convex functions for the more
restricted setting of n = 1 and first-order oracle. The parameterized subset of functions we use
2
(see Scheme 2.1) is g? (x) := ?2 kxk ? R?e>
? ? (0, L]. The corresponding minimizer (as
1 x,
?
a function of ?) is x (?) = Re1 , and in this case we seek to approximate it w.r.t. L2 -norm using
k-degree univariate polynomials whose constant term vanishes. The resulting bound is dimension-free
and improves upon other bounds for this setting (e.g. [5]) in that it applies to deterministic algorithms,
as well as to stochastic algorithms (see A.3 for proof).
Theorem 3. The iteration complexity of any oblivious (possibly stochastic) CLI for L-smooth convex
functions equipped with a first-order oracle, is bounded from below by
1/?
? (L(? ? 2)/)
, ? ? (2, 4).
4
Lower Bound for Dual Regularized Loss Minimization with Linear
Predictors
The form of functions (12) discussed in the previous section does not readily adapt to general RLM
problems with linear predictors, i.e.,
n
?
1X
2
?i (hxi , wi) + kwk ,
(17)
min P (w) :=
d
n i=1
2
w?R
where the loss functions ?i are L-smooth and convex, the samples x1 , . . . , xn are d-dimensional
vectors in Rd and ? is some positive constant. Thus, dual methods which exploit the added structure
of this setting through the dual problem [18],
2
n
n
?
1X ?
1 X
?i (??i ) +
xi ?i
,
minn D(?) =
(18)
??R
n
2
?n
i=1
i=1
such as SDCA and accelerated proximal SDCA, are not covered by Theorem 2. Accordingly, in
this section, we address the iteration complexity of oblivious (possibly stochastic) CLI algorithms
equipped with dual RLM oracles:
O(?; t, j) = ? + t?j D(?)ej , t ? R, j ? [n],
(19)
?
?
O(?; j) = ? + t ej , t = argmin D(?1 , . . . , ?j?1 , ?j + t, ?j+1 , . . . , ?d ), j ? [n],
t?R
Following Scheme 2.1, we first describe the relevant parametrized subset of RLM problems. For the
sake of simplicity, we assume that n is even (the proof for odd n holds mutatis mutandis). We denote
by HRLM the set of all (?1 , . . . , ?n/2 ) ? Rn/2 such that all entries are 0, except for some j ? [n/2],
for which ?j ? [??/2, ?/2]. Now, given ? ? HRLM , we set P? (defined in 17) as follows
1
cos(?(i+1)/2 )ei + sin(?(i+1)/2 )ei+1 i is odd
2
?i (w) = (w + 1) , x?,i =
.
ei
o.w.
2
We state below the corresponding lower bound, whose proof, including logarithmic factors and
constants, can be found in Appendix A.4.
Theorem 4. The iteration complexity of oblivious (possibly stochastic) CLIs for RLM (17) equipped
with dual RLM oracles (19) is bounded from below by
p
? + nL/? ln(1/)).
?(n
This bound is tight w.r.t. the class of oblivious CLIs and is attained by accelerated proximal SDCA. As
?
mentioned earlier, a tighter lower bound of ?((n
+ 1/?) ln(1/)) is known for SDCA [3], suggesting
that a tighter bound might hold for the more restricted set of stationary CLIs (for which the oracle
parameters remain fixed throughout the optimization process).
8
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5,622 | 6,088 | Adversarial Multiclass Classification:
A Risk Minimization Perspective
Rizal Fathony
Anqi Liu
Kaiser Asif
Brian D. Ziebart
Department of Computer Science
University of Illinois at Chicago
Chicago, IL 60607
{rfatho2, aliu33, kasif2, bziebart}@uic.edu
Abstract
Recently proposed adversarial classification methods have shown promising results
for cost sensitive and multivariate losses. In contrast with empirical risk minimization (ERM) methods, which use convex surrogate losses to approximate the
desired non-convex target loss function, adversarial methods minimize non-convex
losses by treating the properties of the training data as being uncertain and worst
case within a minimax game. Despite this difference in formulation, we recast
adversarial classification under zero-one loss as an ERM method with a novel
prescribed loss function. We demonstrate a number of theoretical and practical
advantages over the very closely related hinge loss ERM methods. This establishes
adversarial classification under the zero-one loss as a method that fills the long
standing gap in multiclass hinge loss classification, simultaneously guaranteeing
Fisher consistency and universal consistency, while also providing dual parameter
sparsity and high accuracy predictions in practice.
1
Introduction
A common goal for standard classification problems in machine learning is to find a classifier that
minimizes the zero-one loss. Since directly minimizing this loss over training data via empirical
risk minimization (ERM) [1] is generally NP-hard [2], convex surrogate losses are employed to
approximate the zero-one loss. For example, the logarithmic loss is minimized by the logistic
regression classifier [3] and the hinge loss is minimized by the support vector machine (SVM) [4, 5].
Both are Fisher consistent [6, 7] and universally consistent [8, 9] for binary classification, meaning
they minimize the zero-one loss and are Bayes-optimal classifiers when they learn from any true
distribution of data using a rich feature representation. SVMs provide the additional advantage of dual
parameter sparsity so that when combined with kernel methods, extremely rich feature representations
can be efficiently considered. Unfortunately, generalizing the hinge loss to classification tasks with
more than two labels is challenging and existing multiclass convex surrogates [10?12] tend to lose
their consistency guarantees [13?15] or produce low accuracy predictions in practice [15].
Adversarial classification [16, 17] uses a different approach to tackle non-convex losses like the
zero-one loss. Instead of approximating the desired loss function and evaluating over the training
data, it adversarially approximates the available training data within a minimax game formulation
with game payoffs defined by the desired (zero-one) loss function [18, 19]. This provides promising
empirical results for cost-sensitive losses [16] and multivariate losses such as the F-measure and
the precision-at-k [17]. Conceptually, parameter optimization for the adversarial method forces the
adversary to ?behave like? certain properties of the training data sample, making labels easier to
predict within the minimax prediction game. However, a key bottleneck for these methods has been
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
their reliance on zero-sum game solvers for inference, which are computationally expensive relative
to inference in other prediction methods, such as SVMs.
In this paper, we recast adversarial prediction from an empirical risk minimization perspective by
analyzing the Nash equilibrium value of adversarial zero-one classification games to define a new
multiclass loss1 . This enables us to demonstrate that zero-one adversarial classification fills the long
standing gap in ERM-based multiclass classification by simultaneously: (1) guaranteeing Fisher
consistency and universal consistency; (2) enabling computational efficiency via the kernel trick and
dual parameter sparsity; and (3) providing competitive performance in practice. This reformulation
also provides significant computational efficiency improvements compared to previous adversarial
classification training methods [16].
2
Background and Related Work
2.1
Multiclass SVM generalizations
The multiclass support vector machine (SVM) seeks class-based potentials fy (xi ) for each input
vector x ? X and class y ? Y so that the discriminant function, y?f (xi ) = argmaxy fy (xi ),
minimizes misclassification errors, lossf (xi , yi ) = I(yi 6= y?f (xi )). Unfortunately, empirical risk
minimization (ERM), minf EP? (x,y) [lossf (X, Y )], for the zero-one loss is NP-hard once the set of
potentials is (parametrically) restricted (e.g., as a linear function of input features) [2]. Instead, a
hinge loss approximation is employed by the SVM. In the binary setting, yi ? {?1, +1}, where the
potential of one class can be set to zero (f?1 = 0) with no loss in generality, the hinge loss is defined
as [1 ? yi f+1 (xi )]+ , with the compact definition [g(.)]+ , max(0, g(.)). Binary SVM, which is an
empirical risk minimizer using the hinge loss with L2 regularization,
min EP? (x,y) [lossf? (X, Y )] + ?2 ||?||22 ,
f?
(1)
provides strong theoretical guarantees (Fisher consistency and universal consistency) [8, 21] and
computational efficiency [1].
Many methods have been proposed to generalize SVM to the multiclass setting. Apart from
the one-vs-all and one-vs-one decomposed formulations [22], there are three main joint formulations: the WW model by Weston et al. P
[11], which incorporates the sum of hinge losses
for all alternative labels, lossWW (xi , yi ) =
j6=yi [1 ? (fyi (xi ) ? fj (xi ))]+ ; the CS model
by Crammer and Singer [10], which uses the hinge loss of only the largest alternative label,
lossCS (xi , yi ) = maxj6=yi [1 ? (fyi (xi ) ? fj (xi ))]+ ; and the LLW model by Lee et al. [12], which
P
employs an absolute hinge loss, lossLLW (xi , yi ) =
j6=yi [1 + fj (xi )]+ , and a constraint that
P
j fj (xi ) = 0. The former two models (CS and WW) both utilize the pairwise class-based potential
differences fyi (xi ) ? fj (xi ) and are therefore categorized as relative margin methods. LLW, on the
other hand, is an absolute margin method that only relates to fj (xi )[15]. Fisher consistency, or Bayes
consistency [7, 13] guarantees that minimization of a surrogate loss for the true distribution provides
the Bayes-optimal classifier, i.e., minimizes the zero-one loss. If given any possible distribution of
data, a classifier is Bayes-optimal, it is called universally consistent. Of these, only the LLW method
is Fisher consistent and universally consistent [12?14]. However, as pointed out by Do?gan et al. [15],
LLW?s use of an absolute margin in the loss (rather than the relative margin of WW and CS) often
causes it to perform poorly for datasets with low dimensional feature spaces. From the opposite
direction, the requirements for Fisher consistency have been well-characterized [13], yet this has not
led to a multiclass classifier that is both Fisher consistent and performs well in practice.
2.2
Adversarial prediction games
Building on a variety of diverse formulations for adversarial prediction [23?26], Asif et al. [16]
proposed an adversarial game formulation for multiclass classification with cost-sensitive loss
functions. Under this formulation, the empirical training data is replaced by an adversarially chosen
conditional label distribution P? (?
y |x) that must closely approximate the training data, but otherwise
1
Farnia & Tse independently and concurrently discovered this same loss function [20]. They provide an
analysis focused on generalization bounds and experiments for binary classification.
2
seeks to maximize expected loss, while an estimator player P? (?
y |x) seeks to minimize expected loss.
For the zero-one loss, the prediction game is:
h
i
min
max
EP? (x)P? (?y|x)P? (?y|x) I(Y? 6= Y? ) .
(2)
P?
?
P? :EP (x)P? (y|x)
[?(X,Y? )]=?
?
The vector of feature moments, ?? = EP? (x,y) [?(X, Y )], is measured from sample training data.
Using minimax and strong Lagrangian duality, the optimization of Eq. (2) reduces to minimizing the
equilibrium game values of a new set of zero-sum games characterized by matrix L0xi ,? :
?
?
?1,yi (xi )
? ? ? ?|Y|,yi (xi ) + 1
X
?
?
..
..
..
? Txi L0xi ,? p
? xi ; L0xi ,? = ?
min
max min p
(3)
?;
.
.
.
?
p
?
?
p
i
?1,yi (xi ) + 1 ? ? ?
?|Y|,yi (xi )
? xi is a vector representation of the conditional
where ? is a vector of Lagrangian model parameters, p
? xi = [P? (Y? = 1|xi ) P? (Y? = 2|xi ) . . .]T , and similarly for
label distribution, P? (Y? = k|xi ), i.e. p
? xi . The matrix L0xi ,? is a zero-sum game matrix for each example, with ?j,yi (xi ) = fj (xi ) ?
p
fyi (xi ) = ?T (?(xi , j) ? ?(xi , yi )). This optimization problem (Eq. (3)) is convex in ? and the
inner zero-sum game can be solved using linear programming [16].
3
Risk Minimization Perspective of Adversarial Multiclass Classification
3.1
Nash equilibrium game value
Despite the differences in formulation between adversarial loss minimization and empirical risk
minimization, we now recast the zero-one loss adversarial game as the solution to an empirical
risk minimization problem. Theorem 1 defines the loss function that provides this equivalence by
considering all possible combinations of the adversary?s label assignments with non-zero probability
in the Nash equilibrium of the game.2
Theorem 1. The model parameters ? for multiclass zero-one adversarial classification are equivalently obtained from empirical risk minimization under the adversarial zero-one loss function:
P
j?S ?j,yi (xi ) + |S| ? 1
0-1
ALf (xi , yi ) =
max
,
(4)
|S|
S?{1,...,|Y|}, S6=?
where S is any non-empty member of the powerset of classes {1, 2, . . . , |Y|}.
Thus, AL0-1 is the maximum value over 2|Y| ? 1 linear hyperplanes. For binary prediction tasks, there are three linear
? (x)+?2,y (x)+1
hyperplanes: ?1,y (x), ?2,y (x) and 1,y
. Figure
2
1 shows the loss function in potential difference spaces ? when
the true label is y = 1. Note that AL0-1 combines two hinge
functions at ?2,y (x) = ?1 and ?2,y (x) = 1, rather than
SVM?s single hinge at ?1,y (x) = ?1. This difference from
the hinge loss corresponds to the loss that is realized by randomizing label predictions.3 For three classes, the loss function
has seven facets as shown in Figure 2a. Figures 2a, 2b, and
2c show the similarities and differences between AL0-1 and the
multiclass SVM surrogate losses based on class potential differences. Note that AL0-1 is also a relative margin loss function
that utilizes the pairwise potential difference ?j,y (x).
3.2
Figure 1: AL0-1 evaluated over
the space of potential differences
(?j,y (x) = fj (x) ? fy (x); and
?j,j (x) = 0) for binary prediction
tasks when the true label is y = 1.
Consistency properties
Fisher consistency is a desirable property for a surrogate loss function that guarantees its minimizer,
given the true distribution, P (x, y), will yield the Bayes optimal decision boundary [13, 14]. For
2
3
The proof of this theorem and others in the paper are contained in the Supplementary Materials.
We refer the reader to Appendix H for a comparison of the binary adversarial method and the binary SVM.
3
(a)
(b)
(c)
Figure 2: Loss function contour plots over the space of potential differences for the prediction task
with three classes when the true label is y = 1 under AL0-1 (a), the WW loss (b), and the CS loss (c).
(Note that ?i in the plots refers to ?j,y (x) = fj (x) ? fy (x); and ?j,j (x) = 0.)
multiclass zero-one loss, given that we know Pj (x) , P (Y = j|x), Fisher consistency requires
?
that argmaxj fj? (x) ? argmaxj Pj (x), where f ? (x) = [f1? (x), . . . , f|Y|
(x)]T is the minimizer of
E [lossf (X, Y )|X = x]. Since any constant can be added to all fj? (x) while keeping argmaxj fj? (x)
P|Y|
the same, we employ a sum-to-zero constraint, j=1 fj (x) = 0, to remove redundant solutions. We
establish an important property of the minimizer for AL0-1 in the following theorem.
Theorem 2. The loss for the minimizer f ? of E AL0-1
f (X, Y )|X = x resides on the hyperplane
defined (in Eq. 4) by the complete set of labels, S = {1, . . . , |Y|}.
As an illustration for the case of three classes (Figure 2a), the area described in the theorem
above corresponds to the region in the middle where the hyperplane that supports AL0-1 is
?1,y (x)+?2,y (x)+?3,y (x)+2
1
, and, equivalently, where ? |Y|
? fj (x) ? |Y|?1
3
|Y| , ?j ? {1, . . . , |Y|}
P
with a constraint that j fj (x) = 0. Based on this restriction, we focus on the minimization of
|Y|?1
1
E AL0-1
f (X, Y )|X = x subject to ? |Y| ? fj (x) ?
|Y| , ?j ? {1, . . . , |Y|} and the sum of
potentials equal to zero. This minimization reduces to the following optimization:
max
f
|Y|
X
Py (x)fy (x) subject to: ?
y=1
1
|Y| ? 1
? fj (x) ?
|Y|
|Y|
j ? {1, . . . , |Y|};
|Y|
X
fj (x) = 0.
j=1
The solution for this maximization (a linear program) satisfies fj? (x) = |Y|?1
|Y| if j = argmaxj Pj (x),
1
and ? |Y| otherwise, which therefore implies the Fisher consistency theorem.
Theorem 3. The adversarial zero-one loss, AL0-1 , from Eq. (4) is Fisher consistent.
Theorem 3 implies that AL0-1 (Eq. (4)) is classification calibrated, which indicates minimization
of that loss for all distributions on X ? Y also minimizes the zero-one loss [21, 13]. As proven in
general by Steinwart and Christmann [2], Micchelli et al. [27], since AL0-1 (Eq.(4)) is a Lipschitz
loss with constant 1, the adversarial multiclass classifier is universally consistent under the conditions
specified in Corollary 1.
Corollary 1. Given a universal kernel and regularization parameter ? in Eq. (1) tending to zero
slower than n1 , the adversarial multiclass classifier is also universally consistent.
3.3
Optimization
In the learning process for adversarial classification, Asif et al. [16] requires a linear program to be
solved that finds the Nash equilibrium game value and strategy for every training data point in each
gradient update. This requirement is computationally burdensome compared to multiclass SVMs,
which must simply find potential-maximizing labels. We propose two approaches with improved
4
efficiency by leveraging an oracle for finding the maximization inside AL0-1 and Lagrange duality in
the quadratic programming formulation.
3.3.1
Primal optimization using stochastic sub-gradient descent
TheP
sub-gradient in the empirical risk minimization of AL0-1 includes the mean of feature differences,
0-1
1
j?R [?(xi , j) ? ?(xi , yi )] , where R is the set that maximizes AL . The set R is computed
|R|
by the oracle using a greedy algorithm. Given ? and a sample (xi , yi ), the algorithm calculates all
potentials ?j,yi (xi ) for each label j ? {1, . . . , |Y|} and sorts them in non-increasing order. Starting
with the empty
P set R = ?, it then adds labels to R in sorted order until adding a label would decrease
?j,yi (xi )+|R|?1
the value of j?R |R|
.
Theorem 4. The proposed greedy algorithm used by the oracle is optimal.
3.3.2
Dual optimization
In the next subsections, we focus on the dual optimization technique as it enables us to establish
convergence guarantees. We re-formulate the learning algorithm (with L2 regularization) as a
constrained quadratic program (QP) with ?i specifying the amount of AL0-1 incurred by each of the n
training examples:
n
X
1
?i
min k?k2 + C
? 2
i=1
subject to:
?i ? ?i,k
?i ? {1, . . . n}k ? {1, . . . , 2|Y| ? 1},
(5)
where we denote each of the 2|Y| ?1 possible constraints for example i corresponding
to non-empty elP
j?Y ?j,yi (xi )+|Y|?1
ements of the label powerset as ?i,k (e.g., ?i,1 = ?1,yi (xi ), and ?i,2|Y| ?1 =
).
|Y|
Note also that non-negativity for ?i is enforced since ?i,yi = ?yi ,yi (xi ) = 0.
Theorem 5. Let ?i,k be the partial derivative of ?i,k with respect to ?, i.e., ?i,k =
?1,yi (xi )+?3,yi (xi )+?4,yi (xi )+2
,
3
is the constant part of ?i,k (for example if ?i,k =
then the corresponding dual optimization for the primal minimization (Eq. 5) is:
|Y|
max
?
n 2X
?1
X
i=1
k=1
d?i,k
d?
then ?i,k = 23 ),
|Y|
m 2 ?1
1 X X
?i,k ?i,k ?
?i,k ?j,l [?i,k ? ?j,l ]
2 i,j=1
subject to: ?i,k ? 0,
and ?i,k
(6)
k,l=1
2|Y|
?1
X
?i,k = C, i ? {1, . . . , n}, k ? {1, . . . , 2|Y| ? 1},
k=1
where ?i,k is the dual variable for the k-th constraint of the i-th sample.
Note that the dual formulation above only depends on the dot product of two constraints? partial derivatives (with respect to ?) and the constant part of the constraints. The original primal variable ? can be
Pn P2|Y| ?1
recovered from the dual variables using the formula: ? = ? i=1 k=1 ?i,k ?i,k . Given a new
datapoint x, de-randomized predictions are obtained from argmaxj fj (x) = argmaxj ?T ?(x, j).
3.3.3
Efficiently incorporating rich feature spaces using kernelization
Considering large feature spaces is important for developing an expressive classifier that can learn
from large amounts of training data. Indeed, Fisher consistency requires such feature spaces for its
guarantees to be meaningful. However, na?vely projecting from the original input space, xi , to richer
(or possibly infinite) feature spaces ?(xi ), can be computationally burdensome. Kernel methods
enable this feature expansion by allowing the dot products of certain feature functions to be computed
implicitly, i.e., K(xi , xj ) = ?(xi ) ? ?(xj ). Since our dual formulation only depends on dot products,
we employ kernel methods to incorporate rich feature spaces into our formulation as stated in the
following theorem.
Theorem 6. Let X be the input space and K be a positive definite real valued kernel on X ? X with
a mapping function ?(x) : X ? H that maps the input space X to a reproducing kernel Hilbert
5
space H. Then all the values in the dual optimization of Eq. (6) needed to operate in the Hilbert
space H can be computed in terms of the kernel function K(xi , xj ) as:
|Y|
?i,k ? ?j,l = c(i,k),(j,l) K(xi , xj ),
?i,k = ?
n 2X
?1
X
j=1
?j,l c(j,l),(i,k) K(xj , xi ) + ?i,k ,
(7)
l=1
|Y|
fm (xi ) = ?
n 2X
?1
X
j=1
where c(i,k),(j,l) =
?j,l
l=1
1(m ? Rj,l )
? 1(m = yj ) K(xj , xi ) ,
|Rj,l |
(8)
|Y|
X
1(m ? Ri,k )
1(m ? Rj,l )
? 1(m = yi )
? 1(m = yj ) ,
|Ri,k |
|Rj,l |
m=1
and Ri,k is the set of labels included in the constraint ?i,k (for example if ?i,k =
?1,yi (xi )+?3,yi (xi )+?4,yi (xi )+2
, then Ri,k = {1, 3, 4}), the function 1(j = yi ) returns 1 if j = yi or
3
0 otherwise, and the function 1(j ? Ri,k ) returns 1 if j is a member of set Ri,k or 0 otherwise.
3.3.4
Efficient optimization using constraint generation
The number of constraints in the QP formulation above grows exponentially with the number
of classes: O(2|Y| ). This prevents the na?ve formulation from being efficient for large multiclass problems. We employ a constraint generation method to efficiently solve the dual quadratic
programming formulation that is similar to those used for extending the SVM to multivariate loss
functions [28] and structured prediction settings [29].
Algorithm 1 Constraint generation method
Require: Training data (x1 , y1 ), . . . (xn , yn ), C,
1: ? ? 0
2: A?i ? {?i,k |?i,k = ?yi ,yi (xi )} ?i = 1, . . . , n
. Actual label enforces non-negativity
3: repeat
4:
for i ? 1, n do
. Find the most violated constraint
5:
a ? arg maxk|?i,k ?Ai ?i,k
6:
?i ? maxk|?i,k ?A?i ?i,k
. Compute the example?s current loss estimate
7:
if ?i,a > ?i + then
8:
A?i ? A?i ? {?i,a }
. Add it to the enforced constraints set
9:
? ? Optimize dual over A?P= ?iP
A?i
n
10:
Compute ? from ?: ? = ? i=1 k|?i,k ?A? ?i,k ?i,k
i
11:
end if
12:
end for
13: until no A?i has changed in the iteration
Algorithm 1 incrementally expands the set of enforced constraints, A?i , until no remaining constraint
from the set of all 2|Y| ? 1 constraints (in Ai ) is violated by more than . To obtain the most
violated constraint, we use the greedy algorithm described in the primal optimization. The constraint
generation algorithm?s stopping criterion ensures that a solution close to the optimal is returned
(violating no constraint by more than ). Theorem 7 provides a polynomial run time convergence
bounds for the Algorithm 1.
Theorem 7. For any > 0 and training dataset {(x1 , y1 ), . . . , (xn, yn )} with U = maxi [xi ? xi ],
4nCU
Algorithm 1 terminates after incrementally adding at most max 2n
constraints to the
, 2
?
constraint set A .
The proof of Theorem 7 follows the procedures developed by Tsochantaridis et al. [28] for bounding
the running time of structured support vector machines. We observe that this bound is quite loose in
practice and the algorithm tends to converge much faster in our experiments.
6
4
Experiments
We evaluate the performance of the AL0-1 classifier and compare with the three most popular
multiclass SVM formulations: WW [11], CS [10], and LLW [12]. We use 12 datasets from the UCI
Machine Learning repository [30] with various sizes and numbers of classes (details in Table 1). For
each dataset, we consider the methods using the original feature space (linear kernel) and a kernelized
feature space using the Gaussian radial basis function kernel.
Table 1: Properties of the datasets, the number of constraints considered by SVM models
(WW/CS/LLW), the average number of constraints added to the constraint set for AL0-1 and the
average number of active constraints at the optima under both linear and Gausssian kernels.
Properties
Dataset
(1) iris
(2) glass
(3) redwine
(4) ecoli
(5) vehicle
(6) segment
(7) sat
(8) optdigits
(9) pageblocks
(10) libras
(11) vertebral
(12) breasttissue
# class
# train
# test
# feature
SVM
constraints
3
6
10
8
4
7
7
10
5
15
3
6
105
149
1119
235
592
1617
4435
3823
3831
252
217
74
45
65
480
101
254
693
2000
1797
1642
108
93
32
4
9
11
7
18
19
36
64
10
90
6
9
210
745
10071
1645
1776
9702
26610
34407
15324
3528
434
370
AL0-1 constraints added and active
Linear kernel
Gauss. kernel
213
578
5995
614
1310
4410
11721
7932
9459
1592
344
258
223
490
3811
821
1201
4312
11860
10072
9155
1165
342
271
13
125
1681
117
311
244
1524
597
427
389
78
65
38
252
1783
130
248
469
6269
2315
551
353
86
145
For our experimental methodology, we first make 20 random splits of each dataset into training and
testing sets. We then perform two stage, five-fold cross validation on the training set of the first
split to tune each model?s parameter C and the kernel parameter ? under the kernelized formulation.
In the first stage, the values for C are 2i , i = {0, 3, 6, 9, 12} and the values for ? are 2i , i =
{?12, ?9, ?6, ?3, 0}. We select final values for C from 2i C0 , i = {?2, ?1, 0, 1, 2} and values for
? from 2i ?0 , i = {?2, ?1, 0, 1, 2} in the second stage, where C0 and ?0 are the best parameters
obtained in the first stage. Using the selected parameters, we train each model on the 20 training sets
and evaluate the performance on the corresponding testing set. We use the Shark machine learning
library [31] for the implementation of the three multiclass SVM formulations.
Despite having an exponential number of possible constraints (i.e., n(2|Y| ? 1) for n examples versus
n(|Y| ? 1) for SVMs), a much smaller number of constraints need to be considered by the AL0-1
algorithm in practice to realize a better approximation ( = 0) than Theorem 7 provides. Table 1
shows how the total number of constraints for multiclass SVM compares to the number considered
in practice by our AL0-1 algorithm for linear and Gaussian kernel feature spaces. These range from
a small fraction (0.23) of the SVM constraints for optdigits to a slightly greater number (with a
fraction of 1.06) for iris. More specifically, of the over 3.9 million (= 210 ?3823) possible constraints
for optdigits when training the classifier, fewer than 0.3% (7932 or 10072 depending on the feature
representation) are added to the constraint set during the constraint generation process. Fewer still
(597 or 2315 constraints?less than 0.06%) are constraints that are active in the final classifier
with non-zero dual parameters. The sparsity of the dual parameters provides a key computational
benefit for support vector machines over logistic regression, which has essentially all non-zero dual
parameters. The small number of active constraints shown in Table 1 demonstrate that AL0-1 induces
similar sparsity, providing efficiency when employed with kernel methods.
We report the accuracy of each method averaged over the 20 dataset splits for both linear feature
representations and Gaussian kernel feature representations in Table 2. We denote the results that
are either the best of all four methods or not worse than the best with statistical significance (under
paired t-test with ? = 0.05) using bold font. We also show the accuracy averaged over all of the
datasets for each method and the number of dataset for which each method is ?indistinguishably best?
(bold numbers) in the last row. As we can see from the table, the only alternative model that is Fisher
7
Table 2: The mean and (in parentheses) standard deviation of the accuracy for each model with linear
kernel and Gaussian kernel feature representations. Bold numbers in each case indicate that the result
is the best or not significantly worse than the best (paired t-test with ? = 0.05).
Linear Kernel
D
AL
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
avg
#bold
0-1
Gaussian Kernel
AL
0-1
WW
CS
LLW
WW
CS
96.3 (3.1)
62.5 (6.0)
58.8 (2.0)
86.2 (2.2)
78.8 (2.2)
94.9 (0.7)
84.9 (0.7)
96.6 (0.6)
96.0 (0.5)
74.1 (3.3)
85.5 (2.9)
64.4 (7.1)
96.0 (2.6)
62.2 (3.6)
59.1 (1.9)
85.7 (2.5)
78.8 (1.7)
94.9 (0.8)
85.4 (0.7)
96.5 (0.7)
96.1 (0.5)
72.0 (3.8)
85.9 (2.7)
59.7 (7.8)
96.3 (2.4)
62.5 (3.9)
56.6 (2.0)
85.8 (2.3)
78.4 (2.3)
95.2 (0.8)
84.7 (0.7)
96.3 (0.6)
96.3 (0.5)
71.3 (4.3)
85.4 (3.3)
66.3 (6.9)
79.7 (5.5)
52.8 (4.6)
57.7 (1.7)
74.1 (3.3)
69.8 (3.7)
75.8 (1.5)
74.9 (0.9)
76.2 (2.2)
92.5 (0.8)
34.0 (6.4)
79.8 (5.6)
58.3 (8.1)
96.7 (2.4)
69.5 (4.2)
63.3 (1.8)
86.0 (2.7)
84.3 (2.5)
96.5 (0.6)
91.9 (0.5)
98.7 (0.4)
96.8 (0.5)
83.6 (3.8)
86.0 (3.1)
68.4 (8.6)
96.4 (2.4)
66.8 (4.3)
64.2 (2.0)
84.9 (2.4)
84.4 (2.6)
96.6 (0.5)
92.0 (0.6)
98.8 (0.4)
96.6 (0.4)
83.8 (3.4)
85.3 (2.9)
68.1 (6.5)
96.2 (2.3)
69.4 (4.8)
64.2 (1.9)
85.6 (2.4)
83.8 (2.3)
96.3 (0.6)
91.9 (0.5)
98.8 (0.3)
96.7 (0.4)
85.0 (3.9)
85.5 (3.3)
66.6 (8.9)
95.4 (2.1)
69.2 (4.4)
64.7 (2.1)
86.0 (2.5)
84.4 (2.6)
96.4 (0.5)
91.9 (0.4)
98.9 (0.3)
96.6 (0.4)
83.2 (4.2)
84.4 (2.7)
68.0 (7.2)
LLW
81.59
9
81.02
6
81.25
8
68.80
0
85.14
9
84.82
6
85.00
6
84.93
7
consistent?the LLW model?performs poorly on all datasets when only linear features are employed.
This matches with previous experimental results conducted by Do?gan et al. [15] and demonstrates a
weakness of using an absolute margin for the loss function (rather than the relative margins of all other
methods). The AL0-1 classifier performs competitively with the WW and CS models with a slight
advantages on overall average accuracy and a larger number of ?indistinguishably best? performances
on datasets?or, equivalently, fewer statistically significant losses to any other method.
The kernel trick in the Gaussian kernel case provides access to much richer feature spaces, improving
the performance of all models, and the LLW model especially. In general, all models provide
competitive results in the Gaussian kernel case. The AL0-1 classifier maintains a similarly slight
advantage and only provides performance that is sub-optimal (with statistical significance) in three
of the twelve datasets versus six of twelve and five of twelve for the other methods. We conclude
that the multiclass adversarial method performs well in both low and high dimensional feature
spaces. Recalling the theoretical analysis of the adversarial method, it is a well-motivated (from
the adversarial zero-one loss minimization) multiclass classifier that enjoys both strong theoretical
properties (Fisher consistency and universal consistency) and empirical performance.
5
Conclusion
Generalizing support vector machines to multiclass settings in a theoretically sound manner remains a
long-standing open problem. Though the loss function requirements guaranteeing Fisher-consistency
are well-understood [13], the few Fisher-consistent classifiers that have been developed (e.g., LLW)
often are not competitive with inconsistent multiclass classifiers in practice. In this paper, we
have sought to fill this gap between theory and practice. We have demonstrated that multiclass
adversarial classification under zero-one loss can be recast from an empirical risk minimization
perspective and its surrogate loss, AL0-1 , shown to satisfy the Fisher consistency property, leading
to a universally consistent classifier that also performs well in practice. We believe that this is an
important contribution in understanding both adversarial methods and the generalized hinge loss. Our
future work includes investigating the adversarial methods under the different losses and exploring
other theoretical properties of the adversarial framework, including generalization bounds.
Acknowledgments
This research was supported as part of the Future of Life Institute (futureoflife.org) FLI-RFP-AI1
program, grant#2016-158710 and by NSF grant RI-#1526379.
8
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9
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5,623 | 6,089 | Graphons, mergeons, and so on!
Justin Eldridge Mikhail Belkin Yusu Wang
The Ohio State University
{eldridge, mbelkin, yusu}@cse.ohio-state.edu
Abstract
In this work we develop a theory of hierarchical clustering for graphs. Our modeling assumption is that graphs are sampled from a graphon, which is a powerful
and general model for generating graphs and analyzing large networks. Graphons
are a far richer class of graph models than stochastic blockmodels, the primary
setting for recent progress in the statistical theory of graph clustering. We de?ne
what it means for an algorithm to produce the ?correct" clustering, give su?cient
conditions in which a method is statistically consistent, and provide an explicit
algorithm satisfying these properties.
1
Introduction
A fundamental problem in the theory of clustering is that of de?ning a cluster. There is no single
answer to this seemingly simple question. The right approach depends on the nature of the data
and the proper modeling assumptions. In a statistical setting where the objects to be clustered
come from some underlying probability distribution, it is natural to de?ne clusters in terms of the
distribution itself. The task of a clustering, then, is twofold ? to identify the appropriate cluster
structure of the distribution and to recover that structure from a ?nite sample. Thus we would like
to say that a clustering is good if it is in some sense close to the ideal structure of the underlying
distribution, and that a clustering method is consistent if it produces clusterings which converge to
the true clustering, given larger and larger samples. Proving the consistency of a clustering method
deepens our understanding of it, and provides justi?cation for using the method in the appropriate
setting.
In this work, we consider the setting in which the objects to be clustered are the vertices of a graph
sampled from a graphon ? a very general random graph model of signi?cant recent interest. We
develop a statistical theory of graph clustering in the graphon model; To the best of our knowledge,
this is the ?rst general consistency framework developed for such a rich family of random graphs.
The speci?c contributions of this paper are threefold. First, we de?ne the clusters of a graphon. Our
de?nition results in a graphon having a tree of clusters, which we call its graphon cluster tree. We
introduce an object called the mergeon which is a particular representation of the graphon cluster
tree that encodes the heights at which clusters merge. Second, we develop a notion of consistency
for graph clustering algorithms in which a method is said to be consistent if its output converges to
the graphon cluster tree. Here the graphon setting poses subtle yet fundamental challenges which
di?erentiate it from classical clustering models, and which must be carefully addressed. Third, we
prove the existence of consistent clustering algorithms. In particular, we provide su?cient conditions under which a graphon estimator leads to a consistent clustering method. We then identify a
speci?c practical algorithm which satis?es these conditions, and in doing so present a simple graph
clustering algorithm which provably recovers the graphon cluster tree.
Related work. Graphons are objects of signi?cant recent interest in graph theory, statistics, and
machine learning. The theory of graphons is rich and diverse; A graphon can be interpreted as
a generalization of a weighted graph with uncountably many nodes, as the limit of a sequence of
?nite graphs, or, more importantly for the present work, as a very general model for generating
29th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
unweighted, undirected graphs. Conveniently, any graphon can be represented as a symmetric, measurable function W : [0, 1]2 ? [0, 1], and it is this representation that we use throughout this paper.
The graphon as a graph limit was introduced in recent years by [16], [5], and others. The interested
reader is directed to the book by Lov?sz [15] on the subject. There has also been a considerable
recent e?ort to produce consistent estimators of the graphon, including the work of [20], [8], [2],
[18], and others. We will analyze a simple modi?cation of the graphon estimator proposed by [21]
and show that it leads to a graph clustering algorithm which is a consistent estimator of the graphon
cluster tree.
Much of the previous statistical theory of graph clustering methods assumes that graphs are generated by the so-called stochastic blockmodel. The simplest form of the model generates a graph
with n nodes by assigning each node, randomly or deterministically, to one of two communities. An
edge between two nodes is added with probability ? if they are from the same community and with
probability ? otherwise. A graph clustering method is said to achieve exact recovery if it identi?es
the true community assignment of every node in the graph with high probability as n ? ?. The
blockmodel is a special case of a graphon model, and our notion of consistency will imply exact
recovery of communities.
Stochastic blockmodels are widely studied, and it is known that, for example, spectral methods
like that of [17] are able to recover the communities exactly as n ? ?, provided that ? and ?
remain constant, or that the gap between them does not shrink too quickly. For a summary of
consistency results in the blockmodel, see [1], which also provides information-theoretic thresholds
for the conditions under which exact recovery is possible. In a related direction, [4] examines the
ability of spectral clustering to withstand noise in a hierarchical block model.
The density setting. The problem of de?ning the underlying cluster structure of a probability distribution goes back to Hartigan [12] who considered the setting in which the objects to be clustered
are points sampled from a density f : X ? R+ . In this case, the high density clusters of f are
de?ned to be the connected components of the upper level sets {x : f (x) ? ?} for any ? > 0. The
set of all such clusters forms the so-called density cluster tree. Hartigan [12] de?ned a notion of
consistency for the density cluster tree, and proved that single-linkage clustering is not consistent.
In recent years, [9] and [14] have demonstrated methods which are Hartigan consistent. [10] introduced a distance between a clustering of the data and the density cluster tree, called the merge
distortion metric. A clustering method is said to be consistent if the trees it produces converge in
merge distortion to density cluster tree. It is shown that convergence in merge distortion is stronger
than Hartigan consistency, and that the method of [9] is consistent in this stronger sense.
In the present work, we will be motivated by the approach taken in [12] and [10]. We note, however,
that there are signi?cant and fundamental di?erences between the density case and the graphon
setting. Speci?cally, it is possible for two graphons to be equivalent in the same way that two
graphs are: up to a relabeling of the vertices. As such, a graphon W is a representative of an
equivalence class of graphons modulo appropriately de?ned relabeling. It is therefore necessary to
de?ne the clusters of W in a way that does not depend upon the particular representative used. A
similar problem occurs in the density setting when we wish to de?ne the clusters not of a single
density function, but rather of a class of densities which are equal almost everywhere; Steinwart
[19] provides an elegant solution. But while the domain of a density is equipped with a meaningful
metric ? the mass of a ball around a point x is the same under two equivalent densities ? the ambient
metric on the vertices of a graphon is not useful. As a result, approaches such as that of [19] do not
directly apply to the graphon case, and we must carefully produce our own. Additionally, we will
see that the procedure for sampling a graph from a graphon involves latent variables which are in
principle unrecoverable from data. These issues have no analogue in the classical density setting,
and present very distinct challenges.
Miscellany. Due to space constraints, most of the (rather involved) technical details are in the
appendix. We will use [n] to denote the set {1, . . . , n}, ? for the symmetric di?erence, ? for the
Lebesgue measure on [0, 1], and bold letters to denote random variables.
2
2
The graphon model
In order to discuss the statistical properties of a graph clustering algorithm, we must ?rst model the
process by which graphs are generated. Formally, a random graph model is a sequence of random
variables G1 , G2 , . . . such that the range of Gn consists of undirected, unweighted graphs with node
set [n], and the distribution of Gn is invariant under relabeling of the nodes ? that is, isomorphic
graphs occur with equal probability. A random graph model of considerable recent interest is the
graphon model, in which the distribution over graphs is determined by a symmetric, measurable
function W : [0, 1]2 ? [0, 1] called a graphon. Informally, a graphon W may be thought of as the
weight matrix of an in?nite graph whose node set is the continuous unit interval, so that W(x, y)
represents the weight of the edge between nodes x and y.
Interpreting W(x, y) as a probability suggests the following graph sampling procedure: To draw a
graph with n nodes, we ?rst select n points x1 , . . . , xn at random from the uniform distribution on
[0, 1] ? we can think of these xi as being random ?nodes? in the graphon. We then sample a random
graph G on node set [n] by admitting the edge (i, j) with probability W(xi , x j ); by convention, selfedges are not sampled. It is important to note that while we begin by drawing a set of nodes {xi }
from the graphon, the graph as given to us is labeled by integers. Therefore, the correspondence
between node i in the graph and node xi in the graphon is latent.
It can be shown that this sampling procedure de?nes a distribution on ?nite graphs, such that the
probability of graph G = ([n], E) is given by
?
?
? [
]?
1 ? W(xi , x j )
W(xi , x j )
dxi .
(1)
PW (G = G) =
[0,1]n (i, j)?E
(i, j)?E
i?[n]
A very general class of random graph models may be represented as graphons.
In particular, a random graph model G1 , G2 , . . . is said to be consistent if the
random graph Fk?1 obtained by deleting node k from Gk has the same distribution
as Gk . A random graph model is said to be local if whenever S , T ? [k] are
disjoint, the random subgraphs of Gk induced by S and T are independent random
variables. A result of Lov?sz and Szegedy [16] is that any consistent, local random
graph model is equivalent to the distribution on graphs de?ned by PW for some
graphon W; the converse is true as well. That is, any such random graph model is
equivalent to a graphon.
{
{
{
For a ?xed choice of x1 , . . . , xn ? [0, 1], the integrand represents the likelihood that the graph G is
sampled when the probability of the edge (i, j) is assumed to be W(xi , x j ). By integrating over all
possible choices of x1 , . . . , xn , we obtain the probability of the graph.
(a) Graphon W.
A particular random graph model is not uniquely de?ned by a graphon ? it is clear
from Equation 1 that two graphons W1 and W2 which are equal almost everywhere
(i.e., di?er on a set of measure zero) de?ne the same distribution on graphs. In
fact, the distribution de?ned by W is unchanged by ?relabelings? of W?s nodes.
More formally, if ? is the sigma-algebra of Lebesgue measurable subsets of [0, 1]
and ? is the Lebesgue measure, we say that a relabeling function ? : ([0, 1], ?) ?
([0, 1], ?) is measure preserving if for any measurable set A ? ?, ?(??1 (A)) = ?(A).
?
We de?ne the relabeled graphon W ? by W ? (x, y) = W(?(x), ?(y)). By analogy (b) W weakly
isomorphic to
with ?nite graphs, we say that graphons W1 and W2 are weakly isomorphic if they
W.
are equivalent up to relabeling, i.e., if there exist measure preserving maps ?1 and
?1
?2
?2 such that W1 = W2 almost everywhere. Weak isomorphism is an equivalence
relation, and most of the important properties of a graphon in fact belong to its
equivalence class. For instance, a powerful result of [15] is that two graphons
de?ne the same random graph model if and only if they are weakly isomorphic.
An example of a graphon W is shown in Figure 1a. It is conventional to plot
the graphon as one typically plots an adjacency matrix: with the origin in the (c) An instance
upper-left corner. Darker shades correspond to higher values of W. Figure 1b of a graph addepicts a graphon W ? which is weakly isomorphic to W. In particular, W ? is jacency sampled
the relabeling of W by the measure preserving transformation ?(x) = 2x mod 1. from W.
As such, the graphons shown in Figures 1a and 1b de?ne the same distribution
Figure 1
on graphs. Figure 1c shows the adjacency matrix A of a graph of size n = 50
3
sampled from the distribution de?ned by the equivalence class containing W and W ? . Note that it is
in principle not possible to determine from A alone which graphon W or W ? it was sampled from,
or to what node in W a particular column of A corresponds to.
3
The graphon cluster tree
We now identify the cluster structure of a graphon. We will de?ne a graphon?s clusters such that
they are analogous to the maximally-connected components of a ?nite graph. It turns out that the
collection of all clusters has hierarchical structure; we call this object the graphon cluster tree. We
propose that the goal of clustering in the graphon setting is the recovery of the graphon cluster tree.
Connectedness and clusters. Consider a ?nite weighted graph. It is natural to cluster the graph
into connected components. In fact, because of the weighted edges, we can speak of the clusters of
the graph at various levels. More precisely, we say that a set of nodes A is internally connected ?
or, from now on, just connected ? at level ? if for every pair of nodes in A there is a path between
them such that every node along the path is also in A, and the weight of every edge in the path is at
least ?. Equivalently, A is connected at level ? if and only if for every partitioning of A into disjoint,
non-empty sets A1 and A2 there is an edge of weight ? or greater between A1 and A2 . The clusters at
level ? are then the largest connected components at level ?.
A graphon is, in a sense, an in?nite weighted graph, and we will de?ne the clusters of a graphon
using the example above as motivation. In doing so, we must be careful to make our notion robust to
changes of the graphon on a set of zero measure, as such changes do not a?ect the graph distribution
de?ned by the graphon. We base our de?nition on that of Janson [13], who de?ned what it means for
a graphon to be connected as a whole. We extend the de?nition in [13] to speak of the connectivity
of subsets of the graphon?s nodes at a particular height. Our de?nition is directly analogous to the
notion of internal connectedness in ?nite graphs.
De?nition 1 (Connectedness). Let W be a graphon, and let A ? [0, 1] be a set of positive measure.
We say that A is disconnected at level ? if there exists a measurable S ? A such that 0 < ?(S ) < ?(A),
and W < ? almost everywhere on S ? (A \ S ). Otherwise, we say that A is connected at level ?.
We now identify the clusters of a graphon; as in the ?nite case, we will frame our de?nition in terms
of maximally-connected components. We begin by gathering all subsets of [0, 1] which should
belong to some cluster at level ?. Naturally, if a set is connected at level ?, it should be in a cluster
at level ?; for technical reasons, we will also say that a set which is connected at all levels ?? < ?
(though perhaps not at ?) should be contained in a cluster at level ?, as well. That is, for any ?, the
collection A? of sets which should be contained in some cluster at level ? is A? = { A ? ? : ?(A) >
0 and A is connected at every level ?? < ?}. Now suppose A1 , A2 ? A? , and that there is a set A ? A?
such that A ? A1 ? A2 . Naturally, the cluster to which A belongs should also contain A1 and A2 , since
both are subsets of A. We will therefore consider A1 and A2 to be equivalent, in the sense that they
should be contained in the same cluster at level ?. More formally, we de?ne a relation ?? on A?
by A1 ?? A2 ?? ?A ? A? s.t. A ? A1 ? A2 . It can be veri?ed that ?? is an equivalence relation
on A? ; see Claim 9 in Appendix B.
Each equivalence class A in the quotient space A? /?? . consists of connected sets which should
intuitively be clustered together at level ?. Naturally, we will de?ne the clusters to be the largest
elements of each class; in some sense, these are the maximally-connected components at level ?.
More precisely, suppose A is such an equivalence class. It is clear that in general no single member
A ? A can contain all other members of A , since adding a null set (i.e., a set of measure zero) to A
results in a larger set A? which is nevertheless still a member of A . However, we can ?nd a member
A? ? A which contains all but a null set of every other set in A . More formally, we say that A?
is an essential maximum of the class A if A? ? A and for every A ? A , ?(A \ A? ) = 0. A? is of
course not unique, but it is unique up to a null set; i.e., for any two essential maxima A1 , A2 of A ,
we have ?(A1 ? A2 ) = 0. We will write the set of essential maxima of A as ess max A ; the fact that
the essential maxima are well-de?ned is proven in Claim 10 in Appendix B. We then de?ne clusters
as the maximal members of each equivalence class in A? /?? :
De?nition 2 (Clusters). The set of clusters at level ? in W, written CW (?), is de?ned to be the
countable collection CW (?) = { ess max A : A ? A? /?? } .
4
Note that a cluster C of a graphon is not a subset of the unit interval per se, but rather an equivalence
class of subsets which di?er only by null sets. It is often possible to treat clusters as sets rather
than equivalence classes, and we may write ?(C ), C ? C ? , etc., without ambiguity. In addition, if
? : [0, 1] ? [0, 1] is a measure preserving transformation, then ??1 (C ) is well-de?ned.
For a concrete example of our notion of a cluster, consider the graphon W depicted in Figure 1a. A,
B, and C represent sets of the graphon?s nodes. By our de?nitions there are three clusters at level
?3 : A , B, and C . Clusters A and B merge into a cluster A ? B at level ?2 , while C remains a
separate cluster. Everything is joined into a cluster A ? B ? C at level ?1 .
We have taken care to de?ne the clusters of a graphon in such a way as to be robust to changes of
measure zero to the graphon itself. In fact, clusters are also robust to measure preserving transformations. The proof of this result is non-trivial, and comprises Appendix C.
Claim 1. Let W be a graphon and ? a measure preserving transformation. Then C is a cluster of
W ? at level ? if and only if there exists a cluster C ? of W at level ? such that C = ??1 (C ? ).
Cluster trees and mergeons. The set of all clusters of a graphon at any level has hierarchical
structure in the sense that, given any pair of distinct clusters C1 and C2 , either one is ?essentially?
contained within the other, i.e., C1 ? C2 , or C2 ? C1 , or they are ?essentially? disjoint, i.e., ?(C1 ?
C2 ) = 0, as is proven by Claim 8 in Appendix B. Because of this hierarchical structure, we call the
set CW of all clusters from any level of the graphon W the graphon cluster tree of W. It is this tree
that we hope to recover by applying a graph clustering algorithm to a graph sampled from W.
Nevertheless, consider a measurable function M : [0, 1]2 ? [0, 1] which assigns
a merge height to every pair of points. While the value of M on any given pair is
arbitrary, the value of M on sets of positive measure is constrained. Intuitively, if
C is a cluster at level ?, then we must have M ? ? almost everywhere on C ? C .
If M satis?es this constraint for every cluster C we call M a mergeon for C, as it
is a graphon which determines a particular choice for the merge heights of every
pair of points in [0, 1]. More formally:
De?nition 3 (Mergeon). Let C be a cluster?tree. A mergeon1 of C is a graphon
M such that for all ? ? [0, 1], M ?1 [?, 1] = C ?CW (?) C ? C , where M ?1 [?, 1] =
{(x, y) ? [0, 1]2 : M(x, y) ? ?}.
(a) Cluster tree
CW of W.
{
{
{
We may naturally speak of the height at which pairs of distinct clusters merge in
the cluster tree. For instance, let C1 and C2 be distinct clusters of C. We say that
the merge height of C1 and C2 is the level ? at which they are joined into a single
cluster, i.e., max{? : C1 ? C2 ? C(?)}. However, while the merge height of clusters
is well-de?ned, the merge height of individual points is not. This is because the
cluster tree is not a collection of sets, but rather a collection of equivalence classes
of sets, and so a point does not belong to any one cluster more than any other. Note
that this is distinct from the classical density case considered in [12], [9], and [1],
where the merge height of any pair of points is well-de?ned.
(b) Mergeon M
of CW .
Figure 2
An example of a mergeon and the cluster tree it represents is shown in Figure 2. In fact, the cluster
tree depicted is that of the graphon W from Figure 1a. The mergeon encodes the height at which
clusters A , B, and C merge. In particular, the fact that M = ?2 everywhere on A ? B represents
the merging of A and B at level ?2 in W.
It is clear that in general there is no unique mergeon representing a graphon cluster tree, however,
the above de?nition implies that two mergeons representing the same cluster tree are equal almost
everywhere. Additionally, we have the following two claims, whose proofs are in Appendix B.
Claim 2. Let C be a cluster tree, and suppose M is a mergeon representing C. Then C ? C(?) if
and only if C is a cluster in M at level ?. In other words, the cluster tree of M is also C.
Claim 3. Let W be a graphon and M a mergeon of the cluster tree of W. If ? is a measure preserving
transformation, then M ? is a mergeon of the cluster tree of W ? .
1
The de?nition given here involves a slight abuse of notation. For a precise ? but more technical ? version,
see Appendix A.2.
5
4
Notions of consistency
We have so far de?ned the sense in which a graphon has hierarchical cluster structure. We now
turn to the problem of determining whether a clustering algorithm is able to recover this structure
when applied to a graph sampled from a graphon. Our approach is to de?ne a distance between the
in?nite graphon cluster tree and a ?nite clustering. We will then de?ne consistency by requiring that
a consistent method converge to the graphon cluster tree in this distance for all inputs minus a set of
vanishing probability.
Merge distortion. A hierarchical clustering C of a set S ? or, from now on, just a clustering of
S ? is hierarchical collection of subsets of S such that S ? C and for all C, C ? ? C, either C ? C ? ,
C ? ? C, or C ? C ? = ?. Suppose C is a clustering of a ?nite set S consisting of graphon nodes; i.e,
S ? [0, 1]. How might we measure the distance between this clustering and a graphon cluster tree
C? Intuitively, the two trees are close if every pair of points in S merges in C at about the same level
as they merge in C. But this informal description faces two problems: First, C is a collection of
equivalence classes of sets, and so the height at which any pair of points merges in C is not de?ned.
Recall, however, that the cluster tree has an alternative representation as a mergeon. A mergeon does
de?ne a merge height for every pair of nodes in a graphon, and thus provides a solution to this ?rst
issue. Second, the clustering C is not equipped with a height function, and so the height at which
any pair of points merges in C is also unde?ned. Following [10], our approach is to induce a merge
height function on the clustering using the mergeon in the following way:
De?nition 4 (Induced merge height). Let M be a mergeon, and suppose S is a ?nite subset of
[0, 1]. Let C be a clustering of S . The merge height function on C induced by M is de?ned by
? C (s, s? ) = minu,v?C(s,s? ) M(u, v), for every s, s? ? S ? S , where C(s, s? ) denotes the smallest cluster
M
C ? C which contains both s and s? .
We measure the distance between a clustering C and the cluster tree C using the merge distortion:
De?nition 5. Let M be a mergeon, S a ?nite subset of [0, 1], and C a clustering of S . The merge
? C ) = max s,s? ?S , s?s? |M(s, s? ) ? M
? C (s, s? )|.
distortion is de?ned by dS (M, M
De?ning the induced merge height and merge distortion in this way leads to an especially meaningful
interpretation of the merge distortion. In particular, if the merge distortion between C and C is ?,
then any two clusters of C which are separated at level ? but merge below level ? ? ? are correctly
separated in the clustering C. A similar result guarantees that a cluster in C is connected in C at
within ? of the correct level. For a precise statement of these results, see Claim 5 in Appendix A.4.
The label measure. We will use the merge distortion to measure the distance between C, a hierarchical clustering of a graph, and C, the graphon cluster tree. Recall, however, that the nodes of
a graph sampled from a graphon have integer labels. That is, C is a clustering of [n], and not of a
subset of [0, 1]. Hence, in order to apply the merge distortion, we must ?rst relabel the nodes of the
graph, placing them in direct correspondence to nodes of the graphon, i.e., points in [0, 1].
Recall that we sample a graph of size n from a graphon W by ?rst drawing n
points x1 , . . . , xn uniformly at random from the unit interval. We then generate
a graph on node set [n] by connecting nodes i and j with probability W(xi , xj ).
However, the nodes of the sampled graph are not labeled by x1 , . . . , xn , but rather
by the integers 1, . . . , n. Thus we may think of xi as being the ?true? latent label
of node i. In general the latent node labeling is not recoverable from data, as is
demonstrated by the ?gure to the right. We might suppose that the graph shown is
sampled from the graphon above it, and that node 1 corresponds to a, node 2 to b,
node 3 to c, and node 4 to d. However, it is just as likely that node 4 corresponds
to d? , and so neither labeling is more ?correct?. It is clear, though, that some
labelings are less likely than others. For instance, the existence of the edge (1, 2)
makes it impossible that 1 corresponds to a and 2 to c, since W(a, c) is zero.
Therefore, given a graph G = ([n], E) sampled from a graphon, there are many possible relabelings
of G which place its nodes in correspondence with nodes of the graphon, but some are more likely
than others. The merge distortion depends which labeling of G we assume, but, intuitively, a good
clustering of G will have small distortion with respect to highly probable labelings, and only have
large distortion on improbable labelings. Our approach is to assign a probability to every pair (G, S )
of a graph and possible labeling. We will thus be able to measure the probability mass of the set of
6
pairs for which a method performs poorly, i.e., results in a large merge distortion.
More formally, let Gn denote the set of all undirected, unweighted graphs on node set [n], and let
?n be the sigma-algebra of Lebesgue-measurable subsets of [0, 1]n . A graphon W induces a unique
product measure ?W,n de?ned on the product sigma-algebra 2Gn ? ?n such that for all G ? 2Gn and
S ? ?n :
?W,n (G ? S) =
?
G?G
(?
S
)
LW (S |G) dS , where LW (S | G) =
?
(i, j)?E(G)
W(xi , x j )
? [
(i, j)?E(G)
]
1 ? W(xi , x j ) ,
where E(G) represents the edge set of the graph G. We recognize LW (S | G) as the integrand in
Equation 1 for the probability of a graph as determined by a graphon. If G is ?xed, integrating
LW (S | G) over all S ? [0, 1]n gives the probability of G under the model de?ned by W.
We may now formally de?ne our notion of consistency. First, some notation: If C is a clustering of
[n] and S = (x1 , . . . , xn ), write C ? S to denote the relabeling of C by S , in which i is replaced by xi
in every cluster. Then if f is a hierarchical graph clustering method, f (G) ? S is a clustering of S ,
? f (G)?S denotes the merge function induced on f (G) ? S by M.
and M
De?nition 6 (Consistency). Let W be a graphon and M be a mergeon of W. A hierarchical graph
clustering method f is said ({
to be a consistent estimator of
}) the graphon cluster tree of W if for any
? f (G)?S ) > ? ? 0.
?xed ? > 0, as n ? ?, ?W,n (G, S ) : dS (M, M
The choice of mergeon for the graphon W does not a?ect consistency, as any two mergeons of the
same graphon di?er on a set of measure zero. Furthermore, consistency is with respect to the random
graph model, and not to any particular graphon representing the model. The following claim, the
proof of which is in Appendix B, makes this precise.
Claim 4. Let W be a graphon and ? a measure preserving transformation. A clustering method f
is a consistent estimator of the graphon cluster tree of W if and only if it is a consistent estimator of
the graphon cluster tree of W ? .
Consistency and the blockmodel. If a graph clustering method is consistent in the sense de?ned
above, it is also consistent in the stochastic blockmodel; i.e., it ensures strict recovery of the communities with high probability as the size of the graphs grow large. For instance, suppose W is
a stochastic blockmodel graphon with ? along the block-diagonal and ? everywhere else. W has
two clusters at level ?, merging into one cluster at level ?. When the merge distortion between the
graphon cluster tree and a clustering is less than ? ? ?, which will eventually be the case with high
probability if the method is consistent, the two clusters are totally disjoint in C; this implication is
made precise by Claim 5 in Appendix A.4.
5
Consistent algorithms
We now demonstrate that consistent clustering methods exist. We present two results: First, we
show that any method which is capable of consistently estimating the probability of each edge in a
random graph leads to a consistent clustering method. We then analyze a modi?cation of an existing
algorithm to show that it consistently estimates edge probabilities. As a corollary, we identify a
graph clustering method which satis?es our notion of consistency. Our results will be for graphons
which are piecewise Lipschitz (or weakly isomorphic to a piecewise Lipschitz graphon):
De?nition 7 (Piecewise Lipschitz). We say that B = {B1 , . . . , Bk } is a block partition if each Bi is an
open,?half-open, or closed interval in [0, 1] with positive measure, Bi ? B j is empty whenever i ? j,
and B = [0, 1]. We say that a graphon W is piecewise c-Lipschitz if there exists a set of blocks B
such that for any (x, y) and (x? , y? ) in Bi ? B j , |W(x, y) ? W(x? , y? )| ? c(|x ? x? | + |y ? y? |).
Our ?rst result concerns methods which are able to consistently estimate edge probabilities in the
following sense. Let S = (x1 , . . . , xn ) be an ordered set of n uniform random variables drawn from
the unit interval. Fix a graphon W, and let P be the random matrix whose i j entry is given by
W(xi , xj ). We say that P is the random edge probability matrix. Assuming that W has structure, it
is possible to estimate P from a single graph sampled from W. We say that an estimator P? of P is
consistent in max-norm if, for any ? > 0, limn?? P(maxi? j |Pi j ? P? i j | > ?) = 0. The following nontrivial theorem, whose proof comprises Appendix D, states that any estimator which is consistent in
this sense leads to a consistent clustering algorithm:
7
Theorem 1. Let W be a piecewise c-Lipschitz graphon. Let P? be a consistent estimator of P in
max-norm. Let f be the clustering method which performs single-linkage clustering using P? as a
similarity matrix. Then f is a consistent estimator of the graphon cluster tree of W.
Estimating the matrix of edge probabilities has Algorithm 1 Clustering by nbhd. smoothing
been a direction of recent research, however we
are only aware of results which show consis- Require: Adjacency matrix A, C ? (0, 1)
% Step 1: Compute the estimated edge
tency in mean squared error; That is, the liter2
%
probability matrix P? using neighborhood
2
?
1
ature contains estimators for which /n ?P ? P?F
%
smoothing algorithm based on [21]
tends to zero in probability. One practical
n ? Size(A)
method is the neighborhood smoothing algo?
h ? C (log n)/n
rithm of [21]. The method constructs for each
for i ? j ? [n] ? [n] do
node i in the graph G a neighborhood of nodes
A? ? A after setting row/column j to zero
Ni which are similar to i in the sense that for evfor i? ? [n] \ {i, j} do
ery i? ? Ni , the corresponding column Ai? of the
d j (i, i? ) ? maxk?i,i? , j |(A? 2 /n)ik ? (A? 2 /n)i? k |
adjacency matrix is close to Ai in a particular
end
for
distance. Aij is clearly not a good estimate for
qi j ? hth quantile of {d j (i, i? ) : i? ? i, j}
the probability of the edge (i, j), as it is either
Ni j ? {i? ? i, j : d j (i, i? ) ? qi j (h)}
zero or one, however, if the graphon is piece?
end
for
wise Lipschitz, the average Ai? j over i ? Nij
for
(i,
j) ? [n]
will intuitively tend to the true probability. Like
( ? [n] do
)
?
1 1 ?
others, the method of [21] is proven to be con?
Pi j ? 2 Ni j i? ?Ni j Ai? j + N1ji j? ?N ji Ai j?
sistent in mean squared error. Since Theorem 1
end for
requires consistency in max-norm, we analyze
% Step 2: Cluster P? with single linkage
a slight modi?cation of this algorithm and show
C
? the single linkage clusters of P?
that it consistently estimates P in this stronger
C
return
sense. The technical details are in Appendix E.
Theorem 2. If the graphon W is piecewise Lipschitz, the modi?ed neighborhood smoothing algorithm in Appendix E is a consistent estimator of P in max-norm.
As a corollary, we identify a practical graph clustering algorithm which is a consistent estimator of
the graphon cluster tree. The algorithm is shown in Algorithm 1, and details are in Appendix E.2.
Appendix F contains experiments in which the algorithm is applied to real and synthetic data.
Corollary 1. If the graphon W is piecewise Lipschitz, Algorithm 1 is a consistent estimator of the
graphon cluster tree of W.
6
Discussion
We have presented a consistency framework for clustering in the graphon model and demonstrated
that a practical clustering algorithm is consistent. We now identify two interesting directions of
future research. First, it would be interesting to consider the extension of our framework to sparse
random graphs; many real-world networks are sparse, and the graphon generates dense graphs. Recently, however, sparse models which extend the graphon have been proposed; see [7, 6]. It would
be interesting to see what modi?cations are necessary to apply our framework in these models.
Second, it would be interesting to consider alternative ways of de?ning the ground truth clustering of
a graphon. Our construction is motivated by interpreting the graphon W not only as a random graph
model, but also as a similarity function, which may not be desirable in certain settings. For example,
consider a ?bipartite? graphon W, which is one along the block-diagonal and zero elsewhere. The
cluster tree of W consists of a single cluster at all levels, whereas the ideal bipartite clustering has
two clusters. Therefore, consider applying a transformation S to W which maps it to a ?similarity?
graphon. The goal of clustering then becomes the recovery of the cluster tree of S (W) given a
random graph sampled from W. For instance, let S : W 7? W 2 , where W 2 is the operator square
of the bipartite graphon W. The cluster tree of S (W) has two clusters at all positive levels, and
so represents the desired ground truth. In general, any such transformation S leads to a di?erent
clustering goal. We speculate that, with minor modi?cation, the framework herein can be used to
prove consistency results in a wide range of graph clustering settings.
Acknowledgements. This work was supported by NSF grant IIS-1550757.
8
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5,624 | 609 | Probability Estimation from a Database
Using a Gibbs Energy Model
John W. Miller
Microsoft Research (9/1051)
One Microsoft Way
Redmond, WA 98052
Rodney M. Goodman
Dept. of Electrical Engineering (116-81)
California Institute of Technology
Pasadena, CA 91125
Abstract
We present an algorithm for creating a neural network which produces accurate probability estimates as outputs. The network implements a Gibbs probability distribution model of the training
database. This model is created by a new transformation relating
the joint probabilities of attributes in the database to the weights
(Gibbs potentials) of the distributed network model. The theory
of this transformation is presented together with experimental results. One advantage of this approach is the network weights are
prescribed without iterative gradient descent. Used as a classifier
the network tied or outperformed published results on a variety of
databases.
1
INTRODUCTION
This paper addresses the problem of modeling a discrete database. The database
is viewed as a collection of independent samples from a probability distribution.
This distribution is called the underlying distribution. In contrast, the empirical
distribution is the distribution obtained if you take independent random samples
from the database (with replacement). The task of creating a probability model
can be separated into two parts. The first part is the problem of choosing statistics
of the samples which are expected to accurately represent the underlying distribution. The second part is the problem of choosing a model which is consistent with
these statistics. Under reasonable assumptions, the optimal solution to the second
problem is the method of Maximum Entropy. For a broad class of statistics, the
531
532
Miller and Goodman
Maximum Entropy solution is a Gibbs probability distribution (Slepian, 1972). In
this paper, the background and theoretical result of a transformation from joint
statistics to a Gibbs energy (or network weight) representation is presented. We
then outline the experimental test results of an efficient algorithm implementing
this transform without using gradient descent iteration.
2 BACKGROUND
Define a set T to be the set of attributes (or fields) in a database. For a particular
entry (or record) of the database, define the associated set of attribute values to be
the configuration W of the attributes. The set of attribute values associated with
a subset beT is called a sub configuration Wb. Using this set notation the Gibbs
probability distribution may be defined:
pew) = Z-l . eVT(w)
(1)
where
(2)
bCT
The function V is called the energy. The function Jb, called the potential junction,
defines a real value for every sub configuration of the set b. Z is the normalizing
constant that makes the sum of probabilities of all configurations equal to unity.
Prior work in the neural network literature using the Gibbs distribution (such as
the Boltzmann Machine) has primarily used second order models (Jb = 0 if Ibl > 2)
(Hinton, 1986). By adding new attributes not in the original database, second order
potentials have been used to model complex distributions. The work presented in
this paper, in contrast, uses higher order potentials to model complex probability
distributions. We begin by considering the case where every potential of every order
is used to model the distribution.
The Principle of Inclusion-Exclusion from set theory states that the following two
equations are equivalent:
g(A)
Lf(b)
(3)
b~A
L(-l)IA-b l g(b).
bCA
f(A)
(4)
The method of inverting an equation from the form of (3) into one in the form of
(4) is a special case of Mobius Inversion. Clifford-Hammersley (Kindermann, 1980)
used this relation to invert formula (2):
JA(w) = L(-l)IA-b l Vb(w)
(5)
bCA
Define the probability of a sub configuration p(Wb) to be the probability that the
attributes in set b take on the values defined in the configuration w. Using (1)
to describe the probability distribution of sub configurations, equation (5) can be
written:
JA(w) = L(-l)IA-b l In( p(Wb?
(6)
b~A
Probability Estimation from a Database Using a Gibbs Energy Model
3
A TRANSFORMATION TO GIBBS POTENTIALS
Equation (6) provides a technique for modeling distributions by potential functions
rather than directly through the observable joint statistics of sets of attributes. If
the model is truncated by setting high order potentials to zero, then the energy
model becomes an estimate of the model obtained by collecting the joint statistics,
rather than an exact equivalent. If equation (6) is used directly, the error in the
energy due to setting all potentials of order d to zero grows quickly with d. For
this reason (6) must be normalized if it is going to be used in a truncated modeling
scheme. A normalization version of equation (2) that corrects for the unequal
number of potentials of different orders is:
VA(w)
IAI- 1)-1
= L ( Ibl- 1
Jb(W)
(7)
b~A
This equation can be inverted to show the surprising result, a weight associated
with WA:
(8)
JA(W) In(PA(w? - (IAI- 1)-1
In(Pb(w?
L
=
tEA
b=A-t
For example, with three attribute values {x, y, z}, the following potentials are defined:
J{x}
In(p(x))
=
J{y} = In(p(y))
J{z} = In(p(z))
p(xy) )
J{xy} = In ( p(x)p(y)
p(yz) )
J{yz} = In ( p(y)p(z)
p(xz) )
J{xz} = In ( p(x)p(z)
J{xyz} = In (
p(xyz)
)
ylp(xy)p(yz)p(xz)
For a given database sample, a potential is activated if all of its defined attribute
values are true for the sample. The weighted sum of all activated potentials recovers
an approximation of the probability of the database sample. If all potentials of every
order have been used to create the model, then this approximation is exactly the
probability of the sample in the empirical distribution. The correct weighting is
given by equation (7). For example it is easily verified that:
In(p(xyz))
(22) J{xyz} + (2)-1
1
(J{xy} + J{xz} + J{yz})
2)
+ (0 (J{x} + J{y} + J{z}).
-1
-1
533
534
Miller and Goodman
The Gibbs model truncated to second order potentials would estimate the probability in this example by:
In(p( xyz))
~
~
4
(2) -1 (J{xy} + J{xz} + J{yz}) + (2)-1
? (J{x} +
1
J{y}
+ J{z})'
In Vp(xy)p(yz)p(xz)
PROOF OF THE INVERSION FORMULA
Theorem:
Let T be a finite set . Each element of T will be called an attribute. Each attribute
can take on one of a finite set of states called attribute values. A collection of
attribute values for every element of T is called a configuration w. For all A ~ T
(including both the empty set A 0 and the full set A T), let VA(w) and JA(W)
be functions mapping the states of the elements of A to the real numbers. Define
(r;:) = m!j(( m - n)! . n!) to be "m choose n."
Let V0(w) = 0, J0(W) = 0, and let VA(W) = JA(w) if IAI = l.
Then for IAI > 1:
=
=
(9)
and
L
(IAI- 1)-1
. Vb(w)
(10)
bCA
Ibl=IAI-l
are equivalent in that any assignment of VA and J A values for all A
(9) if and only if they also satisfy (1 0).
~
T will satisfy
Proof:
Let .:7 be any assignment ofthe values JA(W) for all A ~ T. Let V be any assignment
of all the values VA(W) for all A ~ T. Then clearly (9) maps any assignment .:7 to a
unique V. We will represent this mapping by the function I, so (9) is abbreviated
V = 1(.:7). Similarly (10) maps any assignment V to a unique.:7. Equation (10)
will be abbreviated .:7 = g(V). The result of Lemma Cl below, applied with the
value 1) set to n, shows that l(g(V))
V. In Lemma C2 below, it is shown
g(/(.:7)) .:7. Therefore the equations (9) and (10) are inverse one-to-one mappings
and the association of assignments between .:7 and V are identical for the two
Q.E.D.
equations.
=
=
Lemma Cl:
Rather than simply showing l(g(V)) = V, a more general result will be shown. Since
the number of potentials of a given order increases exponentially with the order, it
is useful to approximate the energy of a configuration by defining a maximum order
1) such that all potentials of greater order are assumed to be zero
Jb(W)
=0
\:I b such that
Ibl > 1).
Let VA(W) be the resulting approximation to the energy VA(W). Let
IAI = n.
Probability Estimation from a Database Using a Gibbs Energy Model
Given
L
JA(W) = VA(W) -
(n -
1)-1 . ~(w)
(11)
bCA
Ibl=n-l
and the order V approximation to equation (7):
VA(w) =
Lv
(
1)-1 L
~~1
i=l
then
(12)
Jb(W),
bCA
IbT=.
( 1)
VA(W) = L
nV-I
A
-1
Vb(w).
bCA
Ib!=V
Note:
For the case V = n, the approximation is exact
VA(w)
and so f(g(V?
= V is shown.
= VA(W),
The lemma's result has a simple interpretation. The energy of a configuration is
approximated by a scaled average of the energies of the configurations of order
V. Using equation (1) to relate energies to probabilities, shows that the estimated
probability is a scaled geometric mean of the order V marginal probabilities.
Proof:
We start with the given equation for VA(W)
Lv ( ~ ~ 11)-1 L Jb(w).
i=l
b~A
Ibl=?
Use equation (11) to substitute Jb(W) out of the equation:
VA(W)
t (~~ :)
-1
~ (~(W) - cCt;;~l 1)-1 . Vc(W?)
(i -
Ibl=.
Icl=lbl-l
Separate the term in the first sum where i = V
VA(w)
,E;( ~ 1)-1
1
V.(w)
("-1 (1)-1 ~. V.(w) )
+ ~ : ~1
-2:"(~~11)-1 L 2:
i=l
b~A
(i-1)-1?Vc(w).
cCb,lbl~l
Ibl=. 1c1=lbl-l
By subtracting VA(W) from both sides using equation (12) and noting the second
summation over i has no terms when i = 1 we see that it is sufficient to show
I:(:~;rLV.(W)
i=l
bCA
Ihl=.
t(~~;rL
i=2
L
cCb
Ibl=. Icl=lbl-l
b~A
(i-W'? V,(w).
535
536
Miller and Goodman
The right hand side inner double summation counts a given llc(w) once for every b
such that eC b ~ A with i Ibl lei + 1. This occurs exactly IAI- lei n - i + 1
times. Thus
= =
L
V-I (
~~
i=l
11)-1 L ~(w)
=
=
Lv (1)-1
~~1 L
i=2
"CA
1"1=.
?+1
1 . Vc(w).
~~
n
cCA
Icl=i-l
= i-Ion the right hand side
~(w) = ~ (n ~ 1) -1 L n. j . Vc(w).
Now perform a change of variables. Let j
~ (~ i=l
; ) -1
t -
L
J
j=l
"CA
1"1=.
J
cCA
!cl=j
Clearly both sides are identical since
n-t
Q.E.D.
=
Lemma C2: g(/(:1)) :1
Let IAI = n. It is sufficient to show that substituting
an identity:
JA(W)
=
VA(w) -
L
~
out of (10) using (9) yields
-1
(Ib l - 1)-1
L
lel- 1
Jc(w).
cCb
(n _1)-1 ?1Ib(w)
bCA,n;tl
Ibl=n-l
n - 1)
( IblL
1
bCA
L
-1
Jb(W) -
(n -1)
bCA,n;tl
-
Ibl=n-l-
Separate the term in the first sum for which b
JA(W) = JA(w)
n_l)-l
+ L ( Ibl- 1
h(w) -
bCA
b;tA
=A
L
(n - 1)
-1
"CA,n;tl
(lb l -l)-l
L
lel- 1
Jc(w).
cCb
Ibl=n-l-
Subtract J A (w) from both sides. The right hand side double sum counts a given
Jc(w) once for every b such that e ~ b C A with Ibl = IAI- 1 n - 1. This occurs
IAI- lei n - lei times. It is sufficient to show
=
=
cCA,ctA
Both sides are identical since:
_1)-1
( ~Z -
1
n- z
n-l
n -lei
n-l
(
n-2 )
lel- 1
-1
Jc(w).
_2)-1
( ~Z -
1
Q.E.D.
Probability Estimation from a Database Using a Gibbs Energy Model
5
USING THE INVERSION FORMULA TO SET
NETWORK WEIGHTS
Our method of probability estimation is to first collect empirical frequencies of
patterns (sub configurations) from the database. (An efficient hash table implementation of the algorithm is described in (Miller, 1993). The basic idea is to remove
from the database a pattern with low potential whenever there is a hash collision
which prevents a new pattern count from being stored.) Second, interpreting these
frequencies as probabilities, we convert each pattern frequency to a potential using
equation (8). We assume patterns with unknown or uncalculated frequencies have
zero potential. Low order patterns which never occur are assigned a large negative
potential (this approximation is needed to model events with zero probability in
the empirical distribution). Finally, we calculate the probability of any new pattern
not in the training set using the neural network implementation of equations (7)
and (1).
6
RESULTS
One way to validate the performance of a probability model is to test its performance
as a classifier. The probability model is used as a classifier by calculating the
probabilities of each unknown class value together with the known attribute values.
The most probable combination is then chosen as the predicted class. Used as a
classifier the Gibbs model tied or outperformed published results on a variety of
databases. Table 1 outlines results on three datasets taken from the UC Irvine
archive (Murphy, 1992). The Gibbs model results were collected from the very
first experiment using the algorithm with the datasets. No difficult parameter
adjustment is necessary to get the algorithm to classify at these rates. The iris
database has 4 real value attributes. Each attribute was quantized into a decile
ranking for use by the algorithm.
7
CONCLUSION
A new method of extracting a Gibbs probability model from a database has been
presented. The approach uses the Principle of Inclusion-Exclusion to invert a set of
collected statistics into a set of potentials for a Gibbs energy model. A hash table
implementation is used to efficiently process database records in order to collect the
most important potentials, or weights, which can be stored in the available memory.
Although the model is designed to give accurate probability estimates rather than
simply class labels, the model in practice works well as a classifier on a variety of
databases.
Acknowledgements
This work is funded in part by DARPA and ONR under grant NOOOI4-92-J-1860.
537
538
Miller and Goodman
Table 1: Summary of Classification Results
Database
A
C
R
Train
Test
Trials
Gibbs Rate
House Voting
Iris
Iris
Breast Cancer
Breast Cancer
16
4
4
9
9
2
3
3
2
2
435
150
150
699
369
335
120
149
599
200
100
30
1
100
169
50
100
1000
100
100
95.3%
96.3%
97.1%
97.3%
95.7%
Compare
95%
n.a.
98.0%
n.a.
93.7%
=
=
=
A Attribute count in the database, excluding the class attribute
C = Class count
R Record count
Train Number of records used to create the energy for one trial
Test = Number of records tested in a single trial
Trials = Number of independent train-test trials used to calculate the rate
Gibbs Rate = Gibbs energy model classification rate
Compare = Baseline classification result of other methods (Schlimmer, 1987),
(Weiss, 1992),(Zhang, 1992) respectively
References
D. Slepian, "On Maxentropic Discrete Stationary Processes," Bell System
Technical Journal, 51, pp.629-653, 1972.
G.E. Hinton and T .J. Sejnowski, "Learning and Relearning in Boltzmann
Machines," in Parallel Distributed Processing, Vol. I., pp.282-317, Cambridge
MA: MIT Press, 1986.
R. Kindermann, J .L. Snell, Markov Random Fields and their Applications,
Providence, RI: American Mathematical Society, 1980.
J. W. Miller, "Building Probabilistic Models from Databases" California Institute
of Technology, Ph.D. Thesis 1993.
P. Murphy, and D. Aha, UCI Repository of Machine Learning Databases
[Machine-readable data repository at ics.uci.edu in directory
/pub/machine-Iearning-databases]. Irvine, CA: University of California,
Department of Information and Computer Science, 1992.
Schlimmer, J. C., "Concept Acquisition Through Representational Adjustment"
University of California at Irvine, Ph.D. Thesis 1987.
S. Weiss, and I. Kapouleas, "An Empirical Comparison of Pattern Recognition,
Neural Nets, and Machine Learning Classification Methods," in Proceedings of
the 11th International Joint Conference on Artificial Intelligence Vol. 1,
pp.781-787, Los Gatos, CA: Morgan Kaufmann, 1992.
J. Zhang, "Selecting Typical Instances in Instance-Based Learning," in
Proceedings of the Ninth International Machine Learning Conference Aberdeen,
Scotland, pp.470-479, San Mateo CA: Morgan Kaufmann, 1992.
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5,625 | 6,090 | Backprop KF: Learning Discriminative Deterministic
State Estimators
Tuomas Haarnoja, Anurag Ajay, Sergey Levine, Pieter Abbeel
{haarnoja, anuragajay, svlevine, pabbeel}@berkeley.edu
Department of Computer Science, University of California, Berkeley
Abstract
Generative state estimators based on probabilistic filters and smoothers are one
of the most popular classes of state estimators for robots and autonomous vehicles. However, generative models have limited capacity to handle rich sensory
observations, such as camera images, since they must model the entire distribution
over sensor readings. Discriminative models do not suffer from this limitation,
but are typically more complex to train as latent variable models for state estimation. We present an alternative approach where the parameters of the latent state
distribution are directly optimized as a deterministic computation graph, resulting
in a simple and effective gradient descent algorithm for training discriminative
state estimators. We show that this procedure can be used to train state estimators
that use complex input, such as raw camera images, which must be processed
using expressive nonlinear function approximators such as convolutional neural
networks. Our model can be viewed as a type of recurrent neural network, and
the connection to probabilistic filtering allows us to design a network architecture
that is particularly well suited for state estimation. We evaluate our approach on
synthetic tracking task with raw image inputs and on the visual odometry task in
the KITTI dataset. The results show significant improvement over both standard
generative approaches and regular recurrent neural networks.
1
Introduction
State estimation is an important component of mobile robotic applications, including autonomous
driving and flight [22]. Generative state estimators based on probabilistic filters and smoothers are
one of the most popular classes of state estimators. However, generative models are limited in their
ability to handle rich observations, such as camera images, since they must model the full distribution
over sensor readings. This makes it difficult to directly incorporate images, depth maps, and other
high-dimensional observations. Instead, the most popular methods for vision-based state estimation
(such as SLAM [22]) are based on domain knowledge and geometric principles. Discriminative
models do not need to model the distribution over sensor readings, but are more complex to train
for state estimation. Discriminative models such as CRFs [16] typically do not use latent variables,
which means that training data must contain full state observations. Most real-world state estimation
problem settings only provide partial labels. For example, we might observe noisy position readings
from a GPS sensor and need to infer the corresponding velocities. While discriminative models can
be augmented with latent state [18], this typically makes them harder to train.
We propose an efficient and scalable method for discriminative training of state estimators. Instead of
performing inference in a probabilistic latent variable model, we instead construct a deterministic
computation graph with equivalent representational power. This computation graph can then be
optimized end-to-end with simple backpropagation and gradient descent methods. This corresponds
to a type of recurrent neural network model, where the architecture of the network is informed by the
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
structure of the probabilistic state estimator. Aside from the simplicity of the training procedure, one
of the key advantages of this approach is the ability to incorporate arbitrary nonlinear components
into the observation and transition functions. For example, we can condition the transitions on raw
camera images processed by multiple convolutional layers, which have been shown to be remarkably
effective for interpreting camera images. The entire network, including the observation and transition
functions, is trained end-to-end to optimize its performance on the state estimation task.
The main contribution of this work is to draw a connection between discriminative probabilistic state
estimators and recurrent computation graphs, and thereby derive a new discriminative, deterministic
state estimation method. From the point of view of probabilistic models, we propose a method for
training expressive discriminative state estimators by reframing them as representationally equivalent
deterministic models. From the point of view of recurrent neural networks, we propose an approach
for designing neural network architectures that are well suited for state estimation, informed by
successful probabilistic state estimation models. We evaluate our approach on a visual tracking
problem, which requires processing raw images and handling severe occlusion, and on estimating
vehicle pose from images in the KITTI dataset [8]. The results show significant improvement over
both standard generative methods and standard recurrent neural networks.
2
Related Work
Some of the most successful methods for state estimation have been probabilistic generative state space models (SSMs) based on filtering and smoothing (Figure 1).
Kalman filters are perhaps the best known state estimators,
and can be extended to the case of nonlinear dynamics
through linearization and the unscented transform. Nonparametric filtering methods, such as particle filtering, are
also often used for tasks with multimodal posteriors. For
a more complete review of state estimation, we refer the
reader to standard references on this topic [22].
xt?1
xt
xt+1
ot?1
ot
ot+1
Figure 1: A generative state space model
with hidden states xi and observation ot
generated by the model. ot are observed
at both training and test time.
Generative models aim to estimate the distribution over state observation sequences o1:T as originating
from some underlying hidden state x1:T , which is typically taken to be the state space of the system.
This becomes impractical when the observation space is extremely high dimensional, and when the
observation is a complex, highly nonlinear function of the state, as in the case of vision-based state
estimation, where ot corresponds to an image viewed from a robot?s on-board camera. The challenges
of generative state space estimation can be mitigated by using complex observation models [14] or
approximate inference [15], but building effective generative models of images remains a challenging
open problem.
As an alternative to generative models, discriminative models such as conditional random fields
(CRFs) can directly estimate p(xt |o1:t ) [16]. A number of CRFs and conditional state space models
(CSSMs) have been applied to state estimation [21, 20, 12, 17, 9], typically using a log-linear
representation. More recently, discriminative fine-tuning of generative models with nonlinear neural
network observations [6], as well as direct training of CRFs with neural network factors [7], have
allowed for training of nonlinear discriminative models. However, such models have not been
extensively applied to state estimation. Training CRFs and CSSMs typically requires access to
true state labels, while generative models only require observations, which often makes them more
convenient for physical systems where the true underlying state is unknown. Although CRFs have
also been combined with latent states [18], the difficulty of CRF inference makes latent state CRF
models difficult to train. Prior work has also proposed to optimize SSM parameters with respect to a
discriminative loss [1]. In contrast to this work, our approach incorporates rich sensory observations,
including images, and allows for training of highly expressive discriminative models.
Our method optimizes the state estimator as a deterministic computation graph, analogous to recurrent
neural network (RNN) training. The use of recurrent neural networks (RNNs) for state estimation
has been explored in several prior works [24, 4, 23, 19], but has generally been limited to simple
tasks without complex sensory inputs such as images. Part of the reason for this is the difficulty of
training general-purpose RNNs. Recently, innovative RNN architectures have been successful at
mitigating this problem, through models such as the long short-term memory (LSTM) [10] and the
2
ot?1
ot
ot+1
zt?1
zt
zt+1
xt?1
xt
st?2
ot?1
ot
ot+1
g?
g?
g?
?
zt?1
st?1
yt
?
st?1
xt+1
q
yt?1
zt
yt+1
?yt?1
(a)
zt+1
st
st+1
?
st+1
st
q
q
?yt
?yt+1
(b)
Figure 2: (a) Standard two-step engineering approach for filtering with high-dimensional observations.
The generative part has hidden state xt and two observations, yt and zt , where the latter observation
is actually the output of a second deterministic model zt = g? (ot ), denoted by dashed lines and
trained explicitly to predict zt . (b) Computation graph that jointly optimizes both models in (a),
consisting of the deterministic map g? and a deterministic filter that infers the hidden state given zt .
By viewing the entire model as a single deterministic computation graph, it can be trained end-to-end
using backpropagation as explained in Section 4.
gated recurrent unit (GRU) [5]. LSTMs have been combined with vision for perception tasks such as
activity recognition [3]. However, in the domain of state estimation, such black-box models ignore
the considerable domain knowledge that is available. By drawing a connection between filtering and
recurrent networks, we can design recurrent computation graphs that are particularly well suited to
state estimation and, as shown in our evaluation, can achieve improved performance over standard
LSTM models.
3
Preliminaries
Performing state estimation with a generative model directly using high-dimensional observations
ot , such as camera images, is very difficult, because these observations are typically produced by a
complex and highly nonlinear process. However, in practice, a low-dimensional vector, zt , which can
be extracted from ot , can fully capture the dependence of the observation on the underlying state
of the system. Let xt denote this state, and let yt denote some labeling of the states that we wish
to be able to infer from ot . For example, ot might correspond to pairs of images from a camera
on an automobile, zt to its velocity, and yt to the location of the vehicle. In that case, we can first
train a discriminative model g? (ot ) to predict zt from ot in feedforward manner, and then filter the
predictions to output the desired state labels y1:t . For example, a Kalman filter with hidden state
xt could be trained to use the predicted zt as observations, and then perform inference over xt and
yt at test time. This standard approach for state estimation with high-dimensional observations is
illustrated in Figure 2a.
While this method may be viewed as an engineering solution without a probabilistic interpretation, it
has the advantage that g? (ot ) is trained discriminatively, and the entire model is conditioned on ot ,
with xt acting as an internal latent variable. This is why the model does not need to represent the
distribution over observations explicitly. However, the function g? (ot ) that maps the raw observations
ot to low-dimensional predictions zt is not trained for optimal state estimation. Instead, it is trained
to predict an intermediate variable zt that can be readily integrated into the generative filter.
4
Discriminative Deterministic State Estimation
Our contribution is based on a generalized view of state estimation that subsumes the na?ve, piecewisetrained models discussed in the previous section and allows them to be trained end-to-end using
simple and scalable stochastic gradient descent methods. In the na?ve approach, the observation
function g? (ot ) is trained to directly predict zt , since a standard generative filter model does not
provide for a straightforward way to optimize g? (ot ) with respect to the accuracy of the filter on
the labels y1:T . However, the filter can be viewed as a computation graph unrolled through time, as
shown in Figure 2b. In this graph, the filter has an internal state defined by the posterior over xt . For
3
example, in a Kalman filter with Gaussian posteriors, we can represent the internal state with the
tuple st = (?xt , ?xt ). In general, we will use st to refer to the state of any filter. We also augment
this graph with an output function q(st ) = ?yt that outputs the parameters of a distribution over
labels yt . In the case of a Kalman filter, we would simply have q(st ) = (Cy ?xt , Cy ?xt CT
y ), where
the matrix Cy defines a linear observation function from xt to yt .
Viewing the filter as a computation graph in this way, g? (ot ) can be trained discriminatively
on the entire sequence, rather than individually on single time steps. Let l(?yt ) be a loss function on the output distribution of the computation graph, which might, for example, be given by
l(?yt ) = ? log p?yt (yP
t ), where p?yt is the distribution induced by the parameters ?yt , and yt is
the label. Let L(?) = t l(?yt ) be the loss on an entire sequence with respect to ?. Furthermore,
let ?(st , zt+1 ) denote the operation performed by the filter to compute st+1 based on st and zt+1 .
We can compute the gradient of l(?) with respect to the parameters ? by first recursively computing
the gradient of the loss with respect to the filter state st from the back to the front according to the
following recursion:
d?yt?1 dL
dst dL
dL
=
+
,
dst?1
dst d?yt?1
dst?1 dst
(1)
and then applying the chain rule to obtain
?? L(?) =
T
X
dzt dst dL
.
d? dzt dst
t=1
(2)
All of the derivatives in these equations can be obtained from g? (ot ), ?(st?1 , zt ), q(st ), and l(?yt ):
dst
dst
= ?st?1 ?(st?1 , zt ),
= ?zt ?(st?1 , zt ),
dst?1
dzt
dL
dzt
d?yt
= ?? g? (ot ).
= ??yt l(?yt ),
= ?st q(st ),
d?yt
dst
d?
(3)
The parameters ? can be optimized with gradient descent using these gradients. This is an instance of
backpropagation through time (BPTT), a well known algorithm for training recurrent neural networks.
Recognizing this connection between state-space models and recurrent neural networks allows us to
extend this generic filtering architecture and explore the continuum of models between filters with
a discriminatively trained observation model g? (ot ) all the way to fully general recurrent neural
networks. In our experimental evaluation, we use a standard Kalman filter update as ?(st , zt+1 ), but
we use a nonlinear convolutional neural network observation function g? (ot ). We found that this
provides a good trade-off between incorporating domain knowledge and end-to-end learning for the
task of visual tracking and odometry, but other variants of this model could be explored in future
work.
5
Experimental Evaluation
In this section, we compare our deterministic discriminatively trained state estimator with a set
of alternative methods, including simple feedforward convolutional networks, piecewise-trained
Kalman filter, and fully general LSTM models. We evaluate these models on two tasks that require
processing of raw image input: synthetic task of tracking a red disk in the presence of clutter and
severe occlusion; and the KITTI visual odometry task [8].
5.1
State Estimation Models
Our proposed model, which we call the ?backprop Kalman filter? (BKF), is a computation graph
made up of a Kalman filter (KF) and a feedforward convolutional neural network that distills the
observation ot into a low-dimensional signal zt , which serves as the observation for the KF. The
neural network outputs both a point observation zt and an observation covariance matrix Rt . Since
the network is trained together with the filter, it can learn to use the covariance matrix to communicate
the desired degree of uncertainty about the observation, so as to maximize the accuracy of the final
filter prediction.
4
?xt?1
fc
zt
A?xt?1
reshape
diag exp Lt
h4
fc
fc
h3
ReLU
fc
h2
ReLU
ReLU
max_pool
conv
h1
Kalman filter
resp_norm
ReLU
max_pool
conv
ot
resp_norm
Feedforward network
?t
L
Lt LT
t
?0xt + Kt zt ? Cz ?0xt
Rt
0
T
?0xt CT
z Cz ?xt Cz + Rt
T
A?xt?1 A +
Bw QBT
w
?0xt
yt
?0xt
?1
?xt
Loss
PN PT
i=1
1
t=1 2T N
2
(i)
(i)
Cy ?xt ? yt
2
Kt
(I ? Kt Cz ) ?0xt
?xt
?xt?1
Figure 3: Illustration of the computation graph for the BKF. The graph is composed of a feedforward
?t
part, which processes the raw images ot and outputs intermediate observations zt and a matrix L
that is used to form a positive definite observation covariance matrix Rt , and a recurrent part that
integrates zt through time to produce filtered state estimates. See Appendix A for details.
We compare the backprop KF to three alternative state estimators: the ?feedforward model?, the
?piecewise KF?, and the ?LSTM model?. The simplest of the models, the feedforward model, does
not consider the temporal structure in the task at all, and consists only of a feedforward convolutional
network that takes in the observations ot and outputs a point estimate y
?t of the label yt . This approach
is viable only if the label information can be directly inferred from ot , such as when tracking an
object. On the other hand, tasks that require long term memory, such as visual odometry, cannot
be solved with a plain feedforward network. The piecewise KF model corresponds to the simple
generative approach described in Section 3, which combines the feedforward network with a Kalman
? t . The
filter that filters the network predictions zt to produce a distribution over the state estimate x
piecewise model is based on the same computation graph as the BKF, but does not optimize the filter
and network together end-to-end, instead training the two pieces separately. The only difference
between the two graphs is that the piecewise KF does not implement the additional pathway for
propagating the uncertainty from the feedforward network into the filter, but instead, the filter needs
to learn to handle the uncertainty in zt independently. An example instantiation of BKF is depicted
in Figure 3. A detailed overview of the computational blocks shown in the figure is deferred to
Appendix A.
Finally, we compare to a recurrent neural network based on LSTM hidden units [10]. This model
resembles the backprop KF, except that the filter portion of the graph is replaced with a generic
LSTM layer. The LSTM model learns the dynamics from data, without incorporating the domain
knowledge present in the KF.
5.2
Neural Network Design
A special aspect of our network design is a novel response normalization layer that is applied to the
convolutional activations before applying the nonlinearity. The response normalization transforms
the activations such that the activations of layer i have always mean ?i and variance ?i2 regardless
of the input to the layer. The parameters ?i and ?i2 are learned along with other parameters. This
normalization is used in all of the convolutional networks in our evaluation, and resembles batch
normalization [11] in its behavior. However, we found this approach to be substantially more effective
for recurrent models that require backpropagation through time, compared to the more standard
batch normalization approach, which is known to require additional care when applied to recurrent
networks. It has been since proposed independently from our work in [2], which gives an in-depth
analysis of the method. The normalization is followed by a rectified linear unit (ReLU) and a max
pooling layer.
5.3
Synthetic Visual State Estimation Task
Our state estimation task is meant to reflect some of the typical challenges in visual state estimation:
the need for long-term tracking to handle occlusions, the presence of noise, and the need to process
raw pixel data. The task requires tracking a red disk from image observations, as shown in Figure
4. Distractor disks with random colors and radii are added into the scene to occlude the red disk,
and the trajectories of all disks follow linear-Gaussian dynamics, with a linear spring force that pulls
the disks toward the center of the frame and a drag force that prevents high velocities. The disks
can temporally leave the frame since contacts are not modeled. Gaussian noise is added to perturb
the motion. While these model parameters are assumed to be known in the design of the filter, it is
a straightforward to learn also the model parameters. The difficulty of the task can be adjusted by
increasing or decreasing the number of distractor disks, which affects the frequency of occlusions.
5
Figure 4: Illustration of six consecutive frames of two training sequences. The objective is to track
the red disk (circled in the the first frame for illustrative purposes) throughout the 100-frame sequence.
The distractor disks are sampled for each sequence at random and overlaid on top of the target disk.
The upper row illustrates an easy sequence (9 distractors), while the bottom row is a sequence of high
difficulty (99 distractors). Note that the target is very rarely visible in the hardest sequences.
Table 1: Benchmark Results
State Estimation Model # Parameters RMS test error ??
feedforward model
piecewise KF
LSTM model (64 units)
LSTM model (128 units)
BKF (ours)
0.2322 ? 0.1316
0.1160 ? 0.0330
0.1407 ? 0.1154
0.1423 ? 0.1352
0.0537 ? 0.1235
7394
7397
33506
92450
7493
The easiest variants of the task are solvable with a feedforward estimator, while the hardest variants
require long-term tracking through occlusion. To emphasize the sample efficiency of the models, we
trained them using 100 randomly sampled sequences.
The results in Table 1 show that the BKF outperforms both the standard probabilistic KF-based
estimators and the more powerful and expressive LSTM estimators. The tracking error of the simple
feedforward model is significantly larger due to the occlusions, and the model tends to predict the
mean coordinates when the target is occluded. The piecewise model performs better, but because
the observation covariance is not conditioned on ot , the Kalman filter learns to use a very large
observation covariance, which forces it to rely almost entirely on the dynamics model for predictions.
On the other hand, since the BKF learns to output the observation covariances conditioned on ot that
optimize the performance of the filter, it is able to find a compromise between the observations and
the dynamics model. Finally, although the LSTM model is the most general, it performs worse than
the BKF, since it does not incorporate prior knowledge about the structure of the state estimation
problem.
6
feedforward
piecewise
LSTM
10 0
RMS error
To test the robustness of the estimator to occlusions,
we trained each model on a training set of 1000 sequences of varying amounts of clutter and occlusions.
We then evaluated the models on several test sets,
each corresponding to a different level of occlusion
and clutter. The tracking error as the test set difficulty
is varied is shown Figure 5. Note that even in the
absence of distractors, BKF and LSTM models outperform the feedforward model, since the target occasionally leaves the field of view. The performance
of the piecewise KF does not change significantly as
the difficulty increases: due to the high amount of
clutter during training, the piecewise KF learns to
use a large observation covariance and rely primarily
on feedforward estimates for prediction. The BKF
achieves the lowest error in nearly all cases. At the
same time, the BKF also has dramatically fewer parameters than the LSTM models, since the transitions
correspond to simple Kalman filter updates.
10 -1
10 -2
10 -3
0
20
40
60
80
100
# distractors
Figure 5: The RMS error of various models
trained on a single training set that contained
sequences of varying difficulty. The models
were then evaluated on several test sets of
fixed difficulty.
Figure 6: Example image sequence from the KITTI dataset (top row) and the corresponding difference
image that is obtained by subtracting the RGB values of the previous image from the current image
(bottom row). The observation ot is formed by concatenating the two images into a six-channel
feature map which is then treated as an input to a convolutional neural network. The figure shows
every fifth sample from the original sequence for illustrative purpose.
Table 2: KITTI Visual Odometry Results
Test 100
# training trajectories
Translational Error [m/m]
piecewise KF
LSTM model (128 units)
LSTM model (256 units)
BKF (ours)
Rotational Error [deg/m]
piecewise KF
LSTM model (128 units)
LSTM model (256 units)
BKF (ours)
5.4
Test 100/200/400/800
3
6
10
3
6
10
0.3257
0.5022
0.5199
0.3089
0.2452
0.3456
0.3172
0.2346
0.2265
0.2769
0.2630
0.2062
0.3277
0.5491
0.5439
0.2982
0.2313
0.4732
0.4506
0.2031
0.2197
0.4352
0.4228
0.1804
0.1408
0.5484
0.4960
0.1207
0.1028
0.3681
0.3391
0.0901
0.0978
0.3767
0.2933
0.0801
0.1069
0.4123
0.3845
0.0888
0.0768
0. 3573
0.3566
0.0587
0.0754
0.3530
0.3221
0.0556
KITTI Visual Odometry Experiment
Next, we evaluated the state estimation models on visual odometry task in the KITTI dataset [8]
(Figure 6, top row). The publicly available training set contains 11 trajectories of ego-centric video
sequences of a passenger car driving in suburban scenes, along with ground truth position and
orientation. The dataset is challenging since it is relatively small for learning-based algorithms, and
the trajectories are visually very diverse. For training the Kalman filter variants, we used a simplified
state-space model with three of the state variables corresponding to the vehicle?s 2D pose (two spatial
coordinates and heading) and two for the forward and angular velocities. Because the dynamics
model is non-linear, we equipped our model-based state estimators with extended Kalman filters,
which is a straightforward addition to the BKF framework.
The objective of the task is to estimate the relative change in the pose during fixed-length subsequences. However, because inferring the pose requires integration over all past observations, a simple
feedforward model cannot be used directly. Instead, we trained a feedforward network, consisting of
four convolutional and two fully connected layers and having approximately half a million parameters,
to estimate the velocities from pairs of images at consecutive time steps. In practice, we found it better
to use a difference image, corresponding to the change in the pixel intensities between the images,
along with the current image as an input to the feedforward network (Figure 6). The ground truth
velocities, which were used to train the piecewise KF as well as to pretrain the other models, were
computed by finite differencing from the ground truth positions. The recurrent models?piecewise
KF, the BKF, and the LSTM model?were then fine-tuned to predict the vehicle?s pose. Additionally,
for the LSTM model, we found it crucial to pretrain the recurrent layer to predict the pose from the
velocities before fine-tuning.
We evaluated each model using 11-fold cross-validation, and we report the average errors of the
held-out trajectories over the folds. We trained the models by randomly sampling subsequences of
100 time steps. For each fold, we constructed two test sets using the held-out trajectory: the first set
contains all possible subsequences of 100 time steps, and the second all subsequences of lengths
100, 200, 400, and 800.1 We repeated each experiment using 3, 6, or all 10 of the sequences in each
training fold to evaluate the resilience of each method to overfitting.
1
The second test set aims to mimic the official (publicly unavailable) test protocol. Note, however, that
because the methods are not tested on the same sequences as the official test set, they are not directly comparable
to results on the official KITTI benchmark.
7
Table 2 lists the cross-validation results. As expected, the error decreases consistently as the number
of training sequences becomes larger. In each case, BKF outperforms the other variants in both
predicting the position and heading of the vehicle. Because both the piecewise KF and the BKF
incorporate domain knowledge, they are more data-efficient. Indeed, the performance of the LSTM
degrades faster as the number of training sequences is decreased. Although the models were trained
on subsequences of 100 time steps, they were also tested on a set containing a mixture of different
sequence lengths. The LSTM model generally failed to generalize to longer sequences, while the
Kalman filter variants perform slightly better on mixed sequence lengths.
6
Discussion
In this paper, we proposed a discriminative approach to state estimation that consists of reformulating
probabilistic generative state estimation as a deterministic computation graph. This makes it possible
to train our method end-to-end using simple backpropagation through time (BPTT) methods, analogously to a recurrent neural network. In our evaluation, we present an instance of this approach that
we refer to as the backprop KF (BKF), which corresponds to a (extended) Kalman filter combined
with a feedforward convolutional neural network that processes raw image observations. Our approach to state estimation has two key benefits. First, we avoid the need to construct generative state
space models over complex, high-dimensional observation spaces such as raw images. Second, by
reformulating the probabilistic state-estimator as a deterministic computation graph, we can apply
simple and effective backpropagation and stochastic gradient descent optimization methods to learn
the model parameters. This avoids the usual challenges associated with inference in continuous,
nonlinear conditional probabilistic models, while still preserving the same representational power as
the corresponding approximate probabilistic inference method, which in our experiments corresponds
to approximate Gaussian posteriors in a Kalman filter.
Our approach also can be viewed as an application of ideas from probabilistic state-space models to
the design of recurrent neural networks. Since we optimize the state estimator as a deterministic computation graph, it corresponds to a particular type of deterministic neural network model. However,
the architecture of this neural network is informed by principled and well-motivated probabilistic
filtering models, which provides us with a natural avenue for incorporating domain knowledge into
the system.
Our experimental results indicate that end-to-end training of a discriminative state estimators can
improve their performance substantially when compared to a standard piecewise approach, where
a discriminative model is trained to process the raw observations and produce intermediate lowdimensional observations that can then be integrated into a standard generative filter. The results also
indicate that, although the accuracy of the BKF can be matched by a recurrent LSTM network with
a large number of hidden units, BKF outperforms the general-purpose LSTM when the dataset is
limited in size. This is due to the fact that BKF incorporates domain knowledge about the structure of
probabilistic filters into the network architecture, providing it with a better inductive bias when the
training data is limited, which is the case in many real-world robotic applications.
In our experiments, we primarily focused on models based on the Kalman filter. However, our
approach to state estimation can equally well be applied to other probabilistic filters for which the
update equations (approximate or exact) can be written in closed form, including the information
filter, the unscented Kalman filter, and the particle filter, as well as deterministic filters such as state
observers or moving average processes. As long as the filter can be expressed as a differentiable mapping from the observation and previous state to the new state, we can construct and differentiate the
corresponding computation graph. An interesting direction for future work is to extend discriminative
state-estimators with complex nonlinear dynamics and larger latent state. For example, one could
explore the continuum of models that span the space between simple KF-style state estimators and
fully general recurrent networks. The trade-off between these two extremes is between generality and
domain knowledge, and striking the right balance for a given problem could produce substantially
improved results even with relative modest amounts of training data.
Acknowledgments
This research was funded in part by ONR through a Young Investigator Program award, by the Army
Research Office through the MAST program, and by the Berkeley DeepDrive Center.
8
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9
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5,626 | 6,091 | Operator Variational Inference
Rajesh Ranganath
Princeton University
Jaan Altosaar
Princeton University
Dustin Tran
Columbia University
David M. Blei
Columbia University
Abstract
Variational inference is an umbrella term for algorithms which cast Bayesian inference as optimization. Classically, variational inference uses the Kullback-Leibler
divergence to define the optimization. Though this divergence has been widely
used, the resultant posterior approximation can suffer from undesirable statistical
properties. To address this, we reexamine variational inference from its roots as
an optimization problem. We use operators, or functions of functions, to design
variational objectives. As one example, we design a variational objective with a
Langevin-Stein operator. We develop a black box algorithm, operator variational
inference (opvi), for optimizing any operator objective. Importantly, operators enable us to make explicit the statistical and computational tradeoffs for variational
inference. We can characterize different properties of variational objectives, such
as objectives that admit data subsampling?allowing inference to scale to massive
data?as well as objectives that admit variational programs?a rich class of posterior approximations that does not require a tractable density. We illustrate the
benefits of opvi on a mixture model and a generative model of images.
1
Introduction
Variational inference is an umbrella term for algorithms that cast Bayesian inference as optimization [10]. Originally developed in the 1990s, recent advances in variational inference have scaled
Bayesian computation to massive data [7], provided black box strategies for generic inference in
many models [19], and enabled more accurate approximations of a model?s posterior without sacrificing efficiency [21, 20]. These innovations have both scaled Bayesian analysis and removed the
analytic burdens that have traditionally taxed its practice.
Given a model of latent and observed variables p.x; z/, variational inference posits a family of distributions over its latent variables and then finds the member of that family closest to the posterior,
p.z j x/. This is typically formalized as minimizing a Kullback-Leibler (kl) divergence from the
approximating family q./ to the posterior p./. However, while the kl.q k p/ objective offers many
beneficial computational properties, it is ultimately designed for convenience; it sacrifices many desirable statistical properties of the resultant approximation.
When optimizing kl, there are two issues with the posterior approximation that we highlight. First,
it typically underestimates the variance of the posterior. Second, it can result in degenerate solutions
that zero out the probability of certain configurations of the latent variables. While both of these issues can be partially circumvented by using more expressive approximating families, they ultimately
stem from the choice of the objective. Under the kl divergence, we pay a large price when q./ is
big where p./ is tiny; this price becomes infinite when q./ has larger support than p./.
In this paper, we revisit variational inference from its core principle as an optimization problem. We
use operators?mappings from functions to functions?to design variational objectives, explicitly
trading off computational properties of the optimization with statistical properties of the approximation. We use operators to formalize the basic properties needed for variational inference algorithms.
We further outline how to use them to define new variational objectives; as one example, we design
a variational objective using a Langevin-Stein operator.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
We develop operator variational inference (opvi), a black box algorithm that optimizes any operator
objective. In the context of opvi, we show that the Langevin-Stein objective enjoys two good properties. First, it is amenable to data subsampling, which allows inference to scale to massive data.
Second, it permits rich approximating families, called variational programs, which do not require
analytically tractable densities. This greatly expands the class of variational families and the fidelity
of the resulting approximation. (We note that the traditional kl is not amenable to using variational
programs.) We study opvi with the Langevin-Stein objective on a mixture model and a generative
model of images.
Related Work. There are several threads of research in variational inference with alternative divergences. An early example is expectation propagation (ep) [16]. ep promises approximate minimization of the inclusive kl divergence kl.pjjq/ to find overdispersed approximations to the posterior.
ep hinges on local minimization with respect to subsets of data and connects to work on ?-divergence
minimization [17, 6]. However, it does not have convergence guarantees and typically does not minimize kl or an ?-divergence because it is not a global optimization method. We note that these
divergences can be written as operator variational objectives, but they do not satisfy the tractability
criteria and thus require further approximations. Li and Turner [14] present a variant of ?-divergences
that satisfy the full requirements of opvi. Score matching [9], a method for estimating models by
matching the score function of one distribution to another that can be sampled, also falls into the
class of objectives we develop.
Here we show how to construct new objectives, including some not yet studied. We make explicit the
requirements to construct objectives for variational inference. Finally, we discuss further properties
that make them amenable to both scalable and flexible variational inference.
2
Operator Variational Objectives
We define operator variational objectives and the conditions needed for an objective to be useful
for variational inference. We develop a new objective, the Langevin-Stein objective, and show how
to place the classical kl into this class. In the next section, we develop a general algorithm for
optimizing operator variational objectives.
2.1
Variational Objectives
Consider a probabilistic model p.x; z/ of data x and latent variables z. Given a data set x, approximate Bayesian inference seeks to approximate the posterior distribution p.z j x/, which is applied in
all downstream tasks. Variational inference posits a family of approximating distributions q.z/ and
optimizes a divergence function to find the member of the family closest to the posterior.
The divergence function is the variational objective, a function of both the posterior and the approximating distribution. Useful variational objectives hinge on two properties: first, optimizing the
function yields a good posterior approximation; second, the problem is tractable when the posterior
distribution is known up to a constant.
The classic construction that satisfies these properties is the evidence lower bound (elbo),
Eq.z/ ?log p.x; z/
log q.z/?:
(1)
It is maximized when q.z/ D p.z j x/ and it only depends on the posterior distribution up to a
tractable constant, log p.x; z/. The elbo has been the focus in much of the classical literature. Maximizing the elbo is equivalent to minimizing the kl divergence to the posterior, and the expectations
are analytic for a large class of models [4].
2.2
Operator Variational Objectives
We define a new class of variational objectives, operator variational objectives. An operator objective has three components. The first component is an operator O p;q that depends on p.z j x/ and
q.z/. (Recall that an operator maps functions to other functions.) The second component is a family
of test functions F , where each f .z/ 2 F maps realizations of the latent variables to real vectors
Rd . In the objective, the operator and a function will combine in an expectation Eq.z/ ?.O p;q f /.z/?,
designed such that values close to zero indicate that q is close to p. The third component is a distance
2
function t .a/ W R ! ?0; 1/, which is applied to the expectation so that the objective is non-negative.
(Our example uses the square function t .a/ D a2 .)
These three components combine to form the operator variational objective. It is a non-negative
function of the variational distribution,
L.qI O p;q ; F ; t / D sup t .Eq.z/ ?.O p;q f /.z/?/:
(2)
f 2F
Intuitively, it is the worst-case expected value among all test functions f 2 F . Operator variational
inference seeks to minimize this objective with respect to the variational family q 2 Q.
We use operator objectives for posterior inference. This requires two conditions on the operator and
function family.
1. Closeness. The minimum of the variational objective is at the posterior, q.z/ D p.z j x/. We
meet this condition by requiring that Ep.z j x/ ?.O p;p f /.z/? D 0 for all f 2 F . Thus, optimizing
the objective will produce p.z j x/ if it is the only member of Q with zero expectation (otherwise
it will produce a distribution in the equivalence class: q 2 Q with zero expectation). In practice,
the minimum will be the closest member of Q to p.z j x/.
2. Tractability. We can calculate the variational objective up to a constant without involving the
exact posterior p.z j x/. In other words, we do not require calculating the normalizing constant of
the posterior, which is typically intractable. We meet this condition by requiring that the operator
O p;q ?originally in terms of p.z j x/ and q.z/?can be written in terms of p.x; z/ and q.z/.
Tractability also imposes conditions on F : it must be feasible to find the supremum. Below, we
satisfy this by defining a parametric family for F that is amenable to stochastic optimization.
Equation 2 and the two conditions provide a mechanism to design meaningful variational objectives
for posterior inference. Operator variational objectives try to match expectations with respect to q.z/
to those with respect to p.z j x/.
2.3
Understanding Operator Variational Objectives
Consider operators where Eq.z/ ?.O p;q f /.z/? only takes positive values. In this case, distance to zero
can be measured with the identity t .a/ D a, so tractability implies the operator need only be known
up to a constant. This family includes tractable forms of familiar divergences like the kl divergence
(elbo), R?nyi?s ?-divergence [14], and the -divergence [18].
When the expectation can take positive or negative values, operator variational objectives are closely
related to Stein divergences [2]. Consider a family of scalar test functions F that have expectation
zero with respect to the posterior, Ep.z j x/ ?f .z/? D 0. Using this family, a Stein divergence is
DStein .p; q/ D sup jEq.z/ ?f .z/?
f 2F
Ep.z j x/ ?f .z/?j:
Now recall the operator objective of Equation 2. The closeness condition implies that
L.qI O p;q ; F ; t / D sup t .Eq.z/ ?.O p;q f /.z/?
f 2F
Ep.z j x/ ?.O p;p f /.z/?/:
In other words, operators with positive or negative expectations lead to Stein divergences with a more
generalized notion of distance.
2.4
Langevin-Stein Operator Variational Objective
We developed the operator variational objective. It is a class of tractable objectives, each of which
can be optimized to yield an approximation to the posterior. An operator variational objective is
built from an operator, function class, and distance function to zero. We now use this construction
to design a new type of variational objective.
An operator objective involves a class of functions that has known expectations with respect to an
intractable distribution. There are many ways to construct such classes [1, 2]. Here, we construct an
operator objective from the generator Stein?s method applied to the Langevin diffusion.
3
Let r > f denote the divergence of a vector-valued function f , that is, the sum of its individual gradients. Applying the generator method of Barbour [2] to Langevin diffusion gives the operator
p
.Ols
f /.z/ D rz log p.x; z/> f .z/ C r > f:
(3)
We call this the Langevin-Stein (ls) operator. We obtain the corresponding variational objective by
using the squared distance function and substituting Equation 3 into Equation 2,
p
L.qI Ols
; F / D sup .Eq ?rz log p.x; z/> f .z/ C r > f ?/2 :
(4)
f 2F
The ls operator satisfies both conditions. First, it satisfies closeness because it has expectation zero
under the posterior (Appendix A) and its unique minimizer is the posterior (Appendix B). Second, it
is tractable because it requires only the joint distribution. The functions f will also be a parametric
family, which we detail later.
Additionally, while the kl divergence finds variational distributions that underestimate the variance,
the ls objective does not suffer from that pathology. The reason is that kl is infinite when the support
of q is larger than p; here this is not the case.
We provided one example of a variational objectives using operators, which is specific to continuous variables. In general, operator objectives are not limited to continuous variables; Appendix C
describes an operator for discrete variables.
2.5
The KL Divergence as an Operator Variational Objective
Finally, we demonstrate how classical variational methods fall inside the operator family. For example, traditional variational inference minimizes the kl divergence from an approximating family to
the posterior [10]. This can be construed as an operator variational objective,
p;q
.OKL
f /.z/ D log q.z/
log p.zjx/
8f 2 F :
(5)
This operator does not use the family of functions?it trivially maps all functions f to the same
function. Further, because kl is strictly positive, we use the identity distance t .a/ D a.
The operator satisfies both conditions. It satisfies closeness because KL.pjjp/ D 0. It satisfies
tractability because it can be computed up to a constant when used in the operator objective of Equation 2. Tractability comes from the fact that log p.z j x/ D log p.z; x/ log p.x/.
3
Operator Variational Inference
We described operator variational objectives, a broad class of objectives for variational inference. We
now examine how it can be optimized. We develop a black box algorithm [27, 19] based on Monte
Carlo estimation and stochastic optimization. Our algorithm applies to a general class of models and
any operator objective.
Minimizing the operator objective involves two optimizations: minimizing the objective with respect
to the approximating family Q and maximizing the objective with respect to the function class F
(which is part of the objective).
We index the family Q with variational parameters and require that it satisfies properties typically
assumed by black box methods [19]: the variational distribution q.zI / has a known and tractable
density; we can sample from q.zI /; and we can tractably compute the score function r log q.zI /.
We index the function class F with parameters , and require that f ./ is differentiable. In the
experiments, we use neural networks, which are flexible enough to approximate a general family of
test functions [8].
Given parameterizations of the variational family and test family, operator variational inference
(opvi) seeks to solve a minimax problem,
D inf sup t .E ?.O p;q f /.z/?/:
(6)
We will use stochastic optimization [23, 13]. In principle, we can find stochastic gradients of
by rewriting the objective in terms of the optimized value of , ./. In practice, however, we
4
Algorithm 1: Operator variational inference
Input : Model log p.x; z/, variational approximation q.zI /
Output: Variational parameters
Initialize and randomly.
while not converged do
Compute unbiased estimates of r L from Equation 7.
Compute unbiased esimates of r L from Equation 8.
Update , with unbiased stochastic gradients.
end
simultaneously solve the maximization and minimization. Though computationally beneficial, this
produces saddle points. In our experiments we found it to be stable enough. We derive gradients for
the variational parameters and test function parameters . (We fix the distance function to be the
square t .a/ D a2 ; the identity t .a/ D a also readily applies.)
Gradient with respect to . For a fixed test function with parameters , denote the objective
L D t .E ?.O p;q f /.z/?/:
The gradient with respect to variational parameters is
r L D 2 E ?.O p;q f /.z/? r E ?.O p;q f /.z/?:
Now write the second expectation with the score function gradient [19]. This gradient is
r L D 2 E ?.O p;q f /.z/? E ?r log q.zI /.O p;q f /.z/ C r .O p;q f /.z/?:
(7)
Equation 7 lets us calculate unbiased stochastic gradients. We first generate two sets of independent
samples from q; we then form Monte Carlo estimates of the first and second expectations. For the
second expectation, we can use the variance reduction techniques developed for black box variational
inference, such as Rao-Blackwellization [19].
We described the score gradient because it is general. An alternative is to use the reparameterization
gradient for the second expectation [11, 22]. It requires that the operator be differentiable with respect
to z and that samples from q can be drawn as a transformation r of a parameter-free noise source ,
z D r.; /. In our experiments, we use the reparameterization gradient.
Gradient with respect to . Mirroring the notation above, the operator objective for fixed variational is
L D t .E ?.O p;q f /.z/?/:
The gradient with respect to test function parameters is
r L D 2 E ?.O p;q f /.z/? E ?r O p;q f .z/?:
(8)
Again, we can construct unbiased stochastic gradients with two sets of Monte Carlo estimates. Note
that gradients for the test function do not require score gradients (or reparameterization gradients)
because the expectation does not depend on .
Algorithm. Algorithm 1 outlines opvi. We simultaneously minimize the variational objective with
respect to the variational family q while maximizing it with respect to the function class f . Given
a model, operator, and function class parameterization, we can use automatic differentiation to calculate the necessary gradients [3]. Provided the operator does not require model-specific computation,
this algorithm satisfies the black box criteria.
3.1
Data Subsampling and opvi
With stochastic optimization, data subsampling scales up traditional variational inference to massive
data [7, 26]. The idea is to calculate noisy gradients by repeatedly subsampling from the data set,
without needing to pass through the entire data set for each gradient.
5
An as illustration, consider hierarchical models. Hierarchical models consist of global latent variables ? that are shared across data points and local latent variables zi each of which is associated to
a data point xi . The model?s log joint density is
log p.x1Wn ; z1Wn ; ?/ D log p.?/ C
n h
X
iD1
i
log p.xi j zi ; ?/ C log p.zi j ?/ :
Hoffman et al. [7] calculate unbiased estimates of the log joint density (and its gradient) by subsampling data and appropriately scaling the sum.
We can characterize whether opvi with a particular operator supports data subsampling. opvi relies
on evaluating the operator and its gradient at different realizations of the latent variables (Equation 7
and Equation 8). Thus we can subsample data to calculate estimates of the operator when it derives
from linear operators of the log density, such as differentiation and the identity. This follows as a
linear operator of sums is a sum of linear operators, so the gradients in Equation 7 and Equation 8
decompose into a sum. The Langevin-Stein and kl operator are both linear in the log density; both
support data subsampling.
3.2
Variational Programs
Given an operator and variational family, Algorithm 1 optimizes the corresponding operator objective. Certain operators require the density of q. For example, the kl operator (Equation 5) requires
its log density. This potentially limits the construction of rich variational approximations for which
the density of q is difficult to compute.1
Some operators, however, do not depend on having a analytic density; the Langevin-Stein (ls) operator (Equation 3) is an example. These operators can be used with a much richer class of variational
approximations, those that can be sampled from but might not have analytically tractable densities.
We call such approximating families variational programs.
Inference with a variational program requires the family to be reparameterizable [11, 22]. (Otherwise
we need to use the score function, which requires the derivative of the density.) A reparameterizable variational program consists of a parametric deterministic transformation R of random noise .
Formally, let
Normal.0; 1/;
z D R.I /:
(9)
This generates samples for z, is differentiable with respect to , and its density may be intractable. For
operators that do not require the density of q, it can be used as a powerful variational approximation.
This is in contrast to the standard Kullback-Leibler (kl) operator.
As an example, consider the following variational program for a one-dimensional random variable.
Let i denote the i th dimension of and make the corresponding definition for :
z D .3 > 0/R.1 I 1 /
.3 0/R.2 I 2 /:
(10)
When R outputs positive values, this separates the parametrization of the density to the positive
and negative halves of the reals; its density is generally intractable. In Section 4, we will use this
distribution as a variational approximation.
Equation 9 contains many densities when the function class R can approximate arbitrary continuous
functions. We state it formally.
Theorem 1. Consider a posterior distribution p.z j x/ with a finite number of latent variables and
continuous quantile function. Assume the operator variational objective has a unique root at the
posterior p.z j x/ and that R can approximate continuous functions. Then there exists a sequence
of parameters 1 ; 2 : : : ; in the variational program, such that the operator variational objective
converges to 0, and thus q converges in distribution to p.z j x/.
This theorem says that we can use variational programs with an appropriate q-independent operator
to approximate continuous distributions. The proof is in Appendix D.
1 It
is possible to construct rich approximating families with kl.qjjp/, but this requires the introduction of
an auxiliary distribution [15].
6
4
Empirical Study
We evaluate operator variational inference on a mixture of Gaussians, comparing different choices
in the objective. We then study logistic factor analysis for images.
4.1
Mixture of Gaussians
Consider a one-dimensional mixture of Gaussians as the posterior of interest,
p.z/ D 12 Normal.zI 3; 1/ C 21 Normal.zI 3; 1/. The posterior contains multiple modes.
We seek to approximate it with three variational objectives: Kullback-Leibler (kl) with a Gaussian
approximating family, Langevin-Stein (ls) with a Gaussian approximating family, and ls with a
variational program.
KL
Langevin-Stein
Variational Program
Truth
Truth
Truth
5
0
5
Value of Latent Variable z
5
0
5
Value of Latent Variable z
5
0
5
Value of Latent Variable z
Figure 1: The true posterior is a mixture of two Gaussians, in green. We approximate it with a
Gaussian using two operators (in blue). The density on the far right is a variational program given
in Equation 10 and using the Langevin-Stein operator; it approximates the truth well. The density
of the variational program is intractable. We plot a histogram of its samples and compare this to the
histogram of the true posterior.
Figure 1 displays the posterior approximations. We find that the kl divergence and ls divergence
choose a single mode and have slightly different variances. These operators do not produce good
results because a single Gaussian is a poor approximation to the mixture. The remaining distribution
in Figure 1 comes from the toy variational program described by Equation 10 with the ls operator.
Because this program captures different distributions for the positive and negative half of the real
line, it is able to capture the posterior.
In general, the choice of an objective balances statistical and computational properties of variational
inference. We highlight one tradeoff: the ls objective admits the use of a variational program;
however, the objective is more difficult to optimize than the kl.
4.2
Logistic Factor Analysis
Logistic factor analysis models binary vectors xi with a matrix of parameters W and biases b,
zi Normal.0; 1/
xi;k Bernoulli. .w>
k zi C bk //;
where zi has fixed dimension K and is the sigmoid function. This model captures correlations of
the entries in xi through W .
We apply logistic factor analysis to analyze the binarized MNIST data set [24], which contains 28x28
binary pixel images of handwritten digits. (We set the latent dimensionality to 10.) We fix the model
parameters to those learned with variational expectation-maximization using the kl divergence, and
focus on comparing posterior inferences.
We compare the kl operator to the ls operator and study two choices of variational models: a fully
factorized Gaussian distribution and a variational program. The variational program generates samples by transforming a K-dimensional standard normal input with a two-layer neural network, using
rectified linear activation functions and a hidden size of twice the latent dimensionality. Formally,
7
Inference method
Completed data log-likelihood
Mean-field Gaussian + kl
Mean-field Gaussian + ls
Variational Program + ls
-59.3
-75.3
-58.9
Table 1: Benchmarks on logistic factor analysis for binarized MNIST. The same variational approximation with ls performs worse than kl on likelihood performance. The variational program with
ls performs better without directly optimizing for likelihoods.
the variational program we use generates samples of z as follows:
z0 Normal.0; I /
>
h0 D ReLU.W0q z0 C bq0 /
>
h1 D ReLU.W1q h0 C bq1 /
>
z D W2q h1 C bq2 :
The variational parameters are the weights W q and biases bq . For f , we use a three-layer neural network with the same hidden size as the variational program and hyperbolic tangent activations where
unit activations were bounded to have norm two. Bounding the unit norm bounds the divergence.
We used the Adam optimizer [12] with learning rates 2 10 4 for f and 2 10 5 for the variational
approximation.
There is no standard for evaluating generative models and their inference algorithms [25]. Following
Rezende et al. [22], we consider a missing data problem. We remove half of the pixels in the test set
(at random) and reconstruct them from a fitted posterior predictive distribution. Table 1 summarizes
the results on 100 test images; we report the log-likelihood of the completed image. ls with the
variational program performs best. It is followed by kl and the simpler ls inference. The ls performs
better than kl even though the model parameters were learned with kl.
5
Summary
We present operator variational objectives, a broad yet tractable class of optimization problems for
approximating posterior distributions. Operator objectives are built from an operator, a family of
test functions, and a distance function. We outline the connection between operator objectives and
existing divergences such as the KL divergence, and develop a new variational objective using the
Langevin-Stein operator. In general, operator objectives produce new ways of posing variational
inference.
Given an operator objective, we develop a black box algorithm for optimizing it and show which
operators allow scalable optimization through data subsampling. Further, unlike the popular evidence
lower bound, not all operators explicitly depend on the approximating density. This permits flexible
approximating families, called variational programs, where the distributional form is not tractable.
We demonstrate this approach on a mixture model and a factor model of images.
There are several possible avenues for future directions such as developing new variational objectives,
adversarially learning [5] model parameters with operators, and learning model parameters with
operator variational objectives.
Acknowledgments. This work is supported by NSF IIS-1247664, ONR N00014-11-1-0651,
DARPA FA8750-14-2-0009, DARPA N66001-15-C-4032, Adobe, NSERC PGS-D, Porter Ogden
Jacobus Fellowship, Seibel Foundation, and the Sloan Foundation. The authors would like to thank
Dawen Liang, Ben Poole, Stephan Mandt, Kevin Murphy, Christian Naesseth, and the anonymous
reviews for their helpful feedback and comments.
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9
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5,627 | 6,092 | The Multiple Quantile Graphical Model
Alnur Ali
Machine Learning Department
Carnegie Mellon University
alnurali@cmu.edu
J. Zico Kolter
Computer Science Department
Carnegie Mellon University
zkolter@cs.cmu.edu
Ryan J. Tibshirani
Department of Statistics
Carnegie Mellon University
ryantibs@cmu.edu
Abstract
We introduce the Multiple Quantile Graphical Model (MQGM), which extends
the neighborhood selection approach of Meinshausen and B?hlmann for learning
sparse graphical models. The latter is defined by the basic subproblem of modeling the conditional mean of one variable as a sparse function of all others. Our
approach models a set of conditional quantiles of one variable as a sparse function
of all others, and hence offers a much richer, more expressive class of conditional
distribution estimates. We establish that, under suitable regularity conditions, the
MQGM identifies the exact conditional independencies with probability tending to
one as the problem size grows, even outside of the usual homoskedastic Gaussian
data model. We develop an efficient algorithm for fitting the MQGM using the
alternating direction method of multipliers. We also describe a strategy for sampling from the joint distribution that underlies the MQGM estimate. Lastly, we
present detailed experiments that demonstrate the flexibility and effectiveness of
the MQGM in modeling hetereoskedastic non-Gaussian data.
1
Introduction
We consider modeling the joint distribution Pr(y1 , . . . , yd ) of d random variables, given n independent draws from this distribution y (1) , . . . , y (n) ? Rd , where possibly d n. Later, we generalize
this setup and consider modeling the conditional distribution Pr(y1 , . . . , yd |x1 , . . . , xp ), given n
independent pairs (x(1) , y (1) ), . . . , (x(n) , y (n) ) ? Rp+d . Our starting point is the neighborhood selection method [28], which is typically considered in the context of multivariate Gaussian data, and seen
as a tool for covariance selection [8]: when Pr(y1 , . . . , yd ) is a multivariate Gaussian distribution,
it is a well-known fact that yj and yk are conditionally independent given the remaining variables
if and only if the coefficent corresponding to yk is zero in the (linear) regression of yj on all other
variables (e.g., [22]). Therefore, in neighborhood selection we compute, for each k = 1, . . . , d,
a lasso regression ? in order to obtain a small set of conditional dependencies ? of yk on the
remaining variables, i.e.,
2
n
X
X
(i)
(i)
minimize
yk ?
?kj yj
+ ?k?k k1 ,
(1)
?k ?Rd
i=1
j6=k
for a tuning parameter ? > 0. This strategy can be seen as a pseudolikelihood approximation [4],
Pr(y1 , . . . , yd ) ?
d
Y
Pr(yk |y?k ),
(2)
k=1
where y?k denotes all variables except yk . Under the multivariate Gaussian model for Pr(y1 , . . . , yd ),
the conditional distributions Pr(yk |y?k ), k = 1, . . . , d here are (univariate) Gaussians, and maximizing the pseudolikelihood in (2) is equivalent to separately maximizing the conditionals, as is precisely
done in (1) (with induced sparsity), for k = 1, . . . , d.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Following the pseudolikelihood-based approach traditionally means carrying out three steps: (i) we
write down a suitable family of joint distributions for Pr(y1 , . . . , yd ), (ii) we derive the conditionals
Pr(yk |y?k ), k = 1, . . . , d, and then (iii) we maximize each conditional likelihood by (freely) fitting
the parameters. Neighborhood selection, and a number of related approaches that came after it (see
Section 2.1), can be all thought of in this workflow. In many ways, step (ii) acts as the bottleneck
here, and to derive the conditionals, we are usually limited to a homoskedastic and parameteric family
for the joint distribution.
The approach we take in this paper differs somewhat substantially, as we begin by directly modeling
the conditionals in (2), without any preconceived model for the joint distribution ? in this sense, it
may be seen a type of dependency network [13] for continuous data. We also employ heteroskedastic,
nonparametric models for the conditional distributions, which allows us great flexibility in learning
these conditional relationships. Our method, called the Multiple Quantile Graphical Model (MQGM),
is a marriage of ideas in high-dimensional, nonparametric, multiple quantile regression with those in
the dependency network literature (the latter is typically focused on discrete, not continuous, data).
An outline for this paper is as follows. Section 2 reviews background material, and Section 3 develops
the MQGM estimator. Section 4 studies basic properties of the MQGM, and establishes a structure
recovery result under appropriate regularity conditions, even for heteroskedastic, non-Gaussian data.
Section 5 describes an efficient ADMM algorithm for estimation, and Section 6 presents empirical
examples comparing the MQGM versus common alternatives. Section 7 concludes with a discussion.
2
2.1
Background
Neighborhood selection and related methods
Neighborhood selection has motivated a number of methods for learning sparse graphical models. The
literature here is vast; we do not claim to give a complete treatment, but just mention some relevant
approaches. Many pseudolikelihood approaches have been proposed, see e.g., [35, 33, 12, 24, 17, 1].
These works exploit the connection between estimating a sparse inverse covariance matrix and regression, and they vary in terms of the optimization algorithms they use and the theoretical guarantees
they offer. In a clearly related but distinct line of research, [45, 2, 11, 36] proposed `1 -penalized
likelihood estimation in the Gaussian graphical model, a method now generally termed the graphical
lasso (GLasso). Following this, several recent papers have extended the GLasso in various ways. [10]
examined a modification based on the multivariate Student t-distribution, for robust graphical modeling. [37, 46, 42] considered conditional distributions of the form Pr(y1 , . . . , yd |x1 , . . . , xp ). [23]
proposed a model for mixed (both continuous and discrete) data types, generalizing both GLasso and
pairwise Markov random fields. [25, 26] used copulas for learning non-Gaussian graphical models.
A strength of neighborhood-based (i.e., pseudolikelihood-based) approaches lies in their simplicity;
because they essentially reduce to a collection of univariate probability models, they are in a sense
much easier to study outside of the typical homoskedastic, Gaussian data setting. [14, 43, 44] elegantly studied the implications of using univariate exponential family models for the conditionals in
(2). Closely related to pseudoliklihood approaches are dependency networks [13]. Both frameworks
focus on the conditional distributions of one variable given all the rest; the difference lies in whether
or not the model for conditionals stems from first specifying some family of joint distributions (pseudolikelihood methods), or not (dependency networks). Dependency networks have been thoroughly
studied for discrete data, e.g., [13, 29]. For continuous data, [40] proposed modeling the mean in a
Gaussian neighborhood regression as a nonparametric, additive function of the remaining variables,
yielding flexible relationships ? this is a type of dependency network for continuous data (though it
is not described by the authors in this way). Our method, the MQGM, also deals with continuous
data, and is the first to our knowledge that allows for fully nonparametric conditional distributions, as
well as nonparametric contributions of the neighborhood variables, in each local model.
2.2
Quantile regression
In linear regression, we estimate the conditional mean of y|x1 , . . . , xp from samples. Similarly, in ?quantile regression [20], we estimate the conditional ?-quantile of y|x1 , . . . , xp for a given ? ? [0, 1],
formally Qy|x1 ,...,xp (?)
. . , xp ) ? ?}, by solving the convex optimization
Pn= inf{t :(i)Pr(yP?p t|x1 , .(i)
problem: minimize?
? j=1 ?j xj ), where ?? (z) = max{?z, (? ? 1)z} is the ?i=1 ?? (y
2
quantile loss (also called the ?pinball? or ?tilted absolute? loss). Quantile regression can be useful
when the conditional distribution in question is suspected to be heteroskedastic and/or non-Gaussian,
e.g., heavy-tailed, or if we wish to understand properties of the distribution other than the mean,
e.g., tail behavior. In multiple quantile regression, we solve several quantile regression problems
simultaneously, each corresponding to a different quantile level; these problems can be coupled
somehow to increase efficiency in estimation (see details in the next section). Again, the literature
on quantile regression is quite vast (especially that from econometrics), and we only give a short
review here. A standard text is [18]. Nonparametric modeling of quantiles is a natural extension from
the (linear) quantile regression approach outlined above; in the univariate case (one conditioning
variable), [21] suggested a method using smoothing splines, and [38] described an approach using
kernels. More recently, [19] studied the multivariate nonparametric case (more than one conditioning
variable), using additive models. In the high-dimensional setting, where p is large, [3, 16, 9] studied
`1 -penalized quantile regression and derived estimation and recovery theory for non-(sub-)Gaussian
data. We extend results in [9] to prove structure recovery guarantees for the MQGM (in Section 4.3).
3
The multiple quantile graphical model
Many choices can be made with regards to the final form of the MQGM, and to help in understanding
these options, we break down our presentation in parts. First fix some ordered set A = {?1 , . . . , ?r }
of quantile levels, e.g., A = {0.05, 0.10, . . . , 0.95}. For each variable yk , and each level ?` , we
model the conditional ?` -quantile given the other variables, using an additive expansion of the form:
Qyk |y?k (?` ) = b?`k +
d
X
?
f`kj
(yj ),
(3)
j6=k
?
where b?`k ? R is an intercept term, and f`kj
, j = 1, . . . , d are smooth, but not parametric in form. In
its most general form, the MQGM estimator is defined as a collection of optimization problems, over
k = 1, . . . , d and ` = 1, . . . , r:
X
n
?
X
X
(i)
(i)
f`kj (yj ) +
?1 P1 (f`kj ) + ?2 P2 (f`kj ) . (4)
minimize
??` yk ? b`k ?
b`k , f`kj ?F`kj ,
i=1
j=1,...,d
j6=k
j6=k
Here ?1 , ?2 ? 0 are tuning parameters, F`kj , j = 1, . . . , d are univariate function spaces, ? > 0 is
a fixed exponent, and P1 , P2 are sparsity and smoothness penalty functions, respectively. We give
three examples below; many other variants are also possible.
Example 1: basis expansion model Consider taking F`kj = span{?j1 , . . . , ?jm }, the span of m
basis functions, e.g., radial basis functions (RBFs) with centers placed at appropriate locations across
the domain of variable j, for each j = 1, . . . , d. This means that each f`kj ? F`kj can be expressed
T
as f`kj (x) = ?`kj
?j (x), for a coefficient vector ?`kj ? Rm , where ?j (x) = (?j1 (x), . . . , ?jm (x)).
Also consider an exponent ? = 1, and the sparsity and smoothness penalties
P1 (f`kj ) = k?`kj k2
and
P2 (f`kj ) = k?`kj k22 ,
respectively, which are group lasso and ridge penalties, respectively. With these choices in place, the
MQGM problem in (4) can be rewritten in finite-dimensional form:
X
minimize
??` Yk ? b`k 1 ? ??`k +
?1 k?`kj k2 + ?2 k?`kj k22 .
(5)
b`k , ?`k =(?`k1 ,...,?`kd )
j6=k
Pn
Above, we have used the abbreviation ??` (z) = i=1 ??` (zi ) for a vector z = (z1 , . . . , zn ) ? Rn ,
(1)
(n)
and also Yk = (yk , . . . , yk ) ? Rn for the observations along variable k, 1 = (1, . . . , 1) ? Rn , and
(i)
n?dm
??R
for the basis matrix, with blocks of columns to be understood as ?ij = ?(yj )T ? Rm .
The basis expansion model is simple and tends to work well in practice, so we focus on it for most of
the paper. In principle, essentially all our results apply to the next two models we describe, as well.
Example 2: smoothing splines model Now consider taking F`kj = span{g1j , . . . , gnj }, the span
(1)
(n)
of m = n natural cubic splines with knots at yj , . . . , yj , for j = 1, . . . , d. As before, we can
T
j
then write f`kj (x) = ?`kj g (x) with coefficients ?`kj ? Rn , for f`kj ? F`kj . The work of [27], on
high-dimensional additive smoothing splines, suggests a choice of exponent ? = 1/2, and penalties
P1 (f`kj ) = kGj ?`kj k22
T
and P2 (f`kj ) = ?`kj
?j ?`kj ,
3
for sparsity and smoothness, respectively, where Gj ? Rn?n is a spline basis matrix with entries
(i)
Gjii0 = gij0 (yj ), and ?j is the smoothing spline penalty matrix containing integrated products of
pairs of twice differentiated basis functions. The MQGM problem in (4) can be translated into a
finite-dimensional form, very similar to what we have done in (5), but we omit this for brevity.
Example 3: RKHS model Consider taking F`kj = Hj , a univariate reproducing kernel Hilbert
space (RKHS), with kernel function ?j (?, ?). The representer theorem allows
Pn us to express each
(i)
function f`kj ? Hj in terms of the representers of evaluation, i.e., f`kj (x) = i=1 (?`kj )i ?j (x, yj ),
n
for a coefficient vector ?`kj ? R . The work of [34], on high-dimensional additive RKHS modeling,
suggests a choice of exponent ? = 1, and sparsity and smoothness penalties
q
T Kj?
P1 (f`kj ) = kK j ?`kj k2 and P2 (f`kj ) = ?`kj
`kj ,
(i)
(i0 )
respectively, where K j ? Rn?n is the kernel matrix with entries Kiij 0 = ?j (yj , yj ). Again, the
MQGM problem in (4) can be written in finite-dimensional form, now an SDP, omitted for brevity.
Structural constraints
Several structural constraints can be placed on top of the MQGM optimization problem in order to guide the estimated component functions to meet particular shape
requirements. An important example are non-crossing constraints (commonplace in nonparametric,
multiple quantile regression [18, 38]): here, we optimize (4) jointly over ` = 1, . . . , r, subject to
X
X
(i)
(i)
b`k +
f`kj (yj ) ? b`0 k +
f`0 kj (yj ), for all ?` < ?`0 , and i = 1, . . . , n.
(6)
j6=k
j6=k
This ensures that the estimated quantiles obey the proper ordering, at the observations. For concreteness, we consider the implications for the basis regression model, in Example 1 (similar statements
hold for the other two models). For each ` = 1, . . . , r, denote by F`k (b`k , ?`k ) the criterion in (5).
Introducing the non-crossing constraints requires coupling (5) over ` = 1, . . . , r, so that we now have
the following optimization problems, for each target variable k = 1, . . . , d:
r
X
minimize
F`k (b`k , ?`k ) subject to (1BkT + ??k )DT ? 0,
(7)
Bk ,?k
`=1
where we denote Bk = (b1k , . . . , brk ) ? Rr , ? ? Rn?dm the basis matrix as before, ?k ? Rdm?r
given by column-stacking ?`k ? Rdm , ` = 1, . . . , r, and D ? R(r?1)?r is the usual discrete
difference operator. (The inequality in (7) is to be interpreted componentwise.) Computationally,
coupling the subproblems across ` = 1, . . . , r clearly adds to the overall difficulty of the MQGM, but
statistically this coupling acts as a regularizer, by constraining the parameter space in a useful way,
thus increasing our efficiency in fitting multiple quantile levels from the given data.
(i)
(i0 )
For a triplet
`, k, j, monotonicity constraints are also easy to add, i.e., f`kj (yj ) ? f`kj (yj ) for all
(i)
(i0 )
yj < yj . Convexity constraints, where we require f`kj to be convex over the observations, for a
particular `, k, j, are also straightforward. Lastly, strong non-crossing constraints, where we enforce
(6) over all z ? Rd (not just over the observations) are also possible with positive basis functions.
Exogenous variables and conditional random fields So far, we have considered modeling the
joint distribution Pr(y1 , . . . , yd ), corresponding to learning a Markov random field (MRF). It is not
hard to extend our framework to model the conditional distribution Pr(y1 , . . . , yd |x1 , . . . , xp ) given
some exogenous variables x1 , . . . , xp , corresponding to learning a conditional random field (CRF).
x
To extend the basis regression model, we introduce the additional parameters ?`k
? Rp in (5), and the
n?q
T
x
loss now becomes ??` (Yk ? b`k 1 ? ??`k ? X?`k ), where X ? R
is filled with the exogenous
observations x(1) , . . . , x(n) ? Rq ; the other models are changed similarly.
4
4.1
Basic properties and theory
Quantiles and conditional independence
?
In the model (3), when a particular variable yj has no contribution, i.e., satisfied f`kj
= 0 across all
quantile levels ?` , ` = 1, . . . , r, what does this imply about the conditional independence between yk
and yj , given the rest? Outside of the multivariate normal model (where the feature transformations
need only be linear), nothing can be said in generality. But we argue that conditional independence can
be understood in a certain approximate sense (i.e., in a projected approximation of the data generating
model). We begin with a simple lemma. Its proof is elementary, and given in the supplement.
4
Lemma 4.1. Let U, V, W be random variables, and suppose that all conditional quantiles of U |V, W
do not depend on V , i.e., QU |V,W (?) = QU |W (?) for all ? ? [0, 1]. Then U and V are conditionally
independent given W .
By the lemma, if we knew that QU |V,W (?) = h(?, U, W ) for a function h, then it would follow that
U, V are conditionally independent given W (n.b., the converse is true, as well). The MQGM problem
in (4), with sparsity imposed on the coefficients, essentially aims to achieve such a representation
for the conditional quantiles; of course we cannot use a fully nonparametric representation of the
conditional distribution yk |y?k and instead we use an r-step approximation to the conditional cumulative distribution function (CDF) of yk |y?k (corresponding to estimating r conditional quantiles),
and (say) in the basis regression model, limit the dependence on conditioning variables to be in terms
of an additive function of RBFs in yj , j 6= k. Thus, if at the solution in (5) we find that ??`kj = 0,
` = 1, . . . , r, we may interpret this to mean that yk and yj are conditionally independent given the
remaining variables, but according to the distribution defined by the projection of yk |y?k onto the
space of models considered in (5) (r-step conditional CDFs, which are additive expansions in yj ,
j 6= k). This interpretation is no more tenuous (arguably, less so, as the model space here is much
larger) than that needed when applying standard neighborhood selection to non-Gaussian data.
4.2
Gibbs sampling and the ?joint? distribution
When specifying a form for the conditional distributions in a pseudolikelihood approximation as in
(2), it is natural to ask: what is the corresponding joint distribution? Unfortunately, for a general
collection of conditional distributions, there need not exist a compatible joint distribution, even
when all conditionals are continuous [41]. Still, pseudolikelihood approximations (a special case
of composite likelihood approximations), possess solid theoretical backing, in that maximizing the
pseudolikelihood relates closely to minimizing a certain (expected composite) Kullback-Leibler
divergence, measured to the true conditionals [39]. Recently, [7, 44] made nice progress in describing
specific conditions on conditional distributions that give rise to a valid joint distribution, though their
work was specific to exponential families. A practical answer to the question of this subsection is to
use Gibbs sampling, which attempts to draw samples consistent with the fitted conditionals; this is
precisely the observation of [13], who show that Gibbs sampling from discrete conditionals converges
to a unique stationary distribution, although this distribution may not actually be compatible with the
conditionals. The following result establishes the analogous claim for continuous conditionals; its
proof is in the supplement. We demonstrate the practical value of Gibbs sampling through various
examples in Section 6.
Lemma 4.2. Assume that the conditional distributions Pr(yk |y?k ), k = 1, . . . , d take only positive
values on their domain. Then, for any given ordering of the variables, Gibbs sampling converges to a
unique stationary distribution that can be reached from any initial point. (This stationary distribution
depends on the ordering.)
4.3
Graph structure recovery
When log d = O(n2/21 ), and we assume somewhat standard regularity conditions (listed as A1?A4
in the supplement), the MQGM estimate recovers the underlying conditional independencies with
high probability (interpreted in the projected model space, as explained in Section 4.1). Importantly,
we do not require a Gaussian, sub-Gaussian, or even parametric assumption on the data generating
process; instead, we assume i.i.d. draws y (1) , . . . , y (n) ? Rd , where the conditional distributions
yk |y?k have quantiles specified by the model in (3) for k = 1, . . . , d, ` = 1, . . . , r, and further, each
?
T
?
f`kj
(x) = ?`kj
?j (x)? for coefficients ?`kj
? Rm , j = 1, . . . , d, as in the basis expansion model.
Let E ? denote the corresponding edge set of conditional dependencies from these neighborhood
?
?
models, i.e., {k, j} ? E ? ?? max`=1,...,r max{k?`kj
k2 , |?`jk
k2 } > 0. We define the estimated
?
edge set E in the analogous way, based on the solution
in
(5).
Without
a loss of generality, we assume
?
the features have been scaled to satisfy k?j k ? n for all j = 1, . . . , dm. The following is our
recovery result; its proof is provided in the supplement.
Theorem 4.3. Assume log d = O(n2/21 ), and conditions A1?A4 in the supplement. Assume that
?
the tuning parameters ?1 , ?2 satisfy ?1 (mn log(d2 mr/?) log3 n)1/2 and ?2 = o(n41/42 /?max
),
?
?
where ?max = max`,k,j k?`kj k2 . Then for n sufficiently large, the MQGM estimate in (5) exactly
? = E ? , with probability at least 1 ? ?.
recovers the underlying conditional dependencies, i.e., E
5
The theorem shows that the nonzero pattern in the MQGM estimate identifies, with high probability,
the underlying conditional independencies. But to be clear, we emphasize that the MQGM estimate
is not an estimate of the inverse covariance matrix itself (this is also true of neighborhood regression,
SpaceJam of [40], and many other methods for learning graphical models).
5
Computational approach
By design, the MQGM problem in (5) separates into d subproblems, across k = 1, . . . , d (it therefore
suffices to consider only a single subproblem, so we omit notational dependence on k for auxiliary variables). While these subproblems are challenging for off-the-shelf solvers (even for only
moderately-sized graphs), the key terms here all admit efficient proximal operators [32], which makes
operator splitting methods like the alternating direction method of multipliers [5] a natural choice.
As an illustration, we consider the non-crossing constraints in the basis regression model below.
Reparameterizing our problem, so that we may apply ADMM, yields:
Pr Pd
minimize
?A (Z) + ?1 `=1 j=1 kW`j k2 + ?22 kW k2F + I+ (V DT )
?k ,Bk ,V,W,Z
(8)
subject to
V = 1BkT + ??k , W = ?k , Z = Yk 1T ? 1BkT ? ??k ,
Pr Pd
where for brevity ?A (A) = `=1 j=1 ??` (A`j ), and I+ (?) is the indicator function of the space
of elementwise nonnegative matrices. The augmented Lagrangian associated with (8) is:
L? (?k , Bk , V, W, Z, UV , UW , UZ ) = ?A (Z) + ?1
r X
d
X
`=1 j=1
kW`j k2 +
?2
kW k2F + I+ (V DT )
2
?
+
k1BkT + ??k ? V + UV k2F + k?k ? W + UW k2F + kYk 1T ? 1BkT ? ??k ? Z + UZ k2F ,
2
(9)
where ? > 0 is the augmented Lagrangian parameter, and UV , UW , UZ are dual variables corresponding to the equality constraints on V, W, Z, respectively. Minimizing (9) over V yields:
V ? Piso 1BkT + ??k + UV ,
(10)
where Piso (?) denotes the row-wise projection operator onto the isotonic cone (the space of componentwise nondecreasing vectors), an O(nr) operation here [15]. Minimizing (9) over W`j yields the
update:
(?k )`j + (UW )`j
?1 /?
W`j ?
,
(11)
1?
1 + ?2 /?
k(?k )`j + (UW )`j k2 +
where (?)+ is the positive part operator. This can be seen by deriving the proximal operator of the
function f (x) = ?1 kxk2 + (?2 /2)kxk22 . Minimizing (9) over Z yields the update:
Z ? prox(1/?)?A (Yk 1T ? 1bTk ? ??k + UZ ),
(12)
where proxf (?) denotes the proximal operator of a function f . For the multiple quantile loss function
?A , this is a kind of generalized soft-thresholding. The proof is given in the supplement.
Lemma 5.1. Let P+ (?) and P? (?) be the elementwise positive and negative part operators, respectively, and let a = (?1 , . . . , ?r ). Then proxt?A (A) = P+ (A ? t1aT ) + P? (A ? t1aT ).
Finally, differentiation in (9) with respect to Bk and ?k yields the simultaneous updates:
?1
1 ?T ? + 21 I ?T 1
?k
T
?
[I 0] (W ? UW ) +
BkT
1T ?
1T 1
2
T
[? 1] (Yk 1T ? Z + UZ + V ? UV ) . (13)
A complete description of our ADMM algorithm for solving the MQGM problem is in the supplement.
Gibbs sampling Having fit the conditionals yk |y?k , k = 1, . . . d, we may want to make predictions
or extract joint distributions over subsets of variables. As discussed in Section 4.2, there is no general
analytic form for these joint distributions, but the pseudolikelihood approximation underlying the
MQGM suggests a natural Gibbs sampler. A careful implementation that respects the additive model
in (3) yields a highly efficient Gibbs sampler, especially for CRFs; the supplement gives details.
6
6
Empirical examples
6.1
Synthetic data
We consider synthetic examples, comparing the MQGM to neighborhood selection (MB), the graphical lasso (GLasso), SpaceJam [40], the nonparanormal skeptic [26], TIGER [24], and neighborhood
selection using the absolute loss (Laplace).
Ring example As a simple but telling example, we drew n = 400 samples from a ?ring? distribution
in d = 4 dimensions. Data were generated by drawing a random angle ? ? Uniform(0, 1), a random
radius R ? N (0, 0.1), and then computing the coordinates y1 = R cos ?, y2 = R sin ? and
y3 , y4 ? N (0, 1), i.e., y1 and y2 are the only dependent variables here. The MQGM was used with
m = 10 basis functions (RBFs), and r = 20 quantile levels. The left panel of Figure 1 plots samples
(blue) of the coordinates y1 , y2 as well as new samples from the MQGM (red) fitted to these same
(blue) samples, obtained by using our Gibbs sampler; the samples from the MQGM appear to closely
match the samples from the underlying ring. The main panel of Figure 1 shows the conditional
dependencies recovered by the MQGM, SpaceJam, GLasso, and MB (plots for the other methods are
given in the supplement), when run on the ring data. We visualize these dependencies by forming a
d ? d matrix with the cell (j, k) set to black if j, k are conditionally dependent given the others, and
white otherwise. Across a range of tuning parameters for each method, the MQGM is the only one
that successfully recovers the underlying conditional dependencies, at some point along its solution
path. In the supplement, we present an evaluation of the conditional CDFs given by each method,
when run on the ring data; again, the MQGM performs best in this setting.
Larger examples To investigate performance at larger scales, we drew n ? {50, 100, 300} samples
from a multivariate normal and Student t-distribution (with 3 degrees of freedom), both in d = 100
dimensions, both parameterized by a random, sparse, diagonally dominant d ? d inverse covariance
matrix, following the procedure in [33, 17, 31, 1]. Over the same set of sample sizes, with d = 100, we
also considered an autoregressive setup in which we drew samples of pairs of adjacent variables from
the ring distribution. In all three data settings (normal, t, and autoregressive), we used m = 10 and
r = 20 for the MQGM. To summarize the performances, we considered a range of tuning parameters
for each method, computed corresponding false and true positive rates (in detecting conditional
dependencies), and then computed the corresponding area under the curve (AUC), following, e.g.,
[33, 17, 31, 1]. Table 1 reports the median AUCs (across 50 trials) for all three of these examples; the
MQGM outperforms all other methods on the autoregressive example; on the normal and Student t
examples, it performs quite competitively.
y2
0.5
Truth
?1 =8.00000
?1 =16.00000
?1 =32.00000
?1 =64.00000
?1 =128.00000
Truth
?1 =0.00781
?1 =0.01562
?1 =0.03125
?1 =0.06250
?1 =0.12500
Truth
?1 =0.12500
?1 =0.25000
?1 =0.50000
?1 =1.00000
?1 =2.00000
Truth
?1 =0.06250
?1 =0.12500
?1 =0.25000
?1 =0.50000
?1 =1.00000
MB
truth
MQGM
MQGM
1.5
1.0
0.0
GLasso
?1.5
?1.5
SpaceJam
?0.5
?1.0
?1.0
?0.5
0.0
y1
0.5
1.0
1.5
Figure 1: Left: data from the ring distribution (blue) as well as new samples from the MQGM (red) fitted to
the same (blue) data, obtained by using our Gibbs sampler. Right: conditional dependencies recovered by the
MQGM, MB, GLasso, and SpaceJam on the ring data; black means conditional dependence. The MQGM is the
only method that successfully recovers the underlying conditional dependencies along its solution path.
Table 1: AUC values for the MQGM, MB, GLasso, SpaceJam, the nonparanormal skeptic, TIGER, and
Laplace for the normal, t, and autoregressive data settings; higher is better, best in bold.
MQGM
MB
GLasso
SpaceJam
Nonpara.
TIGER
Laplace
n = 50
0.953
0.850
0.908
0.889
0.881
0.732
0.803
Normal
n = 100
0.976
0.959
0.964
0.968
0.962
0.921
0.931
n = 300
0.988
0.994
0.998
0.997
0.996
0.996
0.989
Student t
n = 100
0.947
0.923
0.605
0.965
0.942
0.873
0.876
n = 50
0.928
0.844
0.691
0.893
0.862
0.420
0.800
7
n = 300
0.981
0.988
0.965
0.993
0.998
0.989
0.991
n = 50
0.726
0.532
0.541
0.624
0.545
0.503
0.530
Autoregressive
n = 100
n = 300
0.754
0.955
0.563
0.725
0.620
0.711
0.708
0.854
0.590
0.612
0.518
0.718
0.554
0.758
1
10
8
2
5
7
Figure 2: Top panel and bottom row,
middle panel: conditional dependencies recovered by the MQGM on the
flu data; each of the first ten cells corresponds to a region of the U.S., and black
means dependence. Bottom row, left
panel: wallclock time (in seconds) for
solving one subproblem using ADMM
versus SCS. Bottom row, right panel:
samples from the fitted marginal distribution of the weekly flu incidence rates
at region 6; samples at larger quantiles
are shaded lighter, and the median is in
darker blue.
3
9
4
6
1
10
8
2
5
7
3
9
4
6
1
6
11
16
1
6
3
10
11
2
10
16
1
6.2
0
50
100
150
Seconds
200
250
% of flu-like symptoms
Objective value
MQGM
SCS
10
Region 6
9
4
10
8
7
6
5
4
3
2
1
0
30
40
50
8
18
28
Week
Modeling flu epidemics
We study n = 937 weekly flu incidence reports from September 28, 1997 through August 30,
2015, across 10 regions in the United States (see the top panel of Figure 2), obtained from [6]. We
considered d = 20 variables: the first 10 encode the current week?s flu incidence (precisely, the
percentage of doctor?s visits in which flu-like symptoms are presented) in the 10 regions, and the last
10 encode the same but for the prior week. We set m = 5, r = 99, and also introduced exogenous
variables to encode the week numbers, so p = 1. Thus, learning the MQGM here corresponds
to learning the structure of a spatiotemporal graphical model, and reduces to solving 20 multiple
quantile regression subproblems, each of dimension (19 ? 5 + 1) ? 99 = 9504. All subproblems
took about 1 minute on a 6 core 3.3 Ghz Core i7 X980 processor.
The bottom left panel in Figure 2 plots the time (in seconds) taken for solving one subproblem using
ADMM versus SCS [30], a cone solver that has been advocated as a reasonable choice for a class
of problems encapsulating (4); ADMM outperforms SCS by roughly two orders of magnitude. The
bottom middle panel of Figure 2 presents the conditional independencies recovered by the MQGM.
Nonzero entries in the upper left 10 ? 10 submatrix correspond to dependencies between the yk
variables for k = 1, . . . , 10; e.g., the nonzero (0,2) entry suggests that region 1 and 3?s flu reports are
dependent. The lower right 10 ? 10 submatrix corresponds to the yk variables for k = 11, . . . , 20,
and the nonzero banded entries suggest that at any region the previous week?s flu incidence (naturally)
influences the next week?s. The top panel of Figure 2 visualizes these relationships by drawing an
edge between dependent regions; region 6 is highly connected, suggesting that it may be a bellwether
for other regions, roughly in keeping with the current understanding of flu dynamics. To draw samples
from the fitted distributions, we ran our Gibbs sampler over the year, generating 1000 total samples,
making 5 passes over all coordinates between each sample, and with a burn-in period of 100 iterations.
The bottom right panel of Figure 2 plots samples from the marginal distribution of the percentages
of flu reports at region 6 (other regions are in the supplement) throughout the year, revealing the
heteroskedastic nature of the data.
For space reasons, our last example, on wind power data, is presented in the supplement.
7
Discussion
We proposed and studied the Multiple Quantile Graphical Model (MQGM). We established theoretical
and empirical backing to the claim that the MQGM is capable of compactly representing relationships
between heteroskedastic non-Gaussian variables. We also developed efficient algorithms for both
estimation and sampling in the MQGM. All in all, we believe that our work represents a step forward
in the design of flexible yet tractable graphical models.
Acknowledgements AA was supported by DOE Computational Science Graduate Fellowship DEFG02-97ER25308. JZK was supported by an NSF Expeditions in Computation Award, CompSustNet,
CCF-1522054. RJT was supported by NSF Grants DMS-1309174 and DMS-1554123.
8
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9
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5,628 | 6,093 | A Consistent Regularization Approach for Structured
Prediction
Carlo Ciliberto ?,1
cciliber@mit.edu
1
Alessandro Rudi ?,1,2
ale_rudi@mit.edu
Lorenzo Rosasco 1,2
lrosasco@mit.edu
Laboratory for Computational and Statistical Learning - Istituto Italiano di Tecnologia, Genova, Italy &
Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
2
Universit? degli Studi di Genova, Genova, Italy.
?
Equal contribution
Abstract
We propose and analyze a regularization approach for structured prediction problems. We characterize a large class of loss functions that allows to naturally
embed structured outputs in a linear space. We exploit this fact to design learning
algorithms using a surrogate loss approach and regularization techniques. We
prove universal consistency and finite sample bounds characterizing the generalization properties of the proposed method. Experimental results are provided to
demonstrate the practical usefulness of the proposed approach.
1
Introduction
Many machine learning applications require dealing with data-sets having complex structures, e.g.
natural language processing, image segmentation, reconstruction or captioning, pose estimation,
protein folding prediction to name a few [1, 2, 3]. Structured prediction problems pose a challenge
for classic off-the-shelf learning algorithms for regression or binary classification. This has motivated
the extension of methods such as support vector machines to structured problems [4]. Dealing
with structured prediction problems is also a challenge for learning theory. While the theory of
empirical risk minimization provides a very general statistical framework, in practice it needs to be
complemented with an ad-hoc analysis for each specific setting. Indeed, in the last few years, an
effort has been made to analyze specific structured problems, such as multiclass classification [5],
multi-labeling [6], ranking [7] or quantile estimation [8]. A natural question is whether a unifying
learning framework can be developed to address a wide range of problems from theory to algorithms.
This paper takes a step in this direction, proposing and analyzing a general regularization approach
to structured prediction. Our starting observation is that for a large class of these problems, we can
define a natural embedding of the associated loss functions into a linear space. This allows to define
a (least squares) surrogate problem of the original structured one, that is cast within a multi-output
regularized learning framework [9, 10, 11]. We prove that by solving the surrogate, we are able to
recover the exact solution of the original structured problem. The corresponding algorithm essentially
generalizes approaches considered in [12, 13, 14, 15, 16]. We study the generalization properties of
the proposed approach, establishing universal consistency as well as finite sample bounds.
The rest of this paper is organized as follows: in Sec. 2 we introduce the structured prediction problem
in its generality and present our algorithm to approach it. In Sec. 3 we introduce and discuss a
surrogate framework for structured prediction, from which we derive our algorithm. In Sec. 4, we
analyze the theoretical properties of the proposed algorithm. In Sec. 5 we draw connections with
previous work in structured prediction. Sec. 6 reports promising experimental results on a variety of
structured prediction problems. Sec. 7 concludes the paper outlining relevant directions for future
research.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
2
A Regularization Approach to Structured prediction
The goal of supervised learning is to learn functional relations f : X ? Y between two sets X , Y,
given a finite number of examples. In particular in this work we are interested to structured prediction,
namely the case where Y is a set of structured outputs (such as histograms, graphs, time sequences,
points on a manifold, etc.). Moreover, structure on Y can be implicitly induced by a suitable loss
4 : Y ? Y ? R (such as edit distance, ranking error, geodesic distance, indicator function of a
subset, etc.). Then, the problem of structured prediction becomes
Z
minimize E(f ),
with
E(f ) =
4(f (x), y) d?(x, y)
(1)
f :X ?Y
X ?Y
and the goal is to find a good estimator for the minimizer of the above equation, given a finite number
of (training) points {(xi , yi )}ni=1 sampled from a unknown probability distribution ? on X ? Y. In
the following we introduce an estimator f? : X ? Y to approach Eq. (1). The rest of this paper is
devoted to prove that f? it a consistent estimator for a minimizer of Eq. (1).
Our Algorithm for Structured Prediction. In this paper we propose and analyze the following
estimator
n
X
f?(x) = argmin
?i (x) 4 (y, yi ) with ?(x) = (K + n?I)?1 Kx ? Rn
(Alg. 1)
y?Y
i=1
given a positive definite kernel k : X ?X ? R and training set {(xi , yi )}ni=1 . In the above expression,
?i (x) is i-th entry in ?(x), K ? Rn?n is the kernel matrix Ki,j = k(xi , xj ), Kx ? Rn the vector
with entires (Kx )i = k(x, xi ), ? > 0 a regularization parameter and I the identity matrix.
From a computational perspective, the procedure in Alg. 1 is divided in two steps: a learning step
where input-dependents weights ?i (?) are computed (which essentially consists in solving a kernel
ridge regression problem) and a prediction step where the ?i (x)-weighted linear combination in
Alg. 1 is optimized, leading to a prediction f?(x) given an input x. The idea of a similar two-steps
strategy goes back to standard approaches for structured prediction and was originally proposed
in [17], where a ?score? function F (x, y) was learned to estimate the ?likelihood? of a pair (x, y)
sampled from ?, and then used in f?(x) = argminy?Y ?F (x, y), to predict the best f?(x) ? Y given
x ? X . This strategy was extended in [4] for the popular SVMstruct and adopted also in a variety of
approaches for structured prediction [1, 12, 14].
Intuition. While providing a principled derivation of Alg. 1 for a large class of loss functions is a
main contribution of this work, it is useful to first consider the special case where 4 is induced by a
reproducing kernel h : Y ? Y ? R on the output set, such that
4(y, y 0 ) = h(y, y) ? 2h(y, y 0 ) + h(y 0 , y 0 ).
(2)
This choice of 4 was originally considered in Kernel Dependency Estimation (KDE) [18]. In
particular, for the special case of normalized kernels (i.e. h(y, y) = 1 ?y ? Y), Alg. 1 essentially
reduces to [12, 13, 14] and recalling their derivation is insightful. Note that, since a kernel can be
written as h(y, y 0 ) = h?(y), ?(y 0 )iHY , with ? : Y ? HY a non-linear map into a feature space
HY [19], then Eq. (2) can be rewritten as
4(f (x), y 0 ) = k?(f (x)) ? ?(y 0 )k2HY .
(3)
Directly minimizing the equation above with respect to f is generally challenging due to the non
linearity ?. A possibility is to replace ? ? f by a function g : X ? HY that is easier to optimize. We
can then consider the regularized problem
n
1X
minimize
kg(xi ) ? ?(yi )k2HY + ?kgk2G
(4)
g?G
n i=1
P
with G a space of functions1 g : X ? HY of the form g(x) = i=1 k(x, xi )ci with ci ? HY and k
a reproducing kernel. Indeed, in this case the solution to Eq. (4) is
n
X
g?(x) =
?i (x)?(yi ) with ?(x) = (K + n?I)?1 Kx ? Rn
(5)
i=1
1
G is the reproducing kernel Hilbert space for vector-valued functions [9] with inner product
hk(xi , ?)ci , k(xj , ?)cj iG = k(xi , xj )hci , cj iHY
2
where the ?i are the same as in Alg. 1. Since we replaced 4(f (x), y) by kg(x) ? ?(y)k2HY , a natural
question is how to recover an estimator f? from g?. In [12] it was proposed to consider
f?(x) = argmin k?(y) ? g?(x)k2HY = argmin h(y, y) ? 2
y?Y
y?Y
n
X
?i (x)h(y, yi ),
(6)
i=1
which corresponds to Alg. 1 when h is a normalized kernel.
The discussion above provides an intuition on how Alg. 1 is derived but raises also a few questions.
First, it is not clear if and how the same strategy could be generalized to loss functions that do not
satisfy Eq. (2). Second, the above reasoning hinges on the idea of replacing f? with g? (and then
recovering f? by Eq. (6)), however it is not clear whether this approach can be justified theoretically.
Finally, we can ask what are the statistical properties of the resulting algorithm. We address the
first two questions in the next section, while the rest of the paper is devoted to establish universal
consistency and generalization bounds for algorithm Alg. 1.
3
Surrogate Framework and Derivation
To derive Alg. 1 we consider ideas from surrogate approaches [20, 21, 7] and in particular [5]. The
idea is to tackle Eq. (1) by substituting 4(f (x), y) with a ?relaxation? L(g(x), y) on a space HY ,
that is easy to optimize. The corresponding surrogate problem is
Z
minimize R(g),
with
R(g) =
L(g(x), y) d?(x, y),
(7)
g:X ?HY
X ?Y
?
and the question is how a solution g for the above problem can be related to a minimizer f ? of
Eq. (1). This is made possible by the requirement that there exists a decoding d : HY ? Y, such that
Fisher Consistency: E(d ? g ? ) = E(f ? ),
Comparison Inequality: E(d ? g) ? E(f ? ) ? ?(R(g) ? R(g ? )),
(8)
(9)
hold for all g : X ? HY , where ? : R ? R is such that ?(s) ? 0 for s ? 0. Indeed, given an
estimator g? for g ? , we can ?decode? it considering f? = d ? g? and use the excess risk R(?
g ) ? R(g ? )
to control E(f?) ? E(f ? ) via the comparison inequality in Eq. (9). In particular, if g? is a datadependent predictor trained on n points and R(?
g ) ? R(g ? ) when n ? +?, we automatically
have E(f?) ? E(f ? ). Moreover, if ? in Eq. (9) is known explicitly, generalization bounds for g? are
automatically extended to f?.
Provided with this perspective on surrogate approaches, here we revisit the discussion of Sec. 2 for
the case of a loss function induced by a kernel h. Indeed, by assuming the surrogate L(g(x), y) =
kg(x) ? ?(y)k2HY , Eq. (4) becomes the empirical version of the surrogate problem at Eq. (7)
and leads to an estimator g? of g ? as in Eq. (5). Therefore, the approach in [12, 14] to recover
f?(x) = argminy L(g(x), y) can be interpreted as the result f?(x) = d ? g?(x) of a suitable decoding
of g?(x). An immediate question is whether the above framework satisfies Eq. (8) and (9). Moreover,
we can ask if the same idea could be applied to more general loss functions.
In this work we identify conditions on 4 that are satisfied by a large family of functions and moreover
allow to design a surrogate framework for which we prove Eq. (8) and (9). The first step in this
direction is to introduce the following assumption.
Assumption 1. There exists a separable Hilbert space HY with inner product h?, ?iHY , a continuous
embedding ? : Y ? HY and a bounded linear operator V : HY ? HY , such that
4(y, y 0 ) = h?(y), V ?(y 0 )iHY
?y, y 0 ? Y
(10)
Asm. 1 is similar to Eq. (3) and in particular to the definition of a reproducing kernel. Note however
that by not requiring V to be positive semidefinite (or even symmetric), we allow for a surprisingly
wide range of functions beyond kernel functions. Indeed, below we give some examples of functions
that satisfy Asm. 1 (see supplementary material Sec. C for more details):
Example 1. The following functions of the form 4 : Y ? Y ? R satisfy Asm. 1:
3
1. Any loss on Y of finite cardinality. Several problems belong to this setting, such as MultiClass Classification, Multi-labeling, Ranking, predicting Graphs (e.g. protein foldings).
2. Regression and Classification Loss Functions. Least-squares, Logistic, Hinge, -insensitive,
? -Pinball.
3. Robust Loss Functions. Most loss functions used for robust estimation [22] such as the
absolute value, Huber, Cauchy, German-McLure, ?Fair? and L2 ? L1. See [22] or the
supplementary material for their explicit formulation.
4. KDE. Loss functions 4 induced by a kernel such as in Eq. (2).
5. Distances on Histograms/Probabilities. The ?2 and the Hellinger distances.
6. Diffusion distances on Manifolds. The squared diffusion distance induced by the heat kernel
(at time t > 0) on a compact Reimannian manifold without boundary [23].
The Least Squares Loss Surrogate Framework. Asm. 1 implicitly defines the space HY similarly
to Eq. (3). The following result motivates the choice of the least squares surrogate and moreover
suggests a possible choice for the decoding.
Lemma 1. Let 4 : Y ? Y ? R satisfy Asm. 1 with ? : Y ? HY bounded. Then the expected risk
in Eq. (1) can be written as
Z
E(f ) =
h?(f (x)), V g ? (x)iHY d?X (x)
(11)
X
for all f : X ? Y, where g ? : X ? HY minimizes
Z
R(g) =
kg(x) ? ?(y)k2HY d?(x, y).
(12)
X ?Y
Lemma 1 shows how Eq. (12) arises naturally as surrogate problem. In particular, Eq. (11) suggests
to choose the decoding
d(h) = argmin h ?(y) , V h iHY
?h ? HY ,
(13)
y?Y
?
since d ? g (x) = arg miny?Y h?(y), V g ? (x)i and therefore E(d ? g ? ) ? E(f ) for any measurable
f : X ? Y, leading to Fisher Consistency. We formalize this in the following result.
Theorem 2. Let 4 : Y ? Y ? R satisfy Asm. 1 with Y a compact set. Then, for every measurable
g : X ? HY and d : HY ? Y satisfying Eq. (13), the following holds
E(d ? g ? ) = E(f ? )
(14)
p
?
E(d ? g) ? E(f ) ? c4 R(g) ? R(g ? ).
(15)
with c4 = kV k maxy?Y k?(y)kHY .
Thm. 2 shows that for all 4 satisfying Asm. 1, the corresponding surrogate framework identified
by the surrogate in Eq. (12) and decoding Eq. (13) satisfies Fisher consistency Eq. (14) and the
comparison inequality in Eq. (15). We recall that a finite set Y is always compact, and moreover,
assuming the discrete topology on Y, we have that any ? : Y ? HY is continuous. Therefore,
Thm. 2 applies in particular to any structured prediction problem on Y with finite cardinality.
Thm. 2 suggest to approach structured prediction by first learning g? and then decoding it to recover
f? = d ? g?. A natural question is how to choose g? in order to compute f? in practice. In the rest of this
section we propose an approach to this problem.
Derivation for Alg. 1. Minimizing R in Eq. (12) corresponds to a vector-valued regression problem
[9, 10, 11]. In this work we adopt an empirical risk minimization approach to learn g? as in Eq. (4).
The following result shows that combining g? with the decoding in Eq. (13) leads to the f? in Alg. 1.
Lemma 3. Let 4 : Y ? Y ? R satisfy Asm. 1 with Y a compact set. Let g? : X ? HY be the
minimizer of Eq. (4). Then, for all x ? X
n
X
d ? g?(x) = argmin
?i (x) 4 (y, yi )
?(x) = (K + n?I)?1 Kx ? Rn
(16)
y?Y
i=1
4
Lemma 3 concludes the derivation of Alg. 1. An interesting observation is that computing f? does not
require explicit knowledge of the embedding ? and the operator V , which are implicitly encoded
within the loss 4 by Asm. 1. In analogy to the kernel trick [24] we informally refer to such assumption
as the ?loss trick?. We illustrate this effect with an example.
Example 2 (Ranking). In ranking problems the goal is to predict ordered sequences of a fixed number
` of labels. For these problems, Y corresponds to the set of all ordered sequences of ` labels and has
cardinality |Y| = `!, which is typically dramatically larger than the number n of training examples
(e.g. for ` = 15, `! ' 1012 ). Therefore, given an input x ? X , directly computing g?(x) ? R|Y| is
impractical. On the opposite, the loss trick allows to express d ? g?(x) only in terms of the n weights
?i (x) in Alg. 1, making the computation of the argmin easier to approach in general. For details on
the rank loss 4rank and the corresponding optimization over Y, we refer to the empirical analysis of
Sec. 6.
In this section we have shown a derivation for the structured prediction algorithm proposed in this
work. In Thm. 2 we have shown how the expected risk of the proposed estimator f? is related to an
estimator g? via a comparison inequality. In the following we will make use of these results to prove
consistency and generalization bounds for Alg. 1.
4
Statistical Analysis
In this section we study the statistical properties of Alg. 1 exploiting of the relation between the
structured and surrogate problems characterized be the comparison inequality in Thm. 2. We begin
our analysis by proving that Alg. 1 is universally consistent.
Theorem 4 (Universal Consistency). Let 4 : Y ? Y ? R satisfy Asm. 1, X and Y be compact sets
and k : X ? X ? R a continuous universal reproducing kernel2 . For any n ? N and any distribution
? on X ? Y let f?n : X ? Y be obtained by Alg. 1 with {(xi , yi )}ni=1 training points independently
sampled from ? and ?n = n?1/4 . Then,
lim E(f?n ) = E(f ? )
n?+?
with probability 1
(17)
Thm. 4 shows that, when the 4 satisfies Asm. 1, Alg. 1 approximates a solution f ? to Eq. (1)
arbitrarily well, given a sufficient number of training examples. To the best of our knowledge this is
the first consistency result for structured prediction in the general setting considered in this work and
characterized by Asm. 1, in particular for the case of Y with infinite cardinality (dense or discrete).
The No Free Lunch Theorem [25] states that it is not possible to prove uniform convergence rates for
Eq. (17). However, by imposing suitable assumptions on the regularity of g ? it is possible to prove
generalization bounds for g? and then, using Thm. 2, extend them to f?. To show this, it is sufficient
to require that g ? belongs to G the reproducing kernel Hilbert space used in the ridge regression of
Eq. (4). Note that in the proofs of Thm. 4 and Thm. 5, our analysis on g? borrows ideas from [10] and
extends their result to our setting for the case of HY infinite dimensional (i.e. when Y has infinite
cardinality). Indeed, note that in this case [10] cannot be applied to the estimator g? considered in this
work (see supplementary material Sec. B.3, Lemma 18 for details).
Theorem 5 (Generalization Bound). Let 4 : Y ? Y ? R satisfy Asm. 1, Y be a compact set and
k : X ? X ? R a bounded continuous reproducing kernel. Let f?n denote the solution of Alg. 1 with
n training points and ? = n?1/2 . If the surrogate risk R defined in Eq. (12) admits a minimizer
g ? ? G, then
1
E(f?n ) ? E(f ? ) ? c? 2 n? 4
(18)
holds with probability 1 ? 8e?? for any ? > 0, with c a constant not depending on n and ? .
The bound in Thm. 5 is of the same order of the generalization bounds available for the least squares
binary classifier [26]. Indeed, in Sec. 5 we show that in classification settings Alg. 1 reduces to least
squares classification. This opens the way to possible improvements, as we discuss in the following.
2
This is a standard assumption for universal consistency (see [21]). An example of continuous universal
kernel is the Gaussian k(x, x0 ) = exp(??kx ? x0 k2 ), with ? > 0.
5
Remark 1 (Better Comparison Inequality). The generalization bounds for the least squares classifier
can be improved by imposing regularity conditions on ? via the Tsybakov condition [26]. This was
observed in [26] for binary classification with the least squares surrogate, where a tighter comparison
inequality than the one in Thm. 2 was proved. Therefore, a natural question is whether the inequality
of Thm. 2 could be similarly improved, consequently leading to better rates for Thm. 5. Promising
results in this direction can be found in [5], where the Tsybakov condition was generalized to the
multi-class setting and led to a tight comparison inequality analogous to the one for the binary setting.
However, this question deserves further investigation. Indeed, it is not clear how the approach in [5]
could be further generalized to the case where Y has infinite cardinality.
Remark 2 (Other Surrogate Frameworks). In this paper we focused on a least squares surrogate
loss function and corresponding framework. A natural question is to ask whether other loss functions
could be considered to approach the structured prediction problem, sharing the same or possibly even
better properties. This question is related also to Remark 1, since different surrogate frameworks
could lead to sharper comparison inequalities. This seems an interesting direction for future work.
5
Connection with Previous Work
Binary and Multi-class Classification. It is interesting to note that in classification settings, Alg. 1
corresponds to the least squares classifier [26]. Indeed, let Y = {1, . . . , `} be a set of labels and
consider the misclassification loss 4(y, y 0 ) = 1 for y 6= y 0 and 0 otherwise. Then 4(y, y 0 ) =
`
`
e>
y V ey 0 with ei ? R the i-the element of the canonical basis of R and V = 1 ? I, where I is the
` ? ` identity matrix and 1 the matrix with all entries equal to 1. In the notation of surrogate methods
adopted in this work, HY = R` and ?(y) = ey . Note that both Least squares classification and our
approach solve the surrogate problem at Eq. (4)
n
1X
kg(xi ) ? eyi k2R` + ? kgk2G
n i=1
(19)
to obtain a vector-valued predictor g? : X ? R` as in Eq. (5). Then, the least squares classifier c? and
the decoding f? = d ? g? are respectively obtained by
f?(x) = argmin V g?(x).
c?(x) = argmax g?(x)
i=1,...,`
(20)
i=1,...,`
However, since V = 1 ? I, it is easy to see that c?(x) = f?(x) for all x ? X .
Kernel Dependency Estimation. In Sec. 2 we discussed the relation between KDE [18, 12] and
Alg. 1. In particular, we have observed that if 4 is induced by a kernel h : Y ? Y ? R as in Eq. (2)
and h is normalized, i.e. h(y, y) = ? ?y ? Y, with ? > 0, then algorithm Eq. (6) proposed in [12]
leads to the same predictor as Alg. 1. Therefore, we can apply Thm. 4 and 5 to prove universal
consistency and generalization bounds for methods such as [12, 14]. Some theoretical properties of
KDE have been previously studied in [15] from a PAC Bayesian perspective. However, the obtained
bounds do not allow to control the excess risk or establish consistency of the method. Moreover, note
that when the kernel h is not normalized, the ?decoding? in Eq. (6) is not equivalent to Alg. 1. In
particular, given the surrogate solution g ? , applying Eq. (6) leads to predictors that do not minimize
Eq. (1). As a consequence, approaches in [12, 13, 14] are not consistent in the general case.
Support Vector Machines for Structured Output. A popular approach to structured prediction
is the Support Vector Machine for Structured Outputs (SVMstruct) [4] that extends ideas from the
well-known SVM algorithm to the structured setting. One of the main advantages of SVMstruct is
that it can be applied to a variety of problems since it does not impose strong assumptions on the loss.
In this view, our approach shares similar properties, and in particular allows to consider Y of infinite
cardinality. Moreover, we note that generalization studies for SVMstruct are available [3] (Ch. 11).
However, it seems that these latter results do not allow to derive universal consistency of the method.
6
Experiments
In this section we report on preliminary experiments showing the performance of the proposed
approach on simulated as well as real structured prediction problems.
6
Rank Loss
Linear [7]
Hinge [27]
Logistic [28]
SVM Struct [4]
Alg. 1
0.430 ? 0.004
0.432 ? 0.008
0.432 ? 0.012
0.451 ? 0.008
0.396 ? 0.003
Table 1: Normalized 4rank for ranking
methods on the MovieLens dataset [29].
Loss
KDE [18]
(Gaussian)
Alg. 1
(Hellinger)
4G
4H
4R
0.149 ? 0.013
0.736 ? 0.032
0.294 ? 0.012
0.172 ? 0.011
0.647 ? 0.017
0.193 ? 0.015
Table 2:
Digit reconstruction using Gaussian
(KDE [18]) and Hellinger loss.
Ranking Movies. We considered the problem of ranking movies in the MovieLens dataset [29]
(ratings (from 1 to 5) of 1682 movies by 943 users). The goal was to predict preferences of a given
user, i.e. an ordering of the 1682 movies, according to the user?s partial ratings. We applied Alg. 1 to
the ranking problem using the rank loss [7]
4rank (y, y 0 ) =
M
1 X
?(y 0 )ij (1 ? sign(yi ? yj )),
2 i,j=1
(21)
where M is the number of movies, y is a re-ordering of the sequence 1, . . . , M . The scalar ?(y)ij
denotes the costs (or reward) of having movie j ranked higher than movie i. Similarly to [7], we set
?(y)ij equal to the difference of ratings provided by user associated to y (from 1 to 5). We chose as k
in Alg. 1, a linear kernel on features similar to those proposed in [7], which were computed based on
users? profession, age, similarity of previous ratings, etc. Since solving Alg. 1 for 4rank is NP-hard
(see [7]) we adopted the Feedback Arc Set approximation (FAS) proposed in [30] to approximate the
f?(x) of Alg. 1. Results are reported in Tab. 1 comparing Alg. 1 (Ours) with surrogate ranking methods
using a Linear [7], Hinge [27] or Logistic [28] loss and Struct SVM [4]. We randomly sampled
n = 643 users for training and tested on the remaining 300. We performed 5-fold cross-validation for
model selection. We report the normalized 4rank , averaged over 10 trials to account for statistical
variability. Interestingly, our approach appears to outperform all competitors, suggesting that Alg. 1
is a viable approach to ranking.
Image Reconstruction with Hellinger Distance. We considered the USPS digits reconstruction
experiment originally proposed in [18]. The goal is to predict the lower half of an image depicting a
digit, given the upper half of the same image in input. The standard approach is to use a Gaussian
kernel kG on images in input and adopt KDE methods such as [18, 12, 14] with loss 4G (y, y 0 ) =
1 ? kG (y, y 0 ). Here we take a different approach and, following [31], we interpret an image depicting
a digit as an histogram and normalize it to sum up to 1. Therefore, Y is the unit simplex in R128
(16 ? 16 images) and we adopt the Hellinger distance 4H
0
4H (y, y ) =
M
X
|(yi )1/2 ? (yi0 )1/2 |
for
y = (yi )M
i=1
(22)
i=1
to measure distances on Y. We used the kernel kG on the input space and compared Alg. 1 using
respectively 4H and 4G . For 4G Alg. 1 correpsponds to [12]. We performed digit reconstruction
experiments by training on 1000 examples evenly distributed among the 10 digits of USPS and
tested on 5000 images. We performed 5-fold cross-validation for model selection. Tab. 2 reports
the performance of Alg. 1 and the KDE methods averaged over 10 runs. Performance are reported
according to the Gaussian loss 4G and Hellinger loss 4H . Unsurprisingly, methods trained with
respect to a specific loss perform better than the competitor with respect to such loss. Therefore, as a
further measure of performance we also introduced the ?Recognition? loss 4R . This loss has to be
intended as a measure of how ?well? a predictor was able to correctly reconstruct an image for digit
recognition purposes. To this end, we trained an automatic digit classifier and defined 4R to be the
misclassification error of such classifier when tested on images reconstructed by the two prediction
algorithms. This automatic classifier was trained using a standard SVM [24] on a separate subset of
USPS images and achieved an average 0.04% error rate on the true 5000 test sets. In this case a clear
difference in performance can be observed between using two different loss functions, suggesting
that 4H is more suited for the reconstruction problem.
7
4
n
Alg. 1
RNW
KRLS
2
50
100
200
500
1000
0
?2
?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
0.8
Alg. 1
RNW
KRR
0.39 ? 0.17
0.21 ? 0.04
0.12 ? 0.02
0.08 ? 0.01
0.07 ? 0.01
0.45 ? 0.18
0.29 ? 0.04
0.24 ? 0.03
0.22 ? 0.02
0.21 ? 0.02
0.62 ? 0.13
0.47 ? 0.09
0.33 ? 0.04
0.31 ? 0.03
0.19 ? 0.02
1
Figure 1: Robust estimation on the regression problem in Sec. 6 by minimizing the Cauchy loss with
Alg. 1 (Ours) or Nadaraya-Watson (Nad). KRLS as a baseline predictor. Left. Example of one run of
the algorithms. Right. Average distance of the predictors to the actual function (without noise and
outliers) over 100 runs with respect to training sets of increasing dimension.
Robust Estimation. We considered a regression problem with many outliers and evaluated Alg. 1
using the Cauchy loss (see Example 1 - (3)) for robust estimation. Indeed, in this setting, Y =
[?M, M ] ? R is not structured, but the non-convexity of 4 can be an obstacle to the learning
process. We generated a dataset according to the model y = sin(6?x) + + ?, where x was sampled
uniformly on [?1, 1] and according to a zero-mean Gaussian with variance 0.1. ? modeled the
outliers and was sampled according to a zero-mean random variable that was 0 with probability
0.90 and a value uniformly at random in [?3, 3] with probability 0.1. We compared Alg. 1 with the
Nadaraya-Watson robust estimator (RNW) [32] and kernel ridge regression (KRR) with a Gaussian
kernel as baseline. To train Alg. 1 we used a Gaussian kernel on the input and performed predictions
(i.e. solved Eq. (16)) using Matlab FMINUNC function for unconstrained minimization. Experiments
were performed with training sets of increasing dimension (100 repetitions each) and test set of 1000
examples. 5-fold cross-validation for model selection. Results are reported in Fig. 1, showing that
our estimator significantly outperforms the others. Moreover, our method appears to greatly benefit
from training sets of increasing size.
7
Conclusions and Future Work
In this work we considered the problem of structured prediction from a Statistical Learning Theory
perspective. We proposed a learning algorithm for structured prediction that is split into a learning
and prediction step similarly to previous methods in the literature. We studied the statistical properties
of the proposed algorithm by adopting a strategy inspired to surrogate methods. In particular, we
identified a large family of loss functions for which it is natural to identify a corresponding surrogate
problem. This perspective allows to prove a derivation of the algorithm proposed in this work.
Moreover, by exploiting a comparison inequality relating the original and surrogate problems we
were able to prove universal consistency and generalization bounds under mild assumption. In
particular, the bounds proved in this work recover those already known for least squares classification,
of which our approach can be seen as a generalization. We supported our theoretical analysis with
experiments showing promising results on a variety of structured prediction problems.
A few questions were left opened. First, we ask whether the comparison inequality can be improved
(under suitable hypotheses) to obtain faster generalization bounds for our algorithm. Second, the
surrogate problem in our work consists of a vector-valued regression (in a possibly infinite dimensional
Hilbert space), we solved this problem by plain kernel ridge regression but it is natural to ask whether
approaches from the multi-task learning literature could lead to substantial improvements in this
setting. Finally, an interesting question is whether alternative surrogate frameworks could be derived
for the setting considered in this work, possibly leading to tighter comparison inequalities. We will
investigate these questions in the future.
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1984.
9
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5,629 | 6,094 | Agnostic Estimation for Misspecified
Phase Retrieval Models
Matey Neykov
Zhaoran Wang
Han Liu
Department of Operations Research and Financial Engineering
Princeton University, Princeton, NJ 08544
{mneykov, zhaoran, hanliu}@princeton.edu
Abstract
The goal of noisy high-dimensional phase retrieval is to estimate an s-sparse parameter ? ? ? Rd from n realizations of the model Y = (X > ? ? )2 + ?. Based
on this model, we propose a significant semi-parametric generalization called misspecified phase retrieval (MPR), in which Y = f (X > ? ? , ?) with unknown f and
Cov(Y, (X > ? ? )2 ) > 0. For example, MPR encompasses Y = h(|X > ? ? |) + ?
with increasing h as a special case. Despite the generality of the MPR model, it
eludes the reach of most existing semi-parametric estimators. In this paper, we propose an estimation procedure, which consists of solving a cascade of two convex
programs and provably recovers the direction of ? ? . Our theory is backed up by
thorough numerical results.
1
Introduction
In scientific and engineering fields researchers often times face the problem of quantifying the
relationship between a given outcome Y and corresponding predictor vector X, based on a sample
{(Yi , Xi> )> }ni=1 of n observations. In such situations it is common to postulate a linear ?working?
model, and search for a d-dimensional signal vector ? ? satisfying the following familiar relationship:
Y = X > ? ? + ?.
(1.1)
When the predictor X is high-dimensional in the sense that d n, it is commonly assumed that
the underlying signal ? ? is s-sparse. In a certain line of applications, such as X-ray crystallography,
microscopy, diffraction and array imaging1 , one can only measure the magnitude of X > ? ? but not
its phase (i.e., sign in the real domain). In this case, assuming model (1.1) may not be appropriate. To
cope with such applications in the high-dimensional setting, [7] proposed the thresholded Wirtinger
flow (TWF), a procedure which consistently estimates the signal ? ? in the following real sparse
noisy phase retrieval model:
Y = (X > ? ? )2 + ?,
(1.2)
where one additionally knows that the predictors have a Gaussian random design X ? N (0, Id ). In
the present paper, taking an agnostic point of view, we recognize that both models (1.1) and (1.2)
represent an idealized view of the data generating mechanism. In reality, the nature of the data could
be better reflected through the more flexible viewpoint of a single index model (SIM):
Y = f (X > ? ? , ?),
(1.3)
?
where f is an unknown link function, and it is assumed that k? k2 = 1 for identifiability. A recent
line of work on high-dimensional SIMs [25, 27], showed that under Gaussian designs, one can apply
`1 regularized least squares to successfully estimate the direction of ? ? and its support. The crucial
condition allowing for the above somewhat surprising application turns out to be:
Cov(Y, X > ? ? ) 6= 0.
(1.4)
While condition (1.4) is fairly generic, encompassing cases with a binary outcome, such as logistic
regression and one-bit compressive sensing [5], it fails to capture the phase retrieval model (1.2).
1
In such applications it is typically assumed that X ? Cd is a complex normal random vector. In this paper
for simplicity we only consider the real case X ? Rd .
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
More generally, it is easy to see that when the link function f is even in its first coordinate, condition
(1.4) fails to hold. The goal of the present manuscript is to formalize a class of SIMs, which includes
the noisy phase retrieval model as a special case in addition to various other additive and non-additive
models with even link functions, and develop a procedure that can successfully estimate the direction
of ? ? up to a global sign. Formally, we consider models (1.3) with Gaussian design that satisfy the
following moment assumption:
Cov(Y, (X > ? ? )2 ) > 0.
(1.5)
Unlike (1.4), one can immediately check that condition (1.5) is satisfied by model (1.2). In ?2 we give
multiple examples, both abstract and concrete, of SIMs obeying this constraint. Our second moment
constraint (1.5) can be interpreted as a semi-parametric robust version of phase-retrieval. Hence, we
will refer to the class of models satisfying condition (1.5) as misspecified phase retrieval (MPR)
models. In this point of view it is worth noting that condition (1.4) relates to linear regression in a
way similar to how condition (1.5) relates to the phase retrieval model. Our motivation for studying
SIMs under such a constraint can ultimately be traced to the vast sufficient dimension reduction
(SDR) literature. In particular, we would like to point out [22] as a source of inspiration.
Contributions. Our first contribution is to formulate a novel and easily implementable two-step
procedure, which consistently estimates the direction of ? ? in an MPR model. In the first step
b , such that |b
we solve a semidefinite program producing a unit vector v
v> ? ? | is sufficiently large.
Once such a pilot estimate is available, we consider solving an `1 regularized least squares on the
b , where Y is the average of Yi ?s, to produce a second
augmented outcome Yei = (Yi ? Y )Xi> v
b which is then normalized to obtain the final refined estimator ?b = b/k
b bk
b 2 . In addition
estimate b,
to being universally applicable to MPR models, our procedure has an algorithmic advantage in that
it relies solely on convex optimization, and as a consequence we can obtain the corresponding global
minima of the two convex programs in polynomial time.
Our second contribution is to rigorously demonstrate that the above procedure consistently estimates
the direction of ? ? . We prove that for a given MPR model, with high probability, one has:
p
min??{?1,1} k?b ? ?? ? k2 . s log d/n,
provided that the sample size n satisfies n & s2 log d. While the same rates (with different constants)
hold for the TWF algorithm of [7] in the special case of noisy phase retrieval model, our procedure
provably achieves these rates over the broader class of MPR models.
Lastly, we propose an optional refinement of our algorithm, which shows improved performance in
the numerical studies.
Related Work. The phase retrieval model has received considerable attention in the recent years by
statistics, applied mathematics as well as signal processing communities. For the non-sparse version
of (1.2), efficient algorithms have been suggested based on both semidefinite programs [8, 10] and
non-convex optimization methods that extend gradient descent [9]. Additionally, a non-traditional
instance of phase retrieval model (which also happens to be a special case of the MPR model) was
considered by [11], where the authors suggested an estimation procedure originally proposed for the
problem of mixed regression. For the noisy sparse version of model (1.2), near optimal solutions
were achieved with a computationally infeasible program by [20]. Subsequently, a tractable gradient
descent approach achieving minimax optimal rates was developed by [7].
Abstracting away from the phase retrieval or linear model settings, we note that inference for SIMs
in the case when d is small or fixed, has been studied extensively in the literature [e.g., 18, 24, 26, 34,
among many others]. In another line of research on SDR, seminal insights shedding light on condition
(1.4) can be found in, e.g., [12, 21, 23]. The modified condition (1.5) traces roots to [22], where the
authors designed a procedure to handle precisely situations where (1.4) fails to hold. More recently,
there have been active developments for high-dimensional SIMs. [27] and later [31] demonstrated that
under condition (1.4), running the least squares with `1 regularization can obtain a consistent estimate
of the direction of ? ? , while [25] showed that this procedure also recovers the signed support of the
direction. Excess risk bounds were derived in [14]. Very recently, [16] extended this observation to
other convex loss functions under a condition corresponding to (1.4) depending implicitly on the loss
function of interest. [28] proposed a non-parametric least squares with an equality `1 constraint to
handle simultaneous estimation of ? ? as well as f . [17] considered a smoothed-out U -process type
of loss function with `1 regularization, and proved their approach works for a sub-class of functions
satisfying condition (1.4). None of the aforementioned works on SIMs can be directly applied to
tackle the MPR class (1.5). A generic procedure for estimating sparse principal eigenvectors was
2
developed in [37]. While in principle this procedure can be applied to estimate the direction in MPR
models, it requires proper initialization, and in addition, it requires knowledge of the sparsity of the
vector ? ? . We discuss this approach in more detail in ?4.
Regularized procedures have also been proposed for specific choices of f and Y . For example, [36]
studied consistent estimation under the model P(Y = 1|X) = (h(X > ? ? ) + 1)/2 with binary Y ,
where h : R 7? [?1, 1] is possibly unknown. Their procedure is based on taking pairs of differences
in the outcome, and therefore replaces condition (1.4) with a different type of moment conditon. [35]
considered the model Y = h(X > ? ? ) + ? with a known continuously differentiable and monotonic
h, and developed estimation and inferential procedures based on the `1 regularized quadratic loss, in
a similar spirit to the TWF algorithm suggested by [7]. In conclusion, although there exists much
prior related work, to the best of our knowledge, none of the available literature discusses the MPR
models in the generality we attempt in the present manuscript.
Notation. We now briefly outline some commonly used notations. Other notations will be defined as
needed throughout the paper. For a (sparse) vector v = (v1 , . . . , vp )> , we let Sv := supp(v) = {j :
vj 6= 0} denote its support, kvkp denote the `p norm (with the usual extension when p = ?) and
v?2 := vv> is a shorthand for the outer product. With a standard abuse of notation we will denote
by kvk0 = |supp(v)| the cardinality of the support of v. We often use Id to denote a d ? d identity
matrix. For a real random variable X, define
kXk?2 = sup p?1/2 (E|X|p )1/p , kXk?1 = sup p?1 (E|X|p )1/p .
p?1
p?1
Recall that a random variable is called sub-Gaussian if kXk?2 < ? and sub-exponential if kXk?1 <
? [e.g., 32]. For any integer k ? N we use the shorthand notation [k] = {1, . . . , k}. We also use
standard asymptotic notations. Given two sequences {an }, {bn } we write an = O(bn ) if there exists
a constant C < ? such that an ? Cbn , and an bn if there exist positive constants c and C such
that c < an /bn < C.
Organization. In ?2 and ?3 we introduce the MPR model class and our estimation procedure, and
?3.1 is dedicated to stating the theoretical guarantees of our proposed algorithm. Simulation results
are given in ?4. A brief discussion is provided in ?5. We defer the proofs to the appendices due to
space limitations.
2
MPR Models
In this section we formally introduce MPR models. In detail, we argue that the class of such models
is sufficiently rich, including numerous models of interest. Motivated by the setup in the sparse noisy
phase retrieval model (1.2), we assume throughout the remainder of the paper that X ? N (0, Id ).
We begin our discussion with a formal definition.
?X
Definition 2.1 (MPR Models). Assume that we are given model (1.3), where X ? N (0, Id ), ? ?
and ? ? ? Rd is an s-sparse unit vector, i.e., k? ? k2 = 1. We call such a model misspecified phase
retrieval (MPR) model, if the link function f and noise ? further satisfy, for Z ? N (0, 1) and K > 0,
c0 := Cov(f (Z, ?), Z 2 ) > 0,
kY k?1 ? K.
(2.1)
(2.2)
Both assumptions (2.1) and (2.2) impose moment restrictions on the random variable Y . Assumption
(2.1) states that Y is positively correlated with the random variable (X > ? ? )2 , while assumption
(2.2) requires Y to have somewhat light-tails. Also, as mentioned in the introduction, the unit norm
constraint on the vector ? ? is required for the identifiability of model (1.3). We remark that the class of
MPR models is convex in the sense that if we have two MPR models f1 (X > ? ? , ?) and f2 (X > ? ? , ?),
all models generated by their convex combinations ?f1 (X > ? ? , ?)+(1??)f2 (X > ? ? , ?) (? ? [0, 1])
are also MPR models. It is worth noting the > direction in (2.1) is assumed without loss of generality.
If Cov(Y, (X > ? ? )2 ) < 0 one can apply the same algorithm to ?Y . However, the knowledge of the
direction of the inequality is important.
In the following, we restate condition (2.1) in a more convenient way, enabling us to easily calculate
the explicit value of the constant c0 in several examples.
Proposition 2.2. Assume that there exists a version of the map ?(z) : z 7? E[f (Z, ?)|Z = z] such
that ED2 ?(Z) > 0, where D2 is the second distributional derivative of ? and Z ? N (0, 1). Then
the SIM (1.3) satisfies assumption (2.1) with c0 = ED2 ?(Z).
We now provide three concrete MPR models as warm up examples for the more general examples
discussed in Proposition 2.3 and Remark 2.3. Consider the models:
3
Y = (X > ? ? )2 + ?, (2.3)
Y = |X > ? ? | + ?,
(2.4)
Y = |X > ? ? + ?|,
(2.5)
where ? ?
? X is sub-exponential noise, i.e., k?k?1 ? K? for some K? > 0. Model (2.3) is the
noisy phase retrieval model considered by [7], while models (2.4) and (2.5) were both discussed
in [11], where the authors proposed a method to solve model (2.5) in the low-dimensional regime.
Below we briefly explain why these models satisfy conditions (2.1) and (2.2). First, observe that in
all three models we have a sum of two sub-exponential random variables, and hence by the triangle
inequality it follows that the random variable Y is also sub-exponential, which implies (2.2). To
understand why (2.1)?
holds, by applying Proposition 2.2 we have c0 = E2 = 2 > 0 for model (2.3),
c0 = E2?0 (Z) = 2/ 2? > 0 for model (2.4), and c0 = E2?0 (Z + ?) = 2E?(?) > 0 for model
(2.5), where ?0 (?) is the Dirac delta function centered at zero, and ? is the density of the standard
normal distribution.
Admittedly, calculating the second distributional derivative could be a laborious task in general. In
the remainder of this section we set out to find a simple to check generic sufficient condition on the
link function f and error term ?, under which both (2.1) and (2.2) hold. Before giving our result note
that the condition ED2 ?(Z) > 0 is implied whenever ? is strictly convex and twice differentiable.
However, strictly convex functions ? may violate assumption (2.2) as they can inflate the tails of Y
arbitrarily (consider, e.g., f (x, ?) = x4 + ?). Moreover, the functions in example (2.4) and (2.5) fail
to be twice differentiable. In the following result we handle those two problems, and in addition we
provide a more generic condition than convexity, which suffices to ensure the validity of (2.1).
Proposition 2.3. The following statements hold.
(i) Let the function ? defined in Proposition 2.2 be such that the map z 7? ?(z) + ?(?z)
is non-decreasing on R+
0 and and there exist z1 > z2 > 0 such that ?(z1 ) + ?(?z1 ) >
?(z2 ) + ?(?z2 ). Then (2.1) holds.
(ii) A sufficient condition for (i) to hold, is that z 7? ?(z) is convex and sub-differentiable at
every point z ? R, and there exists a point z0 ? R+
0 satisfying ?(z0 ) + ?(?z0 ) > 2?(0).
(iii) Assume that there exist functions g1 , g2 such that f (Z, ?) ? g1 (Z) + g2 (?), and g1 is
essentially quadratic in the sense that there exists a closed interval I = [a, b] with 0 ? I,
such that for all z satisfying g1 (z) ? I c we have |g1 (z)| ? Cz 2 for a sufficiently large
constant C > 0, and let g2 (?) be sub-exponential. Then (2.2) holds.
Remark 2.4. Proposition 2.3 shows that the class of MPR models is sufficiently broad. By (i) and
(ii) it immediately follows that the additive models
Y = h(X > ? ? ) + ?,
(2.6)
R+
0
where the link function h is even and increasing on
or convex, satisfy the covariance condition
(2.1) by (i) and (ii) of Proposition 2.3 respectively. If h is also essentially quadratic and ? is subexponentially distributed, using (iii) we can deduce that Y in (2.6) is a sub-exponential random
variable, and hence under these restrictions model (2.6) is an MPR model. Both examples (2.3) and
(2.4) take this form.
Additionally, Proposition 2.3 implies that the model
Y = h(X > ? ? + ?)
(2.7)
satisfies (2.1), whenever the link h is a convex sub-differentiable function, such that h(z0 )+h(?z0 ) >
2h(0) for some z0 > 0, E|h(z + ?)| < ? for all z ? R and E|h(Z + ?)| < ?. This conclusion
follows because under the latter conditions the function ?(z) = Eh(z + ?) satisfies (ii), which is
proved in Appendix C under Lemma C.1. Moreover, if it turns out that h is essentially quadratic and
h(2?) is sub-exponential, then by Jensen?s inequality we have 2h(Z +?) ? h(2Z)+h(2?) and hence
(iii) implies that (2.2) is also satisfied. Model (2.5) is of the type (2.7). Unlike the additive noise
models (2.6), models (2.7) allow noise corruption even within the argument of the link function. On
the negative side, it should be apparent that (2.1) fails to hold in cases where ? is an odd function, i.e.,
?(z) = ??(?z). In many such cases (e.g. when ? is monotone or non-constant and non-positive/nonnegative on R+ ), one would have Cov(Y, X > ? ? ) = E[?(Z)Z] 6= 0, and hence direct application
of the `1 regularized least squares algorithm is possible as we discussed in the introduction.
3
Agnostic Estimation for MPR
In this section we describe and motivate our two-step procedure, which consists of a convex relaxation
and an `1 regularized least squares program, for performing estimation in the MPR class of models
4
described by Definition 2.1. We begin our motivation by noting that any MPR model satisfies the
following inequality
Cov((Y ? ?)X > ? ? , X > ? ? ) = E{(Y ? ?)(X > ? ? )2 } = Cov(f (Z, ?), Z 2 ) = c0 > 0, (3.1)
where we have denoted ? := EY . This simple observation plays a major role in the motivation
of our procedure. Notice that in view of condition (1.4), inequality (3.1) implies that if instead of
observing Y we had observed Y? = g(X > ? ? , ?) = (Y ? ?)X > ? ? . However, there is no direct way
of generating the random variable Y? , as doing so would require the knowledge of ? ? and the mean ?.
b first, use an empirical estimate Y of ?, and
Here, we propose to roughly estimate ? ? by a vector v
b
then obtain the `1 regularized least squares estimate on the augmented variable Ye = (Y ? Y )X > v
to sharpen the convergence rate. At first glance it might appear counter-intuitive that introducing a
noisy estimate of ? ? can lead to consistent estimates, as the so-defined Ye variable depends on the
b }. Decompose
projection of X on span{? ? , v
b = (b
v
v> ? ? )? ? + ?b? ,
(3.2)
where ?b? ? ? ? . To better motivate this proposal, in the following we analyze the population least
b for some fixed unit vector v
b with
squares fit, based on the augmented variable Y? = (Y ? ?)X > v
decomposition (3.2). Writing out the population solution for least squares yields:
[EX ?2 ]?1 E[X Y? ] = E[X(Y ? ?)X > (b
v> ? ? )? ? ] + E[X(Y ? ?)X > ?b? ] .
(3.3)
|
{z
} |
{z
}
I1
I2
We will now argue that left hand side of (3.3) is proportional to ? ? . First, we observe that
I1 = c0 (b
v> ? ? )? ? , since multiplying by any vector b ? ? ? yields b> I1 = 0 by independence.
Second, and perhaps more importantly, we have that I2 = 0. To see this, we first take a vector b ? span{? ? , ?b? }? . Since the three variables b> X, Y ? ? and ?b? X are independent, we have
b> I2 = 0. Multiplying by ? ? we have ? ? > I2 = 0 since ? ? > X(Y ? ?) is independent of X > ?b? .
Finally, multiplying by ?b? yields I2> ?b? = 0, since (X > ?b? )2 is independent of Y ? ?.
(a) Initialization
(b) Second Step
b and ?b produced by the first and second steps of Algorithm
Figure 1: An illustration of the estimates v
1. After the first step we can guarantee that the vector ? ? belongs to one of two spherical caps which
contain all vectors w such that |b
v> w| ? ? for some constant ? > 0, provided that the sample size
n & s2 log d is sufficiently large. After the second step we can guarantee that the vector ? ? belongs
to one of two spherical caps in (b), which are shrinking with (n, s, d) at a faster rate.
It is noteworthy to mention that the above derivation crucially relies on the fact that the Y variable
b was fixed. In what follows we formulate a pilot procedure which
was centered, and the vector v
b such that |b
produces an estimate v
v> ? ? | ? ? > 0. A proper initialization algorithm can be achieved
by using a spectral method, such as the Principal Hessian Directions (PHD) proposed by [22]. Cast
into the framework of SIM, the PHD framework implies the following simple observation:
Lemma 3.1. If we have an MPR model, then argmaxkvk2 =1 v> E[Y (X ?2 ? I)]v = ?? ? .
A proof of this fact can be found in Appendix C. Lemma 3.1 encourages us to look into the following
sample version maximization problem
Pn
argmaxkvk2 =1,kvk0 =s n?1 v> i=1 [Yi (Xi?2 ? I)]v,
(3.4)
which targets a restricted (s-sparse) principal eigenvector. Unfortunately, solving such a problem is a
computationally intensive task, and requires knowledge of s. Here we take a standard route of relaxing
the above problem to a convex program, and solving it efficiently via semidefinite programming
(SDP). A similar in spirit SDP relaxation for solving sparse PCA problems, was originally proposed
b = n?1 Pn Yi (X ?2 ?I), and solve the following convex
by [13]. Instead of solving (3.4), define ?
i
i=1
5
program:
Pd
b = argmaxtr(A)=1,A?Sd tr(?A)
b
A
? ?n i,j=1 |Aij |,
+
(3.5)
where Sd+ is the convex cone of non-negative semidefinite matrices, and ?n is a regularization parameter encouraging element-wise sparsity in the matrix A. The hopes of introducing the optimization
b will be a good first estimate of ? ??2 . In practice it could turn out that the
program above are that A
b
b In theory we
b as the principal eigenvector of A.
matrix A is not rank one, hence we suggest taking v
b
show that with high probability the matrix A will indeed be of rank one. Observation (3.3), Lemma
3.1 and the SDP formulation motivate the agnostic two-step estimation procedure for misspecified
phase retrieval in Algorithm 1.
Algorithm 1
input :(Yi , Xi )ni=1 : data, ?n , ?n : tuning parameters
1. Split the sample into two approximately equal sets S1 , S2 , with |S1 | = bn/2c, |S2 | = dn/2e.
?2
b := |S1 |?1 P
b
b be the first eigenvector of A.
2. Let ?
? Id ). Solve (3.5). Let v
i?S1 Yi (Xi
P
?1
3. Let Y = |S2 |
i?S2 Yi . Solve the following program:
>
b = argmin (2|S2 |)?1 P
b ? Xi> b)2 + ?n kbk1 .
b
(3.6)
b
i?S2 ((Yi ? Y )Xi v
b bk
b 2.
4. Return ?b := b/k
The sample split is required to ensure that after decomposition (3.2), the vector ?b? and the value
b > ? ? are independent of the remaining sample.
v
In ?3.1 we demonstrate that Algorithm 1 succeeds
p
with optimal (in the noisy regime) `2 rate s log d/n, provided that s2 log d . n. The latter requireb of optimization program (3.5) is
ment on the sample size suffices to guarantee that the solution A
rank one. Figure 1 illustrates the two steps of Algorithm 1. In addition to our main procedure, we
propose an optional refinement step (Algorithm 2) in which one applies steps 3. and 4. of Algorithm
1 on the full dataset using the output vector ?b of Algorithm 1. Doing so can potentially result in
additional stability and further refinements of the rate constant.
Algorithm 2 Optional Refinement
0
b
input :(Yi , Xi )ni=1
P: data, ?n : tuning parameter, output ? from the Algorithm 1
?1
5. Let Y = n
i?[n] Yi . Solve the following program:
b = argmin (2n)?1 Pn ((Yi ? Y )X > ?b ? X > b)2 + ? 0 kbk1 .
b
b
i
i
n
i=1
(3.7)
b bk
b 2.
6. Return ?b0 := b/k
3.1
Theoretical Guarantees
In this section we present our main theoretical results, which consist of theoretical justification of our
procedures, as well as lower bounds for certain types of SIM (1.3). To simplify the presentation for
this section, we slightly change the notation and assume that the sample size is 2n and S1 = [n] and
S2 = {n + 1, . . . , 2n}. Of course this abuse of notation does not restrict our analysis to only even
sample size cases.
b which is
Our first result shows that the optimization program (3.5) succeeds in producing a vector v
close to the vector ? ? .
p
Proposition 3.2. Assume that n is large enough so that s log d/n < (1/6 ? ?/4)c0 /(C1 + C2 )
for some smallp
but fixed ? > 0 and constants C1 , C2 (depending on f and ?). Then there exists a
b the solution of (3.5), satisfies
b of A,
value of ?n log d/n such that the principal eigenvector v
|b
v> ? ? | ? ? > 0,
?1
with probability at least 1 ? 4d ? O(n ).
Proposition 3.2 shows that the first step of Algorithm 1 narrows down the search for the direction
b satisfies |b
of ? ? to a union of two spherical caps (i.e., the estimate v
v> ? ? | ? ? for some constant
? > 0, see also Figure 1a). Our main result below, demonstrates that in combination with program
(3.6) this suffices to recover the direction of ? ? at an optimal rate with high probability.
?1
6
p
Theorem 3.3. There
exist values of ?n , ?n log d/n and a constant R > 0 depending on f and
p
?, such that if s log d/n < R and log(d) log2 (n)/n = o(1), the output of Algorithm 1 satisfies:
r
s log d
?
sup
P??
? O(d?1 ? n?1 ),
(3.8)
min k?b ? ?? k2 > L
n
??{1,?1}
k? ? k2 =1,k? ? k0 ?s
where L is a constant depending solely on f and ?.
p
We remark
that
although
the
estimation
rate
is
of
the
order
s log d/n, our procedure still requires
p
that s log d/n is sufficiently small. This phenomenon is similar to what has been observed by [7],
and it is our belief that this requirement cannot be relaxed for computationally feasible algorithms.
We would further like to mention that while in bound (3.8) we control the worst case probability of
failure, it is less clear whether the estimate ?b is universally consistent (i.e., whether the sup can be
moved inside the probability in (3.8)).
4
Numerical Experiments
In this section we provide numerical experiments based on the three models (2.3), (2.4) and (2.5)
where the random variable ? ? N (0, 1). All models are compared with the Truncated Power Method
(TPM), proposed in [37]. For model (2.3) we also compare the results of our approach to the ones
given by the TWF algorithm of [7]. Our setup is as follows. In all scenarios the vector ? ? was held
fixed at ? ? = (?s?1/2 , s?1/2 , . . . , s?1/2 , 0, . . . 0). Since our theory requires that n & s2 log d, we
|
{z
} | {z }
s
d?s
have setup four different sample sizes n ? ?s2 log d, where ? ? {4, 8, 12, 16}. We let s depend on
the dimension d and we take s ? log d. In addition to the suggested approach in Algorithm 1, we
also provide results using the refinement procedure (see Algorithm 3.7). We also provide the values
of two ?warm? starts of our algorithm, produced by solving program (3.5) with half and full data
correspondingly. It is evident that for all scenarios the second step of Algorithms 1 and 2 outperform
the warm start from SDP, except in Figure 2 (b), (c), when the sample size is simply two small to for
the warm start on half of the data to be accurate. All values we report are based on an average over
100 simulations.
The SDP parameter was kept at a constant value (0.015) throughout all simulations, and we observed
that varying this parameter had little influence on the final SDP solution. To select the ?n parameter
for (3.6) a pre-specified grid of parameters {? 1 , . . . , ? l } was selected, and the following heuristic
procedure based on K-fold cross-validation was used. We divide S2 into K = 5 approximately
equally sized non-intersecting sets S2 = ?j?[K] Se2j . For each j ? [K] and k ? [l] we run (3.6) on the
set ?r?[K],r6=j Se2r with a tuning parameter ?n = ? k to obtain an estimate ?bk,?Sej . Lemma 3.1 then
2
justifies the following criteria to select the optimal index for selecting ?bn = ? l where
X X
b
l = argmax
Yi (X > ?b ej )2 .
b
i
k?[l]
k,?S2
ej
j?[K] i?S
2
Our experience suggests this approach works well in practice provided
that the values {? 1 , . . . , ? l }
p
are selected within appropriate range and are of the magnitude log d/n.
Since the TPM algorithm requires an estimate of the sparsity s, we tuned it as suggested in Section
4.1.2 of [37]. In particular, for each scenario we considered the set of possible sparsities K =
{s, 2s, 4s, 8s}. For each k ? K the algorithm is ran on the first part of the data S1 , to obtain an
estimate ?bk , and the final estimate is taken to be ?bbk where b
k is given by
X
?1
b
k = argmax ?bk> |S2 |
Yi (Xi?2 ? Id )?bk .
k?K
i?S2
The TPM is ran for 2000 iterations. In the case of phase retrieval, the TWF algorithm was also ran
at a total number of 2000 iterations, using the tuning parameters originally suggested in [7]. As
expected the TWF algorithm which targets the sparse phase retrieval model in particular outperforms
our approach in the case when the sample size n is small, however our approach performs very
comparatively to the TWF, and in fact even slightly better once we increase the sample size. It is
possible that the TWF algorithm can perform better if it is ran for a longer than 2000 iterations,
though in most cases it appeared to have converged to its final value. The results are visualized on
Figure 2 above. The TPM algorithm, has performance comparable to that of Algorithm 1, is always
7
?
?
?
?
?
?
?
2.0
Init
Second Step
Init full data
Refined
TPM
1.5
?
?=4
?=8
? = 12
? = 16
?
?
Init
Second Step
Init full data
Refined
TPM
?
?
1.0
2.0
1.5
?=4
?=8
? = 12
? = 16
^
||? ? ?*||2
?
1.0
^
||? ? ?*||2
?
Init
Second Step
Init full data
Refined
TPM
TWF full data
1.0
?
^
||? ? ?*||2
2.0
1.5
?=4
?=8
? = 12
? = 16
?
?
0.0
?
?
?
?
?
(d) Model (2.3), d = 400
0.0
?
?
?
?
2.0
1.5
?
?
?
?
?
?
?
?
?
1.0
2.0
?
?=4
?=8
? = 12
? = 16
Init
Second Step
Init full data
Refined
TPM
?
?
0.5
?
(c) Model (2.5), d = 200
Init
Second Step
Init full data
Refined
TPM
?
0.5
?
?
?
?
?
?
?
?
?
0.0
?
?=4
?=8
? = 12
? = 16
^
||? ? ?*||2
1.0
?
0.5
^
||? ? ?*||2
?
1.5
?
(b) Model (2.4), d = 200
Init
Second Step
Init full data
Refined
TPM
TWF full data
1.0
?=4
?=8
? = 12
? = 16
^
||? ? ?*||2
1.5
2.0
(a) Model (2.3), d = 200
?
?
?
?
0.0
?
?
?
?
0.0
0.0
?
?
?
0.5
0.5
0.5
?
?
?
?
?
(e) Model (2.4), d = 400
(f) Model (2.5), d = 400
Figure 2: Simulation results for the three examples considered in ?2, in two different settings for the
n
dimension d = 200, 400. Here the parameter ? ? s2 log
d describes the relationship between sample
size, dimension and sparsity of the signal. Algorithm 2 dominates in most settings, with exceptions
when ? is too small, in which case none of the approaches provides meaningful results.
worse than the estimate produced by Algorithm 2, and it needs an initialization (for the first step
of Algorithm 1 is used) and further requires a rough knowledge of the sparsity s, whereas both
Algorithms 1 and 2 do not require an estimate of s.
5
Discussion
In this paper we proposed a two-step procedure for estimation of MPR models with standard Gaussian
designs. We argued that the MPR models form a rich class including numerous additive SIMs (i.e.,
Y = h(X > ? ? ) + ?) with an even and increasing on R+ link function h. Our algorithm is based
solely on convex optimization, and achieves optimal rates of estimation.
Our procedure does require that the sample size n & s2 log d to ensure successful initialization. The
same condition has been exhibited previously, e.g., in [7] for the phase retrieval model, and in works
on sparse principal components analysis [see, e.g., 3, 15, 33]. We anticipate that for a certain subclass
of MPR models, the sample size requirement n & s2 log d is necessary for computationally efficient
algorithms to exist. We conjecture that models (2.3)-(2.5) are such models. It is however certainly
not true that this sample size requirement holds for all models from the MPR class. For example, the
following model can be solved efficiently by applying the Lasso algorithm, without requiring the
sample size scaling n & s2 log d
Y = sign(X > ? ? + c),
where c < 0 is fixed. This discussion leads to the important question under what conditions of the
(known) link and error distribution (f, ?) one can efficiently solve the SIM Y = f (X > ? ? , ?) with
an optimal sample complexity. We would like to investigate this issue further in our future work.
Acknowledgments: The authors would like to thank the reviewers and meta-reviewers for carefully
reading the manuscript and their helpful suggestions which improved the presentation. The authors
would also like to thank Professor Xiaodong Li for kindly providing the code for the TWF algorithm.
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9
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5,630 | 6,095 | Lifelong Learning with Weighted Majority Votes
Anastasia Pentina
IST Austria
apentina@ist.ac.at
Ruth Urner
Max Planck Institute for Intelligent Systems
rurner@tuebingen.mpg.de
Abstract
Better understanding of the potential benefits of information transfer and representation learning is an important step towards the goal of building intelligent
systems that are able to persist in the world and learn over time. In this work, we
consider a setting where the learner encounters a stream of tasks but is able to retain
only limited information from each encountered task, such as a learned predictor.
In contrast to most previous works analyzing this scenario, we do not make any
distributional assumptions on the task generating process. Instead, we formulate a
complexity measure that captures the diversity of the observed tasks. We provide a
lifelong learning algorithm with error guarantees for every observed task (rather
than on average). We show sample complexity reductions in comparison to solving every task in isolation in terms of our task complexity measure. Further, our
algorithmic framework can naturally be viewed as learning a representation from
encountered tasks with a neural network.
1
Introduction
Machine learning has made significant progress in understanding both theoretical and practical
aspects of solving a single prediction problem from a set of annotated examples. However, if we
aim at building autonomous agents, capable to persist in the world, we need to establish methods
for continuously learning various tasks over time [25, 26]. There is no hope to initially provide, for
example, an autonomous robot with sufficiently rich prior knowledge to solve any problem that it may
encounter during the course of its life. Therefore, an important goal of machine learning research
is to replicate humans? ability to learn from experience and to reuse knowledge from previously
encountered tasks for solving new ones more efficiently. This is aimed at in lifelong learning or
learning to learn, where a learning algorithm is assumed to encounter a stream of tasks and is aiming
to exploit commonalities between them by transferring information from earlier tasks to later ones.
The first theoretical formulation of this framework was proposed by Baxter [4]. In that model, tasks
are generated by a probability distribution and the goal, given a sample of tasks from this distribution,
is to perform well in expectation over tasks. Under certain assumptions, such as a shared good
hypothesis set, this model allows for sample complexity savings [4]. However, good performance in
expectation is often too weak a requirement. To stay with the robot example, failure on a single task
may cause severe malfunction ? and the end of the robot?s life. Moreover, the theoretical analysis of
this model relies on the assumption that the learner maintains access to training data for all previously
observed tasks, which allows to formulate a joint optimization problem. However, it is unlikely that
an autonomous robot is able to keep all this data.
Thus, we instead focus on a streaming setting for lifelong learning, where the learner can only
retain learned models from previously encountered tasks. These models have a much more compact
description than the joint training data. Specifically, we are interested in analysis and performance
guarantees in the scenario that 1) tasks arrive one at a time without distributional or i.i.d. assumptions,
2) the learner can only keep the learned hypotheses from previously observed tasks, 3) error bounds
are required for every single task, rather than on average.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
The first analysis of this challenging setting was recently provided by Balcan et al. [3]. That work
demonstrates sample complexity improvements for learning linear halfspaces (and some boolean
function classes) in the lifelong learning setting in comparison to solving each task in isolation under
the assumption that the tasks share a common low dimensional representation. However, the analysis
relies on the marginal distributions of all tasks being isotropic log-concave. It was stated as an open
challenge in that work whether similar guarantees (error bounds for every task, while only keeping
limited information from earlier tasks) were possible under less restrictive distributional assumptions.
In this work, we (partially) answer this question in the positive. We do so by proposing to learn with
weighted majority votes rather than linear combinations over linear predictors. We show that the
shift from linear combinations to majority votes introduces stability to the learned ensemble that
allows exploiting it for later tasks. Additionally, we show that this stability is achieved for any ground
hypothesis class. We formulate a relatedness assumption on the sequence of tasks (similar to one used
in [3]) that captures how suitable to lifelong learning a sequence of tasks is. With this, we prove that
sample complexity savings through lifelong learning are obtained for arbitrary marginal distributions
(provided that these marginal distributions are related in terms of their discrepancy [5, 17]). This is a
significant generalization towards more practically relevant scenarios.
Summary of our work We employ a natural algorithmic paradigm, similar to the one in [3]. The
algorithm maintains a set of base hypotheses from some fixed ground hypothesis class H. These base
hypotheses are predictors learned on previous tasks. For each new task, the algorithm first attempts to
achieve good prediction performance with a weighted majority vote over the current base hypotheses,
and uses this predictor for the task if successful. Otherwise (if no majority vote classifier achieves
high accuracy), the algorithm resorts to learning a classifier from the ground class for this task. This
classifier is then added to the set of base hypotheses, to be used for subsequent tasks. We describe
this algorithm Section 4.1.
If the ground class is the class of linear predictors, this algorithm is actually learning a neural network.
Each base classifier becomes a node in a hidden middle layer, which represents a learned feature
representation of the neural net. A new task is then either solved by employing the representation
learned from previous tasks (the current middle layer), and just learning task specific weights for the
last layer, or, in case this is not possible, it extends the current representation. See also Section 4.2.
This paradigm yields sample complexity savings, if the tasks encountered are related in the sense
that for many tasks, good classification accuracy can be achieved with a weighted majority vote
over previously learned models. We formally capture this property as an effective dimension of
the sequence of tasks. We prove in Section 4.3 that if this effective dimension is bounded by
k, then the total samplecomplexity for learning n tasks with this paradigm is upper bounded by
? (nk + VC(H)k 2 )/ , a reduction from O
? (nVC(H)/), the sample complexity of learning n
O
tasks individually without retaining information from previous tasks.
The main technical difficulty is to control the propagation of errors. Since every task is represented
by a finite training set, the learner has access only to approximations of the true labeling functions,
which may degrade the quality of this collection of functions as a ?basis? for new tasks. Balcan et
al. [3] control this error propagation using elegant geometric arguments for linear predictors under
isotropic log-concave distributions. We show that moving from linear combinations to majority votes
yields the required stability for the quality of the representation under arbitrary distributions.
Finally, while we first present our algorithm and results for known upper bounds of k base tasks and
n tasks in total, we also provide a variation of the algorithm that does not need to know the number
of tasks and the complexity parameter of the task sequence. We show that similar sample complexity
improvements are achievable in this setting in Section 5.
2
Related Work
Lifelong learning. While there are many ways in which prediction tasks may be related [20], most
of the existing approaches to transfer or lifelong learning are exploiting possible similarities between
the optimal predictors for the considered tasks. In particular, one widely used relatedness assumption
is that these predictors can be described as linear or sparse combinations of some common metafeatures and the corresponding methods aim at learning this representations [10, 2, 15, 3]. Though
this idea was originally used in the multi-task setting, it was later extended to lifelong learning by
2
Eaton et al. [9], who proposed a method for sequentially updating the underlying representation as
new tasks arrive. These settings were theoretically analyzed in a series of works [18, 19, 23, 21]
that have demonstrated that information transfer can lead to provable sample complexity reductions
compared to solving each task independently. However, all these results rely on Baxter?s model of
lifelong learning and therefore assume access to the training data for all (observed) tasks and provide
guarantees only on average performance over all tasks. An exception is [6], where the authors provide
error guarantees for every task in the multi-task scenario. However, these guarantees are due to the
relatedness assumption used which implies that all tasks have the same expected error. The task
relatedness assumption that we employ is related to the one used in [1] for multi-task learning with
expert advice. There the authors consider a setting where there exists a small subset of experts that
perform well on all tasks. Similarly, we assume that there is a small subset of base tasks, such that
the remaining ones can be solved well using majority votes over the corresponding base hypotheses.
Majority votes. Weighted majority votes are a theoretically well understood and widely used in
practice type of ensemble predictor. In particular, they are employed in boosting [11]. They are
also often considered in works that utilize PAC-Bayesian techniques [22, 12]. Majority votes are
also used in the concept drift setting [14]. The corresponding method, conceptually similar to the
one proposed here, dynamically updates a set of experts and uses their weighted majority votes for
making predictions.
3
3.1
Formal Setting
General notation and background
We let X ? Rd denote a domain set and let Y denote a label set. A hypothesis is a function
h : X ? Y, and a hypothesis class H is a set of hypotheses. We model learning tasks as pairs
hD, h? i of a distribution D over X and a labeling function h? : X ? Y. The quality of a hypothesis
is measured by a loss function ` : Y ? Y ? R+ . We deal with binary classification tasks, that is,
Y = {?1, 1}, under the 0/1-loss function, that is, `(y, y 0 ) = Jy 6= y 0 K (we let J?K denote the indicator
function). The risk of a hypothesis h with respect to task hD, h? i is defined as its expected loss:
LD,h? (h) := Ex?D [`(h(x), h? (x))].
Given a sample S = {(x1 , y1 ), (x2 , y2 ), . . . , (xn , yn )}, the empirical risk of h with respect to S is
n
1X
LS (h) :=
`(h(xi ), yi ).
n i=1
For binary classification, the sample complexity of learning a hypothesis class is characterized (that
is, upper and lower bounded), by the VC-dimension of the class [27]. We will employ the following
generalization bounds for classes of finite VC-dimension:
Theorem 1 (Corollaries 5.2 and 5.3 in [7]). Let H be a class of binary functions with a finite
VC-dimension. There exists a constant C, such that for any ? ? (0, 1), for any task hD, h? i, with
probability at least 1 ? ? over a training set S of size n, sampled i.i.d. from hD, h? i:
q
?
?
? ? ? + ?,
?
LD,h (h) ? LS (h) + LS (h)
(1)
q
? ? LD,h? (h)
? + LD,h? (h)
? ? ? + ?,
LS (h)
(2)
q
? ? inf LD (h) + inf LD (h) ? ? + ?,
LD,h? (h)
(3)
h?H
h?H
? ? arg minh?H LS (h) is an empirical risk minimizer and
where h
?=C
VC(H) log(n) + log(1/?)
.
n
(4)
In therealizable case (h? ? H), the above bounds imply that the sample complexity is upper bounded
? VC(H)+log(1/?) .
by O
3
Weighted majority votes Given a hypothesis class H, we define the class of k-majority votes as
(
!)
k
X
MV(H, k) = g : X ? Y | ?h1 , . . . , hk ? H, ?w1 , . . . , wk ? R : g(x) = sign
wi hi (x)
.
i=1
We will omit k in the above notation if clear from the context. The VC-dimension of MV(H, k) is
upper bounded as
VC(MV(H, k)) ? k(VC(H) + 1)(3 log(VC(H) + 1) + 2)
(5)
(see Theorem 10.3 in [24]). This implies in particular, that the VC-dimension of majority votes over
? log(k)).
a fixed set of k functions is upper bounded by O(k
3.2
Lifelong learning
In the lifelong learning setting the learner encounters a stream of prediction problems
hD1 , h?1 i, . . . , hDn , h?n i, one at a time. In this work we focus on the realizable case, i.e. h?i ? H for
every i and some fixed H. In contrast to most works on lifelong learning, we assume that the only
information the learner is able to store about the already observed tasks is the obtained predictors, i.e.
it does not have access to the training data of already solved tasks. The need to keep little information
from previous tasks has also been argued for in the context of domain adaptation [16].
Possible benefits of information transfer depend on how related or, in other words, how diverse the
observed tasks are. Moreover, since we do not make any assumptions on the task generation process
(in contrast to Baxter?s i.i.d. model [4]), we will formulate our relatedness assumption in terms of a
sequence of tasks. Intuitively, one would expect that the information transfer is beneficial if only a
few times throughout the course of learning information obtained from the already solved tasks will
not be sufficient to solve the current one. In order to formalize this intuition, we use the following
(pseudo-)metric over the hypothesis class with respect to a marginal distribution D:
dD (h, h0 ) = E Jh(x) 6= h0 (x)K.
x?D
(6)
Further, we can define a distance of a hypothesis to a hypothesis space as
dD (h, H0 ) = min
dD (h, h0 )
0
0
h ?H
(7)
and the distance between two sets of hypotheses as
dD (H, H0 ) = max dD (h, H0 ) = max min
dD (h, h0 ).
0
0
h?H h ?H
h?H
(8)
Note that the latter is not necessarily a metric over subsets of the hypothesis space. However, it does
satisfy the triangle inequality (see Section 1 in the supplementary material).
Now we can formulate the diversity measure for a sequence of learning tasks that we will employ.
Note that the concepts below are closely related to the ones used in [3] for the case of linear predictors
and linear combinations over these.
Definition 1. A sequence of learning tasks hD1 , h?1 i, . . . , hDn , h?n i is ?-separated, if for every i
dDi (h?i , MV(h?1 , . . . , h?i?1 )) > ?.
Definition 2. A sequence of learning tasks hD1 , h?1 i, . . . , hDn , h?n i has ?-effective dimension k, if
the largest ?-separated subsequence of these tasks has length k.
Formally, we will assume that the ?-effective dimension k of the observed sequence of tasks is
relatively small for a sufficiently small ?. Note that this assumption can also be seen as a relaxation of
the one used in [8]. There the authors assumed that there exists a set of k hypothesis such that every
task can be well solved by one of them. This would correspond to substituting the sets of weighted
majority votes MV(h?1 , . . . , h?i?1 ) by just the collections {h?1 , . . . , h?i?1 } in the above definitions.
Moreover, we will assume that the marginal distributions have small discrepancy with respect to the
hypothesis set H:
discH (Di , Dj ) = max
|dDi (h, h0 ) ? dDj (h, h0 )|.
(9)
0
h,h ?H
This is a measure of task relatedness that has been introduced in [13] and shown to be beneficial in
the context of domain adaptation [5, 17]. Note, however, that we do not make any assumptions on the
marginal distributions D1 , . . . , Dn themselves.
4
4
4.1
Algorithm and complexity guarantees
The algorithm
We employ a natural algorithmic paradigm, which is similar to the one in [3]. Algorithm 1 below
provides pseudocode for our procedure. The algorithm takes as parameters a class H, which we call
the ground class, accuracy and confidence parameters and ?, as well as a task horizon n (the number
of tasks to be solved) and a parameter k (a guessed upper bound on the number of tasks that will not
be solvable as majority votes over earlier tasks). In Section 5 we present a version that does not need
to know n and k in advance.
Algorithm 1 Lifelong learning of majority votes
1: Input parameters H, n, k, , ?
2: set ? 0 = ?/(2n), 0 = /(8k)
3: draw a training set S1 from hD1 , h?1 i, such that ?1 := ?(VC(H), ? 0 , |S1 |) ? 0
4: g1 = arg minh?H LS1 (h)
? = 1, i1 = 1, h
? 1 = g1
5: set k
6: for i = 2 to n do
? 1, . . . , h
? ? )), ? 0 , |Si |) ?
7:
draw a training set Si from hDi , h?i i, such that ?i := ?(VC(MV(h
k
8:
gi = arg minh?MV(h? 1 ,...,h? ? ) LSi (h)
k
p
9:
if LSi (gi ) + LSi (gi ) ? ?i + ?i > then
10:
draw a training set Si from hDi , h?i i, such that ?i := ?(VC(H), ? 0 , |Si |) ? 0
11:
gi = arg minh?H LSi (h)
? ? = gi , i? = i
12:
set k? = k? + 1, h
k
k
13:
end if
14: end for
15: return g1 , . . . , gn
40
? 1, . . . , h
? ? ) from
During the course of its ?life?, the algorithm maintains a set of base hypotheses (h
k
the ground class, which are predictors learned on previous tasks. In order to solve the first task, it
0
uses the hypothesis set H and a large enough training set S1 to ensure the error guarantee ? /8k
? 1 ? H is the first member
with probability at least 1 ? ? 0 , where ? 0 = ?/2n. The learned hypothesis h
of the set of base hypotheses. For each new task i, the algorithm first attempts to achieve good
prediction performance (up to error ) with a weighted majority vote over the base hypotheses, i.e. it
? 1, . . . , h
? ? ), and uses the obtained predictor for the
attempts to learn this task using the class MV(h
k
task if successful. Otherwise (if no majority vote classifier achieves high accuracy), the algorithm
resorts to learning a classifier from the base class for this task, which is then added to the set of base
hypotheses, to be used for subsequent tasks. The error guarantees are ensured with Theorem 1 by
choosing the training sets Si large enough so that
?i := ?(VC(Hi ), ? 0 , |Si |) := C
VC(Hi ) log(|Si |) + log(1/? 0 )
? c,
|Si |
where Hi is either the ground class H or the set of weighted majority votes over the current set of
? 1, . . . , h
? ? ), and constant c is set according to case, see pseudocode.
base hypotheses MV(h
k
While this approach is very natural, the challenge is to analyze it and to specify the parameters.
In particular, we need to ensure that the algorithm will not have to search over (potentially large)
hypothesis set H too often and, consequently, will lead to provable sample complexity reductions
over solving each task independently. The following theorem summarizes the performance guarantees
for Algorithm 1 (the proof is in Section 4.3).
Theorem 2. Consider running Algorithm 1 on a sequence of tasks with ?-effective dimension at most
k and discH (Di , Dj ) ? ? for all i, j. Then, if ? ? /4 and k? < /8, with probability at least 1 ? ?:
? The error of every task is bounded: LDi ,h?i (gi ) ? for every i = 1, . . . , n.
2
? nk+VC(H)k .
? The total number of labeled examples used is O
5
Discussion Note that if we assume that all tasks
by H, independently learning them
are realizable
VC(H)n
?
. The sample complexity of learning n
up to error would have sample complexity O
2
? nk+VC(H)k . This is a
tasks in the lifelong learning regime with our paradigm in contrast is O
significant reduction if the effective dimension of the task sequence k is small in comparison to the
total number n of tasks, as well as the complexity measure VC(H) of the ground class. That is,
if most tasks are learnable as combination of previously stored base predictors, much less data is
required overall.
Note that for all those tasks that are solved as majority votes, our algorithm and analysis actually require realizability only by the class of k-majority votes over H and not by the ground
class
tasks independently under this assumption, has sample complexity
H. Learning the n
? VC(H)k + (n?k)VC(H)k . In contrast, the lifelong learning method gradually identifies the
O
relevant set of base predictors and thereby reduces the number of required examples.
4.2
Neural networks
If the ground class is the class of linear predictors, our algorithm is actually learning a neural network
(with sign() as the activation function). Each base classifier becomes a new node in a hidden middle
layer. Thus, the maintained set of base classifiers can be viewed as feature representation in the
neural net, which was learned based on the encountered tasks. A new task is then either solved by
employing the representation learned from previous tasks (the current middle layer), and just learning
task specific weights for the last layer; or, in case this is not possible, a fresh linear classifier is
learned, and added as a node to the middle layer. Thus, in this case, the feature representation is
extended.
4.3
Analysis
We start with presenting the following two lemmas that show how to control the error propagation of
the learned representations (sets of base classifiers). We then proceed to the proof of Theorem 2.
Lemma 1. Let V = MV(h1 , . . . , hk , g) and V? = MV(h1 , . . . , hk , g?). Then, for any distribution D:
dD (V, V? ) ? dD (g, g?).
(10)
Proof. By the definition of dD (V, V? ) there exists u ? V such that:
dD (V, V? ) = dD (u, V? ).
(11)
Pk
Pk
We can represent u as u = sign( i=1 ?i hi + ?g) and let u1 = i=1 ?i hi . Note that while all hi -s,
g and g? are assumed to take values in {?1, 1}, u1 can take values in R. Then:
? ?
dD (u, V? ) = min dD (u, h)
? V?
h?
?
max
min
?
h?MV(u
g)
1 ,?
min
?
h?MV(u1 ,g) h?MV(u
g)
1 ,?
?
dD (u, h)
? = dD (MV(u1 , g), MV(u1 , g?)).
dD (h, h)
Now we show that for any ?1 u1 + ?2 g ? MV(u1 , g) there exists a close hypothesis in MV(u1 , g?).
In particular, this hypothesis is ?1 u1 + ?2 g?:
dD (?1 u1 + ?2 g, ?1 u1 + ?2 g?) =
=
E Jsign(?1 u1 (x) + ?2 g(x)) 6= sign(?1 u1 (x) + ?2 g?(x))K
x?D
E J?12 u21 (x) + ?1 ?2 u1 (x)g(x) + ?1 ?2 u1 (x)?
g (x) + ?22 g(x)?
g (x) < 0K.
x?D
Note that for every x on which g and g? agree, i.e. g(x)?
g (x) = 1, we obtain:
?12 u21 (x) + ?1 ?2 u1 (x)g(x) + ?1 ?2 u1 (x)?
g (x) + ?22 g(x)?
g (x) = (?1 u1 (x) + ?2 g(x))2 ? 0.
Therefore:
dD (?1 u1 + ?2 g, ?1 u1 + ?2 g?) ? E Jg(x) 6= g?(x)K = dD (g, g?).
x?D
6
(12)
? 1, . . . , h
? k ). For any distribution D, if
Lemma 2. Let Vk = MV(h1 , . . . , hk ) and V?k = MV(h
P
k
? i ) ? i for every i = 1, . . . , k, then dD (Vk , V?k ) ?
dD (hi , h
i .
i=1
For the proof see Section 2 in the supplementary material.
Proof of Theorem 2. 1. First, note that for every task Algorithm 1 solves at most 2 estimation
problems with a probability of failure ? 0 for each of them. Therefore, with a union bound argument,
the probability of any of these estimations being wrong is at most 2 ? n ? ? 0 = ?. Thus, from now we
assume that all the estimations were correct, that is, the high probability events of Theorem 1 hold.
2. To see that the error of every encountered task is bounded by , note that there are
p two cases. For
tasks i that are solved by a majority vote over previous tasks, we have LSi (gi ) + LSi (gi ) ? ?i +
?i ? . In this case, Equation (1) in Theorem 1 implies LDi ,h?i (gi ) ? . For tasks i that are not solved
? 1, . . . , h
? ? )), ? 0 , m) ? /8k.
as a majority vote over previous tasks, we have ?i = ?(VC(MV(h
k
Since task i is realizable by the base class H, we have inf h?H LDi ,h?i (h) = 0, and thus Equation (3)
of Theorem 1 implies LDi ,h?i (gi ) ? /8k < .
3. To upper bound the sample complexity we first prove that the number k? of tasks, which are not
learned as majority votes over previous tasks, is at most k. For that we use induction showing that for
? when we create a new h
? ? from the i? -th task, we have that
every k? ? k,
k
k
dDi? (h?ik? , MV(h?i1 , . . . , h?ik?1
)) > ?.
(13)
?
k
This implies k? ? k by invoking that the ?-effective dimension of the sequence of encountered tasks
is at most k.
To proceed to the induction, note that for k? = 1, the claim follows immediately. Consider k? > 1. If
? ? , it means that the condition in line 9 is true, which is:
we create a new h
k
q
LSi? (gik? ) + LSi? (gik? ) ? ?i + ?i > .
(14)
k
k
Therefore LSi? (gik? ) > 0.83. Consequently, due to (2), LDi? ,h?i (gik? ) > 0.67. Finally, by (3),
?
k
k
k
? 1, . . . , h
? ? that
inf g LD ,h? (g) > 0.5. Therefore there is no majority vote predictor based on h
i?
k
k?1
i?
k
leads to error less than /2 on the problem ik? . In other words:
? 1, . . . , h
? ? )) > /2.
dD (h? , MV(h
ik
?
i?
k
k?1
(15)
Now, by way of contradiction, suppose that dDi? (h?ik? , MV(h?i1 , . . . , h?ik?1
)) ? ?. By construction
?
k
? ?
0
?
for every j = 1, . . . , k ? 1 dD (h , hj ) ? ? /8k. By the definition of discrepancy and the
ij
ij
assumption on the marginal distributions it follows that for all j:
? j ) ? dD (h? , h
? j ) + discH (Di , Di ) ? 0 + ?.
dDi? (h?ij , h
ij
ij
j
?
k
k
(16)
Therefore by Lemma 2:
? 1, . . . , h
? ? )) ? k(0 + ?).
dDi? (MV(h?i1 , . . . , h?ik?1
), MV(h
k
?
k
Consequently, by using the triangle inequality:
? 1, . . . , h
? ? )) ? ? + k(0 + ?) ? /4 + /8 + /8 = /2,
dD (h? , MV(h
i?
k
ik
?
k?1
(17)
(18)
which is in contradiction with (15).
4. The total sample complexity of Algorithm 1 consists of two parts. First, for every task Algorithm 1
checks, whether it can be solved by a majority vote over the base, at most k? predictors. For that it
employs Theorem 1 and therefore needs the following number of samples:
!
? log(2n/?)
nk? log k? log(k? log k)
nk
?
?
O
=O
.
(19)
Second, there are at most k? tasks that satisfy the condition in line 9 and are learned using the
0
hypothesis
= /(8k). Therefore the corresponding sample complexity
? set H with estimation
error 2
kVC(H) log(2n/?)
VC(H)k
?
is: O
=O
.
/(8k)
7
5
Lifelong learning with unknown horizon
In this section we present a modification of Algorithm 1 for the case when the total number of
tasks n and the complexity of the task sequence k are not known in advance. The main difference
between Algorithm 2 and Algorithm 1 is that with unknown n and k the learner has to adopt the
parameters ? 0 and 0 on the fly. We show that this can be done by the doubling trick that is often used
in online learning. Theorem 3 summarizes the resulting guarantees (the proof can be found in the
supplementary material, Section 3).
Algorithm 2 Lifelong learning of majority votes with unkown horizon
1: Input parameters H, , ?
2: set ?1 = ?/2, 01 = /16
3: draw a training set S1 from hD1 , h?1 i of size m, such that ?(VC(H), ?1 , m) ? 01 (see (4))
4: g1 = arg minh?H LS1 (h)
? = 1, i1 = 1, h
? 1 = g1
5: set k
6: for i = 2 to n do
7:
set l = blog ic, m = blog k? + 1c
?
8:
set ?i = 22l+2
, 0i = 22m+4
? 1, . . . , h
? ? )), ?i , m) ?
draw a training set Si from hDi , h?i i of size m, such that ?(VC(MV(h
9:
k
/40 (see (4))
10:
gi = arg minh?MV(h? 1 ,...,h? ? ) LSi (h)
k
p
11:
if LSi (gi ) + LSi (gi ) ? ? + ? > then
12:
draw a training set Si from hDi , h?i i of size m, such that ?(VC(H), ?i , m) ? 0i (see (4))
13:
gi = arg minh?H LSi (h)
? ? = gi , i? = i
14:
set k? = k? + 1, h
k
k
15:
end if
16: end for
17: return g1 , . . . , gn
Theorem 3. Consider running Algorithm 2 on a sequence of tasks with ?-effective dimension at most
k and discH (Di , Dj ) ? ? for all i, j. Then, if ? ? /4 and k? < /8, with probability at least 1 ? ?:
? The error of every task is bounded: LDi ,h?i (gi ) ? for every i = 1, . . . , n.
3
? nk+VC(H)k .
? The total number of labeled examples used is O
6
Conclusion
In this work, we have shown sample complexity improvements with lifelong learning in the challenging, yet as argued important setting, where tasks arrive in a stream (without assumptions on the tasks
generating process), where the learner is only allowed to maintain limited amounts of information
from previously encountered tasks, and where high performance is required for every single task,
rather than on average. While such improvements have been established in very specific settings [3],
our work shows they are possible in much more general and realistic scenarios. We hope that this will
open the door for more work in this area of machine lifelong learning and lead to better understanding
of how and when learning machines can benefit from past experience. An intriguing direction is to
investigate whether there exists a more general characterization of ensemble methods and/or data
distributions that would lead to benefits with lifelong learning. Another one is to better understand
lifelong learning with neural networks, analyzing cases of more complex network structures and
activation functions, an area where current machine learning practice yields exciting successes, but
little is understood.
Acknowledgments
This work was in parts funded by the European Research Council under the European Union?s
Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 308036.
8
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9
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5,631 | 6,096 | Learning a Probabilistic Latent Space of Object
Shapes via 3D Generative-Adversarial Modeling
Jiajun Wu*
MIT CSAIL
Chengkai Zhang*
MIT CSAIL
William T. Freeman
MIT CSAIL, Google Research
Tianfan Xue
MIT CSAIL
Joshua B. Tenenbaum
MIT CSAIL
Abstract
We study the problem of 3D object generation. We propose a novel framework,
namely 3D Generative Adversarial Network (3D-GAN), which generates 3D objects from a probabilistic space by leveraging recent advances in volumetric convolutional networks and generative adversarial nets. The benefits of our model are
three-fold: first, the use of an adversarial criterion, instead of traditional heuristic
criteria, enables the generator to capture object structure implicitly and to synthesize high-quality 3D objects; second, the generator establishes a mapping from
a low-dimensional probabilistic space to the space of 3D objects, so that we can
sample objects without a reference image or CAD models, and explore the 3D
object manifold; third, the adversarial discriminator provides a powerful 3D shape
descriptor which, learned without supervision, has wide applications in 3D object
recognition. Experiments demonstrate that our method generates high-quality 3D
objects, and our unsupervisedly learned features achieve impressive performance
on 3D object recognition, comparable with those of supervised learning methods.
1
Introduction
What makes a 3D generative model of object shapes appealing? We believe a good generative model
should be able to synthesize 3D objects that are both highly varied and realistic. Specifically, for
3D objects to have variations, a generative model should be able to go beyond memorizing and
recombining parts or pieces from a pre-defined repository to produce novel shapes; and for objects to
be realistic, there need to be fine details in the generated examples.
In the past decades, researchers have made impressive progress on 3D object modeling and synthesis [Van Kaick et al., 2011, Tangelder and Veltkamp, 2008, Carlson, 1982], mostly based on meshes
or skeletons. Many of these traditional methods synthesize new objects by borrowing parts from
objects in existing CAD model libraries. Therefore, the synthesized objects look realistic, but not
conceptually novel.
Recently, with the advances in deep representation learning and the introduction of large 3D CAD
datasets like ShapeNet [Chang et al., 2015, Wu et al., 2015], there have been some inspiring attempts
in learning deep object representations based on voxelized objects [Girdhar et al., 2016, Su et al.,
2015a, Qi et al., 2016]. Different from part-based methods, many of these generative approaches
do not explicitly model the concept of parts or retrieve them from an object repository; instead,
they synthesize new objects based on learned object representations. This is a challenging problem
because, compared to the space of 2D images, it is more difficult to model the space of 3D shapes due
to its higher dimensionality. Their current results are encouraging, but often there still exist artifacts
(e.g., fragments or holes) in the generated objects.
In this paper, we demonstrate that modeling volumetric objects in a general-adversarial manner could
be a promising solution to generate objects that are both novel and realistic. Our approach combines
? indicates equal contributions. Emails: {jiajunwu, ckzhang, tfxue, billf, jbt}@mit.edu
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
the merits of both general-adversarial modeling [Goodfellow et al., 2014, Radford et al., 2016] and
volumetric convolutional networks [Maturana and Scherer, 2015, Wu et al., 2015]. Different from
traditional heuristic criteria, generative-adversarial modeling introduces an adversarial discriminator
to classify whether an object is synthesized or real. This could be a particularly favorable framework
for 3D object modeling: as 3D objects are highly structured, a generative-adversarial criterion, but
not a voxel-wise independent heuristic one, has the potential to capture the structural difference of
two 3D objects. The use of a generative-adversarial loss may also avoid possible criterion-dependent
overfitting (e.g., generating mean-shape-like blurred objects when minimizing a mean squared error).
Modeling 3D objects in a generative-adversarial way offers additional distinctive advantages. First, it
becomes possible to sample novel 3D objects from a probabilistic latent space such as a Gaussian
or uniform distribution. Second, the discriminator in the generative-adversarial approach carries
informative features for 3D object recognition, as demonstrated in experiments (Section 4). From
a different perspective, instead of learning a single feature representation for both generating and
recognizing objects [Girdhar et al., 2016, Sharma et al., 2016], our framework learns disentangled
generative and discriminative representations for 3D objects without supervision, and applies them
on generation and recognition tasks, respectively.
We show that our generative representation can be used to synthesize high-quality realistic objects,
and our discriminative representation can be used for 3D object recognition, achieving comparable
performance with recent supervised methods [Maturana and Scherer, 2015, Shi et al., 2015], and
outperforming other unsupervised methods by a large margin. The learned generative and discriminative representations also have wide applications. For example, we show that our network can
be combined with a variational autoencoder [Kingma and Welling, 2014, Larsen et al., 2016] to
directly reconstruct a 3D object from a 2D input image. Further, we explore the space of object
representations and demonstrate that both our generative and discriminative representations carry
rich semantic information about 3D objects.
2
Related Work
Modeling and synthesizing 3D shapes 3D object understanding and generation is an important
problem in the graphics and vision community, and the relevant literature is very rich [Carlson,
1982, Tangelder and Veltkamp, 2008, Van Kaick et al., 2011, Blanz and Vetter, 1999, Kalogerakis
et al., 2012, Chaudhuri et al., 2011, Xue et al., 2012, Kar et al., 2015, Bansal et al., 2016, Wu et al.,
2016]. Since decades ago, AI and vision researchers have made inspiring attempts to design or learn
3D object representations, mostly based on meshes and skeletons. Many of these shape synthesis
algorithms are nonparametric and they synthesize new objects by retrieving and combining shapes and
parts from a database. Recently, Huang et al. [2015] explored generating 3D shapes with pre-trained
templates and producing both object structure and surface geometry. Our framework synthesizes
objects without explicitly borrow parts from a repository, and requires no supervision during training.
Deep learning for 3D data The vision community have witnessed rapid development of deep
networks for various tasks. In the field of 3D object recognition, Li et al. [2015], Su et al. [2015b],
Girdhar et al. [2016] proposed to learn a joint embedding of 3D shapes and synthesized images, Su
et al. [2015a], Qi et al. [2016] focused on learning discriminative representations for 3D object recognition, Wu et al. [2016], Xiang et al. [2015], Choy et al. [2016] discussed 3D object reconstruction
from in-the-wild images, possibly with a recurrent network, and Girdhar et al. [2016], Sharma et al.
[2016] explored autoencoder-based networks for learning voxel-based object representations. Wu
et al. [2015], Rezende et al. [2016], Yan et al. [2016] attempted to generate 3D objects with deep
networks, some using 2D images during training with a 3D to 2D projection layer. Many of these
networks can be used for 3D shape classification [Su et al., 2015a, Sharma et al., 2016, Maturana
and Scherer, 2015], 3D shape retrieval [Shi et al., 2015, Su et al., 2015a], and single image 3D
reconstruction [Kar et al., 2015, Bansal et al., 2016, Girdhar et al., 2016], mostly with full supervision.
In comparison, our framework requires no supervision for training, is able to generate objects from a
probabilistic space, and comes with a rich discriminative 3D shape representation.
Learning with an adversarial net Generative Adversarial Nets (GAN) [Goodfellow et al., 2014]
proposed to incorporate an adversarial discriminator into the procedure of generative modeling. More
recently, LAPGAN [Denton et al., 2015] and DC-GAN [Radford et al., 2016] adopted GAN with
convolutional networks for image synthesis, and achieved impressive performance. Researchers have
also explored the use of GAN for other vision problems. To name a few, Wang and Gupta [2016]
discussed how to model image style and structure with sequential GANs, Li and Wand [2016] and
Zhu et al. [2016] used GAN for texture synthesis and image editing, respectively, and Im et al. [2016]
2
512?4?4?4
z
256?8?8?8
128?16?16?16
64?32?32?32
G(z) in 3D Voxel Space
64?64?64
Figure 1: The generator in 3D-GAN. The discriminator mostly mirrors the generator.
developed a recurrent adversarial network for image generation. While previous approaches focus on
modeling 2D images, we discuss the use of an adversarial component in modeling 3D objects.
3
Models
In this section we introduce our model for 3D object generation. We first discuss how we build
our framework, 3D Generative Adversarial Network (3D-GAN), by leveraging previous advances
on volumetric convolutional networks and generative adversarial nets. We then show how to train
a variational autoencoder [Kingma and Welling, 2014] simultaneously so that our framework can
capture a mapping from a 2D image to a 3D object.
3.1 3D Generative Adversarial Network (3D-GAN)
As proposed in Goodfellow et al. [2014], the Generative Adversarial Network (GAN) consists of
a generator and a discriminator, where the discriminator tries to classify real objects and objects
synthesized by the generator, and the generator attempts to confuse the discriminator. In our 3D
Generative Adversarial Network (3D-GAN), the generator G maps a 200-dimensional latent vector z,
randomly sampled from a probabilistic latent space, to a 64 ? 64 ? 64 cube, representing an object
G(z) in 3D voxel space. The discriminator D outputs a confidence value D(x) of whether a 3D
object input x is real or synthetic.
Following Goodfellow et al. [2014], we use binary cross entropy as the classification loss, and present
our overall adversarial loss function as
L3D-GAN = log D(x) + log(1 ? D(G(z))),
(1)
where x is a real object in a 64 ? 64 ? 64 space, and z is a randomly sampled noise vector from a
distribution p(z). In this work, each dimension of z is an i.i.d. uniform distribution over [0, 1].
Network structure Inspired by Radford et al. [2016], we design an all-convolutional neural
network to generate 3D objects. As shown in Figure 1, the generator consists of five volumetric fully
convolutional layers of kernel sizes 4 ? 4 ? 4 and strides 2, with batch normalization and ReLU
layers added in between and a Sigmoid layer at the end. The discriminator basically mirrors the
generator, except that it uses Leaky ReLU [Maas et al., 2013] instead of ReLU layers. There are no
pooling or linear layers in our network. More details can be found in the supplementary material.
Training details A straightforward training procedure is to update both the generator and the
discriminator in every batch. However, the discriminator usually learns much faster than the generator,
possibly because generating objects in a 3D voxel space is more difficult than differentiating between
real and synthetic objects [Goodfellow et al., 2014, Radford et al., 2016]. It then becomes hard
for the generator to extract signals for improvement from a discriminator that is way ahead, as all
examples it generated would be correctly identified as synthetic with high confidence. Therefore,
to keep the training of both networks in pace, we employ an adaptive training strategy: for each
batch, the discriminator only gets updated if its accuracy in the last batch is not higher than 80%. We
observe this helps to stabilize the training and to produce better results. We set the learning rate of
G to 0.0025, D to 10?5 , and use a batch size of 100. We use ADAM [Kingma and Ba, 2015] for
optimization, with ? = 0.5.
3.2 3D-VAE-GAN
We have discussed how to generate 3D objects by sampling a latent vector z and mapping it to the
object space. In practice, it would also be helpful to infer these latent vectors from observations. For
example, if there exists a mapping from a 2D image to the latent representation, we can then recover
the 3D object corresponding to that 2D image.
3
Following this idea, we introduce 3D-VAE-GAN as an extension to 3D-GAN. We add an additional
image encoder E, which takes a 2D image x as input and outputs the latent representation vector z.
This is inspired by VAE-GAN proposed by [Larsen et al., 2016], which combines VAE and GAN by
sharing the decoder of VAE with the generator of GAN.
The 3D-VAE-GAN therefore consists of three components: an image encoder E, a decoder (the
generator G in 3D-GAN), and a discriminator D. The image encoder consists of five spatial
convolution layers with kernel size {11, 5, 5, 5, 8} and strides {4, 2, 2, 2, 1}, respectively. There
are batch normalization and ReLU layers in between, and a sampler at the end to sample a 200
dimensional vector used by the 3D-GAN. The structures of the generator and the discriminator are
the same as those in Section 3.1.
Similar to VAE-GAN [Larsen et al., 2016], our loss function consists of three parts: an object
reconstruction loss Lrecon , a cross entropy loss L3D-GAN for 3D-GAN, and a KL divergence loss LKL
to restrict the distribution of the output of the encoder. Formally, these loss functions write as
L = L3D-GAN + ?1 LKL + ?2 Lrecon ,
(2)
where ?1 and ?2 are weights of the KL divergence loss and the reconstruction loss. We have
L3D-GAN = log D(x) + log(1 ? D(G(z))),
LKL = DKL (q(z|y) || p(z)),
Lrecon = ||G(E(y)) ? x||2 ,
(3)
(4)
(5)
where x is a 3D shape from the training set, y is its corresponding 2D image, and q(z|y) is the
variational distribution of the latent representation z. The KL-divergence pushes this variational
distribution towards to the prior distribution p(z), so that the generator can sample the latent representation z from the same distribution p(z). In this work, we choose p(z) a multivariate Gaussian
distribution with zero-mean and unit variance. For more details, please refer to Larsen et al. [2016].
Training 3D-VAE-GAN requires both 2D images and their corresponding 3D models. We render 3D
shapes in front of background images (16, 913 indoor images from the SUN database [Xiao et al.,
2010]) in 72 views (from 24 angles and 3 elevations). We set ?1 = 5, ?2 = 10?4 , and use a similar
training strategy as in Section 3.1. See our supplementary material for more details.
4
Evaluation
In this section, we evaluate our framework from various aspects. We first show qualitative results
of generated 3D objects. We then evaluate the unsupervisedly learned representation from the
discriminator by using them as features for 3D object classification. We show both qualitative and
quantitative results on the popular benchmark ModelNet [Wu et al., 2015]. Further, we evaluate
our 3D-VAE-GAN on 3D object reconstruction from a single image, and show both qualitative and
quantitative results on the IKEA dataset [Lim et al., 2013].
4.1 3D Object Generation
Figure 2 shows 3D objects generated by our 3D-GAN. For this experiment, we train one 3D-GAN
for each object category. For generation, we sample 200-dimensional vectors following an i.i.d.
uniform distribution over [0, 1], and render the largest connected component of each generated object.
We compare 3D-GAN with Wu et al. [2015], the state-of-the-art in 3D object synthesis from a
probabilistic space, and with a volumetric autoencoder, whose variants have been employed by
multiple recent methods [Girdhar et al., 2016, Sharma et al., 2016]. Because an autoencoder does not
restrict the distribution of its latent representation, we compute the empirical distribution p0 (z) of the
latent vector z of all training examples, fit a Gaussian distribution g0 to p0 , and sample from g0 . Our
algorithm produces 3D objects with much higher quality and more fine-grained details.
Compared with previous works, our 3D-GAN can synthesize high-resolution 3D objects with detailed
geometries. Figure 3 shows both high-res voxels and down-sampled low-res voxels for comparison.
Note that it is relatively easy to synthesize a low-res object, but is much harder to obtain a high-res
one due to the rapid growth of 3D space. However, object details are only revealed in high resolution.
A natural concern to our generative model is whether it is simply memorizing objects from training
data. To demonstrate that the network can generalize beyond the training set, we compare synthesized
objects with their nearest neighbor in the training set. Since the retrieval objects based on `2 distance in
the voxel space are visually very different from the queries, we use the output of the last convolutional
4
Our results (64 ? 64 ? 64)
NN
Gun
Chair
Car
Sofa
Table
Objects generated by Wu et al. [2015] (30 ? 30 ? 30)
Car
Table
Objects generated by a volumetric autoencoder (64 ? 64 ? 64)
Chair
Table
Sofa
Figure 2: Objects generated by 3D-GAN from vectors, without a reference image/object. We show,
for the last two objects in each row, the nearest neighbor retrieved from the training set. We see that
the generated objects are similar, but not identical, to examples in the training set. For comparison,
we show objects generated by the previous state-of-the-art [Wu et al., 2015] (results supplied by the
authors). We also show objects generated by autoencoders trained on a single object category, with
latent vectors sampled from empirical distribution. See text for details.
High-res Low-res
High-res Low-res
High-res Low-res
High-res Low-res
Figure 3: We present each object at high resolution (64 ? 64 ? 64) on the left and at low resolution
(down-sampled to 16 ? 16 ? 16) on the right. While humans can perceive object structure at a
relatively low resolution, fine details and variations only appear in high-res objects.
layer in our discriminator (with a 2x pooling) as features for retrieval instead. Figure 2 shows that
generated objects are similar, but not identical, to the nearest examples in the training set.
4.2 3D Object Classification
We then evaluate the representations learned by our discriminator. A typical way of evaluating
representations learned without supervision is to use them as features for classification. To obtain
features for an input 3D object, we concatenate the responses of the second, third, and fourth
convolution layers in the discriminator, and apply max pooling of kernel sizes {8, 4, 2}, respectively.
We use a linear SVM for classification.
Data We train a single 3D-GAN on the seven major object categories (chairs, sofas, tables, boats,
airplanes, rifles, and cars) of ShapeNet [Chang et al., 2015]. We use ModelNet [Wu et al., 2015] for
testing, following Sharma et al. [2016], Maturana and Scherer [2015], Qi et al. [2016].? Specifically,
we evaluate our model on both ModelNet10 and ModelNet40, two subsets of ModelNet that are often
?
For ModelNet, there are two train/test splits typically used. Qi et al. [2016], Shi et al. [2015], Maturana
and Scherer [2015] used the train/test split included in the dataset, which we also follow; Wu et al. [2015], Su
5
Supervision
Pretraining
ModelNet40
ModelNet10
ImageNet
MVCNN [Su et al., 2015a]
MVCNN-MultiRes [Qi et al., 2016]
90.1%
91.4%
-
None
3D ShapeNets [Wu et al., 2015]
DeepPano [Shi et al., 2015]
VoxNet [Maturana and Scherer, 2015]
ORION [Sedaghat et al., 2016]
77.3%
77.6%
83.0%
-
83.5%
85.5%
92.0%
93.8%
-
SPH [Kazhdan et al., 2003]
LFD [Chen et al., 2003]
T-L Network [Girdhar et al., 2016]
VConv-DAE [Sharma et al., 2016]
3D-GAN (ours)
68.2%
75.5%
74.4%
75.5%
83.3%
79.8%
79.9%
80.5%
91.0%
Category labels
Unsupervised
Classification (Accuracy)
Method
Table 1: Classification results on the ModelNet dataset. Our 3D-GAN outperforms other unsupervised
learning methods by a large margin, and is comparable to some recent supervised learning frameworks.
Accuracy (%)
85
80
Figure 5: The effects of individual dimensions of the object vector
75
70
3D-GAN
VoxNet
VConv-DAE
65
10
20
40
80
160
full
# objects per class in training
Figure 4: ModelNet40 classification with limited training data
Figure 6: Intra/inter-class interpolation between object vectors
used as benchmarks for 3D object classification. Note that the training and test categories are not
identical, which also shows the out-of-category generalization power of our 3D-GAN.
Results We compare with the state-of-the-art methods [Wu et al., 2015, Girdhar et al., 2016,
Sharma et al., 2016, Sedaghat et al., 2016] and show per-class accuracy in Table 1. Our representation
outperforms other features learned without supervision by a large margin (83.3% vs. 75.5% on
ModelNet40, and 91.0% vs 80.5% on ModelNet10) [Girdhar et al., 2016, Sharma et al., 2016].
Further, our classification accuracy is also higher than some recent supervised methods [Shi et al.,
2015], and is close to the state-of-the-art voxel-based supervised learning approaches [Maturana and
Scherer, 2015, Sedaghat et al., 2016]. Multi-view CNNs [Su et al., 2015a, Qi et al., 2016] outperform
us, though their methods are designed for classification, and require rendered multi-view images and
an ImageNet-pretrained model.
3D-GAN also works well with limited training data. As shown in Figure 4, with roughly 25
training samples per class, 3D-GAN achieves comparable performance on ModelNet40 with other
unsupervised learning methods trained with at least 80 samples per class.
4.3 Single Image 3D Reconstruction
As an application, our show that the 3D-VAE-GAN can perform well on single image 3D reconstruction. Following previous work [Girdhar et al., 2016], we test it on the IKEA dataset [Lim et al., 2013],
and show both qualitative and quantitative results.
Data The IKEA dataset consists of images with IKEA objects. We crop the images so that the
objects are centered in the images. Our test set consists of 1, 039 objects cropped from 759 images
(supplied by the author). The IKEA dataset is challenging because all images are captured in the
wild, often with heavy occlusions. We test on all six categories of objects: bed, bookcase, chair, desk,
sofa, and table.
Results We show our results in Figure 7 and Table 2, with performance of a single 3D-VAE-GAN
jointly trained on all six categories, as well as the results of six 3D-VAE-GANs separately trained on
et al. [2015a], Sharma et al. [2016] used 80 training points and 20 test points in each category for experiments,
possibly with viewpoint augmentation.
6
Method
Bed
Bookcase
Chair
Desk
Sofa
Table
Mean
AlexNet-fc8 [Girdhar et al., 2016]
AlexNet-conv4 [Girdhar et al., 2016]
T-L Network [Girdhar et al., 2016]
29.5
38.2
56.3
17.3
26.6
30.2
20.4
31.4
32.9
19.7
26.6
25.8
38.8
69.3
71.7
16.0
19.1
23.3
23.6
35.2
40.0
3D-VAE-GAN (jointly trained)
3D-VAE-GAN (separately trained)
49.1
63.2
31.9
46.3
42.6
47.2
34.8
40.7
79.8
78.8
33.1
42.3
45.2
53.1
Table 2: Average precision for voxel prediction on the IKEA dataset.?
Figure 7: Qualitative results of single image 3D reconstruction on the IKEA dataset
each class. Following Girdhar et al. [2016], we evaluate results at resolution 20 ? 20 ? 20, use the
average precision as our evaluation metric, and attempt to align each prediction with the ground-truth
over permutations, flips, and translational alignments (up to 10%), as IKEA ground truth objects
are not in a canonical viewpoint. In all categories, our model consistently outperforms previous
state-of-the-art in voxel-level prediction and other baseline methods.?
5
Analyzing Learned Representations
In this section, we look deep into the representations learned by both the generator and the discriminator of 3D-GAN. We start with the 200-dimensional object vector, from which the generator produces
various objects. We then visualize neurons in the discriminator, and demonstrate that these units
capture informative semantic knowledge of the objects, which justifies its good performance on
object classification presented in Section 4.
5.1 The Generative Representation
We explore three methods for understanding the latent space of vectors for object generation. We
first visualize what an individual dimension of the vector represents; we then explore the possibility
of interpolating between two object vectors and observe how the generated objects change; last, we
present how we can apply shape arithmetic in the latent space.
Visualizing the object vector To visualize the semantic meaning of each dimension, we gradually
increase its value, and observe how it affects the generated 3D object. In Figure 5, each column
corresponds to one dimension of the object vector, where the red region marks the voxels affected by
changing values of that dimension. We observe that some dimensions in the object vector carries
semantic knowledge of the object, e.g., the thickness or width of surfaces.
Interpolation We show results of interpolating between two object vectors in Figure 6. Earlier
works demonstrated interpolation between two 2D images of the same category [Dosovitskiy et al.,
2015, Radford et al., 2016]. Here we show interpolations both within and across object categories.
We observe that for both cases walking over the latent space gives smooth transitions between objects.
Arithmetic Another way of exploring the learned representations is to show arithmetic in the latent
space. Previously, Dosovitskiy et al. [2015], Radford et al. [2016] presented that their generative
nets are able to encode semantic knowledge of chair or face images in its latent space; Girdhar et al.
[2016] also showed that the learned representation for 3D objects behave similarly. We show our
shape arithmetic in Figure 8. Different from Girdhar et al. [2016], all of our objects are randomly
sampled, requiring no existing 3D CAD models as input.
5.2 The Discriminative Representation
We now visualize the neurons in the discriminator. Specifically, we would like to show what input
objects, and which part of them produce the highest intensity values for each neuron. To do that,
?
For methods from Girdhar et al. [2016], the mean values in the last column are higher than the originals in
their paper, because we compute per-class accuracy instead of per-instance accuracy.
7
Figure 8: Shape arithmetic for chairs and tables. The left images show the obtained ?arm? vector can
be added to other chairs, and the right ones show the ?layer? vector can be added to other tables.
Figure 9: Objects and parts that activate specific neurons in the discriminator. For each neuron, we
show five objects that activate it most strongly, with colors representing gradients of activations with
respect to input voxels.
for each neuron in the second to last convolutional layer of the discriminator, we iterate through
all training objects and exhibit the ones activating the unit most strongly. We further use guided
back-propagation [Springenberg et al., 2015] to visualize the parts that produce the activation.
Figure 9 shows the results. There are two main observations: first, for a single neuron, the objects
producing strongest activations have very similar shapes, showing the neuron is selective in terms of
the overall object shape; second, the parts that activate the neuron, shown in red, are consistent across
these objects, indicating the neuron is also learning semantic knowledge about object parts.
6
Conclusion
In this paper, we proposed 3D-GAN for 3D object generation, as well as 3D-VAE-GAN for learning
an image to 3D model mapping. We demonstrated that our models are able to generate novel objects
and to reconstruct 3D objects from images. We showed that the discriminator in GAN, learned
without supervision, can be used as an informative feature representation for 3D objects, achieving
impressive performance on shape classification. We also explored the latent space of object vectors,
and presented results on object interpolation, shape arithmetic, and neuron visualization.
Acknowledgement This work is supported by NSF grants #1212849 and #1447476, ONR MURI
N00014-16-1-2007, the Center for Brain, Minds and Machines (NSF STC award CCF-1231216),
Toyota Research Institute, Adobe, Shell, IARPA MICrONS, and a hardware donation from Nvidia.
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5,632 | 6,097 | Learning Sparse Gaussian Graphical Models with
Overlapping Blocks
1
Mohammad Javad Hosseini1
Su-In Lee1,2
Department of Computer Science & Engineering, University of Washington, Seattle
2
Department of Genome Sciences, University of Washington, Seattle
{hosseini, suinlee}@cs.washington.edu
Abstract
We present a novel framework, called GRAB (GRaphical models with overlApping
Blocks), to capture densely connected components in a network estimate. GRAB
takes as input a data matrix of p variables and n samples and jointly learns both
a network of the p variables and densely connected groups of variables (called
?blocks?). GRAB has four major novelties as compared to existing network estimation methods: 1) It does not require blocks to be given a priori. 2) Blocks
can overlap. 3) It can jointly learn a network structure and overlapping blocks. 4)
It solves a joint optimization problem with the block coordinate descent method
that is convex in each step. We show that GRAB reveals the underlying network
structure substantially better than four state-of-the-art competitors on synthetic data.
When applied to cancer gene expression data, GRAB outperforms its competitors
in revealing known functional gene sets and potentially novel cancer driver genes.
1
Introduction
Many real-world networks contain subsets of variables densely connected to one another, a property
called modularity (Fig 1A); however, standard network inference methods do not incorporate this
property. As an example, biologists are increasingly interested in understanding how thousands of
genes interact with each other on the basis of gene expression data that measure expression levels of
p genes across n samples. This has stimulated considerable research into the structure estimation
of a network from high-dimensional data (p
n). It is well-known that the network structure
corresponds to the non-zero pattern of the inverse covariance matrix, ? 1 [1]. Thus, obtaining a
sparse estimate of ? 1 by using `1 penalty has been a standard approach to inferring a network, a
method called graphical lasso [2]. However, applying an `1 penalty to each edge fails to reflect the
fact that genes involved in similar functions are more likely to be connected with each other and that
how genes are organized into functional modules are often not known.
We present a novel structural prior, called GRAB prior, which encourages the network estimate to
be dense within a block (i.e, a subset of variables) and sparse between blocks, where blocks are not
given a priori. Fig 1B illustrates the effectiveness of the GRAB prior (bottom) in a high-dimensional
setting (p = 200 and n = 100), where it is difficult to reveal the true underlying network by using
the graphical lasso (GLasso) (top). The major novelty of GRAB is four-fold:
First, unlike previous work [3, 4, 5], GRAB allows each variable to belong to more than one block,
which is an important property of many real-world networks. For example, genes important in disease
processes are often involved in multiple functional modules [6], and identifying such genes would be
of great scientific interest (Section 4.2). Although existing methods to learn non-overlapping blocks
allow edges between different blocks, they use stronger regularization parameters for between-block
edges, which decreases the power to detect variables associated with multiple blocks.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Second, GRAB jointly learns the network structure and the assignment of variables into overlapping
blocks (Fig 2). Existing methods to incorporate blocks in network learning either use blocks given a
priori or use a sequential approach to learn blocks and then learn a network given the blocks held
fixed. Interestingly, the GRAB algorithm can be viewed as a generalization of the joint learning of the
distance metric among p variables and graph-cut clustering of p variables into blocks (Section 3.4)
Third, GRAB solves a joint optimization problem with the block coordinate descent method that is
convex in each step. This is a powerful feature that is difficult to be achieved by existing methods
to cluster variables into blocks. This property guarantees the convergence of the learning algorithm
(Section 3).
Finally, the GRAB framework we presented in this paper uses the Gaussian graphical model as a
baseline model. However, the GRAB prior, formulated as tr ZZ| |?| (Section 2.2), can be used in
any kind of network models such as pairwise Markov random fields.
In the following sections, we show that GRAB outperforms the graphical lasso [2] and existing
methods to learn blocks and network estimates [3, 4] on synthetic data and cancer gene expression
data. We also demonstrate GRAB?s potential to identify novel genes that drive cancer.
(B)$
x4
x5
Z:#assignment#matrix#
x2
x3
x1
Zi#
x7
(B)#
p#
!#
x6
ZT #
K#
(A)#
K#
p#
ZjT# (ZZT)ij#
x8
ZZT#
?1#
p#
?2#
Learning#Z##
Z#
(C)#
ZT #
K#
p#
SBM#
Learning#?##
(D)# ?#
?w#
p#
?b#
!#
x1
x2
x3
x4
x5
x6
x7
x8
Block#2#
Block#3#
?.step#
x1
x2
x3
x4
x5
x6
x7
x8
(E)#GRAB#
p#
Block#1#
The#similarity#between#ith#
and#jth#variables##
Z.step#
(A)$
K#
p#
p#
Sparsity#paBern#
encouraged#by#ZZT#
?1,#?2:#based#on#Z#
?w:#within?block#
?b:#between?block#
Figure 1: (A) A network with overlapping
blocks (top) and its adjacency matrix (bot- Figure 2: The GRAB framework ? an iterative
tom). (B) Network estimates of GLasso (top) algorithm that jointly learns ? and Z.
and GRAB (bottom) in a toy example.
2
2.1
GGM with Overlapping Blocks
Background: High-Dimensional Gaussian Graphical Model (GGM)
We aim to learn a GGM of p variables on the basis of n observations (p
n). That is, suppose that
X(1) , . . . , X(n) are i.i.d. N (?, ?), where ? 2 Rp and ? is a p ? p positive definite matrix. It is well
known that the sparsity pattern of ? 1 determines the conditional independence structure of the p
variables; there is an edge between the ith and jth variables if and only if the (i, j) element of ? 1 is
non-zero [1]. A number of authors have proposed to estimate ? 1 using the graphical lasso [2, 7, 8]:
maximize log det ?
tr(S?)
??0
k?k1 ,
(1)
b is an estimate of ? 1 , S denotes the empirical covariance matrix, and is a
where the solution ?
nonnegative tuning parameter that controls the strength of the `1 penalty applied to the elements of
?. This amounts to maximizing a penalized log-likelihood.
2
2.2
GGM with the Overlapping Block Prior
Here, we present the GRAB prior, formulated as tr ZZ| |?| , that encourages ? to have overlapping
blocks. Let X = {X1 , . . . , Xp } be variables in the network and Z be a real matrix of size p ? K,
where K is the total number of blocks. Each element 1 ? Zik ? 1 can be interpreted as a score
representing how likely the ith variable Xi belongs to the kth block Bk . The ith row of Z, denoted
by Zi , can be interpreted as a low-rank embedding for the variable Xi showing its block assignment
scores. Then, the (i, j) element (ZZ| )ij = ?K
k=1 Zik Zjk (the dot product of Zi and Zj ) represents
the similarity between variables Xi and Xj in their embeddings.
To more clearly understand the impact of the GRAB prior on the sparsity structure of ?, let us
assume a hard assignment model in which we assign variables to blocks. Then, Z becomes a binary
matrix and the sparsity pattern of ZZ| would indicate the region covered by all K blocks (Fig 2A-B).
Then, jointly learning Z and ? to increase ?i,j (ZZ| )ij |?ij | would encourage ? to have a sparsity
structure imposed by (ZZ| ). In the continuous case, it would encourage |?ij | to be non-zero when
Xi and Xj have similar embeddings (i.e., a dot product of Zi and Zj is large).
Incorporating the GRAB prior into Eq (1) as a structural prior leads to:
?
?
maximize
log det ? tr(S?)
k?k1 tr ZZ| |?| ,
??0,Z2D
where
is a non-negative tuning parameter. We can re-write Eq (2) as:
?
X ?
maximize log det ? tr(S?)
1 (ZZ| )ij |?ij |.
??0,Z2D
i,j
(2)
(3)
We use the value of the sparsity tuning parameter 1 (ZZ| )ij for each (i, j) element ?ij . A
network edge that corresponds to two variables with similar embeddings would be penalized less.
p?K
The set D ? [ 1, 1]
contains matrices Z satisfying the following constraints: (a) kZi k2 ? 1,
where Zi denotes the ith row of Z. This constraint ensures the regularization parameters of all (i, j)
pairs of variables are non-negative. (b) kZkF ? . In addition to the variable specific constraint
on each Zi in (a), we need a global constraint on Z to prevent all regularization parameters from
becoming zero (8i, j : (ZZ T )ij = 1). (c) kZk2 ? ? , where k.k2 of a matrix is its maximum singular
value. This constraint prevents the case where all variables are assigned to one block.
There are two hyperparameters, and ? ; however we describe below that we set ? = pK and that
has an effect to guarantee
p that there are at least K non-empty blocks. In our experiments, we set the
hyper-parameter = p2 , which, intuitively, would allow each variable to get on average half of
Pp
its largest possible squared norm. Given that kZk2F = i=1 i2 where i is the ith singular value
Pp
of Z, from the constraint (b), i=1 i2 ? 2 . We set ? = pK , where ? means the upper bound
of the maximum singular value, given the constraint (c). This means that there would be at least
K non-empty blocks given that the constraint (b) is tight. We show in Section 3 that this choice of
hyperparameters makes our learning algorithm simpler (see Lemma 3.2).
2.3
Probabilistic Interpretation
The joint distribution over X, ? and Z is as: P (X, ?, Z) = P (X|?)P (?|Z)P (Z). The first two
terms, log det(?) trace(S?), in Eq (3) correspond to log P (X|?), the log-likelihood of GGM
given a particular
parameter ? (i.e., an estimate of ? 1 ), as described in Section 2.1. For ? ? 0,
Q
P (?|Z) = P (?ij |Z), where P (?ij |Z) represents a conditional probability over ?ij given the
block assignment scores of Xi and Xj . We useQthe Laplacian prior with the sparsity parameter value
1
(1 (ZZ| )ij ). For ? ? 0, P (?|Z) is: D
(ZZ| )ij ))|?ij |), where D is the
(i,j) exp( ( (1
normalization constant. The prior probability P (Z) is proportional to D.
2.4
Related Work
To our knowledge, GRAB is the first attempt to jointly learn the overlapping blocks and the structure
of a conditional dependence network such as a GGM. Related work consists of 3 categories:
3
1) Learning blocks with a network held fixed: This category includes (a) stochastic block model
(SBM) [9], (b) spectral clustering [10], and (c) a screening rule to identify non-overlapping blocks
based on the empirical covariance matrix [11].
2) Learning a network with blocks given a priori and held fixed: This category includes a) a method
to solve graphical lasso with group `1 penalty to encourage group sparsity of edges within pairs of
blocks [12], and b) an efficient learning algorithm for GGMs given a set of overlapping blocks [13].
3) Learning non-overlapping blocks first and then the network given the blocks: (a) Marlin et
al. (2009) extend the prior work [12] to identify non-overlapping blocks which are then used to
learn a network [3]. (b) Another method assigns each variable to one block, and use different
regularization parameters for within-block and between-block edges [14]. (c) Tan et al. (2015)
propose to use hierarchical clustering (complete-linkage and average-linkage) to cluster variables
into non-overlapping blocks, and apply graphical lasso to each block [4].
3
GRAB Learning Algorithm
3.1
Overview
Our learning algorithm jointly learns the block assignment scores Z and the network estimate ? by
solving Eq (2). We adopt the block coordinate descent (BCD) method to iteratively learn Z and ?. Our
learning algorithm essentially performs adaptive distance (similarity) metric learning and clustering
of variables into blocks simultaneously (Section 3.4). Given the current assignment of variables into
blocks, Z, we learn a network among variables, ?. Then, |?| is used as a similarity matrix among
variables to update the assignment of variables to blocks, Z. We iterate until convergence.
Convergence is theoretically guaranteed. Since our objective function is continuous on a compact
level set, based on Theorem 4.1 in [15], the solution sequence of our method is defined and bounded.
Every coordinate block found by the ?-step and Z-step is a stationary point of GRAB. We indeed
observed the value of the objective function monotonically increases until convergence.
In the following, we show that the BCD method will be convex in each step. We first re-write Eq (2)
with all the constraints explicitly:
maximize log det ?
??0,Z
?
tr(S?)
k?k1
tr ZZ| |?|
?
subject to kZk2 ? ?, kZi k2 ? 1, kZkF ? , (i 2 {1, . . . p})).
Now, we state the following lemma, the proof of which can be found in the Appendix.
Lemma 3.1 Eq (4) is equivalent to the following:
maximize log det ?
??0,W?0
tr(S?)
subject to rank(W) ? K, W
?
k?k1
tr W|?|
?
? 2 I, diag(W) ? 1, tr(W) ?
(4)
(5)
2
,
where W is a p ? p matrix, K means the number of blocks, and I is the identity matrix of size p.1
Corollary 3.1.1 Suppose
that (?? , W? ) is the optimal solution of the optimization problem (5).
p
?
?
Then, ? , Z = U D is the optimal solution of problem 4, where U 2 Rp?K is a matrix with
columns containing K eigenvectors of W corresponding to the largest eigenvalues and D is a
diagonal matrix of the corresponding eigenvalues.
3.2
Learning ? (?-step)
To estimate ? given Z, based on Eq (3), we solve the following problem:
P
maximize
log det ? tr(S?)
(i,j) ?ij |?ij |,
??0
|
(ZZ )ij ).
(6)
where ?ij = (1
This is the graphical lasso with edge-specific regularization parameters
?ij . Eq (6) is a convex problem and we solve it by adopting a standard solver for graphical lasso [16].
1
In this paper, we assume diag is an operator that maps a vector to a diagonal matrix with the vector as its
diagonal, and maps a matrix to a vector containing its diagonal.
4
3.3
Learning Z (Z-step)
Here we describe how to learn Z given ?. Instead of solving (4), we solve (5) because (5) is a
convex optimization problem with respect to W. Interestingly, we can remove the rank constraint,
rank(W) ? K; in Lemma 3.2, we show that with the choice of ? = pK , the rank constraint is
automatically satisfied. This leads to the following optimization problem:
maximize tr W|?|
W?0
(7)
subject to W ? 2 I, diag(W) ? 1, tr(W) ? 2 .
This W-step is a semi-definite programming problem. We solve the dual of Eq (7) that leads to an
efficient optimization problem.2 We introduce three dual variables: 1) a matrix Y ? 0 for the `2
norm constraint, 2) a vector v 2 Rp+ for the constraints on the diagonal and 3) a scalar y 0 for the
constraint on trace. The Lagrangian is:
L(W, Y, v , y) = tr W|?| + tr (? 2 I W)Y + y( 2 tr(W)) + v T (1 diag(W)). (8)
The dual function is as:
sup tr W|?| + tr (? 2 I W)Y + y( 2 tr W)) + v | (1 diag(W))
W?0
= sup tr W(|?|
Y
yI
diag(v)) + ? 2 tr(Y ) + y + v | 1
W?0
?
? tr(Y ) + y + v 1 if Y ? |?| yI
+1
otherwise
consequently, we get the following dual problem for Eq (7):
minimize ? 2 tr(Y ) + y 2 + v | 1
=
2
T
diag(v)
(9)
.
Y,y,v
(10)
subject to Y ? |?| yI diag(v) , Y ? 0, y 0, v 0.
Eq (10) has a closed form solution in Y and y given that v is fixed. The dual problem boils down to:
K
X
(11)
minimize g(v) = minimize ? 2
C +,i + v | 1,
v 0
v 0
i=1
p
where we have replaced ? 2 with K (because ? = / K). We define C = (|?| diag(v)) and
assume it has eigenvalues ( 1 , . . . p ) in descending order and (C)+,i = max(0, i ). We solve
Eq (11) by projected subgradient descent method where the subgradient direction is:
(12)
rv g(v) = ? 2 diag UC 1K (DC )UC| + 1.
DC is the diagonal matrix of eigenvalues in descending order and UC is the matrix containing
orthonormal eigenvectors of C as its columns. We define 1K (DC ) as a binary vector of size p with
jth element equal to 1 if and only if j ? K and j > 0.
2
After finding the optimal v ? , the optimal solution W? can be obtained by:
W? = argmax tr W(|?| diag(v ? ))
(13)
subject to W ? 2 I, tr(W) ? 2 .
One can see that the solution of problem (13) is W ? = ? 2 UC ? 1 2 /? 2 (DC ? )UC| ? =
? 2 UC ? 1K (DC ? )UC| ? , where C ? , UC ? , DC ? and 1K (DC ? ) are defined similarly to (12). By definition, 1K (.) is a diagonal matrix with at most K nonzeros elements. Therefore, W ? will have rank at
most K, which means that we do not need the rank constraint on W. This leads to the following
lemma.
W?0
Lemma 3.2 If we set ? =
p
K
in (5), the constraint rank(W) ? K will be automatically satisfied.
p
Finally, we construct Z? = U D as instructed in corollary 3.1.1. Note that in the intermediate
iterations, we do not need to compute Z; we need to construct the matrix Z? to find the overlapping
blocks after the learning algorithm will converge3 .
2
The primal problem has a strictly feasible solution ?I, where ? is a small number and I is the identity matrix;
therefore strong duality holds.
3
The source code is available at: http://suinlee.cs.washington.edu/software/grab
5
3.4
A special case: K-way graph cut algorithm
Here, we show that GRAB algorithm generalizes the K-way graph cut algorithm in two ways: 1)
GRAB allows each variable to be in multiple blocks with soft membership; and 2) GRAB updates a
network structure ?, used as a similarity matrix, in each iteration. The proof is in the Appendix.
Lemma 3.3 Say that we use a binary matrix Z (hard assignment) with the following constraints: a)
For all variables i, kZi k2 ? 1, where Zi denotes the ith row of Z. b) For all blocks k, kZ k k2 >= 1,
where Z k denotes the kth column of Z. This means that each variable can belong to only one block
(i.e., non-overlapping blocks), and each block has at least one variable. Then GRAB is equivalent to
iterating between K-way graph-cut on |?| to find Z and solving graphical lasso problem to find ?.
4
Experimental Results
We present results on synthetically generated data and real data.
Comparison. Three state-of-the-art competitors are considered: UGL1 - unknown group `1 regularization [3]; CGL - cluster graphical lasso [4]; and GLasso - standard graphical lasso [2]. CGL
has two variants depending on the type of hierarchical clustering used: average linkage clustering
(CGL:ALC) and complete linkage clustering (CGL:CLC). Each method selects the regularization
parameter using the standard cross-validation (or held out validation) procedure.
CGL and UGL1 have their own ways of selecting the number of blocks K [4, 3]. GRAB selects
K based on the validation-set log-likelihood in initialization. We initialize GRAB by constructing
the Z matrix. We first perform spectral clustering on |S|, where S denotes the empirical covariance
matrix, then add overlap by assigning a random subset of variables to clusters with the highest
average correlation. Then, we project the Z matrix into the convex set defined in Section
2.2 and
p
form W = ZZ| . In the Z-step of the GRAB learning algorithm, we use step size 1/ t, where t is
the iteration number and iterate until the relative change in the objective function is less than 10 6
(Section 3.3). We use the warm-start technique between the BCD iterations.
Evaluation criteria. In the synthetic data experiments (sectoin 4.1), we evaluate each method based
on the learned network with the optimal regularization parameter chosen for each method based on
only training-set. For the AML dataset (Section 4.2), we evaluate the learned blocks for varying
regularization parameters (x-axis) to better illustrate the difference among the methods in terms of
their performances. In all experiments, we standardize the data and show the average results over 10
runs and the standard deviations as error bars.
4.1
Synthetic Data Experiments
Data generation. We first generate ? overlapping blocks forming a chain, a random tree or a lattice.
In each case, two neighboring blocks overlap each other by o (the ratio of the variables shared between
two overlapping blocks). Then, we randomly generate a true underlying network of p variables with
density of 20%, and convert it to the precision matrix following the procedure of [17]. We generate
100 training samples and 50 validation samples from the multivariate Gaussian distribution with
mean zero and the covariance matrix equal to the inverse of the precision matrix.
We consider a varying number of true blocks ? 2 {9, 25, 49} and overlap ratio o = .25. For
? = 25, we consider o 2 {.1, .25, .4}. We vary the number of variables p 2 {400, 800} for the
lattice-structured blocks. The results on the chain and random tree blocks are similar and so we
provide only the results for p = 400 for these block structures. For all methods, we considered the
regularization parameter 2 [.02, .4] with step size .02.
Results. Fig 3 compares five methods when a regularization parameter was selected for each method
based on the 50 validation samples. Each of the four plots correspond to different block structure or
number of variables. Each bar group corresponds to a particular (?, o, ?), in which we computed the
modularity measure ? as (fraction of edges that fall within groups - expected fraction if edges were
distributed at random), as was done by [18]. Fig 3A shows how accurately each method recovers
the true network. For each method m, we compared the learned edges (EZ,m ) and that from the
underlying network (EZ ). By comparing EZ,m and EZ , we can compute the precision and recall
6
La/ce:'p=400,n=100'
?=9' ?=25' ?=25' ?=25' ?=49'
o=.25' o=.4' o=.25' o=.1' o=.25'
?=0.85'?=0.88' ?=0.91' ?=0.96' ?=0.93'
'
Chain:'p=400,n=100'
La/ce:'p=800,n=100'
?=9' ?=25' ?=25' ?=25' ?=49'
o=.25' o=.4' o=.25' o=.1' o=.25'
?=0.85' ?=0.88' ?=0.91' ?=0.96' ?=0.93'
'
?=10' ?=30' ?=30' ?=30' ?=50'
o=.25' o=.4' o=.25' o=.1' o=.25'
?=0.85'?=0.94' ?=0.95' ?=0.96' ?=0.97'
'
Random:'p=400,n=100'
?=10' ?=30' ?=30' ?=30' ?=50'
o=.25' o=.4' o=.25' o=.1' o=.25'
?=0.87' ?=0.93' ?=0.96' ?=0.97' ?=0.96'
'
Figure 3: Comparison based on average network recovery F1 on synthetic data from lattice blocks,
when p = 400 (first panel) and p = 800 (second panel), chain blocks (third panel) and random
blocks (fourth panel) when p = 400. Each bar group corresponds to a particular (number of blocks ?,
overlap ratio o, modularity ?).
of network recovery. Since it is not enough to get only high precision or recall, we use the F1 (or
pr?rec
F-measure) = 2 pr+rec
as an evaluation metric.
A number of authors have shown that identifying the underlying network structure is very challenging
in the high-dimensional setting, resulting in low accuracies even on synthetic data [14, 19, 4]. Our
results also show that the F1 scores for network are lower than 0.40. Despite that, GRAB identifies
network edges much more accurately than its competitors.
4.2
Cancer Gene Expression Data
We consider the MILE data [20] that measure the mRNA expression levels of 16,853 genes in 541
patients with acute myeloid leukemia (AML), an aggressive blood cancer. For a better visualization
of the network in limited space (Fig 5), we selected 500 genes4 , consisting of 488 highest varying
genes in MILE and 12 genes highly associated with AML: FLT3, NPM1, CEBPA, KIT, N-RAS,
MLL, WT1, IDH1/2, TET2, DNMT3A, and ASXL1. These genes are identified by [21] in a large
study on 1,185 patients with AML to be significantly mutated in these AML patients. These genes
are well-known to have significant role in driving AML.
Here, we evaluate GRAB and the other methods qualitatively in terms of how useful each method is
for cancer biologists to make discovery from data. For that, we fix the number of blocks to be K = 10
across all methods such that we get average of over 50 variables per block, which is considered close
to the average number of genes in known pathways [22]. We varied K and obtained similar results.
Genes in the same block are likely to share similar functions. Statistical significance of the overlap
between gene clusters (here, blocks) and known functional gene sets have been widely used as an
evaluation criteria [23, 5]. We show how to obtain blocks from the learned Z.
Obtaining blocks from Z. After the GRAB algorithm converges, we obtain a network estimate ?
and a block membership matrix Z. We find K overlapping blocks satisfying two constraints: a)
maximum number of assignments is C; and b) each variable is assigned to 1 block. Here, we
used C = 1.3p. We perform the following greedy procedure: 1) We first run k-means clustering
5
algorithm on
Pthe p rows of the matrix Z. . 2) We compute the similarity of variables i to blocks
Bk as |B1k | j2Bk (ZZ| )ij , where |Bk | is the number of variables in Bk . Then, we add overlap by
assigning C p variables to blocks with highest similarity.
To evaluate the blocks, we used 4,722 curated gene sets from the molecular signature database [24]
and computed a p-value to measure the significance of the overlap between each block and each
gene set. We consider the (block, gene set) pairs with false discovery rate (FDR)-corrected p < 0.05
to be significantly overlapping pairs. When a block is significantly overlapped with a gene set,
we consider the gene set to be revealed by the corresponding block. We compare GRAB with the
4
GRAB runs for 0.5-1.5 hours for 500 genes and up to 20 hours for 2,000 genes on a computer with 2.5 GHz
Intel Core i5 processor
5
This resembles spectral clustering (equivalently, kmeans on eigenvectors of Laplacian matrix)
7
methods introduced in section 4.1. Since we only need the blocks for this experiment, we added
two more competitors: k-means and spectral clustering methods applied to |S|, where S denotes the
empirical covariance matrix. Fig 4 shows the number of gene sets that are revealed by any block
(FDR-corrected p < 0.05) in each method. GRAB significantly outperforms, which indicates the
importance of learning overlapping blocks; GRAB?s overlapping blocks reveal known functional
organization of genes better than other methods. Fig 4 shows the average results of 10 random
initializations.
Fig 5 compares the learned networks ? by GLasso (A) and GRAB (B) when the regularization
parameters are set such that the networks show a similar level of sparsity. For GRAB, we removed
the between-block edges and reordered genes such that the genes in the same blocks tend to appear
next to each other. GRAB shows more interpretable network structure, highlighting the genes that
belong to multiple blocks.
The key innovation of GRAB is to allow for overlap between blocks. Interestingly, the 12 well-known
AML genes are significantly enriched for the genes assigned to 3 or more blocks: FLT3, NPM1, TET2
and DNMT3A belong to 3 blocks while there are only 24 such genes out of 500 genes (p-value: 0.001)
(Fig 5B). This supports our claim that variables assigned to multiple blocks are likely important. Out
of the 24 genes assigned to 3 blocks, 12 are known to be involved in myeloid differentiation (the
process impaired in AML) or other types of cancer. This can lead to new discovery on the genes that
drive AML.
These genes include CCNA1 that has shown to be significantly differentially expressed in AML
(A)$
(B)$ of some (C)$
patients [25]. TSPAN7 is expressed in acute myelocytic
leukemia
patients6 . Several genes
are associated with other types of cancer. For example, CCL20 is associated with pancreatic cancer
[26]. ELOVL7 is involved in prostate cancer growth [27]. SCRN1 is a novel marker for prognosis
in colorectal cancer [28]. These genes assigned to many blocks and have been implicated in other
cancers or leukemias can lead the discovery of novel AML driver genes.
(A)$
(B)$
NPM1$
DNMT3A$
TET2$
FLT3$
Figure 5: Learned networks of: (A) GLasso and
(B)
GRAB. For GRAB, we have sorted the genes
Figure 4: Average number of gene sets highly
based
on the blocks and highlighted the following
associated with blocks at a varying regulariza4
genes
(out of the 12 highly associated genes with
tion parameter. The cross-validation results
AML)
that
belong to many blocks: NPM1, FLT3,
are consistent with these results.
DNMT3A and TET2.
5
Discussion and Future Work
We present a novel general framework, called GRAB, that can explicitly model densely connected
network components that can overlap with each other in a graphical model. The novel GRAB
structural prior encourages the network estimate to be dense within each block (i.e., a densely
connected group of variables) and sparse between the variables in different blocks. The GRAB
learning algorithm adopts BCD and is convex in each step. We demonstrate the effectiveness of our
framework in synthetic data and cancer gene expression dataset. Our framework is general and can
be applied to other kinds of graphical models, such as pairwise Markov random fields.
Acknowledgements: We give warm thanks to Reza Eghbali and Amin Jalali for many useful
discussions. This work was supported by the National Science Foundation grant DBI-1355899 and
the American Cancer Society Research Scholar Award 127332-RSG-15-097-01-TBG.
6
http://www.genecards.org/cgi-bin/carddisp.pl?gene=TSPAN7
8
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9
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5,633 | 6,098 | Discriminative Gaifman Models
Mathias Niepert
NEC Labs Europe
Heidelberg, Germany
mathias.niepert@neclabs.eu
Abstract
We present discriminative Gaifman models, a novel family of relational machine
learning models. Gaifman models learn feature representations bottom up from
representations of locally connected and bounded-size regions of knowledge bases
(KBs). Considering local and bounded-size neighborhoods of knowledge bases
renders logical inference and learning tractable, mitigates the problem of overfitting, and facilitates weight sharing. Gaifman models sample neighborhoods
of knowledge bases so as to make the learned relational models more robust to
missing objects and relations which is a common situation in open-world KBs. We
present the core ideas of Gaifman models and apply them to large-scale relational
learning problems. We also discuss the ways in which Gaifman models relate to
some existing relational machine learning approaches.
1
Introduction
Knowledge bases are attracting considerable interest both from industry and academia [2, 6, 15, 10].
Instances of knowledge bases are the web graph, social and citation networks, and multi-relational
knowledge graphs such as Freebase [2] and YAGO [11]. Large knowledge bases motivate the
development of scalable machine learning models that can reason about objects as well as their
properties and relationships. Research in statistical relational learning (SRL) has focused on particular
formalisms such as Markov logic [22] and P ROB L OG [8] and is often concerned with improving the
efficiency of inference and learning [14, 28]. The scalability problems of these statistical relational
languages, however, remain an obstacle and have prevented a wider adoption. Another line of work
focuses on efficient relational machine learning models that perform well on a particular task such
as knowledge base completion and relation extraction. Examples are knowledge base factorization
and embedding approaches [5, 21, 23, 26] and random-walk based ML models [15, 10]. We aim to
advance the state of the art in relational machine learning by developing efficient models that learn
knowledge base embeddings that are effective for probabilistic query answering on the one hand, and
interpretable and widely applicable on the other.
Gaifman?s locality theorem [9] is a result in the area of finite model theory [16]. The Gaifman graph
of a knowledge base is the undirected graph whose nodes correspond to objects and in which two
nodes are connected if the corresponding objects co-occur as arguments of some relation. Gaifman?s
locality theorem states that every first-order sentence is equivalent to a Boolean combination of
sentences whose quantifiers range over local neighborhoods of the Gaifman graph. With this paper,
we aim to explore Gaifman locality from a machine learning perspective. If every first-order sentence
is equivalent to a Boolean combination of sentences whose quantifiers range over local neighborhoods
only, we ought to be able to develop models that learn effective representations from these local
neighborhoods. There is increasing evidence that learning representations that are built up from
local structures can be highly successful. Convolutional neural networks, for instance, learn features
over locally connected regions of images. The aim of this work is to investigate the effectiveness
and efficiency of machine learning models that perform learning and inference within and across
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
locally connected regions of knowledge bases. This is achieved by combining relational features that
are often used in statistical relatinal learning with novel ideas from the area of deep learning. The
following problem motivates Gaifman models.
Problem 1. Given a knowledge base (relational structure, mega-example, knowledge graph) or a
collection of knowledge bases, learn a relational machine learning model that supports complex
relational queries. The model learns a probability for each tuple in the query answer.
Note that this is a more general problem than knowledge base completion since it includes the
learning of a probability distribution for a complex relational query. The query corresponding to
knowledge base completion is r(x, y) for logical variables x and y, and relation r. The problem also
touches on the problem of open-world probabilistic KBs [7] since tuples whose prior probability is
zero will often have a non-zero probability in the query answer.
2
Background
We first review some important concepts and notation in first-order logic.
2.1
Relational First-order Logic
An atom r(t1 , ..., tn ) consists of predicate r of arity n followed by n arguments, which are either
elements from a finite domain D = {a, b, ...} or logical variables {x, y, ...}. We us the terms domain
element and object synonymously. A ground atom is an atom without logical variables. Formulas are
built from atoms using the usual Boolean connectives and existential and universal quantification. A
free variable in a first-order formula is a variable x not in the scope of a quantifier. We write ?(x, y)
to denote that x, y are free in ?, and free(?) to refer to the free variables of ?. A substitution
replaces all occurrences of logical variable x by t in some formula ? and is denoted by ?[x/t].
A vocabulary consists of a finite set of predicates R and a domain D. Every predicate r is associated
with a positive integer called the arity of r. A R-structure (or knowledge base) D consists of the
domain D, a set of predicates R, and an interpretation. The Herbrand base of D is the set of all
ground atoms that can be constructed from R and D. The interpretation assigns a truth value to
every atom in the Herbrand base by specifying rD ? Dn for each n-ary predicate r ? R. For a
formula ?(x1 , ..., xn ) and a structure D, we write D |= ?(d1 , ..., dn ) to say that D satisfies ? if
the variables x1 , ..., xn are substituted with the domain elements d1 , ...., dn . We define ?(D) :=
{(d1 , ..., dn ) ? Dn | D |= ?(d1 , ..., dn )}. For the R-structure D and C ? D, hCiD denotes the
substructure induced by C on D, that is, the R-structure C with domain C and rC := rD ? Cn for
every n-ary r ? R.
2.2
Gaifman?s Locality Theorem
The Gaifman graph of a R-structure D is the graph GD with vertex set D and an edge between
two vertices d, d0 ? D if and only if there exists an r ? R and a tuple (d1 , ..., dk ) ? rD such that
d, d0 ? {d1 , ..., dk }. Figure 1 depicts a fragment of a knowledge base and the corresponding Gaifman
graph. The distance dD (d1 , d2 ) between two elements d1 , d2 ? D of a structure D is the length of
the shortest path in GD connecting d1 and d2 . For r ? 1 and d ? D, we define the r-neighborhood
of d to be Nr (d) := {x ? D | dD (d, x) ? r}. We refer to r also as the depth of the neighborhood.
Let d = (d1 , ..., dn ) ? Dn . The r-neighborhood of d is defined as
Nr (d) =
n
[
Nr (di ).
i=1
For the Gaifman graph in Figure 1, we have that N1 (d4 ) = {d1 , d2 , d5 } and N1 ((d1 , d2 )) =
{d1 , ..., d6 }. ?Nr (x) is the formula obtained from ?(x) by relativizing all quantifiers to Nr (x), that
is, by replacing every subformula of the form ?y?(x, y, z) by ?y(dD (x, y) ? r ? ?(x, y, z)) and
every subformula of the form ?y?(x, y, z) by ?y(dD (x, y) ? r ? ?(x, y, z)). A formula ?(x) of
the form ?Nr (x), for some ?(x), is called r-local. Whether an r-local formula ?(x) holds depends
only on the r-neighborhood of x, that is, for every structure D and every d ? D we have D |= ?(d)
2
103
d2
d5
number of nodes
locatedIn(d6, d5)
d6
livesIn(d2, d5)
worksAt(d2, d6)
studentOf(d1, d2)
studentAt(d1, d6)
d3 d1
bornIn(d1, d3)
studentOf(d4, d2)
introducedBy(d1, d4, d2)
livesIn(d4, d5)
d4
102
101
100
10?1 0
10
101
102
103
104
degree
Figure 1: A knowledge base fragment for the pair
(d1 , d2 ) and the corresponding Gaifman graph.
Figure 2: The degree distribution of the Gaifman
graph for the Freebase fragment F B 15 K.
if and only if hNr (d)i |= ?(d). For r, k ? 1 and ?(x) being r-local, a local sentence is of the form
?
?
^
^
?x1 ? ? ? ?xk ?
dD (xi , xj ) > 2r ?
?(xi )? .
1?i<j?k
1?i?k
We can now state Gaifman?s locality theorem.
Theorem 1. [9] Every first-order sentence is equivalent to a Boolean combination of local sentences.
Gaifman?s locality theorem states that any first-order sentence can be expressed as a Boolean
combination of r-local sentences defined for neighborhoods of objects that are mutually far apart
(have distance at least 2r + 1). Now, a novel approach to (statistical) relational learning would be to
consider a large set of objects (or tuples of objects) and learn models from their local neighborhoods
in the Gaifman graphs. It is this observation that motivates Gaifman models.
3
Learning Gaifman Models
Instead of taking the costly approach of applying relational learning and inference directly to entire
knowledge bases, the representations of Gaifman models are learned bottom up, by performing
inference and learning within bounded-size, locally connected regions of Gaifman graphs. Each
Gaifman model specifies the data generating process from a given knowledge base (or collection of
knowledge bases), a set of relational features, and a ML model class used for learning.
Definition 1. Given a R-structure D, a discriminative Gaifman model for D is a tuple (q, r, k, ?, M)
as follows:
? q is a first-order formula called the target query with at least one free variable;
? r is the depth of the Gaifman neighborhoods;
? k is the size-bound of the Gaifman neighborhoods;
? ? is a set of first-order formulas (the relational features);
? M is the base model class (loss, hyper-parameters, etc.).
Throughout the rest of the paper, we will provide detailed explanations of the different parameters of
Gaifman models and their interaction with data generation, learning, and inference.
During the training of Gaifman models, neighborhoods are generated for tuples of objects d ? Dn
based on the parameters r and k. We first describe the procedure for arbitrary tuples d of objects
and will later explain where these tuples come from. For a given tuple d the r-neighborhood of d
within the Gaifman graph is computed. This results in the set of objects Nr (d). Now, from this
neighborhood we sample w neighborhoods consisting of at most k objects. Sampling bounded-size
sub-neighborhoods from Nr (d) is motivated as follows:
3
1. The degree distribution of Gaifman graphs is often skewed (see Figure 2), that is, the
number of other objects a domain element is related to varies heavily. Generating smaller,
bounded-size neighborhoods allows the transfer of learned representations between more
and less connected objects. Moreover, the sampling strategy makes Gaifman models more
robust to object uncertainty [19]. We show empirically that larger values for k reduce the
effectiveness of the learned models for some knowledge bases.
2. Relational learning and inference is performed within the generated neighborhoods. Nr (d)
can be very large, even for r = 1 (see Figure 2), and we want full control over the complexity
of the computational problems.
3. Even for a single object tuple d we can generate a large number of training examples if
|Nr (d)| > k. This mitigates the risk of overfitting. The number of training examples per
tuple strongly influences the models? accuracy.
We can now define the set of (r, k)-neighborhoods generated from a r-neighborhood.
{N | N ? Nr (d) and |N| = k} if |Nr (d)| ? k
Nr,k (d) :=
{Nr (d)}
otherwise.
For a given tuple of objects d, Algorithm 1 returns a set of w neighborhoods drawn from Nr,k (d)
such that the number of objects for each di is the same in expectation.
The formulas in the set ? are indexed and of the form ?i (s1 , ..., sn , u1 , ..., um ) with sj ? free(q)
and uj 6? free(q). For every tuple d = (d1 , ..., dn ), generated neighborhood N ? Nr,k (d),
and ?i ? ?, we perform the substitution [s1 /d1 , ..., sn /dn ] and relativize ?i ?s quantifiers to N,
N
resulting in ?N
i [s1 /d1 , ..., sn /dn ] which we write as ?i [s/d]. Let hNi be the substructure induced
by N on D. For every formula ?i (s1 , ..., sn , u1 , ..., um ) and every n ? Nm , we now have that
N
D |= ?N
i [s/d, u/n] if and only if hNi |= ?i [s/d, u/n]. In other words, satisfaction is now checked
locally within the neighborhoods N, by deciding whether hNi |= ?N
i [s/d, u/n]. The relational
semantics of Gaifman models is based on the set of formulas ?. The feature vector v = (v1 , ..., v|?| )
for tuple d, and neighborhood N ? Nr,k (d), written as vN , is constructed as follows
?
N
? ?N
i [s/d](hNi) if free(?i [s/d]) > 0
vi :=
1
if hNi |= ?N
i [s/d]
? 0
otherwise.
That is, if ?N
i [s/d] has free variables, vi is equal to the number of groundings of ?i [s/d] that are
satisfied within the neighborhood substructure hNi; if ?i [s/d] has no free variables, vi = 1 if
and only if ?i [s/d] is satisfied within the neighborhod substructure hNi; and vi = 0 otherwise.
The neighborhood representations v capture r-local formulas and help the model learn formula
combinations that are associated with negative and positive examples. For the right choices of the
parameters r and k, the neighborhood representations of Gaifman models capture the relational
structure associated with positive and negative examples.
Deciding D |= ? for a structure D and a first-order formula ? is referred to as model checking and
computing ?(D) is called ?-counting. The combined complexity of model checking is PSPACEcomplete [29] and there exists a ||D||O(||?||) algorithm for both problems where || ? || is the size of an
encoding. Clearly, for most real-world KBs this is not feasible. For Gaifman models, however, where
the neighborhoods are bounded-size, typically 10 ? |N| = k ? 100, the above representation can
be computed very efficiently for a large class of relational features. We can now state the following
complexity result.
Theorem 2. Let D be a relational structure (knowledge base), let d be the size of the largest rneighborhood of D?s Gaifman graph, and let s be the greatest encoding size of any formula in ?.
For a Gaifman model with parameters r and k, the worst-case complexity for computing the feature
representations of N neighborhoods is O(N (d + |?|k s )).
Existing SRL approaches could be applied to the generated neighborhoods, treating each as a possible
world for structure and parameter learning. However, our goal is to learn relational models that utilize
embeddings computed by multi-layered neural networks.
4
Algorithm 1 G EN N EIGHS: Computes a list of w
neighborhoods of size k for an input tuple d.
?
1: input: tuple d ? Dn , parameters r, k, and w
2: S = [ ]
3: while |S| < w do
4:
S=?
5:
N = Nr (d)
6:
for all i ? {1, ..., n} do
7:
U = min(bk/nc, |Nr (di )|) elements
...
...
Wn
!
M
Figure 3: Learning of a Gaifman model.
sampled uniformly from Nr (di )
N =N \U
S =S?U
U = min(|S| ? k, |N |) elements sampled
uniformly from N
11:
S =S?U
12:
S=S+S
13: return S
8:
9:
10:
3.1
W1
?
?
W1
...
Wn
!
M
Figure 4: Inference with a Gaifman model.
Learning Distributions for Relational Queries
Let q be a first-order formula (the relational query) and S(q) the result set of the query, that is, all
groundings that render the formula satisfied in the knowledge base. The feature representations
generated for tuples of objects d ? S(q) serve as positive training examples. The Gaifman models?
aim is to learn neighborhood embeddings that capture local structure of tuples for which we know
that the target query evaluates to true. Similar to previous work, we generate negative examples by
corrupting tuples that correspond to positive examples. The corruption mechanism takes a positive
input tuple d = (d1 , ..., dn ) and substitutes, for each i ? {1, ..., n}, the domain element di with
objects sampled from D while keeping the rest of the tuple fixed.
The discriminative Gaifman model performs the following steps.
1. Evaluate the target query q and compute the result set S(q)
2. For each tuple d in the result set S(q):
? ? Nr,k (d) with Algorithm 1; each
? Compute N , a multiset of w neighborhoods N
such neighborhood serves as a positive training example
? , a multiset of w
? for corrupted versions of d
? Compute N
? neighborhoods N ? Nr,k (d)
with Algorithm 1; each such neighborhood serves as a negative training example
? Perform model checking and counting within the neighborhoods to compute the feature
?
?
representations vN and vN
? for each N ? N and N ? N , respectively
3. Learn a ML model with the generated positive and negative training examples.
Learning the final Gaifman model depends on the base ML model class M and its loss function.
We obtained state of the art results with neural networks, gradient-based learning, and categorical
cross-entropy as loss function
?
?
X
X
L = ??
log pM (vN ) +
log(1 ? pM (vN
? ))? ,
? N
?
N?
N?N
where pM (vN ) is the probability the model returns on input vN . However, other loss functions are
possible. The probability of a particular substitution of the target query to be true is now
P (q[s/d] = True) =
E
[pM (vN )].
N?N(r,k) (d)
The expected probability of a representation of a neighborhood drawn uniformly at random from
N(r,k) (d). It is now possible to generate several neighborhoods N and their representations vN to
5
estimate P (q[s/d] = True), simply by averaging the neighborhoods? probabilities. We have found
experimentally that a single neighborhood already leads to highly accurate results but also that more
neighborhood samples further improve the accurracy.
Let us emphasize again the novel semantics of Gaifman models. Gaifman models generate a large
number of small, bounded-size structures from a large structure, learn a representation for these
bounded-size structures, and use the resulting representation to answer queries concerning the
original structure as a whole. The advantages are model weight sharing across a large number of
neighborhoods and efficiency of the computational problems. Figure 3 and Figure 4 illustrate learning
from bounded-size neighborhood structures and inference in Gaifman models.
3.2
Structure Learning
Structure learning is the problem of determining the set of relational features ?. We provide some
directions and leave the problem to future work. Given a collection of bounded-size neighborhoods
of the Gaifman graph, the goal is to determine suitable relational features for the problem at hand.
There is a set of features which we found to be highly effective. For example, formulas of the form
?x r(s1 , x), ?x r(s1 , x) ? r(x, s2 ), and ?x, y r1 (s1 , x) ? r2 (x, y) ? r3 (y, s2 ) for all relations. The
latter formulas capture fixed-length paths between s1 and s2 in the neighborhoods. Hence, Path
Ranking type features [15] can be used in Gaifman models as a particular relational feature class. For
path formulas with several different relations we cannot include all |R|3 combinations and, hence,
we have to determine a subset occurring in the training data. Fortunately, since the neighborhood
size is bounded, it is computationally feasible to compute frequent paths in the neighborhoods and
to use these as features. The complexity of this learning problem is in the number of elements
in the neighborhood and not in the number of all objects in the knowledge base. Relation paths
that do not occur in the data can be discarded. Gaifman models can also use features of the form
?x, y r(x, y) ? r(y, x), ?x, y r(x, y), and ?x, y, z r(x, y) ? r(y, z) ? r(x, z), to name but a few.
Moreover, features with free variables, such as r(s1 , x) are counting features (here: the r out-degree
of s1 ). It is even computationally feasible to include specific second-order features (for instance,
quantifiers ranging over R) and aggregations of feature values.
3.3
Prior Confidence Values, Types, and Numerical Attributes
Numerous existing knowledge bases assign confidence values (probabilities, weights, etc.) to their
statements. Gaifman models can incorporate confidence values during the sampling and learning
process. Instead of adding random noise to the representations, which we have found to be beneficial,
noise can be added inversely proportional to the confidence values. Statements for which the prior
confidence values are lower are more likely to be dropped out during training than statements with
higher confidence values. Furthermore, Gaifman models can directly incorporate object types such as
Actor and Action Movie as well as numerical features such as location and elevation. One simply
has to specify a fixed position in the neighborhood representation v for each object position within
the input tuples d.
4
Related Work
Recent work on relational machine learning for knowledge graphs is surveyed in [20]. We focus on
a select few methods we deem most related to Gaifman models and refer the interested reader to
the above article. A large body of work exists on learning inference rules from knowledge bases.
Examples include [31] and [1] where inference rules of length one are learned; and [25] where general
inference rules are learned by applying a support threshold. Their method does not scale to large KBs
and depends on predetermined thresholds. Lao et al. [15] train a logistic regression classifier with
path features to perform KB completion. The idea is to perform a random walk between objects and
to exploit the discovered paths as features. SFE [10] improves PRA by making the generation of
random walks more efficient. More recent embedding methods have combined paths in KBs with
KB embedding methods [17]. Gaifman models support a much broader class of relational features
subsuming path features. For instance, Gaifman models incorporate counting features that have
shown to be beneficial for relational models.
6
Latent feature models learn features for objects and relations that are not directly observed in
the data. Examples of latent feature models are tensor factorization [21, 23, 26] and embedding
models [5, 3, 4, 18, 13, 27]. The majority of these models can be understood as more or less complex
neural networks operating on object and relation representations. Gaifman models can also be used
to learn knowledge base embeddings. Indeed, one can show that it generalizes or complements
existing approaches. For instance, the universal schema [23] considers pairs of objects where relation
membership variables comprise the model?s features. We have the following interesting relationship
between universal schemas [23] and Gaifman
models. Given a knowledge base D. The Gaifman
S
model for D with r = 0, k = 2, ? = r?R {r(s1 , s2 ), r(s2 , s1 )}, w = 1 and w
? = 0 is equivalent
to the Universal Schema [23] for D up to the base model class M. More recent methods combine
embedding methods and inference-based logical approaches for relation extraction [24]. Contrary
to most existing multi-relational ML models [20], Gaifman models natively support higher-arity
relations, functional and type constraints, numerical features, and complex target queries.
5
Experiments
Table 1: The statistics of the data sets.
The aim of the experiments is to understand the
efficiency and effectiveness of Gaifman models for
Dataset
|D|
|R|
# train
# test
typical knowledge base inference problems. We
WN18 40,943
18
141,442
5,000
evaluate the proposed class of models with two data
FB15k 14,951 1,345 483,142 59,071
sets derived from the knowledge bases W ORD N ET
and F REEBASE [2]. Both data sets consist of a list of statements r(d1 , d2 ) that are known to be true.
For a detailed description of the data sets, whose statistics are listed in Table 1, we refer the reader to
previous work [4].
After training the models, we perform entity prediction as follows. For each statement r(d1 , d2 ) in
the test set, d2 is replaced by each of the KB?s objects in turn. The probabilities of the resulting
statements are predicted and sorted in descending order. Finally, the rank of the correct statement
within this ordered list is determined. The same process is repeated now with replacements of d1 . We
compare Gaifman models with q = r(x, y) to state of the art knowledge base completion approaches
which are listed in Table 2. We trained Gaifman models with r = 1 and different values for k, w,
and w.
? We use a neural network architecture with two hidden layers, each having 100 units and
sigmoid activations, dropout of 0.2 on the input layer, and a softmax layer. Dropout makes the model
more robust to missing relations between objects. We trained one model per relation and left the
hyper-parameters fixed across models. We did not perform structure learning and instead used the
following set of relational features
[
r(s1 , s2 ), r(s2 , s1 ), ?x r(x, si ), ?x r(si , x),
.
? :=
?x r(s1 , x) ? r(x, s2 ), ?x r(s2 , x) ? r(x, s1 )
r?R, i?{1,2}
We performed runtime experiments to evaluate the
models? efficiency. Embedding models have the
advantage that one dot product for every candidate
object is sufficient to compute the score for the
corresponding statement and we need to assess the
performance of Gaifman models in this context.
All experiments were run on commodity hardware
with 64G RAM and a single 2.8 GHz CPU.
Query answers per second
To compute the probabilities, we averaged the probabilities of N = 1, 2, or 3 generated (r, k)neighborhoods.
?104
1.5
WN18
FB15k
1.0
0.5
Table 2 lists the experimental results for different
parameter settings [N, k, w, w].
? The Gaifman mod5
10
20
50
100
inf
els achieve the highest hits@10 and hits@1 values
for both data sets. As expected, the more neighborhood samples are used to compute the probability Figure 5: Query answers per second rates for
estimate (N = 1, 2, 3) the better the result. When different values of the parameter k.
the entire 1-neighborhood is considered (k = ?),
the performance for WN18 does not deteriorate as it does for FB15k. This is due to the fact that
7
Table 2: Results of the entity prediction experiments.
Data Set
Metric
RESCAL[21]
SE[5]
LFM[12]
TransE[4]
TransR[18]
DistMult[30]
Gaifman [1, ?, 1, 5]
Gaifman [1, 20, 1, 2]
Gaifman [1, 20, 5, 25]
Gaifman [2, 20, 5, 25]
Gaifman [3, 20, 5, 25]
Mean rank
1,163
985
456
251
219
902
298
357
392
378
352
WN18
Hits@10
52.8
80.5
81.6
89.2
91.7
93.7
93.9
88.1
93.6
93.9
93.9
Hits@1
8.9
76.1
75.8
66.8
76.4
76.7
76.1
Mean rank
683
162
164
51
78
97
124
114
97
84
75
FB15K
Hits@10
44.1
39.8
33.1
71.5
65.5
82.8
78.1
79.2
82.1
83.4
84.2
Hits@1
28.1
44.3
59.8
60.1
65.6
68.5
69.2
objects in WN18 have on average few neighbors. FB15k has more variance in the Gaifman graph?s
degree distribution (see Figure 2) which is reflected in the better performance for smaller k values.
The experiments also show that it is beneficial to generate a large number of representations (both
positive and negative ones). The performance improves with larger number of training examples.
The runtime experiments demonstrate that Gaifman models perform inference very efficiently for
k ? 20. Figure 5 depicts the number of query answers the Gaifman models are able to serve per
second, averaged over relation types. A query answer returns the probability for one object pair.
These numbers include neighborhood generation and network inference. The results are promising
with about 5000 query answers per second (averaged across relation types) as long as k remains
small. Since most object pairs of WN18 have a 1-neighborhood whose size is smaller than 20, the
answers per second rates for k > 20 is not reduced as drastically as for FB15k.
6
Conclusion and Future Work
Gaifman models are a novel family of relational machine learning models that perform learning
and inference within and across locally connected regions of relational structures. Future directions
of research include structure learning, more sophisticated base model classes, and application of
Gaifman models to additional relational ML problems.
Acknowledgements
Many thanks to Alberto Garc?a-Dur?n, Mohamed Ahmed, and Kristian Kersting for their helpful
feedback.
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9
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5,634 | 6,099 | Professor Forcing: A New Algorithm for Training
Recurrent Networks
Anirudh Goyal?, Alex Lamb? , Ying Zhang, Saizheng Zhang,
Aaron Courville and Yoshua Bengio1
MILA, Universit? de Montr?al, 1 CIFAR
{anirudhgoyal9119, alex6200, ying.zhlisa, saizhenglisa,
aaron.courville, yoshua.umontreal}@gmail.com
Abstract
The Teacher Forcing algorithm trains recurrent networks by supplying observed
sequence values as inputs during training and using the network?s own one-stepahead predictions to do multi-step sampling. We introduce the Professor Forcing
algorithm, which uses adversarial domain adaptation to encourage the dynamics of
the recurrent network to be the same when training the network and when sampling
from the network over multiple time steps. We apply Professor Forcing to language
modeling, vocal synthesis on raw waveforms, handwriting generation, and image
generation. Empirically we find that Professor Forcing acts as a regularizer, improving test likelihood on character level Penn Treebank and sequential MNIST.
We also find that the model qualitatively improves samples, especially when sampling for a large number of time steps. This is supported by human evaluation of
sample quality. Trade-offs between Professor Forcing and Scheduled Sampling are
discussed. We produce T-SNEs showing that Professor Forcing successfully makes
the dynamics of the network during training and sampling more similar.
1
Introduction
Recurrent neural networks (RNNs) have become to be the generative models of choice for sequential
data (Graves, 2012) with impressive results in language modeling (Mikolov, 2010; Mikolov and
Zweig, 2012), speech recognition (Bahdanau et al., 2015; Chorowski et al., 2015), Machine Translation (Cho et al., 2014a; Sutskever et al., 2014; Bahdanau et al., 2014), handwriting generation (Graves,
2013), image caption generation (Xu et al., 2015; Chen and Lawrence Zitnick, 2015), etc.
The RNN models the data via a fully-observed directed graphical model: it decomposes the distribution over the discrete time sequence y1 , y2 , . . . yT into an ordered product of conditional distributions
over tokens
T
Y
P (y1 , y2 , . . . yT ) = P (y1 )
P (yt | y1 , . . . yt?1 ).
t=1
By far the most popular training strategy is via the maximum likelihood principle. In the RNN
literature, this form of training is also known as teacher forcing (Williams and Zipser, 1989), due to
the use of the ground-truth samples yt being fed back into the model to be conditioned on for the
prediction of later outputs. These fed back samples force the RNN to stay close to the ground-truth
sequence.
When using the RNN for prediction, the ground-truth sequence is not available conditioning and
we sample from the joint distribution over the sequence by sampling each yt from its conditional
?
Indicates first authors. Ordering determined by coin flip.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
distribution given the previously generated samples. Unfortunately, this procedure can result in
problems in generation as small prediction error compound in the conditioning context. This can
lead to poor prediction performance as the RNN?s conditioning context (the sequence of previously
generated samples) diverge from sequences seen during training.
Recently, (Bengio et al., 2015) proposed to remedy that issue by mixing two kinds of inputs during
training: those from the ground-truth training sequence and those generated from the model. However,
when the model generates several consecutive yt ?s, it is not clear anymore that the correct target
(in terms of its distribution) remains the one in the ground truth sequence. This is mitigated in
various ways, by making the self-generated subsequences short and annealing the probability of
using self-generated vs ground truth samples. However, as remarked by Husz?r (2015), scheduled
sampling yields a biased estimator, in that even as the number of examples and the capacity go
to infinity, this procedure may not converge to the correct model. It is however good to note that
experiments with scheduled sampling clearly showed some improvements in terms of the robustness
of the generated sequences, suggesting that something indeed needs to be fixed (or replaced) with
maximum-likelihood (or teacher forcing) training of generative RNNs.
In this paper, we propose an alternative way of training RNNs which explicitly seeks to make the
generative behavior and the teacher-forced behavior match as closely as possible. This is particularly
important to allow the RNN to continue generating robustly well beyond the length of the sequences
it saw during training. More generally, we argue that this approach helps to better model long-term
dependencies by using a training objective that is not solely focused on predicting the next observation,
one step at a time.
Our work provides the following contributions regarding this new training framework:
? We introduce a novel method for training generative RNNs called Professor Forcing, meant
to improve long-term sequence sampling from recurrent networks. We demonstrate this
with human evaluation of sample quality by performing a study with human evaluators.
? We find that Professor Forcing can act as a regularizer for recurrent networks. This is
demonstrated by achieving improvements in test likelihood on character-level Penn Treebank,
Sequential MNIST Generation, and speech synthesis. Interestingly, we also find that
training performance can also be improved, and we conjecture that it is because longer-term
dependencies can be more easily captured.
? When running an RNN in sampling mode, the region occupied by the hidden states of the
network diverges from the region occupied when doing teacher forcing. We empirically
study this phenomenon using T-SNEs and show that it can be mitigated by using Professor
Forcing.
? In some domains the sequences available at training time are shorter than the sequences
that we want to generate at test time. This is usually the case in long-term forecasting tasks
(climate modeling, econometrics). We show how using Professor Forcing can be used to
improve performance in this setting. Note that scheduled sampling cannot be used for this
task, because it still uses the observed sequence as targets for the network.
2
Proposed Approach: Professor Forcing
The basic idea of Professor Forcing is simple: while we do want the generative RNN to match the
training data, we also want the behavior of the network (both in its outputs and in the dynamics of
its hidden states) to be indistinguishable whether the network is trained with its inputs clamped to
a training sequence (teacher forcing mode) or whether its inputs are self-generated (free-running
generative mode). Because we can only compare the distribution of these sequences, it makes sense
to take advantage of the generative adversarial networks (GANs) framework (Goodfellow et al., 2014)
to achieve that second objective of matching the two distributions over sequences (the one observed
in teacher forcing mode vs the one observed in free-running mode).
Hence, in addition to the generative RNN, we will train a second model, which we call the discriminator, and that can also process variable length inputs. In the experiments we use a bidirectional
RNN architecture for the discriminator, so that it can combine evidence at each time step t from the
past of the behavior sequence as well as from the future of that sequence.
2
2.1
Definitions and Notation
Let the training distribution provide (x, y) pairs of input and output sequences (possibly there are
no inputs at all). An output sequence y can also be generated by the generator RNN when given an
input sequence x, according to the sequence to sequence model distribution P?g (y|x). Let ?g be the
parameters of the generative RNN and ?d be the parameters of the discriminator. The discriminator
is trained as a probabilistic classifier that takes as input a behavior sequence b derived from the
generative RNN?s activity (hiddens and outputs) when it either generates or is constrained by a
sequence y, possibly in the context of an input sequence x (often but not necessarily of the same
length). The behavior sequence b is either the result of running the generative RNN in teacher forcing
mode (with y from a training sequence with input x), or in free-running mode (with y self-generated
according to P?g (y|x), with x from the training sequence). The function B(x, y, ?g ) outputs the
behavior sequence (chosen hidden states and output values) given the appropriate data (where x
always comes from the training data but y either comes from the data or is self-generated). Let D(b)
be the output of the discriminator, estimating the probability that b was produced in teacher-forcing
mode, given that half of the examples seen by the discriminator are generated in teacher forcing mode
and half are generated in the free-running mode.
Note that in the case where the generator RNN does not have any conditioning input, the sequence
x is empty. Note also that the generated output sequences could have a different length then the
conditioning sequence, depending of the task at hand.
2.2
Training Objective
The discriminator parameters ?d are trained as one would expect, i.e., to maximize the likelihood of
correctly classifying a behavior sequence:
Cd (?d |?g ) = E(x,y)?data [? log D(B(x, y, ?g ), ?d )+Ey?P?g (y|x) [? log(1?D(B(x, y, ?g ), ?d )]].
(1)
Practically, this is achieved with a variant of stochastic gradient descent with minibatches formed by
combining N sequences obtained in teacher-forcing mode and N sequences obtained in free-running
mode, with y sampled from P?g (y|x). Note also that as ?g changes, the task optimized by the
discriminator changes too, and it has to track the generator, as in other GAN setups, hence the notation
Cd (?d |?g ).
The generator RNN parameters ?g are trained to (a) maximize the likelihood of the data and (b) fool
the discriminator. We considered two variants of the latter. The negative log-likelihood objective (a)
is the usual teacher-forced training criterion for RNNs:
N LL(?g ) = E(x,y)?data [? log P?g (y|x)].
(2)
Regarding (b) we consider a training objective that only tries to change the free-running behavior so
that it better matches the teacher-forced behavior, considering the latter fixed:
Cf (?g |?d ) = Ex?data,y?P?g (y|x) [? log D(B(x, y, ?g ), ?d )].
(3)
In addition (and optionally), we can ask the teacher-forced behavior to be indistinguishable from the
free-running behavior:
Ct (?g |?d ) = E(x,y)?data [? log(1 ? D(B(x, y, ?g ), ?d ))].
(4)
In our experiments we either perform stochastic gradient steps on N LL + Cf or on N LL + Cf + Ct
to update the generative RNN parameters, while we always do gradient steps on Cd to update the
discriminator parameters.
3
Related Work
Professor Forcing is an adversarial method for learning generative models that is closely related to
Generative Adversarial Networks (Goodfellow et al., 2014) and Adversarial Domain Adaptation
Ajakan et al. (2014); Ganin et al. (2015). Our approach is similar to generative adversarial networks
(GANs) because both use a discriminative classifier to provide gradients for training a generative
model. However, Professor Forcing is different because the classifier discriminates between hidden
3
Teacher
Forcing
...
Distributions of
hidden states are
forced to be close
to each other by
Discriminator
Share
parameters
Discriminator
...
Free
Running
Figure 1: Architecture of the Professor Forcing - Learn correct one-step predictions such as to to
obtain the same kind of recurrent neural network dynamics whether in open loop (teacher forcing)
mode or in closed loop (generative) mode. An open loop generator that does one-step-ahead prediction
correctly. Recursively composing these outputs does multi-step prediction (closed-loop) and can
generate new sequences. This is achieved by train a classifier to distinguish open loop (teacher
forcing) vs. closed loop (free running) dynamics, as a function of the sequence of hidden states and
outputs. Optimize the closed loop generator to fool the classifier. Optimize the open loop generator
with teacher forcing. The closed loop and open loop generators share all parameters
states from sampling mode and teacher forcing mode, whereas the GAN?s classifier discriminates
between real samples and generated samples. One practical advantage of Professor Forcing over
GANs is that Professor Forcing can be used to learn a generative model over discrete random variables
without requiring to approximate backpropagation through discrete spaces Bengio et al. (2013).
The Adversarial Domain Adaptation uses a classifier to discriminate between the hidden states of the
network with inputs from the source domain and the hidden states of the network with inputs from
the target domain. However this method was not applied in the context of generative models, more
specifically, was not applied to the task of improving long-term generation from recurrent networks.
Alternative non-adversarial methods have been explored for improving long-term generation from
recurrent networks. The scheduled sampling method Bengio et al. (2015), which is closely related
to SEARN (Daum? et al., 2009) and DAGGER Ross et al. (2010), involves randomly using the
network?s predictions as its inputs (as in sampling mode) with some probability that increases over the
course of training. This forces the network to be able to stay in a reasonable regime when receiving
the network?s predictions as inputs instead of observed inputs. While Scheduled Sampling shows
improvement on some tasks, it is not a consistent estimation strategy. This limitation arises because
the outputs sampled from the network could correspond to a distribution that is not consistent with
the sequence that the network is trained to generate. This issue is discussed in detail in Husz?r (2015).
A practical advantage of Scheduled Sampling over Professor Forcing is that Scheduled Sampling
does not require the additional overhead of having to train a discriminator network.
Finally, the idea of matching the behavior of the model when it is generating in a free-running way
with its behavior when it is constrained by the observed data (being clamped on the "visible units")
is precisely that which one obtains when zeroing the maximum likelihood gradient on undirected
graphical models with latent variables such as the Boltzmann machine. Training Boltzmann machines
amounts to matching the sufficient statistics (which summarize the behavior of the model) in both
"teacher forced" (positive phase) and "free-running" (negative phase) modes.
4
4.1
Experiments
Networks Architecture and Professor Forcing Setup
The neural networks and Professor Forcing setup used in the experiments is the following. The
generative RNN has single hidden layer of gated recurrent units (GRU), previously introduced
by (Cho et al., 2014b) as a computationally cheaper alternative to LSTM units (Hochreiter and
Schmidhuber, 1997). At each time step, the generative RNN reads an element xt of the input
4
sequence (if any) and an element of the output sequence yt (which either comes from the training
data or was generated at the previous step by the RNN). It then updates its state ht as a function of
its previous state ht?1 and of the current input (xt , yt ). It then computes a probability distribution
P?g (yt+1 |ht ) = P?g (yt+1 |x1 , . . . , xt , y1 , . . . , yt ) over the next element of the output. For discrete
outputs this is achieved by a softmax / affine layer on top of ht , with as many outputs as the size of
the set of values that yt can take. In free-running mode, yt+1 is then sampled from this distribution
and will be used as part of the input for the next time step. Otherwise, the ground truth yt is used.
The behavior function B used in the experiments outputs the pre-tanh activation of the GRU states
for the whole sequence considered, and optionally the softmax outputs for the next-step prediction,
again for the whole sequence.
The discriminator architecture we used for these experiments is based on a bidirectional recurrent
neural network, which comprises two RNNs (again, two GRU networks), one running forward in
time on top of the input sequence b, and one running backwards in time, with the same input. The
hidden states of these two RNNs are concatenated at each time step and fed to a multi-layer neural
network shared across time (the same network is used for all time steps). That MLP has three layers,
each composing an affine transformation and a rectifier (ReLU). Finally, the output layer composes
an affine transformation and a sigmoid that outputs D(b).
When the discriminator is too poor, the gradient it propagates into the generator RNN could be
detrimental. For this reason, we back-propagate from the discriminator into the generator RNN only
when the discriminator classification accuracy is greater than 75%. On the other hand, when the
discriminator is too successful at identifying fake inputs, we found that it would also hurt to continue
training it. So when its accuracy is greater than 99%, we do not update the discriminator.
Both networks are trained by minibatch stochastic gradient descent with adaptive learning rates and
momentum determined by the Adam algorithm (Kingma and Ba, 2014). All of our experiments were
implemented using the Theano framework (Al-Rfou et al., 2016).
4.2
Character-Level Language Modeling
We evaluate Professor Forcing on character-level language modeling on Penn-Treebank corpus,
which has an alphabet size of 50 and consists of 5059k characters for training, 396k characters for
validation and 446k characters for test. We divide the training set into non-overlapping sequences
with each length of 500. During training, we monitor the negative log-likelihood (NLL) of the output
sequences. The final model are evaluated by bits-per-character (BPC) metric. The generative RNN
Figure 2: Penn Treebank Likelihood Curves in terms of the number of iterations. Training Negative
Log-Likelihood (left). Validation BPC (Right)
implements an 1 hidden layer GRU with 1024 hidden units. We use Adam algorithm for optimization
with a learning rate of 0.0001. We feed both the hidden states and char level embeddings into the
discriminator. All the layers in the discriminator consists of 2048 hidden units. Output activation of
the last layer is clipped between -10 and 10. We see that training cost of Professor Forcing network
decreases faster compared to teacher forcing network. The training time of our model is 3 times
more as compared to teacher forcing, since our model includes sampling phase, as well as passing
the hidden distributions corresponding to free running and teacher forcing phase to the discriminator.
The final BPC on validation set using our baseline was 1.50 while using professor forcing it is 1.48.
On word level Penn Treebank we did not observe any difference between Teacher Forcing and
Professor Forcing. One possible explanation for this difference is the increased importance of
long-term dependencies in character-level language modeling.
5
Figure 3: T-SNE visualization of hidden states, left: with teacher forcing, right: with professor
forcing. Red dots correspond to teacher forcing hidden states, while the gold dots correspond to
free running mode. At t = 500, the closed-loop and open-loop hidden states clearly occupy distinct
regions with teacher forcing, meaning that the network enters a region during sampling distinct from
the region seen during teacher forcing training. With professor forcing, these regions now largely
overlap. We computed 30 T-SNEs for Teacher Forcing and 30 T-SNEs for Professor Forcing and
found that the mean centroid distance was reduced from 3000 to 1800 (40% relative reduction). The
mean distance from a hidden state in the training network to a hidden state in the sampling network
was reduced from 22.8 with Teacher Forcing to 16.4 with Professor Forcing (vocal synthesis).
Method
MNIST NLL
DBN 2hl (Germain et al., 2015)
NADE (Larochelle and Murray, 2011)
EoNADE-5 2hl (Raiko et al., 2014)
DLGM 8 leapfrog steps (Salimans et al., 2014)
DARN 1hl (Gregor et al., 2015)
DRAW (Gregor et al., 2015)
Pixel RNN (van den Oord et al., 2016)
Professor Forcing (ours)
? 84.55
88.33
84.68
? 85.51
? 84.13
? 80.97
79.2
79.58
Table 1: Test set negative log-likelihood evaluations on Sequential MNIST.
4.3
Sequential MNIST
We evaluated Professor Forcing on the task of sequentially generating the pixels in MNIST digits.
We use the standard binarized MNIST dataset Murray and Salakhutdinov (2009). We selected
hyperparameters for our model on the validation set and elected to use 512 hidden states and a
learning rate of 0.0001. For all experiments we used a 3-layer GRU as our generator. Unlike our other
experiments, we used a convolutional network for the discriminator instead of a bi-directional RNN,
as the pixels have a 2D spatial structure. In Table 1, We note that our model achieves the second best
reported likelihood on this task, after the PixelRNN, which used a significantly more complicated
architecture for its generator van den Oord et al. (2016). Combining Professor Forcing with the
PixelRNN would be an interesting area for future research. However, the PixelRNN parallelizes
computation in the teacher forcing network in a way that doesn?t work in the sampling network.
Because Professor Forcing requires running the sampling network during training, naively combining
Professor Forcing with the PixelRNN would be very slow.
Figure 4: Samples with Teacher Forcing (left) and Professor Forcing (right) on Sequential MNIST.
6
Response
Percent
Count
Professor Forcing Much Better
Professor Forcing Slightly Better
Teacher Forcing Slightly Better
Teacher Forcing Much Better
19.7
57.2
18.9
4.3
151
439
145
33
Total
100.0
768
Table 2: Human Evaluation Study Results for Handwriting Generation.
4.4
Handwriting Generation
With this task we wanted to investigate if Professor Forcing could be used to perform domain
adaptation from a training set with short sequences to sampling much longer sequences. We train
the Teacher Forcing model on only 50 steps of text-conditioned handwriting (corresponding to a few
letters) and then sample for 1000 time steps . We let the model learn a sequence of (x, y) coordinates
together with binary indicators of pen-up vs. pen-down, using the standard handwriting IAM-OnDB
dataset, which consists of 13,040 handwritten lines written by 500 writers Liwicki and Bunke (2005).
For our teacher forcing model, we use the open source implementation Brebisson (2016) and use their
hyperparameters which is based on the model in Graves (2013). For the professor forcing model, we
sample for 1000 time steps and run a separate discriminator on non-overlapping segments of length
50 (the number of steps used in the teacher forcing model).
We performed a human evaluation study on handwriting samples. We gave 48 volunteers 16 randomly
selected Prof. Forcing samples randomly paired with 16 Teacher Forcing samples and asked them to
indicate which sample was higher quality and whether it was ?much better? or ?slightly better?. Both
models had equal training time and samples were drawn using the same procedure. Volunteers were
not aware of which samples came from which model, see Table 2 for results.
4.5
Music Synthesis on Raw Waveforms
We considered the task of vocal synthesis on raw waveforms. For this task we used three hours of monk
chanting audio scraped from YouTube (https://www.youtube.com/watch?v=9-pD28iSiTU).
We sampled the audio at a rate of 1 kHz and took four seconds for each training and validation
example. On each time step of the raw audio waveform we binned the signal?s value into 8000 bins
with boundaries drawn uniformly between the smallest and largest signal values in the dataset. We
then model the raw audio waveform as a 4000-length sequence with 8000 potential values on each
time step.
Figure 6: Music Synthesis. Left: training likelihood curves. Right: validation likelihood curves.
We evaluated the quality of our vocal synthesis model using two criteria. First, we demonstrated a
regularizing effect and improvement in negative log-likelihood. Second, we observed improvement in
the quality of samples. We included a few randomly selected samples in the supplementary material
and also performed human evaluation of the samples.
Visual inspection of samples is known to be a flawed method for evaluating generative models,
because a generative model could simply memorize a small number of examples from the training set
(or slightly modified examples from the training set) and achieve high sample quality. This issue was
discussed in Theis et al. (2015). However, this is unlikely to be an issue with our evaluation because
our method also improved validation set likelihood, whereas a model that achieves quality samples
by dropping coverage would have poorer validation set likelihood.
7
We performed human evaluation by asking 29 volunteers to listen to five randomly selected teacher
forcing samples and five randomly selected professor forcing samples (included in supplementary
materials and then rate each sample from 1-3 on the basis of quality. The annotators were given
the samples in random order and were not told which samples came from which algorithm. The
human annotators gave the Professor Forcing samples an average score of 2.20, whereas they gave
the Teacher Forcing samples an average score of 1.30.
Figure 7: Human evaluator ratings for vocal synthesis samples (higher is better). The height of the
bar is the mean of the ratings and the error bar shows the spread of one standard deviation.
5
Conclusion
The idea of matching behavior of a model when it is running on its own, making predictions,
generating samples, etc. vs when it is forced to be consistent with observed data is an old and
powerful one. In this paper we introduce Professor Forcing, a novel instance of this idea when the
model of interest is a recurrent generative one, and which relies on training an auxiliary model, the
discriminator to spot the differences in behavior between these two modes of behavior. A major
motivation for this approach is that the discriminator can look at the statistics of the behavior and not
just at the single-step predictions, forcing the generator to behave the same when it is constrained by
the data and when it is left generating outputs by itself for sequences that can be much longer than the
training sequences. This naturally produces better generalization over sequences that are much longer
than the training sequences, as we have found. We have also found that it helped to generalize better
in terms of one-step prediction (log-likelihood), even though we are adding a possibly conflicting
term to the log-likelihood training objective. This suggests that it acts like a regularizer but a very
interesting one because it can also greatly speed up convergence in terms of number of training
updates. We validated the advantage of Professor Forcing over traditional teacher forcing on a variety
of sequential learning and generative tasks, with particularly impressive results in acoustic generation,
where the training sequences are much shorter (because of memory constraints) than the length of the
sequences we actually want to generate.
Acknowledgments
We thank Martin Arjovsky, Dzmitry Bahdanau, Nan Rosemary Ke, Jos? Manuel Rodr?guez Sotelo,
Alexandre de Br?bisson, Olexa Bilaniuk, Hal Daum? III, Kari Torkkola, and David Krueger.
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5,635 | 61 | 402
HOW
THE
PROCESSING
CATFISH TRACKS ITS PREY: AN INTERACTIVE "PIPELINED"
SYSTEM MAY DIRECT FORAGING VIA RETlCULOSPINAL NEURONS.
Jagmeet S. Kanwal
Dept. of Cellular & Structural Biology, Univ. of Colorado, Sch. of
Medicine, 4200 East, Ninth Ave., Denver, CO 80262.
ABSTRACT
Ictalurid catfish use a highly developed gustatory system to
localize, track and acquire food from their aquatic environment.
The neural organization of the gustatory system illustrates well
the
importance
of
the
four
fundamental
ingredients
(representation, architecture, search and knowledge) of an
"intelligent" system.
In addition, the "pipelined" design of
architecture illustrates how a goal-directed system effectively
utilizes interactive feedback from its environment.
Anatomical
analysis
of
neural networks involved in
target-tracking
indicated that reticular neurons within the medullary region of
the
brainstem,
mediate connections between the gustatory
(sensory) inputs and the motor outputs of the spinal cord.
Electrophysiological
analysis suggested that these neurons
integrate selective spatio-temporal patterns of sensory input
transduced through a rapidly adapting-type peripheral filter
(responding tonically only to a continuously increasing stimulus
concentration).
The connectivity and response patterns of
reticular cells and the nature of the peripheral taste response
suggest a unique "gustation-seeking" fUnction of reticulospinal
cells, which may enable a catfish to continuously track a
stimulus source once its directionality has been computed.
INTRODUCTION
Food search is an example of a broad class of behaviors
generally classified as goal-directed behaviors.
Goal-directed
behavior is frequently exhibited by animals, humans and some
machines.
Although a preprogrammed, hard-wired machine may achieve
a particular goal in a relatively short time, the general and
heuristic nature of complex goal-directed tasks, however, is best
exhibited by animals and best studied in some of the less advanced
animal species,
such as fishes,
where anatomical,
electrophysiological and behavioral analyses can be performed relatively
accurately and easily.
Food search, which may lead to food acquisition and ingestion,
is critical for the survival of an organism and, therefore, only
highly successful systems are selected during the evolution of a
species. The act of food search may be classified into two distinct
phases, (i) orientation, and (ii) tracking (navigation and homing).
In the channel catfish (the animal model utilized for this study),
locomotion (swimming) is primarily controlled by the large forked
caudal fin, which also mediates turning and directional swimming.
@ American Institute of Physics 1988
403
Both these forms of movement, which constitute the essential
movements
of
target-tracking,
involve
control
of
the
hypaxial/epiaxial muscles of the flank.
The alternate contraction
of these muscles causes caudal fin undulations.
Each cycle of the
caudal
fin undulation provides either a symmetrical
or
an
asymmetrical bilateral thrust.
The former provides a net thrust
forward, along the longitudinal axis of the fish causing it to move
ahead, while the latter biases the direction of movement towards the
right or left side of the fish.
HRP injection site
1 /recording site
*A
................................... ........................... ......................... NEUROBIOLOGY ...... ....................................
fEEDING BEHAVIOR
MUSCLE SET
MOTOR POOLS PREMOTOR NEURONS
GUSTATORY INPUTS
....................................... ..............................................................................................................................
food Search
1
Pick Up
flank and
Tail Fi n
Muscles
Caudal
Reticular
Spinal
VCord > ....."...... f orma tl on
:
:.... u.
flank
Mu:sculature
Ja'W Muscles
I
facial lobe
> ......_..............
I
II .... UIlIl ..... " _....................... , . , . .. UIl. U Il ........... _WI ......... 111111 ... ' .
Rostral
Spinal Cord
Reticular
f
t?
orma Ion
facial lobe
Vagal lobe
Intrl nsic
Interneurons
Vagal lobe
>.........
I .....
> ........_..............
facial and/or > ..........................................._..._J....._. . .__?
Trigeminal
Motor Nucleus
,~
Selective
Ingestion
Oral and
Pharyngeal
Musculature
Vagal Motor
Nuclei
Fig. 1.
Schematic representation of possible pathways
gustatory modulation of foraging in the catfish.
for
the
404
Ictalurid catfishes possess a well developed gustatory system
and
use it to locate and acquire food from their
aquatic
environmentl,2,~ Behavioral evidence also indicates that ictalurid
catfishes can
detect small intensity (stimulus
concentration)
differences
across
their
barbels
(interbarbel
intensity
differences), and may use this or other extraoral taste information
to compute the directionality in space and track a gustatory
stimulus source 1
In other words, based upon the analysis of
locomotion, it may be inferred that during food search, the
gustatory sense of the catfish influences the duration and degree of
asymmetrical or symmetrical undulations of the caudal fin, besides
controlling reflex turns of the head and flank.
Since directional
swimming is ultimately dependent upon movement of the large caudal
fin it may be postulated that, if the gustatory system is to
coordinate food tracking, gustato-spinal connections exist upto the
level of the caudal fin of the catfish (fig. 1).
The objectives of this study were (i) to reconsider the
functional
organization
of the gustatory system within
the
costraints of the four fundamental ingredients (representation,
architecture, search and knowledge) of a naturally or artificially
"intelligent" agent,
(ii) to test the existence of the postulated
gustato-spinal connections, and (iii) to delineate as far as
possible, using neuroanatomical and electrophysiological techniques,
the neural mechanism/s involved in the control of goal-directed
(foraging) behavior.
ORGANIZATIONAL CONSIDERATIONS
I. REPRESENTATION
Representation refers to the translation of a particular task
into information structures and information processes and determines
to a great extent the efficiency and efficacy with which a solution
to the task can be generate<i4.
The elaborate and highly sensitive
taste system of an ictalurid catfish consists of an extensive array
of chemo- and mechanosensory receptors distributed over most of the
extraoral
as
well
as oral regions of
the
epithelium2, 5.
Peripherally, branches of the facial nerve (which innervates all
extraoral taste buds) re~ond to a wide range of stimulus (amino
acids) concentrationEP, 7 , l:Si..e. from 10-% to 10-3 M.
The taste
activity however, adapts rapidly (phasic response) to ongoing
stimulation of the same concentration (Fig.
2) and responds
tonically only to continuously increasing concentrations of stimuli,
such as L-arginine and L-alanine.
rp
ros
Fig.:t. Integrated, facial taste recordings to continuous application of amino acids to the palate and nasal barbel showing the
phasic nature of the taste responses of the ramus palatinus (rp)
and ramus ophthalmicus superficialis (ros), respectively .
405
Gustatory information from the extraoral and oral epithelium is
"pipelined"
into
two
separate
subsystems,
facial
and
glossopharyngeal-vagal, respectively.
Each sUbsystem processes a
subset
of the incoming information
(extraoral or oral) and
coordinates a different component of food acquisition.
Food search
is accomplished by the extraoral subsystem,
while
selective
2
ingestion is accomplished by the oral subsystem
(Fig. 3).
The
extraoral gustatory information terminates in the facial lobe where
it is represented as a well-defin~d topographic map 9,10 , while the
oral information terminates in the adjacent vagal lobe where it is
represented as a relatively diffuse map 11.
II. ARCHITECTURE
The
information represented in an information
structure
eventually requires an operating frame (architecture) within which
to select and carry out the various processes. In ictalurid catfish,
partially processed information from the primary gustatory centers
(facial and vagal lobes) in the medullary region of the brainstem
converges along ascending and descending pathways (Fig. 4). One of
the centers in the ascending pathways is the secondary gustatory
nucleus
in
the isthmic region which is connected
to
the
corresponding
nucleus
of
the opposite side
via
a
large
commissurel 2 ,13.
Facial and vagal gustatory information crosses
over to the opposite side via this commissure thus making it
possible for neurons to extract information about interbarbel or
interflank intensity differences.
Although neurons in this region
are known to have large receptive fields 14 , the exact function of
this large commissural nucleus is not yet clearly established.
It is quite clear, however, that gustatory information is at
first "pipelined" into separate regions where it is processed in
parallel 15 before converging onto neurons in the ascending (isthmic)
and descending (reticular) processors as well as other regions
within the medulla.
The "pipelined" architecture underscores the
need for differential processing of subsets of sensory inputs which
are consequently integrated to coordinate temporal transitions
between the various components of goal-directed behavior.
III. SEARCH
An important task underlying all "intelligent" goal directed
activity is that of search.
In artificial systems this involves
application of several general problem-solving methods such as
means-end analysis, generate and test methods and heuristic search
methods.
No attempt, as yet, has been made to fit any of these
models to the food-tracking behavior of the catfish.
However,
behavioral
observations
suggest
that the
catfish
uses
a
combinatorial approach resulting in a different yet optimal foraging
strategy each time ~
What is interesting about biological models is that the
intrinsic search strategy is expressed extrinsically by the behavior
of the animal which, with a few precautions, can be observed ~uite
easily. In addition, simple manipulations of either the animal or
its environment can provide interesting data about the search
406
SENSORY
BRA IN
INPUT
Ie xtra
5
t
e
-=-=b
(0 r a I)
-
..?: .-
~
~I'- ?
-.
t
-.
,
\
__
",to
u
d
s
Jr
VII
~ ac l al
lobe
p
IX
X
vagal
lobe
~~~~~
~~v~ ?(!l](L~~
[pl~??~~~?(:l
? __
~!pa~Cil[1.
!p[3C!J(!J~~~C!J(3
Fig. 4.
OUT PUT
r--:r
a
I-ora I
Fig. 3.
BEHAVIORAL
F ISH
food
searc h
and
DIe k uO
selectIve
IngestIon
I
407
strategy/ies being used by the animal, which in turn can highlight
some of the computational (neuronal) search strategies adopted by
the brain e.g. the catfish seems to minimize the probability of
failure by continuously interacting with the environment so as to be
able to correct any computational or knowledge-based errors.
IV. KNOWLEDGE
If
an "intelligent" goal-directed system resets to zero
knowledge before each search trial, its success would depend
entirely upon the information obtained over the time period of a
search.
Such a system would also require a labile architecture to
process the varying sets of information generated during each
search. For such a system, the solution space can become very large
and given the constraints of time (generally an important cri te:don
in biological systems) this can lead to continuous failure.
For
these reasons, knowledge becomes an important ingredient of an
"intelligent" agent since it can keep the search under control.
For the gustatory system of the catfish
too,
randomly
accessable knowledge, in combination with the immediately available
information about the target, may playa critical role in the
adoption of a successful search strategy.
Although a significant
portion of this knowledge is probably learned, it is not yet clear
where and how this knowledge is stored in the catfish brain.
The
reduction in the solution space for a catfish which has gradually
learned to find food in its environment may be attributed to the
increase in the amount of knowledge, which to some extent may
involve a restructuring of the neural networks during development.
EXPERIMENTAL METHODS
The methods employed for the present study are only briefly
introduced here.
Neuroanatomical tracing techniques exploit the
phenomenon of axonal transport. Crystals of the enzyme, horseradish
peroxidase (HRP) or some other substance, when injected at a small
locus in the brain, are taken up by the damaged neurons and
transported anterogradely and retrogradely from cell bodies and/or
axons at the injection site.
In the present study,
small
superficial injections of HRP (Sigma, Type VI) were made at various
loci in the facial lobe (FL) in separate animals.
After a survival
period of 3 to 5 days, the animals were sacrificed and the brains
sectioned and reacted for visualization of the neuronal tracer.
In
this manner, complex neural circuits can be gradually delineated.
Electrophysiological recordings from neurons in the central
nervous system were obtained using heat-pulled glass micropipettes.
These glass electrodes had a tip diameter of approximately 1 um and
an impedance of less than 1 megohm when filled with an electrolyte
(3M KCl or 3M Nacl).
Chemical stimulation of the receptive fields was accomplished
by injection of stimuli (amino acids, amino acid mixtures and liver
or bait-extract solutions) into a continuous flow of well-water over
the receptive epithelium.
Tactile stimulation was performed by
gentle strokes of a sable hair brush or a glass probe.
408
EXPERIMENTAL OBSERVATIONS
Injections of HRP into the spinal cord labelled two relevant
populations of cells, (i) in the ipsilateral reticular formation at
the level of the facial lobe (FL), and (ii) a few large scattered
cells within the ipsilateral, rostral portion OI the lateral lobule
of the FL (Fig. 5). Injection of HRP at several sites within the FL
resulted in the identification of a small region in the FL from
where anterogradely filled fibers project to the reticular formation
(Fig. 5).
Superimposition of these injection sites onto the
anatomical map OI the extraoral surface of the catfish indicated
that this small region, within the facial lobe, corresponds to the
snout region of the extraoral surface.
FACIO-RETICULAR PROJECTIONS
FACIO- & RETICULO -SPINAL PROJECTIONS
injection site
?r
1
aID
SpC
1
,
injection site
2
3
CB =cerebellum
LL =lateral line lobe
Fig. 5. Schematic chartings showing
labelled-cell bodies(squares) and fibers
transverse sections through the medulla.
4
-
(dots)
in
409
FL
facial lobe
RF = reticular formation
SpC= spinal cord
VL = vagal lobe
Fl
=
VL
\.
Qjg LlP~ ~PC
FLANK
RF
SNOUT
~---------~
Fig. 6A.
WATER SaUIRT -HEAD ?
+
GLIDING TOUCH -FLANK
!III1/1 III11I11 J11I11II
LIVER EXTRACT
-SNOUT
IIHI~I J+~ ,. 'I I I I ! II
r
1-,'1 1?1l(I
UJi!
-iJ,l" ILL !~~ I I I I
'?n
ill'!Pll r~1"r MIl?l "", i II
I I , I? J.
I L
I~
if
I,II
1/' II"11'
I'//1I I,. II
AMINO ACID MIXTURE ?
(Receptive
fields)
(Sample unit responses)
LI.J 11111,ltll!,llllliltUII.I~I'III~UII AU) Jlldll,IIIl~lIijkml!II,1.
:
CONTROL
IJ,[ I!11111,11II1".!L
?
til.llluI1Inlll.h!LIIIII,hll, 1,1/L . I LI ~ iLL.! L..IJ" ,1.
AMINO ACID MIXTURE
J ./,
,I lJ
,1,1 \ L [1.. 1" 1,1,1./ d ,II "I..I1LLLlL/.i.LLU~LUt
" ijl,ldqllljl,lJJ\lL,~Lld 1"ltH! I~d II~ 1,,1
TOUCH -SNOUT
I ... ? .
, 11,1" I Lt J iJ,tt.i
? -SNOUT
LlLlII t LI
?
J
"
1 I
?.d.?' ,_J. L.._,I ... ,:!...,J. ..... 1...?.... 1, 1...... 1 UJ"">1.-I",,JrlHk', ,Llk.-I
t
Fig. 6B.
" .+1.1.4.
d II I
Jl;!d.t4.l, .lui,
410
Multiunit
electrophysiological
recordings
from
various
anteroposterior levels of the reticular formation indicated that the
snout region (upper lip and proximal portion of the maxillary
barbels) of the catfish project to a disproportionately large region
of the reticular formation along with a mixed representation of the
flank (Fig. 6A).
Single
unit recordings indicated that some neurons have
receptive fields restricted to a bilateral portion of the snout
region, while others had large receptive fields extending over the
whole flank or over an anteroposterior half of the body (Fig. 6B).
DISCUSSION
The experimental results obtained here suggest that facial lobe
projections to the reticular formation form a functional connection.
The reticular neurons project to the spinal cord and, most likely,
influence
the general cycle of swimming-related activity
of
motoneurons within the spinal cord 16.
The disproportionately large representation of the snout region
within
the
medullary
reticular
formation,
as
determined
electrophysiologically,
is consistent with the anatomical data
indicating that most of the fibers projecting to the reticular
formatirnl originate from cells in that portion of the facial lobe
where the snout region is mapped.
The lateral lobule of the spinal
cord has a second pathway which projects directly into the spinal
cord upto the level of the anterior end of the caudal fin and may
coordinate reflexive turning.
The significance of the present results is best understood when
considered together with previously known information about the
anatomy and electrophysiology of the gustatory
system.
The
information presented above is used to propose a model (Fig. 7) for
a mechanism that may be involved during the homing phase of target
tracking by the catfish.
During homing, which refers to the last
phase of target-tracking during food search, it may be assumed that
the fish is rapidly approaching its target or moving through a steep
signal intensity (stimulus concentration) gradient.
The data
presented above suggest that a neuronal mechanism exists which helps
the catfish to lock on to the target during homing.
This proposal
is based upon the following considerations:
1. Owing to the rapidly adapting response of the peripheral filter,
a tonic level of activity in the facial lobe input can occur only
when the animal is moving through an increasing concentration
gradient of the gustatory stimUlUS.
2. Facial lobe neurons, which receive inputs from the snout region,
project to a group of cells in the reticular formation. Activity in
the facio-reticular pathway causes a suppression in the spontaneous
activity of the reticular neurons.
3. Direct and/or indirect spinal projections from the reticular
neurons are involved in the modulation of activity of those spinal
motoneurons which coordinate swimming. Thus, it may be hypothesized
that during complete suppression of activity in a specific reticulospinal pathway, the fish swims straight ahead, but during excitation
411
of certain reticulospinal neurons the fish
dictated by the pattern of activation.
Fig. 7.
The snout region of
the catfish has special significance
because
of
its
extensive
representation
in the
reticular
formation. In case the fish makes a
random or computational error, while
approaching its target, the snout is
the first region to move out of the
stimulus gradient.
as
.'.
: .:
~.
Thus, the spinal motoneurons, teleologically speaking, "seek" a
gustatory
stimulus in order to suppress activity of certain
reticulospinal neurons, which in turn reduce variations in the
pattern
of
activity of swimming-related spinal
motoneurons.
Accordingly, in a situation where the fish is rapidly approaching ~
target, ie. under the specific conditions of a continuously rising
stimulus concentration at the snout region and an absence of a
stimulus intensity difference across the barbels, there is a locking
of the movement of the body (of the fish) towards the stationary or
moving target (food or prey).
It should be pointed out, however, that the empirical data
available so far, only offers clues to the target-tracking mechanism
proposed here. Clearly, more research is needed to validate this
proposal
and to identify other mechanisms of target-tracking
utilized by this biological system.
This research
T.E. Finger.
was supported in part by NIH Grant
NS15258
to
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whole:1 mediate:1 amino:6 body:4 neuronal:3 site:7 fig:17 tl:1 elaborate:1 scattered:1 cil:1 axon:1 aid:1 reticulospinal:4 herrick:3 ix:1 specific:2 substance:1 showing:2 neurol:3 survival:2 evidence:1 essential:1 intrinsic:1 exists:1 teeter:1 effectively:1 importance:1 te:1 illustrates:2 t4:1 vii:1 lt:1 electrophysiology:1 likely:1 expressed:1 restructuring:1 tracking:9 partially:1 reflex:1 brush:1 corresponds:1 newell:1 determines:1 lth:1 goal:9 trigeminal:1 consequently:1 towards:2 labelled:2 absence:1 hard:1 directionality:2 determined:1 lui:1 specie:2 secondary:1 experimental:3 east:1 indicating:1 select:1 latter:1 liiiii:1 ongoing:1 dept:1 phenomenon:1 |
5,636 | 610 | Parameterising Feature Sensitive Cell
Formation in Linsker Networks in the
Auditory System
Lance C. Walton
University of Kent at Canterbury
Canterbury
Kent
England
David L. Bisset
University of Kent at Canterbury
Canterbury
Kent
England
Abstract
This paper examines and extends the work of Linsker (1986) on
self organising feature detectors. Linsker concentrates on the visual processing system, but infers that the weak assumptions made
will allow the model to be used in the processing of other sensory
information. This claim is examined here, with special attention
paid to the auditory system, where there is much lower connectivity and therefore more statistical variability. On-line training is
utilised, to obtain an idea of training times. These are then compared to the time available to pre-natal mammals for the formation
of feature sensitive cells.
1
INTRODUCTION
Within the last thirty years, a great deal of research has been carried out in an
attempt to understand the development of cells in the pathways between the sensory
apparatus and the cortex in mammals. For example, theories for the development of
feature detectors were forwarded by Nass and Cooper (1975), by Grossberg (1976)
and more recently Obermayer et al (1990).
Hubel and Wiesel (1961) established the existence of several different types of feature sensitive cell in the visual cortex of cats. Various subsequent experiments have
1007
1008
Walton and Bisset
shown that a considerable amount of development takes place before birth (i.e.
without environmental input). This must either be dependent on a genetic predispostion for individual cells to develop in an appropriate way without external
influence, or some low level rules sufficient to create the required cell morphologies
in the presence of random action potentials.
Although there is a great deal of a priori information concerning axon growth
and synapse arborisation (governed by chemical means in the brain), it is difficult
to conceive of a biological system that could use genetic information to directly
manipulate the spatial information about the pre-synaptic target with respect to
the axon with which the synapse is made. However, there is considerable random
activity in the sensory apparatus that could be used to effect synaptic development .
Various authors have constructed models that deal with different aspects of selforganisation of this kind and some have pointed out the value of these types of
cells in pattern classification problems (Grossberg 1976), but either the biological
plausibility of these models is questionable, or the subject of pre-natal development
is not addressed (i.e. without environmental input).
In this paper, the networks of Linsker (1986) will be examined. Although these
networks have been analysed quite extensively by Linsker, and also by Mackay and
Miller (1990), the biological aspects of parameter ranges and choices have only
been touched upon. It is our aim in this paper, to add further detail in this area
by examining the one-dimensional case which represents the auditory pathways.
2
LINSKER NETWORKS
The network is based on a Multi Layer Perceptron, with feed forward connections in
all layers, and la.teral connections (inhibition and excitation) in higher layers. The
neural outputs are sums of the weighted inputs, and the weights develop according
to a constrained Hebbian Rule. Each layer is lettered for reference starting from A
and subsequent layers are lettered B,C,D etc. The superscript M will be used to
refer to an arbitrary layer, and L is used to refer to the previous layer. Each layer has
a set of parameters which are the same for all neurons in that layer . Connectivity
is random but is based on a Gaussian density distribution (exp( -r2/rll?' where
rM is the arbor radius for layer M.
Each layer is a rectangular array of neurons (or vector of neurons for the one dimensional case). The layers are assumed to be large enough so that edge effects
are not important or do not occur . Layers develop one at a time starting from the
B layer. The A layer is an input layer, which is divided into boxes, within each of
which activity is uniform . This is biologically realistic, since sensory neurons fan
out to a number of cells (an average of lOin the cochlea) each of which only take
input from one sensory cell. Hence the input layer for the network acts like a layer
of tonotopically organised neurons.
Parameterising Feature Sensitive Cell Formation in Linsker Networks in the Auditory System
3
NETWORK DEVELOPMENT
The output of a neuron in layer M is given by
F~1rr
= Ra + Rb. L
CnjFp~:(nj)
(1)
j
Where,
7r
indexes a pattern presentation,
The subscript n is used to index the M layer neurons,
R a , Rb are layer parameters,
Fp~:(nj) is the output of the L layer neuron which is pre-synaptic to the j'th input
of the n'th M layer neuron.
The synaptic weights develop according to a constrained Hebbian learning rule,
(.6cn i)7I' = ka + k b .(FnM7r - Ftt).(Fp~:(ni) - Fl)
(2)
Where,
(.6cn d7l' is the change in the i'th weight of neuron n,
k a , kb' FtJ , Frf are layer parameters.
Synaptic weights are constrained to lie within t.he ra.nge (nem - 1, n em ). (In this
work, nem = 0.5)
Linsker (1986a) derives an Ensemble Averaged Development equation which shows
how development depends on the parameters, and how correlations develop between
spatially proximate neurons in layers beyond the first. In so doing, the number of
parameters is reduced from five per layer to two per layer, and therefore the equation
is a very useful aid in understanding the self-organising nature of this model. The
development equation is
QL
(.)
(,),c nJ'
. }., +}' _ + ~.
~)
pren, .pre nJ
(3)
Cni = '\1
\.::!'Cn
NM
Qt ==
< (Fl7r - FL).(F/ 7r - fL)
(4)
f6
Where,
N M is the number of synaptic connections to an M layer neuron,
FL is the a.verage output activity in the L layer,
ka + kb.(Ra - F~1) . (FL - Frf)
K1
2
NMkbRbfo
=
(5)
K'l = fL .(FL'l- Frf)
~
f&
is a unit of activity used to normalise the two point correlation function
this work
is chosen to set Qt = 1
fJ
(6)
fa
Qb.
Angle brackets denote an a.verage ta.ken over the ensemble of input patterns.
In
1009
1010
Walton and Bisset
4
MORPHOLOGICAL REGIMES
From equation 3, an expression can be found for the average weight value c in a
layer, and therefore certain properties of the system can be described. Although
Mackay and Miller (1990) have described the regimes with the aid of eigenvalues
and eigenfunctions, there is a much simpler method which will provide the same
information.
For an all-excitatory (AE) layer, the average weight value is equal to 71. em . Since
all weights are equal to llem, the summation in equation 3 can be re-written
QL
N
- h
rB
n em ? ~
L.Jj
pre(ni).pre(nj) = ll em ? M?q, were q = 2 .N c? J rc
2 + 2 ?
rB
A similar expression can be found for all-inhibitory (AI) layers, and therefore the
J(l - 1<2 plane can be sub-divided into three regions which will yield AE cells, AI
cells, and mixed-mode cells (see figure 1).
The plane can be divided further for the mixed-mode cell type in the C layer. Oncenter and off-center cells develop close to the AE and AI boundaries respectively.
Mackay and Miller have shown why these cells develop and have placed a theoretical
lower bound on c which agrees with experimental data. However, in so doing the
effect of the intercept on the /{2 axis was deemed small, due to a large number
of synaptic connections. This approximation depends upon the large number of
connections between the Band C layers. In the auditory case, the number of
connections is smaller, and it is possible that this assumption no longer holds.
From equation 3, it can be seen that movement into the On-Centre region from
the AE region, causes the value of ~.
(.).c n )? t.o decrease. This has the
L.J} QL
pre (.)
7H .pre n}
effect of moving the intercept of the constant c line from J{:? = q towards /{ 2 = O.
J{2 finally reaches 0 when c 0, and then begins to move back towards ij as the AI
regime is approached.
=
This has two potentially important effects. Firstly, it means that the tolerance of
/{ 2 varies with J( 1; for a particular value of /{l, there are upper and lower limits on
the value of K2 which will allow maturation of on-center cells. This range of values
(i.e. the difference between the limits) varies in a linear way with K I, but the ratio
of the range to a value of K2 which is within the range (i.e. the center value) is not
linear with /{l. Here, tolerance is defined as that ratio. Secondly, there is a region
of negative /{2 where the nature of the cell morphology which will be produced is
unknown.
It is therefore important that K'2 should be larger than this value in order to produce
On-Center or Off-Center cells reliably. Mackay and Miller use IK21 -+ 00 in their
analysis. Unfortunately, this would require the fundamental network parameter
Fl -+ 00 from equation 6, and therefore is an unsuitable choice. It is reasonable
to assume that Fl is of the same order as FL, and hence an order for 1<2 can be
established. For a concrete example, assume inputs are binary (giving Qfi = 0.25)
and Fl
FL x 1.2, this will ensure K2 < 0 (equation 6) while adhering to the
assumption made above. Equation 6 now gives the order for K2 = 0.2.
=
To find the value of ij, which will place a lower bound on IK2L a particular system
should be chosen. The auditory system is chosen here.
Parameterising Feature Sensitive Cell Formation in Linsker Networks in the Auditory System
Kl
~~~~~~-----------------------------------.K2
o
Figure 1: Graph of Morphological Regions for C Layer
There are approximately 3000 inner hair cells in the cochlea, each of which fans out
to an average of 10 neurons (which sets our box size p = 10). These neurons take
input from only one hair cell. The anteroventral cochlea nucleus takes input from
1000 in Linsker
this layer of cells, with a fan in N B ~ 50 (c.f. the value of N B
(1986a)). The assumption is made that the three sections of the cochlea nucleus
each contain approximately the same number of cells. With this smaller number of
connections, the correlation function for this layer is somewhat coarser, and does
not follow the theoretical curve for the continuum limit so well.
=
In addition, the on-center cells found in the posteroventral cochlea nucleus and
the dorsal nucleus have centres with a tuning curve response Q of about 2.5 which
corresponds to about 2000 B layer cells . If it is assumed that the surround of the cell
is half the width of the core, then there is a total "c ~ 3000 neurons. Simulations
here use Nc = 100 which is a realistic number of connections in the context of a
one-dimensional network .
In general, the arbor radius increases as layers become closer to the cortex. From
Linsker, rc /"B = 3. "B is therefore equal to 1000. This yields the average number
of connections to a given B cell from a. particular A box being approximately unity,
which agrees well with the condition expressed by Linskel'.
Using the expression above, if can be calculated as approximately 1.5 x 10- 3 . This
value is certainly insignificant with respect to the value of f{2 = 0.2 quoted earlier,
and therefore any effects due to the summation term in equation 3 can be ignored
in the calculation of c for this system. This means that the original approximation
still holds even in this low connectivity case.
1011
1012
Walton and Bisset
5
SIMULATION RESULTS
A network was trained using the connectivity sta.ted above to give various values of
C with f{2 = 0.2. To obtain an idea of the total number of presentations that were
required to train the network , without any artifacts that might be produced as a
result of batch training, the original network equations were used. In all of these
simulations , R a , Ftt = 0 so that the value of f{ 1 could be easily controlled.
The findings were that the maximum value of kb was about 10- 3 which required
2.5 million pattern presentations to mature the network. With this value, on-center
cells with an average weight value less than about 0.3 would not mature. However
as the value of kb was decreased (keeping f{l constant), the value of c could be
made lower, at the expense of more pattern presentations. The figures obtained
for the maturation of feature sensitive cells are extremely biologically realistic in
the light of the number of pattern presentations available to an average mammal.
For example , the foetal cat has sufficient time for about 25 million presentations
(assuming 10 presentations per second) .
6
CONCLUSION
We have shown that the class of network developed by Linsker is extendable to
the auditory system where the number and density of synapses is considerably
smaller than in the visual case. It has also been shown that the time for layer
maturation by this method is sufficiently short even for mammals with a relatively
short gestation period. and therefore should also be sufficient in mammals with
longer foetal development times . We conclude that the model is therefore a good
representation of feature detector development in the pre-natal mammal.
References
Grossberg S. (1976) - On the Development of Feature Detectors in the Visual
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Grossberg S. (1976) - Adaptive Pattern Classification and Universal Recoding : 1
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Hu bel D. H. and Wiesel T . N. (1961) - Receptive Fields, Binocular Interaction and
Functional Architechture in the Cat 's Visual Cortex, Journal of Physiology, 160,
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Kalil R. E. (1989) - Synapse Formation In The Developing Brain, Scientific American, Decembe r 1989, 38 - 45
Klinke R . (1986) - Physiology of Hearing. In Schmidt R . 'N . (ed .), Fundamentals
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MacKay D. J. C. and Miller K. D. (1990) - Analysis of Linsker 's Simulations of
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C. (1979) - Development of Ocularity Domains and Growth
Parameterising Feature Sensitive Cell Formation in Linsker Networks in the Auditory System
Behaviour of Axon Terminals, Biological Cybernetics, :]2, 49 - 62
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(USA), 83, 7508 - 7512
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Emergence of Orientation-Selective Cells, Proceedings of the National Academy of
Sciences (USA), 83, 8390 - 8394
Linsker R. (1986c) - From Basic Network Principles To Neural Architecture: Emergence of Orientation-Columns, Proceedings of the National Academy of Sciences
(USA), 83, 8779 - 8783
Nass M. M. and Cooper L. N. (1975) - A Theory for the Development of Feature
Detecting Cells in the Visua.l Cortex, Biological Cybernetics, 19, 1 - 18
Obermayer K. Ritter H. and Schulten K. (1990) - Development and Spatial
Structure of Cortical Feature Maps: A Model Study NIPS, 3, 11 - 17
Sloman A. (1989) - On Designing a Visual System (Towards a Gibsonian Computational Model of Vision) Journal of Experimental and Theoretical Artificial Intelligence, 1, 289 - 337
Tanaka S. (1990) - Intera.ction among Ocularity, Retinotopy and On-Center/Off
Center Pathways During Development NIPS, 3, 18 - 25
1013
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5,637 | 6,100 | Active Nearest-Neighbor Learning in Metric Spaces
Aryeh Kontorovich
Department of Computer Science
Ben-Gurion University of the Negev
Beer Sheva 8499000, Israel
Sivan Sabato
Department of Computer Science
Ben-Gurion University of the Negev
Beer Sheva 8499000, Israel
Ruth Urner
Max Planck Institute for Intelligent Systems
Department for Empirical Inference
T?bingen 72076, Germany
Abstract
We propose a pool-based non-parametric active learning algorithm for general metric spaces, called MArgin Regularized Metric Active Nearest Neighbor
(MARMANN), which outputs a nearest-neighbor classifier. We give prediction
error guarantees that depend on the noisy-margin properties of the input sample,
and are competitive with those obtained by previously proposed passive learners.
We prove that the label complexity of MARMANN is significantly lower than
that of any passive learner with similar error guarantees. Our algorithm is based
on a generalized sample compression scheme and a new label-efficient active
model-selection procedure.
1
Introduction
In this paper we propose a non-parametric pool-based active learning algorithm for general metric
spaces, which outputs a nearest-neighbor classifier. The algorithm is named MArgin Regularized
Metric Active Nearest Neighbor (MARMANN). In pool-based active learning [McCallum and Nigam,
1998] a collection of random examples is provided, and the algorithm can interactively query an
oracle to label some of the examples. The goal is good prediction accuracy, while keeping the label
complexity (the number of queried labels) low. MARMANN receives a pool of unlabeled examples
in a general metric space, and outputs a variant of the nearest-neighbor classifier. The algorithm
obtains a prediction error guarantee that depends on a noisy-margin property of the input sample, and
has a provably smaller label complexity than any passive learner with a similar guarantee.
The theory of active learning has received considerable attention in the past decade [e.g., Dasgupta,
2004, Balcan et al., 2007, 2009, Hanneke, 2011, Hanneke and Yang, 2015]. Active learning has
been mostly studied in a parametric setting (that is, learning with respect to a fixed hypothesis class
with a bounded capacity). Various strategies have been analyzed for parametric classification [e.g.,
Dasgupta, 2004, Balcan et al., 2007, Gonen et al., 2013, Balcan et al., 2009, Hanneke, 2011, Awasthi
et al., 2013].An active model selection procedure has also been developed for the parametric setting
Balcan et al. [2010]. However, the number of labels used there depends quadratically on the number
of possible model classes, which is prohibitive in our non-parametric setting.
The potential benefits of active learning for non-parametric classification in metric spaces are less well
understood. The paradigm of cluster-based active learning [Dasgupta and Hsu, 2008] has been shown
to provide label savings under some distributional clusterability assumptions [Urner et al., 2013,
Kpotufe et al., 2015]. Certain active learning methods for nearest neighbor classification are known
to be Bayes consistent [Dasgupta, 2012], and an active querying rule, based solely on information in
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
the unlabeled data, has been shown to be beneficial for nearest neighbors under covariate shift [Berlind
and Urner, 2015]. Castro and Nowak [2007] analyze minimax rates for a class of distributions in
Euclidean space, characterized by decision boundary regularity and noise conditions. However, no
active non-parametric strategy for general metric spaces, with label complexity guarantees for general
distributions, has been proposed so far. Here, we provide the first such algorithm and guarantees.
The passive nearest-neighbor classifier is popular among theorists and practitioners alike [Fix and
Hodges, 1989, Cover and Hart, 1967, Stone, 1977, Kulkarni and Posner, 1995]. This paradigm is
applicable in general metric spaces, and its simplicity is an attractive feature for both implementation
and analysis. When appropriately regularized [e.g. Stone, 1977, Devroye and Gy?rfi, 1985, von
Luxburg and Bousquet, 2004, Gottlieb et al., 2010, Kontorovich and Weiss, 2015] this type of learner
can be made Bayes consistent. Another desirable property of nearest-neighbor-based methods is
their ability to generalize at a rate that scales with the intrinsic data dimension, which can be much
lower than that of the ambient space [Kpotufe, 2011, Gottlieb et al., 2014a, 2016a, Chaudhuri and
Dasgupta, 2014]. Furthermore, margin-based regularization makes nearest neighbors ideally suited
for sample compression, which yields a compact representation, faster classification runtime, and
improved generalization performance [Gottlieb et al., 2014b, Kontorovich and Weiss, 2015]. The
resulting error guarantees can be stated in terms of the sample?s noisy-margin, which depends on the
distances between differently-labeled examples in the input sample.
Our contribution. We propose MARMANN, a non-parametric pool-based active learning algorithm
that obtains an error guarantee competitive with that of a noisy-margin-based passive learner, but
can provably use significantly fewer labels. This is the first non-parametric active learner for general
metric spaces that achieves prediction error that is competitive with passive learning for general
distributions, and provably improves label complexity.
Our approach. Previous passive learning approaches to classification using nearest-neighbor rules
under noisy-margin assumptions [Gottlieb et al., 2014b, 2016b] provide statistical guarantees using
sample compression bounds [Graepel et al., 2005]. The finite-sample guarantees depend on the
number of noisy labels relative to an optimal margin scale. A central challenge in the active setting is
performing model selection (selecting the margin scale) with a low label complexity. A key insight
that we exploit in this work is that by designing a new labeling scheme for the compression set, we
can construct the compression set and estimate its error with label-efficient procedures. We obtain
statistical guarantees for this approach using a generalized sample compression analysis. We derive a
label-efficient (as well as computationally efficient) active model-selection procedure. This procedure
finds a good scale by estimating the sample error for some scales, using a small number of active
querying rounds. Crucially, unlike cross-validation, our model-selection procedure does not require a
number of labels that depends on the worst possible scale, nor does it test many scales. This allows
our label complexity bounds to be low, and to depend only on the final scale selected by the algorithm.
Our error guarantee is a constant factor over the error guarantee of the passive learner of Gottlieb
et al. [2016b]. An approach similar to Gottlieb et al. [2016b], proposed in Gottlieb et al. [2014a], has
been shown to be Bayes consistent [Kontorovich and Weiss, 2015]. The Bayes-consistency of the
passive version of our approach is the subject of ongoing work.
Paper outline. We define the setting and notations in Section 2. In Section 3 we provide our main
result, Theorem 3.2, giving error and label complexity guarantees for MARMANN. Section 4 shows
how to set the nearest neighbor rule for a given scale, and Section 5 describes the model selection
procedure. Some of the analysis is omitted due to lack of space. The full analysis is available at
Kontorovich et al. [2016].
2
Setting and notations
We consider learning in a general metric space (X , ?), where X is a set and ? is the metric on X .
Our problem setting is that of classification of the instance space X into some finite label set Y.
Assume that there is some distribution D over X ? Y, and let S ? Dm be a labeled sample of size m,
where m is an integer. Denote the sequence of unlabeled points in S by U(S). We sometimes treat S
and U(S) as multisets, since the order is unimportant. The error of a classifier h : X ? Y on D is
denoted err(h, D) := P[h(X) 6= P
Y ], where (X, Y ) ? D. The empirical error on a labeled sample
1
S instantiates to err(h, S) = |S|
I[h(X) 6= Y ]. A passive learner receives a labeled sample Sin
as input. An active learner receives the unlabeled part of the sample Uin := U(Sin ) as input, and
2
is allowed to adaptively select examples from Uin and request their label from Sin . When either
? : X ? Y, with the goal of achieving a low err(h,
? D). An
learner terminates, it outputs a classifier h
additional goal of the active learner is to achieve a performance competitive with that of the passive
learner, while querying considerably fewer labels.
The diameter of a set A ? X is defined by diam(A) := supa,a0 ?A ?(a, a0 ). Denote the index of the
closest point in U to x ? X by ?(x, U ) := argmini:xi ?U ?(x, xi ). We assume here and throughout
this work that when there is more than one minimizer for ?(x, xi ), ties are broken arbitrarily (but
in a consistent fashion). For a set Z ? X , denote ?(Z, U ) := {?(z, U ) | z ? Z}. Any labeled
sample S = ((xi , yi ))i?[k] naturally induces the nearest-neighbor classifier hnn
S : X ? Y, via
hnn
(x)
:=
y
.
?(x,U(S))
S
For x ? X , and t > 0, denote by ball(x, t) the (closed) ball of radius t around x: ball(x, t) :=
{x0 ? X | ?(x, x0 ) ? t}. The doubling dimension, the effective dimension of the metric space, which
controls generalization and runtime performance of nearest-neighbors [Kpotufe, 2011, Gottlieb et al.,
2014a], is defined as follows. Let ? = ?(X ) be the smallest number such that every ball in X
can be covered by ? balls of half its radius, where all balls are centered at points of X . Formally,
?(X ) := min{? ? N : ?x ? X , r > 0, ?x1 , . . . , x? ? X : ball(x, r) ? ??i=1 ball(xi , r/2)}.
Then the doubling dimension of X is defined by ddim(X ) := log2 ?. In line with modern literature,
we work in the low-dimension, big-sample regime, where the doubling dimension is assumed to
be constant and hence sample complexity and algorithmic runtime may depend on it exponentially.
This exponential dependence is unavoidable, even under margin assumptions, as previous analysis
[Kpotufe, 2011, Gottlieb et al., 2014a] indicates.
0
A set A ? X is t-separated if inf a,a0 ?A:a6
S =a0 ?(a, a ) ? t. For A ? B ? X , the set A is a
t-net of B if A is t-separated and B ? a?A ball(a, t). Constructing a minimum size t-net for
a general set B is NP-hard [Gottlieb and Krauthgamer, 2010], however efficient procedures exist
for constructing some t-net [Krauthgamer and Lee, 2004, Gottlieb et al., 2014b]. The size of
any t-net is at most 2ddim(B) times the smallest possible size (see Kontorovich et al. [2016]). In
ddim(X )+1
addition, the size of any t-net is at most ddiam(B)/te
[Krauthgamer and Lee, 2004].
Throughout the paper, we fix a deterministic procedure for constructing a t-net, and denote its
output for a multiset U ? X by Net(U, t). Let Par(U, t) be a partition of X into regions induced
by Net(U, t), that is: for Net(U, t) = {x1 , . . . , xN }, define Par(U, t) := {P1 , . . . , PN }, where
Pi = {x ? X | ?(x, Net(U, t)) = i}. For t > 0, denote N (t) := |Net(Uin , t)|. For a labeled
multiset S ? X ? Y and y ? Y, denote S y := {x | (x, y) ? S}; in particular, U(S) = ?y?Y S y .
3
Main results
Non-parameteric binary classification admits performance guarantees that scale with the sample?s
noisy-margin [von Luxburg and Bousquet, 2004, Gottlieb et al., 2010, 2016b]. We say that a labeled
multiset S is (?, t)-separated, for ? ? [0, 1] and t > 0 (representing a margin t with noise ?), if one
can remove a ?-fraction of the points in S, and in the resulting multiset, points with different labels
are at least t-far from each other. Formally, S is (?, t)-separated if there exists a subsample S? ? S
? ? ?|S| and ?y1 6= y2 ? Y, a ? S?y1 , b ? S?y2 , we have ?(a, b) ? t. For a given
such that |S \ S|
labeled sample S, denote by ?(t) the smallest value ? such that S is (?, t)-separated. Gottlieb et al.
[2016b] propose a passive learner with the following guarantees as a function of the separation of S.
Setting ? := m/(m ? N ), define the following form of a generalization bound:
s
?((N + 1) log(mk) + log( 1? ))
2 (N + 1) log(mk) + log( 1? )
3
GB(, N, ?, m, k) := ? +
+?
.
3
m?N
m?N
2
Theorem 3.1 (Gottlieb et al. [2016b]). Let m be an integer, Y = {0, 1}, ? ? (0, 1). There exists a
passive learning algorithm that returns a nearest-neighbor classifier hnn
Spas , where Spas ? Sin , such
that, with probability 1 ? ?,
err(hnn
Spas , D) ?
GB(?(t), N (t), ?, m, 1).
min
t>0:N (t)<m
The passive algorithm of Gottlieb et al. [2016b] generates Spas of size approximately N (t) for
the optimal scale t > 0 (found by searching over all scales), removing the |Sin |?(t) points that
3
obstruct the t-separation between different labels in Sin , and then selecting a subset of the remaining
labeled examples to form Spas , so that the examples are a t-net for Sin . We propose a different
approach for generating a compression set for a nearest-neighbor rule. This approach, detailed in the
following sections, does not require finding and removing all the obstructing points in Sin , and can
be implemented in an active setting using a small number of labels. The resulting active learning
algorithm, MARMANN, has an error guarantee competitive with that of the passive learner and a
label complexity that can be significantly lower. Our main result is the following guarantee for
MARMANN.
Theorem 3.2. Let Sin ? Dm , where m ? max(6, |Y|), ? ? (0, 14 ). Let S? be the output of
? Sin ), and
? := hnn and ? := err(h,
? := |S|.
? Let h
MARMANN(Uin , ?), where S? ? X ? Y, and let N
?
S
? := GB(?
? , ?, m, 1). With a probability of 1 ? ? over Sin and randomness of MARMANN,
denote G
, N
? D) ? 2G
??O
err(h,
min
GB(?(t), N (t), ?, m, 1) ,
t>0:N (t)<m
and the number of labels from Sin requested by MARMANN is at most
1
1
3 m
?
O log ( )
log( ) + mG
.
?
?
?
G
G
Here O(?) hides only universal numerical constants.
To observe the advantages of MARMANN over a passive learner, consider a scenario in which
?
the upper bound GB of Theorem 3.1, as well as the Bayes error of D, are of order ?(1/ m).
?
? = ?(1/ m) as well. Therefore, MARMANN obtains a prediction error guarantee of
Then G
?
? ?m) labels instead of m. Moreover,
?(1/ m), similarly to the passive learner, but it uses only ?(
no learner that selects labels randomly from Sin can compete with MARMANN: In Kontorovich
et al. [2016] we adapt an argument of Devroye et al. [1996] to show that for any passive learner
? ?m) random labels from Sin , there exists a distribution D with the above properties,
that uses ?(
? ?1/4 ), a decay rate which is
for which the prediction error of the passive
? learner in this case is ?(m
almost quadratically slower than the O(1/ m) rate achieved by MARMANN. Thus, the guarantees
of MARMANN cannot be matched by any passive learner.
MARMANN operates as follows. First, a scale t? > 0 is selected, by calling t? ? SelectScale(?),
where SelectScale is our model selection procedure. SelectScale has access to Uin , and queries labels
from Sin as necessary. It estimates the generalization error bound GB for several different scales,
and executes a procedure similar to binary search to identify a good scale. The binary search keeps
the number of estimations (and thus requested labels) small. Crucially, our estimation procedure
is designed to prevent the search from spending a number of labels that depends on the net size
of the smallest possible scale t, so that the total label complexity of MARMANN depends only
on error of the selected t?. Second, the selected scale t? is used to generate the compression set by
calling S? ? GenerateNNSet(t?, [N (t?)], ?), where GenerateNNSet is our compression set generation
procedure. For clarity of presentation, we first introduce in Section 4 the procedure GenerateNNSet,
which determines the compression set for a given scale, and then in Section 5, we describe how
SelectScale chooses the appropriate scale.
4
Active nearest-neighbor at a given scale
The passive learner of Gottlieb et al. [2014a, 2016b] generates a compression set by first finding
and removing from Sin all points that obstruct (?, t)-separation at a given scale t > 0. We propose
below a different approach for generating a compression set, which seems more conducive to active
learning: as we show below, it also generates a low-error nearest neighbor rule, just like the passive
approach. At the same time, it allows us to estimate the error on many different scales using few
label queries. A small technical difference, which will be evident below, is that in this new approach,
examples in the compression set might have a different label than their original label in Sin . Standard
sample compression analysis [e.g. Graepel et al., 2005] assumes that the classifier is determined by a
small number of labeled examples from Sin . This does not allow the examples in the compression set
to have a different label than their original label in Sin . Therefore, we require a slight generalization
of previous compression analysis, which allows setting arbitrary labels for examples that are assigned
to the compression set. The following theorem quantifies the effect of this change on generalization.
4
Theorem 4.1. Let m ? |Y| be an integer, ? ? (0, 14 ). Let Sin ? Dm . With probability at least 1 ? ?,
1
if there exist N < m and S ? (X ? Y)N such that U(S) ? Uin and := err(hnn
S , Sin ) ? 2 , then
nn
err(hS , D) ? GB(, N, ?, m, |Y|) ? 2GB(, N, 2?, m, 1).
The proof is similar to that of standard sample compression schemes. If the compression set
includes only the original labels, the compression analysis of Gottlieb et al. [2016b] gives the bound
GB(, N, ?, m, 1). Thus the effect of allowing the labels to change is only logarithmic in |Y|, and
does not appreciably degrade the prediction error.
We now describe the generation of the compression set for a given scale t > 0. Recall that ?(t) is
the smallest value for which Sin is (?, t)-separated. We define two compression sets. The first one,
denoted Sa (t), represents an ideal compression set, which induces an empirical error of at most ?(t),
but calculating it might require many labels. The second compression set, denoted S?a (t), represents
an approximation to Sa (t), which can be constructed using a small number of labels, and induces a
sample error of at most 4?(t) with high probability. MARMANN constructs only S?a (t), while Sa (t)
is defined for the sake of analysis only.
We first define the ideal set Sa (t) := {(x1 , y1 ), . . . , (xN , yN )}. The examples in Sa (t) are the
points in Net(Uin , t/2), and the label of each example is the majority label out of the examples
in Sin to which xi is closest. Formally, {x1 , . . . , xN } := Net(Uin , t/2), and for i ? [N ], yi :=
argmaxy?Y |S y ? Pi |, where Pi = {x ? X | ?(x, Net(U, t/2)) = i} ? Par(Uin , t/2). For i ? [N ],
let ?i := S yi ? Pi . The following lemma bounds the empirical error of hnn
Sa (t) .
Lemma 4.2. For every t > 0, err(hnn
Sa (t) , Sin ) ? ?(t).
Proof. Since Net(Uin , t/2) is a t/2-net, diam(P ) ? t for any P ? Par(Uin , t/2). Let S? ? S
? ? m(1 ? ?(t)), and for
be a subsample that witnesses the (?(t), t)-separation of S, so that |S|
? := U(S).
? Since
? if ?(x, x0 ) ? t then y = y 0 . Denote U
any two points (x, y), (x0 , y 0 ) ? S,
?
maxP ?Par(Uin ,t/2) diam(P ) ? t, for any i ? [N ] all the points in U ? Pi must have the same label in
? Therefore, ?y ? Y such that U
? ? Pi ? S?y ? Pi . Hence |U
? ? Pi | ? |?i |. It follows
S.
X
X
? ? Pi | = |S| ? |S|
? = m ? ?(t).
m ? err(hnn
|?i | ? |S| ?
|U
Sa (t) , Sin ) ? |S| ?
i?[N ]
i?[N ]
Dividing by m we get the statement of the theorem.
Now, calculating Sa (t) requires knowing most of the labels in Sin . MARMANN constructs instead
an approximation S?a (t), in which the examples are the points in Net(Uin , t/2) (so that U(S?a (t)) =
U(Sa (t)) ), but the labels are determined using a bounded number of labels requested from Sin . The
labels in S?a (t) are calculated by the simple procedure GenerateNNSet given in Alg. 1. The empirical
error of the output of GenerateNNSet is bounded in Theorem 4.3 below.1
A technicality in Alg. 1 requires explanation: In MARMANN, the generation of S?a (t) will be split
into several calls to GenerateNNSet, so that different calls determine the labels of different points in
S?a (t). Therefore GenerateNNSet has an additional argument I, which specifies the indices of the
points in Net(Uin , t/2) for which the labels should be returned this time. Crucially, if during the run of
MARMANN, GenerateNNSet is called again for the same scale t and the same point in Net(Uin , t/2),
then GenerateNNSet returns the same label that it returned before, rather than recalculating it using
fresh labels from Sin . This guarantees that despite the randomness in GenerateNNSet, the full
S?a (t) is well-defined within any single run of MARMANN, and is distributed like the output of
GenerateNNSet(t, [N (t/2)], ?), which is convenient for the analysis.
Theorem 4.3. Let S?a (t) be the output of GenerateNNSet(t, [N (t/2)], ?). With a probability at least
?
nn
1 ? 2m
2 , we have err(hS , Sin ) ? 4?(t). Denote this event by E(t).
1
In the case of binary labels (|Y| = 2), the problem of estimating Sa (t) can be formulated as a special case
of the benign noise setting for parametric active learning, for which tight lower and upper bounds are provided
in Hanneke and Yang [2015]. However, our case is both more general (as we allow multiclass labels) and more
specific (as we are dealing with a specific hypothesis class). Thus we provide our own procedure and analysis.
5
Algorithm 1 GenerateNNSet(t, I, ?)
input Scale t > 0, a target set I ? [N (t/2)], confidence ? ? (0, 1).
output A labeled set S ? X ? Y of size |I|
{x1 , . . . , xN } ? Net(Uin , t/2), {P1 , . . . , PN } ? Par(Uin , t/2), S ? ()
for i ? I do
if y?i has not already
been calculated
for Uin with this values of t then
Draw Q := 18 log(2m3 /?) points uniformly at random from Pi and query their labels.
Let y?i be the majority label observed in these Q queries.
end if
S ? S ? {(xi , y?i )}.
end for
Output S
Proof. By Lemma 4.2, err(hnn
Sa (t) , Sin ) ? ?(t). In Sa (t), the labels assigned to each point in
Net(Uin , t/2) are the majority labels (based on Sin ) of the points in the regions in Par(Uin , t/2).
Denote the majority label for region Pi by yi := argmaxy?Y |S y ? Pi |. We now compare these
labels to the labels y?i assigned by Alg. 1. Let p(i) = |?i |/|Pi | be the fraction of points in Pi which
are labeled by the majority label yi . Let p?(i) be the fraction of labels equal to yi out of those queried
by Alg. 1 in round i. Let ? := 1/6. By Hoeffding?s inequality and union bounds, we have that with a
Q
?
probability of at least 1 ? N (t/2) exp(? 18
p(i) ? p(i)| ? ?.
) ? 1 ? 2m
2 , we have maxi?[N (t/2)] |?
0
0
Denote this ?good? event by E . We now prove that E ? E(t). Let J ? [N (t/2)] = {i | p?(i) > 12 }.
It can be easily seen that y?i = yi for all i ? J. Therefore, for all x such that ?(x, U(Sa (t))) ? J,
nn
nn
hnn
/ J] + err(hnn
S (x) = hSa (t) (x), and hence err(hS , Uin ) ? PX?Uin [?(X, U(Sa (t))) ?
Sa (t) , Uin ).
0
The second term is at most ?(t), and it remains
P to bound the 0first term, on the condition that E 1holds.
1
We have PX?U [?(X, U(Sa (t))) ?
/ J] = m
|P
|.
If
E
holds,
then
for
any
i
?
/
J,
p(i)
?
i
i?J
/
2 + ?,
therefore |Pi | ? |?i | = (1 ? p(i))|Pi | ? ( 12 ? ?)|Pi |. Therefore
1?
1 X
1 X
|?i | ?
|Pi |( 12 ? ?) = PX?U [?(X, U(Sa (t))) ?
/ J]( 12 ? ?).
m
m
i?J
/
i?J
/
On the other hand, as in the proof of Lemma 4.2, 1 ?
PX?U [?(X, S) ?
/ J] ?
5
?(t)
1
2 ??
1
m
P
= 3?(t). It follows that under
i?[N (t/2)] |?i | ? ?(t). Thus,
E 0 , err(hnn
S , Uin ) ? 4?(t).
under E 0 ,
Model Selection
We now show how to select the scale t? that will be used to generate the output nearest-neighbor rule.
The main challenge is to do this with a low label complexity: Generating the full classification rule
for scale t requires a number of labels that depends on N (t), which might be very large. We would
like the label complexity of MARMANN to depend only on N (t?) (where t? is the selected scale),
? Therefore, during model selection we can only invest a bounded number
which is of the order mG.
of labels in each tested scale. In addition, to keep the label complexity low, we cannot test all scales.
For t > 0, let S?a (t) be the model that MARMANN would generate if the selected scale were set to t.
Our model selection procedure performs a search, similar to binary search, over the possible scales.
For each tested scale t, the procedure estimates (t) := err(hnn
?a (t) , S) within a certain accuracy, using
S
an estimation procedure we call EstimateErr. EstimateErr outputs an estimate ?(t) of (t), up to a
given accuracy ? > 0, using labels requested from Sin . It draws random examples from Sin , asks for
their label, and calls GenerateNNSet (which also might request labels) to find the prediction error
of hnn
?(t) is set to this prediction error. The number
?a (t) on these random examples. The estimate
S
of random examples drawn by EstimateErr is determined based on the accuracy ?, using empirical
Bernstein bounds [Maurer and Pontil, 2009]. Theorem 5.1 gives a guarantee for the accuracy and
label complexity of EstimateErr. The full implementation of EstimateErr and the proof of Theorem
5.1 can be found in the long version of this paper Kontorovich et al. [2016].
6
Theorem 5.1. Let t, ? > 0 and ? ? (0, 1), and let ?(t) ? EstimateErr(t, ?, ?). Let Q be as defined
in Alg. 1. The following properties (which we denote below by V (t)) hold with a probability of
?
?
1 ? 2m
2 over the randomness of EstimateErr (and conditioned on Sa (t)).
1. If ?(t) ? ?, then (t) ? 5?/4. Otherwise,
4(t)
5
? ?(t) ?
4(t)
3 .
2
2. EstimateErr requests at most
520(Q+1) log( 1040m
)
?? 0
?0
labels, where ? 0 := max(?, (t)).
The model selection procedure SelectScale, given in Alg. 2, implements its search based on the guarantees in Theorem 5.1. First, we introduce some notation. Let G? = mint GB(?(t), N (t), ?, m, 1).
?
We would like MARMANN to obtain a generalization guarantee that is competitive
p with G . Denote
1
2
3
?
?(t) := ((N (t) + 1) log(m) + log( ? ))/m, and let G(, t) := + 3 ?(t) + 2 ?(t). Note that for
all , t,
m
GB(, N (t), ?, m, 1) =
G(, t).
m ? N (t)
When referring to G(?(t), t), G((t), t), or G(?
(t), t) we omit the second t for brevity. Instead of
directly optimizing GB, we will select a scale based on our estimate G(?
(t)) of G((t)).
Let Dist denote the set of pairwise distances in the unlabeled dataset Uin (note that |Dist| < m
2 ).
We remove from Dist some distances, so that the remaining distances have a net size N (t) that is
monotone non-increasing in t. We also remove values with a very large net size. Concretely, define
Distmon := Dist \ {t | N (t) + 1 > m/2} \ {t | ?t0 ? Dist, t0 < t and N (t0 ) < N (t)}.
Then for all t, t0 ? Distmon such that t0 < t, we have N (t0 ) ? N (t). The output of SelectScale is
always a value in Distmon . The following lemma shows that it suffices to consider these scales.
Lemma 5.2. Assume m ? 6 and let t?m ? argmint?Dist G(?(t)). If G? ? 1/3 then t?m ? Distmon .
Proof. Assume by way of contradiction that t?m ? Dist \ Distmon . First, since G(?(t?m )) ? G? ?
N (t? )+1
1/3 we have m?Nm(t? ) log(m) ? 12 . Therefore, since m ? 6, it is easy to verify N (t?m ) + 1 ? m/2.
m
Therefore, by definition of Distmon there exists a t ? t?m with ?(t) < ?(t?m ). Since ?(t) is monotone
over all of t ? Dist, we also have ?(t) ? ?(t?m ). Now, ?(t) < ?(t?m ) and ?(t) ? ?(t?m ) together
imply that G(?(t)) < G(?(t?m )), a contradiction. Hence, t?m ? Distmon .
SelectScale follows a search similar to binary search, however the conditions for going right and for
going left are not complementary. The search ends when either none of these two conditions hold, or
when there is nothing left to try. The final output of the algorithm is based on minimizing G(?
(t))
over some of the values tested during search.
For c > 0, define ?(c) := 1 +
implications
2
3c
+
?3
2c
and ?? (c) :=
? c?(t) ? ?(c) ? G(, t)
and
1
c
+
2
3
+
?3 .
2c
For all t, > 0 we have the
?(t) ? c ? ?? (c)?(t) ? G(, t).
(1)
The following lemma uses Eq. (1) to show that the estimate G(?
(t)) is close to the true G((t)).
Lemma 5.3. Let t > 0, ? ? (0, 1), and suppose that SelectScale calls ?(t) ?
EstimateErr(t, ?(t), ?). Suppose that V (t) as defined in Theorem 5.1 holds. Then 16 G(?
(t)) ?
G((t)) ? 6.5G(?
(t)).
Proof. Under V (t), we have that if ?(t) < ?(t) then (t) ? 54 ?(t). In this case, G((t)) ?
?? (4/5)?(t) ? 4.3?(t), by Eq. (1). Therefore G((t)) ? 3?4.3
(t)). In addition, G((t)) ? 32 ?(t)
2 G(?
(t)). Therefore G((t)) ?
(from the definition of G), and by Eq. (1) and ?? (1) ? 4, ?(t) ? 14 G(?
1
4
G(?
(t)).
On
the
other
hand,
if
?
(t)
?
?(t),
then
by
Theorem
5.1
(t)
? ?(t) ? 43 (t). Therefore
6
5
4
5
G(?
(t)) ? 3 G((t)) and G((t)) ? 4 G(?
(t)). Taking the worst-case of both possibilities, we get
the bounds in the lemma.
The next theorem bounds the label complexity of SelectScale. Let Ttest ? Distmon be the set of
scales that are tested during SelectScale (that is, their ?(t) was estimated).
7
Algorithm 2 SelectScale(?)
input ? ? (0, 1)
output Scale t?
T ? Distmon ,
# T maintains the current set of possible scales
while T =
6 ? do
t ? the median value in T
# break ties arbitrarily
?(t) ? EstimateErr(t, ?(t), ?).
if ?(t) < ?(t) then
T ? T \ [0, t] # go right in the binary search
11
else if ?(t) > 10
?(t) then
T ? T \ [t, ?) # go left in the binary search
else
t0 ? t, T0 ? {t0 }.
break from loop
end if
end while
if T0 was not set yet then
If the algorithm ever went to the right, let t0 be the last value for which this happened, and let
T0 := {t0 }. Otherwise, T0 := ?.
end if
Let TL be the set of all t that were tested and made the search go left
Output t? := argmint?TL ?T0 G(?
(t))
Theorem 5.4. Suppose that the event V (t) defined in Theorem 5.1 holds for all t ? Ttest for the calls
?(t) ? EstimateErr(t, ?(t), ?). If the output of SelectScale is t?, then the number of labels requested
by SelectScale is at most
19240|Ttest |(Q + 1)
38480m2
1
log(
),
G((t?))
?G((t?))
where Q is as defined in Alg. 1.
The following theorem provides a competitive error guarantee for the selected scale t?.
Theorem 5.5. Suppose that V (t) and E(t), defined in Theorem 5.1 and Theorem 4.3, hold for all
values t ? Ttest , and that G? ? 1/3. Then SelectScale outputs t? ? Distmon such that
GB((t?), N (t?), ?, m, 1) ? O(G? ),
where O(?) hides numerical constants only.
The idea of the proof is as follows: First, we show (using Lemma 5.3) that it suffices to prove that
G(?(t?m )) ? O(G(?
(t?))) to derive the bound in the theorem. Now, SelectScale ends in one of two
cases: either T0 is set within the loop, or T = ? and T0 is set outside the loop. In the first case,
neither of the conditions for turning left and turning right holds for t0 , so we have ?(t0 ) = ?(?(t0 ))
(where ? hides numerical constants). We show that in this case, whether t?m ? t0 or t?m ? t0 ,
G(?(t?m )) ? O(G(?
(t0 ))). In the second case, there exist (except for edge cases, which are also
handled) two values t0 ? T0 and t1 ? TL such that t0 caused the binary search to go right, and t1
caused it to go left, and also t0 ? t1 , and (t0 , t1 ) ? Distmon = ?. We use these facts to show that for
t?m ? t1 , G(?(t?m )) ? O(G(?
(t1 ))), and for t?m ? t0 , G(?(t?m )) ? O(G(?
(t0 ))). Since t? minimizes
over a set that includes t0 and t1 , this gives G(?(t?m )) ? O(G(?
(t?))) in all cases.
The proof of the main theorem, Theorem 3.2, which gives the guarantee for MARMANN, is almost
immediate from Theorem 4.1, Theorem 4.3, Theorem 5.5 and Theorem 5.4.
Acknowledgements
Sivan Sabato was partially supported by the Israel Science Foundation (grant No. 555/15). Aryeh
Kontorovich was partially supported by the Israel Science Foundation (grants No. 1141/12 and
755/15) and a Yahoo Faculty award. We thank Lee-Ad Gottlieb and Dana Ron for helpful discussions.
8
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9
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5,638 | 6,101 | Relevant sparse codes with variational information
bottleneck
Matthew Chalk
IST Austria
Am Campus 1
A - 3400 Klosterneuburg, Austria
Olivier Marre
Institut de la Vision
17, Rue Moreau
75012, Paris, France
Gasper Tkacik
IST Austria
Am Campus 1
A - 3400 Klosterneuburg, Austria
Abstract
In many applications, it is desirable to extract only the relevant aspects of data.
A principled way to do this is the information bottleneck (IB) method, where
one seeks a code that maximizes information about a ?relevance? variable, Y ,
while constraining the information encoded about the original data, X. Unfortunately however, the IB method is computationally demanding when data are
high-dimensional and/or non-gaussian. Here we propose an approximate variational scheme for maximizing a lower bound on the IB objective, analogous to
variational EM. Using this method, we derive an IB algorithm to recover features
that are both relevant and sparse. Finally, we demonstrate how kernelized versions
of the algorithm can be used to address a broad range of problems with non-linear
relation between X and Y .
1
Introduction
An important problem, for both humans and machines, is to extract relevant information from complex
data. To do so, one must be able to define which aspects of data are relevant and which should be
discarded. The ?information bottleneck? (IB) approach, developed by Tishby and colleagues [1],
provides a principled way to approach this problem. The idea behind the IB approach is to use
additional ?variables of interest? to determine which aspects of a signal are relevant. For example, for
speech signals, variables of interest could be the words being pronounced, or alternatively, the speaker
identity. One then seeks a coding scheme that retains maximal information about these variables of
interest, constrained on the information encoded about the input.
The IB approach has been used to tackle a wide variety of problems, including filtering, prediction and
learning [2-5]. However, it quickly becomes intractable with high-dimensional and/or non-gaussian
data. Consequently, previous research has primarily focussed on tractable cases, where the data
comprises a countably small number of discrete states [1-5], or is gaussian [6].
Here, we extend the IB algorithm of Tishby et al. [1] using a variational approximation. The algorithm
maximizes a lower bound on the IB objective function, and is closely related to variational EM.
Using this approach, we derive an IB algorithm that can be effectively applied to ?sparse? data in
which input and relevance variables are generated by sparsely occurring latent features. The resulting
solutions share many properties with previous sparse coding models, used to model early sensory
processing [7]. However, unlike these sparse coding models, the learned representation depends on:
(i) the relation between the input and variable of interest; (ii) the trade-off between encoding quality
and compression. Finally, we present a kernelized version of the algorithm, that can be applied to a
large range of problems with non-linear relation between the input data and variables of interest.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
2
Variational IB
Let us define an input variable X, as well as a ?relevance variable?, Y , with joint distribution
p (y, x). The goal of the IB approach is to compress the variable X through another variable R,
while conserving information about Y . Mathematically, we seek an encoding model, p (r|x), that
maximizes:
Lp(r|x)
= I (R; Y ) ? ?I (R; X)
?
hlog p (y|r) ? log p (y) + ? log p (r) ? ? log p (r|x)ip(r,x,y) ,
(1)
where 0 < ? < 1 is a Lagrange multiplier that determines the strength of the bottleneck.
Tishby and colleagues showed that
i be optimizedR by applying iterative
h theRIB loss function can
updates: pt+1 (r|x) ? pt (r) exp ? ?1 y p (y|x) log pp(y|x)
, pt+1 (r) = x p (x) pt+1 (r|x) and
t (y|r)
R
pt+1 (y|r) = x p (y|x) pt+1 (x|r) [1]. Unfortunately however, when p (x, y) is high-dimensional
and/or non-gaussian these updates become intractable, and approximations are required.
Due to the positivity of the KL divergence, we can write, hlog q (?)ip(?) ? hlog p (?)ip(?) for any
approximative distribution q(?). This allows us to formulate a variational lower bound for the IB
objective function:
? p(r|x),q(y|r),q(r)
L
=
N
1 X
hlog q (yn |r) + ? log q (r) ? ? log p (r|xn )ip(r|xn )
N n=1
?
Lp(r|x) ,
(2)
where q (yn |r) and q (r) are variational distributions, and we have replaced the expectation over
p (x, y) with the empirical expectation over training data. (Note that, for notational simplicity we
have also omitted the constant term, HY = ? hlog p (y)ip(y) .)
? = L), and leads to
Setting q (yn |r) ? p (yn |r) and q (r) ? p (r) fully tightens the bound (so that L
the iterative algorithm of Tishby et al. However, when these exact updates are not possible, one can
instead choose a restricted class of distributions q (y|r) ? Qy|r and q (r) ? Qr for which inference is
? with respect to parameters ? of the encoding distribution p (r|x, ?),
tractable. Thus, to maximize L
we repeat the following steps until convergence:
?
? For fixed ?, find {q new (y|r) , q new (r)} = arg max{q(y|r),q(r)}?{Qy|r ,Qr } L
?
? For fixed q (y|r) and q (r), find ? = arg max? L.
We note that using a simple approximation for the decoding distribution, q(y|r), can carry additional
benefits, besides rendering the IB algorithm tractable. Specifically, while an advantage of mutual
information is its generality, in certain cases this can also be a drawback. That is, because Shannon
information does not make any assumptions about the code, it is not always apparent how information
should be best extracted from the responses: just because information is ?there? does not mean we
know how to get at it.
In contrast, using a simple approximation for the decoding distribution, q(y|r) (e.g. linear gaussian),
constrains the IB algorithm to find solutions where information about Y can be easily extracted from
the responses (e.g. via linear regression).
3
Sparse IB
In previous work on gaussian IB [6], responses were equal to a linear projection of the input, plus
noise: r = W x + ?, where W is an Nr ? Nx matrix of encoding weights, and ? ? N (?|0, ?),
where ? is an Nr ? Nr covariance matrix. When the joint distribution, p (x, y), is gaussian, it follows
that the marginal and decoding distributions, p (r) and p (y|r), are also gaussian, and the parameters
of the encoding distribution, W and ?, can be found analytically.
To illustrate the capabilities of the variational algorithm, while permitting comparison to gaussian
IB, we begin by adding a single degree of complexity. In common with gaussian IB, we consider
2
a linear gaussian encoder, p (r|x) = N (r|W x, ?), and decoder, q (y|r) = N (y|U r, ?). However,
unlike
gaussian IB, we use
the response marginal: q (r) =
a student-t distribution to approximate
Q
2
2
i Student ri |0, ?i , ?i , with scale and shape parameters, ?i and ?i , respectively. When the shape
parameter, ?i , is small then the student-t distribution is heavy-tailed, or ?sparse?, compared to a
gaussian distribution. Thus, we call the resulting algorithm ?sparse IB?. Unlike gaussian IB, the
introduction of a student-t marginal means the IB algorithm cannot be solved analytically, and one
requires approximations.
3.1
Iterative algorithm
Recall that the IB objective function consists of two terms: I (R; Y ), and I (R; X). We begin by
describing how to optimize the lower and upper bound of each of these two terms with respect to the
variational distributions q(y|r) and q(r), respectively.
The first term of the IB objective function is bounded from below by:
E
1
1 XD
T
I (R; Y ) ? ? log |?| ?
(yn ? U r) ??1 (yn ? U r)
+ const.
2
2N n
p(r|xn )
(3)
Maximizing the lower bound on I (R; Y ) with respect to the decoding parameters, U and ?, gives:
?1
T
? = Cyy ? U W Cxy , U = Cxy
W T W Cxx W T + ?
(4)
P
P
P
1
1
1
T
T
T
where Cyy = N n yn yn , Cxy = N n xn yn , and Cxx = N n xn xn .
Unfortunately, it is not straightforward to express the bound on I (R; X) in closed form. Instead, we
use an additional variational approximation, utilising the fact that
can be ex the
R student-t distribution
pressed as an infinite mixture of gaussians: Student r|0, ? 2 , ? = ? N r|0, ? 2 Gamma ?| ?2 , ?2
[8]. Following a standard
EM procedure[9], one can thus write a tractable lower bound on the log
likelihood, l ? log Student r|0, ? 2 , ? , which corresponds to an upper-bound on the bottleneck
term:
X
I (R; X) ?
h? log q (ri ) + log p (ri |xn )ip(ri |xn )
(5)
i,n
#
"
N
X 1
2
1
1 X
2
?ni rni + f (?i , ?i , ai ) ? log |?| + const.
?
log ?i +
2
2
2N
?
2
i n=1
i
where ?ni , and ai denote
for the ith unit and nth data instance. We used the
2 variationalTparameters
T
2
shorthand notation, rni = wi xn xn wi + ?i , where ?i2 is the ith diagonal element of ? and wi is
the ith row of W . For notational simplicity, terms that do not depend on the encoding parameters
were pushed into the function, f (?i , ?i , ai )1 .
Minimizing the upper bound on I (R; X) with respect to ?i2 , ?ni and ai (for fixed ?i ) gives:
?i2 =
N
2
?i + 1
1 X
1
, ?ni =
?ni rni
, ai = (?i + 1),
2
2
N n=1
?i + hrni i /?i
2
(6)
The shape parameter, ?i , is then found numerically on each iteration (for fixed ?ni and ai ), by solving:
N
?
?
ai
1 X
i
i
?(ai ) ? log
?
? log
=1+
? ?ni ,
(7)
2
2
N n=1
?ni
where ?(?) is the digamma function [9].
? with respect to the encoding distribution,
Next we maximize the full variational objective function L
? with respect to the encoding noise covariance, ?,
p (r|x) (for fixed q(y|r) and q(r)). Maximizing L
gives:
N
X
1
1
??1 = U T ??1 U + ??1
?n ,
(8)
?
N
n=1
1
f (?i , ?i , ai ) = log ?
?i
2
?
?i
2
log
?i
2
?
1
N
P h ?i ?1
n
2
?(ai ) ? ln
ai
?ni
?
?i
?
2 ni
i
+ Hni , where Hni
is the entropy of a gamma distribution with shape and rate parameters: ai , and ai /?ni , respectively [9].
3
Y?
R
W
B
U
gaussian IB
encoding filters
D
prob density
X
gaussian IB
sparse IB
1
0.1
0.01
0.001
?10
decoding filters
0
10
response (a.u.)
E
Ilin (Y ; R) (nats)
A
100
0
0
50
100
150
I(X; R) (nats)
sparse IB
encoding filters
gaussian IB
sparse IB
null model
decoding filters
F
1
2
s
2+ 2
s
n
C
50
0.5
0
0
gaussian IB
sparse IB
20
40
units
60
80
Figure 1: Behaviour of sparse IB and gaussian IB algorithms, on denoising task. (A) Artificial image
patches were constructed from combinations of orientated edge-like features. Patches were corrupted
with white noise to generate the input, X. The goal of the IB algorithm is to learn a linear code that
maximized information about the original patches, Y , constrained on information encoded about
the input, X. (B) A selection of linear encoding (left), and decoding (right) filters obtained with the
gaussian IB algorithm. (C) Same as B, but for the sparse IB algorithm. (D) Response histograms
for the 10 units with highest variance, for the gaussian (red) and sparse (blue) IB algorithms. (E)
Information curves for the gaussian (red) and sparse (blue) algorithms, alongside a ?null? model,
where responses were equal to the original input, plus white noise. (F) Fraction of response variance
attributed to signal fluctuations, for each unit. Solid and dashed curves correspond to strong and weak
bottlenecks, respectively (corresponding to the vertical dashed lines in panel E).
where ? and ?n are Nr ? Nr diagonal covariance matrices with diagonal elements ?ii = ?i2 , and
(?n )ii = ?ni , respectively.
? with respect to the encoding weights, W , gives:
Finally, taking the derivative of L
?
?L
1 X
T
= U T ??1 Cxy
? U T ??1 U W Cxx ? ???1
?n W xn xTn ,
?W
N n
(9)
Setting the derivative to zero, we can solve for W directly. One may verify that, when variational
parameters, ?ni , are unity, the above iterative updates are identical to the iterative gaussian IB
algorithm described in [6].
3.2
Simulations
In our framework, the approximation of the response marginal, q (r), plays an analogous role to
the prior distribution in a probabilistic generative model. Thus, we hypothesized that a sparse
approximation for the response marginal, q(r), would permit the IB algorithm to recover sparsely
occurring input features, analogous to the effect of using a sparse prior.
4
Figure 2: spatially correlated noise
B
Y
C
D
stim.
reconstruction
10
no. of units
A X
5
0
?90
?45
0
45
90
encoded orientation
(relative to vertical)
Figure 2: Variant of the task in figure 1, in which the input noise is spatically correlated. (A) Example
input X and patch, Y . Spatial noise correlations were aligned along the vertical direction. (B) Subset
of decoding filters obtained with the sparse IB algorithm. (C) Distribution of encoded orientations.
(D) Example stimulus (left) and reconstruction (right) of bars presented at variable orientations
(presented with zero input noise, so that X ? Y for this example).
To show this, we constructed artificial 9 ? 9 image patches from combinations of orientated bar
features. Each bar had a gaussian cross-section, with maximum amplitude drawn from a standard
normal distribution of width 1.2 pixels. Patches were constructed by linearly combining 3 bars, with
uniformly random orientation and position.
Initially, we considered a simple de-noising task, where the input, X, was a noisy version of the
original image patches (gaussian noise, with variance ? 2 = 0.005; figure 1A). Training data consisted
of 10,000 patches. Figure 1B and 1C show a selection of encoding (W ) and decoding (U ) filters
obtained with the gaussian and sparse IB models, respectively. As predicted, only the sparse IB model
was able to recover the original bar features. In addition, response histograms were considerably
more heavy-tailed for the sparse IB model (fig. 1D).
The relevant information, I(R; Y ), encoded by the sparse model was greater than for the gaussian
model, over a range of bottleneck strengths (fig. 1E). While the difference may appear small, it
is consistent with work showing that sparse coding models achieve only a small improvement in
log-likelihood for natural image patches [10]. We also plotted the information curve for a ?null
model?, with responses sampled from p(r|x) = N (r|x, ? 2 I). Interestingly, the performance of this
null model was almost identical to the gaussian IB model.
w C
wT
i
Figure 1F plots the fraction of response variance due to the signal, for each unit ( wi Cixxxx
). Solid
wiT +?i2
and dashed curves denote strong and weak bottlenecks, respectively. In both cases, the gaussian model
gave a smooth spectrum of response magnitudes, while the sparse model was more ?all-or-nothing?.
One way the sparse IB algorithm differs qualitatively from traditional sparse coding algorithms, is
that the learned representation depends on the relation between X and Y , rather than just the input
statistics. To illustrate this, we conducted simulations with patches corrupted by spatially correlated
noise, aligned along the vertical direction (fig. 2A). The spatial covariance of the noise was described
by a gaussian envelope, with standard deviation 3 pixels in the vertical direction and 1 pixel in
horizontal direction.
Figure 2B shows a selection of decoding filters obtained from the sparse IB model, with correlated input noise. The shape of individual filters was qualitatively similar to those obtained with uncorrelated
noise (fig. 1C). However, with this stimulus, the IB model avoided ?wasting? bits by representing
features co-orientated with the noise (fig. 2C). Consequently, it was not possible to reconstruct vertical
bars from the responses, when bars were presented alone, even with zero noise (fig. 2D).
4
Kernel IB
One way to improve the IB algorithm is to consider non-linear encoders. A general choice is:
p (r|x) = N (r|W ?(x), ?), where ?(x) is an embedding to a high-dimensional non-linear feature
space.
5
D
f (X)
R
U
sparse kIB
B
E
1 gaussian kIB
sparse kIB
20
prob density
Y?
X
Ilin (Y ; R) (nats)
15
10
gaussian kIB
sparse kIB
sparse IB
5
0
0
20
0.1
0.01
0.001
0.0001
40
?10
I(X; R) (nats)
F
C
gaussian kIB
sparse IB
gaussian IB
stim 1
0
10
response (a.u.)
stim 2
stim 3
recon. patches
A
G
?5
st
im
st 1
im
st 2
im
3
0
st
im
st 1
im
st 2
im
3
response (a.u.)
5
Figure 3: Behaviour of kernel IB algorithm on occlusion task. (A) Image patches were the same as
for figure 1. However, the input, X, was restricted to 2 columns to either side of the patch. The target
variable, Y , was the central region. (B) Subset of decoding filters, U , for the sparse kernel IB (?sparse
kIB?) algorithm. (C) As for B, for other versions of the IB algorithm. (D) Information curves for
the gaussian kIB (blue) sparse kIB (green) and sparse IB algorithms (red). The bottleneck strength
for the other panels in this figure is indicated by a vertical dashed line. (E) Response histogram for
the 10 units with highest variance, for the gaussian and sparse kIB models. (F) (above) Three test
stimuli, used to demonstrate the non-linear properties of the sparse KIB code. (below) Reconstruction
obtained from responses to test stimulus. (G) Responses of two units which showed strong responses
to stimulus 3. The decoding filters for these units are shown above the bar plots.
The variational objective functions for both gaussian and sparse IB algorithms are quadratic in the
responses, and thus can be expressed in terms of dot products of the row vector, ?(x). Consequently,
PN
every solution for wi can be expressed as an expansion of mapped training data, wi = n=1 ain ?(xn )
[11]. It follows that the variational IB algorithm can be expressed in?dual space?, with responses to the
nth input drawn from r ? N (r|Akn , ?), where A is an Nr ? N matrix of expansion coefficients, and
kn is the nth column of the N ? N kernel-gram matrix, K, with elements Knm = ?(xn )?(xm )T .
In this formulation, the problem of finding the linear encoding weights, W , is replaced by finding the
expansion coefficients, A.
The advantage of expressing the algorithm in the dual space is that we never have to deal with
?(x) directly, so are free to consider high- (or even infinite) dimensional feature spaces. However,
without additional constraints on the expansion coefficients, A, the IB algorithm becomes degenerate
(i.e. the solutions are independent of the input, X). A standard way to deal with this is to add an L2
regularization term that favours solutions with small expansion coefficients. Here, this is achieved
here by replacing ?Tn ?n with ?Tn ?n + ?I, where ? is a fixed regularization parameter. Doing so, the
? with respect to A becomes:
derivative of L
X
?
?L
= U T ??1 Y K ?
U T ??1 U + ???1 ?n A kn knT + ?K
(10)
?A
n
Setting the derivative to zero and solving for A directly requires inverting an N Nr ? N Nr matrix,
which is expensive. Instead, one can use an iterative solver (we used the conjugate gradients squared
6
Figure 5: handwritten digits
Y?
X
f (X)
C
R
B
U
sparse kIB
gaussian kIB
sparse kIB
1
prob density
A
0.1
0.01
0.001
?10
Dgaussian kIB
0
10
response (a.u.)
sparse IB
gaussian IB
Figure 4: Behaviour of kernel IB algorithm on handwritten digit data. (A) As with figure 4, we
considered an occlusion task. This time, units were provided with the left hand side of the image
patch, and had to reconstruct the right hand side. (B) Response distribution for 10 neurons with
highest variance, for the gaussian (blue) and sparse (green) kIB algorithms. (C) Decoding filters for a
subset of units, obtained with the sparse kIB algorithm. Note that, for clearer visualization, we show
here the decoding filter for the entire image patch, not just the occluded region. (D) A selection of
decoding filters obtained with the alternative IB algorithms.
method). In addition, the computational complexity can be reduced by restricting the solution to lie
PM
on a subspace of training instances, such that, wi = n=1 ain ?(xn ), where M < N . The derivation
does not change, only now K has dimensions M ? N [11].
When q(r) is gaussian (equivalent to setting ?n = I), solving for A gives:
?1 T ?1
A = U T ??1 U + ???1
U ? AKRR
(11)
where AKRR = Y (K + ?I)?1 are the coefficients obtained from kernel ridge-regression (KRR).
This suggests the following two stage algorithm: first, we learn the regularisation constant, ?, and
parameters of the kernel matrix, K, to maximize KRR performance on hold-out data; next, we
perform variational IB, with fixed K and ?.
4.1
Simulations
To illustrate the capabilities of the kernel IB algorithm, we considered an ?occlusion? task, with the
outer columns of each patch presented as input, X (2 columns to the far left and right), and the inner
columns as the relevance variable Y , to be reconstructed. Image patches were as before. Note that
performing the occlusion task optimally requires detecting combinations of features presented to
either side of the occluded region, and is thus inherently nonlinear.
We used gaussian kernels, with scale parameter, ?, and regularisation constant, ?, chosen to maximize
KRR performance on test data. Both test and training data consisted of 10,000 images. However, A
was restricted to lie on a subset of 1000 randomly chosen training patches (see earlier).
Figure 3B shows a selection of decoding filters (U ) learned by the sparse kernel IB algorithm (?sparse
kIB?). A large fraction of filters resembled near-horizontal bars, traversing the occluded region. This
was not the case for the sparse linear IB algorithm, which recovered localized blobs either side of the
occluded region, nor the gaussian linear or kernelized models, which recovered non-local features
(fig. 3C). Figure 3D shows a small but significant improvement in performance for the sparse kIB
versus the gaussian kIB model. Most noticeable, however, is the distribution of responses, which are
much more heavy tailed for the sparse kIB algorithm (fig. 3E).
To demonstrate the non-linear behaviour of the sparse kIB model, we presented bar segments: first
to either side of the occluded patch, then to both sides simultaneously. When bar segments were
presented to both sides simultaneously, the sparse KIB model ?filled in? the missing bar segment,
7
in contrast to the reconstruction obtained with single bar segments (fig. 3F). This behaviour was
reflected in the non-linear responses of certain encoding units, which were large when two segments
were presented together, but near zero when one segment was presented alone (fig. 3G).
Finally, we repeated the occlusion task with handwritten digits, taken from the USPS dataset (www.
gaussianprocess.org/gpml/data). We used 4649 training and 4649 test patches, of 16?16
pixels. However, expansion coeffecients were restricted to a lie on subset of 500 randomly patches.
We set X and Y , to be the left and right side of each patch, respectively (fig. 4A).
In common with the artificial data, the response distributions achieved with the sparse kIB algorithm
were more heavy-tailed than for the gaussian kIB algorithm (fig. 4B). Likewise, recovered decoding
filters closely resembled handwritten digits, and extended far into the occluded region (fig. 4C). This
was not the case for the alternative IB algorithms (fig. 4D).
5
Discussion
Previous work has shown close parallels between the IB framework and maximum-likelihood
estimation in a latent variable model [12, 13]. For the sparse IB algorithm presented here, maximizing
the IB objective function is closely related to maximizing the likelihood of a ?sparse coding? latent
variable model, with student-t prior and linear gaussian likelihood function. However, unlike
traditional sparse coding models, the encoding (or ?recognition?) model p(r|x) is conditioned on a
seperate set of inputs, X, distinct from the image patches themselves. Thus, the solutions depend on
the relation between X and Y , not just the image statistics (e.g. see fig. 2). Second, an additional
parameter, ?, not present in sparse coding models, controls the trade-off between encoding and
compression. Finally, in contrast to traditional sparse coding algorithms, IB gives an unambiguous
ordering of features, which can be arranged according to the response variance of each unit (fig. 1F).
Our work is also closely related to the IM algorithm, proposed by Barber et al. to solve the information
maximization (?infomax?) problem [14]. However, a general issue with infomax problems is that they
are usually ill-posed, necessitating additional ad hoc constraints on the encoding weights or responses
[15]. In contrast, in the IB approach, such constraints emerge automatically from the bottleneck term.
A related method to find low-dimensional projections of X/Y pairs is canonical correlation analysis
(?CCA?), and its kernel analogue [16]. In fact, the features obtained with gaussian IB are identical
to those obtained with CCA [6]. However, unlike CCA, the number and ?scale? of the features are
not specified in advance, but determined by the bottleneck parameter, ?. Secondly, kernel CCA
is symmetric in X and Y , and thus performs nonlinear embedding of both X and Y . In contrast,
the IB problem is assymetric: we are interested in recovering Y from an input X. Thus, only X is
kernelized, while the decoder remains linear. Finally, the features obtained from gaussian IB (and
thus, CCA) differ qualitatively from the sparse IB algorithm, which recovers sparse features that
account jointly for X and Y .
Sparse IB can be extended to the nonlinear regime using a kernel expansion. For the gaussian
model, the expansion coefficients, A, are a linear projection of the coefficients used for kernel-ridgeregression (?KRR?). A general disadvantage of KRR, is that it can be difficult to know which aspects
of X are relied on to perform the regression. In contrast, the kernel IB framework provides an
intermediate representation, allowing one to visualize the features that jointly account for both X and
Y (figs. 3B & 4C). Furthermore, this learned representation permits generalisation across different
tasks that rely on the same set of latent features; something not possible with KRR.
Finally, the IB approach has important implications for models of early sensory processing [17, 18].
Notably, ?efficient coding? models typically consider the low-noise limit, where the goal is to reduce
the neural response redundancy [7]. In contrast, the IB approach provides a natural way to explore the
family of solutions that emerge as one varies internal coding constraints (by varying ?) and external
constraints (by varying the input, X) [19, 20]. Further, our simulations suggest how the framework
can be used to go beyond early sensory processing: for example to explain higher-level cognitive
phenomena such as perceptual filling in (fig. 3G). In future, it would be interesting to explore how
the IB framework can be used to extend the efficient coding theory, by accounting for modulations
in sensory processing that occur due to changing task demands (i.e. via changes to the relevance
variable, Y ), rather than just the input statistics (X).
8
References
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[3] Slonim, N. (2003) Information bottleneck theory and applications. PhD thesis, Hebrew University of
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[14] Barber, D. & Agakov, F. (2004) The IM algorithm: a variational approach to information maximization. In
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noisy spiking neurons. Proceedings of the National Academy of Sciences 107(32), pp. 14419-14424.
9
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5,639 | 6,102 | Multistage Campaigning in Social Networks
Mehrdad Farajtabar?
Xiaojing Ye?
Sahar Harati?
Le Song?
Hongyuan Zha?
Georgia Institute of Technology?
Georgia State University?
Emory University?
mehrdad@gatech.edu
xye@gsu.edu
sahar.harati@emory.edu
{lsong,zha}@cc.gatech.edu
Abstract
We consider the problem of how to optimize multi-stage campaigning over social
networks. The dynamic programming framework is employed to balance the high
present reward and large penalty on low future outcome in the presence of extensive uncertainties. In particular, we establish theoretical foundations of optimal
campaigning over social networks where the user activities are modeled as a multivariate Hawkes process, and we derive a time dependent linear relation between
the intensity of exogenous events and several commonly used objective functions
of campaigning. We further develop a convex dynamic programming framework
for determining the optimal intervention policy that prescribes the required level
of external drive at each stage for the desired campaigning result. Experiments on
both synthetic data and the real-world MemeTracker dataset show that our algorithm can steer the user activities for optimal campaigning much more accurately
than baselines.
1
Introduction
Obama was the first US president in history who successfully leveraged online social media in presidential campaigning, which has been popularized and become a ubiquitous approach to electoral
politics (such as in the on-going 2016 US presidential election) in contrast to the decreasing relevance of traditional media such as TV and newspapers [1, 2]. The power of campaigning via social
media in modern politics is a consequence of online social networking being an important part of
people?s regular daily social lives. It has been quite common that individuals use social network sites
to share their ideas and comment on other people?s opinions. In recent years, large organizations,
such as governments, public media, and business corporations, also start to announce news, spread
ideas, and/or post advertisements in order to steer the public opinion through social media platform.
There has been extensive interest for these entities to influence the public?s view and manipulate
the trend by incentivizing influential users to endorse their ideas/merits/opinions at certain monetary
expenses or credits. To obtain most cost-effective trend manipulations, one needs to design an optimal campaigning strategy or policy such that quantities of interests, such as influence of opinions,
exposure of a campaign, adoption of new products, can be maximized or steered towards the target
amount given realistic budget constraints.
The key factor differentiating social networks from traditional media is peer influence. In fact, events
in an online social network can be categorized roughly into two types: endogenous events where
users just respond to the actions of their neighbors within the network, and exogenous events where
users take actions due to drives external to the network. Then it is natural to raise the following
fundamental questions regarding optimal campaigning over social networks: can we model and
exploit those event data to steer the online community to a desired exposure level? More specifically,
can we drive the overall exposure to a campaign to a certain level (e.g., at least twice per week per
user) by incentivizing a small number of users to take more initiatives? What about maximizing the
overall exposure for a target group of people?
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
More importantly, those exposure shaping tasks are more effective when the interventions are implemented in multiple stages. Due to the inherent uncertainty in social behavior, the outcome of each
intervention may not be fully predictable but can be anticipated to some extent before the next intervention happens. A key aspect of such situations is that interventions can?t be viewed in isolation
since one must balance the desire for high present reward with the penalty of low future outcome.
In this paper, the dynamic programming framework [3] is employed to tackle the aforementioned
issues. In particular, we first establish the fundamental theory of optimal campaigning over social
networks where the user activities are modeled as a multivariate Hawkes process (MHP) [4, 5] since
MHP can capture both endogenous and exogenous event intensities. We also derive a time dependent
linear relation between the intensity of exogenous events and the overall exposure to the campaign.
Exploiting this connection, we develop a convex dynamic programming framework for determining
the optimal intervention policy that prescribes the required level of external drive at each stage in
order for the campaign to reach a desired exposure profile. We propose several objective functions
that are commonly considered as campaigning criteria in social networks. Experiments on both
synthetic data and real world network of news websites in the MemeTracker dataset show that our
algorithms can shape the exposure of campaigns much more accurately than baselines.
2
Basics and Background
An n-dimensional temporal point process is a random process whose realization consists of a
list of discrete events in time and their associated dimension, {(tk , dk )} with tk ? R+ and
dk ? {1, . . . , n}. Many different types of data produced in online social networks can be represented as temporal point processes, such as likes and tweets. A temporal point process can be
equivalently represented as a counting process, N (t) = (N 1 (t), . . . , N n (t))? associated to n users
in the social network. Here, N i (t) records the number of events user i performs before time t for
1 ? i ? n. Let the history Hi (t) be the list of times of events {t1 , t2 , . . . , tk } of the i-th user up
to time t. Then, the number of observed events in a small time window [t, t + dt) of length dt is
"t
!
dN i (t) = tk ?Hi (t) ?(t ? tk ) dt, and hence N i (t) = 0 dN i (s), where ?(t) is a Dirac delta function. The point process representation of temporal data is fundamentally different from the discrete
time representation typically used in social network analysis. It directly models the time interval
between events as random variables, avoids the need to pick a time window to aggregate events, and
allows temporal events to be modeled in a fine grained fashion. Moreover, it has a remarkably rich
theoretical support [6].
An important way to characterize temporal point processes is via the conditional intensity function
? a stochastic model for the time of the next event given all the times of previous events. Formally,
the conditional intensity function ?i (t) (intensity, for short) of user i is the conditional
probability
#
$
of observing an event in a small window [t, t + dt) given the history H(t) = H1 (t), . . . , Hn (t) :
?i (t)dt := P {user i performs event in [t, t + dt) | H(t)} = E[dN i (t) | H(t)],
(1)
where one typically assumes that only one event can happen in a small window of size dt. The
functional form of the intensity ?i (t) is often designed to capture the phenomena of interests.
The Hawkes process [7] is a class of self and mutually exciting point process models,
n & t
%
%
?i (t) = ?i (t) +
?idk (t, tk ) = ?i (t) +
?ij (t, s)dN j (s),
j=1
k:tk <t
(2)
0
where the intensity is history dependent. ?ij (t, s) is the impact function capturing the temporal
influence of an event by user j at time s to the future events of user j at time t ! s. Here, the first term
?i (t) is the exogenous!
event intensity modeling drive outside the network and indecent of the history,
and the second term k:tk <t ?idk (t, tk ) is the endogenous event intensity modeling interactions
within the network [8]. Defining ?(t, s) = [?ij (t, s)]i,j=1...n , and ?(t) = (?1 (t), . . . , ?n (t))? , and
?(t) = (?1 (t), . . . , ?n (t))? we can compactly rewrite Eq 2 in matrix form:
& t
?(t) = ?(t) +
?(t, s)dN (s).
(3)
0
In practice it is standard to employ shift-invariant impact function, i.e., ?(t, s) = ?(t ? s). Then,
"t
by using notation of convolution f (t) ? g(t) = 0 f (t ? s)g(s)ds we have
?(t) = ?(t) + ?(t) ? dN (t).
2
(4)
3
From Intensity to Average Activity
In this section we will develop a closed form relation between the expected total intensity E[?(t)]
and the intensity ?(t) of exogenous events. This relation establish the basis of our campaigning
framework. First, define the mean function as M(t) := E[N (t)] = EH(t) [E(N (t)|H(t))]. Note that
M(t) is history independent, and it gives the average number of events up to time t for each of the
dimension. Similarly, the rate function ?(t) is given by ?(t)dt := dM(t). On the other hand,
dM(t) = dE[N (t)] = EH(t) [E(dN (t)|H(t))] = EH(t) [?(t)|H(t)]dt = E[?(t)]dt.
(5)
Therefore ?(t) = E[?(t)] which serves as a measure of activity in the network. In what follows we
will find an analytical form for the average activity. Proofs are presented in Appendix C.
Lemma 1. Suppose ? : [0, T ] ? Rn?n is a non-increasing matrix function, then for every fixed
constant intensity ?(t) = c ? Rn+ , ?c (t) := ?(t)c solves the semi-infinite integral equation
& t
?(t) = c +
?(t ? s)?(s)ds, ?t ? [0, T ],
(6)
0
if and only if ?(t) satisfies
?(t) = I +
&
t
?(t ? s)?(s)ds,
0
?t ? [0, T ].
(7)
In particular, if ?(t) = Ae??t 1?0 (t) = [aij e??t 1?0 (t)]ij where 0 ? ? ?
/ Spectrum(A), then
?(t) = e(A??I)t + ?(A ? ?I)?1 (e(A??I)t ? I)
(8)
for t ? [0, T ], where, 1?0 (t) is an indicator function for t ? 0.
Let ? : [0, T ] ? Rn+ be a right-continuous piecewise constant function
?(t) =
M
%
cm 1[?m?1 ,?m ) (t),
(9)
m=1
where 0 = ?0 < ?1 < ? ? ? < ?M = T is a finite partition of time interval [0, T ] and function
1[?m?1 ,?m ) (t) indicates ?m?1 ? t < ?m . The next theorem shows that if ?(t) satisfies (7), then one
can calculate ?(t) for piecewise constant intensity ? : [0, T ] of form (9).
Theorem 2. Let ?(t) satisfy (7) and ?(t) be a right-continuous piecewise constant intensity function
of form (9), then the rate function ?(t) is given by
?(t) =
m
%
k=0
?(t ? ?k )(ck ? ck?1 ),
(10)
for all t ? (?m?1 , ?m ] and m = 1, . . . , M , where c?1 := 0 by convention.
Using the above lemma, for the first time, we derive the average intensity for a general exogenous
intensity. Appendix E includes a few experiments to investigate these results empirically.
Theorem 3. If ? ? C 1 ([0, T ]) and satisfies (7), and exogenous intensity ? is bounded and piecewise absolutely continuous on [0, T ] where ?(t+) = ?(t) at all discontinuous points t, then ? is
differentiable almost everywhere, and the semi-indefinite integral
& t
?(t) = ?(t) +
?(t ? s)?(s)ds, ?t ? [0, T ],
(11)
0
yields a rate function ? : [0, T ] ?
given by
& t
?(t) =
?(t ? s)d?(s).
Rn+
(12)
0
Corollary 4. Suppose ? and ? satisfy the same conditions as in Thm. 3, and define ? = ?? , then
the rate function is ?(t) = (? ? ?)(t). In particular, if ?(t) = Ae??t 1?0 (t) = [aij e??t 1?0 (t)]ij
"t
then the rate function ?(t) = ?(t) + A 0 e(A?wI)(t?s) ?(s)ds.
3
4
Multi-stage Closed-loop Control Problem
Given the analytical relation between exogenous intensity and expected overall intensity (rate function), one can solve a single one-stage campaigning problem to find the optimal constant intervention
intensity [8]. Alternatively, the time window can be partitioned into multiple stages and one can impose different levels of interventions in these stages. This yields an open-loop optimization of the
cost function where one selects all the intervention actions at initial time 0. More effectively, we
tackle the campaigning problem in a dynamic and adaptive manner where we can postpone deciding
the intervention by observing the process until the next stage begins. This is called the closed-loop
optimization of the objective function.
In this section, we establish the foundation to formulate the problem as a multi-stage closed-loop optimal control problem. We assume that n users are generating events according to multi-dimensional
Hawkes process with exogenous intensity ?(t) ? Rn and impact function ?(t, s) ? Rn?n .
Event exposure. Event exposure is the quantity of major interests in campaigning. The exposure
process is mathematically represented as a counting process, E(t) = (E 1 (t), . . . , E n (t))? : Here,
E i (t) records the number of times user i is exposed (she or one of her neighbors performs an activity)
to the campaign by time t. Let B be the adjacency matrix of the user network, i.e., bij = 1 if user
i follows user j or equivalently user j influences user i. We assume bii = 1 for all i. Then the
exposure process is given by E(t) = B N (t).
Stages and interventions. Let [0, T ] be the time horizon and 0 = ?0 < ?1 < . . . < ?M ?1 <
?M = T be a partition into the M stages. In order to steer the activities of network towards a
desired level (criteria given below) at these stages, we impose a constant intervention um ? Rn to
the existing exogenous intensity ? during time [?m , ?m+1 ) for each stage m = 0, 1, . . . , M ? 1. The
"t
activity intensity at the m-th stage is ?m (t) = ? + um + 0 ?(t, s) dN (s) for ?m ? t < ?m+1
where N (t) tracks the counting process of activities since t = 0. Note that the intervention itself
exhibits a stochastic nature: adding uim to ?i is equivalent to incentivizing user i to increase her
activity rate but it is still uncertain when she will perform an activity, which appropriately mimics
the randomness in real-world campaigning.
States and state evolution. Note that the Hawkes process is non-Markov and one needs complete
knowledge of the history to characterize the entire process. However, the conditional intensity ?(t)
only depends on the state of process at time t when the standard exponential kernel ?(t, s) =
Ae??(t?s) 1?0 (t ? s) is employed. In this case, the activity rate at stage m is
& ?m
& t
??(t?s)
?m (t) = ? + um +
Ae
dN (s) +
Ae??(t?s) dN (s)
(13)
0
?m
'
()
* '
()
*
from previous stages
current stage
Define xm := ?m?1 (?m ) ? um?1 ? ? (and
" ?x0 = 0 by convention) then the intensity due to events
of all previous m stages can be written as 0 m Ae??(t?s) dN (s) = xm e??(t??m ) . In other words,
xm is sufficient to encode the information of activity in the past m stages that is relevant to future.
This is in sharp contrast to the general case where the state space grows with the number of events.
!M ?1
Objective function. For a sequence of controls u(t) = m=0 um 1[?m ,?m+1 ) (t), the activity count"t
ing process N (t) is generated by intensity ?(t) = ? + u(t) + 0 Ae??(t?s) dN (s). For each stage
m from 0 to M ? 1, xm encodes the effects from previous
" t m stages as above and um is the current
i
control imposed at this stage. Let Em
(t; xm , um ) := B ?m dN i (s) be the number of times user i is
exposed to the campaign by time t ? [?m , ?m+1 ) in stage m, then the goal is to steer the expected
i
i
total number of exposure E?m
(xm , um ) := E[Em
(?m+1 ; xm , um )] to a desired level. In what follows,
i
we introduce several instances of the objective function g(xm , um ) in terms of {E?m
(xm , um )}ni=1
in each stage m that characterize different exposure shaping tasks. Then the overall control problem
!M ?1
is to find u(t) that optimizes the total objective m=0 gm (xm , um ).
? Capped Exposure Maximization (CEM): In real networks, there is a cap on the exposure each user
i
can tolerate due to the limited attention of a user. Suppose we know the upper bound ?m
, on user
i?s exposure tolerance over which the extra exposure is not counted towards the objective. Then,
we can form the following capped exposure maximization
n
# i
$
1%
i
gm (xm , um ) =
min E?m
(xm , um ), ?m
(14)
n i=1
4
Algorithm 1: Closed-loop Multi-stage Dynamic Programming
Input: Intervention constraints: c0 . . . cM ?1 , C0 . . . CM ?1 , ?0 . . . ?M ?1 ,
Input: Objective-specific constraints: ?0 . . . ?M ?1 for CEM and ?0 . . . ?M ?1 for LES
Input: Time: T , Hawkes parameters: A, ?
Output: Optimal intervention u0 . . . uM ?1 , Optimal cost: Cost
Set x0 ? 0 and Cost ? 0
for l ? 0 : M ? 1 do
(vl . . . vM ?1 ) = open loop(xl ) (Problems (24), (25), (26) for CEM, MEM, LES respectively)
Set ul ? vl and drop vl+1 . . . vM ?1
Update next state xl+1 ? fl (xl , ul ) and Cost = Cost + gl (xl , ul )
? Minimum Exposure Maximization (MEM): Suppose our goal is instead to maintain the exposure
of campaign on each user above a certain minimum level, at each stage or, alternatively to make
the user with the minimum exposure as exposed as possible, we can consider the following cost
function:
i
gm (xm , um ) = min E?m
(xm , um )
(15)
i
? Least-squares Exposure Shaping (LES): Sometimes we want to achieve a pre-specified target exposure levels, ?m ? Rn , for the users. For example, we may like to divide users into groups and
desire a different level of exposure in each group. To this end, we can perform least-squares campaigning task with the following cost function where D encodes potentially additional constraints
(e.g., group partitions):
1
gm (xm , um ) = ? ?DE?m (xm , um ) ? ?m ?2
(16)
n
Policy and actions. By observing the counting process in previous stages (summarized in a sequence of xm ) and taking the future uncertainty into account, the control problem is to design a
policy ? = {?m : Rn ? Rn : m = 0, . . . , M ? 1} such that the controls um = ?m (xm ) can maxi!M ?1
mize the total objective m=0 gm (xm , um ). In addition, we may have constraints on the amount of
control. For example, a budget constraint on the sum of all interventions to users at each stage, or, a
cap over the amount of intensity a user can handle.# A feasible set or an action space over which
we
$
find the best intervention is represented as Um := um ? Rn |c?
m um ? Cm , 0 " um " ?m . Here,
cm ? Rn+ contains the price of each person per unit increase of exogenous intensity and Cm ? R+
is the total budget at stage m. Also, ?m ? Rn+ is the cap on the amount of activities of the users.
To summarize, the following problem is formulated to find the optimal control policy ?:
maximize
?
5
M
?1
%
m=0
gm (xm , ?m (xm )), subject to ?m (xm ) ? Um , for m = 0, . . . , M ? 1.
(17)
Closed-loop Dynamic Programming Solution
We have formulated the control problem as an optimization in (17). However, when control
policy ?m is to be implemented, only xm is observed and there are still uncertainties in future
{xm+1 , . . . , xM ?1 }. For instance, when ?m is implemented according to xm starting from time
?m , the intensity xm+1 := f (xm , ?m (xm )) at time ?m+1 depends on xm and the control ?m (xm ),
but is also random due to the stochasticity of the process during time [?m , ?m+1 ). Therefore, the
design of ? needs to take future uncertainties into considerations.
Suppose we have arrived at stage M at time ?M ?1 with observation xM ?1 , then the optimal policy
?M ?1 satisfies gM ?1 (xM ?1 , ?M ?1 (xM ?1 )) = maxu?UM ?1 gM ?1 (xM ?1 , u) =: JM ?1 (xM ?1 ).
We then repeat this procedure for m from M ? 1 to 0 backward to find the sequence of controls via
dynamic programming such that the control ?m (xm ) ? Um yields optimal objective value
Jm (xm ) = max E[gm (xm , um ) + Jm+1 (f (xm , um ))]
um ?Um
(18)
Approximate Dynamic Programming. Solving (18) for finding Jm (xm ) analytically is intractable.
Therefore, we will adopt an approximate dynamic programming scheme. In fact approximate control is as essential part of dynamic programming as the optimization is usually intractable due to
5
curse of dimensionality except a few especial cases [3]. Here we adopt a suboptimal control scheme,
certainty equivalent control (CEC), which applies at each stage the control that would be optimal
if the uncertain quantities were fixed at some typical values like the average behavior. It results in
an optimal control sequence, the first component of which is used at the current stage, while the remaining components are discarded. The procedure is repeated for the remaining stages. Algorithm 1
summarizes the dynamic programing steps. This algorithm has two parts: (i) certainty equivalence
which the random behavior is replaced by its average; and (ii) the open-loop optimization. Let?s
assume we are at the beginning of stage l of the Alg. 1 with state vector xl at ?l .
Certainty equivalence. We use the machinery developed in Sec. 3 to compute the average of
exposure at any stage m = l, l + 1, . . . , M ? 1.
+& ?m+1
,
& ?m+1
E?m (xm , um ) = BE[N (?m+1 ) ? N (?m )] = BE
dN (s) = B
?m (s) ds (19)
?m
?m
"t
where ?m (t) = E[?m (t)] and ?m (t) = ? + um + xl e??(t??l ) + ?l Ae??(t?s) dN (s) for t ?
[?m , ?m+1 ). Now, we use the superposition property of point processes [4] to decompose the process
as N (t) = N c (t) + N v (t) corresponding to ?m (t) = ?cm (t) + ?vm (t) where the first ?cm (t) =
"t
? + um + ?l Ae??(t?s) dN c (s) consists of events caused by exogenous intensity at current stage m
"t
and the second ?vm (t) = xl e??(t??l ) + ?l Ae??(t?s) dN v (s) is due to activities in previous stages.
According to Thm. 2 we have
c
?m
(t) := E[?cm (t)] = ?(t ? ?l )? + ?(t ? ?l )ul +
k=l+1
and according to Thm. 3 we have
v
?m
(t) := E[?vm (t)] =
&
t
?l
m?1
%
?(t ? ?k )(uk ? uk?1 ),
?(t ? s) d(xl e??(s??l ) 1[?l ,?) (s)).
(20)
(21)
From now on, for simplicity, we assume stages are based on equal partition of [0, T ] to M segments
c
v
where each has length ?M . Combining Eq. (19) and ?m (t) = ?m
(t) + ?m
(t) yields:
E?m (xm , um ) =?((m ? l + 1)?M )ul + ?((m ? l)?M )(ul+1 ? ul ) + . . .
(22)
+ ?(?M )(um ? um?1 ) + ?((m ? l + 1)?M )? + ?((m ? l + 1)?M )xl
where ?(t) and ?(t) are matrices independent of um ?s and are defined in Appendix D. Note the
linear relation between average exposure E?m (xm , um ) and intervention values ul , . . . , um?1 .
Open-loop optimization. Having found the average exposure at stages m = l, . . . , M ?1 we formulate an open-loop optimization to find optimal ul , ul+1 , . . . , uM ?1 . Defining u
?l = (ul ; . . . ; uM ?1 )
and E?l = (E?l (xl , ul ); . . . ; E?M ?1 (xM ?1 , uM ?1 )) we can write
Xl u
?l + Yl ? + Wl xl = E?l
where Zl u
? l ? zl
(23)
and Xl , Yl , Wl , Zl , and zl are independent of u
?l , ?, and xl as defined in Appendix D.
Defining the expanded form of constraint variables as c?l = (cl ; . . . ; cM ?1 ), C?l = (Cl ; . . . ; CM ?1 ),
and ?
? l = (?l ; . . . ; ?M ?1 ) we provide the optimization from of the above exposure shaping tasks.
For CEM consider ??l = (?l ; . . . , ?M ?1 ). Then the problem
1 ??
? ??l ? h,
? Zl u
maximizeh,?
?l + Yl ? + Wl xl ? h,
?l ? zl ,
? ul n 1 h subject to Xl u
(24)
solves CEM where h is an auxiliary vector of size n(M ? l).
? a vector of size n(M ? 1).
For MEM consider the auxiliary h as a vector of size M ? l and h
?
h = (h(1); . . . ; h(1); h(2); . . . , h(2); . . . , h(M ? l); . . . ; h(M ? l)) where each h(k) is repeated n
times. Then MEM is equivalent to
??
? ??l ? h,
? Zl u
maximizeh,?
?l + Yl ? + Wl xl ? h,
? l ? zl
? ul 1 h subject to Xl u
6
(25)
5
300
250
200
150
100
CLL OPL RND PRK WEI
1.5
4.5
average distance
minimum exposure
sum of exposure
350
4
3.5
3
2.5
2
1.5
CLL
OPL RND WFL PRP
?104
1
0.5
CLL
OPL RND GRD REL
a) Capped maximization b) Minimum maximization c) Least-squares shaping
Figure 1: The objective on simulated events and synthetic network; n = 300, M = 6, T = 40
? l = diag(D, . . . , D), then
For LES let ??l = (?l ; . . . ; ?M ?1 ) and D
? l (Xl u
(26)
minimizeu?l n1 ?D
?l + Yl ? + Wl xl ) ? ??l ?2 subject to Zl u
?l ? zl
All the three tasks involve convex (and linear) objective function with linear constraints which impose a convex feasible set. Therefore, one can use the rich and well-developed literature on convex
optimization and linear programming to find the optimum intervention.
6
Experiments
We evaluate our campaigning framework using both simulated and real world data and show that
our approach significantly outperforms several baselines1 .
Campaigning results on synthetic networks. In this section, we experiment with a synthetic network of 300 nodes. Details of the experimental setup and parameter setting are found in appendix
F. We focus on three tasks: capped exposure maximization, minimax exposure shaping, and least
square exposure shaping. To compare the methods we simulate the network with the prescribed
intervention intensity and compute the objective function based on the events happened during the
simulation. The mean and standard deviation of the objective function out of 10 runs are reported.
Fig. 1 summarizes the performance of the proposed algorithm (CLL) and 4 other baselines on different campaigning tasks. For CEM, our approach consistently outperforms the others by at least
10. This means it exposes each user to the campaign at least 10 times more than the rest consuming
the same budget and within the same constraints. The extra 20 units of exposures of over OPL or
value of information shows how much we gain by incorporating a dynamic closed-loop solution
as opposed to open-loop one-time optimization over all stages. For MEM, the proposed method
outperforms the others by a smaller margin, however, the 0.1 exposure difference with the second
best method is not trifling. This is expected as lifting the minimum exposure is a difficult task [8].
For LES, results demonstrate the superiority of CLL by a large margin. The 103 difference with the
second best algorithm aggregated over 6 stages roughly is translated to 103 /6 ? 13 difference in
the number of exposures per user. Given the heterogeneity of the network activity and target shape,
this is a significant improvement over the baselines. Appendix F includes further results on varying
number of nodes, number of stages, and duration of each stage.
Campaigning results on real world networks. We also evaluate the proposed framework on real
world data. To this end, we utilize the MemeTracker dataset [9] which contains the information flows
captured by hyperlinks between different sites with timestamps during 9 months. This data has been
previously used to validate Hawkes process models of social activity [5, 10]. For the real data, we
utilize two evaluation procedures. First, similar to the synthetic case, we simulate the network, but
now on a network based on the learned parameters from real data. However, the more interesting
evaluation scheme would entail carrying out real intervention in a social media platform. Since this
is very challenging to do, instead, in this evaluation scheme we used held-out data to mimic such
procedure. Second, we form 10 pairs of clusters/cascades by selecting any 2 combinations of 5
largest clusters in the Memetracker data. Each is a cascade of events around a common subject. For
any of these 10 pairs, the methods are faced to the question of predicting which cascade will reach
the objective function better. They should be able to answer this by measuring how similar their
prescription is to the real exogenous intensity. The key point here is that the real events happened
are used to evaluate the objective function of the methods. Then the results are reported on average
prediction accuracy on all stages over 10 runs of random constraint and parameter initialization on
10 pairs of cascades. The details of the experimental setup is further explained in Appendix F.
Fig. 2, left column illustrates the performance with respect to increasing the number of users in the
network. The performance drops slightly with the network size. This means that prediction becomes
1
codes are available at http://www.cc.gatech.edu/~mfarajta/
7
CLL OPL RND PRK WEI
0.65
150
200
250
0.7
0.65
0.6
2
prediction accuracy
0.55
0.5
150
200
250
0.65
0.6
0.55
0.5
0.45
0.4
2
4
6
8
10
0.6
0.55
150
200
250
OPL RND PRK
WEI
1
0.5
0
CLL
OPL RND WFL PRP
7000
0.65
0.6
0.55
0.5
0.45
2
1.5
methods
CLL OPL RND GRD REL
network size
CLL
methods
CLL OPL RND GRD REL
100
5
10
intervention points
0.65
0.5
50
8
network size
prediction accuracy
prediction accuracy
0.6
100
6
10
CLL OPL RND WFL PRP
CLL OPL RND WFL PRP
0.65
0.45
50
4
15
intervention points
minimum exposure
100
0.75
average distance
0.6
50
sum of exposure
prediction accuracy
prediction accuracy
0.7
20
0.8
network size
prediction accuracy
Least-squares Shaping
Minimum Maximization Capped Maximization
CLL OPL RND PRK WEI
0.75
4
6
8
intervention points
10
6500
6000
5500
5000
CLL OPL RND GRD REL
methods
Performance vs. # users
Performance vs. # points
Objective function
Figure 2: real world dataset results; n = 300, M = 6, T = 40
more difficult as more random variables are involved. The middle panel shows the performance with
respect to increasing the number of intervention points. Here, a slight increase in the performance
is apparent. As the number of intervention points increases the algorithm has more control over the
outcome and can reach the objective function better.
Fig. 2 top row summarizes the results of CEM. The left panel demonstrates the predictive performance of the algorithms. CLL consistently outperforms the rest. With 65-70 % of accuracy in
predicting the optimal cascade. The right panel shows the objective function simulated 10 times
with the learned parameters for network of n = 300 users on 6 intervention points. The extra 2.5
extra exposure per user compared to the second best method with the same budget and constraint
would be a significant advertising achievement. Among the competitors OPL and RND seem to
perform good. If there where no cap over the resultant exposure, all methods would perform comparably because of the linearity of sum of exposure. However, the successful method is the one who
manage to maximize exposure considering the cap. Failure of PRK and WEI indicates that structural
properties are not enough to capture the influence. Compared to these two, RND performs better in
average, however exhibits a larger variance as expected.
Fig. 2 middle row summarizes the results for MEM and shows CLL outperforms others consistently.
CLL still is the best algorithm and OPL and RND are the significant baselines. Failure of WFL and
PRP shows the network structure plays a significant role in the activity and exposure processes.
The bottom row in Fig. 2 demonstrates the results of LES. CLL is still the best method. OPL is still
strong but RND is not performing well. The objective function is summation of the square of the
gap between target and current exposure. This explains why GRD is showing a comparable success,
since, it starts with the highest gap in the exposure and greedily allocates the budget.
Conclusion. In this paper, we introduced the optimal multistage campaigning problem, which is a
generalization of the activity shaping and influence maximization problems, and it allows for more
elaborate goal functions. Our model of social activity is based on multivariate Hawkes process,
and for the first time, we manage to derive a linear connection between a time-varying exogenous
intensity and the overall network exposure of the campaign.
Acknowledgement. The work is supported in part by NSF/NIH BIGDATA R01 GM108341, NSF
IIS-1639792, NSF DMS-1620345, and NSF DMS-1620342.
8
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9
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5,640 | 6,103 | Coordinate-wise Power Method
1
Qi Lei 1
Kai Zhong 1
Inderjit S. Dhillon 1,2
Institute for Computational Engineering & Sciences 2 Department of Computer Science
University of Texas at Austin
{leiqi, zhongkai}@ices.utexas.edu, inderjit@cs.utexas.edu
Abstract
In this paper, we propose a coordinate-wise version of the power method from
an optimization viewpoint. The vanilla power method simultaneously updates
all the coordinates of the iterate, which is essential for its convergence analysis.
However, different coordinates converge to the optimal value at different speeds.
Our proposed algorithm, which we call coordinate-wise power method, is able
to select and update the most important k coordinates in O(kn) time at each
iteration, where n is the dimension of the matrix and k ? n is the size of the
active set. Inspired by the ?greedy? nature of our method, we further propose a
greedy coordinate descent algorithm applied on a non-convex objective function
specialized for symmetric matrices. We provide convergence analyses for both
methods. Experimental results on both synthetic and real data show that our
methods achieve up to 23 times speedup over the basic power method. Meanwhile,
due to their coordinate-wise nature, our methods are very suitable for the important
case when data cannot fit into memory. Finally, we introduce how the coordinatewise mechanism could be applied to other iterative methods that are used in machine
learning.
1
Introduction
Computing the dominant eigenvectors of matrices and graphs is one of the most fundamental tasks
in various machine learning problems, including low-rank approximation, principal component
analysis, spectral clustering, dimensionality reduction and matrix completion. Several algorithms are
known for computing the dominant eigenvectors, such as the power method, Lanczos algorithm [14],
randomized SVD [2] and multi-scale method [17]. Among them, the power method is the oldest and
simplest one, where a matrix A is multiplied by the normalized iterate x(l) at each iteration, namely,
x(l+1) = normalize(Ax(l) ).
The power method is popular in practice due to its simplicity, small memory footprint and robustness,
and particularly suitable for computing the dominant eigenvector of large sparse matrices [14]. It has
applied to PageRank [7], sparse PCA [19, 9], private PCA [4] and spectral clustering [18]. However,
its convergence rate depends on | 2 |/| 1 |, the ratio of magnitude of the top two dominant eigenvalues
[14]. Note that when | 2 | ? | 1 |, the power method converges slowly.
In this paper, we propose an improved power method, which we call coordinate-wise power method,
to accelerate the vanilla power method. Vanilla power method updates all n coordinates of the iterate
simultaneously even if some have already converged to the optimal. This motivates us to develop new
algorithms where we select and update a set of important coordinates at each iteration. As updating
each coordinate costs only n1 of one power iteration, significant running time can be saved when n is
very large. We raise two questions for designing such an algorithm.
The first question: how to select the coordinate? A natural idea is to select the coordinate that will
change the most, namely,
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Ax
x,
(1)
xT Ax
where xTAx
is a scaled version of the next iterate given by power method, and we will explain this
Ax
special scaling factor in Section 2. Note that ci denotes the i-th element of the vector c. Instead of
choosing only one coordinate to update, we can also choose k coordinates with the largest k changes
in {|ci |}ni=1 . We will justify this selection criterion by connecting our method with greedy coordinate
descent algorithm for minimizing a non-convex function in Section 3. With this selection rule, we
are able to show that our method has global convergence guarantees and faster convergence rate
compared to vanilla power method if k satisfies certain conditions.
Another key question: how to choose these coordinates without too much overhead? How to efficiently
select important elements to update is of great interest in the optimization community. For example,
[1] leveraged nearest neighbor search for greedy coordinate selection, while [11] applied partially
biased sampling for stochastic gradient descent. To calculate the changes in Eq (1) we need to know
all coordinates of the next iterate. This violates our previous intention to calculate a small subset of
the new coordinates. We show, by a simple trick, we can use only O(kn) operations to update the
most important k coordinates. Experimental results on dense as well as sparse matrices show that our
method is up to 8 times faster than vanilla power method.
Relation to optimization. Our method reminds us of greedy coordinate descent method. Indeed,
we show for symmetric matrices our coordinate-wise power method is similar to greedy coordinate
descent for rank-1 matrix approximation, whose variants are widely used in matrix completion [8]
and non-negative matrix factorization [6]. Based on this interpretation, we further propose a faster
greedy coordinate descent method specialized for symmetric matrices. This method achieves up to 23
times speedup over the basic power method and 3 times speedup over the Lanczos method on large
real graphs. For this non-convex problem, we also provide convergence guarantees when the initial
iterate lies in the neighborhood of the optimal solution.
Extensions. With the coordinate-wise nature, our methods are very suitable to deal with the case
when data cannot fit into memory. We can choose a k such that k rows of A can fit in memory, and
then fully process those k rows of data before loading the RAM (random access memory) with a new
partition of the matrix. This strategy helps balance the data processing and data loading time. The
experimental results show our method is 8 times faster than vanilla power method for this case.
The paper is organized as follows. Section 2 introduces coordinate-wise power method for computing
the dominant eigenvector. Section 3 interprets our strategy from an optimization perspective and
proposes a faster algorithm. Section 4 provides theoretical convergence guarantee for both algorithms.
Experimental results on synthetic or real data are shown in Section 5. Finally Section 6 presents
the extensions of our methods: dealing with out-of-core cases and generalizing the coordinate-wise
mechanism to other iterative methods that are useful for the machine learning community.
argmaxi |ci |, where c =
2
Coordinate-wise Power Method
The classical power method (PM) iteratively multiplies the iterate x 2 Rn by the matrix A 2 Rn?n ,
which is inefficient since some coordinates may converge faster than others. To illustrate this
(a) The percentage of unconverged coordinates versus the number of operations
(b) Number of updates of each coordinate
Figure 1: Motivation for the Coordinate-wise Power Method.
Figure 1(a) shows how the percentage
of unconverged coordinates decreases with the number of operations. The gradual decrease demonstrates the
unevenness of each coordinate as the iterate converges to the dominant eigenvector. In Figure 1(b), the X-axis
is the coordinate indices of iterate x sorted by their frequency of updates, which is shown on the Y-axis. The
area below each curve approximately equals the total number of operations.The given matrix is synthetic with
| 2 |/| 1 | = 0.5, and terminating accuracy ? is set to be 1e-5.
2
phenomenon, we conduct an experiment with the power method; we set the stopping criterion as
kx v 1 k1 < ?, where ? is the threshold for error, and let v i denote the i-th dominant eigenvector
(associated with the eigenvalue of the i-th largest magnitude) of A in this paper. During the iterative
process, even if some coordinates meet the stopping criterion, they still have to be updated at every
iteration until uniform convergence. In Figure 1(a), we count the number of unconverged coordinates,
which we define as {i : i 2 [n] |xi v1,i | > ?}, and see it gradually decreases with the iterations,
which implies that the power method makes a large number of unnecessary updates. In this paper, for
computing the dominant eigenvector, we exhibit a coordinate selection scheme that has the ability to
select and update ?important? coordinates with little overhead. We call our method Coordinate-wise
Power Method (CPM). As shown in Figure 1(a) and 1(b), by selecting important entries to update,
the number of unconverged coordinates drops much faster, leading to an overall fewer flops.
Algorithm 1 Coordinate-wise Power Method
1: Input: Symmetric matrix A 2 Rn?n , number of selected coordinates k, and number of iterations, L.
(0)
2: Initialize x(0) 2 Rn and set z (0) = Ax(0) . Set coordinate selecting criterion c(0) = x(0) (x(0)z)T z(0) .
3: for l = 1 to L do
Let ?(l) be a set containing k coordinates of c(l 1) with the largest magnitude. Execute the following
4:
8
updates:
(l 1)
zj
<
, j 2 ?(l)
(l)
(x(l 1) )T z (l 1)
yj
=
(2)
(l
1)
:
xj
, j2
/ ?(l)
z (l)
z
(l)
c(l)
=
=
=
z (l
z
(l)
1)
(l)
+ A(y ?(l)
/ky
x(l)
(l)
k, x
z
(x(l
(l)
(l 1)
(3)
x?(l) )
=y
(l)
(l)
/ky
(l)
k
1) )T z (l 1)
5: Output: Approximate dominant eigenvector x(L)
Algorithm 1 describes our coordinate-wise power method that updates k entries at a time for computing the dominant eigenvector for a symmetric input matrix, while a generalization to asymmetric
cases is straightforward. The algorithm starts from an initial vector x(0) , and iteratively performs
updates xi
aTi x/xT Ax with i in a selected set of coordinates ? ? [n] defined in step 4, where
ai is the i-th row of A. The set of indices ? is chosen to maximize the difference between the current
coordinate value xi and the next coordinate value aTi x/xT Ax. z (l) and c(l) are auxiliary vectors.
Maintaining z (l) ? Ax(l) saves much time, while the magnitude of c represents importance of each
coordinate and is used to select ?.
We use the Rayleigh Quotient xT Ax (x is normalized) for scaling, different from kAxk in the power
method. Our intuition is as follows: on one hand, it is well known that Rayleigh Quotient is the
best estimate for eigenvalues. On the other hand, the limit point using xT Ax scaling will satisfy
? = A?
x
x/?
xT A?
x, which allows both negative or positive dominant eigenvectors, while the scaling
kAxk is always positive, so its limit point only lies in the eigenvectors associated with positive
eigenvalues, which rules out the possibility of converging to the negative dominant eigenvector.
2.1
Coordinate Selection Strategy
An initial understanding for our coordinate selection strategy is that we select coordinates with the
largest potential change. With a current iterate x and an arbitrary active set ?, let y ? be a potential
next iterate with only coordinates in ? updated, namely,
? T
ai x
?
, i2?
xT Ax
(y )i =
xi , i 2
/?
According to our algorithm, we select active set ? to maximize the iterate change. Therefore:
(
)
(
)
2
2
Ax
Ax
def
I
2
I
2
? = arg max
(x
)I = ky
xk
= arg min
y
= kgk
xT Ax
xT Ax
I?[n],|I|=k
I?[n],|I|=k
This is to say, with our updating rule, our goal of maximizing iteration gap is equivalent to minimizing
the difference between the next iterate y (l+1) and Ax(l) /(x(l) )T Ax(l) , where this difference could
be interpreted as noise g (l) . A good set ? ensures a sufficiently small noise g (l) , thus achieving a
3
similar convergence rate in O(kn) time (analyzed later) as the power method does in O(n2 ) time.
More formal statement for the convergence analysis is given in Section 4.
Another reason for this selection rule is that it incurs little overhead. For each iteration, we maintain a
vector z ? Ax with kn flops by the updating rule in Eq.(3). And the overhead consists of calculating
c and choosing ?. Both parts cost O(n) operations. Here ? is chosen by Hoare?s quick selection
algorithm [5] to find the k th largest entry in |c|. Thus the overhead is negligible compared with O(kn).
Thus CPM spends as much time on each coordinate as PM does on average, while those updated k
coordinates are most important. For sparse matrices, the time complexity is O(n + nk nnz(A)) for
each iteration, where nnz(A) is the number of nonzero elements in matrix A.
Although the above analysis gives us a good intuition on how our method works, it doesn?t directly
show that our coordinate selection strategy has any optimal properties. In next section, we give
another interpretation of our coordinate-wise power method and establish its connection with the
optimization problem for low-rank approximation.
3
Optimization Interpretation
The coordinate descent method [12, 6] was popularized due to its simplicity and good performance.
With all but one coordinates fixed, the minimization of the objective function becomes a sequence of
subproblems with univariate minimization. When such subproblems are quickly solvable, coordinate
descent methods can be efficient. Moreover, in different problem settings, a specific coordinate
selecting rule in each iteration makes it possible to further improve the algorithm?s efficiency.
The power method reminds us of the rank-one matrix factorization
arg min
x2Rn ,y2Rd
f (x, y) = kA
xy T k2F
(4)
Ay
With alternating minimization, the update for x becomes x
kyk2 and vice versa for y. Therefore
for symmetric matrix, alternating minimization is exactly PM apart from the normalization constant.
Meanwhile, the above similarity between PM and alternating minimization extends to the similarity
between CPM and greedy coordinate descent. A more detailed interpretation is in Appendix A.5,
where we show the equivalence in the following coordinate selecting rules for Eq.(4): (a) largest coordinate value change, denoted as | xi |; (b) largest partial gradient (Gauss-Southwell rule), |ri f (x)|;
(c) largest function value decrease, |f (x + xi ei ) f (x)|. Therefore, the coordinate selection rule
is more formally testified in optimization viewpoint.
3.1
Symmetric Greedy Coordinate Descent (SGCD)
We propose an even faster algorithm based on greedy coordinate descent. This method is designed
for symmetric matrices and additionally requires to know the sign of the most dominant eigenvalue.
We also prove its convergence to the global optimum with a sufficiently close initial point.
A natural alternative objective function specifically for the symmetric case would be
arg min f (x) = kA
x2Rn
xxT k2F .
(5)
2
Notice that the stationary
p points of f (x), which require rf (x) = 4(kxk x Ax) = 0, are obtained
?
at eigenvectors: xi =
i v i , if the eigenvalue i is positive. The global minimum for Eq. (5) is the
eigenvector corresponding to the largest positive eigenvalue, not the one with the largest magnitude.
For most applications like PageRank we know 1 is positive, but if we want to calculate the negative
eigenvalue with the largest magnitude, just optimize on f = kA + xxT k2F instead.
Now we introduce Algorithm 2 that optimizes Eq. (5). With coordinate descent, we update the i-th
(l+1)
(l)
coordinate by xi
arg min? f (x(l) + (? xi )ei ), which requires the partial derivative of
f (x) in i-th coordinate to be zero, i.e.,
()
ri f (x) = 4(xi kxk22
x3i
aTi x) = 0.
+ pxi + q = 0, where p = kxk2
(6)
x2i
aii , and q =
aTi x + aii xi
(7)
Similar to CPM, the most time consuming part comes from maintaining z (? Ax), as the calculation
for selecting the criterion c and the coefficient q requires it. Therefore the overall time complexity for
one iteration is the same as CPM.
4
Notice that c from Eq.(6) is the partial gradient of f , so we are using the Gauss-Southwell rule to
choose the active set. And it is actually the only effective and computationally cheap selection rule
among previously analyzed rules (a), (b) or (c). For calculating the iterate change | xi |, one needs to
obtain roots for n equations. Likewise, the function decrease | fi | requires even more work.
Remark: for an unbiased initializer, x(0) should be scaled by a constant ? such that
s
(x(0) )T Ax(0)
? = arg min kA (ax(0) )(ax(0) )T kF =
kx(0) k4
a 0
Algorithm 2 Symmetric greedy coordinate descent (SGCD)
1: Input: Symmetric matrix A 2 Rn?n , number of selected coordinate, k, and number of iterations, L.
(0)
2: Initialize x(0) 2 Rn and set z (0) = Ax(0) . Set coordinate selecting criterion c(0) = x(0) kxz(0) k2 .
3: for l = 0 to L 1 do
4:
Let ?(l) be a set containing k coordinates of c(l) with the largest magnitude. Execute the following
updates:
?
?
(
(l)
arg min? f x(l) + (? xj )ej , if j 2 ?(l) ,
(l+1)
xj
=
(l)
xj ,
if j 2
/ ?(l) .
(l+1)
z (l+1)
=
z (l) + A(x?(l)
c(l+1)
=
x(l+1)
(l)
x?(l) )
z (l+1)
kx(l+1) k2
5: Output: vector x(L)
4
Convergence Analysis
In the previous section, we propose coordinate-wise power method (CPM) and symmetric greedy
coordinate descent (SGCD) on a non-convex function for computing the dominant eigenvector.
However, it remains an open problem to prove convergence of coordinate descent methods for general
non-convex functions. In this section, we show that both CPM and SGCD converge to the dominant
eigenvector under some assumptions.
4.1
Convergence of Coordinate-wise Power Method
Consider a positive semidefinite matrix A, and let v 1 be its leading eigenvector. For any sequence
(l)
(l)
(x(0) , x(1) , ? ? ? ) generated
q by Algorithm 1, let ? to be the angle between vector x and v 1 ,
P
def
(l) 2
(l)
(l)
(l)
and (l) (k) = min|?|=k
i2?
/ (ci ) /kc k2 = kg k/kc k. The following lemma illustrates
(l)
convergence of the tangent of ? .
Lemma 4.1. Suppose k is large enough such that
(l)
Then
tan ?(l+1)
?
(k) <
1
2
(1 + tan ?(l) )
tan ?(l) (
2
1
.
(l)
+
(8)
1
(k))
< tan ?(l)
cos ?(l)
(9)
With the aid of Lemma 4.1, we show the following iteration complexity:
Theorem 4.2. For any sequence (x(0) , x(1) , ? ? ? ) generated by Algorithm 1 with k satisfying
(0)
(l)
1
2
(k) < 2 1 (1+tan
, if x(0) is not orthogonal to v 1 , then after T = O( 1 1 2 log( tan"? ))
? (l) )
iterations we have tan ?(T ) ? ".
The iteration complexity shown is the same as the power method, but since it requires less operations
(O(knnz(A)/n) instead of O(nnz(A)) per iteration, we have
Corollary 4.2.1. If the requirements in Theorem 4.2 apply and additionally k satisfies:
k < n log(( 1 + 2 )/(2 1 ))/ log( 2 / 1 ),
(10)
CPM has a better convergence rate than PM in terms of the number of equivalent passes over the
coordinates.
5
The RHS of (10) ranges from 0.06n to 0.5n when 21 goes from 10 5 to 1 10 5 . Meanwhile,
experiments show that the performance of our algorithms isn?t too sensitive to the choice of k. Figure
6 in Appendix A.6 illustrates that a sufficiently large range of k guarantees good performances. Thus
n
we use a prescribed k = 20
throughout our experiments in this paper, which saves the burden of
tuning parameters and is a theoretically and experimentally favorable choice.
Part of the proof is inspired by the noisy power method [3] in that we consider the unchanged part
g as noise. For the sake of a neat proof we require our target matrix to be positive semidefinite,
although experimentally a generalization to regular matrices is also valid for our algorithm. Details
can be found in Appendix A.1 and A.3.
4.2
Local Convergence for Optimization on kA
xxT k2F
As the objective in Problem (5) is non-convex, it is hard to show global convergence. Clearly, with
exact coordinate descent, Algorithm 2 will converge to some stationary point. In the following, we
show that Algorithm 2 converges to the global minimum with a starting point sufficiently close to it.
(0)
Theorem 4.3. (Local Linear Convergence) For any sequence of iterates (xp
, x(1) , ? ? ? ) generated
(0)
by Algorithm 2, assume the starting
1 v 1 with radius r =
p point x is in a ball centered by
1
2
O( p
), or formally, x(0) 2 Br ( 1 v 1 ), then (x0 , x1 , ? ? ? ) converges to the optima linearly.
1
?
(0)
maxi |aii |
Specifically, when k = 1, then after T = 14 1 2 2 +4
log f (x ") f iterations, we have
?
p
f (x(T ) ) f ? ? ", where f ? = f ( 1 v 1 ) is the global minimum of the objective function f , and
rf (y)k1
? = inf x,y2Br (p 1 v1 ) krf (x)
2 [ 3( 1n 2 ) , 3( 1
2 )].
kx yk1
We prove this by showing that the objective (5) is strongly convex and coordinate-wise Lipschitz
continuous in a neighborhood of the optimum. The proof is given in Appendix A.4.
Remark: For real-life graphs, the diagonal values aii = 0, and the coefficient in the iteration
complexity could be simplified as 14 1? 2 2 when k = 1.
10 7
6
10
1
10
0
10 2
CPM
SGCD
PM
Lanczos
VRPCA
10 5
time (sec)
flops/n
time (sec)
10
10 2
CPM
SGCD
PM
Lanczos
VRPCA
10
1
10
0
CPM
SGCD
PM
Lanczos
VRPCA
10 -1
10
4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
?2 /?1
(a) Convergence flops vs
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
?2 /?1
2
1
(b) Convergence time vs
1
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
n
2
1
(c) Convergence time vs dimension
Figure 2: Matrix properties affecting performance. Figure 2(a), 2(b) show the performance of five methods
with 21 ranging from 0.01 to 0.99 and fixed matrix size n = 5000. In Figure 2(a) the measurement is FLOPs
while in Figure 2(b) Y-axis is CPU time. Figure 2(c) shows how the convergence time varies with the dimension
when fixing 21 = 2/3. In all figures Y-axis is in log scale for better observation. Results are averaged over from
20 runs.
5
Experiments
In this section, we compared our algorithms with PM, Lanczos method [14], and VRPCA [16] on
dense as well as sparse dataset. All the experiments were executed on Intel(R) Xeon(R) E5430
machine with 16G RAM and Linux OS. We implement all the five algorithms in C++ with Eigen
library.
5.1
Comparison on Dense and Simulated Dataset
We compare PM with our CPM and SGCD methods to show how coordinate-wise mechanism
improves the original method. Further, we compared with a state-of-the-art algorithm Lanczos
method. Besides, we also include a recent proposed stochastic SVD algorithm, VRPCA, that enjoys
exponential convergence rate and shows similar insight in viewing the data in a separable way.
With dense and synthetic matrices, we are able to test the condition that our methods are preferable,
and how the properties of the matrix, like 2 / 1 or the dimension, affect the performance. For each
algorithm, we start from the same random vector, and set stopping condition to be cos ? 1 ?, ? =
10 6 , where ? is the angle between the current iterate and the dominant eigenvector.
6
First we compare the performances with number of FLOPs (Floating Point Operations), which could
better illustrate how greediness affects the algorithm?s efficiency. From Figure 2(a) we can see our
method shows much better performance than PM, especially when 2 / 1 ! 1, where CPM and
SGCD respectively achieve more than 2 and 3 times faster than PM. Figure 2(b) shows running time
using five methods under different eigenvalue ratios 2 / 1 . We can see that only in some extreme
cases when PM converges in less than 0.1 second, PM is comparable to our methods. In Figure 2(c)
the testing factor is the dimension, which shows the performance is independent of the size of n.
Meanwhile, in most cases, SGCD is better than Lanczos method. And although VRPCA has better
convergence rate, it requires at least 10n2 operations for one data pass. Therefore in real applications,
it is not even comparable to PM.
5.2
Comparison on Sparse and Real Dataset
Table 1: Six datasets and the performance of three methods on them.
Dataset
n
nnz(A)
nnz/n
com-Orkut
soc-LiveJournal
soc-Pokec
web-Stanford
ego-Gplus
ego-Twitter
3.07M
4.85M
1.63M
282K
108K
81.3K
234M
86M
44M
3.99M
30.5M
2.68M
76.3
17.8
27.3
14.1
283
33
2
1
Time (sec)
CPM SGCD
31.5
19.3
17.9
13.7
26.5
5.2
1.05
0.54
0.57
0.61
0.15
0.11
PM
109.6
58.5
118
8.15
0.99
0.31
0.71
0.78
0.95
0.95
0.51
0.65
Lanczos
63.6
25.8
14.2
0.69
1.01
0.19
VRPCA
189.7
88.1
596.2
7.55
5.06
0.98
To test the scalability of our methods, we further test and compare our methods on large and sparse
datasets. We use the following real datasets:
1) com-Orkut: Orkut online social network
2) soc-LiveJournal: On-line community for maintaining journals, individual and group blogs
3) soc-Pokec: Pokec, most popular on-line social network in Slovakia
4) web-Stanford: Pages from Stanford University (stanford.edu) and hyperlinks between them
5) ego-Gplus (Google+): Social circles from Google+
6) ego-Twitter: Social circles from Twitter
The statistics of the datasets are summarized in Table 1, which includes the essential properties of the
datasets that affect the performances and the average CPU time for reaching cos ?x,v1 1 10 6 .
Figure 3 shows tan ?x,v1 against the CPU time for the four methods with multiple datasets.
From the statistics in Table 1 we can see that in all the cases, either CPM or SGCD performs the best.
CPM is roughly 2-8 times faster than PM, while SGCD reaches up to 23 times and 3 times faster
than PM and Lanczos method respectively. Our methods show their privilege in the soc-Pokec(3(c))
and web-Stanford(3(d)), the most ill-conditioned cases ( 2 / 1 ? 0.95), achieving 15 or 23 times
of speedup on PM with SGCD. Meanwhile, when the condition number of the datasets is not too
small (see 3(a),3(b),3(e),3(f)), both CPM and SGCD outperform PM as well as Lanczos method. And
10 1
10 1
CPM
SGCD
PM
Lanczos
VRPCA
10 -1
10
-2
10 -3
10 -1
10
0
20
40
60
80
100
120
140
-2
10 -3
160
0
10
20
30
60
10 -3
70
10 -1
10
2
3
4
5
40
60
6
7
8
time (sec)
(d) Performance on web-Stanford
9
-2
10 -3
80
100
120
140
160
10 1
CPM
SGCD
PM
Lanczos
VRPCA
10 -1
10
0
0.5
1
1.5
2
2.5
time (sec)
(e) Performance on Google+
3
CPM
SGCD
PM
Lanczos
VRPCA
10 0
tan ?x,v1
tan ?x,v1
-2
1
20
time (sec)
10 0
10 -1
0
0
(c) Performance on soc-Pokec
10 1
CPM
SGCD
PM
Lanczos
VRPCA
10 0
tan ?x,v1
50
(b) Performance on LiveJournal
10 1
10 -3
40
-2
time (sec)
(a) Performance on com-Orkut
10
10 -1
10
time (sec)
CPM
SGCD
PM
Lanczos
VRPCA
10 0
tan ?x,v1
10 0
tan ?x,v1
tan ?x,v1
10 0
10 1
CPM
SGCD
PM
Lanczos
VRPCA
-2
10 -3
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
time (sec)
(f) Performance on ego-Twitter
Figure 3: Time comparison for sparse dataset. X-axis shows the CPU time while Y-axis is log scaled tan ?
between x and v 1 . The empirical performance shows all three methods have linear convergence.
7
similar to the reasoning in the dense case, although VRPCA requires less iterations for convergence,
the overall CPU time is much longer than others in practice.
In summary of performances on both dense and sparse datasets, SGCD is the fastest among others.
6
Other Application and Extensions
6.1
Comparison on Out-of-core Real Dataset
10 2
CPM
SGCD
10 1
PM
An important application for coordinate-wise
power method is the case when data can not fit
into memory. Existing methods can?t be easily
applied to out-of-core dataset. Most existing
methods don?t indicate how we can update part
of the coordinates multiple times and fully reuse
part of the matrix corresponding to those active
coordinates. Therefore the data loading and data
time (sec)
processing time are highly unbalanced. A naive
Figure
4:
A
pseudograph
for time comparison of
way of using PM would be repetitively loading
out-of-core
dataset
from
Twitter.
Each "staircase" ilpart of the matrix from the disk and calculating
that part of matrix-vector multiplication. But lustrates the performance of one data pass. The flat part
indicates the stage of loading data, while the downward
from Figure 4 we can see reading from the disk part shows the phase of processing data. As we only
costs much more time than the process of com- updated auxiliary vectors instead of the iterate every
putation, therefore we will waste a lot of time time we load part of the matrix, we could not test perif we cannot fully use the data before dumping formances until a whole data pass. Therefore for the
it. For CPM, as we showed in Theorem 4.1 that sake of clear observation, we group together the loading
updating only k coordinates of iterate x may phase and the processing phase in each data pass.
still enhance the target direction, we could do
matrix vector multiplication multiple times after one single loading. As with SGCD, optimization on
part of x for several times will also decrease the function value.
We did experiments on the dataset from Twitter [10] using out-of-core version of the three algorithms
shown in Algorithm 3 in Appendix A.7. The data, which contains 41.7 million user profiles and 1.47
billion social relations, is originally 25.6 GB and then separated into 5 files. In Figure 4, we can see
that after data pass, our methods can already reach rather high precision, which compresses hours of
processing time to 8 minutes.
tan ?x,v1
10 0
10 -1
10 -2
10 -3
10 -4
10 -5
6.2
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Extension to other linear algebraic methods
With the interpretation in optimization, we could apply a coordinate-wise mechanism to PM and get
good performance. Meanwhile, for some other iterative methods in linear algebra, if the connection to
optimization is valid, or if the update is separable for each coordinate, the coordinate-wise mechanism
may also be applicable, like Jacobi method.
For diagonal dominant matrices, Jacobi iteration [15] is a classical method for solving linear system
Ax = b with linear convergence rate. The iteration procedure is:
Initialize: A ! D + R, where D =Diag(A), and R = A
Iterations: x+
D 1 (b Rx).
D.
This method is similar to the vanilla power method, which includes a matrix vector multiplication
Rx with an extra translation b and a normalization step D 1 . Therefore, a potential similar
realization of greedy coordinate-wise mechanism is also applicable here. See Appendix A.8 for more
experiments and analyses, where we also specify its relation to Gauss-Seidel iteration [15].
7
Conclusion
In summary, we propose a new coordinate-wise power method and greedy coordinate descent
method for computing the most dominant eigenvector of a matrix. This problem is critical to many
applications in machine learning. Our methods have convergence guarantees and achieve up to 23
times of speedup on both real and synthetic data, as compared to the vanilla power method.
Acknowledgements
This research was supported by NSF grants CCF-1320746, IIS-1546452 and CCF-1564000.
8
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9
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5,641 | 6,104 | Fast learning rates with heavy-tailed losses
1
Vu Dinh1 Lam Si Tung Ho2 Duy Nguyen3 Binh T. Nguyen4
Program in Computational Biology, Fred Hutchinson Cancer Research Center
2
Department of Biostatistics, University of California, Los Angeles
3
Department of Statistics, University of Wisconsin-Madison
4
Department of Computer Science, University of Science, Vietnam
Abstract
We study fast learning rates when the losses are not necessarily bounded and may
have a distribution with heavy tails. To enable such analyses, we introduce two
new conditions: (i) the envelope function supf ?F |` ? f |, where ` is the loss function and F is the hypothesis class, exists and is Lr -integrable, and (ii) ` satisfies
the multi-scale Bernstein?s condition on F. Under these assumptions, we prove
that learning rate faster than O(n?1/2 ) can be obtained and, depending on r and
the multi-scale Bernstein?s powers, can be arbitrarily close to O(n?1 ). We then
verify these assumptions and derive fast learning rates for the problem of vector
quantization by k-means clustering with heavy-tailed distributions. The analyses enable us to obtain novel learning rates that extend and complement existing
results in the literature from both theoretical and practical viewpoints.
1
Introduction
The rate with which a learning algorithm converges as more data comes in play a central role in
machine learning. Recent progress has refined our theoretical understanding about setting under
which fast learning rates are possible, leading to the development of robust algorithms that can
automatically adapt to data with hidden structures and achieve faster rates whenever possible. The
literature, however, has mainly focused on bounded losses and little has been known about rates of
learning in the unbounded cases, especially in cases when the distribution of the loss has heavy tails
[van Erven et al., 2015].
Most of previous work about learning rate for unbounded losses are done in the context of density
estimation [van Erven et al., 2015, Zhang, 2006a,b], of which the proofs of fast rates implicitly
employ the central condition [Gr?unwald, 2012] and cannot be extended to address losses with polynomial tails [van Erven et al., 2015]. Efforts to resolve this issue include Brownlees et al. [2015],
which proposes using some robust mean estimators to replace empirical means, and Cortes et al.
[2013], which derives relative deviation and generalization bounds for unbounded losses with the
assumption that Lr -diameter of the hypothesis class is bounded. However, results about fast learning
rates were not obtained in both approaches. Fast learning rates are derived in Lecu?e and Mendelson [2013] for sub-Gaussian losses and in Lecu?e and Mendelson [2012] for hypothesis classes that
have sub-exponential envelope functions. To the best of our knowledge, no previous work about fast
learning rates for heavy-tailed losses has been done in the literature.
The goal of this research is to study fast learning rates for the empirical risk minimizer when the
losses are not necessarily bounded and may have a distribution with heavy tails. We recall that
heavy-tailed distributions are probability distributions whose tails are not exponentially bounded:
that is, they have heavier tails than the exponential distribution. To enable the analyses of fast
rates with heavy-tailed losses, two new assumptions are introduced. First, we assume the existence
and the Lr -integrability of the envelope function F = supf ?F |f | of the hypothesis class F for
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
some value of r ? 2, which enables us to use the results of Lederer and van de Geer [2014] on
concentration inequalities for suprema of empirical unbounded processes. Second, we assume that
the loss function satisfies the multi-scale Bernstein?s condition, a generalization of the standard
Bernstein?s condition for unbounded losses, which enables derivation of fast learning rates.
Building upon this framework, we prove that if the loss has finite moments up to order r large
enough and if the hypothesis class satisfies the regularity conditions described above, then learning
rate faster than O(n?1/2 ) can be obtained. Moreover, depending on r and the multi-scale Bernstein?s
powers, the learning rate can be arbitrarily close to the optimal rate O(n?1 ). We then verify these
assumptions and derive fast learning rates for the k-mean clustering algorithm and prove that if the
distribution of observations has finite moments up to order r and satisfies the Pollard?s regularity
conditions, then fast learning rate can be derived. The result can be viewed as an extension of
the result from Antos et al. [2005] and Levrard [2013] to cases when the source distribution has
unbounded support, and produces a more favorable convergence rate than that of Telgarsky and
Dasgupta [2013] under similar settings.
2
Mathematical framework
Let the hypothesis class F be a class of functions defined on some measurable space X with values
in R. Let Z = (X, Y ) be a random variable taking values in Z = X ?Y with probability distribution
P where Y ? R. The loss ` : Z ? F ? R+ is a non-negative function. For a hypothesis f ? F and
n iid samples {Z1 , Z2 , . . . , Zn } of Z, we define
n
P `(f ) = EZ?P [`(Z, f )]
and
Pn `(f ) =
1X
`(Zi , f ).
n i=1
For unsupervised learning frameworks, there is no output (Y = ?) and the loss has the form `(X, f )
depending on applications. Nevertheless, P `(f ) and Pn `(f ) can be defined in a similar manner. We
will abuse the notation to denote the losses `(Z, f ) by `(f ). We also denote the optimal hypothesis
f ? be any function for which P `(f ? ) = inf f ?F P `(f ) := P ? and consider the empirical risk
minimizer (ERM) estimator f?n = arg minf ?F Pn `(f ).
We recall that heavy-tailed distributions are probability distributions whose tails are not exponentially bounded. Rigorously, the distribution of a random variable V is said to have a heavy right
tail if limv?? e?v P[V > v] = ? for all ? > 0 and the definition is similar for heavy left tail. A
learning problem is said to be with heavy-tailed loss if the distribution of `(f ) has heavy tails from
some or all hypotheses f ? F.
For a pseudo-metric space (G, d) and > 0, we denote by N (, G, d) the covering number of (G, d);
that is, N (, G, d) is the minimal number of balls of radius needed to cover G. The universal metric
entropy of G is defined by H(, G) = supQ log N (, G, L2 (Q)), where the supremum is taken over
the set of all probability measures Q concentrated on some finite subset of G. For convenience, we
define G = ` ? F the class of all functions g such that g = `(f ) for some f ? F and denote by F a
finite subset of F such that G is contained in the union of balls of radius with centers in G = `?F .
We refer to F and G as an -net of F and G, respectively.
To enable the analyses of fast rates for learning problems with heavy-tailed losses, throughout the
paper, we impose the following regularity conditions on F and `.
Assumption 2.1 (Multi-scale Bernstein?s condition). Define F ? = arg minF P `(f ). There exist a
finite partition of F = ?i?I Fi , positive constants B = {Bi }i?I , constants ? = {?i }i?I in (0, 1],
?
and f ? = {fi? }i?I ? F ? such that E[(`(f ) ? `(fi? ))2 ] ? Bi (E[`(f ) ? `(fi? )]) i for all i ? I and
f ? Fi .
Assumption 2.2 (Entropy bounds). The hypothesis class F is separable and there exist C ? 1,
K ? 1 such that ? ? (0, K], the L2 (P )-covering numbers and the universal metric entropies of G
are bounded as log N (, G, L2 (P )) ? C log(K/) and H(, G) ? C log(K/).
Assumption 2.3 (Integrability of the envelope function). There exists W > 0, r ? C + 1 such that
1/r
E supg?G |g|r
? W.
The multi-scale Bernstein?s condition is more general than the Bernstein?s condition. This entails
that the multi-scale Bernstein?s condition holds whenever the Bernstein?s condition does, thus al2
lows us to consider a larger class of problems. In other words, our results are also valid with the
Bernstein?s condition. The multi-scale Bernstein?s condition is more proper to study unbounded
losses since it is able to separately consider the behaviors of the risk function on microscopic and
macroscopic scales, for which the distinction can only be observed in an unbounded setting.
We also recall that if G has finite VC-dimension, then Assumption 2.2 is satisfied [Boucheron et al.,
2013, Bousquet et al., 2004]. Both Bernstein?s condition and the assumption of separable parametric
hypothesis class are standard assumptions frequently used to obtain faster learning rates in agnostic
settings. A review about the Bernstein?s condition and its applications is Mendelson [2008], while
fast learning rates for bounded losses on hypothesis classes satisfying Assumptions 2.2 were previously studied in Mehta and Williamson [2014] under the stochastic mixability condition. Fast
learning rate for hypothesis classes with envelope functions were studied in Lecu?e and Mendelson
[2012], but under a much stronger assumption that the envelope function is sub-exponential.
Under these assumptions, we illustrate that fast rates for heavy-tailed losses can be obtained.
Throughout the analyses, two recurrent analytical techniques are worth mentioning. The first comes
from the simple observation that in the standard derivation of fast learning rates for bounded losses,
the boundedness assumption is used in multiple places only to provide reverse-Holder-type inequalities, where the L2 -norm are upper bounded by the L1 -norm. This use of the boundedness assumption
can be simply relieved by the assumption that the Lr -norm of the loss is bounded, which implies
(r?2)/(2r?2)
kukL2 ? kukL1
r/(2r?2)
kukLr
.
The second technique relies on the following results of Lederer and van de Geer [2014] on concentration inequalities for suprema of empirical unbounded processes.
Lemma 2.1. If {Vk : k ? K} is a countable family of non-negative functions such that
E sup |Vk |r ? M r
k?K
? 2 = sup EVk2
and
V := sup Pn Vk ,
k?K
k?K
then for all ?, x > 0, we have
P[V ? (1 + ?)EV + x] ? min (1/x)l
1?l?r
64/? + ? + 7) (l/n)
1?l/r
M + 4?
p
l
l/n .
An important notice from this result is that the failure probability is a polynomial in the deviation
x. As we will see later, for a given level of confidence ?, this makes the constant in the convergence
rate a polynomial function of (1/?) instead of log(1/?) as in sub-exponential cases. Thus, more
careful examinations of the order of the failure probability are required for the derivation of any
generalization bound with heavy-tailed losses.
3
Fast learning rates with heavy-tailed losses
The derivation of fast learning rate with heavy tailed losses proceeds as follows. First, we will use
the assumption of integrable envelope function to prove a localization-based result that allows us to
reduce the analyses from the separable parametric classes F to its finite -net F . The multi-scale
Bernstein?s condition is then employed to derive a fast-rate inequality that helps distinguish the
optimal hypothesis from alternative hypotheses in F . The two results are then combined to obtain
fast learning rates.
3.1
Preliminaries
Throughout this section, let G be an -net for G in the L2 (P )-norm, with = n?? for some
1 ? ? > 0. Denote by ? : G ? G an L2 (P )-metric projection from G to G . For any g0 ? G , we
denote K(g0 ) = {|g0 ? g| : g ? ? ?1 (g0 )}. We have
(i) the constant zero function is an element of K(g0 ),
(ii) E[supu?K(g0 ) |u|r ] ? (2W )r ; and supu?K(g0 ) kukL2 (P ) ? ,
(iii) N (t, K(g0 ), L2 (P )) ? (K/t)C for all t > 0.
3
Given a sample Z = (Z1 , . . . , Zn ), we denote by KZ the projection of K(g0 ) onto the sample Z
and by D(KZ ) half of the radius of (KZ , k ? k2 ), that is D(KZ ) = supu,v?KZ ku ? vk/4. We have
the following preliminary lemma, for which the proofs are provided in the Appendix.
r?2
2(r?1)
r
Lemma 3.1. ?2n ED(KZ ) ? + E supu?K(g0 ) (Pn ? P )u
(2W ) 2(r?1) .
Lemma 3.2. Given 0 < ? < 1, there exist constant C1 , C2 > 0 depending only on ? such that for
all x > 0, if x ? ax? + b then x ? C1 a1/(1??) + C2 b.
Lemma 3.3. Define
A(l, r, ?, C, ?) = max l2 /r ? (1 ? ?)l + ?C, [? (1 ? ?/2) ? 1/2] l + ?C .
Assuming that r ? 4C and ? ? 1, if we choose l = r (1 ? ?) /2 and
p
0 < ? < (1 ? 2 C/r)/(2 ? ?),
(3.1)
(3.2)
p
then 1 ? l ? r and A(l, r, ?, C, ?) < 0. This also holds if ? ? 1 and 0 < ? < 1 ? 2 C/r.
3.2
Local analysis of the empirical loss
The preliminary lemmas enable us to locally bound E supu?K(g0 ) (Pn ? P )u as follows:
Lemma 3.4. If ? < (r ? 1)/r, there exists c1 > 0 such that E supu?K(g0 ) (Pn ? P )u ? c1 n?? for
all n.
Proof. Without loss of generality, we assume that K(g0 ) is countable. The arguments to extend
the bound from countable classes to separable classes are standard (see, for example, Lemma 12
of Mehta and Williamson [2014]). Denote Z? = supu?K(g0 ) (Pn ? P )u and let = 1/n? , R =
(R1 , R2 , . . . , Rn ) be iid Rademacher random variables, using standard results about symmetrization
and chaining of Rademacher process (see, for example, Corollary 13.2 in Boucheron et al. [2013]),
we have
?
?
n
X
Rj u(Xj )?
nE sup (Pn ? P )g ? 2E ?ER sup
u?K(g0 )
u?K(g0 ) j=1
Z
? 24E
D(KX )?
Z
p
log N (t, KX , k ? k2 )dt ? 24E
0
D(KX )?
q
?
H t/ n, K(g0 ) dt,
0
where ER denotes the expected value with respect to the random variables R1 , R2 , . . . , Rn . By
Assumption 2.2, we deduce that
p
nEZ? ? C0 (K, n, ?, C)( + ED(KX ))
where
C0 = O( log n).
If we define
p
p
?
r
? b = C0 /n = O( log n/n?+1 ), a = C0 n?1/2 (2W ) 2(r?1)
x = + EZ,
/2 = O( log n/ n),
then by Lemma 3.1, we have x ? ax(r?2)/(2r?2) + b + . Using lemma 3.2, we have
x ? C1 a2(r?1)/r + C2 (b + ) ? C3 n?? ,
which completes the proof.
p
Lemma 3.5. Assuming that r ? 4C, if ? < 1 ? 2 C/r, there exist c1 , c2 > 0 such that for all n
and ? > 0
sup Pn u ? 9c1 + (c2 /?)1/[r(1??)] n??
?g0 ? G
u?K(g0 )
with probability at least 1 ? ?.
4
Proof. Denote Z = supu?K(g0 ) Pn u and Z? = supu?K(g0 ) (Pn ? P )u. We have
Z = sup Pn u ? Z? + sup P u ? Z? + sup kukL2 (P ) = Z? + .
u?K(g0 )
u?K(g0 )
u?K(g0 )
? using the facts that
Applying Lemma 2.1 for ? = 8 and x = y/n for Z,
p
? = sup
E[u(X)]2 ? = 1/n? ,
and
E[ sup |u|r ] ? (2W )r ,
?
u?Kg0
u?Kg0
we have
p l
1?l/r ?
?
?l
?
?
P Z ? 9EZ + y/n ? min y
46 (l/n)
n W + 4 l/n
:= ?(y, n).
1?l?r
To provide a union bound for all g0 ? G , we want the total failure probability ?(y, n)(n? K)C ? ?.
This failure probability, as a function of n, is of order A(l,
p r, ?, C, ?) (as define in Lemma 3.3) with
? = 2 . By choosing l = r(1 ? ?)/2 and ? < 1 ? 2 C/r, we deduce that there exist c2 , c3 > 0
such that ?(y, n)(n? K)C ? c2 /(nc3 y l ) ? c2 /y r(1??)/2 . The proof is completed by choosing
p
2/[r(1??)]
y = (c2 /?)
and using the fact that EZ? ? c1 /n? (note that 1 ? 2 C/r ? (r ? 1)/r
and we can apply Lemma 3.4 to get the bound).
A direct consequence of this Lemma is the following localization-based result.
Theorem 3.1 (Local analysis). Under Assumptions p
2.1, 2.2 and 2.3, let G be a minimal -net for G
??
in the L2 (P )-norm, with = n where ? < 1 ? 2 C/r. Then there exist c1 , c2 > 0 such that for
all ? > 0,
Pn g ? Pn (?(g)) ? 9c1 + (c2 /?)2/[r(1??)] n??
?g ? G
with probability at least 1 ? ?.
3.3
Fast learning rates with heavy-tailed losses
Theorem 3.2. Given a0 , ? > 0. Under the multi-scale (B, ?, I)-Bernstein?s condition and the
assumption that r ? 4C, consider
p
0 < ? < (1 ? 2 C/r)/(2 ? ?i )
?i ? I.
(3.3)
Then there exist Na0 ,?,r,B,? > 0 such that ?f ? F and n ? Na0 ,?,r,B,? , we have
P `(f ) ? P ? ? a0 /n?
with probability at least 1 ? ?.
implies
?f ? ? F ? :
Pn `(f ) ? Pn `(f ? ) ? a0 /(4n? )
Proof. Define a = [P `(f ) ? P ? ]n? . Assuming that f ? Fi , applying Lemma 2.1 for ? = 1/2 and
x = a/4n? for a single hypothesis f , we have
P [Pn `(f ) ? Pn `(fi? ) ? (P `(f ) ? P `(fi? ))/4] ? h(a, n)
where
p l
1?l/r
h(a, n, i) = min (4/a)l 50n? (l/n)
W + 4n? Bi a?i /2 /n??i /2 l/n
1?l?r
?
using the fact that ? = E[`(f )?`(fi? )]2 ? Bi [E(`(f ) ? `(fi? ))] i = Bi a?i /n??i if f ? Fi . Since
?i ? 1, h(a, n, i) is a non-increasing function in a. Thus,
P [Pn `(f ) ? Pn `(fi? ) ? (P `(f ) ? P `(fi? ))/4] ? h(a0 , n, i).
2
To provide a union bound for all f ? F such that P `(f ) ? P `(fi? ) ? a0 /n? , we want the total
failure probability to be small. This is guaranteed if h(a0 , n, i)(n? K)C ? ?. This failure probability,
as a function of n, is of order A(l, r, ?, C, ?i ) as defined in equation (3.1). By choosing r, l as in
Lemma 3.3 and ? as in equation (3.3), we have 1 ? l ? r and A(l, r, ?, C, ?i ) < 0 for all i. Thus,
there exists c4 , c5 , c6 > 0 such that
?c (1?? /2)
i
h(a0 , n, i)(n? K)C ? c6 a0 5
n?c4
?n, i.
1/c4
?c (1??
? /2)
Hence, when n ? Na,?,r,B,? =
c6 ?a0 5
where ?? = max{?}1{a0 ?1} +
min{?}1{a0 <1} , we have: ?f ? F , P `(f ) ? P ? ? a0 /n?
Pn `(f ? ) ? a0 /(4n? ) with probability at least 1 ? ?.
5
implies
?f ? ? F ? , Pn `(f ) ?
Theorem 3.3. Under Assumptions 2.1, 2.2 and 2.3, consider ? as in equation (3.3) and c1 , c2 as in
previous theorems. For all ? > 0, there exists N?,r,B,? such that if n ? N?,r,B,? , then
2/[r(1??)]
P `(f?z ) ? P `(f ? ) + 36c1 + 1 + 4 (2c2 /?)
n??
with probability at least 1 ? ?.
Proof of Theorem 3.3. Let F by an -net of F with = 1/n? such that f ? ? F . We denote the
projection of f?z to F by f1 = ?(f?z ). For a given ? > 0, define
o
n
2/[r(1??)]
A1 = ?f ? F : Pn f ? Pn (?(f )) ? 9c1 + (c3 /?)
n?? ,
A2 = ?f ? F : Pn `(?(f )) ? Pn `(f ? ) ? a0 /(4n? ) and P `(?(f )) ? P `(f ? ) ? a0 /n? ,
where c1 , c2 is defined as in previous theorem, a0 /4 = 9c1 + (c3 /?)2/[r(1??)] and n ? Na0 ,?,r,? .
We deduce that A1 and A2 happen with probability at most ?. On the other hand, under the event
that A1 and A2 do not happen, we have
2/[r(1??)]
Pn `(f1 ) ? Pn `(f?z ) + 9c1 + (c3 /?)
n?? ? Pn `(f ? ) + a0 /(4n? ).
By definition of F , we have P `(f?z ) ? P `(f1 ) + ? P `(f ? ) + (a0 + 1)/n? .
3.4
Verifying the multi-scale Bernstein?s condition
In practice, the most difficult condition to verify for fast learning rates is the multi-scale Bernstein?s
condition. We derive in this section some approaches to verify the condition. We first extend the result of Mendelson [2008] to prove that the (standard) Bernstein?s condition is automatically satisfied
for functions that are relatively far way from f ? under the integrability condition of the envelope
function (proof in the Appendix). We recall that R(f ) = E`(f ) is referred to as the risk function.
Lemma 3.6. Under Assumption 2.3, we define M = W r/(r?2) and ? = (r ? 2)/(r ? 1). Then, if
? > M and R(f ) ? ?/(? ? M )R(f ? ), then E(`(f ) ? `(f ? ))2 ? 2?? E(`(f ) ? `(f ? ))? .
This allows us to derive the following result, for which the proof is provided in the Appendix.
Lemma 3.7. If F is a subset of a vector space with metric d and the risk function R(f ) = E`(f )
has a unique minimizer on F at f ? in the interior of F and
(i) There exists L > 0 such that E(`(f ) ? `(g))2 ? Ld(f, g)2 for all f, g ? F.
(ii) There exists m ? 2, c > 0 and a neighborhood U around f ? such that
R(f ) ? R(f ? ) ? cd(f, f ? )m for all f ? U .
Then the multi-scale Bernstein?s condition holds for ? = ((r ? 2)/(r ? 1), 2/m).
Corollary 3.1. Suppose that (F, d) is a pseudo-metric space, ` satisfies condition (i) in Lemma 3.7
and the risk function is strongly convex with respect to d, then the Bernstein?s condition holds with
? = 1.
Remark 3.1. If the risk function is analytic at f ? , then condition (ii) in Lemma 3.7 holds. Similarly,
if the risk function is continuously differentiable up to order 2 and the Hessian of R(f ) is positive
definite at f ? , then condition (ii) is valid with m = 2.
Corollary 3.2. If the risk function R(f ) = E`(f ) has a finite number of global minimizers
f1 , f2 , . . . , fk , ` satisfies condition (i) in Lemma 3.7 and there exists mi ? 2, ci > 0 and neighborhoods Ui around fi such that R(f ) ? R(fi ) ? ci d(f, fi )mi for all f ? Ui , i = 1, . . . , k, then the
multi-scale Bernstein?s condition holds for ? = ((r ? 2)/(r ? 1), 2/m1 , . . . , 2/mk ).
3.5
Comparison to related work
Theorem 3.3 dictates that under our settings, the problem of learning with heavy-tailed losses can
obtain convergence rates up to order
?
O n?(1?2 C/r)/(2?min{?})
(3.4)
6
where ? is the multi-scale Bernstein?s order and r is the degree of integrability of the loss. We
recall that convergence rate of O(n?1/(2??) ) is obtained in Mehta and Williamson [2014] under the
same setting but for bounded losses. (The analysis there was done under the ?-weakly stochastic
mixability condition, which is equivalent with the standard ?-Bernstein?s condition for bounded
losses [van Erven et al., 2015]). We note that if the loss is bounded, r = ? and (3.4) reduces to the
convergence rate obtained in Mehta and Williamson [2014].
Fast learning rates for unbounded loses are previously derived in Lecu?e and Mendelson [2013]
for sub-Gaussian losses and in Lecu?e and Mendelson [2012] for hypothesis classes that have subexponential envelope functions. In Lecu?e and Mendelson [2013], the Bernstein?s condition is not
directly imposed, but is replaced by condition (ii) of Lemma 3.7 with m = 2 on the whole hypothesis
class, while the assumption of sub-Gaussian hypothesis class validates condition (i). This implies
the standard Bernstein?s condition with ? = 1 and makes the convergence rate O(n?1 ) consistent
with our result (note that for sub-Gaussian losses, r can be chosen arbitrary large). The analysis of
Lecu?e and Mendelson [2012] concerns about non-exact oracle inequalities (rather than the sharp
oracle inequalities we investigate in this paper) and can not be directly compared with our results.
4
Application: k-means clustering with heavy-tailed source distributions
k-means clustering is a method of vector quantization aiming to partition n observations into k ? 2
clusters in which each observation belongs to the cluster with the nearest mean. Formally, let X be
a random vector taking values in Rd with distribution P . Given a codebook (set of k cluster centers)
C = {yi } ? (Rd )k , the distortion (loss) on an instant x is defined as `(C, x) = minyi ?C kx ?
yi k2 and k-means clustering method aims at finding a minimizer C ? of R(`(C)) = P `(C) via
minimizing the empirical distortion Pn `(C).
The rate of convergence of k-means clustering has drawn considerable attention in the statistics and
machine learning literatures [Pollard, 1982, Bartlett et al., 1998, Linder et al., 1994, Ben-David,
2007]. Fast learning rates for k-means clustering (O(1/n)) have also been derived by Antos et al.
[2005] in the case when the source distribution is supported on a finite set of points, and by Levrard [2013] under the assumptions that the source distribution has bounded support and satisfies the
so-called Pollard?s regularity condition, which dictates that P has a continuous density with respect
to the Lebesgue measure and the Hessian matrix of the mapping C ? R(C) is positive definite
at C ? . Little is known about the finite-sample performance of empirically designed quantizers under possibly heavy-tailed distributions. In Telgarsky and Dasgupta [2013], a convergence rate of
O(n?1/2+2/r ) are derived, where r is the number of moments of X that are assumed to be finite.
Brownlees et al. [2015] uses some robust mean estimators to replace empirical means and derives a
convergence rate of O(n?1/2 ) assuming only that the variance of X is finite.
The results from previous sections enable us to prove that with proper setting, the convergence
rate of k-means clustering for heavy-tailed source distributions can be arbitrarily close to O(1/n).
Following the framework of Brownlees et al. [2015], we consider
G = {`(C, x) = min kx ? yi k2 , C ? F = (??, ?)d?k }
yi ?C
for some ? > 0 with the regular Euclidean metric. We let C ? , C?n be defined as in the previous
sections.
Theorem 4.1. If X has finite moments up to order r ? 4k(d + 1), P has a continuous density with
respect to the Lebesgue measure, the risk function has a finite number of global minimizers and the
Hessian matrix of C ? R(C) is positive definite at the every optimal C ? in the interior of F, then
for all ? that satisfies
p
r?1
0<?<
(1 ? 2 k(d + 1)/r),
r
there exists c1 , c2 > 0 such that for all ? > 0, with probability at least 1 ? ?, we have
2/r
R(C?n ) ? R(C ? ) ? c1 + 4 (c2 /?)
n??
Moreover, when r ? ?, ? can be chosen arbitrarily close to 1.
7
Proof. We have
1/r
1/r
1/r
1
1
1 2r
2
2 r
r
2r
?
E[kXk + ? ]
E sup `(C, X)
?
EkXk + ?
? W < ?,
2r
2
2
C?F
while standard results about VC-dimension of k-means clustering hypothesis class guarantees that
C ? k(d + 1) [Linder et al., 1994]. On the other hand, we can verify that
E[`(C, X) ? `(C 0 , X)]2 ? L? kC ? C 0 k22 ,
which validates condition (i) in Lemma 3.7. The fact that the Hessian matrix of C ? R(C) is
positive definite at C ? prompts R(C?n )?R(C ? ) ? ckC?n ?C ? k2 for some c > 0 in a neighborhood U
around any optimal codebook C ? . Thus, Lemma 3.6 confirms the multi-scale Bernstein?s condition
with ? = ((r ? 2)/(r ? 1), 1, . . . , 1). The inequality is then obtained from Theorem 3.3.
5
Discussion and future work
We have shown that fast learning rates for heavy-tailed losses can be obtained for hypothesis classes
with an integrable envelope when the loss satisfies the multi-scale Bernstein?s condition. We then
verify those conditions and obtain new convergence rates for k-means clustering with heavy-tailed
losses. The analyses extend and complement existing results in the literature from both theoretical
and practical points of view. We also introduce a new fast-rate assumption, the multi-scale Bernstein?s condition, and provide a clear path to verify the assumption in practice. We believe that the
multi-scale Bernstein?s condition is the proper assumption to study fast rates for unbounded losses,
for its ability to separate the behaviors of the risk function on microscopic and macroscopic scales,
for which the distinction can only be observed in an unbounded setting.
There are several avenues for improvement. First, we would like to consider hypothesis class with
polynomial entropy bounds. Similarly, the condition of independent and identically distributed observations can be replaced with mixing properties [Steinwart and Christmann, 2009, Hang and Steinwart, 2014, Dinh et al., 2015]. While the condition of integrable envelope is an improvement from
the condition of sub-exponential envelope previously investigated in the literature, it would be interesting to see if the rates retain under weaker conditions, for example, the assumption that the
Lr -diameter of the hypothesis class is bounded [Cortes et al., 2013]. Finally, the recent work of
Brownlees et al. [2015], Hsu and Sabato [2016] about robust estimators as alternatives of ERM to
study heavy-tailed losses has yielded more favorable learning rates under weaker conditions, and we
would like to extend the result in this paper to study such estimators.
Acknowledgement
Vu Dinh was supported by DMS-1223057 and CISE-1564137 from the National Science Foundation
and U54GM111274 from the National Institutes of Health. Lam Si Tung Ho was supported by NSF
grant IIS 1251151.
8
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9
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5,642 | 6,105 | Guided Policy Search via Approximate Mirror
Descent
William Montgomery
Dept. of Computer Science and Engineering
University of Washington
wmonty@cs.washington.edu
Sergey Levine
Dept. of Computer Science and Engineering
University of Washington
svlevine@cs.washington.edu
Abstract
Guided policy search algorithms can be used to optimize complex nonlinear policies, such as deep neural networks, without directly computing policy gradients
in the high-dimensional parameter space. Instead, these methods use supervised
learning to train the policy to mimic a ?teacher? algorithm, such as a trajectory
optimizer or a trajectory-centric reinforcement learning method. Guided policy
search methods provide asymptotic local convergence guarantees by construction,
but it is not clear how much the policy improves within a small, finite number of
iterations. We show that guided policy search algorithms can be interpreted as an
approximate variant of mirror descent, where the projection onto the constraint
manifold is not exact. We derive a new guided policy search algorithm that is simpler and provides appealing improvement and convergence guarantees in simplified
convex and linear settings, and show that in the more general nonlinear setting, the
error in the projection step can be bounded. We provide empirical results on several
simulated robotic navigation and manipulation tasks that show that our method is
stable and achieves similar or better performance when compared to prior guided
policy search methods, with a simpler formulation and fewer hyperparameters.
1
Introduction
Policy search algorithms based on supervised learning from a computational or human ?teacher? have
gained prominence in recent years due to their ability to optimize complex policies for autonomous
flight [16], video game playing [15, 4], and bipedal locomotion [11]. Among these methods, guided
policy search algorithms [6] are particularly appealing due to their ability to adapt the teacher to
produce data that is best suited for training the final policy with supervised learning. Such algorithms
have been used to train complex deep neural network policies for vision-based robotic manipulation
[6], as well as a variety of other tasks [19, 11]. However, convergence results for these methods
typically follow by construction from their formulation as a constrained optimization, where the
teacher is gradually constrained to match the learned policy, and guarantees on the performance of
the final policy only hold at convergence if the constraint is enforced exactly. This is problematic in
practical applications, where such algorithms are typically executed for a small number of iterations.
In this paper, we show that guided policy search algorithms can be interpreted as approximate variants
of mirror descent under constraints imposed by the policy parameterization, with supervised learning
corresponding to a projection onto the constraint manifold. Based on this interpretation, we can
derive a new, simplified variant of guided policy search, which corresponds exactly to mirror descent
under linear dynamics and convex policy spaces. When these convexity and linearity assumptions do
not hold, we can show that the projection step is approximate, up to a bound that depends on the step
size of the algorithm, which suggests that for a small enough step size, we can achieve continuous
improvement. The form of this bound provides us with intuition about how to adjust the step size in
practice, so as to obtain a simple algorithm with a small number of hyperparameters.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Algorithm 1 Generic guided policy search method
1: for iteration k ? {1, . . . , K} do
2:
C-step: improve each pi (ut |xt ) based on surrogate cost `?i (xt , ut ), return samples Di
3:
S-step: train ?? (ut |xt ) with supervised learning on the dataset D = ?i Di
4:
Modify `?i (xt , ut ) to enforce agreement between ?? (ut |xt ) and each p(ut |xt )
5: end for
The main contribution of this paper is a simple new guided policy search algorithm that can train
complex, high-dimensional policies by alternating between trajectory-centric reinforcement learning
and supervised learning, as well as a connection between guided policy search methods and mirror
descent. We also extend previous work on bounding policy cost in terms of KL divergence [15, 17]
to derive a bound on the cost of the policy at each iteration, which provides guidance on how
to adjust the step size of the method. We provide empirical results on several simulated robotic
navigation and manipulation tasks that show that our method is stable and achieves similar or better
performance when compared to prior guided policy search methods, with a simpler formulation and
fewer hyperparameters.
2
Guided Policy Search Algorithms
We first review guided policy search methods and background. Policy search algorithms aim
to optimize a parameterized policy ?? (ut |xt ) over actions ut conditioned on the state xt . Given
stochastic dynamics p(xt+1 |xt , ut ) and cost `(xt , ut ), the goal is to minimize the expected cost under
PT
the policy?s trajectory distribution, given by J(?) = t=1 E?? (xt ,ut ) [`(xt , ut )], where we overload
QT
notation to use ?? (xt , ut ) to denote the marginals of ?? (? ) = p(x1 ) t=1 p(xt+1 |xt , ut )?? (ut |xt ),
where ? = {x1 , u1 , . . . , xT , uT } denotes a trajectory. A standard reinforcement learning (RL)
approach to policy search is to compute the gradient ?? J(?) and use it to improve J(?) [18, 14]. The
gradient is typically estimated using samples obtained from the real physical system being controlled,
and recent work has shown that such methods can be applied to very complex, high-dimensional
policies such as deep neural networks [17, 10]. However, for complex, high-dimensional policies,
such methods tend to be inefficient, and practical real-world applications of such model-free policy
search techniques are typically limited to policies with about one hundred parameters [3].
Instead of directly optimizing J(?), guided policy search algorithms split the optimization into a
?control phase? (which we?ll call the C-step) that finds multiple simple local policies pi (ut |xt ) that
can solve the task from different initial states xi1 ? p(x1 ), and a ?supervised phase? (S-step) that
optimizes the global policy ?? (ut |xt ) to match all of these local policies using standard supervised
learning. In fact, a variational formulation of guided policy search [7] corresponds to the EM
algorithm, where the C-step is actually the E-step, and the S-step is the M-step. The benefit of this
approach is that the local policies pi (ut |xt ) can be optimized separately using domain-specific local
methods. Trajectory optimization might be used when the dynamics are known [19, 11], while local
RL methods might be used with unknown dynamics [5, 6], which still requires samples from the real
system, though substantially fewer than the direct approach, due to the simplicity of the local policies.
This sample efficiency is the main advantage of guided policy search, which can train policies with
nearly a hundred thousand parameters for vision-based control using under 200 episodes [6], in
contrast to direct deep RL methods that might require orders of magnitude more experience [17, 10].
A generic guided policy search method is shown in Algorithm 1. The C-step invokes a local policy
optimizer (trajectory optimization or local RL) for each pi (ut |xt ) on line 2, and the S-step uses
supervised learning to optimize the global policy ?? (ut |xt ) on line 3 using samples from each
pi (ut |xt ), which are generated during the C-step. On line 4, the surrogate cost `?i (xt , ut ) for each
pi (ut |xt ) is adjusted to ensure convergence. This step is crucial, because supervised learning does not
in general guarantee that ?? (ut |xt ) will achieve similar long-horizon performance to pi (ut |xt ) [15].
The local policies might not even be reproducible by a single global policy in general. To address
this issue, most guided policy search methods have some mechanism to force the local policies to
agree with the global policy, typically by framing the entire algorithm as a constrained optimization
that seeks at convergence to enforce equality between ?? (ut |xt ) and each pi (ut |xt ). The form of the
2
overall optimization problem resembles dual decomposition, and usually looks something like this:
min
N X
T
X
?,p1 ,...,pN
Epi (xt ,ut ) [`(xt , ut )] such that pi (ut |xt ) = ?? (ut |xt ) ?xt , ut , t, i.
(1)
i=1 t=1
PN PT
Since xi1 ? p(x1 ), we have J(?) ?
i=1
t=1 Epi (xt ,ut ) [`(xt , ut )] when the constraints are
enforced exactly. The particular form of the constraint varies depending on the method: prior works
have used dual gradient descent [8], penalty methods [11], ADMM [12], and Bregman ADMM [6].
We omit the derivation of these prior variants due to space constraints.
2.1
Efficiently Optimizing Local Policies
A common and simple choice for the local policies pi (ut |xt ) is to use time-varying linear-Gaussian
controllers of the form pi (ut |xt ) = N (Kt xt + kt , Ct ), though other options are also possible
[12, 11, 19]. Linear-Gaussian controllers represent individual trajectories with linear stabilization and
Gaussian noise, and are convenient in domains where each local policy can be trained from a different
(but consistent) initial state xi1 ? p(x1 ). This represents an additional assumption beyond standard
RL, but allows for an extremely efficient and convenient local model-based RL algorithm based on
iterative LQR [9]. The algorithm proceeds by generating N samples on the real physical system
from each local policy pi (ut |xt ) during the C-step, using these samples to fit local linear-Gaussian
dynamics for each local policy of the form pi (xt+1 |xt , ut ) = N (fxt xt + fut ut + fct , Ft ) using
linear regression, and then using these fitted dynamics to improve the linear-Gaussian controller via a
modified LQR algorithm [5]. This modified LQR method solves the following optimization problem:
min
T
X
Kt ,kt ,Ct
Epi (xt ,ut ) [`?i (xt , ut )] such that DKL (pi (? )k?
pi (? )) ? ,
(2)
t=1
where we again use pi (? ) to denote the trajectory distribution induced by pi (ut |xt ) and the fitted
dynamics pi (xt+1 |xt , ut ). Here, p?i (ut |xt ) denotes the previous local policy, and the constraint
ensures that the change in the local policy is bounded, as proposed also in prior works [1, 14, 13].
This is particularly important when using linearized dynamics fitted to local samples, since these
dynamics are not valid outside of a small region around the current controller. In the case of linearGaussian dynamics and policies, the KL-divergence constraint DKL (pi (? )k?
pi (? )) ? can be shown
to simplify, as shown in prior work [5] and Appendix A:
DKL (pi (? )k?
pi (? )) =
T
X
DKL (pi (ut |xt )k?
pi (ut |xt )) =
t=1
T
X
?Epi (xt ,ut ) [log p?i (ut |xt )]?H(pi (ut |xt )),
t=1
and the resulting Lagrangian of the problem in Equation (2) can be optimized with respect to the primal
variables using the standard LQR algorithm, which suggests a simple method for solving the problem
in Equation (2) using dual gradient descent [5]. The surrogate objective `?i (xt , ut ) = `(xt , ut )+?i (?)
typically includes some term ?i (?) that encourages the local policy pi (ut |xt ) to stay close to the
global policy ?? (ut |xt ), such as a KL-divergence of the form DKL (pi (ut |xt )k?? (ut |xt )).
2.2
Prior Convergence Results
Prior work on guided policy search typically shows convergence by construction, by framing the
C-step and S-step as block coordinate ascent on the (augmented) Lagrangian of the problem in
Equation (1), with the surrogate cost `?i (xt , ut ) for the local policies corresponding to the (augmented)
Lagrangian, and the overall algorithm being an instance of dual gradient descent [8], ADMM
[12], or Bregman ADMM [6]. Since these methods enforce the constraint pi (ut |xt ) = ?? (ut |xt )
at convergence (up to linearization or sampling error, depending on the method), we know that
PN
1
1
i=1 Epi (xt ,ut ) [`(xt , ut )] ? E?? (xt ,ut ) [`(xt , ut )] at convergence. However, prior work does
N
not say anything about ?? (ut |xt ) at intermediate iterations, and the constraints of policy search in
the real world might often preclude running the method to full convergence. We propose a simplified
variant of guided policy search, and present an analysis that sheds light on the performance of both
the new algorithm and prior guided policy search methods.
As mentioned previously, the initial state xi1 of each local policy pi (ut |xt ) is assumed to be drawn from
p(x1 ), hence the outer sum corresponds to Monte Carlo integration of the expectation under p(x1 ).
1
3
Algorithm 2 Mirror descent guided policy search (MDGPS): convex linear variant
1: for iteration k ? {1, . . . , K} do h
i
PT
2:
C-step: pi ? arg minpi Epi (? )
t=1 `(xt , ut ) such that DKL (pi (? )k?? (? )) ?
3:
S-step: ?? ? arg min?
4: end for
3
P
i
DKL (pi (? )k?? (? )) (via supervised learning)
Mirror Descent Guided Policy Search
In this section, we propose our new simplified guided policy search, which we term mirror descent
guided policy search (MDGPS). This algorithm uses the constrained LQR optimization in Equation (2) to optimize each of the local policies, but instead of constraining each local policy pi (ut |xt )
against the previous local policy p?i (ut |xt ), we instead constraint it directly against the global policy
?? (ut |xt ), and simply set the surrogate cost to be the true cost, such that `?i (xt , ut ) = `(xt , ut ). The
method is summarized in Algorithm 2. In the case of linear dynamics and a quadratic cost (i.e. the
LQR setting), and assuming that supervised learning can globally solve a convex optimization problem, we can show that this method corresponds to an instance of mirror descent [2] on the objective
J(?). In this formulation, the optimization is performed on the space of trajectory distributions, with
a constraint that the policy must lie on the manifold of policies with the chosen parameterization. Let
?? be the set of all possible policies ?? for a given parameterization, where we overload notation to
also let ?? denote the set of trajectory distributions that are possible under the chosen parameteriPT
zation. The return J(?) can be optimized according to ?? ? arg min???? E?(? ) [ t=1 `(xt , ut )].
Mirror descent solves this
" Toptimization
# by alternating between two steps at each iteration k:
X
pk ? arg min Ep(? )
`(xt , ut ) s. t. D p, ? k ? ,
? k+1 ? arg min D pk , ? .
p
????
t=1
The first step finds a new distribution pk that minimizes the cost and is close to the previous policy
? k in terms of the divergence D p, ? k , while the second step projects this distribution onto the
constraint set ?? , with respect to the divergence D(pk , ?). In the linear-quadratic case with a convex
supervised learning phase, this corresponds exactly to Algorithm 2: the C-step optimizes pk , while the
S-step is the projection. Monotonic improvement of the global policy ?? follows from the monotonic
improvement of mirror descent [2]. In the case of linear-Gaussian dynamics and policies, the S-step,
which minimizes KL-divergence between trajectory distributions, in fact only requires minimizing
the KL-divergence between policies. Using the identity in Appendix A, we know that
DKL (pi (? )k?? (? )) =
T
X
Epi (xt ) [DKL (pi (ut |xt )k?? (ut |xt ))] .
(3)
t=1
3.1
Implementation for Nonlinear Global Policies and Unknown Dynamics
In practice, we aim to optimize complex policies for nonlinear systems with unknown dynamics. This
requires a few practical considerations. The C-step requires a local quadratic cost function, which
can be obtained via Taylor expansion, as well as local linear-Gaussian dynamics p(xt+1 |xt , ut ) =
N (fxt xt + fut ut + fct , Ft ), which we can fit to samples as in prior work [5]. We also need a local
time-varying linear-Gaussian approximation to the global policy ?? (ut |xt ), denoted ?
??i (ut |xt ). This
can be obtained either by analytically differentiating the policy, or by using the same linear regression
method that we use to estimate p(xt+1 |xt , ut ), which is the approach in our implementation. In both
cases, we get a different global policy linearization around each local policy. Following prior work
[5], we use a Gaussian mixture model prior for both the dynamics and global policy fit.
The S-step can be performed approximately in the nonlinear case by using the samples collected for
dynamics fitting to also train the global policy. Following prior work [6], our S-step minimizes2
X
1 X
Epi (xt ) [DKL (?? (ut |xt )kpi (ut |xt ))] ?
DKL (?? (ut |xt,i,j )kpi (ut |xt,i,j )),
|Di | i,t,j
i,t
2
Note that we flip the KL-divergence inside the expectation, following [6]. We found that this produced better
results. The intuition behind this is that, because log pi (ut |xt ) is proportional to the Q-function of pi (ut |xt )
(see Appendix B.1), DKL (?? (ut |xt,i,j )kpi (ut |xt,i,j ) minimizes the cost-to-go under pi (ut |xt ) with respect to
?? (ut |xt ), which provides for a more informative objective than the unweighted likelihood in Equation (3).
4
Algorithm 3 Mirror descent guided policy search (MDGPS): unknown nonlinear dynamics
1: for iteration k ? {1, . . . , K} do
2:
Generate samples Di = {?i,j } by running either pi or ??i
3:
Fit linear-Gaussian dynamics pi (xt+1 |xt , ut ) using samples in Di
4:
Fit linearized global policy ?
??i (ut |xt ) using samples in Di
PT
5:
C-step: pi ? arg minpiP
Epi (? ) [ t=1 `(xt , ut )] such that DKL (pi (? )k?
??i (? )) ?
6:
S-step: ?? ? arg min? t,i,j DKL (?? (ut |xt,i,j )kpi (ut |xt,i,j )) (via supervised learning)
7:
Adjust (see Section 4.2)
8: end for
where xt,i,j is the j th sample from pi (xt ) obtained by running pi (ut |xt ) on the real system. For
linear-Gaussian pi (ut |xt ) and (nonlinear) conditionally Gaussian ?? (ut |xt ) = N (?? (xt ), ?? (xt )),
where ?? and ?? can be any function (such as a deep neural network), the KL-divergence
DKL (?? (ut |xt,i,j )kpi (ut |xt,i,j )) can easily be evaluated and differentiated in closed form [6]. However, in the nonlinear setting, minimizing this objective no longer minimizes the KL-divergence
between trajectory distributions DKL (?? (? )kpi (? )) exactly, which means that MDGPS does not
correspond exactly to mirror descent: although the C-step can still be evaluated exactly, the S-step
now corresponds to an approximate projection onto the constraint manifold. In the next section, we
discuss how we can bound the error in this projection. A summary of the nonlinear MDGPS method
is provided in Algorithm 4, and additional details are in Appendix B. The samples for linearizing
the dynamics and policy can be obtained by running either the last local policy pi (ut |xt ), or the last
global policy ?? (ut |xt ). Both variants produce good results, and we compare them in Section 6.
3.2
Analysis of Prior Guided Policy Search Methods as Approximate Mirror Descent
The main distinction between the proposed method and prior guided policy search methods is that
the constraint DKL (pi (? )k?
??i (? )) ? is enforced on the local policies at each iteration, while in
prior methods, this constraint is iteratively enforced via a dual descent procedure over multiple
iterations. This means that the prior methods perform approximate mirror descent with step sizes
that are adapted (by adjusting the Lagrange multipliers) but not constrained exactly. In our empirical
evaluation, we show that our approach is somewhat more stable, though sometimes slower than these
prior methods. This empirical observation agrees with our intuition: prior methods can sometimes be
faster, because they do not exactly constrain the step size, but our method is simpler, requires less
tuning, and always takes bounded steps on the global policy in trajectory space.
4
Analysis in the Nonlinear Case
Although the S-step under nonlinear dynamics is not an optimal projection onto the constraint manifold, we can bound the additional cost incurred by this projection in terms of the KL-divergence
between pi (ut |xt ) and ?? (ut |xt ). This analysis also reveals why prior guided policy search algorithms, which only have asymptotic convergence guarantees, still attain good performance in practice
even after a small number of iterations. We will drop the subscript i from pi (ut |xt ) in this section
for conciseness, though the same analysis can be repeated for multiple local policies pi (ut |xt ).
4.1
Bounding the Global Policy Cost
The analysis in this section is based on the following lemma, which we prove in Appendix C.1,
building off of earlier results by Ross et al. [15] and Schulman et al. [17]:
Lemma 4.1 Let t = maxxt DKL (p(ut |xt )k?? (ut |xt ). Then DTV (p(xt )k?? (xt )) ? 2
PT
t=1
?
2t .
This means that if we can bound the KL-divergence between the policies, then the total variation
divergence between their state marginals (given by DTV (p(xt )k?? (xt )) = 12 kp(xt ) ? ?? (xt )k1 ) will
also be bounded. This bound allows us in turn to relate the total expected costs of the two policies to
each other according to the following lemma, which we prove in Appendix C.2:
5
Lemma 4.2 If DTV (p(xt )k?? (xt )) ? 2
T
X
PT
t=1
?
2t , then we can bound the total cost of ?? as
T
X
?
?
E?? (xt ,ut ) [`(xt , ut )] ?
Ep(xt ,ut ) [`(xt , ut )] + 2t max `(xt , ut ) + 2 2t Qmax,t
t=1
where Qmax,t =
xt ,ut
t=1
PT
t0 =t
maxxt0 ,ut0 `(xt0 , ut0 ), the maximum total cost from time t to T .
This bound on the cost of ?? (ut |xt ) tells us that if we update p(ut |xt ) so as to decrease its total cost
or decrease its KL-divergence against ?? (ut |xt ), we will eventually reduce the cost of ?? (ut |xt ).
For the MDGPS algorithm, this bound suggests that we can ensure improvement of the global policy
within a small number of iterations by appropriately choosing the constraint during the C-step.
PT
Recall that the C-step constrains t=1 t ? , so if we choose to be small enough, we can close
the gap between the local and global policies. Optimizing the bound directly turns out to produce
very slow learning in practice, because the bound is very loose. However, it tells us that we can either
decrease toward the end of the optimization process or if we observe the global policy performing
much worse than the local policies. We discuss how this idea can be put into action in the next
section.
4.2
Step Size Selection
Setting the local policy step size is important for proper convergence of guided policy search
methods. Since we are approximating the true unknown dynamics with time-varying linear dynamics,
setting too large can produce unstable local policies which cause the method to fail. However,
setting too small will prevent the local policies from improving significantly between iterations,
leading to slower learning rates.
In prior work [8], the step size in the local policy optimization is dynamically adjusted by considering
the difference between the predicted change in the cost of the local policy p(ut |xt ) under the fitted
dynamics, and the actual cost obtained when sampling from that policy. The intuition is that, because
the linearized dynamics are local, we incur a larger cost the further we deviate from the previous
policy. We can adjust the step size by estimating the rate at which the additional cost is incurred and
choosing the optimal tradeoff. In Appendix B.3 we describe the step size adjustment rule used for
BADMM in prior work, and use it to derive two step size adjustment rules for MDGPS: ?classic? and
?global.? The classic step size adjustment is a direct reintrepretation of the BADMM step rule for
MDGPS, while the global step rule is a more conservative rule that takes the difference between the
global and local policies into account.
5
Relation to Prior Work
While we?ve discussed the connections between MDGPS and prior guided policy search methods, in
this section we?ll also discuss the connections between our method and other policy search methods.
One popular supervised policy learning methods is DAGGER [15], which also trains the policy using
supervised learning, but does not attempt to adapt the teacher to provide better training data. MDGPS
removes the assumption in DAGGER that the supervised learning stage has bounded error against an
arbitrary teacher policy. MDGPS does not need to make this assumption, since the teacher can be
adapted to the limitations of the global policy learning. This is particularly important when the global
policy has computational or observational limitations, such as when learning to use camera images
for partially observed control tasks or, as shown in our evaluation, blind peg insertion.
When we sample from the global policy ?? (ut |xt ), our method resembles policy gradient methods
with KL-divergence constraints [14, 13, 17]. However, policy gradient methods update the policy
?? (ut |xt ) at each iteration by linearizing with respect to the policy parameters, which often requires
small steps for complex, nonlinear policies, such as neural networks. In contrast, we linearize in the
space of time-varying linear dynamics, while the policy is optimized at each iteration with many steps
of supervised learning (e.g. stochastic gradient descent). This makes MDGPS much better suited for
quickly and efficiently training highly nonlinear, high-dimensional policies.
6
Figure 1: Results for MDGPS variants and BADMM on each task. MDGPS is tested with local policy (?off
policy?) and global policy (?on policy?) sampling (see Section 3.1), and both the ?classic? and ?global? step
sizes (see Section 4.2). The vertical axis for the obstacle task shows the average distance between the point mass
and the target. The vertical axis for the peg tasks shows the average distance between the bottom of the peg
and the hole. Distances above 0.1, which is the depth of the hole (shown as a dotted line) indicate failure. All
experiments are repeated ten times, with the average performance and standard deviation shown in the plots.
6
Experimental Evaluation
We compare several variants of MDGPS and a prior guided policy search method based on Bregman
ADMM (BADMM) [6]. We evaluate all methods on one simulated robotic navigation task and
two manipulation tasks. For MDGPS, during training we sample from either the local policies
(?off-policy? sampling) or the global policy (?on-policy? sampling), and we use both forms of the
step rule described in Section 4.2 (?classic? and ?global?). 3
Obstacle Navigation. In this task, a 2D point mass (grey) must navigate
around obstacles to reach a target (shown in green), using velocities and
positions relative to the target. We use N = 5 initial states, with 5
samples per initial state per iteration. The target and obstacles are fixed,
but the starting position varies.
Peg Insertion. This task, which is more complex, requires controlling
a 7 DoF 3D arm to insert a tight-fitting peg into a hole. The hole can
be in different positions, and the state consists of joint angles, velocities,
and end-effector positions relative to the target. This task is substantially
more challenging physically. We use N = 9 different hole positions, with
5 samples per initial state per iteration.
Blind Peg Insertion. The last task is a blind variant of the peg insertion
task, where the target-relative end effector positions are provided to the
local policies, but not to the global policy ?? (ut |xt ). This requires the
global policy to search for the hole, since no input to the global policy
can distinguish between the different initial state xi1 . This makes it much
more challenging to adapt the global and local policies to each other, and
makes it impossible for the global learner to succeed without adaptation
of the local policies. We use N = 4 different hole positions, with 5
samples per initial state per iteration.
The global policy for each task consists of a fully connected neural network with two hidden layers
with 40 rectified linear units. The same settings are used for MDGPS and the prior BADMM-based
method, except for the difference in surrogate costs, constraints, and step size adjustment methods
discussed in the paper. Results are presented in Figure 1 and Table 1. On the easier point mass
navigation task all methods achieve similar performance, but the on-policy variants of MDGPS
outperform the off-policy variants. This suggests that we can benefit from directly sampling from
the global policy during training, which is not possible in the BADMM formulation. Although
performance is similar among all methods, the MDGPS methods are all substantially easier to
apply to these tasks, since they have very few free hyperparameters. An initial step size must be
selected, but the adaptive step size adjustment rules make this choice less important. In contrast,
3
Guided policy search code, including BADMM and MDGPS methods,
https://www.github.com/cbfinn/gps.
7
is available at
Peg
Blind Peg
Itr.
3
6
9
12
3
6
9
12
BADMM
1.1% ? 3.3%
51.1% ? 10.2%
72.2% ? 14.3%
74.4% ? 19.3%
20.0% ? 31.2%
65.0% ? 22.9%
82.5% ? 25.1%
82.5% ? 16.1%
Off/Classic
11.1 ? 9.9%
62.2 ? 17.4%
82.2 ? 11.3%
83.3 ? 11.4%
2.5 ? 7.5%
62.5 ? 32.1%
80.0 ? 24.5%
95.0 ? 10.0%
Off/Global
6.7% ? 7.4%
64.4% ? 19.1%
71.1% ? 24.0%
84.4% ? 15.1%
7.5% ? 16.0%
70.0% ? 21.8%
60.0% ? 32.0%
85.0% ? 22.9%
On/Classic
6.7% ? 7.4%
68.9% ? 18.5%
90.0% ? 10.5%
90.0% ? 11.6%
2.5% ? 7.5%
72.5% ? 28.4%
80.0% ? 35.0%
85.0% ? 20.0%
On/Global
6.7% ? 7.4%
63.3% ? 20.0%
85.6% ? 8.7%
87.8% ? 13.6%
15.0% ? 30.0%
70.0% ? 35.0%
82.5% ? 19.5%
85.0% ? 12.2%
Table 1: Success rates of each method on each peg insertion task. Success is defined as inserting the peg into the
hole with a final distance of less than 0.06. Results are averaged over ten runs.
the BADMM method requires choosing an initial weight on the augmented Lagrangian term, an
adjustment schedule for this term, a step size on the dual variables, and a step size for local policies,
all of which have a substantial impact on the final performance of the method (the reported results are
for the best setting of these parameters, identified with a hyperparameter sweep).
On the peg insertion tasks, all variants MDGPS consistently outperform BADMM as shown by
the success rates in Table 1, which shows that the MDGPS policies succeed at actually inserting
the peg into the hole more often and on more conditions. This suggests that our method is better
able to improve global policies, particularly in situations where informational or representational
constraints make na?ve imitation of the local policies insufficient to solve the task. On both tasks, we
see faster learning from the on-policy variants, although this is less noticeable on the harder blind peg
insertion task, where the best final policy is the off-policy variant with classic step size adjustment.
Sampling from the global policies may be desirable in practice, since the global policies can directly
use observations at runtime instead of requiring access to the state [6]. The global step size also
tends to be more conservative than the classic step size, but produces more consistent and monotonic
improvement.
7
Discussion and Future Work
We presented a new guided policy search method that corresponds to mirror descent under linearity
and convexity assumptions, and showed how prior guided policy search methods can be seen as
approximating mirror descent. We provide a bound on the return of the global policy in the nonlinear
case, and argue that an appropriate step size can provide improvement of the global policy in this
case also. Our analysis provides us with the intuition to design an automated step size adjustment
rule, and we illustrate empirically that our method achieves good results on a complex simulated
robotic manipulation task while requiring substantially less tuning and hyperparameter optimization
than prior guided policy search methods. Manual tuning and hyperparameter searches are a major
challenge across a range of deep reinforcement learning algorithms, and developing scalable policy
search methods that are simple and reliable is vital to enable further progress.
As discussed in Section 5, MDGPS has interesting connections to other policy search methods. Like
DAGGER [15], MDGPS uses supervised learning to train the policy, but unlike DAGGER, MDGPS
does not assume that the learner is able to reproduce an arbitrary teacher?s behavior with bounded
error, which makes it very appealing for tasks with partial observability or other limits on information,
such as learning to use camera images for robotic manipulation [6]. When sampling directly from
the global policy, MDGPS also has close connections to policy gradient methods that take steps of
fixed KL-divergence [14, 17], but with the steps taken in the space of trajectories rather than policy
parameters, followed by a projection step. In future work, it would be interesting to explore this
connection further, so as to develop new model-free policy gradient methods.
Acknowledgments
We thank the anonymous reviewers for their helpful and constructive feedback. This research was
supported in part by an ONR Young Investigator Program award.
8
References
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9
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5,643 | 6,106 | Learning Additive Exponential Family Graphical
Models via ?2,1-norm Regularized M-Estimation
Xiao-Tong Yuan? Ping Li?? Tong Zhang? Qingshan Liu? Guangcan Liu?
?B-DAT Lab, Nanjing University of Info. Sci.&Tech.
Nanjing, Jiangsu, 210044, China
?Depart. of Statistics and ?Depart. of Computer Science, Rutgers University
Piscataway, NJ, 08854, USA
{xtyuan,qsliu, gcliu}@nuist.edu.cn, {pingli,tzhang}@stat.rutgers.edu
Abstract
We investigate a subclass of exponential family graphical models of which the
sufficient statistics are defined by arbitrary additive forms. We propose two ?2,1 norm regularized maximum likelihood estimators to learn the model parameters
from i.i.d. samples. The first one is a joint MLE estimator which estimates all
the parameters simultaneously. The second one is a node-wise conditional MLE
estimator which estimates the parameters for each node individually. For both
estimators, statistical analysis shows that under mild conditions the extra flexibility gained by the additive exponential family models comes at almost no cost of
statistical efficiency. A Monte-Carlo approximation method is developed to efficiently optimize the proposed estimators. The advantages of our estimators over
Gaussian graphical models and Nonparanormal estimators are demonstrated on
synthetic and real data sets.
1
Introduction
As an important class of statistical models for exploring the interrelationship among a large number
of random variables, undirected graphical models (UGMs) have enjoyed popularity in a wide range
of scientific and engineering domains, including statistical physics, computer vision, data mining,
and computational biology. Let X = [X1 , ..., Xp ]? be a p-dimensional random vector with each
variable Xi taking values in a set X . Suppose G = (V, E) is an undirected graph consists of a set
of vertices V = {1, ..., p} and a set of unordered pairs E representing edges between the vertices.
The pairwise UGMs over X corresponding to G can be written as the following exponential family
distribution:
?
?
?
??
?
(1)
?s ?s (Xs ) +
?st ?st (Xs , Xt ) .
P(X; ?) ? exp
?
?
s?V
(s,t)?E
In such a pairwise model, (Xs , Xt ) are conditionally independent (given the rest of the variables)
if and only if the weight ?st is zero. The most popular instances of pairwise UGMs are Gaussian
graphical models (GGMs) [19, 2] for real-valued random variables and Ising (or Potts) models [15]
for binary or finite nominal discrete random variables. More broadly, in order to derive multivariate graphical models from univariate exponential family distributions (such as the Gaussian, binomial/multinomial, Poisson, exponential distributions, etc.), the exponential family graphical models
(EFGMs) [27, 21] were proposed as a unified framework to learn UGMs with node-wise conditional
distributions arising from generalized linear models (GLMs).
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
1.1
Overview of contribution
A fundamental issue that arises in UGMs is to specify sufficient statistics, i.e.,
{?s (Xs ), ?st (Xs , Xt )}, for modeling the interactions among variables. It is noteworthy that
most prior pairwise UGMs use pairwise product of variables (or properly transformed variables)
as pairwise sufficient statistics [16, 11, 27]. This is clearly restrictive in modern data analysis
tasks where the underlying pairwise interactions among variables are more often than not highly
complex and unknown a prior. The goal of this work is to remove such a restriction and explore
the feasibility (in theory and practice) of defining sufficient statistics in an additive formulation to
approximate the underlying unknown sufficient statistics. To this end, we consider the following
Additive Exponential Family Graphical Model (AdEFGM) distribution with joint density function:
?
?
??
?
?
P(X; f ) = exp
fs (Xs ) +
fst (Xs , Xt ) ? A(f ) ,
(2)
?
?
s?V
(s,t)?E
where fs : X ? R and{ fst (?, ?) : X 2 ? R are respectively node-wise
and pairwise statistics, and
}
?
?
?
A(f ) := log X p exp
f
(X
)
+
f
(X
,
X
)
dX
is
the log-partition function.
s
s
t
s?V s
(s,t)?E st
We require the condition A(f ) < ? holds so that the definition of probability is valid. In this paper,
we assume the formulations of sufficient statistics fs and fst are unknown but they admit linear
representations over two sets of pre-fixed basis functions {?k (?), k = 1, 2, ..., q} and {?l (?, ?), l =
1, 2, ..., r}, respectively. That is,
fs (Xs ) =
q
?
?s,k ?k (Xs ),
fst (Xs , Xt ) =
k=1
r
?
?st,l ?l (Xs , Xt ),
(3)
l=1
where q and r are the truncation order parameters. In the formulation (3), the choice of basis and
their sizes is flexible and task-dependent. For instance, if the mapping functions fs and fst are
periodic, then we can choose {?k (?)} as 1-D Fourier basis and {?l (?, ?)} as 2-D Fourier basis. As
another instance, the basis {?l } can be chosen as multiple kernels which are commonly used in
computer vision tasks. Specially, when q = r = 1, ?l (Xs , Xt ) = Xs Xt and ?k (Xs ) is fixed as
certain parametric function, AdEFGM reduces to the standard EFGM [27, 21]. In general cases, by
imposing an additive structure on the sufficient statistics fs and fst , AdEFGM is expected to be able
to capture more complex interactions among variables beyond pairwise product.
As the core contribution of this paper, we propose two ?2,1 -norm regularized maximum likelihood
estimation (MLE) estimators to learn the weights of AdEFGM in high dimensional settings. The
first estimator is formulated as an ?2,1 -norm regularized MLE to jointly estimate all the parameters
in the model. The second estimator is formulated as an ?2,1 -norm regularized node-wise conditional
MLE to estimate the parameters associated with each node individually. Theoretically,
we prove that
?
under mild conditions the joint MLE estimator achieves convergence rate O(( (2|E|
? + p) ln p/n)
where |E| while the node-wise conditional estimator achieves convergence rate O( (d + 1) ln p/n)
in which d is the degree of the underlying graph G. Computationally, we propose a Monte-Carlo
approximation scheme to efficiently optimize the estimators via proximal gradient descent methods.
We conduct numerical studies on simulated and real data to support our claims. The simulation
results confirm that, when the data are drawn from an underlying UGMs with highly nonlinear sufficient statistics, our estimators significantly outperform GGMs and Nonparanormal [10] estimators
in most cases. The experimental results on a stock price data show that our estimators are able to
recover more accurate category links among stocks than GMMs and Nonparanormal estimators.
1.2
Related work
In order to model random variables beyond parametric UGMs such as GGMs and Ising models, researchers recently investigated semi-parametric/nonparametric extensions of these parametric models. The Nonparanormal [11] and copula-based methods [5] are semi-parametric graphical models
which assume that data is Gaussian after applying a monotone transformation. More broadly, one
could learn transformations of the variables and then fit any parametric UGMs (like EFGMs) over
the transformed variables. In [10, 26], two rank-based estimators were used to estimate correlation
matrix and then fit the GGMs. In [24], a semi-parametric method was proposed to fit the conditional
2
means of the features with an arbitrary additive formulation. The Semi-EFGM proposed in [28] is
a semi-parametric rank-based conditional estimator for exponential family graphical models. In [1],
a kernel method was proposed for learning the structure of graphical models by treating variables
as Gaussians in a mapped high-dimensional feature space. In [7], Gu proposed a functional minimization framework to estimate the nonparametric model (1) over a Reproducing Hilbert Kernel
Space (RKHS). Nonparametric exponential family graphical models based on score matching loss
were investigated in [9, 20]. The forest density estimation [8] is a fully nonparametric method for
estimating UGMs with structure restricted to be a forest. In contrast to all these existing semiparametric/nonparametric models, our approach is novel in model definition and computation: we
impose a simple additive structure on sufficient statistics to describe complex interactions between
variables and use Monte-Carlo approximation to estimate the intractable normalization constant for
efficient optimization.
1.3
Notation and organization
Notation Let ? = {?s,k , ?st,l : s ? V, k = 1, 2, .., , (s, t) ? V 2 , s ?= t, l = 1, 2, ...} be a vector
of parameters associated with AdEFGM and G = {{(s, k)}k , {(st, l)}l : s ? V, (s, t) ? V 2 , s ?=
t} be a group induced by the additive structures of nodes?
and edges. We conventionally define
the following grouped-norm related notations: ???2,1 =
g?G ??g ?, ???2,? = maxg?G ??g ?,
supp(?, G) = {g ? G : ??g ? ?= 0} and ???2,0 = |supp(?, G)|. For any S ? G, these notations can
be defined restrictively over ?S . We denote S? = G \ S the complement of S in G.
Organization. The remaining of this paper is organized as follows: In ?2, we present two maximum
likelihood estimators for learning the model parameters of AdEFGM. The statistical guarantees of
the proposed estimators are analyzed in ?3. Monte-Carlo simulations and experimental results on
real stock price data are presented in ?4. Finally, we conclude this paper in ?5. Due to space limit,
all the technical proofs of theoretical results are deferred to an appendix section which is included
in the supplementary material.
2 ?2,1 -norm Regularized MLE for AdEFGM
In this section, we investigate the problem of estimating the parameters of AdEFGM in high dimensional settings. By substituting (3) into (2), the distribution of an AdEFGM can be converted to the
following form:
P(X; ?) = exp {B(X; ?) ? A(?)} ,
(4)
where ? = {?s,k , ?st,l }, and
?
?
?
?st,l ?l (Xs , Xt ), A(?) := log
exp {B(X; ?)} dX.
?s,k ?k (Xs ) +
B(X; ?) :=
s?V,k
Xp
(s,t)?E,l
Suppose we have n i.i.d. samples Xn = {X (i) }ni=1 drawn from the following AdEFGM with true
parameters ?? :
P(X; ?? ) = exp {B(X; ?? ) ? A(?? )} .
(5)
An important goal of graphical model learning is to estimate the true parameters ?? from the observed data Xn . The more accurate parameter estimation is, the more accurate we are able to recover
the underlying true graph structure. We next propose two ?2,1 -norm regularized maximum likelihood
estimation (MLE) methods for joint and node-conditional learning of parameters, respectively.
2.1
Joint MLE estimation
Given the sample set Xn = {X (i) }ni=1 , the negative log-likelihood of the joint distribution (5) is:
1?
B(X (i) ; ?) + A(?).
n i=1
n
L(?; Xn ) = ?
It is trivial to verify L(?; Xn ) has the following first order derivative (see, e.g., [25]):
1?
?L
= E? [?k (Xs )] ?
?k (Xs(i) ),
??s,k
n i=1
n
1?
?L
(i)
= E? [?l (Xs , Xt )] ?
?l (Xs(i) , Xt ), (6)
??st,l
n i=1
n
3
where the expectation E? [?] is taken over the joint distribution (2). Also, it is well known that
L(?; Xn ) is convex in ?.
In order to estimate the parameters which are expected to be sparse in edge level due to the potential
sparse structure of graph, we consider the following ?2,1 -norm regularized MLE estimator:
??n = arg min {L(?; Xn ) + ?n ???2,1 } ,
(7)
?
(?
)1/2 ?
(?
)1/2
?
q
r
2
2
where ???2,1 =
?
+
?
is the ?2,1 -norm with
2
s?V
k=1 s,k
(s,t)?V ,s?=t
l=1 st,l
respect to the basis statistics and ?n > 0 is the regularization strength parameter dependent on n.
The ?2,1 -norm penalty is used to promote edgewise sparsity as the graph structure is expected to be
sparse in high dimensional settings.
2.2
Node-conditional MLE estimation
Recent state of the art methods for learning UGMs suggest a natural procedure deriving multivariate
graphical models from univariate distributions [12, 15, 27]. The common idea in these methods is
to learn the graph structure by estimating node-neighborhoods, or by fitting the node-conditional
distribution of each individual node conditioned on the rest of the nodes. Indeed, these node-wise
fitting methods have been shown to have strong statistical guarantees and attractive computational
performance. Inspired by these approaches, we propose an alternative estimator to estimate the
weights of sufficient statistics associated with each individual node. With a slight abuse of notation,
we denote ?s a subvector of ? associated with node s, i.e.,
?s := {?s,k | k = 1, ..., q} ? {?st,l | t ? N (s), l = 1, ..., r},
where N (s) is the neighborhood of s. Given the joint distribution (4), it is easy to show that the
conditional distribution of Xs given the rest variables, X\s , is written by:
{
}
P(Xs | X\s ; ?s ) = exp C(Xs | X\s ; ?s ) ? D(X\s ; ?s ) ,
(8)
?
?
? ? (X ) + t?N (s),l ?st,l ?l (Xs , Xt ), and D(X\s ; ?s ) :=
where C(Xs | X\s ; ?s ) :=
{
} k s,k k s
?
log X exp C(Xs | X\s ; ?s ) dXs is the log-partition function which ensures normalization. We
note that the condition A(?) < ? for the joint log-partition function implies D(X\s ; ?s ) < ?.
In order to estimate the parameters associated with a node, we consider using the sparsity regularized
conditional maximum likelihood estimation. Given n independent samples Xn drawn from (5), we
can write the negative log-likelihood of the conditional distribution as:
}
?{
(i)
(i)
? s ; Xn ) = 1
L(?
?C(Xs(i) | X\s ; ?s ) + D(X\s ; ?s ) .
n i=1
n
? s ; Xn ) is convex with respect to ?s and it has the following first-order derivative:
It is standard that L(?
}
? s ; Xn ) 1 ? {
? L(?
(i)
=
??k (Xs(i) ) + E?s [?k (Xs ) | X\s ] ,
??s,k
n i=1
n
n {
}
? s ; Xn ) 1 ?
? L(?
(i)
(i)
(i)
=
??l (Xs(i) , Xt ) + E?s [?l (Xs , Xt ) | X\s ] ,
??st,l
n i=1
(9)
where the expectation E?s [? | X\s ] is taken over the node-wise conditional distribution (8).
Let us consider the following ?2,1 -norm regularized conditional MLE formulation associated with
the variable Xs :
{
}
? s ; Xn ) + ?n ??s ?2,1 ,
??sn = arg min L(?
(10)
(?
?s
)1/2 ?
(?
)1/2
q
r
2
2
where ??s ?2,1 =
?
+
?
is the grouped ?2,1 -norm with respect to
k=1 s,k
t?=s
l=1 st,l
the node-wise and pairwise basis associated with s and ?n > 0 controls the regularization strength.
4
2.3
Computation via Monte-Carlo approximation
We consider using proximal gradient descent methods [22] to solve the composite optimization
problems in (7) and (10). For both estimators, the major computational overhead is to iteratively
? s ; Xn ). In general,
calculate the expectation terms involved in the gradients ?L(?; Xn ) and ?L(?
these expectation terms have no close-form for exact calculation and sampling methods such as importance sampling and MCMC are usually needed for approximate estimation. There are, however,
two challenging issues with such a sampling based optimization procedure: (1) the multivariate
sampling methods typically suffer from high computational cost even when the dimensionality p
is moderately large; and (2) the non-vanishing sampling error of gradient will accumulate during
the iteration which according to the results in [18] will deteriorate the overall convergence performance. Obviously, the main source of these challenges is caused by the intractable log-partition
terms appeared in the estimators.
To more efficiently apply the first-order methods without suffering from iterative sampling and error
accumulation, it is a natural idea to replace the log-partition terms by a Monte-Carlo approximation
and minimize the resulting approximated formulation. Taking the joint estimator (7) as an example,
we resort to the
? basic importance sampling method to approximately estimate the log-partition term
A(?) = log X p exp {B(X; ?)} dX. Assume we have m i.i.d. samples Ym = {Y (j) }m
j=1 drawn
from a random vector Y ? X p with known probability density P(Y ). Given ?, an importance
sampling estimate of exp{A(?)} is given by
{
}
m
1 ? exp B(Y (j) ; ?)
?
exp{A(?; Ym )} =
.
m j=1
P(Y (j) )
We consider the following Monte-Carlo approximation to the estimator (7):
{
}
?
? Xn , Ym ) + ?n ???2,1 ,
??n = arg min L(?;
(11)
?
? Ym ). Since the random samples Ym are fixed
? Xn , Ym ) = ? 1 ?n B(X (i) ; ?) + A(?;
where L(?;
i=1
n
? Xn , Ym ). Concerning
in (11), the sampling operation can be avoided in the computation of ?L(?;
the accuracy of the approximate estimator (11), the following result guarantees that, with high probability, the minimizer of the
? approximate estimator (11) is suboptimal to the population estimator (7)
with suboptimality O(1/ m). A proof of this proposition is provided in A.1 (see the supplementary
material).
Proposition 1. Assume that P(Y ) > 0. Then the following inequality holds with high probability:
(
)
?
? ??n ; Ym )}
2.58?
? exp{?A(??n } + exp{?A(
?
?
?
,
L(??n ; Xn )+?n ???n ?2,1 ? L(??n ; Xn )+?n ???n ?2,1 +
m
(
)2
?m
exp{B(Y (j) ;??n )}
1
?
?
where ?
?n = m j=1
? exp{A(?n ; Ym )} .
P(Y (j) )
A similar Monte-Carlo approximation strategy can be applied to the node-wise MLE estimator (10).
3
Statistical Analysis
In this section, we provide statistical guarantees on parameter estimation error for the joint MLE
estimator (7) and the node-conditional estimator (10). In large picture, our analysis follows the
techniques presented in [13, 30] by specifying the conditions under which these techniques can be
applied to our setting.
3.1
Analysis of the joint estimator
We are interested in the concentration bounds of the random variables defined by
Zs,k := ?k (Xs ) ? E?? [?k (Xs )],
Zst,l := ?l (Xs , Xt ) ? E?? [?l (Xs , Xt )],
5
where the expectation E?? [?] is taken over the underlying true distribution (5). By the ?law of the
unconscious statistician? we have E[Zs,k ] = E[Zst,l ] = 0. That is, {Zs,k } and {Zst,l } are zeromean random variables. We introduce the following technical condition on {Zs,k , Zst,l } which we
will show to guarantee the gradient ?L(?? ; Xn ) vanishes exponentially fast, with high probability,
as sample size increases.
Assumption 1. For all (s, k) and all (s, t, l), we assume that there exist constants ? > 0 and ? > 0
such that for all |?| ? ?,
{
}
{
}
E[exp{?Zs,k }] ? exp ? 2 ? 2 /2 , E[exp{?Zst,l }] ? exp ? 2 ? 2 /2 .
This assumption essentially imposes an exponential-type bound on the moment generating function
of the random variables Zs,k , Zst,l .
It is well known that the Hessian ?2 L(?; Xn ) is positive semidefinite at any ? and it is independent
on the sample set Xn . We also need the following condition which guarantees the restricted positive
definiteness of ?2 L(?; Xn ) over certain low dimensional subspace when ? is in the vicinity of ?? .
Assumption 2 (Locally Restricted Positive Definite Hessian). Let S = supp(?? ; G). There exist
constants ? > 0 and ? > 0 such that for any ? ? {?? ? ?? ? ? ?}, the inequality ?? ?2 L(?; Xn )? ?
????2 holds for any ? ? CS := {??S? ?2,1 ? 3??S ?2,1 }.
Assumption 2 requires that the Hessian ?2 L(?; Xn ) is positive definite in the cone CS when ? lies
in a local ball centered at ?? . This condition is a specification of the concept restricted strong
convexity [30] to AdEFGM.
Remark 1 (Minimal Representation). We say an AdEFGM has minimal representation if there is a
unique parameter vector ? associated with the distribution (4). This condition equivalently requires
that there exists no non-zero ? such that B(X; ?) is equal to an absolute constant. This implies that
for any ? and for all non-zero ?,
Var? [B(X; ?)] = ?? ?2 L(?; Xn )? > 0.
If AdEFGM has minimal representation at ?? , then there must exist sufficiently small constants
? > 0 and ? > 0 such that for any ? ? {?? ? ?? ? ? ?}, ?? ?2 L(?; Xn )? ? ????2 . Therefore,
Assumption 2 holds true when AdEFGM has minimal representation at ?? .
The following theorem is our main result on the estimation error of the joint MLE estimator (7). A
proof of this result is provided in Appendix A.2 in the supplementary material.
Theorem 1. Assume that the conditions in Assumption 1 and Assumption 2 hold. If sample size n
satisfies
{
}
6 max{q, r} ln p 54c20 ? 2 max{q, r}??? ?2,0 ln p
n > max
,
,
?2 ? 2
?2 ? 2
then with probability at least 1 ? 2 max{q, r}p?1 , the following inequality holds:
?
???n ? ?? ? ? 3c0 ? ?1 ? 6 max{q, r}??? ?2,0 ln p/n.
Remark 2. The main message Theorem 1 conveys
? is that when n is sufficiently large, the estimation
error ???n ? ?? ? vanishes at the order of O( max{q, r}(2|E| + p) ln p/n) with high probability.
This convergence rate matches the results obtained in [17, 16] for GGMs and the results in [10, 26]
for Nonparanormal.
3.2
Analysis of the node-conditional estimator
For the node-conditional estimator (10), we study the rate of convergence of the parameter estimation error ???sn ? ?s? ? as a function of sample size n. We need Assumption 1 and the following
assumption in our analysis.
Assumption 3. For any node s, let S = supp(?s? ; G). There exist constants ?? > 0 and ?? > 0 such
? the inequality ?? ?2 L(?
? s ?2 holds for any
? s ; Xn )?s ? ???
that for any ?s ? {??s ? ?s? ? < ?},
s
?s ? C?S := {?(?s )S? ?2,1 ? 3?(?s )S ?2,1 }.
6
The following is our main result on the convergence rate of node-conditional estimation error ???sn ?
?s? ?. A proof of this result is provided in Appendix A.3 in the supplementary material.
Theorem 2. Assume that the conditions in Assumption 1 and Assumption 3 hold. If sample size n
satisfies
}
{
c20 ?
6 max{q, r} ln p 216?
? 2 max{q, r}??s? ?2,0 ln p
,
n > max
,
?2 ? 2
? 2 ??2
then with probability at least 1 ? 4 max{q, r}p?2 , the following inequality holds:
?
???sn ? ?s? ? ? 6?
c0 ???1 ? 6 max{q, r}??s? ?2,0 ln p/n.
Remark 3. Theorem
? 2 indicates that with overwhelming probability, the estimation error
???sn ? ?s? ? = O( (d + 1) ln p)/n) where d is the degree of the underlying graph, i.e., d =
maxs?V ??s? ?2,0 ? 1. We may combine the parameter estimation errors from all the nodes as
a global measurement of?accuracy. Indeed, by Theorem 2 and union of probability we get that
maxs?V ???sn ??s? ? = O( (d + 1) ln p/n) holds with probability at least 1?4 max{q, r}p?1 . This
estimation error bound matches those for GGMs with neighborhood-selection-type estimators [29].
4
Experiments
This section is devoted to showing the actual learning performance of AdEFGM. We first investigate
graph structure recovery accuracy using simulation data (for which we know the ground truth), and
then we apply our method to a stock price data for inferring the statistical dependency among stocks.
4.1
Monte-Carlo simulation
This is a proof-of-concept experiment. The purpose is to confirm that when the pairwise interactions
of the underlying graphical models are highly nonlinear and unknown a prior, our additive estimator
will be significantly superior to existing parametric/semi-parametric graphical models for inferring
the structure of graphs. The numerical results of AdEFGM reported in this experiment are obtained
by the joint MLE estimator in (7).
Simulated data Our simulation study employs a graphical model of which the edges are generated
independently with probability P . We will consider the model under different levels of sparsity
by adjusting the probability P . For simplicity purpose, we assume fs (Xs ) ? 1 and consider a
nonlinear pairwise interaction function fst (Xs , Xt ) = cos(?(Xs ? Xt )/5). We fit the data to
the additive model (4) with a 2-D Fourier basis of size 8. Using Gibbs sampling, we generate a
training sample of size n from the true graphical model, and an independent sample of the same
size from the same distribution for tuning the strength parameter ?n . We compare performance for
n = 200, varying values of p ? {50, 100, 150, 200, 250, 300}, and different sparsity levels under
P = {0.02, 0.05, 0.1}, replicated 10 times for each configuration.
Baselines We compare the performance of our estimator to Graphical Lasso [6] as a GGM estimator
and SKEPTIC [10] as a Nonparanormal estimator. In our implementation, we use a version of
SKEPTIC with Kendall?s tau to infer the correlation.
Evaluation metric To evaluate the support recovery performance, we use the standard F-score from
the information retrieval literature. The larger F-score is, the better the support recovery performance. The numerical values over 10?3 in magnitude are considered to be nonzero.
Results Figure 1 shows the support recovery F-scores of the considered methods on the synthetic
data. From this group of results we can observe that by using 2-D Fourier basis to approximate the
unknown cosine distance function, AdEFGM is able to more accurately recover the underlying graph
structure than the other two considered methods. The advantage of AdEFGM illustrated here is as
expected because it is designed to automatically learn the unknown complex pairwise interactions
while GGM and Nonparanormal are restrictive to certain UGMs with known sufficient statistics.
4.2
Stock price data
We further study the performance of AdEFGM on a stock price data. This data contains the historical
prices of S&P500 stocks over 5 years, from January 1, 2008 to January 1, 2013. By taking out the
7
1
0.8
0.6
0.4
1
0.8
0.6
0.4
0.2
0.2
0
0
50
100
150
200
250
Cosine Distance
AdEFGM
GGM
Nonparanormal
Recovery F?score
Cosine Distance
AdEFGM
GGM
Nonparanormal
Recovery F?score
Recovery F?score
Cosine Distance
AdEFGM
GGM
Nonparanormal
1
0.8
0.6
0.4
0.2
0
300
50
100
Dimension p
150
200
250
300
50
100
Dimension p
150
200
250
300
Dimension p
Figure 1: Simulated data: Support recovery F-score curves. Left panels: P = 0.02, Middle panels:
P = 0.05, Right panels: P = 0.1.
Link Precision
0.8
0.6
0.4
0.2
1
0.8
AdEFGM
GGM
Nonparanormal
AdEFGM
GGM
Nonparanormal
0.6
Link F?score
AdEFGM
GGM
Nonparanormal
Link Recall
1
0.6
0.4
0.4
0.2
0.2
0
0
2
4
6
Number of links
8
10
4
x 10
0
2
4
6
Number of links
8
10
4
x 10
2
4
6
Number of links
8
10
4
x 10
Figure 2: Stock price data S&P500: Category link precision, recall and F-score curves.
stocks with less than 5 years of history, we end up with 465 stocks, each having daily closing
prices over 1,260 trading days. The prices are first adjusted for dividends and splits and the used
to calculate daily log returns. Each day?s return can be represented as a point in R465 . To apply
AdEFGM to this data, we consider the general model (4) with the 2-D Fourier basis being used to
approximate the pairwise interaction between stocks Xs and Xt . Since the category information of
S&P500 is available, we measure the performance by Precision, Recall and F-score of the top k
links (edges) on the constructed graph. A link is regarded as true if and only if it connects two nodes
belonging to the same category. We use the joint MLE estimator for this experiment. Figure 2 shows
the curves of precision, recall and F-score as functions of k in a wide range [103 , 105 ]. It is apparent
that AdEFGM significantly outperforms GGM and Nonparanormal for identifying correct category
links. This result suggests that the interactions among the S&P500 stocks are highly nonlinear.
5
Conclusions
In this paper, we proposed and analyzed AdEFGMs as a generic class of additive undirected graphical models. By expressing node-wise and pairwise sufficient statistics as linear representations over
a set of basis statistics, AdEFGM is able to capture complex interactions among variables which
are not uncommon in modern engineering applications. We investigated two types of ?2,1 -norm
regularized MLE estimators for joint and node-conditional high dimensional estimation. Based on
our theoretical justification and empirical observation, we can draw the following two conclusions:
1) the ?2,1 -norm regularized AdEFGM learning is a powerful tool for inferring pairwise exponential family graphical models with unknown arbitrary sufficient statistics; and 2) the extra flexibility
gained by AdEFGM comes at almost no cost of statistical and computational efficiency.
Acknowledgments
Xiao-Tong Yuan and Ping Li were partially supported by NSF-Bigdata-1419210, NSF-III-1360971,
ONR-N00014-13-1-0764, and AFOSR-FA9550-13-1-0137. Xiao-Tong Yuan is also partially supported by NSFC-61402232, NSFC-61522308, and NSFJP-BK20141003. Tong Zhang is supported
by NSF-IIS-1407939 and NSF-IIS-1250985. Qingshan Liu is supported by NSFC-61532009.
Guangcan Liu is supported by NSFC-61622305, NSFC-61502238 and NSFJP-BK20160040.
8
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9
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5,644 | 6,107 | Observational-Interventional Priors for
Dose-Response Learning
Ricardo Silva
Department of Statistical Science and Centre for Computational Statistics and Machine Learning
University College London
ricardo@stats.ucl.ac.uk
Abstract
Controlled interventions provide the most direct source of information for learning
causal effects. In particular, a dose-response curve can be learned by varying the
treatment level and observing the corresponding outcomes. However, interventions
can be expensive and time-consuming. Observational data, where the treatment is
not controlled by a known mechanism, is sometimes available. Under some strong
assumptions, observational data allows for the estimation of dose-response curves.
Estimating such curves nonparametrically is hard: sample sizes for controlled
interventions may be small, while in the observational case a large number of
measured confounders may need to be marginalized. In this paper, we introduce
a hierarchical Gaussian process prior that constructs a distribution over the doseresponse curve by learning from observational data, and reshapes the distribution
with a nonparametric affine transform learned from controlled interventions. This
function composition from different sources is shown to speed-up learning, which
we demonstrate with a thorough sensitivity analysis and an application to modeling
the effect of therapy on cognitive skills of premature infants.
1
Contribution
We introduce a new solution to the problem of learning how an outcome variable Y varies under
different levels of a control variable X that is manipulated. This is done by coupling different
Gaussian process priors that combine observational and interventional data. The method outperforms
estimates given by using only observational or only interventional data in a variety of scenarios and
provides an alternative way of interpreting related methods in the design of computer experiments.
Many problems in causal inference [14] consist of having a treatment variable X and and outcome Y ,
and estimating how Y varies as we control X at different levels. If we have data from a randomized
controlled trial, where X and Y are not confounded, many standard modeling approaches can be
used to learn the relationship between X and Y . If X and Y are measured in an observational study,
the corresponding data can be used to estimate the association between X and Y , but this may not be
the same as the causal relationship of these two variables because of possible confounders.
To distinguish between the observational regime (where X is not controlled) and the interventional
regime (where X is controlled), we adopt the causal graphical framework of [16] and [19]. In Figure
1 we illustrate the different regimes using causal graphical models. We will use p(? | ?) to denote
(conditional) density or probability mass functions. In Figure 1(a) we have the observational, or
?natural,? regime where common causes Z generate both treatment variable X and outcome variable
Y . While the conditional distribution p(Y = x | X = x) can be learned from this data, this quantity is
not the same as p(Y = y | do(X = x)): the latter notation, due to Pearl [16], denotes a regime where
X is not random, but a quantity set by an intervention performed by an external agent. The relation
between these regimes comes from fundamental invariance assumptions: when X is intervened upon,
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Y
X
ZO
Z
Z
Y
X
(a)
(b)
ZH
Y
X
(c)
Figure 1: Graphs representing causal graphical models. Circles represent random variables, squares
represent fixed constants. (a) A system where Z is a set of common causes (confounders), common
parents of X and Y here represented as a single vertex. (b) An intervention overrides the value of X
setting it to some constant. The rest of the system remains invariant. (c) ZO is not a common cause
of X and Y , but blocks the influence of confounder ZH .
?all other things are equal,? and this invariance is reflected by the fact that the model in Figure 1(a)
and Figure 1(b) share the same conditional distribution p(Y = x|X = x, Z = z) and marginal
distribution p(Z = z). If we observe Z, p(Y = y | do(X = x)) can be learned from observational
data, as we explain in the next section.
Our goal is to learn the relationship
f (x) ? E[Y | do(X = x)], x ? X ,
(1)
where X ? {x1 , x2 , . . . , xT } is a pre-defined set of treatment levels. We call the vector f (X ) ?
[f (x1 ); . . . ; f (xT )]> the response curve for the ?doses? X . Although the term ?dose? is typically
associated with the medical domain, we adopt here the term dose-response learning in its more
general setup: estimating the causal effect of a treatment on an outcome across different (quantitative)
levels of treatment. We assume that the causal structure information is known, complementing
approaches for structure learning [19, 9] by tackling the quantitative side of causal prediction.
In Section 2, we provide the basic notation of our setup. Section 3 describes our model family.
Section 4 provides a thorough set of experiments assessing our approach, including sensitivity to
model misspecification. We provide final conclusions in Section 5.
2
Background
The target estimand p(Y = y | do(X = x)) can be derived from the structural assumptions of Figure
1(b) by standard conditioning and marginalization operations:
Z
p(Y = y | do(X = x)) = p(Y = y | X = x, Z = z)p(Z = z) dz.
(2)
Notice the important difference between the above and p(Y = y | X = x), which can be derived from
the assumptions in Figure 1(a) by marginalizing over p(Z = z | X = x) instead. The observational
and interventional distributions can be very different. The above formula is sometimes known as the
back-door adjustment [16] and it does not require measuring all common causes of treatment and
outcome. It suffices that we measure variables Z that block all ?back-door paths? between X and Y ,
a role played by ZO in Figure 1(c). A formal description of which variables Z will validate (2) is
given by [20, 16, 19]. We will assume that the selection of which variables Z to adjust for has been
decided prior to our analysis, although in our experiments in Section 4 we will assess the behavior
of our method under model misspecification. Our task is to estimate (1) nonparametrically given
observational and experimental data, assuming that Z satisfies the back-door criteria.
One possibility for estimating (1) from observational data Dobs ? {(Y (i) , X (i) , Z(i) )}, 1 ? i ? N ,
is by first estimating g(x, z) ? E[Y | X = x, Z = z]. The resulting estimator,
N
1 X
f?(x) ?
g?(x, z(i) ),
N i=1
(3)
is consistent under some general assumptions on f (?) and g(?, ?). Estimating g(?, ?) nonparametrically
seems daunting, since Z can in principle be high-dimensional. However, as shown by [5], under
2
some conditions the problem of estimating f?(?) nonparametrically via (3) is no harder than a onedimensional nonparametric regression problem. There is however one main catch: while observational
data can be used to choose the level of regularization for g?(?), this is not likely to be an optimal choice
for f?(?) itself. Nevertheless, even if suboptimal smoothing is done, the use of nonparametric methods
for estimating causal effects by back-door adjustment has been successful. For instance, [7] uses
Bayesian classification and regression trees for this task.
Although of practical use, there are shortcomings in this idea even under the assumption that Z
provides a correct back-door adjustment. In particular, Bayesian measures of uncertainty should be
interpreted with care: a fully Bayesian equivalent to (3) would require integrating over a model for
p(Z) instead of the empirical distribution for Z in Dobs ; evaluating a dose x might require combining
many g(x, z(i) ) where the corresponding training measurements x(i) are far from x, resulting on
possibly unreliable extrapolations with poorly calibrated credible intervals. While there are well
established approaches to deal with this ?lack of overlap? problem in binary treatments or linear
responses [18, 8], it is less clear what to do in the continuous case with nonlinear responses.
In this paper, we focus on a setup where it is possible to collect interventional data such that treatments
are controlled, but where sample sizes might be limited due to financial and time costs. This is related
to design of computer experiments, where (cheap, but biased) computer simulations are combined
with field experiments [2, 6]. The key idea of combining two sources of data is very generic, the
value of new methods being on the design of adequate prior families. For instance, if computer
simulations are noisy, it is may not be clear how uncertainty at that level should be modeled. We
leverage knowledge of adjustment techniques for causal inference, so that it provides a partially
automated recipe to transform observational data into informed priors. We leverage knowledge
of the practical shortcomings of nonparametric adjustment (3) so that, unlike the biased but low
variance setup of computer experiments, we try to improve the (theoretically) unbiased but possibly
oversmooth structure of such estimators by introducing a layer of pointwise affine transformations.
Heterogeneous effects and stratification. One might ask why marginalize Z in (2), as it might be
of greater interest to understand effects at the finer subpopulation levels conditioned on Z. In fact, (2)
should be seen as the most general case, where conditioning on a subset of covariates (for instance,
gender) will provide the possibly different average causal effect for each given strata (different levels
of gender) marginalized over the remaining covariates. Randomized fine-grained effects might be
hard to estimate and require stronger smoothing and extrapolation assumptions, but in principle
they could be integrated with the approaches discussed here. In practice, in causal inference we are
generally interested in marginal effects for some subpopulations where many covariates might not be
practically measurable at decision time, and for the scientific purposes of understanding total effects
[5] at different levels of granularity with weaker assumptions.
3
Hierarchical Priors via Inherited Smoothing and Local Affine Changes
The main idea is to first learn from observational data a Gaussian process over dose-response curves,
then compose it with a nonlinear transformation biased toward the identity function. The fundamental
innovation is the construction a nonstationary covariance function from observational data.
3.1
Two-layered Priors for Dose-responses
Given an observational dataset Dobs of size N , we fit a Gaussian process to learn a regression model
of outcome Y on (uncontrolled) treatment X and covariates Z. A Gaussian likelihood for Y given X
and Z is adopted, with conditional mean g(x, z) and variance ?g2 . A Mat?rn 3/2 covariance function
with automatic relevance determination priors is given to g(?, ?), followed by marginal maximum
likelihood to estimate ?g2 and the covariance hyperparameters [12, 17]. This provides a posterior
distribution over functions g(?, ?) in the input space of X and Z. We then define fobs (X ), x ? X , as
fobs (x) ?
N
1 X
g(x, z(i) ),
N i=1
(4)
where set {g(x, z(i) )} is unknown. Uncertainty about fobs (?) comes from the joint predictive distribution of {g(x, z(i) )} learned from Dobs , itself a Gaussian distribution with a T N ? 1 mean vector
3
??g and a T N ? T N covariance matrix, T ? |X |. Since (4) is a linear function of {g(x, z(i) )}, this
PN
implies fobs (X ) is also a (nonstationary) Gaussian process with mean ?obs (x) = N1 i=1 ??g (x, z (i) )
for each x ? X . The motivation for (4) is that ?obs is an estimator of the type (3), inheriting its
desirable properties and caveats.
The cost of computing the covariance matrix Kobs of fobs (X ) is O(T 2 N 2 ), potentially expensive. In
many practical applications, however, the size of X is not particularly large as it is a set of intervention
points to be decided according to practical real-world constraints. In our simulations in Section 4, we
chose T = |X | = 20. Approximating such covariance matrix, if necessary, is a future research topic.
(i)
(i)
(i)
Assume interventional data Dint ? {(Yint , xint )}, 1 ? i ? M , is provided (with assignments xint
chosen by some pre-defined design in X ). We assign a prior to f (?) according to the model
fobs (X )
a(X )
b(X )
f (X )
(i)
Yint
?
?
?
=
?
N (?obs , Kobs )
N (1, Ka )
N (0, Kb )
a(X ) fobs (X ) + b(X )
(i)
2
N (f (xint ), ?int
), 1 ? i ? M,
(5)
where N (m, V) is the multivariate normal distribution with mean m and covariance matrix V, is
the elementwise product, a(?) is a vector which we call the distortion function, and b(?) the translation
function. The role of the ?elementwise affine? transform a fobs + b is to bias f toward fobs with
uncertainty that varies depending on our uncertainty about fobs . The multiplicative component
a fobs also induces a heavy-tail prior on f . In the Supplementary Material, we discuss briefly the
alternative of using the deep Gaussian process of [4] in our observational-interventional setup.
3.2
Hyperpriors
We parameterize Ka as follows. Every entry ka (x, x0 ) of Ka , (x, x0 ) ? X ? X , assumes the shape
of a squared exponential kernel modified according to the smoothness and scale information obtained
from Dobs . First, define ka (x, x0 ) as
1 (?
x?x
?0 )2 + (?
yx ? y?x0 )2
0
ka (x, x ) ? ?a ? vx ? vx0 ? exp ?
+ ?(x ? x0 )10?5 ,
(6)
2
?a
where (?a , ?h ) are hyperparameters, ?(?) is the delta function, vx is a rescaling of Kobs (x, x)1/2 ,
x
? is a rescaling of X to the [0, 1] interval, y?x is a rescaling of ?obs (x) to the [0, 1] interval. More
precisely,
s
x ? min(X )
Kobs (x, x)
?obs (x) ? min(?obs (X ))
, y?x ?
, vx =
.
x
??
max(X ) ? min(X )
max(?obs (X )) ? min(?obs (X ))
maxx0 Kobs (x0 , x0 )
(7)
Equation (6) is designed to borrow information from the (estimated) smoothness of f (X ), by
decreasing the correlation of the distortion factors a(x) and a(x0 ) as a function of the Euclidean
distance between the 2D points (x, ?obs (x)) and (x0 , ?obs (x0 )), properly scaled. Hyperparameter
?a controls how this distance is weighted. (6) also captures information about the amplitude
of the distortion signal, making it proportional to the ratios of the diagonal entries of Kobs (X ).
Hyperparameter ?a controls how this amplitude is globally adjusted. Nugget 10?5 brings stability to
the sampling of a(X ) within Markov chain Monte Carlo (MCMC) inference. Hyper-hyperpriors on
?a and ?a are set as
log(?a ) ? N (0, 0.5),
log(?a ) ? N (0, 0.1).
(8)
That is, ?a follows a log-Normal distribution with median 1, approximately 90% of the mass below
2.5, and a long tail to the right. The implied distribution for a(x) where sx = 1 will have most
of its mass within a factor of 10 from its median. The prior on ?a follows a similar shape, but
with a narrower allocation of mass. Covariance matrix Kb is defined in the same way, with its own
2
hyperparameters ?b and ?b . Finally, the usual Jeffrey?s prior for error variances is given to ?int
.
Figure 2 shows an example of inference obtained from synthetic data, generated according to the
protocol of Section 4. In this example, the observational relationship between X and Y has the
opposite association of the true causal one, but after adjusting for 15 of the 25 confounders that
4
Observational data (N = 1000)
Interventional data (M = 200)
4
4
3
3
2
2
K
obs
2
Outcome Y
Outcome Y
4
1
0
?1
6
1
8
10
0
12
?1
14
?2
?2
?3
?3
16
?4
?4
?3
?2
?1
0
1
2
?4
3
18
20
?4
?3
?2
Treatment X
0
1
2
3
2
Prior: distortion only
3
8
2.5
2
6
1.5
1
0.5
4
2
0
0
?4
?0.5
0
14
16
18
20
1
?2
?1
12
0.5
0
?2
10
1.5
?0.5
?3
8
2
Outcome Y
Distortion H
10
?4
6
Prior on dose?response
3
2.5
?1
4
Treatment X
Prior: observational only
Outcome Y
?1
1
2
?6
3
?4
?3
?2
Treatment X
?1
0
1
2
?1
3
?4
?3
?2
Treatment X
Posterior: observational only
Posterior: distortion only
3
?1
0
1
2
3
Treatment X
Posterior on dose?response
4
3
2.5
2.5
3
1.5
1
0.5
Outcome Y
2
Outcome Y
Outcome Y
2
2
1
0
0
1.5
1
0.5
0
?1
?0.5
?1
?0.5
?4
?3
?2
?1
0
Treatment X
1
2
3
?2
?4
?3
?2
?1
0
1
Treatment X
2
3
?1
?4
?3
?2
?1
0
1
2
3
Treatment X
Figure 2: An example with synthetic data (|Z| = 25), from priors to posteriors. Figure best seen in
color. Top row: scatterplot of observational data, with true dose-response function in solid green,
adjusted ?obs in dashed red, and the unadjusted Gaussian process regression of Y on X in dashedand-circle magenta (which is a very badly biased estimate in this example); scatterplot in the middle
shows interventional data, 20 dose levels uniformly spread in the support of the observational data
and 10 outputs per level ? notice that the sign of the association is the opposite of the observational
regime; matrix Kobs is depicted at the end, where the nonstationarity of the process is evident. Middle
row: priors constructed on fobs (X ) and a(X ) with respective means; plot at the end corresponds to
the implied prior on a fobs + b. Bottom row: the respective posteriors obtained by Gibbs sampling.
generated the data (10 confounders are randomly ignored to mimic imperfect prior knowledge), a
reasonable initial estimate for f (X ) is obtained. The combination with interventional data results
in a much better fit, but imperfections still exist at the strongest levels of treatment: the green curve
drops at x > 2 stronger than the expected posterior mean. This is due to having both a prior derived
from observational data that got the wrong direction of the dose-response curve at x > 1.5, and being
unlucky at drawing several higher than expected values in the interventional regime for x = 3. The
model then shows its strength on capturing much of the structure of the true dose-response curve
even under misspecified adjustments, but the example provides a warning that only so much can be
done given unlucky draws from a small interventional dataset.
3.3
Inference, Stratified Learning and Active Learning
In our experiments, we infer posterior distributions by Gibbs sampling, alternating the sampling of
2
latent variables f (X ), a(X ), b(X ) and hyperparameters ?a , ?a , ?b , ?b , ?int
, using slice sampling
[15] for the hyperparameters. The meaning of the individual posterior distribution over fobs (X )
might also be of interest. This quantity is potentially identifiable by considering
a joint model
R
for (Dobs , Dint ): in this case, fobs (X ) learns the observational adjustment g(x, z)p(z) dz. This
suggests that the posterior distribution for fobs (X ) will change little according to model (5), which
5
is indeed observed in practice and illustrated by Figure 2. Learning the hyperparameters for Kobs
could be done jointly with the remaining hyperparameters, but the cost per iteration would be high
due to the update of Kobs . The MCMC procedure for (5) is relatively inexpensive assuming that |X |
is small. Learning the hyperparameters of Kobs separately is a type of ?modularization? of Bayesian
inference [10].
As we mentioned in Section 2, it is sometimes desirable to learn dose-response curves conditioned
on a few covariates S ? Z of interest. In particular, in this paper we will consider the case of
straightforward stratification: given a set S of discrete covariates assuming instantiations s, we have
functions f s (X ) to be learned. Different estimation techniques can be used to borrow statistical
s
strength across levels of S, both for f s (X ) and fobs
(X ). However, in our implementation, where we
assume |S| is very small (a realistic case for many experimental designs), we construct independent
s
priors for the different fobs
(X ) with independent affine transformations.
Finally, in the Supplementary Material we also consider simple active learning schemes [11], as
suggested by the fact that prior information already provides different estimates of uncertainty across
X (Figure 2), which is sometimes dramatically nonstationary.
4
Experiments
Assessing causal inference algorithms requires fitting and predicting data generated by expensive
randomized trials. Since this is typically unavailable, we will use simulated data where the truth is
known. We divide our experiments in two types: first, one where we generate random dose-response
functions, which allows us to control the difficulty of the problem in different directions; second, one
where we start from a real world dataset and generate ?realistic? dose-response curves from which
simulated data can be given as input to the method.
4.1
Synthetic Data Studies
We generate studies where the observational sample has N = 1000 data points and |Z| = 25
confounders. Interventional data is generated at three different levels of sample size, M = 40,
100 and 200 where the intervention space X is evenly distributed within the range shown by the
observational data, with |X | = 20. Covariates Z are generated from a zero-mean, unit variance
Gaussian with correlation of 0.5 for all pairs. Treatment X is generated by first sampling a function
fi (zi ) for every covariate from a Gaussian process, summing over 1 ? i ? 25 and adding Gaussian
noise. Outcome Y is generated by first sampling linear coefficients and one intercept to weight the
contribution of confounders Z, and then passing the linear combination through a quadratic function.
The dose-response function of X on Y is generated as a polynomial, which is added to the contribution
of Z and a Gaussian error. In this way, it is easy to obtain the dose-response function analytically.
Besides varying M , we vary the setup in three other aspects: first, the dose-response is either a
quadratic or cubic polynomial; second, the contribution of X is scaled to have its minimum and
maximum value spam either 50% or 80% of the range of all other causes of Y , including the Gaussian
noise (a spam of 50% already generates functions of modest impact to the total variability of Y );
third, the actual data given to the algorithm contains only 15 of the 25 confounders. We either
discard 10 confounders uniformly at random (the R ANDOM setup), or remove the ?top 10 strongest?
confounders, as measured by how little confounding remains after adjusting for that single covariate
alone (the A DVERSARIAL setup). In the interest of space, we provide a fully detailed description of
the experimental setup in the Supplementary Material. Code is also provided to regenerate our data
and re-run all of these experiments.
Evaluation is done in two ways. First, by the normalized absolute difference between an estimate
f?(x) and the true f (x), averaged over X . The normalization is done by dividing the difference by the
gap between the maximum and minimum true values of f (X ) within each simulated problem1 . The
second measure is the log density of each true f (x), averaged over x ? X , according to the inferred
posterior distribution approximated as a Gaussian distribution, with mean and variance estimated by
MCMC. We compare our method against: I. a variation of it where a and b are fixed at 1 and 0, so the
only randomness is in fobs ; II. instead of an affine transformation, we set f (X ) = fobs (X ) + r(X ),
1
Data is also normalized to a zero mean, unit variance according to the empirical mean and variance of the
observational data, in order to reduce variability across studies.
6
Table 1: For each experiment, we have either quadratic (Q) or cubic (C) ground truth, with a signal
range of 50% or 80%, and an interventional sample size of M = 40, 100 and 200. Ei denotes the
difference between competitor i and our method regarding mean error, see text for a description of
competitors. The mean absolute error for our method is approximately 0.20 for M = 40 and 0.10 for
M = 200 across scenarios. Li denotes the difference between our method and competitor i regarding
log-likelihood (differences > 10 are ignored, see text). That is, positive values indicate our method
is better according to the corresponding criterion. All results are averages over 50 independent
simulations, italics indicate statistically significant differences by a two-tailed t-test at level ? = 0.05.
EI
EII
EIII
LI
LII
LIII
EI
EII
EIII
LI
LII
LIII
Q50% R ANDOM
40
100
200
0.00 0.02 0.01
0.05 0.02
0.01
0.11 0.07 0.03
2.33 2.31 2.18
0.78 0.28 0.17
> 10 > 10 0.43
C50% R ANDOM
40
100
200
0.01 0.02 0.03
0.05 0.03 0.02
0.08 0.04 0.04
> 10 > 10 > 10
3.49 0.83 0.41
> 10 > 10 > 10
Q50% A DV
Q80% R ANDOM
100
200
40
100
200
0.07
0.05
0.00 0.00 0.01
0.00
0.00 0.04 0.03 0.02
0.01
0.01 0.11 0.06 0.03
6.68
6.23
0.62 0.53 0.45
-0.17 -0.16 0.53 0.42 0.20
> 10
-0.06 0.74 0.44 0.36
C50% A DV
C80% R ANDOM
40
100
200
40
100
200
0.08 0.08
0.07
0.03 0.05
0.05
0.05 0.02
0.01 0.05 0.03
0.02
0.03 0.04
0.02
0.11 0.06
0.03
9.62 9.05
8.68 > 10 > 10 > 10
4.45
0.43 -0.10 1.07 0.64
-0.04
> 10 > 10 > 10 > 10 0.79
0.03
40
0.07
0.04
0.05
7.16
0.44
> 10
Q80% A DV
100
200
0.04
0.03
0.02
0.00
0.03
0.01
1.79
1.50
0.07
-0.09
-0.01 -0.10
C80% A DV
40
100
200
0.09 0.09
0.08
0.07 0.03
0.02
0.09 0.05
0.02
> 10 > 10 > 10
0.96
0.30
0.14
0.45
0.18 -0.03
40
0.05
0.04
0.08
2.16
0.25
0.33
where r is given a generic squared exponential Gaussian process prior, which is fit by marginal
maximum likelihood; III. Gaussian process regression with squared exponential kernel applied to
the interventional data only and hyperparameters fitted by marginal likelihood. The idea is that
competitors I and II provide sensitivity analysis of whether our more specialized prior is adding
value. In particular, competitor II would be closer to the traditional priors used in computer-aided
experimental design [2] (but for our specialized Kobs ). Results are shown in Table 1, according to
the two assessment criteria, using E for average absolute error, and L for average log-likelihood.
Our method demonstrated robustness to varying degrees of unmeasured confounding. Compared to
Competitor I, the mean obtained without any further affine transformation already provides a competitive estimator of f (X ), but this suffers when unmeasured confounding is stronger (A DVERSARIAL
setup). Moreover, uncertainty estimates given by Competitor I tend to be overconfident. Competitor
II does not make use of our special covariance function for the correction, and tends to be particularly
weak against our method in lower interventional sample sizes. In the same line, our advantage over
Competitor III starts stronger at M = 40 and diminishes as expected when M increases. Competitor
III is particularly bad at lower signal-to-noise ratio problems, where sometimes it is overly confident
that f (X ) is zero everywhere (hence, we ignore large likelihood discrepancies in our evaluation).
This suggests that in order to learn specialized curves for particular subpopulations, where M will
invariably be small, an end-to-end model for observational and interventional data might be essential.
4.2
Case Study
We consider an adaptation of the study analyzed by [7]. Targeted at premature infants with low
birth weight, the Infant Health and Development Program (IHDP) was a study of the efficacy of
?educational and family support services and pediatric follow-up offered during the first 3 years of
life? [3]. The study originally randomized infants into those that received treatment and those that
did not. The outcome variable was an IQ test applied when infants reached 3 years. Within those
which received treatment, there was a range of number of days of treatment. That dose level was not
randomized, and again we do not have ground truth for the dose-response curve. For our assessment,
we fit a dose-response curve using Gaussian processes with Gaussian likelihood function and the
back-door adjustment (3) on available covariates. We then use the model to generate independent
synthetic ?interventional data.? Measured covariates include birth weight, sex, whether the mother
smoked during pregnancy, among other factors detailed by [7, 3]. The Supplementary Material
goes in detail about the preprocessing, including R/MATLAB scripts to generate the data. The
7
Posterior on dose?response (college)
Posterior on dose?response (high school)
Posterior on dose?response (all)
130
115
120
115
110
120
110
100
95
90
85
Outcome Y
Outcome Y
Outcome Y
105
110
100
90
80
100
95
90
85
80
80
75
70
105
75
0
50
100
150
200
250
300
Treatment X
(a)
350
400
450
70
0
50
100
150
200
250
300
Treatment X
(b)
350
400
450
70
0
50
100
150
200
250
300
350
400
450
Treatment X
(c)
Figure 3: An illustration of a problem generated from a model fitted to real data. That is, we generated
data from ?interventions? simulated from a model that was fitted to an actual study on premature
infant development [3], where the dose is the number of days that an infant is assigned to follow a
development program and the outcome is an IQ test at age 3. (a) Posterior distribution for the stratum
of infants whose mothers had up to some high school education, but no college. The red curve is the
posterior mean of our method, and the blue curve the result of Gaussian process fit with interventional
data only. (b) Posterior distributions for the infants whose mothers had (some) college education. (c)
The combined strata.
observational sample contained 347 individuals (corresponding only to those which were eligible for
treatment and had no missing outcome variable) and 21 covariates. This sample included 243 infants
whose mother attended (some) high school but not college, and 104 with at least some college.
We generated 100 synthetic interventional datasets stratified by mother?s education, (some) highschool vs. (some) college. 19 treatment levels were pre-selected (0 to 450 days, increments of 25
days). All variables were standardized to zero mean and unit standard deviation according to the
observational distribution per stratum. Two representative simulated studies are shown in Figure 3,
depicting dose-response curves which have modest evidence of non-linearity, and differ in range
per stratum2 . On average, our method improved over the fitting of a Gaussian process with squared
exponential covariance function that was given interventional data only. According to the average
normalized absolute differences, the improvement was 0.06, 0.07 and 0.08 for the high school,
college and combined data, respectively (where error was reduced in 82%, 89% and 91% of the runs,
respectively), each in which 10 interventional samples were simulated per treatment level per stratum.
5
Conclusion
We introduced a simple, principled way of combining observational and interventional measurements
and assessed its accuracy and robustness. In particular, we emphasized robustness to model misspecification and we performed sensitivity analysis to assess the importance of each individual component
of our prior, contrasted to off-the-shelf solutions that can be found in related domains [2].
We are aware that many practical problems remain. For instance, we have not discussed at all the
important issue of sample selection bias, where volunteers for an interventional study might not come
from the same p(Z) distribution as in the observational study. Worse, neither the observational nor
the interventional data might come from the population in which we want to enforce a policy learned
from the combined data. While these essential issues were ignored, our method can in principle
be combined with ways of assessing and correcting for sample selection bias [1]. Moreover, if
unmeasured confounding is too strong, one cannot expect to do well. Methods for sensitivity analysis
of confounding assumptions [13] can be integrated with our framework. A more thorough analysis of
active learning using our approach, particularly in the light of possible model misspecification, is
needed as our results in the Supplementary Material only superficially covers this aspect.
Acknowledgments
Thanks to Jennifer Hill for clarifications about the IHDP data, and Robert Gramacy for several useful discussions.
2
We do not claim that these curves represent the true dose-response curves: confounders are very likely to
exist, as the dose level was not decided at the beginning of the trial and is likely to have been changed ?on the
fly? as the infant responded. It is plausible that our covariates cannot reliably account for this feedback effect.
8
References
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data-fusion problem. Proceedings of the National Academy of Sciences, in press, 2016.
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A framework for validation of computer models. Technometrics, 49:138?154, 2007.
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5,645 | 6,108 | Blind Regression: Nonparametric Regression for
Latent Variable Models via Collaborative Filtering
Christina E. Lee
Yihua Li
Devavrat Shah
Dogyoon Song
Laboratory for Information and Decision Systems
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
{celee, liyihua, devavrat, dgsong}@mit.edu
Abstract
We introduce the framework of blind regression motivated by matrix completion
for recommendation systems: given m users, n movies, and a subset of user-movie
ratings, the goal is to predict the unobserved user-movie ratings given the data,
i.e., to complete the partially observed matrix. Following the framework of nonparametric statistics, we posit that user u and movie i have features x1 (u) and
x2 (i) respectively, and their corresponding rating y(u, i) is a noisy measurement of
f (x1 (u), x2 (i)) for some unknown function f . In contrast with classical regression,
the features x = (x1 (u), x2 (i)) are not observed, making it challenging to apply
standard regression methods to predict the unobserved ratings.
Inspired by the classical Taylor?s expansion for differentiable functions, we provide a prediction algorithm that is consistent for all Lipschitz functions. In fact,
the analysis through our framework naturally leads to a variant of collaborative
filtering, shedding insight into the widespread success of collaborative filtering in
practice. Assuming each entry is sampled independently with probability at least
max(m?1+? , n?1/2+? ) with ? > 0, we prove that the expected fraction of our
estimates with error greater than is less than ? 2 /2 plus a polynomially decaying
term, where ? 2 is the variance of the additive entry-wise noise term.
Experiments with the MovieLens and Netflix datasets suggest that our algorithm
provides principled improvements over basic collaborative filtering and is competitive with matrix factorization methods.
1
Introduction
In this paper, we provide a statistical framework for performing nonparametric regression over latent
variable models. We are initially motivated by the problem of matrix completion arising in the
context of designing recommendation systems. In the popularized setting of Netflix, there are m
users, indexed by u ? [m], and n movies, indexed by i ? [n]. Each user u has a rating for each
movie i, denoted as y(u, i). The system observes ratings for only a small fraction of user-movie
pairs. The goal is to predict ratings for the rest of the unknown user-movie pairs, i.e., to complete
the partially observed m ? n rating matrix. To be able to obtain meaningful predictions from the
partially observed matrix, it is essential to impose a structure on the data.
We assume each user u and movie i is associated to features x1 (u) ? X1 and x2 (i) ? X2 for some
compact metric spaces X1 , X2 equipped with Borel probability measures. Following the philosophy
of non-parametric statistics, we assume that there exists some function f : X1 ? X2 ? R such that
the rating of user u for movie i is given by
y(u, i) = f (x1 (u), x2 (i)) + ?ui ,
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
(1)
where ?ui is some independent bounded noise. We observe ratings for a subset of the user-movie
pairs, and the goal is to use the given data to predict f (x1 (u), x2 (i)) for all (u, i) ? [m] ? [n] whose
rating is unknown. In classical nonparametric regression, we observe input features x1 (u), x2 (i)
along with the rating y(u, i) for each datapoint, and thus we can approximate the function f well
using local approximation techniques as long as f satisfies mild regularity conditions. However, in
our setting, we do not observe the latent features x1 (u), x2 (i), but instead we only observe the indices
(u, i). Therefore, we use blind regression to refer to the challenge of performing regression with
unobserved latent input variables. This paper addresses the question, does there exist a meaningful
prediction algorithm for general nonparametric regression when the input features are unobserved?
Related Literature. Matrix completion has received enormous attention in the past decade. Matrix
factorization based approaches, such as low-rank approximation, and neighborhood based approaches,
such as collaborative filtering, have been the primary ways to address the problem. In the recent
years, there has been exciting intellectual development in the context of matrix factorization based
approaches. Since any matrix can be factorized, its entries can be described by a function f in (1) with
the form f (x1 , x2 ) = xT1 x2 , and the goal of factorization is to recover the latent features for each row
and column. [25] was one of the earlier works to suggest the use of low-rank matrix approximation,
observing that a low-rank matrix has a comparatively small number of free parameters. Subsequently,
statistically efficient approaches were suggested using optimization based estimators, proving that
matrix factorization can fill in the missing entries with sample complexity as low as rn log n, where
r is the rank of the matrix [5, 23, 11, 21, 10]. There has been an exciting line of ongoing work to
make the resulting algorithms faster and scalable [7, 17, 4, 15, 24, 20].
Many of these approaches are based on the structural assumption that the underlying matrix is
low-rank and the matrix entries are reasonably ?incoherent?. Unfortunately, the low-rank assumption
may not hold in practice. The recent work [8] makes precisely this observation, showing that a simple
non-linear, monotonic transformation of a low-rank matrix could easily produce an effectively highrank matrix, despite few free model parameters. They provide an algorithm and analysis specific to
the form of their model, which achieves sample complexity of O((mn)2/3 ). However, their algorithm
only applies to functions f which are a nonlinear monotonic transformation of the inner product of
the latent features. [6] proposes the universal singular value thresholding estimator (USVT), and
they provide an analysis under a similar model in which they assume f to be a bounded Lipschitz
function. They achieve a sample complexity, or the required fraction of measurements over the total
mn entries, which scales with the latent space dimension q according to ? m?2/(q+2) for a square
matrix, whereas we achieve a sample complexity of ?(m?1/2+? ) (which is independent of q) as long
as the latent dimension scales as o(log n).
The term collaborative filtering was coined in [9], and this technique is widely used in practice due to
its simplicity and ability to scale. There are two main paradigms in neighborhood-based collaborative
filtering: the user-user paradigm and the item-item paradigm. To recommend items to a user in the
user-user paradigm, one first looks for similar users, and then recommends items liked by those
similar users. In the item-item paradigm, in contrast, items similar to those liked by the user are
found and subsequently recommended. Much empirical evidence exists that the item-item paradigm
performs well in many cases [16, 14, 22], however the theoretical understanding of the method has
been limited. In recent works, Latent mixture models or cluster models have been introduced to
explain the collaborative filtering algorithm as well as the empirically observed superior performance
of item-item paradigms, c.f. [12, 13, 1, 2, 3]. However, these results assume a specific parametric
model, such as a mixture distribution model for preferences across users and movies. We hope that
by providing an analysis for collaborative filtering within our broader nonparametric model, we can
provide a more complete understanding of the potentials and limitations of collaborative filtering.
The algorithm that we propose in this work is inspired by local functional approximations, specifically Taylor?s approximation and classical kernel regression, which also relies on local smoothed
approximations, c.f. [18, 26]. However, since kernel regression and other similar methods use explicit
knowledge of the input features, their analysis and proof techniques do not extend to our context of
Blind regression, in which the features are latent. Although our estimator takes a similar form of
computing a convex combination of nearby datapoints weighted according to a function of the latent
distance, the analysis required is entirely different.
2
Contributions. The key contribution of our work is in providing a statistical framework for nonparametric regression over latent variable models. We refrain from any specific modeling assumptions
on f , keeping mild regularity conditions aligned with the philosophy of non-parametric statistics.
We assume that the latent features are drawn independently from an identical distribution (IID) over
bounded metric spaces; the function f is Lipschitz with respect to the latent spaces; entries are observed independently with some probability p; and the additive noise in observations is independently
distributed with zero mean and bounded support. In spite of the minimal assumptions of our model,
we provide a consistent matrix completion algorithm with finite sample error bounds. Furthermore,
as a coincidental by-product, we find that our framework provides an explanation of the practical
mystery of ?why collaborative filtering algorithms work well in practice?.
There are two conceptual parts to our algorithm. First, we derive an estimate of f (x1 (u), x2 (i)) for
an unobserved index pair (u, i) by using first order local Taylor approximation expanded around the
points corresponding to (u, i0 ), (u0 , i), and (u0 , i0 ). This leads to estimation that
y?(u, i) ? y(u0 , i) + y(u, i0 ) ? y(u0 , i0 ) ? f (x1 (u), x2 (i)),
(2)
as long as x1 (u0 ) is close to x1 (u) or x2 (i0 ) is close to x2 (i). In kernel regression, distances between
input features are used to upper bound the error of individual estimates, but since the latent features
are not observed, we need another method to determine which of these estimates are reliable.
Secondly, under mild regularity conditions, we upper bound the squared error of the estimate in (2)
by the the variance of the squared difference between commonly observed entries in rows (u, v) or
columns (i, j). We empirically estimate this quantity and use it similarly to distance in the latent
space in order to appropriately weight individual estimates to a final prediction. If we choose only the
datapoints with minimum empirical row variance, we recover user-user nearest neighbor collaborative
filtering. Inspired by kernel regression, we also propose using computing the weights according to a
Gaussian kernel applied to the minimum of the row or column sample variances.
As the main technical result, we show that the user-user nearest neighbor variant of collaborative
filtering method with our similarity metric yields a consistent estimator for any Lipschitz function
as long as we observe max(m?1+? , n?1/2+? ) fraction of the matrix with ? > 0. In the process, we
obtain finite sample error bounds, whose details are stated in Theorem 1. We compared the Gaussian
kernel variant of our algorithm to classic collaborative filtering algorithms and a matrix factorization
based approach (softImpute) on predicting user-movie ratings for the Netflix and MovieLens datasets.
Experiments suggest that our method improves over existing collaborative filtering methods, and
sometimes outperforms matrix-factorization-based approaches depending on the dataset.
2
Setup
Operating assumptions. There are m users and n movies. The rating of user u ? [m] for movie
i ? [n] is given by (1), taking the form y(u, i) = f (x1 (u), x2 (i)) + ?u,i . We make the following
assumptions.
(a) X1 and X2 are compact metric spaces endowed with metric dX1 and dX2 respectively:
dX1 (x1 , x01 ) ? BX , ? x1 , x01 ? X1 , and dX2 (x2 , x02 ) ? BX , ? x2 , x02 ? X2 .
(3)
(b) f : X1 ? X2 ? R is L?Lipschitz with respect to ?-product metric:
|f (x1 , x2 ) ? f (x01 , x02 )| ? L max {dX1 (x1 , x01 ), dX2 (x2 , x02 )} , ?x1 , x01 ? X1 , x2 , x02 ? X2 .
(c) The latent features of each user u and movie i, x1 (u) and x2 (i), are sampled independently
according to Borel probability measures PX1 and PX2 on (X1 , TX1 ) and (X2 , TX2 ), where
TX denotes the Borel ?-algebra of a metric space X .
(d) The additive noise for all data points are independent and bounded with mean zero and
variance ? 2 : for all u ? [m], i ? [n],
?u,i ? [?B? , B? ],
E[?u,i ] = 0,
Var[?u,i ] = ? 2 .
(e) Rating of each entry is revealed (observed) with probability p, independently.
3
(4)
Notation. Let random variable Mui = 1 if the rating of user u and movie i is revealed and 0
otherwise. Mui is an independent Bernoulli random variable with parameter p. Let N1 (u) denote the
set of column indices of observed entries in row u. Similarly, let N2 (i) denote the set of row indices
of observed entries in column i. That is,
N1 (u) , {i : M (u, i) = 1} and N2 (i) , {u : M (u, i) = 1}.
(5)
For rows v 6= u, N1 (u, v) , N1 (u) ? N1 (v) denotes column indices of commonly observed entries
of rows (u, v). For columns i 6= j, N2 (i, j) , N2 (i) ? N2 (j) denotes row indices of commonly
observed entries of columns (i, j). We refer to this as the overlap between two rows or columns.
3
Algorithm Intuition
Local Taylor Approximation. We propose a prediction algorithm for unknown ratings based on
insights from the classical Taylor approximation of a function. Suppose X1 ?
= X2 ?
= R, and we wish
to predict unknown rating, f (x1 (u), x2 (i)), of user u ? [m] for movie i ? [n]. Using the first order
Taylor expansion of f around (x1 (v), x2 (j)) for some u 6= v ? [m], i 6= j ? [n], it follows that
2 (j))
2 (j))
f (x1 (u), x2 (i)) ? f (x1 (v), x2 (j)) + (x1 (u) ? x1 (v)) ?f (x1 (v),x
+ (x2 (i) ? x2 (j)) ?f (x1 (v),x
.
?x1
?x2
We are not able to directly compute this expression, as we do not know the latent features, the
function f , or the partial derivatives of f . However, we can again apply Taylor expansion for
f (x1 (v), x2 (i)) and f (x1 (u), x2 (j)) around (x1 (v), x2 (j)), which results in a set of equations with
the same unknown terms. It follows from rearranging terms and substitution that
f (x1 (u), x2 (i)) ? f (x1 (v), x2 (i)) + f (x1 (u), x2 (j)) ? f (x1 (v), x2 (j)),
as long as the first order Taylor approximation is accurate. Thus if the noise term in (1) is small, we
can approximate f (x1 (u), x2 (i)) by using observed ratings y(v, j), y(u, j) and y(v, i) according to
y?(u, i) = y(u, j) + y(v, i) ? y(v, j).
(6)
Reliability of Local Estimates. We will show that the variance of the difference between two rows
or columns upper bounds the estimation error. Therefore, in order to ensure the accuracy of the above
estimate, we use empirical observations to estimate the variance of the difference between two rows
or columns, which directly relates to an error bound. By expanding (6) according to (1), the error
f (x1 (u), x2 (i)) ? y?(u, i) is equal to
(f (x1 (u), x2 (i)) ? f (x1 (v), x2 (i))) ? (f (x1 (u), x2 (j)) ? f (x1 (v), x2 (j))) ? ?vi + ?vj ? ?uj .
If we condition on x1 (u) and x1 (v),
h
i
2
E (Error) | x1 (u), x1 (v) = 2 Varx?X2 [f (x1 (u), x) ? f (x1 (v), x) | x1 (u), x1 (v)] + 3? 2 .
Similarly, if we condition on x2 (i) and x2 (j) it follows that the expected squared error is bounded by
the variance of the difference between the ratings of columns i and j. This theoretically motivates
weighting the estimates according to the variance of the difference between the rows or columns.
4
Algorithm Description
We provide the algorithm for predicting an unknown entry in position (u, i) using available data.
Given a parameter ? ? 2, define ?-overlapping neighbors of u and i respectively as
Su? (i) = {v s.t. v ? N2 (i), v 6= u, |N1 (u, v)| ? ?},
Si? (u) = {j s.t. j ? N1 (u), j 6= i, |N2 (i, j)| ? ?}.
For each v ? Su? (i), compute the empirical row variance between u and v,
X
1
2
s2uv =
((y(u, i) ? y(v, i)) ? (y(u, j) ? y(v, j))) .
2|N1 (u, v)|(|N1 (u, v)| ? 1)
i,j?N1 (u,v)
4
(7)
Similarly, compute empirical column variances between i and j, for all j ? Si? (u),
X
1
2
((y(u, i) ? y(u, j)) ? (y(v, i) ? y(v, j))) .
s2ij =
2|N2 (i, j)|(|N2 (i, j)| ? 1)
(8)
u,v?N2 (i,j)
Let B ? (u, i) denote the set of positions (v, j) such that the entries y(v, j), y(u, j) and y(v, i) are
observed, and the commonly observed ratings between (u, v) and between (i, j) are at least ?.
n
o
B ? (u, i) = (v, j) ? Su? (i) ? Si? (u) s.t. M (v, j) = 1 .
Compute the final estimate as a convex combination of estimates derived in (6) for (v, j) ? B ? (u, i),
P
(v,j)?B ? (u,i) wui (v, j) (y(u, j) + y(v, i) ? y(v, j))
P
y?(u, i) =
,
(9)
(v,j)?B ? (u,i) wui (v, j)
where the weights wui (v, j) are defined as a function of (7) and (8). We proceed to discuss a few
choices for the weight function, each of which results in a different algorithm.
User-User or Item-Item Nearest Neighbor Weights. We can evenly distribute the weights only
among entries in the nearest neighbor row, i.e., the row with minimal empirical variance,
wvj = I(v = u? ), for u? ? arg min s2uv .
?
v?Su
(i)
If we substitute these weights in (9), we recover an estimate which is asymptotically equivalent to the
mean-adjusted variant of the classical user-user nearest neighbor (collaborative filtering) algorithm,
y?(u, i) = y(u? , i) + muu? ,
where muu? is the empirical mean of the difference of ratings between rows u and u? . For any u, v,
X
1
muv =
(y(u, j) ? y(v, j)).
|N1 (u, v)|
j?N1 (u,v)
Equivalently, we can evenly distribute the weights among entries in the nearest neighbor columns, i.e.,
the column with minimal empirical variance, recovering the classical mean-adjusted item-item nearest
neighbor collaborative filtering algorithm. Theorem 1 proves that this simple algorithm produces
a consistent estimator, and we provide the finite sample error analysis. Due to the similarities, our
analysis also directly implies the proof of correctness and consistency for the classic user-user and
item-item collaborative filtering method.
User-Item Gaussian Kernel Weights. Inspired by kernel regression, we introduce a variant of
the algorithm which computes the weights according to a Gaussian kernel function with bandwith
parameter ?, substituting in the minimum row or column sample variance as a proxy for the distance,
wvj = exp(?? min{s2uv , s2ij }).
When ? = ?, the estimate only depends on the basic estimates whose row or column has the
minimum sample variance. When ? = 0, the algorithm equally averages all basic estimates. We
applied this variant of our algorithm to both movie recommendation and image inpainting data, which
show that our algorithm improves upon user-user and item-item classical collaborative filtering.
Connections to Cosine Similarity Weights. In our algorithm, we determine reliability of estimates
as a function of the sample variance, which is equivalent to the squared distance of the meanadjusted values. In classical collaborative filtering, cosine similarity is commonly used, which can be
approximated as a different choice of the weight kernel over the squared difference.
5
Main Theorem
Let E ? [m] ? [n] denote the set of user-movie pairs for which the algorithm predicts a rating. For
? > 0, the overall ?-risk of the algorithm is the fraction of estimates whose error is larger than ?,
1 X
Risk? =
I(|f (x1 (u), x2 (i)) ? y?(u, i)| > ?).
(10)
|E|
(u,i)?E
5
In Theorem 1, we upper bound the expected ?-Risk, proving that the user-user nearest neighbor
estimator is consistent, i.e., in the presence of no noise, estimates converge to the true values as m, n
go to infinity. We may assume m ? n without loss of generality.
Theorem 1. For a fixed ? > 0, as long as p ? max{m?1+? , n?1/2+? } (where ? > 0), for any
? = ?(n?2?/3 ), the user-user nearest-neighbor variant of our method with ? = np2 /2 achieves
3 ? 21/3 ? 2 ?
3? + ? 2
1
1 2?
?
?
3
3
1+
E[Risk? ] ?
n
+ O exp ? Cm + m exp ? 2 n
.
?2
?
4
5B
p? 1
where B = 2(LBX + B? ), and C = h
L2 ? 6 for h(r) := inf x0 ?X1 Px?PX1 (dX1 (x, x0 ) ? r).
For a generic ?, we can also provide precise error bounds of a similar form, with modified rates of
convergence. Choosing ? to grow with np2 ensures that as n goes to infinity, the required overlap
between rows also goes to infinity, thus the empirical mean and variance computed in the algorithm
converge precisely to the true mean and variance. The parameter ? in Theorem 1 is introduced purely
for the purpose of analysis, and is not used within the implementation of the the algorithm.
The function h behaves as a lower bound of the cumulative distribution function of PX1 , and it always
exists under our assumptions that X1 is compact. It is used to ensure that for any u ? [m], with high
probability, there exists another row v ? Su? (i) such that dX1 (x1 (u), x1 (v)) is small, implying by
the Lipschitz condition that we can use the values of row v to approximate the values of row u well.
For example, if PX1 is a uniform distribution over a unit cube in q dimensional Euclidean space,
then h(r) = min(1, r)q , and our error bound becomes meaningful for n ? (L2 /?)q/2? . On the other
hand, if PX1 is supported over finitely many points, then h(r) = minx?supp(PX1 ) PX1 (x) is a positive
constant, and the role of the latent dimension becomes irrelevant. Intuitively, the ?geometry? of PX1
through h near 0 determines the impact of the latent space dimension on the sample complexity, and
our results hold as long as the latent dimension q = o(log n).
6
Proof Sketch
For any evaluation set of unobserved entries E, the expectation of ?-risk is
1 X
P(|f (x1 (u), x2 (i)) ? y?(u, i)| > ?) = P(|f (x1 (u), x2 (i)) ? y?(u, i)| > ?),
E[Risk? ] =
|E|
(u,i)?E
because the indexing of the entries are exchangeable and identically distributed. To bound the
expected risk, it is sufficient to provide a tail bound for the probability of the error. For any fixed
a, b ? X1 , and random variable x ? PX2 , we denote the mean and variance of the difference
f (a, x) ? f (b, x) by
?ab , Ex [f (a, x) ? f (b, x)] = E[muv |x1 (u) = a, x1 (v) = b],
2
?ab
, Varx [f (a, x) ? f (b, x)] = E[s2uv |x1 (u) = a, x1 (v) = b] ? 2? 2 ,
which we point out is also equivalent to the expectation of the empirical means and variances
computed by the algorithm when we condition on the latent representations of the users. The
computation of y?(u, i) involves two steps: first the algorithm determines the neighboring row with the
minimum sample variance, u? = arg minv?Su? (i) s2uv , and then it computes the estimate by adjusting
according to the empirical mean, y?(u, i) := y(u? , i) + muu? .
The proof involves three key steps, each stated within a lemma. Lemma 1 proves that with high
probability the observations are dense enough such that there is sufficient number of rows with
overlap of entries larger than ?, i.e., the number of the candidate rows, |Su? (i)|, concentrates around
(m ? 1)p. This relies on concentration of Binomial random variables via Chernoff?s bound.
Lemma 1. Given p > 0, 2 ? ? ? np2 /2 and ? > 0, for any (u, i) ? [m] ? [n],
?2 (m ? 1)p
np2
?
+ (m ? 1) exp ?
.
P |Su (i)| ?
/ (1 ? ?)(m ? 1)p ? 2 exp ?
3
8
Lemma 2 proves that since the latent features are sampled iid from a bounded metric space, for any
index pair (u, i), there exists a ?good? neighboring row v ? Su? (i), whose ?x2 1 (u)x1 (v) is small.
6
Lemma 2. Consider u ? [n] and set S ? [n] \ {u}. Then for any ? > 0,
r |S|
?
,
P min ?x2 1 (u)x1 (v) > ? ? 1 ? h
v?S
L2
where h(r) := inf x0 ?X1 Px?PX1 (dX1 (x, x0 ) ? r).
Subsequently, conditioned on the event that |Su? (i)| ? (m ? 1)p, Lemmas 3 and 4 prove that the
sample mean and sample variance of the differences between two rows concentrate around the
true mean and true variance with high probability. This involves using the Lipschitz and bounded
assumptions on f and X1 , as well as the Bernstein and Maurer-Pontil inequalities.
Lemma 3. Given u, v ? [m], i ? [n] and ? ? 2, for any ? > 0,
P ?x1 (u)x1 (v) ? muv > ? | v ? Su? (i) ? exp ?
3??2
6B 2 + 2B?
,
where recall that B = 2(LBX + B? ).
Lemma 4. Given u ? [m], i ? [n], and ? ? 2, for any ? > 0,
2
?
2
2
P suv ? (?x1 (u)x1 (v) + 2? ) > ? v ? Su (i) ? 2 exp ?
??2
2 + 4? 2 + ?)
4B 2 (2LBX
,
where recall that B = 2(LBX + B? ).
Given that there exists a neighbor v ? Su? (i) whose true variance ?x2 1 (u)x1 (v) is small, and conditioned
on the event that all the sample variances concentrate around the true variance, it follows that the true
variance between u and its nearest neighbor u? is small with high probability. Finally, conditioned
on the event that |Su? (i)| ? (m ? 1)p and the true variance between the target row and the nearest
neighbor row is small, we provide a bound on the tail probability of the estimation error by using
Chevyshev inequalities. The only term in the error probability which does not decay to zero is the
error from Chebyshev?s inequality, which dominates the final expression, leading to the final result.
7
Experiments
We evaluated the performance of our algorithm to predict user-movie ratings on the MovieLens 1M
and Netflix datasets. For the implementation of our method, we used user-item Gaussian kernel
weights for the final estimator. We chose overlap parameter ? = 2 to ensure the algorithm is able
to compute an estimate for all missing entries. When ? is larger, the algorithm enforces rows (or
columns) to have more commonly rated movies (or users). Although this increases the reliability of
the estimates, it also reduces the fraction of entries for which the estimate is defined. We optimized
the ? bandwidth parameter of the Gaussian kernel by evaluating the method with multiple values for
? and choosing the value which minimizes the error.
We compared our method with user-user collaborative filtering, item-item collaborative filtering,
and softImpute from [20]. We chose the classic mean-adjusted collaborative filtering method, in
which the weights are proportional to the cosine similarity of pairs of users or items (i.e. movies).
SoftImpute is a matrix-factorization-based method which iteratively replaces missing elements in the
matrix with those obtained from a soft-thresholded SVD.
For both MovieLens and Netflix data sets, the ratings are integers from 1 to 5. From each dataset, we
generated 100 smaller user-movie rating matrices, in which we randomly subsampled 2000 users and
2000 movies. For each rating matrix, we randomly select and withhold a percentage of the known
ratings for the test set, while the remaining portion of the data set is revealed to the algorithm for
computing the estimates. After the algorithm computes its predictions for unrevealed movie-user
pairs, we evaluate the Root Mean Squared Error (RMSE) of the predictions compared with the
withheld test set, where RMSE is defined as the square root of the mean of squared prediction error
over the evaluation set. Figure 1 plots the RMSE of our method along with classic collaborative
filtering and softImpute evaluated against 10%, 30%, 50%, and 70% withheld test sets. The RMSE is
averaged over 100 subsampled rating matrices, and 95% confidence intervals are provided.
7
Figure 1: Performance of algorithms on Netflix and MovieLens datasets with 95% confidence interval.
? values used by our algorithm are 2.8 (10%), 2.3 (30%), 1.7 (50%), 1 (70%) for MovieLens, and 1.8
(10%), 1.7 (30%), 1.6 (50%), 1.5 (70%) for Netflix.
Figure 1 suggests that our algorithm achieves a systematic improvement over classical user-user
and item-item collaborative filtering. SoftImpute performs the worst on the MovieLens dataset,
but it performs the best on the Netflix dataset. This behavior could be due to different underlying
assumptions of low rank for matrix factorization methods as opposed to Lipschitz for collaborative
filtering methods, which could lead to dataset dependent performance outcomes.
8
Discussion
We introduced a generic framework of blind regression, i.e., nonparametric regression over latent
variable models. We allow the model to be any Lipschitz function f over any bounded feature space
X1 , X2 , while imposing the limitation that the input features are latent. This is applicable to a wide
variety of problems, including recommendation systems, but also includes social network analysis,
community detection, crowdsourcing, and product demand prediction. Many parametric models (e.g.
low rank assumptions) can be framed as a specific case of our model.
Despite the generality and limited assumptions of our model, we present a simple similarity based
estimator, and we provide theoretical guarantees bounding its error within the noise level ? 2 . The
analysis provides theoretical grounds for the popularity of similarity based methods. To the best of
our knowledge, this is the first provable guarantee on the performance of neighbor-based collaborative
filtering within a fully nonparametric model. Our algorithm and analysis follows from local Taylor
approximation, along with an observation that the sample variance between rows or columns is a good
indicator of ?closeness?, or the similarity of their function values. The algorithm essentially estimates
the local metric information between the latent features from observed data, and then performs local
smoothing in a similar manner as classical kernel regression.
Due to the local nature of our algorithm, our sample complexity does not depend on the latent
dimension, whereas Chatterjee?s USVT estimator [6] requires sampling almost every entry when
the latent dimension is large. This difference is due to the fact that Chatterjee?s result stems from
showing that a Lipschitz function can be approximated by a piecewise constant function, which upper
bound the rank of the target matrix. This discretization results in a large penalty with regards to the
dimension of the latent space. Since our method follows from local approximations, we only require
sufficent sampling such that locally there are enough close neighbor points.
The connection of our framework to regression implies many natural future directions. We can
extend model (1) to multivariate functions f , which translates to the problem of higher order tensor
completion. Variations of the algorithm and analysis that we provide for matrix completion can
extend to tensor completion, due to the flexible and generic assumptions of our model. It would also
be useful to extend the results to capture general noise models, sparser sampling regimes, or mixed
models with both parametric and nonparametric or both latent and observed variables.
Acknowledgements: This work is supported in parts by ARO under MURI award 133668-5079809,
by NSF under grants CMMI-1462158 and CMMI-1634259, and additionally by a Samsung Scholarship, Siebel Scholarship, NSF Graduate Fellowship, and Claude E. Shannon Research Assistantship.
8
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9
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5,646 | 6,109 | SEBOOST ? Boosting Stochastic Learning Using
Subspace Optimization Techniques
Elad Richardson*1 Rom Herskovitz*1 Boris Ginsburg2 Michael Zibulevsky1
1
Technion, Israel Institute of Technology 2 Nvidia INC
{eladrich,mzib}@cs.technion.ac.il {fornoch,boris.ginsburg}@gmail.com
Abstract
We present SEBOOST, a technique for boosting the performance of existing stochastic optimization methods. SEBOOST applies a secondary optimization process
in the subspace spanned by the last steps and descent directions. The method
was inspired by the SESOP optimization method, and has been adapted for the
stochastic learning. It can be applied on top of any existing optimization method
with no need to tweak the internal algorithm. We show that the method is able
to boost the performance of different algorithms, and make them more robust to
changes in their hyper-parameters. As the boosting steps of SEBOOST are applied
between large sets of descent steps, the additional subspace optimization hardly
increases the overall computational burden. We introduce hyper-parameters that
control the balance between the baseline method and the secondary optimization
process. The method was evaluated on several deep learning tasks, demonstrating
significant improvement in performance. Video presentation is given in [15]
1
Introduction
Stochastic Gradient Descent (SGD) based optimization methods are widely used for many different
learning problems. Given some objective function that we want to optimize, a vanilla gradient
descent method would simply take some fixed step in the direction of the current gradient. In
many learning problems the objective, or loss, function is averaged over the set of given training
examples. In that scenario calculating the loss over the entire training set would be expensive, and is
therefore approximated on a small batch, resulting in a stochastic algorithm that requires relatively
few calculations per step. The simplicity and efficiency of SGD algorithms have made them a
standard choice for many learning tasks, and specifically for deep learning [9, 6, 5, 10] . Although
the vanilla SGD has no memory of previous steps, they are usually utilized in some way, for example
using momentum [13]. Alternatively, the AdaGrad method uses the previous gradients in order to
normalize each component in the new gradient adaptively [3], while the ADAM method uses them to
estimate an adaptive moment [8]. In this work we utilize the knowledge of previous steps in spirit
of the Sequential Subspace Optimization (SESOP) framework [11]. The nature of SESOP allows
it to be easily merged with existing algorithms. Several such extensions were introduced over the
years to different fields, such as PCD-SESOP and SSF-SESOP, showing state-of-the-art results in
their matching fields [4, 17, 16].
The core idea of our method is as follows. At every outer iteration we first perform several steps
of a baseline stochastic optimization algorithm which are then summed up as an inner cumulative
stochastic step. Afterwards, we minimize the objective function over the affine subspace spanned
by the cumulative stochastic step, several previous outer steps and optional other directions. The
subspace optimization boosts the performance of the baseline algorithm, therefore our method is
called the Sequential Subspace Optimization Boosting method (SEBOOST).
*Equal contribution
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
2
The algorithm
As our algorithm tries to find the balance between SGD and SESOP, we start by a brief review of the
original algorithms, and then move to the SEBOOST algorithm.
2.1
Vanilla SGD
In many different large-scale optimization problems, applying complex optimization methods is not
practical. Thus, popular optimization methods for those problems are usually based on a stochastic
estimation of the gradient. Let minx?Rn f (x) be some minimization problem, and let g(x) be the
gradient of f (x). The general stochastic approach applies the following optimization rule
xk+1 = xk ? ?g ? (xk )
where xi is the result of the ith iteration, ? is the learning rate and g ? (xk ) is an approximation of
g(xk ) obtained using only a small subset (mini-batch) of the training data. These stochastic descent
methods have proved themselves in many different problems, specifically in the context of deep
learning algorithms, providing a combination of simplicity and speed. Notice that the vanilla SGD
algorithm has no memory of previous iterations. Different optimization methods which are based on
SGD usually utilize the previous iterations in order to make a more informed descent process.
2.2
Vanilla SESOP
The SEquential Subspace OPtimization Method [11, 16] is an optimization technique used for large
scale optimization problems. The core idea of SESOP is to perform the optimization of the objective
function in the subspace spanned by the current gradient direction and a set of directions obtained
from the previous optimization steps. Following the notations in Section 2.1, a subspace structure for
SESOP is usually defined based on the following directions:
1. Gradients: Current gradient and [optionally] older ones {g (xi ) : i = k, k ? 1, . . . k ? s1 }
2. Previous directions: {pi = xi ? xi?1 : i = k, k ? 1, . . . k ? s2 }
In the SESOP formulation the current gradient and the last step are mandatory and any other set can
be used to enrich the subspace. From a theoretical point of view, one can enrich the subspace by two
Nemirovsky directions: A weighted average of the previous gradients and the direction to the starting
point. This will provide optimal worst case complexity of the method (see also [12].) Denoting Pk as
the set of directions at iteration k, the SESOP algorithm would solve the minimization problem
?k = arg min f (xk + Pk ?)
?
xk+1 = xk + Pk ?k
Thus SESOP reduces the optimization problem to the subspace spanned by Pk at each iteration. This
means that instead of solving an optimization problem in Rn the dimensionality of the subspace is
governed by the size of Pk and can be controlled.
2.3
The SEBOOST algorithm
As explained in Section 2.1, when dealing with large-scale optimization problems, stochastic learning
methods are usually better fitted to the task then many more involved optimization methods. However,
when applied correctly those methods can still be used to boost the optimization process and achieve
faster convergence rates. We propose to start with some SGD algorithm as a baseline, and then apply
a SESOP-like optimization method over it in an alternating manner. The subspace for the SESOP
algorithm arises from the descent directions of the baseline, utilizing the previous iterations.
A description of the method is given in Algorithm 1. Note that the subset of the training data used for
the secondary optimization in step 7 isn?t necessarily the same as that of the baseline in step 2, as will
be shown in Section 3. Also, note that in step 8 the last added direction is changed, that is done in
order to incorporate the step performed by the secondary optimization into the subspace.
2
Algorithm 1 The SEBOOST algorithm
1: for k = 1, . . . do
2:
Perform ` steps of baseline stochastic optimization method to get from xk0 to xk`
3:
Add the direction of the cumulative step xk` ? xk0 to the optimization subspace P
4:
if Subspace dimension exceeded the limit: dim(P ) > M then
5:
Remove oldest direction from the optimization subspace P
6:
end if
7:
Perform optimization over subspace P to get from xk` to xk+1
0
8:
Change the last added direction to p = xk+1
? xk0
0
9: end for
It is clear that SEBOOST offers an attractive balance between the baseline stochastic steps and the
more costly subspace optimizations. Firstly, as the number ` of stochastic steps grows, the effect of
subspace optimization over the result subsides, where taking ` ? ? reduces the algorithm back to
the baseline method. Secondly, the dimensionality of the subspace optimization problem is governed
by the size of P and can be reduced to as few parameters as desired. Notice also that as SEBOOST is
added on top of baseline stochastic optimization method, it does not require any internal changes to
be made to the original algorithm. Thus, it can be applied on top of any such method with minimal
implementation cost, while potentially boosting the base method.
2.4
Enriching the subspace
Although the core elements of our optimization subspace are the directions of last M ? 1 external
steps and the new stochastic cumulative direction, many more elements can be added to enrich the
subspace.
Anchor points As only the last (M ? 1) directions are saved in our subspace, the subspace has
knowledge only of recent history of the optimization process. The subspace might benefit from
directions dependent on preceding directions as well. For example, one could think of the overall
descent achieved by the algorithm p = xk0 ? x00 as a possible direction, or the descent over the second
k/2
half of the optimization process p = xk0 ? x0 .
We formulate this idea by defining anchor points. Anchors points are locations chosen throughout
the descent process which we fix and update only rarely. For each anchor point ai the direction
p = xk0 ? ai is added to the subspace. Different techniques can be chosen for setting and changing
the anchors. In our formulation each point is associated with a parameter ri which describes the
number of boosting steps between each update of the point. After every ri steps the corresponding
point ai is initialized back to the current x. That way we can control the number of iterations before
an anchor point becomes irrelevant and is initialized again. Algorithm 2 shows how the anchor points
can be added to Algorithm 1, by incorporating it before step 7.
Current gradient As in the SESOP formulation, the gradient at the current point can be added to
the subspace.
Momentum Similarly to the idea of momentum in SGD methods one can save a weighted average
of the previous updates and add it to the optimization subspace. Denoting the current momentum as
mk and the last step as p = xk+1
? xk0 , the momentum is updated as mk+1 = ??mk + p, where ? is
0
some hyper-parameter, as in regular SGD momentum.
Algorithm 2 Controlling anchors in SEBOOST
1: for i = 1, . . . , #anchors do
2:
if ri %k == 0 then
3:
Change the anchor ai to xk`
4:
end if
5:
Normalize the direction p = xk` ? ai and add it to the subspace
6: end for
3
Simple Regression
-2.5
SGD
SEBOOST-SGD
NAG
SEBOOST-NAG
ADAGRAD
SEBOOST-ADAGRAD
-3
-3.5
logarithmic test error
logarithmic test error
-3
Simple Regression
-2.5
SGD
SEBOOST-SGD
NAG
SEBOOST-NAG
ADAGRAD
SEBOOST-ADAGRAD
-4
-4.5
-5
-3.5
-4
-4.5
-5
-5.5
-5.5
0
5
10
15
20
25
0
train time in seconds
50
100
150
200
250
300
350
400
number of epochs
Figure 1: Results for experiment 3.1. The baseline parameters was set as lrSGD = 0.5, lrN AG = 0.1,
lrAdaGrad = 0.05, which provided good convergence. SEBOOST?s parameters were fixed at M = 50
and ` = 100 with 50 function evaluations for the secondary optimization.
3
Experiments
Following the recent rise of interest in deep learning tasks we focus our evaluation on different neural
networks problems. We start with a small, yet challenging, regression problem and then proceed
to the known problems of the MNIST autoencoder and CIFAR-10 classifier. For each problem we
compare the results of baseline stochastic methods with our boosted variants, showing that SEBOOST
can give significant improvement over the base method. Note that the purpose of our work is not to
directly compete with existing methods, but rather to show that SEBOOST can improve each learning
method compared to its? original variant, while preserving the original qualities of these algorithms.
The chosen baselines were SGD with momentum, Nesterov?s Accelerated Gradient (NAG) [13] and
AdaGrad [3]. The Conjugate Gradient (CG) [7] was used for the subspace optimization.
Our algorithm was implemented and evaluated using the Torch7 framework [1], and is publicly
available 1 . The main hyper-parameters that were altered during the experiments were:
? lrmethod - The learning rate of a baseline method.
? M - Maximal number of old directions.
? ` - Number of baseline steps between each subspace optimization.
For all experiments the weight decay was set at 0.0001 and the momentum was fixed at 0.9 for SGD
and NAG. Unless stated otherwise, the number of function evaluations for CG was set at 20. The
baseline method used a mini-batch of size 100, while the subspace optimization was applied with
a mini-batch of size 1000. Note that subspace optimization is applied over a significantly larger
batch. That is because while a ?bad? stochastic step will be canceled by the next ones, a single
secondary step has a bigger effect on the overall result and therefore requires better approximation of
the gradient. As the boosting step is applied only between large sets of the base method, the added
cost does not hinder the algorithm.
For each experiment a different architecture will be defined. We will use the notation a ?L b to
denote a classic linear layer with a inputs and b outputs followed by a non-linear Tanh function.
Notice that when presenting our results we show two different graphs. The right one always shows
the error as a function of the number of passes of the baseline algorithms over the data (i.e. epochs),
while the left one shows the error as a function of the actual processor time, taking into account the
additional work required by the boosted algorithms.
3.1
Simple regression
We will start by evaluating our method on a small regression problem. The dataset in question is
a set of 20,000 values simulating some continuous function f : R6 ? R. The dataset was divided
1
https://github.com/eladrich/seboost
4
into 18,000 training examples and 2,000 test examples. The problem was solved using a tiny neural
network with the architecture 6 ?L 12 ?L 8 ?L 4 ?L 1. Although the network size is very small
the resulting optimization problem remains challenging and gives clear indication of SEBOOST?s
behavior. Figure 1 shows the optimization process for the different methods. In all examples the
boosted variant converged faster. Note that the different variants of SEBOOST behave differently,
governed by the corresponding baseline.
3.2
MNIST autoencoder
One of the classic neural network formulation is that of an autoencoder, a network that tries to learn
efficient representation for a given set of data. An autoencoder is usually composed of two parts,
the encoder which takes the input and produces the compact representation and the decoder which
takes the representation and tries to reconstruct the original input. In our experiment the MNIST
dataset was used, with 60,000 training images of size 28 ? 28 and 10,000 test images. The encoder
was defined as three layer network with an architecture of form 784 ?L 200 ?L 100 ?L 64, with
a matching decoder 64 ?L 100 ?L 200 ?L 784.
Figure 3 shows the optimization process for the autoencoder problem. A similar trend can be seen
to that of experiment 3.1, SEBOOST is able to significantly improve SGD and NAG and shows
some improvement over AdaGrad, although not as noticeable. A nice byproduct of working with
an autoencoding problem is that one can visualize the quality of the reconstructions as a function of
the iterations. Figure 2 shows the change in reconstructions quality for SGD and SESOP-SGD, and
shows that the boosting achieved is significant in terms on the actual results.
Original
#10
#30
#100
#200
Original
#10
#30
#100
#200
Figure 2: Reconstruction Results. The first row shows results of the SGD algorithm, while the second
row shows results of SESOP-SGD. The last row gives the number of passes over the data.
3.3
CIFAR-10 classifier
For classification purposes a standard benchmark is the CIFAR-10 dataset. The dataset is composed
of 60,000 images of size 32 ? 32 from 10 different classes, where each class has 6,000 different
images. 50,000 images are used for training and 10,000 for testing. In order to check SEBOOST?s
ability to deal with large and modern networks the ResNet [6] architecture, winner of the ILSVRC
2015 classification task, is used.
MNIST Autoencoder
MNIST Autoencoder
0.3
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SGD
SEBOOST-SGD
NAG
SEBOOST-NAG
ADAGRAD
SEBOOST-ADAGRAD
0.25
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Figure 3: Results for experiment 3.2. The baseline parameters was set at lrSGD = 0.1, lrN AG = 0.01,
lrAdaGrad = 0.01. SEBOOST?s parameters were fixed at M = 10 and ` = 200.
5
CIFAR-10 Classification
60
CIFAR-10 Classification
SGD
SEBOOST-SGD
NAG
SEBOOST-NAG
ADAGRAD
SEBOOST-ADAGRAD
60
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test error (%)
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train time in seconds
Figure 4: Results for experiment 3.3. All baselines were set with lr = 0.1 and a mini-batch of size
128. SEBOOST?s parameters were fixed at M = 10 and ` = 391, with a mini-batch of size 1024.
Figure 4 shows the optimization process and the achieved accuracy for ResNet of depth 32. Note that
we did not manually tweak the learning rate as was done in the original paper. While AdaGrad is not
boosted for this experiment, SGD and NAG achieve significant boosting and reach a better minimum.
The boosting step was applied only once every epoch, applying too frequent boosting steps resulted
in a less stable optimization and higher minima, while applying infrequent steps also lead to higher
minima. Experiment 3.4 shows similar results for MNIST and discusses them.
3.4
Understanding the hyper-parameters
SEBOOST introduces two hyper-parameters: ` the number of baseline steps between each subspace
optimization and M the number of old directions to use. The purpose of the following two experiments
is to measure the effect of those parameters on the achieved result and to give some intuition as to
their meaning. All experiments are based on the MNIST autoencoder problem defined in Section 3.2.
First, let us consider the parameter `, which controls the balance between the baseline SGD algorithm
and the more involved optimization process. Taking small values of ` results in more steps of the
secondary optimization process, however each direction in the subspace is then composed of fewer
steps from the stochastic algorithm, making it less stable. Furthermore, recalling that our secondary
optimization is more costly than regular optimization steps, applying it too often would hinder the
algorithm?s performance. On the other hand, taking large values of ` weakens the effect of SEBOOST
over the baseline algorithm.
Figure 5a shows how ` affects the optimization process. One can see that applying the subspace
optimization too frequently increases the algorithm?s runtime and reaches an higher minimum than
the other variants, as expected. Although taking a large value of ` reaches a better minimum, taking
a value which is too large slows the algorithm. We can see that for this experiment taking ` = 200
balances correctly the trade-offs.
MNIST Autoencoder - Baseline Steps
0.3
NAG
SEBOOST-NAG-5
SEBOOST-NAG-10
SEBOOST-NAG-20
SEBOOST-NAG-50
0.25
MSE test error
MSE test error
0.25
MNIST Autoencoder - History Size
0.3
NAG
SEBOOST-NAG-50
SEBOOST-NAG-200
SEBOOST-NAG-800
0.2
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40
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0.05
120
0
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20
40
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(a)
(b)
Figure 5: Experiment 3.4, analyzing different changes in SEBOOST?s hyper-parameters
6
MNIST Autoencoder - Learning Rate
0.3
NAG
SEBOOST-NAG-basic
SEBOOST-NAG-momentum
SEBOOST-NAG-anchors
SESOP-NAG-both
0.25
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MSE test error
MNIST Autoencoder - Extra Directions
0.3
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SEBOOST-NAG-0.05
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SEBOOST-NAG-0.01
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Figure 6: Experiment 3.5, analyzing different changes in SEBOOST?s subspace
Let us now consider the effect of M , which governs the size of the subspace in which the secondary
optimization is applied. Although taking large values of M allows us to hold more directions and
apply the optimization in a larger subspace it also makes the optimization process more involved.
Figure 5b shows how M affects the optimization process. Interestingly, the lower M is, the faster the
algorithm starts descending. However, larger M values tend to reach better minima. For M = 20 the
algorithm reaches the same minimum as M = 50, but starts the descent process faster, making it a
good choice for this experiment.
To conclude, the introduced hyper-parameters M and ` affect the overall boosting effect achieved by
SEBOOST. Both parameters incorporate different trade-offs of the optimization problem and should
be considered when using the algorithm. Our own experiments show that a good initialization would
be to set ` so the algorithm runs about once or twice per epoch, and to set M between 10 to 20.
3.5
Investigating the subspace
One of the key components of SEBOOST is the structure of the subspace in which the optimization
is applied. The purpose of the following two experiments is to see how changes in the baseline
algorithm, or the addition of more directions, affect the algorithm. All experiments are based on the
MNIST autoencoder problem defined in Section 3.2.
In the basic formulation of SEBOOST the subspace is composed only from the directions of the
baseline algorithm. In Section 3.2 we saw how choosing different baselines affect the algorithm.
Another experiment of interest is to see how our algorithm is influenced by changes in the hyperparameters of the baseline algorithm. Figure 6a shows the effect of the learning rate over the baseline
algorithms and their boosted variants. It can be seen that the change in the original baseline affects our
algorithm, however the impact is noticeably smaller, showing that the algorithm has some robustness
to the original learning rate.
In Section 2.4 a set of additional directions which can be added to the subspace were defined, these
directions can possibly enrich the subspace and improve the optimization process. Figure 6b shows the
influence of those directions on the overall result. In SEBOOST-anchors a set of anchor points were
added with the r values of 500, 250, 100, 50 and 20. In SEBOOST-momnetum a momentum vector
with ? = 0.9 was used. It can be seen that using the proposed anchor directions can significantly
boost the algorithm. The momentum direction is less useful, giving a small boost on its own and
actually slightly hinders the performance when used in conjunction with the anchor directions.
4
Conclusion
In this paper we presented SEBOOST, a technique for boosting stochastic learning algorithms via a
secondary optimization process. The secondary optimization is applied in the subspace spanned by
the preceding descent steps, which can be further extended with additional directions. We evaluated
SEBOOST on different deep learning tasks, showing the achieved results of our methods compared to
their original baselines. We believe that the flexibility of SEBOOST could make it useful for different
learning tasks. One can easily change the frequency of the secondary optimization step, ranging from
7
frequent and more risky steps, to the more stable one step per epoch. Changing the baseline algorithm
and the structure of the subspace allows us to further alter SEBOOST?s behavior.
Although this is not the focus of our work, an interesting research direction for SEBOOST is that of
parallel computing. Similarly to [2, 14], one can look at a framework composed of a single master
and a set of workers, where each worker optimizes a local model and the master saves a global set of
parameters which is based on the workers. Inspired by SEBOOST, one can take the descent directions
from each of the workers and apply a subspace optimization in the spanned subspace, allowing the
master to take a more efficient step based on information from each of its workers.
Another interesting direction for future work is the investigation of pruning techniques. In our work,
when the subspace if fully occupied the oldest direction is simply removed. One might consider more
advanced pruning techniques, such as eliminating the direction which contributed the least for the
secondary optimization step, or even randomly removing one of the subspace directions. A good
pruning technique can potentially have a significant effect on the overall result. These two ideas will
be further researched in future work. Overall, we believe SEBOOST provides a promising balance
between popular stochastic descent methods and more involved optimization techniques.
Acknowledgements
The research leading to these results has received funding from the European Research Council under
European Unions Seventh Framework Program, ERC Grant agreement no. 320649 and was supported
by the Intel Collaborative Research Institute for Computational Intelligence (ICRI-CI).
References
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5,647 | 611 | Learning to See Where and What:
Training a Net to Make Saccades and Recognize
Handwritten Characters
Gale Martin, Mosfeq Rashid, David Chapman, and James Pittman
MCC, 3500 Balcones Center Drive, Austin, Texas 78759
ABSTRACT
This paper describes an approach to integrated segmentation and
recognition of hand-printed characters. The approach, called Saccade,
integrates ballistic and corrective saccades (eye movements) with
character recognition. A single backpropagation net is trained to make
a classification decision on a character centered in its input window, as
well as to estimate the distance of the current and next character from the
center of the input window. The net learns to accurately estimate these
distances regardless of variations in character width, spacing between
characters, writing style and other factors. During testing, the system
uses the net~xtracted classification and distance information, along
with a set of jumping rules, to jump from character to character.
The ability to read rests on multiple foundation skills. In learning how to read, people learn
how to recognize individual characters centered in the visual field. They also learn how
to move their eyes along a line of text, sequentially centering the visual field on successive
characters. We believe that the key to developing optical character recognition (OCR) systems that can mimic human reading capabilities, is to develop systems that can learn these
and other skills in an integrated fashion. In this paper, we demonstrate that a backpropagation net can learn to naVigate along a line of handwritten characters, as well as to recognize
the characters centered in its visual field. The system, called Saccade, extends the current
state of the art in OCR technology by using a single classifier to accurately and efficiently
locate and recognize characters, regardless of whether they touch each other or are separate. The Saccade system was described briefly at the last NIPS conference (Martin & Rashid, 1992). In this paper, we describe it mcx-e fully and report on results demonstrating
its accuracy and efficiency in recognizing handwritten digits.
The Saccade system takes a cue from the ballistic and corrective saccades (eye movements)
of natural vision systems. Natural saccades make it possible to efficiently move from one
informative area to another by jumping. The eye typically initiates a ballistic saccade to
441
442
Martin, Rashid, Chapman, and Pittman
move the center of focus to the general area of interest. followed. if necessary. by one or
more ccnective saccades for fme-grained position corrections. Recognition processes are
applied only at these multiple fixation points.
We have copioo some of these aspects in the artificial Saccade system by training a neural
netwcn to know a1>oot the locations of characters in its input window. as well as to know
about the identity of the character centered in its input window. During run-time, the Saccade system accesses this information computed by the net for successive input windows,
along with a set of simple jumping rules, to yield an OCR system that jumps from character
to character, classifying each character in a sequence.
1 TRAINING DETAILS
As shown in Figure 1, the Saccade system has a wide input window, large enough to contain
~i~ii3a~i~ }
No Centered Character
Current character
Distance to current character I
Distance to next character
I I f .. W ..,
I I I I
4 Part
Output
Vector
Backprop Net
Figure 1.
The Saccade system uses an enlarged input window and a 4-part
output vector.
several characters. Prior to training, each field image of a line of characters is labeled with
the hcrizontal center position of each character in the field, as well as with the category of
each character. During training, the input window slides hcriwntally across a field of characters, and at each position, the contents of the input window are paired with a four-part
target ootput vector, the values of which are computed from the labeled information. The
target values answer the following four questions about the contents of the input window:
1.
2.
3.
4.
Is a character centered in the input window?
What character is closest to the center of the window?
How far off~nter (horizontally) is the centennost character?
How far is the next character to the right from the center of the window?
The first node in the output vector represents the no-centered-character state. It's target
value is set high (e.g .? 1.0) when the center of the input window falls between characters,
Training a Net to Make Saccades and Recognize Handwritten Characters
and set low (e.g .? 0.0) when the center of the input window falls 00 the center of a character.
When the net is trained. the value of the rw-cenlered~haracter node indicates whether the
input window is centered over a character. oc whether a corrective saccade is needed to better center the character.
The second part of the output vector contains a node foc each character category. When
the center of the input window falls on a character. the target value for its cocresponding
node is set high; otherwise it is set low. When the net is trained. the values in these nodes
are used to classify the centered character. The target values for bdh the rw-centeredcharacter and the character-category nodes are defmed continuously across the hcrizontal
dimension as trapewidal functions. such that there are plateaus surrounding the off and on
positions. with linearly increasing and decreasing values connecting the plateaus.
The third and fourth components of the output vector represent distance values. each encoded in a distributed fashion across multiple nodes. using localized receptive fields
(Moody & Darken. 1988). The frrst of these two parts represents the distance by which the
character closest to the center of the window is off~nter. The target value can be positive.
indicating that the center of the window is to the left of the center of character. or it can be
negative. indicating that the center of the window has passed over the character. to the right
of it's center. When trained, the value of the current-character-distance set of nodes is accessed to determine the magnitude of a ccxrective saccade. to make a fme-grained position
adjustment.
The fourth component represems the distance from the center of the window to the center
of the next character to the right. The target value can only be positive. When trained, the
value of this set of nodes is accessed to detennine the magnitude of a ballistic saccade, to
jump to the next character to the right.
It is imponant to note that for bdh distance components, the maximum target value can
not exceed half the window width. The net is never trained to make a distance judgment
that extends beyond its field of view, since it is not given any infonnation about what exists
outside of it's input window. For example. when the center of the next character to the right
is positioned outside of the current input window. the distance value is set to the maximum
value of half the window width. Since the distance values vary with different characters.
different writers. and of course, at different positions with respect to a character. the net is
forced to learn to use the visual characteristics particular to each window to estimate the
distance values. In other words. the net does NOT simply learn average values for each
of the two distance metrics. Moreover. as the results will show, the trained net does not
seem to use simple density histogram cues to estimate the distance values. It is able to reliably estimate the distance values even when characters overlap. and hence would appear
as a single clump in a density histogram.
2 RUN-TIME SACCADE RULES
During run-time, the labeled values are. of course, not available. The system uses the computed values in the character classification and distance components of the output vector,
and some heuristics, to navigate horizontally along a character field. jumping from one
character to the next. and occasionally making a corrective saccade to improve its ability
to classify a character. When the net recognizes a character, it executes a ballistic saccade
443
444
Martin, Rashid, Chapman, and Pittman
to the next character. obtaining the distance to jump by reading the next-character-disUlnCe component of the output vector. When this actioo fails to center a character. as indicated by a low value in the no-centered-character output node, the system executes a corrective saccade to better center the character. It obtains the distance and direction to jump
by reading the current-character-distance component of the output vector. Multiple corrective saccades can be executed.
3 TFSTING ON NIST HANDWRITTEN DIGIT FIELDS
We tested the perfonnance of the system on a set of hand-printed digits collected and distributed by the National Institute of Standards and Technology (NIST). This is a database
containing 273,000 samples of handwritten numerals. Each of2100 Census workers filled
in a fonn with 33 fields, 28 fields of which only contain handwritten digits. The scanning
resolution of the samples was 300 pixels/inch. The neural net was trained on about 80,000
characters from 20,000 fields, written by 800 different individuals. The fields varied in
length from 2 characters per field to 6 characters per field. The horiwntal positions of each
of the characters in these training-data fields were extracted by a person. The test data contained about 20,000 digits from 5,000 fields, written by a different group of 200 individuals.
The test set was chosen to be this large because use of smaller test sets (e.g., 5,000 digits,
1250 fields) yielded significant between-set variations in reported accuracy. Each field
image was preprocessed to remove the box around the field of characters, and any surrounding white space. Each field image was size normalized, with respect to the vertical
axis. to a height of 20 pixels. Aspect ratio was maintained. An input pattern generator was
then passed over the field to create input windows fa training the net. The input window
size was 36 pixels wide and 20 pixels high. The input window scanned the field at 2-pixel
increments during training. Subsequent experiments have shown that training can be
speeded up considerably by training on the character centers and at random points between
the character centers. without causing decreased accuracy.
The backpropagation netwak architecture is described more fully in Martin & Rashid
(1992). It has 2 hidden layers with local, shared connections in the first hidden layer. and
local connections in the second hidden layer. Shared weights are not used in the second
hidden layer because early experiments showed that this retards learning, presumably because extending the ~ition invariance to second-hidden-Iayer nodes inhibits the net in
learning the position specific information regarding what is centered in its input window.
The learning rate of the net was initially set at .05. and then successivel y lowered as training
reached an asymptcxe. The momentum term was set at .9 throughout training. All nodes
in the net used logistic activation functions.
Table I reports on the test results in terms of field-based reject rates. for 1% and .5% percent of the fields rejected. The error rates are field-based in the sense that if the net misclassifies one character in the field. the entire field is considered as mis-classified. Error
rates pertain to the fields remaining after rejection. Rejections are based on placing a
threshold for the acceptable distance between the highest and next highest running activation total. In this way. by varying the threshold, the error rate can be traded off against the
percentage of rejections. In addition. recognized fields were also rejected if the number
of recognized digits differed from the expected number of cligits.
Training a Net to Make Saccades and Recognize Handwritten Characters
Thble 1: Field - Based Error Rates For Saccade System
Field
Field Size
Field
Error Rate
Reject Rate
2-dIalt.
10K
0.5111
6.a
9.J'11
3-di<.
10K
0.5111
12."..
19.6'11
4-dJalt.
1.1111
0.5111
19.5111
350K
5-dJalu
1.1'"
0.5111
23.2111
28.J'11
6-di<.
1.1111
0.5111
26.K
35.0111
Figure 2 presents some of the fields of connected characters that the system correctly recognized. Conventional OCR systems typically fail on connected characters because they
employ an independent character segmentation stage. in which the character is isolated
from its surround using features. such as intervening white spaces. This character segmentation stage typically fails when characters are coonected. The Saccade system goes
beyond conventional OCR systems by integrating segmentation and recognition. and
thereby is able to recognize touching characters.
The Saccade system is also efficient in the sense that it typically jumps from one character
to the next without making a corrective saccades. Corrective saccades tend to be mere likely when characters are touching. In addition. there is almost always a corrective saccade
for the first character in the field. since the system starts at the beginning of the field. with
?1
41" ,. ~.
G9?
~1
I o~71
~ ~J ~~~I
? 1~9___q_30__
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Figure 2. Examples of connected and broken characters that the Saccade
system correctly recognizes.
445
446
Martin, Rashid, Chapman, and Pittman
no knowledge of the location of the character. For fields containing two digits, the average
number of passes of the net on a small test set was about 2 saccades per character. For field
containing five digits, the average number of saccades per character was 1.3.
4 COMPARISONS WITH OTHER SYSTEMS
The Saccade system is an extension of a related integrated segmentation and recognition
system we reported on at last years NIPS conference (Martin & Rashid, 1992). That system
employed an exhaustive scan technique, rather than saccades, to navigate along a line of
text. Essentially, the net was convolved hociwntally across a field image at a scan increment of 2 pixels. The net architecture was very similar to that of the Saccade system, except
that it did not have the two distance components in the output vector. The accuracy rates
of the two systems are essentially equivalent. However, the exhaustive scan version was
considerably less efficient, requiring a forward pass of the netwock at every 2-pixel incremental scan position. On average it required about 5.5 forward passes per character, rather
than the 1.3 focward passes per character required by the Saccade system.
Over the past two years, an approach similar to the exhaustive scan method has been advanced by a number of researchers (Keeler & Rumelhart, 1992; Matan. Burges. Le CuD.
& Denker. 1992). This approacb also involves convolving a network across a field
image. but uses a time-delay-neural-net (1DNN). or completely local. shared
weight. architecture. a smaller input window, and no explicit position labeling of characters. The IDNN approacb has algorithmic advantages over the exhaustive scan version described in the previous paragraph. because the completely shared -weight architecture enables the number of forward passes of the net to be reduced considerably.
5 CONCLUSIONS AND FUTURE WORK
As stated at the beginning of this paper, we believe that the key to developing optical character recognition (OCR) systems that can mimic human reading capabilities is to develop
systems that can learn the multiple foundation skills underlying human reading. This paper
has repocted some progress in this regard. We have demonstrated that a relatively simple
back propagation network can integrate its learning of position and category information.
thereby enabling efficient navigatioo along a field of text through ballistic and corrective
saccades, and accurate recognition of touching or broken characters.
There is however, a long way to go before we can claim a system with capabilities similar
to human reading. The present StJccade system only moves horizontally, in one dimension.
Human reading operates in two-<iimensions. and in a sense. it operates in three-dimensions
because it automatically operates across different scales. Human vision also employs auter
matic contrast adjusunent; the Saccade system does not. Human vision has a wider field
of view and employs a foveal transform. such that objects centered in the field of vision
are represented at a higher resolution than objects in the periphery. This effectively expands the field of vision beyond what would be estimated simply by the size of the receptive
area on the retina. As a resUlt. saccades enable very effective means of scanning a large
visual area. The present artificial Saccade system has only a small field of vision. and no
foveal transform. so it's saccades must necessarily be limited in size. The present system
is also only oriented toward recognizing a single character centered in its input window at
Training a Net to Make Saccades and Recognize Handwritten Characters
a time. Human reading typically only makes one or two saccades per woo1. Finally, human
reading capabilities clearly integrate recognition processes with higher-level processes, to
enable the redundancies of natural language to constrain the recognition decisions.
References
Keeler, J, & Rumelhart, D. E. (1992) A self-aganizing integrated segmentation and
recognition neural network.. In Moody, J.E., Hansoo, S.1., and Lippmann, R.P., (eds.)
A.dvances in Neural Information Processing Systems 4. San Mateo. CA: Mocgan
Kaufmann Publishers.
Malan, 0., Burges, J. C., Le Cun, Y., and Denker. J. S. (1992) Multi-Digit Recognition
Using a Space Displacement Neural Netwm. In Moody. J.E .? Hanson. S.1 .? and Lippmann.
R.P.? (eds.) Advances in Neural Injormation Processing Systems 4. San Mateo.
CA: Mocgan Kaufmann Publishers. 488-495.
Martin. O. L. & Rashid. M. (1992) Recognizing overlapping hand-printed characters by
centered-OOject integrated segmentation and recognitioo. In Moody, J.E .? Hanson, S.1.,
and Lippmann, R.P.. (eds.) Advances in Neural Injormation Processing Systems 4. San
Mateo. CA: Morgan Kaufmann Publishers.
Moody. J. & Darken. C. (1988) Learning with localized receptive fields. Technical Report
Yaleu/DCS/RR-649.
447
PART V
STOCHASTIC
LEARNING AND
ANALYSIS
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5,648 | 6,110 | Unsupervised Domain Adaptation with Residual
Transfer Networks
Mingsheng Long? , Han Zhu? , Jianmin Wang? , and Michael I. Jordan]
?
KLiss, MOE; TNList; School of Software, Tsinghua University, China
]
University of California, Berkeley, Berkeley, USA
{mingsheng,jimwang}@tsinghua.edu.cn, zhuhan10@gmail.com, jordan@berkeley.edu
Abstract
The recent success of deep neural networks relies on massive amounts of labeled
data. For a target task where labeled data is unavailable, domain adaptation can
transfer a learner from a different source domain. In this paper, we propose a new
approach to domain adaptation in deep networks that can jointly learn adaptive
classifiers and transferable features from labeled data in the source domain and
unlabeled data in the target domain. We relax a shared-classifier assumption made
by previous methods and assume that the source classifier and target classifier
differ by a residual function. We enable classifier adaptation by plugging several
layers into deep network to explicitly learn the residual function with reference
to the target classifier. We fuse features of multiple layers with tensor product
and embed them into reproducing kernel Hilbert spaces to match distributions for
feature adaptation. The adaptation can be achieved in most feed-forward models by
extending them with new residual layers and loss functions, which can be trained
efficiently via back-propagation. Empirical evidence shows that the new approach
outperforms state of the art methods on standard domain adaptation benchmarks.
1
Introduction
Deep networks have significantly improved the state of the art for a wide variety of machine-learning
problems and applications. Unfortunately, these impressive gains in performance come only when
massive amounts of labeled data are available for supervised training. Since manual labeling of
sufficient training data for diverse application domains on-the-fly is often prohibitive, for problems
short of labeled data, there is strong incentive to establishing effective algorithms to reduce the
labeling consumption, typically by leveraging off-the-shelf labeled data from a different but related
source domain. However, this learning paradigm suffers from the shift in data distributions across
different domains, which poses a major obstacle in adapting predictive models for the target task [1].
Domain adaptation [1] is machine learning under the shift between training and test distributions. A
rich line of approaches to domain adaptation aim to bridge the source and target domains by learning
domain-invariant feature representations without using target labels, so that the classifier learned from
the source domain can be applied to the target domain. Recent studies have shown that deep networks
can learn more transferable features for domain adaptation [2, 3], by disentangling explanatory factors
of variations behind domains. Latest advances have been achieved by embedding domain adaptation
in the pipeline of deep feature learning which can extract domain-invariant representations [4, 5, 6, 7].
The previous deep domain adaptation methods work under the assumption that the source classifier can
be directly transferred to the target domain upon the learned domain-invariant feature representations.
This assumption is rather strong as in practical applications, it is often infeasible to check whether
the source classifier and target classifier can be shared or not. Hence we focus in this paper on a more
general, and safe, domain adaptation scenario in which the source classifier and target classifier differ
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
by a small perturbation function. The goal of this paper is to simultaneously learn adaptive classifiers
and transferable features from labeled data in the source domain and unlabeled data in the target
domain by embedding the adaptations of both classifiers and features in a unified deep architecture.
Motivated by the state of the art deep residual learning [8], winner of the ImageNet ILSVRC 2015
challenge, we propose a new Residual Transfer Network (RTN) approach to domain adaptation in
deep networks which can simultaneously learn adaptive classifiers and transferable features. We relax
the shared-classifier assumption made by previous methods and assume that the source and target
classifiers differ by a small residual function. We enable classifier adaptation by plugging several
layers into deep networks to explicitly learn the residual function with reference to the target classifier.
In this way, the source classifier and target classifier can be bridged tightly in the back-propagation
procedure. The target classifier is tailored to the target data better by exploiting the low-density
separation criterion. We fuse features of multiple layers with tensor product and embed them into
reproducing kernel Hilbert spaces to match distributions for feature adaptation. The adaptation can be
achieved in most feed-forward models by extending them with new residual layers and loss functions,
and can be trained efficiently using standard back-propagation. Extensive evidence suggests that the
RTN approach outperforms several state of art methods on standard domain adaptation benchmarks.
2
Related Work
Domain adaptation [1] builds models that can bridge different domains or tasks, which mitigates
the burden of manual labeling for machine learning [9, 10, 11, 12], computer vision [13, 14, 15]
and natural language processing [16]. The main technical problem of domain adaptation is that the
domain discrepancy in probability distributions of different domains should be formally reduced.
Deep neural networks can learn abstract representations that disentangle different explanatory factors
of variations behind data samples [17] and manifest invariant factors underlying different populations
that transfer well from original tasks to similar novel tasks [3]. Thus deep neural networks have been
explored for domain adaptation [18, 19, 15], multimodal and multi-task learning [16, 20], where
significant performance gains have been witnessed relative to prior shallow transfer learning methods.
However, recent advances show that deep networks can learn abstract feature representations that
can only reduce, but not remove, the cross-domain discrepancy [18, 4]. Dataset shift has posed a
bottleneck to the transferability of deep features, resulting in statistically unbounded risk for target
tasks [21, 22]. Some recent work addresses the aforementioned problem by deep domain adaptation,
which bridges the two worlds of deep learning and domain adaptation [4, 5, 6, 7]. They extend deep
convolutional networks (CNNs) to domain adaptation either by adding one or multiple adaptation
layers through which the mean embeddings of distributions are matched [4, 5], or by adding a fully
connected subnetwork as a domain discriminator whilst the deep features are learned to confuse
the domain discriminator in a domain-adversarial training paradigm [6, 7]. While performance was
significantly improved, these state of the art methods may be restricted by the assumption that under
the learned domain-invariant feature representations, the source classifier can be directly transferred
to the target domain. In particular, this assumption may not hold when the source classifier and target
classifier cannot be shared. As theoretically studied in [22], when the combined error of the ideal
joint hypothesis is large, then there is no single classifier that performs well on both source and target
domains, so we cannot find a good target classifier by directly transferring from the source domain.
This work is primarily motivated by He et al. [8], the winner of the ImageNet ILSVRC 2015 challenge.
They present deep residual learning to ease the training of very deep networks (hundreds of layers),
termed residual nets. The residual nets explicitly reformulate the layers as learning residual functions
?F (x) with reference to the layer inputs x, instead of directly learning the unreferenced functions
F (x) = ?F (x) + x. The method focuses on standard deep learning in which training data and test
data are drawn from identical distributions, hence it cannot be directly applied to domain adaptation.
In this paper, we propose to bridge the source classifier fS (x) and target classifier fT (x) by the
residual layers such that the classifier mismatch across domains can be explicitly modeled by the
residual functions ?F (x) in a deep learning architecture. Although the idea of adapting source
classifier to target domain by adding a perturbation function has been studied by [23, 24, 25], these
methods require target labeled data to learn the perturbation function, which cannot be applied to
unsupervised domain adaptation, the focus of this study. Another distinction is that their perturbation
function is defined in the input space x, while the input to our residual function is the target classifier
fT (x), which can capture the connection between the source and target classifiers more effectively.
2
3
Residual Transfer Networks
s
In unsupervised domain adaptation problem, we are given a source domain Ds = {(xsi , yis )}ni=1
of ns
t
labeled examples and a target domain Dt = {xtj }nj=1
of nt unlabeled examples. The source domain
and target domain are sampled from different probability distributions p and q respectively, and p 6= q.
The goal of this paper is to design a deep neural network that enables learning of transfer classifiers
y = fs (x) and y = ft (x) to close the source-target discrepancy, such that the expected target risk
Rt (ft ) = E(x,y)?q [ft (x) 6= y] can be bounded by leveraging the source domain supervised data.
The challenge of unsupervised domain adaptation arises in that the target domain has no labeled
data, while the source classifier fs trained on source domain Ds cannot be directly applied to the
target domain Dt due to the distribution discrepancy p(x, y) 6= q(x, y). The distribution discrepancy
may give rise to mismatches in both features and classifiers, i.e. p(x) 6= q(x) and fs (x) 6= ft (x).
Both mismatches should be fixed by joint adaptation of features and classifiers to enable effective
domain adaptation. Classifier adaptation is more difficult than feature adaptation because it is directly
related to the labels but the target domain is fully unlabeled. Note that the state of the art deep feature
adaptation methods [5, 6, 7] generally assume classifiers can be shared on adapted deep features. This
paper assumes fs 6= ft and presents an end-to-end deep learning framework for classifier adaptation.
Deep networks [17] can learn distributed, compositional, and abstract representations for natural data
such as image and text. This paper addresses unsupervised domain adaptation within deep networks
for jointly learning transferable features and adaptive classifiers. We extend deep convolutional
networks (CNNs), i.e. AlexNet [26], to novel residual transfer networks (RTNs) as shown in Figure 1.
Denote by fs (x) the source classifier, and the empirical error of CNN on source domain data Ds is
min
fs
ns
1 X
L (fs (xsi ) , yis ),
ns i=1
(1)
where L(?, ?) is the cross-entropy loss function. Based on the quantification study of feature transferability in deep convolutional networks [3], convolutional layers can learn generic features that are
transferable across domains [3]. Hence we opt to fine-tune, instead of directly adapt, the features of
convolutional layers when transferring pre-trained deep models from source domain to target domain.
3.1
Feature Adaptation
Deep features learned by CNNs can disentangle explanatory factors of variations behind data distributions to boost knowledge transfer [19, 17]. However, the latest literature findings reveal that deep
features can reduce, but not remove, the cross-domain distribution discrepancy [3], which motivates
the state of the art deep feature adaptation methods [5, 6, 7]. Deep features in standard CNNs must
eventually transition from general to specific along the network, and the transferability of features and
classifiers will decrease when the cross-domain discrepancy increases [3]. In other words, the shifts
in the data distributions linger even after multilayer feature abstractions. In this paper, we perform
feature adaptation by matching the feature distributions of multiple layers ` ? L across domains. We
reduce feature dimensions by adding a bottleneck layer f cb on top of the last feature layer of CNNs,
and then fine-tune CNNs on source labeled examples such that the feature distributions of the source
and target are made similar under new feature representations in multiple layers L = {f cb, f cc}, as
shown in Figure 1. To adapt multiple feature layers effectively, we propose the tensor product between
features of multiple layers to perform lossless multi-layer feature fusion, i.e. zsi , ?`?L xs`
i and
ztj , ?`?L xt`
.
We
then
perform
feature
adaptation
by
minimizing
the
Maximum
Mean
Discrepancy
j
(MMD) [27] between source and target domains using the fusion features (dubbed tensor MMD) as
ns X
ns
ns X
nt
nt X
nt
X
X
X
k zsi , zsj
k zsi , ztj
k zti , ztj
min DL (Ds , Dt ) =
+
?2
,
(2)
fs ,ft
n2s
n2t
ns nt
i=1 j=1
i=1 j=1
i=1 j=1
0
2
where the characteristic kernel k(z, z0 ) = e?kvec(z)?vec(z )k /b is the Gaussian kernel function
defined on the vectorization of tensors z and z0 with bandwidth parameter b. Different from DAN [5]
that adapts multiple feature layers using multiple MMD penalties, this paper adapts multiple feature
layers by first fusing them and then adapting the fused features. The advantage of our method against
DAN [5] is that our method can capture full interactions across multilayer features and facilitate
easier model selection, while DAN [5] needs |L| independent MMD penalties for adapting |L| layers.
3
Xs
fcb
AlexNet,
ResNet?
Xt
fcb
?
fcc
fc2
fT ( x )
MMD
?
fc1
?f ( x )
fS ( x ) = fT ( x ) + ?f ( x )
fcc
?f
S
(x)
Ys
fcb
Xs
?
fcc
Xs
Zs
weight layer
?F ( x )
x
MMD
fT ( x )
entropy minimization
Yt
fcb
Xt
Zt
?
weight layer
fcc
Xt
F (x) =
+
?F ( x ) + x
Figure 1: (left) Residual Transfer Network (RTN) for domain adaptation, based on well-established
architectures. Due to dataset shift, (1) the last-layer features are tailored to domain-specific structures
that are not safely transferable, hence we add a bottleneck layer f cb that is adapted jointly with the
classifier layer f cc by the tensor MMD module; (2) Supervised classifiers are not safely transferable,
hence we bridge them by the residual layers f c1?f c2 so that fS (x) = fT (x)+?f (x). (middle) The
tensor MMD module for multi-layer feature adaptation. (right) The building block for deep residual
learning; Instead of using the residual block to model feature mappings, we use it to bridge the source
classifier fS (x) and target classifier fT (x) with x , fT (x), F (x) , fS (x), and ?F (x) , ?f (x).
3.2
Classifier Adaptation
As feature adaptation cannot remove the mismatch in classification models, we further perform
classifier adaptation to learn transfer classifiers that make domain adaptation more effective. Although
the source classifier fs (x) and target classifier ft (x) are different, fs (x) 6= ft (x), they should be
related to ensure the feasibility of domain adaptation. It is reasonable to assume that fs (x) and ft (x)
differ only by a small perturbation function ?f (x). Prior work on classifier adaptation [23, 24, 25]
assumes that ft (x) = fs (x) + ?f (x), where the perturbation ?f (x) is a function of input feature x.
However, these methods require target labeled data to learn the perturbation function, which cannot
be applied to unsupervised domain adaptation where target domain has no labeled data. How to
bridge fs (x) and ft (x) in a framework is a key challenge of unsupervised domain adaptation. We
postulate that the perturbation function ?f (x) can be learned jointly from the source labeled data
and target unlabeled data, given that the source classifier and target classifier are properly connected.
To enable classifier adaptation, consider fitting F (x) as an original mapping by a few stacked layers
(convolutional or fully connected layers) in Figure 1 (right), where x denotes the inputs to the first of
these layers [8]. If one hypothesizes that multiple nonlinear layers can asymptotically approximate
complicated functions, then it is equivalent to hypothesize that they can asymptotically approximate
the residual functions, i.e. F (x) ? x. Rather than expecting stacked layers to approximate F (x), one
explicitly lets these layers approximate a residual function ?F (x) , F (x) ? x, with the original
function being ?F (x) + x. The operation ?F (x) + x is performed by a shortcut connection and an
element-wise addition, while the residual function is parameterized by residual layers within each
residual block. Although both forms are able to asymptotically approximate the desired functions, the
ease of learning is different. In reality, it is unlikely that identity mappings are optimal, but it should
be easier to find the perturbations with reference to an identity mapping, than to learn the function
as new. The residual learning is the key to the successful training of very deep networks. The deep
residual network (ResNet) framework [8] bridges the inputs and outputs of the residual layers by the
shortcut connection (identity mapping) such that F (x) = ?F (x) + x, which eases the learning of
residual function ?F (x) (similar to the perturbation function across the source and target classifiers).
Based on this key observation, we extend the CNN architectures (Figure 1, left) by plugging in the
residual block (Figure 1, right). We reformulate the residual block to bridge the source classifier
fS (x) and target classifier fT (x) by letting x , fT (x), F (x) , fS (x), and ?F (x) , ?f (x). Note
that fS (x) is the outputs of the element-wise addition operator and fT (x) is the outputs of the targetclassifier layer f cc, both before softmax activation ?(?), fs (x) , ? (fS (x)) , ft (x) , ? (fT (x)).
We can connect the source classifier and target classifier (before activation) by the residual block as
fS (x) = fT (x) + ?f (x) ,
(3)
where we use functions fS and fT before softmax for residual block to ensure that the final classifiers
fs and ft will output probabilities. Residual layers f c1?f c2 are fully-connected layers with c ? c
units, where c is the number of classes. We set the source classifier fS as the outputs of the residual
block to make it better trainable from the source-labeled data by deep residual learning [8]. In other
words, if we set fT as the outputs of the residual block, then we may be unable to learn it successfully
as we do not have target labeled data and thus standard back-propagation will not work. Deep residual
learning [8] ensures to output valid classifiers |?f (x)| |fT (x)| ? |fS (x)|, and more importantly,
4
makes the perturbation function ?f (x) dependent on both the target classifier fT (x) (due to the
functional dependency) as well as the source classifier fS (x) (due to the back-propagation pipeline).
Although we successfully cast the classifier adaptation into the residual learning framework while
the residual learning framework tends to make the target classifier ft (x) not deviate much from the
source classifier fs (x), we still cannot guarantee that ft (x) will fit the target-specific structures well.
To address this problem, we further exploit the entropy minimization principle [28] for refining the
classifier adaptation, which encourages the low-density separation between classes by minimizing
the entropy of class-conditional distribution fjt (xti ) = p(yit = j|xti ; ft ) on target domain data Dt as
min
ft
nt
1 X
H ft xti ,
nt i=1
(4)
t
t
where
Pc H(?) is the entropy function of class-conditional distribution ft (xi ) defined as H (ft (xi )) =
? j=1 fjt (xti ) log fjt (xti ), c is the number of classes, and fjt (xti ) is the probability of predicting
point xti to class j. By minimizing entropy penalty (4), the target classifier ft (x) is made directly
accessible to target-unlabeled data and will amend itself to pass through the target low-density regions.
3.3
Residual Transfer Network
To enable effective unsupervised domain adaptation, we propose Residual Transfer Network (RTN),
which jointly learns transferable features and adaptive classifiers by integrating deep feature learning
(1), feature adaptation (2), and classifier adaptation (3)?(4) in an end-to-end deep learning framework,
min
fS =fT +?f
+
ns
1 X
L (fs (xsi ) , yis )
ns i=1
nt
? X
H ft xti
nt i=1
(5)
+ ? DL (Ds , Dt ),
where ? and ? are the tradeoff parameters for the tensor MMD penalty (2) and entropy penalty (4)
respectively. The proposed RTN model (5) is enabled to learn both transferable features and adaptive
classifiers. As classifier adaptation proposed in this paper and feature adaptation studied in [5, 6] are
tailored to adapt different layers of deep networks, they can complement each other to establish better
performance. Since training deep CNNs requires a large amount of labeled data that is prohibitive for
many domain adaptation applications, we start with the CNN models pre-trained on ImageNet 2012
data and fine-tune it as [5]. The training of RTN mainly follows standard back-propagation, with the
residual transfer layers for classifier adaptation as [8]. Note that, the optimization of tensor MMD
penalty (2) requires carefully-designed algorithm to establish linear-time training, as detailed in [5].
We also adopt bilinear pooling [29] to reduce the dimensions of fusion features in tensor MMD (2).
4
Experiments
We evaluate the residual transfer network against state of the art transfer learning and deep learning
methods. Codes and datasets will be available at https://github.com/thuml/transfer-caffe.
4.1
Setup
Office-31 [13] is a benchmark for domain adaptation, comprising 4,110 images in 31 classes collected
from three distinct domains: Amazon (A), which contains images downloaded from amazon.com,
Webcam (W) and DSLR (D), which contain images taken by web camera and digital SLR camera
with different photographical settings, respectively. To enable unbiased evaluation, we evaluate all
methods on all six transfer tasks A ? W, D ? W, W ? D, A ? D, D ? A and W ? A as in [5, 7].
Office-Caltech [14] is built by selecting the 10 common categories shared by Office-31 and Caltech256 (C), and is widely used by previous methods [14, 30]. We can build 12 transfer tasks: A ? W,
D ? W, W ? D, A ? D, D ? A, W ? A, A ? C, W ? C, D ? C, C ? A, C ? W, and C
? D. While Office-31 has more categories and is more difficult for domain adaptation algorithms,
5
Office-Caltech provides more transfer tasks to enable an unbiased look at dataset bias [31]. We adopt
DeCAF7 [2] features for shallow transfer methods and original images for deep adaptation methods.
We compare with both conventional and the state of the art transfer learning and deep learning methods:
Transfer Component Analysis (TCA) [9], Geodesic Flow Kernel (GFK) [14], Deep Convolutional
Neural Network (AlexNet [26]), Deep Domain Confusion (DDC) [4], Deep Adaptation Network
(DAN) [5], and Reverse Gradient (RevGrad) [6]. TCA is a conventional transfer learning method
based on MMD-regularized Kernel PCA. GFK is a manifold learning method that interpolates across
an infinite number of intermediate subspaces to bridge domains. DDC is the first method that
maximizes domain invariance by adding to AlexNet an adaptation layer using linear-kernel MMD
[27]. DAN learns more transferable features by embedding deep features of multiple task-specific
layers to reproducing kernel Hilbert spaces (RKHSs) and matching different distributions optimally
using multi-kernel MMD. RevGrad improves domain adaptation by making the source and target
domains indistinguishable for a discriminative domain classifier via an adversarial training paradigm.
To go deeper with the efficacy of classifier adaptation (residual transfer block) and feature adaptation
(tensor MMD module), we perform ablation study by evaluating several variants of RTN: (1) RTN
(mmd), which adds the tensor MMD module to AlexNet; (2) RTN (mmd+ent), which further adds
the entropy penalty to AlexNet; (3) RTN (mmd+ent+res), which further adds the residual module to
AlexNet. Note that RTN (mmd) improves DAN [5] by replacing the multiple MMD penalties in DAN
by a single tensor MMD penalty in RTN (mmd), which facilitates much easier parameter selection.
We follow standard protocols and use all labeled source data and all unlabeled target data for domain
adaptation [5]. We compare average classification accuracy of each transfer task using three random
experiments. For MMD-based methods (TCA, DDC, DAN, and RTN), we use Gaussian kernel with
bandwidth b set to median pairwise squared distances on training data, i.e. median heuristic [27]. As
there are no target labeled data in unsupervised domain adaptation, model selection proves difficult.
For all methods, we perform cross-valuation on labeled source data to select candidate parameters,
then conduct validation on transfer task A ? W by requiring one labeled example per category from
target domain W as the validation set, and fix the selected parameters throughout all transfer tasks.
We implement all deep methods based on the Caffe deep-learning framework, and fine-tune from
Caffe-provided models of AlexNet [26] pre-trained on ImageNet. For RTN, We fine-tune all the
feature layers, train bottleneck layer f cb, classifier layer f cc and residual layers f c1?f c2, all through
standard back-propagation. Since these new layers are trained from scratch, we set their learning rate
to be 10 times that of the other layers. We use mini-batch stochastic gradient descent (SGD) with
momentum of 0.9 and the learning rate annealing strategy implemented in RevGrad [6]: the learning
rate is not selected through a grid search due to high computational cost?it is adjusted during SGD
?0
using the following formula: ?p = (1+?p)
? , where p is the training progress linearly changing from
0 to 1, ?0 = 0.01, ? = 10 and ? = 0.75, which is optimized for low error on the source domain.
As RTN can work stably across different transfer tasks, the MMD penalty parameter ? and entropy
penalty ? are first selected on A ? W and then fixed as ? = 0.3, ? = 0.3 for all other transfer tasks.
4.2
Results
The classification accuracy results on the six transfer tasks of Office-31 are shown in Table 1, and the
results on the twelve transfer tasks of Office-Caltech are shown in Table 2. The RTN model based
on AlexNet (Figure 1) outperforms all comparison methods on most transfer tasks. In particular,
RTN substantially improves the accuracy on hard transfer tasks, e.g. A ? W and C ? W, where the
source and target domains are very different, and achieves comparable accuracy on easy transfer tasks,
D ? W and W ? D, where source and target domains are similar [13]. These results suggest that
RTN is able to learn more adaptive classifiers and transferable features for safer domain adaptation.
From the results, we can make interesting observations. (1) Standard deep-learning methods (AlexNet)
perform comparably with traditional shallow transfer-learning methods with deep DeCAF7 features
as input (TCA and GFK). The only difference between these two sets of methods is that AlexNet can
take the advantage of supervised fine-tuning on the source-labeled data, while TCA and GFK can
take benefits of their domain adaptation procedures. This result confirms the current practice that
supervised fine-tuning is important for transferring source classifier to target domain [19], and sustains
the recent discovery that deep neural networks learn abstract feature representation, which can only
reduce, but not remove, the cross-domain discrepancy [3]. This reveals that the two worlds of deep
6
Table 1: Accuracy on Office-31 dataset using standard protocol [5] for unsupervised adaptation.
Method
TCA [9]
GFK [14]
AlexNet [26]
DDC [4]
DAN [5]
RevGrad [6]
RTN (mmd)
RTN (mmd+ent)
RTN (mmd+ent+res)
A?W
59.0?0.0
58.4?0.0
60.6?0.4
61.0?0.5
68.5?0.3
73.0?0.6
70.0?0.4
71.2?0.3
73.3?0.3
D?W
90.2?0.0
93.6?0.0
95.4?0.2
95.0?0.3
96.0?0.1
96.4?0.4
96.1?0.3
96.4?0.2
96.8?0.2
W?D
88.2?0.0
91.0?0.0
99.0?0.1
98.5?0.3
99.0?0.1
99.2?0.3
99.2?0.3
99.2?0.1
99.6?0.1
A?D
57.8?0.0
58.6?0.0
64.2?0.3
64.9?0.4
66.8?0.2
67.6?0.4
69.8?0.2
71.0?0.2
D?A
51.6?0.0
52.4?0.0
45.5?0.5
47.2?0.5
50.0?0.4
49.8?0.4
50.2?0.3
50.5?0.3
W?A
47.9?0.0
46.1?0.0
48.3?0.5
49.4?0.6
49.8?0.3
50.0?0.3
50.7?0.2
51.0?0.1
Avg
65.8
66.7
68.8
69.3
71.7
72.1
72.9
73.7
Table 2: Accuracy on Office-Caltech dataset using standard protocol [5] for unsupervised adaptation.
Method
A?W D?W W?D A?D D?A W?A A?C W?C D?C C?A C?W C?D Avg
TCA [9]
84.4 96.9 99.4 82.8 90.4 85.6 81.2 75.5 79.6 92.1 88.1 87.9 87.0
GFK [14]
89.5 97.0 98.1 86.0 89.8 88.5 76.2 77.1 77.9 90.7 78.0 77.1 85.5
AlexNet [26]
79.5 97.7 100.0 87.4 87.1 83.8 83.0 73.0 79.0 91.9 83.7 87.1 86.1
DDC [4]
83.1 98.1 100.0 88.4 89.0 84.9 83.5 73.4 79.2 91.9 85.4 88.8 87.1
DAN [5]
91.8 98.5 100.0 91.7 90.0 92.1 84.1 81.2 80.3 92.0 90.6 89.3 90.1
RTN (mmd)
93.2 98.5 100.0 91.7 88.0 90.7 84.0 81.3 80.4 91.0 89.8 90.4 90.0
RTN (mmd+ent)
93.8 98.6 100.0 92.9 93.6 92.7 87.8 84.8 83.4 93.2 96.6 93.9 92.6
RTN (mmd+ent+res) 95.2 99.2 100.0 95.5 93.8 92.5 88.1 86.6 84.6 93.7 96.9 94.2 93.4
learning and domain adaptation cannot reinforce each other substantially in the two-step pipeline,
which motivates carefully-designed deep adaptation architectures to unify them. (2) Deep-transfer
learning methods that reduce the domain discrepancy by domain-adaptive deep networks (DDC, DAN
and RevGrad) substantially outperform standard deep learning methods (AlexNet) and traditional
shallow transfer-learning methods with deep features as the input (TCA and GFK). This confirms
that incorporating domain-adaptation modules into deep networks can improve domain adaptation
performance. By adapting source-target distributions in multiple task-specific layers using optimal
multi-kernel two-sample matching, DAN performs the best in general among the prior deep-transfer
learning methods. (3) The proposed residual transfer network (RTN) performs the best and sets up a
new state of the art result on these benchmark datasets. Different from all the previous deep-transfer
learning methods that only adapt the feature layers of deep neural networks to learn more transferable
features, RTN further adapts the classifier layers to bridge the source and target classifiers in an
end-to-end residual learning framework, which can correct the classifier mismatch more effectively.
To go deeper into different modules of RTN, we show the results of RTN variants in Tables 1 and 2. (1)
RTN (mmd) slightly outperforms DAN, but RTN (mmd) has only one MMD penalty parameter while
DAN has two or three. Thus the proposed tensor MMD module is effective for adapting multiple
feature layers using a single MMD penalty, which is important for easy model selection. (2) RTN
(mmd+ent) performs substantially better than RTN (mmd). This highlights the importance of entropy
minimization for low-density separation, which exploits the cluster structure of target-unlabeled
data such that the target-classifier can be better adapted to the target data. (3) RTN (mmd+ent+res)
performs the best across all variants. This highlights the importance of residual transfer of classifier
layers for learning more adaptive classifiers. This is critical as in practical applications, there is no
guarantee that the source classifier and target classifier can be safely shared. It is worth noting that,
the entropy penalty and the residual module should be used together, otherwise the residual function
tends to learn useless zero mapping such that the source and target classifiers are nearly identical [8].
4.3
Discussion
Predictions Visualization: We respectively visualize in Figures 2(a)?2(d) the t-SNE embeddings [2]
of the predictions by DAN and RTN on transfer task A ? W. We can make the following observations.
(1) The predictions made by DAN in Figure 2(a)?2(b) show that the target categories are not well
discriminated by the source classifier, which implies that target data is not well compatible with the
source classifier. Hence the source and target classifiers should not be assumed to be identical, which
has been a common assumption made by all prior deep domain adaptation methods [4, 5, 6, 7]. (2)
The predictions made by RTN in Figures 2(c)?2(d) show that the target categories are discriminated
7
(a) DAN: Source=A
(b) DAN: Target=W
(c) RTN: Source=A
(d) RTN: Target=W
6
f S (x )
f T (x )
?f(x )
4
2
0
mean
deviation
Statistics
(a) Layer Responses
-0.5
-1
fs
-1.5
10
ft
20
Class
(b) Classifier Shift
30
Average Accuracy (%)
Layer Responses
8
Weight Parameters
Figure 2: Visualization: (a)-(b) t-SNE of DAN predictions; (c)-(d) t-SNE of RTN predictions.
100
90
80
70
60
A ?W
50
0.01 0.04 0.07 0.1
C?W
0.4
0.7
1
?
(c) Accuracy w.r.t. ?
Figure 3: (a) layer responses; (b) classifier shift; (c) sensitivity of ? (dashed lines show best baselines).
better by the target classifier (larger class-to-class distances), which suggests that residual transfer of
classifiers is a reasonable extension to previous deep feature adaptation methods. RTN simultaneously
learns more adaptive classifiers and more transferable features to enable effective domain adaptation.
Layer Responses: We show in Figure 3(a) the means and standard deviations of the layer responses
[8], which are the outputs of fT (x) (f cc layer), ?f (x) (f c2 layer), and fS (x) (after element-wise
sum operator), respectively. This exposes the response strength of the residual functions. The results
show that the residual function ?f (x) have generally much smaller responses than the shortcut
function fT (x). These results support our motivation that the residual functions are generally smaller
than the non-residual functions, as they characterize the small gap between the source classifier and
target classifier. The small residual function can be learned effectively via deep residual learning [8].
Classifier Shift: To justify that there exists a classifier shift between source classifier fs and target
classifier ft , we train fs on source domain and ft on target domain, both provided with labeled data.
By taking A as source domain and W as target domain, the weight parameters of the classifiers (e.g.
softmax regression) are shown in Figure 3(b), which shows that fs and ft are substantially different.
Parameter Sensitivity: We check the sensitivity of entropy parameter ? on transfer tasks A ? W
(31 classes) and C ? W (10 classes) by varying the parameter in {0.01, 0.04, 0.07, 0.1, 0.4, 0.7, 1.0}.
The results are shown in Figures 3(c), with the best results of the baselines shown as dashed lines.
The accuracy of RTN first increases and then decreases as ? varies and demonstrates a desirable
bell-shaped curve. This justifies our motivation of jointly learning transferable features and adaptive
classifiers by the RTN model, as a good trade-off between them can promote transfer performance.
5
Conclusion
This paper presented a novel approach to unsupervised domain adaptation in deep networks, which
enables end-to-end learning of adaptive classifiers and transferable features. Similar to many prior
domain adaptation techniques, feature adaptation is achieved by matching the distributions of features
across domains. However, unlike previous work, the proposed approach also supports classifier
adaptation, which is implemented through a new residual transfer module that bridges the source
classifier and target classifier. This makes the approach a good complement to existing techniques. The
approach can be trained by standard back-propagation, which is scalable and can be implemented by
most deep learning package. Future work constitutes semi-supervised domain adaptation extensions.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (61502265, 61325008),
National Key R&D Program of China (2016YFB1000701, 2015BAF32B01), and TNList Key Project.
8
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5,649 | 6,111 | Learning What and Where to Draw
Scott Reed1,?
reedscot@google.com
Zeynep Akata2
akata@mpi-inf.mpg.de
Santosh Mohan1
santoshm@umich.edu
Samuel Tenka1
samtenka@umich.edu
Bernt Schiele2
schiele@mpi-inf.mpg.de
Honglak Lee1
honglak@umich.edu
1
2
University of Michigan, Ann Arbor, USA
Max Planck Institute for Informatics, Saarbr?cken, Germany
Abstract
Generative Adversarial Networks (GANs) have recently demonstrated the capability to synthesize compelling real-world images, such as room interiors, album
covers, manga, faces, birds, and flowers. While existing models can synthesize
images based on global constraints such as a class label or caption, they do not
provide control over pose or object location. We propose a new model, the Generative Adversarial What-Where Network (GAWWN), that synthesizes images
given instructions describing what content to draw in which location. We show
high-quality 128 ? 128 image synthesis on the Caltech-UCSD Birds dataset, conditioned on both informal text descriptions and also object location. Our system
exposes control over both the bounding box around the bird and its constituent
parts. By modeling the conditional distributions over part locations, our system
also enables conditioning on arbitrary subsets of parts (e.g. only the beak and tail),
yielding an efficient interface for picking part locations.
1
Introduction
Generating realistic images from informal descriptions would have a wide range of applications.
Modern computer graphics can already generate remarkably realistic scenes, but it still requires the
substantial effort of human designers and developers to bridge the gap between high-level concepts
and the end product of pixel-level details. Fully automating this creative process is currently out of
reach, but deep networks have shown a rapidly-improving ability for controllable image synthesis.
In order for the image-generating system to be useful, it should support high-level control over the
contents of the scene to be generated. For example, a user might provide the category of image to be
generated, e.g. ?bird?. In the more general case, the user could provide a textual description like ?a
yellow bird with a black head?.
Compelling image synthesis with this level of control has already been demonstrated using convolutional Generative Adversarial Networks (GANs) [Goodfellow et al., 2014, Radford et al., 2016].
Variational Autoencoders also show some promise for conditional image synthesis, in particular
recurrent versions such as DRAW [Gregor et al., 2015, Mansimov et al., 2016]. However, current
approaches have so far only used simple conditioning variables such as a class label or a non-localized
caption [Reed et al., 2016b], and did not allow for controlling where objects appear in the scene.
To generate more realistic and complex scenes, image synthesis models can benefit from incorporating
a notion of localizable objects. The same types of objects can appear in many locations in different
scales, poses and configurations. This fact can be exploited by separating the questions of ?what?
?
Majority of this work was done while first author was at U. Michigan, but completed while at DeepMind.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
and ?where? to modify the image at each step of computation. In addition to parameter efficiency,
this yields the benefit of more interpretable image samples, in the sense that we can track what the
network was meant to depict at each location.
For many image datasets, we have not only
global annotations such as a class label but
also localized annotations, such as bird part
keypoints in Caltech-USCD birds (CUB) [Wah
et al., 2011] and human joint locations in the
MPII Human Pose dataset (MHP) [Andriluka
et al., 2014]. For CUB, there are associated
text captions, and for MHP we collected a new
dataset of 3 captions per image.
This bird is completely black.
Beak
Belly
Right leg
This bird is bright blue.
Head
Our proposed model learns to perform locationand content-controllable image synthesis on the
above datasets. We demonstrate two ways to
encode spatial constraints (though there could
a man in an orange jacket, black pants and a black cap wearing sunglasses skiing
be many more). First, we show how to condiFigure 1: Text-to-image examples. Locations can
tion on the coarse location of a bird by incorbe specified by keypoint or bounding box.
porating spatial masking and cropping modules
into a text-conditional GAN, implemented using spatial transformers. Second, we can condition
on part locations of birds and humans in the form of a set of normalized (x,y) coordinates, e.g.
beak@(0.23,0.15). In the second case, the generator and discriminator use a multiplicative gating
mechanism to attend to the relevant part locations.
The main contributions are as follows: (1) a novel architecture for text- and location-controllable
image synthesis, yielding more realistic and higher-resolution CUB samples, (2) a text-conditional
object part completion model enabling a streamlined user interface for specifying part locations, and
(3) exploratory results and a new dataset for pose-conditional text to human image synthesis.
2
Related Work
In addition to recognizing patterns within images, deep convolutional networks have shown remarkable capability to generate images. Dosovitskiy et al. [2015] trained a deconvolutional network to
generate 3D chair renderings conditioned on a set of graphics codes indicating shape, position and
lighting. Yang et al. [2015] followed with a recurrent convolutional encoder-decoder that learned
to apply incremental 3D rotations to generate sequences of rotated chair and face images. Oh et al.
[2015] used a similar approach in order to predict action-conditional future frames of Atari games.
Reed et al. [2015] trained a network to generate images that solved visual analogy problems.
The above models were all deterministic (i.e. conventional feed-forward and recurrent neural
networks), trained to learn one-to-one mappings from the latent space to pixel space. Other recent
works take the approach of learning probabilistic models with variational autoencoders [Kingma and
Welling, 2014, Rezende et al., 2014]. Kulkarni et al. [2015] developed a convolutional variational
autoencoder in which the latent space was ?disentangled? into separate blocks of units corresponding
to graphics codes. Gregor et al. [2015] created a recurrent variational autoencoder with attention
mechanisms for reading and writing portions of the image canvas at each time step (DRAW).
In addition to VAE-based image generation models, simple and effective Generative Adversarial
Networks [Goodfellow et al., 2014] have been increasingly popular. In general, GAN image samples
are notable for their relative sharpness compared to samples from the contemporary VAE models.
Later, class-conditional GAN [Denton et al., 2015] incorporated a Laplacian pyramid of residual
images into the generator network to achieve a significant qualitative improvement. Radford et al.
[2016] proposed ways to stabilize deep convolutional GAN training and synthesize compelling
images of faces and room interiors.
Spatial Transformer Networks (STN) [Jaderberg et al., 2015] have proven to be an effective visual
attention mechanism, and have already been incorporated into the latest deep generative models.
Eslami et al. [2016] incorporate STNs into a form of recurrent VAE called Attend, Infer, Repeat (AIR),
that uses an image-dependent number of inference steps, learning to generate simple multi-object
2D and 3D scenes. Rezende et al. [2016] build STNs into a DRAW-like recurrent network with
impressive sample complexity visual generalization properties.
2
Larochelle and Murray [2011] proposed the Neural Autoregressive Density Estimator (NADE) to
tractably model distributions over image pixels as a product of conditionals. Recently proposed
spatial grid-structured recurrent networks [Theis and Bethge, 2015, van den Oord et al., 2016] have
shown encouraging image synthesis results. We use GANs in our approach, but the same principle of
separating ?what? and ?where? conditioning variables can be applied to these types of models.
3
Preliminaries
3.1
Generative Adversarial Networks
Generative adversarial networks (GANs) consist of a generator G and a discriminator D that compete
in a two-player minimax game. The discriminator?s objective is to correctly classify its inputs as
either real or synthetic. The generator?s objective is to synthesize images that the discriminator will
classsify as real. D and G play the following game with value function V (D, G):
min max V (D, G) = Ex?pdata (x) [log D(x)] + Ex?pz (z) [log(1 ? D(G(z)))]
G
D
where z is a noise vector drawn from e.g. a Gaussian or uniform distribution. Goodfellow et al.
[2014] showed that this minimax game has a global optimium precisely when pg = pdata , and that
when G and D have enough capacity, pg converges to pdata .
To train a conditional GAN, one can simply provide both the generator and discriminator with the
additional input c as in [Denton et al., 2015, Radford et al., 2016] yielding G(z, c) and D(x, c). For
an input tuple (x, c) to be intepreted as ?real?, the image x must not only look realistic but also match
its context c. In practice G is trained to maximize log D(G(z, c)).
3.2
Structured joint embedding of visual descriptions and images
To encode visual content from text descriptions, we use a convolutional and recurrent text encoder to
learn a correspondence function between images and text features, following the approach of Reed
et al. [2016a] (and closely related to Kiros et al. [2014]). Sentence embeddings are learned by
optimizing the following structured loss:
N
1 X
?(yn , fv (vn )) + ?(yn , ft (tn ))
N n=1
(1)
where {(vn , tn , yn ), n = 1, ..., N } is the training data set, ? is the 0-1 loss, vn are the images, tn
are the corresponding text descriptions, and yn are the class labels. fv and ft are defined as
fv (v) = arg max Et?T (y) [?(v)T ?(t))], ft (t) = arg max Ev?V(y) [?(v)T ?(t))]
y?Y
(2)
y?Y
where ? is the image encoder (e.g. a deep convolutional network), ? is the text encoder, T (y) is
the set of text descriptions of class y and likewise V(y) for images. Intuitively, the text encoder
learns to produce a higher compatibility score with images of the correspondong class compared to
any other class, and vice-versa. To train the text encoder we minimize a surrogate loss related to
Equation 1 (see Akata et al. [2015] for details). We modify the approach of Reed et al. [2016a] in a
few ways: using a char-CNN-GRU [Cho et al., 2014] instead of char-CNN-RNN, and estimating the
expectations in Equation 2 using the average of 4 sampled captions per image instead of 1.
4
Generative Adversarial What-Where Networks (GAWWN)
In the following sections we describe the bounding-box- and keypoint-conditional GAWWN models.
4.1 Bounding-box-conditional text-to-image model
Figure 2 shows a sketch of the model, which can be understood by starting from input noise z ? RZ
and text embedding t ? RT (extracted from the caption by pre-trained 2 encoder ?(t)) and following
the arrows. Below we walk through each step.
First, the text embedding (shown in green) is replicated spatially to form a M ? M ? T feature
map, and then warped spatially to fit into the normalized bounding box coordinates. The feature map
2
Both ? and ? could be trained jointly with the GAN, but pre-training allows us to use the best available
image features from higher resolution images (224 ? 224) and speeds up GAN training.
3
entries outside the box are all zeros.3 The diagram shows a single object, but in the case of multiple
localized captions, these feature maps are averaged. Then, convolution and pooling operations are
applied to reduce the spatial dimension back to 1 ? 1. Intuitively, this feature vector encodes the
coarse spatial structure in the image, and we concatenate this with the noise vector z.
Spatial replicate,
crop to bbox
replicate
spatial
= Deconv
16
1
1
1
16
crop to
bbox
A red bird
with a black face
A red bird
with a black face
depth
concat
= Conv
16
1
16
depth
concat
crop to
bbox
128
Local
16
Local
16
16
16
16
16
{ 0, 1 }
128
Global
128
Global
128
16
16
16
128
16
Discriminator Network
Generator Network
Figure 2: GAWWN with bounding box location control.
In the next stage, the generator branches into local and global processing stages. The global pathway
is just a series of stride-2 deconvolutions to increase spatial dimension from 1 ? 1 to M ? M . In
the local pathway, upon reaching spatial dimension M ? M , a masking operation is applied so that
regions outside the object bounding box are set to 0. Finally, the local and global pathways are
merged by depth concatenation. A final series of deconvolution layers are used to reach the final
spatial dimension. In the final layer we apply a Tanh nonlinearity to constrain the outputs to [?1, 1].
In the discriminator, the text is similarly replicated spatially to form a M ? M ? T tensor. Meanwhile
the image is processed in local and global pathways. In the local pathway, the image is fed through
stride-2 convolutions down to the M ? M spatial dimension, at which point it is depth-concatenated
with the text embedding tensor. The resulting tensor is spatially cropped to within the bounding box
coordinates, and further processed convolutionally until the spatial dimension is 1 ? 1. The global
pathway consists simply of convolutions down to a vector, with additive contribution of the orignal
text embedding t. Finally, the local and global pathway output vectors are combined additively and
fed into the final layer producing the scalar discriminator score.
4.2 Keypoint-conditional text-to-image model
Figure 3 shows the keypoint-conditional version of the GAWWN, described in detail below.
part
locs
part
locs
A red bird with
a black face
= Deconv
1
depth
concat
A red bird with
a black face
replicate
spatial
max,
replicate
depth
pointwise
multiply
Local
max,
replicate
depth
depth
concat
16
depth
concat
16
depth
concat
128
Local
128
{ 0, 1 }
16
16
16
16
depth
concat
16
Global
= Conv
1
16
128
Global
16
16
Generator Network
16
128
16
128
Discriminator Network
Figure 3: Text and keypoint-conditional GAWWN.. Keypoint grids are shown as 4 ? 4 for clarity of
presentation, but in our experiments we used 16 ? 16.
The location keypoints are encoded into a M ? M ? K spatial feature map in which the channels
correspond to the part; i.e. head in channel 1, left foot in channel 2, and so on. The keypoint
tensor is fed into several stages of the network. First, it is fed through stride-2 convolutions to produce
a vector that is concatenated with noise z and text embedding t. The resulting vector provides
coarse information about content and part locations. Second, the keypoint tensor is flattened into a
binary matrix with a 1 indicating presence of any part at a particular spatial location, then replicated
depth-wise into a tensor of size M ? M ? H.
In the local and global pathways, the noise-text-keypoint vector is fed through deconvolutions to
produce another M ? M ? H tensor. The local pathway activations are gated by pointwise multiplication with the keypoint tensor of the same size. Finally, the original M ? M ? K keypoint tensor is
3
For details of how to apply this warping see equation 3 in [Jaderberg et al., 2015]
4
depth-concatenated with the local and global tensors, and processed with further deconvolutions to
produce the final image. Again a Tanh nonlinearity is applied.
In the discriminator, the text embedding t is fed into two stages. First, it is combined additively with
the global pathway that processes the image convolutionally producing a vector output. Second, it
is spatially replicated to M ? M and then depth-concatenated with another M ? M feature map
in the local pathway. This local tensor is then multiplicatively gated with the binary keypoint mask
exactly as in the generator, and the resulting tensor is depth-concatenated with the M ? M ? T
keypoints. The local pathway is fed into further stride-2 convolutions to produce a vector, which
is then additively combined with the global pathway output vector, and then into the final layer
producing the scalar discriminator score.
4.3 Conditional keypoint generation model
From a user-experience perspective, it is not optimal to require users to enter every single keypoint of
the parts of the object they wish to be drawn (e.g. for birds our model would require 15). Therefore,
it would be very useful to have access to all of the conditional distributions of unobserved keypoints
given a subset of observed keypoints and the text description. A similar problem occurs in data
imputation, e.g. filling in missing records or inpainting image occlusions. However, in our case we
want to draw convincing samples rather than just fill in the most likely values.
Conditioned on e.g. only the position of a bird?s beak, there could be several very different plausible
poses that satisfy the constraint. Therefore, a simple approach such as training a sparse autoencoder
over keypoints would not suffice. A DBM [Salakhutdinov and Hinton, 2009] or variational autoencoder [Rezende et al., 2014] could in theory work, but for simplicity we demonstrate the results
achieved by applying the same generic GAN framework to this problem.
The basic idea is to use the assignment of each object part as observed (i.e. conditioning variable) or
unobserved as a gating mechanism. Denote the keypoints for a single image as ki := {xi , yi , vi }, i =
1, ..., K, where x and y indicate the row and column position, respectively, and v is a bit set to 1 if the
part is visible and 0 otherwise. If the part is not visible, x and y are also set to 0. Let k ? [0, 1]K?3
encode the keypoints into a matrix. Let the conditioning variables (e.g. a beak position specified
by the user) be encoded into a vector of switch units s ? {0, 1}K , with the i-th entry set to 1 if the
i-th part is a conditioning variable and 0 otherwise. We can formulate the generator network over
keypoints Gk , conditioned on text t and a subset of keypoints k, s, as follows:
Gk (z, t, k, s) := s k + (1 ? s) f (z, t, k)
(3)
where denotes pointwise multiplication and f : RZ+T +3K ? R3K is an MLP. In practice we
concatenated z, t and flattened k and chose f to be a 3-layer fully-connected network.
The discriminator Dk learns to distinguish real keypoints and text (kreal , treal ) from synthetic.
In order for Gk to capture all of the conditional distributions over keypoints, during training we
randomly sample switch units s in each mini-batch. Since we would like to usually specify 1 or 2
keypoints, in our experiments we set the ?on? probability to 0.1. That is, each of the 15 bird parts
only had a 10% chance of acting as a conditioning variable for a given training image.
5
Experiments
In this section we describe our experiments on generating images from text descriptions on the
Caltech-UCSD Birds (CUB) and MPII Human Pose (MHP) datasets.
CUB [Wah et al., 2011] has 11,788 images of birds belonging to one of 200 different species. We also
use the text dataset from Reed et al. [2016a] including 10 single-sentence descriptions per bird image.
Each image also includes the bird location via its bounding box, and keypoint (x,y) coordinates for
each of 15 bird parts. Since not all parts are visible in each image, the keypoint data also provides an
additional bit per part indicating whether the part can be seen.
MHP Andriluka et al. [2014] has 25K images with 410 different common activities. For each image,
we collected 3 single-sentence text descriptions using Mechanical Turk. We asked the workers to
describe the most distinctive aspects of the person and the activity they are engaged in, e.g. ?a man
in a yellow shirt preparing to swing a golf club?. Each image has potentially multiple sets of (x,y)
keypoints for each of the 16 joints. During training we filtered out images with multiple people, and
for the remaining 19K images we cropped the image to the person?s bounding box.
5
We encoded the captions using a pre-trained char-CNN-GRU as described in [Reed et al., 2016a].
During training, the 1024-dimensional text embedding for a given image was taken to be the average
of four randomly-sampled caption encodings corresponding to that image. Sampling multiple captions
per image provides further information required to draw the object. At test time one can average
together any number of description embeddings, including a single caption.
For both CUB and MHP, we trained our GAWWN using the ADAM solver with batch size 16 and
learning rate 0.0002 (See Alg. 1 in [Reed et al., 2016b] for the conditional GAN training algorithm).
The models were trained on all categories and we show samples on a set of held out captions. For the
spatial transformer module, we used a Torch implementation provided by Oquab [2016]. Our GAN
implementation is loosely based on dcgan.torch4 .
In experiments we analyze how accurately the GAWWN samples reflect the text and location
constraints. First we control the location of the bird by interpolation via bounding boxes and
keypoints. We consider both the case of (1) ground-truth keypoints from the data set, and (2) synthetic
keypoints generated by our model, conditioned on the text. Case (2) is advantageous because it
requires less effort from a hypothetical user (i.e. entering 15 keypoint locations). We then compare
our CUB results to representative samples from the previous work. Finally, we show samples on textand pose-conditional generation of images of human actions.
5.1 Controlling bird location via bounding boxes
We first demonstrate sampling from the text-conditional model while varying the bird location. Since
location is specified via bounding box coordinates, we can also control the size and aspect ratio of
the bird. This is shown in Figure 4 by interpolating the bounding box coordinates while at the same
time fixing the text and noise conditioning variables.
Caption
GT
Shrinking
Translation
Stretching
This bird has
a black head,
a long orange
beak and
yellow body
This large
black bird has
a pointy beak
and black eyes
This small blue
bird has a
short pointy
beak and
brown patches
on its wings
Figure 4: Controlling the bird?s position using bounding box coordinates. and previously-unseen text.
With the noise vector z fixed in every set of three frames, the background is usually similar but not
perfectly invariant. Interestingly, as the bounding box coordinates are changed, the direction the bird
faces does not change. This suggests that the model learns to use the the noise distribution to capture
some aspects of the background and also non-controllable aspects of ?where? such as direction.
5.2 Controlling individual part locations via keypoints
In this section we study the case of text-conditional image generation with keypoints fixed to the
ground-truth. This can give a sense of the performance upper bound for the text to image pipeline,
because synthetic keypoints can be no more realistic than the ground-truth. We take a real image and
its keypoint annotations from the CUB dataset, and a held-out text description, and draw samples
conditioned on this information.
This large black bird has a long neck and tail
GT feathers.
This bird is mostly white with a thick black
GT eyebrow, small and black beak and a long tail.
GT This large white bird has an orange-tipped beak.
GT
This bird has a bright red crown and black
wings and beak.
GT
This is a small yellowish green bird with a pointy
black beak, black eyes and gray wings.
This pale pink bird has a black eyebrow and a
GT black pointy beak, gray wings and yellow underparts.
Figure 5: Bird generation conditioned on fixed groundtruth keypoints (overlaid in blue) and previously
unseen text. Each sample uses a different random noise vector.
4
https://github.com/soumith/dcgan.torch
6
Figure 5 shows several image samples that accurately reflect the text and keypoint constraints. More
examples including success and failure are included in the supplement. We observe that the bird pose
respects the keypoints and is invariant across the samples. The background and other small details,
such as thickness of the tree branch or the background color palette do change with the noise.
Caption
GT
Shrinking
Translation
Stretching
This bird has
a black head,
a long orange
beak and
yellow body
This large
black bird has
a pointy beak
and black eyes
This small blue
bird has a
short pointy
beak and
brown patches
on its wings
Figure 6: Controlling the bird?s position using keypoint coordinates. Here we only interpolated the
beak and tail positions, and sampled the rest conditioned on these two.
The GAWWN model can also use keypoints to shrink, translate and stretch objects, as shown
in Figure 6. We chose to specify beak and tail positions, because in most cases these define an
approximate bounding box around the bird.
Unlike in the case of bounding boxes, we can now control which way the bird is pointing; note that
here all birds face left, whereas when we use bounding boxes (Figure 4) the orientation is random.
Elements of the scene, even outside of the controllable location, adjust in order to be coherent with
the bird?s position in each frame although in each set of three frames we use the same noise vector z.
5.3 Generating both bird keypoints and images from text alone
Although ground truth keypoint locations lead to visually plausible results as shown in the previous
sections, the keypoints are costly to obtain. In Figure 7, we provide examples of accurate samples
using generated keypoints. Compared to ground-truth keypoints, on average we did not observe
degradation in quality. More examples for each regime are provided in the supplement.
GT
This white bird has gray wings, red webbed
feet and a long, curved and yellow beak.
This bird has a yellow head, black eyes, a gray
GT pointy beak and orange lines on its breast.
This bird is completely red with a red and
GT cone-shaped beak, black face and a red nape.
GT
This water bird has a long white neck,
black body, yellow beak and black head.
GT
This small bird has a blue and gray head,
pointy beak and a white belly.
This bird is large, completely black, with a
GT long pointy beak and black eyes.
Figure 7: Keypoint- and text-conditional bird generation in which the keypoints are generated
conditioned on unseen text. The small blue boxes indicate the generated keypoint locations.
5.4
Comparison to previous work
In this section we compare our results with previous text-to-image results on CUB. In Figure 8 we
show several representative examples that we cropped from the supplementary material of [Reed et al.,
2016b]. We compare against the actual ground-truth and several variants of GAWWN. We observe
that the 64 ? 64 samples from [Reed et al., 2016b] mostly reflect the text description, but in some
cases lack clearly defined parts such as a beak. When the keypoints are zeroed during training, our
GAWWN architecture actually fails to generate any plausible images. This suggests that providing
additional conditioning variables in the form of location constraints is helpful for learning to generate
high-resolution images. Overall, the sharpest and most accurate results can be seen in the 128 ? 128
samples from our GAWWN with real or synthetic keypoints (bottom two rows).
5.5
Beyond birds: generating images of humans
Here we apply our model to generating images of humans conditioned on a description of their
appearance and activity, and also on their approximate pose. This is a much more challenging task
than generating images of birds due to the larger variety of scenes and pose configurations.
This small bird has a blue and gray head,
pointy beak, black and white patterns on its
wings and a white belly.
7
Groundtruth image
and text
caption
A small sized bird that
has tones of brown and
dark red with a short
stout bill
The bird is solid black with
white eyes and a black
beak.
This bird has a yellow
breast and a dark grey
face
GAN-INT-CLS
(Reed et. al,
2016b)
GAWWN
trained
without key
points
GAWWN
Key points
given
GAWWN
Key points
generated
Figure 8: Comparison of GAWWN to GAN-INT-CLS from Reed et al. [2016b] and also the groundtruth images. For the ground-truth row, the first entry corresonds directly to the caption, and the
second two entries are sampled from the same species.
Caption
GT
Samples
Caption
a woman in a yellow
tank top is doing yoga.
a woman in grey shirt
is doing yoga.
the man wearing the red
shirt and white pants
play golf on the green
grass
a man in green shirt
and white pants is
swinging his golf club.
a man in a red sweater
and grey pants swings a
golf club with one hand.
GT
Samples
a man in an orange
jacket, black pants and a
black cap wearing
sunglasses skiing.
a woman wearing
goggles swimming
through very murky
water
a man is skiing and
competing for the
olympics on the slopes.
Figure 9: Generating humans. Both the keypoints and the image are generated from unseen text.
The human image samples shown in Figure 9 tend to be much blurrier compared to the bird images,
but in many cases bear a clear resemblance to the text query and the pose constraints. Simple captions
involving skiing, golf and yoga tend to work, but complex descriptions and unusual poses (e.g.
upside-down person on a trampoline) remain especially challenging. We also generate videos by
(1) extracting pose keypoints from a pre-trained pose estimator from several YouTube clips, and
(2) combining these keypoint trajectories with a text query, fixing the noise vector z over time and
concatenating the samples (see supplement).
6
Discussion
In this work we showed how to generate images conditioned on both informal text descriptions and
object locations. Locations can be accurately controlled by either bounding box or a set of part
keypoints. On CUB, the addition of a location constraint allowed us to accurately generate compelling
128 ? 128 images, whereas previous models could only generate 64 ? 64. Furthermore, this location
conditioning does not constrain us during test time, because we can also learn a text-conditional
generative model of part locations, and simply generate them at test time.
An important lesson here is that decomposing the problem into easier subproblems can help generate
realistic high-resolution images. In addition to making the overall text to image pipeline easier to
train with a GAN, it also yields additional ways to control image synthesis. In future work, it may
be promising to learn the object or part locations in an unsupervised or weakly supervised way. In
addition, we show the first text-to-human image synthesis results, but performance on this task is
clearly far from saturated and further architectural advances will be required to solve it.
Acknowledgements This work was supported in part by NSF CAREER IIS-1453651, ONR
N00014-13-1-0762, and a Sloan Research Fellowship.
8
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9
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5,650 | 6,112 | Deep Learning without Poor Local Minima
Kenji Kawaguchi
Massachusetts Institute of Technology
kawaguch@mit.edu
Abstract
In this paper, we prove a conjecture published in 1989 and also partially address
an open problem announced at the Conference on Learning Theory (COLT) 2015.
With no unrealistic assumption, we first prove the following statements for the
squared loss function of deep linear neural networks with any depth and any
widths: 1) the function is non-convex and non-concave, 2) every local minimum is
a global minimum, 3) every critical point that is not a global minimum is a saddle
point, and 4) there exist ?bad? saddle points (where the Hessian has no negative
eigenvalue) for the deeper networks (with more than three layers), whereas there
is no bad saddle point for the shallow networks (with three layers). Moreover, for
deep nonlinear neural networks, we prove the same four statements via a reduction
to a deep linear model under the independence assumption adopted from recent
work. As a result, we present an instance, for which we can answer the following
question: how difficult is it to directly train a deep model in theory? It is more difficult than the classical machine learning models (because of the non-convexity),
but not too difficult (because of the nonexistence of poor local minima). Furthermore, the mathematically proven existence of bad saddle points for deeper models
would suggest a possible open problem. We note that even though we have advanced the theoretical foundations of deep learning and non-convex optimization,
there is still a gap between theory and practice.
1
Introduction
Deep learning has been a great practical success in many fields, including the fields of computer
vision, machine learning, and artificial intelligence. In addition to its practical success, theoretical
results have shown that deep learning is attractive in terms of its generalization properties (Livni
et al., 2014; Mhaskar et al., 2016). That is, deep learning introduces good function classes that
may have a low capacity in the VC sense while being able to represent target functions of interest
well. However, deep learning requires us to deal with seemingly intractable optimization problems.
Typically, training of a deep model is conducted via non-convex optimization. Because finding a
global minimum of a general non-convex function is an NP-complete problem (Murty & Kabadi,
1987), a hope is that a function induced by a deep model has some structure that makes the nonconvex optimization tractable. Unfortunately, it was shown in 1992 that training a very simple
neural network is indeed NP-hard (Blum & Rivest, 1992). In the past, such theoretical concerns in
optimization played a major role in shrinking the field of deep learning. That is, many researchers
instead favored classical machining learning models (with or without a kernel approach) that require
only convex optimization. While the recent great practical successes have revived the field, we do
not yet know what makes optimization in deep learning tractable in theory.
In this paper, as a step toward establishing the optimization theory for deep learning, we prove a
conjecture noted in (Goodfellow et al., 2016) for deep linear networks, and also address an open
problem announced in (Choromanska et al., 2015b) for deep nonlinear networks. Moreover, for
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
both the conjecture and the open problem, we prove more general and tighter statements than those
previously given (in the ways explained in each section).
2
Deep linear neural networks
Given the absence of a theoretical understanding of deep nonlinear neural networks, Goodfellow
et al. (2016) noted that it is beneficial to theoretically analyze the loss functions of simpler models,
i.e., deep linear neural networks. The function class of a linear multilayer neural network only
contains functions that are linear with respect to inputs. However, their loss functions are nonconvex in the weight parameters and thus nontrivial. Saxe et al. (2014) empirically showed that
the optimization of deep linear models exhibits similar properties to those of the optimization of
deep nonlinear models. Ultimately, for theoretical development, it is natural to start with linear
models before working with nonlinear models (as noted in Baldi & Lu, 2012), and yet even for
linear models, the understanding is scarce when the models become deep.
2.1
Model and notation
We begin by defining the notation. Let H be the number of hidden layers, and let (X, Y ) be the
training data set, with Y ? Rdy ?m and X ? Rdx ?m , where m is the number of data points.
Here, dy ? 1 and dx ? 1 are the number of components (or dimensions) of the outputs and
inputs, respectively. Let ? = Y X T (XX T )?1 XY T . We denote the model (weight) parameters by
W , which consists of the entries of the parameter matrices corresponding to each layer: WH+1 ?
Rdy ?dH , . . . , Wk ? Rdk ?dk?1 , . . . , W1 ? Rd1 ?dx . Here, dk represents the width of the k-th layer,
where the 0-th layer is the input layer and the (H + 1)-th layer is the output layer (i.e., d0 = dx
and dH+1 = dy ). Let Idk be the dk ? dk identity matrix. Let p = min(dH , . . . , d1 ) be the smallest
width of a hidden layer. We denote the (j, i)-th entry of a matrix M by Mj,i . We also denote the
j-th row vector of M by Mj,? and the i-th column vector of M by M?,i .
We can then write the output of a feedforward deep linear model, Y (W, X) ? Rdy ?m , as
Y (W, X) = WH+1 WH WH?1 ? ? ? W2 W1 X.
We consider one of the most widely used loss functions, squared error loss:
m
X
1
? )= 1
kY (W, X)?,i ? Y?,i k22 = kY (W, X) ? Y k2F ,
L(W
2 i=1
2
2 ?
where k?kF is the Frobenius norm. Note that m
L(W ) is the usual mean squared error, for which
? ) by a constant in W results in an equivalent
all of our results hold as well, since multiplying L(W
optimization problem.
2.2
Background
Recently, Goodfellow et al. (2016) remarked that when Baldi & Hornik (1989) proved Proposition
2.1 for shallow linear networks, they stated Conjecture 2.2 without proof for deep linear networks.
Proposition 2.1 (Baldi & Hornik, 1989: shallow linear network) Assume that H = 1 (i.e.,
Y (W, X) = W2 W1 X), assume that XX T and XY T are invertible, assume that ? has dy distinct eigenvalues, and assume that p < dx , p < dy and dy = dx (e.g., an autoencoder). Then, the
? ) has the following properties:
loss function L(W
(i) It is convex in each matrix W1 (or W2 ) when the other W2 (or W1 ) is fixed.
(ii) Every local minimum is a global minimum.
Conjecture 2.2 (Baldi & Hornik, 1989: deep linear network) Assume the same set of conditions as
? ) has the following properties:
in Proposition 2.1 except for H = 1. Then, the loss function L(W
(i) For any k ? {1, . . . , H + 1}, it is convex in each matrix Wk when for all k 0 6= k, Wk0 is
fixed.
(ii) Every local minimum is a global minimum.
2
Baldi & Lu (2012) recently provided a proof for Conjecture 2.2 (i), leaving the proof of Conjecture
2.2 (ii) for future work. They also noted that the case of p ? dx = dx is of interest, but requires
further analysis, even for a shallow network with H = 1. An informal discussion of Conjecture 2.2
can be found in (Baldi, 1989). In Appendix D, we provide a more detailed discussion of this subject.
2.3
Results
We now state our main theoretical results for deep linear networks, which imply Conjecture 2.2 (ii)
as well as obtain further information regarding the critical points with more generality.
Theorem 2.3 (Loss surface of deep linear networks) Assume that XX T and XY T are of full rank
with dy ? dx and ? has dy distinct eigenvalues. Then, for any depth H ? 1 and for any layer
widths and any input-output dimensions dy , dH , dH?1 , . . . , d1 , dx ? 1 (the widths can arbitrarily
? ) has the following properties:
differ from each other and from dy and dx ), the loss function L(W
(i) It is non-convex and non-concave.
(ii) Every local minimum is a global minimum.
(iii) Every critical point that is not a global minimum is a saddle point.
(iv) If rank(WH ? ? ? W2 ) = p, then the Hessian at any saddle point has at least one (strictly)
negative eigenvalue.1
Corollary 2.4 (Effect of deepness on the loss surface) Assume the same set of conditions as in
? ). For three-layer networks (i.e., H = 1), the
Theorem 2.3 and consider the loss function L(W
Hessian at any saddle point has at least one (strictly) negative eigenvalue. In contrast, for networks
deeper than three layers (i.e., H ? 2), there exist saddle points at which the Hessian does not have
any negative eigenvalue.
The assumptions of having full rank and distinct eigenvalues in the training data matrices in Theorem
2.3 are realistic and practically easy to satisfy, as discussed in previous work (e.g., Baldi & Hornik,
1989). In contrast to related previous work (Baldi & Hornik, 1989; Baldi & Lu, 2012), we do not
assume the invertibility of XY T , p < dx , p < dy nor dy = dx . In Theorem 2.3, p ? dx is allowed,
as well as many other relationships among the widths of the layers. Therefore, we successfully
proved Conjecture 2.2 (ii) and a more general statement. Moreover, Theorem 2.3 (iv) and Corollary
2.4 provide additional information regarding the important properties of saddle points.
Theorem 2.3 presents an instance of a deep model that would be tractable to train with direct greedy
optimization, such as gradient-based methods. If there are ?poor? local minima with large loss values
everywhere, we would have to search the entire space,2 the volume of which increases exponentially
with the number of variables. This is a major cause of NP-hardness for non-convex optimization. In
contrast, if there are no poor local minima as Theorem 2.3 (ii) states, then saddle points are the main
? ) is Lipschitz continuous, if
remaining concern in terms of tractability.3 Because the Hessian of L(W
the Hessian at a saddle point has a negative eigenvalue, it starts appearing as we approach the saddle
point. Thus, Theorem 2.3 and Corollary 2.4 suggest that for 1-hidden layer networks, training can
be done in polynomial time with a second order method or even with a modified stochastic gradient
decent method, as discussed in (Ge et al., 2015). For deeper networks, Corollary 2.4 states that
there exist ?bad? saddle points in the sense that the Hessian at the point has no negative eigenvalue.
However, we know exactly when this can happen from Theorem 2.3 (iv) in our deep models. We
leave the development of efficient methods to deal with such a bad saddle point in general deep
models as an open problem.
3
Deep nonlinear neural networks
Now that we have obtained a comprehensive understanding of the loss surface of deep linear models,
we discuss deep nonlinear models. For a practical deep nonlinear neural network, our theoretical
results so far for the deep linear models can be interpreted as the following: depending on the
If H = 1, to be succinct, we define WH ? ? ? W2 = W1 ? ? ? W2 , Id1 , with a slight abuse of notation.
Typically, we do this by assuming smoothness in the values of the loss function.
3
Other problems such as the ill-conditioning can make it difficult to obtain a fast convergence rate.
1
2
3
nonlinear activation mechanism and architecture, training would not be arbitrarily difficult. While
theoretical formalization of this intuition is left to future work, we address a recently proposed open
problem for deep nonlinear networks in the rest of this section.
3.1
Model
We use the same notation as for the deep linear models, defined in the beginning of Section 2.1. The
output of deep nonlinear neural network, Y? (W, X) ? Rdy ?m , is defined as
?
Y(W,
X) = q?H+1 (WH+1 ?H (WH ?H?1 (WH?1 ? ? ? ?2 (W2 ?1 (W1 X)) ? ??))),
where q ? R is simply a normalization factor, the value of which is specified later. Here, ?k :
Rdk ?m ? Rdk ?m is the element-wise rectified linear function:
??
?? ?
?
? (b1m)
?
? (b11 ) . . . ?
b11 . . . b1m
??
?
.. ?? = ? ..
..
..
..
?k ?? ...
?,
.
.
. ?? ? .
.
bdk 1 ? ? ? bdk m
? (bdk m )
?
? (bdk 1 ) ? ? ? ?
where ?
? (bij ) = max(0, bij ). In practice, we usually set ?H+1 to be an identity map in the last layer,
in which case all our theoretical results still hold true.
3.2
Background
Following the work by Dauphin et al. (2014), Choromanska et al. (2015a) investigated the connection between the loss functions of deep nonlinear networks and a function well-studied via random
matrix theory (i.e., the Hamiltonian of the spherical spin-glass model). They explained that their
theoretical results relied on several unrealistic assumptions. Later, Choromanska et al. (2015b) suggested at the Conference on Learning Theory (COLT) 2015 that discarding these assumptions is an
important open problem. The assumptions were labeled A1p, A2p, A3p, A4p, A5u, A6u, and A7p.
In this paper, we successfully discard most of these assumptions. In particular, we only use a weaker
version of assumptions A1p and A5u. We refer to the part of assumption A1p (resp. A5u) that
corresponds only to the model assumption as A1p-m (resp. A5u-m). Note that assumptions A1p-m
and A5u-m are explicitly used in the previous work (Choromanska et al., 2015a) and included in
A1p and A5u (i.e., we are not making new assumptions here).
As the model Y? (W, X) ? Rdy ?m represents a directed acyclic graph, we can express an output
from one of the units in the output layer as
?
H+1
X
Y (k)
[Xi ](j,p) [Zi ](j,p)
w(j,p) .
(1)
Y? (W, X)j,i = q
p=1
k=1
Here, ? is the total number of paths from the inputs to each j-th output in the directed acyclic graph.
In addition, [Xi ](j,p) ? R represents the entry of the i-th sample input datum that is used in the
(k)
p-th path of the j-th output. For each layer k, w(j,p) ? R is the entry of Wk that is used in the p-th
path of the j-th output. Finally, [Zi ](j,p) ? {0, 1} represents whether the p-th path of the j-th output
is active ([Zi ](j,p) = 1) or not ([Zi ](j,p) = 0) for each sample i as a result of the rectified linear
activation.
Assumption A1p-m assumes that the Z?s are Bernoulli random variables with the same probability
of success, Pr([Zi ](j,p) = 1) = ? for all i and (j, p). Assumption A5u-m assumes that the Z?s are
independent from the input X?s and parameters w?s. With assumptions A1p-m and A5u-m, we can
P?
QH+1 (k)
write EZ [Y? (W, X)j,i ] = q p=1 [Xi ](j,p) ? k=1 w(j,p) .
Choromanska et al. (2015b) noted that A6u is unrealistic because it implies that the inputs are not
shared among the paths. In addition, Assumption A5u is unrealistic because it implies that the
activation of any path is independent of the input data. To understand all of the seven assumptions
(A1p, A2p, A3p, A4p, A5u, A6u, and A7p), we note that Choromanska et al. (2015b,a) used these
seven assumptions to reduce their loss functions of nonlinear neural networks to:
H+1
?
?
X
Y
1
1X 2
Xi1 ,i2 ,...,iH+1
wik subject to
w = 1,
Lprevious (W ) = H/2
? i=1 i
?
i ,i ,...,i
=1
1
2
k=1
H+1
4
where ? ? R is a constant related to the size of the network. For our purpose, the detailed definitions
of the symbols are not important (X and w are defined in the same way as in equation 1). Here,
we point out that the target function Y has disappeared in the loss Lprevious (W ) (i.e., the loss value
does not depend on the target function). That is, whatever the data points of Y are, their loss values
are the same. Moreover, the nonlinear activation function has disappeared in Lprevious (W ) (and the
nonlinearity is not taken into account in X or w). In the next section, by using only a strict subset
of the set of these seven assumptions, we reduce our loss function to a more realistic loss function
of an actual deep model.
Proposition 3.1 (High-level description of a main result in Choromanska et al., 2015a) Assume
A1p (including A1p-m), A2p, A3p, A4p, A5u (including A5u-m), A6u, and A7p (Choromanska
et al., 2015b). Furthermore, assume that dy = 1. Then, the expected loss of each sample datum,
Lprevious (W ), has the following property: above a certain loss value, the number of local minima
diminishes exponentially as the loss value increases.
3.3
Results
We now state our theoretical result, which partially address the aforementioned open problem. We
consider loss functions for all the data points and all possible output dimensionalities (i.e., vectoredvalued output). More concretely, we consider the squared error loss with expectation, L(W ) =
1
2
?
2 kEZ [Y (W, X) ? Y ]kF .
Corollary 3.2 (Loss surface of deep nonlinear networks) Assume A1p-m and A5u-m. Let q = ??1 .
Then, we can reduce the loss function of the deep nonlinear model L(W ) to that of the deep linear
? ). Therefore, with the same set of conditions as in Theorem 2.3, the loss function of the
model L(W
deep nonlinear model has the following properties:
(i)
(ii)
(iii)
(iv)
It is non-convex and non-concave.
Every local minimum is a global minimum.
Every critical point that is not a global minimum is a saddle point.
The saddle points have the properties stated in Theorem 2.3 (iv) and Corollary 2.4.
Comparing Corollary 3.2 and Proposition 3.1, we can see that we successfully discarded assumptions A2p, A3p, A4p, A6u, and A7p while obtaining a tighter statement in the following sense:
Corollary 3.2 states with fewer unrealistic assumptions that there is no poor local minimum, whereas
Proposition 3.1 roughly asserts with more unrealistic assumptions that the number of poor local minimum may be not too large. Furthermore, our model Y? is strictly more general than the model analyzed in (Choromanska et al., 2015a,b) (i.e., this paper?s model class contains the previous work?s
model class but not vice versa).
4
Proof Idea and Important lemmas
In this section, we provide overviews of the proofs of the theoretical results. Our proof approach
largely differs from those in previous work (Baldi & Hornik, 1989; Baldi & Lu, 2012; Choromanska
et al., 2015a,b). In contrast to (Baldi & Hornik, 1989; Baldi & Lu, 2012), we need a different
approach to deal with the ?bad? saddle points that start appearing when the model becomes deeper
(see Section 2.3), as well as to obtain more comprehensive properties of the critical points with
more generality. While the previous proofs heavily rely on the first-order information, the main
parts of our proofs take advantage of the second order information. In contrast, Choromanska et al.
(2015a,b) used the seven assumptions to relate the loss functions of deep models to a function
previously analyzed with a tool of random matrix theory. With no reshaping assumptions (A3p, A4p,
and A6u), we cannot relate our loss function to such a function. Moreover, with no distributional
assumptions (A2p and A6u) (except the activation), our Hessian is deterministic, and therefore, even
random matrix theory itself is insufficient for our purpose. Furthermore, with no spherical constraint
assumption (A7p), the number of local minima in our loss function can be uncountable.
One natural strategy to proceed toward Theorem 2.3 and Corollary 3.2 would be to use the first-order
and second-order necessary conditions of local minima (e.g., the gradient is zero and the Hessian is
5
positive semidefinite).4 However, are the first-order and second-order conditions sufficient to prove
Theorem 2.3 and Corollary 3.2? Corollaries 2.4 show that the answer is negative for deep models
with H ? 2, while it is affirmative for shallow models with H = 1. Thus, for deep models, a simple
use of the first-order and second-order information is insufficient to characterize the properties of
each critical point. In addition to the complexity of the Hessian of the deep models, this suggests that
we must strategically extract the second order information. Accordingly, in section 4.2, we obtain
an organized representation of the Hessian in Lemma 4.3 and strategically extract the information
in Lemmas 4.4 and 4.6. With the extracted information, we discuss the proofs of Theorem 2.3 and
Corollary 3.2 in section 4.3.
4.1
Notations
Let M ? M 0 be the Kronecker product of M and M 0 . Let Dvec(WkT ) f (?) =
?f (?)
?vec(W T )
be the partial
k
din
derivative of f with respect to vec(WkT ) in the numerator layout. That is, if f : R
? Rdout , we
dout ?(dk dk?1 )
. Let R(M ) be the range (or the column space) of a matrix
have Dvec(WkT ) f (?) ? R
M . Let M ? be any generalized inverse of M . When we write a generalized inverse in a condition
or statement, we mean it for any generalized inverse (i.e., we omit the universal quantifier over
generalized inverses, as this is clear). Let r = (Y (W, X) ? Y )T ? Rm?dy be an error matrix.
Let C = WH+1 ? ? ? W2 ? Rdy ?d1 . When we write Wk ? ? ? Wk0 , we generally intend that k > k 0
and the expression denotes a product over Wj for integer k ? j ? k 0 . For notational compactness,
two additional cases can arise: when k = k 0 , the expression denotes simply Wk , and when k < k 0 ,
it denotes Idk . For example, in the statement of Lemma 4.1, if we set k := H + 1, we have that
WH+1 WH ? ? ? WH+2 , Idy .
In Lemma 4.6 and the proofs of Theorems 2.3, we use the following additional notation. We denote an eigendecomposition of ? as ? = U ?U T , where the entries of the eigenvalues are ordered
as ?1,1 > ? ? ? > ?dy ,dy with corresponding orthogonal eigenvector matrix U = [u1 , . . . , udy ].
For each k ? {1, . . . dy }, uk ? Rdy ?1 is a column eigenvector. Let p? = rank(C) ?
{1, . . . , min(dy , p)}. We define a matrix containing the subset of the p? largest eigenvectors as
Up? = [u1 , . . . , up?]. Given any ordered set Ip? = {i1 , . . . , ip? | 1 ? i1 < ? ? ? < ip? ? min(dy , p)},
we define a matrix containing the subset of the corresponding eigenvectors as UIp? = [ui1 , . . . , uip? ].
Note the difference between Up? and UIp? .
4.2
Lemmas
As discussed above, we extracted the first-order and second-order conditions of local minima as
the following lemmas. The lemmas provided here are also intended to be our additional theoretical
results that may lead to further insights. The proofs of the lemmas are in the appendix.
? ) if and
Lemma 4.1 (Critical point necessary and sufficient condition) W is a critical point of L(W
only if for all k ? {1, ..., H + 1},
T
T
? )
Dvec(WkT ) L(W
= WH+1 WH ? ? ? Wk+1 ? (Wk?1 ? ? ? W2 W1 X)T vec(r) = 0.
? ), then
Lemma 4.2 (Representation at critical point) If W is a critical point of L(W
WH+1 WH ? ? ? W2 W1 = C(C T C)? C T Y X T (XX T )?1 .
? ) in a block form
Lemma 4.3 (Block Hessian with Kronecker product) Write the entries of ?2 L(W
as
T
T ?
?
?
?
T
T
T
Dvec(WH+1
? ? ? Dvec(W1T ) Dvec(WH+1
) Dvec(WH+1
) L(W )
) L(W )
?
?
?
..
..
..
? )=?
?2 L(W
?
?.
.
.
.
?
T
T ?
?
? )
T
Dvec(WH+1
? ? ? Dvec(W1T ) Dvec(W1T ) L(W
) Dvec(W1T ) L(W )
4
For a non-convex and non-differentiable function, we can still have a first-order and second-order necessary
condition (e.g., Rockafellar & Wets, 2009, theorem 13.24, p. 606).
6
Then, for any k ? {1, ..., H + 1},
T
? )
Dvec(WkT ) Dvec(WkT ) L(W
= (WH+1 ? ? ? Wk+1 )T (WH+1 ? ? ? Wk+1 ) ? (Wk?1 ? ? ? W1 X)(Wk?1 ? ? ? W1 X)T ,
and, for any k ? {2, ..., H + 1},
T
? )
Dvec(WkT ) Dvec(W1T ) L(W
= C T (WH+1 ? ? ? Wk+1 ) ? X(Wk?1 ? ? ? W1 X)T +
[(Wk?1 ? ? ? W2 )T ? X] [Idk?1 ? (rWH+1 ? ? ? Wk+1 )?,1
...
Idk?1 ? (rWH+1 ? ? ? Wk+1 )?,dk ] .
? ) is positive semidefinite or negLemma 4.4 (Hessian semidefinite necessary condition) If ?2 L(W
ative semidefinite at a critical point, then for any k ? {2, ..., H + 1},
R((Wk?1 ? ? ? W3 W2 )T ) ? R(C T C) or XrWH+1 WH ? ? ? Wk+1 = 0.
? ) is positive semidefinite or negative semidefinite at a critical point, then
Corollary 4.5 If ?2 L(W
for any k ? {2, ..., H + 1},
rank(WH+1 WH ? ? ? Wk ) ? rank(Wk?1 ? ? ? W3 W2 ) or XrWH+1 WH ? ? ? Wk+1 = 0.
? ) is positive semidefinite
Lemma 4.6 (Hessian positive semidefinite necessary condition) If ?2 L(W
at a critical point, then
C(C T C)? C T = Up?Up?T or Xr = 0.
4.3
Proof sketches of theorems
We now provide the proof sketch of Theorem 2.3 and Corollary 3.2. We complete the proofs in the
appendix.
4.3.1
Proof sketch of Theorem 2.3 (ii)
By case analysis, we show that any point that satisfies the necessary conditions and the definition of
a local minimum is a global minimum.
Case I: rank(WH ? ? ? W2 ) = p and dy ? p: If dy < p, Corollary 4.5 with k = H + 1 implies
the necessary condition of local minima that Xr = 0. If dy = p, Lemma 4.6 with k = H + 1
and k = 2, combined with the fact that R(C) ? R(Y X T ), implies the necessary condition that
Xr = 0. Therefore, we have the necessary condition of local minima, Xr = 0 . Interpreting
condition Xr = 0, we conclude that W achieving Xr = 0 is indeed a global minimum.
Case II: rank(WH ? ? ? W2 ) = p and dy > p: From Lemma 4.6, we have the necessary condition that C(C T C)? C T = Up?Up?T or Xr = 0. If Xr = 0, using the exact same proof as in
Case I, it is a global minimum. Suppose then that C(C T C)? C T = Up?Up?T . From Lemma 4.4
with k = H + 1, we conclude that p? , rank(C) = p. Then, from Lemma 4.2, we write
WH+1 ? ? ? W1 = Up UpT Y X T (XX T )?1 , which is the orthogonal projection onto the subspace
spanned by the p eigenvectors corresponding to the p largest eigenvalues following the ordinary
least square regression matrix. This is indeed the expression of a global minimum.
Case III: rank(WH ? ? ? W2 ) < p: We first show that if rank(C) ? min(p, dy ), every local minimum is a global minimum. Thus, we consider the case where rank(WH ? ? ? W2 ) < p and
rank(C) < min(p, dy ). In this case, by induction on k = {1, . . . , H +1}, we prove that we can have
rank(Wk ? ? ? W1 ) ? min(p, dy ) with arbitrarily small perturbation of each entry of Wk , . . . , W1
? ). Once this is proved, along with the results of Case I and Case
without changing the value of L(W
II, we can immediately conclude that any point satisfying the definition of a local minimum is a
global minimum.
We first prove the statement for the base case with k = 1 by using an expression of W1 that is
obtained by a first-order necessary condition: for an arbitrary L1 ,
W1 = (C T C)? C T Y X T (XX T )?1 + (I ? (C T C)? C T C)L1 .
7
By using Lemma 4.6 to obtain an expression of C, we deduce that we can have rank(W1 ) ?
min(p, dy ) with arbitrarily small perturbation of each entry of W1 without changing the loss value.
For the inductive step with k ? {2, . . . , H + 1}, from Lemma 4.4, we use the following necessary
condition for the Hessian to be (positive or negative) semidefinite at a critical point: for any k ?
{2, . . . , H + 1},
R((Wk?1 ? ? ? W2 )T ) ? R(C T C) or XrWH+1 ? ? ? Wk+1 = 0.
We use the inductive hypothesis to conclude that the first condition is false, and thus the second
condition must be satisfied at a candidate point of a local minimum. From the latter condition, with
extra steps, we can deduce that we can have rank(Wk Wk?1 ? ? ? W1 ) ? min(p, dx ) with arbitrarily
small perturbation of each entry of Wk while retaining the same loss value.
We conclude the induction, proving that we can have rank(C) ? rank(WH+1 ? ? ? W1 ) ?
min(p, dx ) with arbitrarily small perturbation of each parameter without changing the value of
? ). Upon such a perturbation, we have the case where rank(C) ? min(p, dy ), for which we
L(W
have already proven that every local minimum is a global minimum. Summarizing the above, any
point that satisfies the definition (and necessary conditions) of a local minimum is indeed a global
minimum. Therefore, we conclude the proof sketch of Theorem 2.3 (ii).
4.3.2 Proof sketch of Theorem 2.3 (i), (iii) and (iv)
We can prove the non-convexity and non-concavity of this function simply from its Hessian (Theorem 2.3 (i)). That is, we can show that in the domain of the function, there exist points at which
the Hessian becomes indefinite. Indeed, the domain contains uncountably many points at which the
Hessian is indefinite.
We now consider Theorem 2.3 (iii): every critical point that is not a global minimum is a saddle
point. Combined with Theorem 2.3 (ii), which is proven independently, this is equivalent to the
statement that there are no local maxima. We first show that if WH+1 ? ? ? W2 6= 0, the loss function
always has some strictly increasing direction with respect to W1 , and hence there is no local maximum. If WH+1 ? ? ? W2 = 0, we show that at a critical point, if the Hessian is negative semidefinite
(i.e., a necessary condition of local maxima), we can have WH+1 ? ? ? W2 6= 0 with arbitrarily small
perturbation without changing the loss value. We can prove this by induction on k = 2, . . . , H + 1,
similar to the induction in the proof of Theorem 2.3 (ii). This means that there is no local maximum.
Theorem 2.3 (iv) follows Theorem 2.3 (ii)-(iii) and the analyses for Case I and Case II in the proof
? ) 0 at a critical point, W is a global
of Theorem 2.3 (ii); when rank(WH ? ? ? W2 ) = p, if ?2 L(W
minimum.
4.3.3 Proof sketch of Corollary 3.2
Since the activations are assumed to be random and independent, the effect of nonlinear activations
? ).
disappear by taking expectation. As a result, the loss function L(W ) is reduced to L(W
5
Conclusion
In this paper, we addressed some open problems, pushing forward the theoretical foundations of
deep learning and non-convex optimization. For deep linear neural networks, we proved the aforementioned conjecture and more detailed statements with more generality. For deep nonlinear neural
networks, when compared with the previous work, we proved a tighter statement (in the way explained in section 3) with more generality (dy can vary) and with strictly weaker model assumptions
(only two assumptions out of seven). However, our theory does not yet directly apply to the practical situation. To fill the gap between theory and practice, future work would further discard the
remaining two out of the seven assumptions made in previous work. Our new understanding of the
deep linear models at least provides the following theoretical fact: the bad local minima would arise
in a deep nonlinear model but only as an effect of adding nonlinear activations to the corresponding
deep linear model. Thus, depending on the nonlinear activation mechanism and architecture, we
would be able to efficiently train deep models.
Acknowledgments
The author would like to thank Prof. Leslie Kaelbling, Quynh Nguyen, Li Huan and Anirbit Mukherjee for their thoughtful comments on the paper. We gratefully acknowledge support from NSF grant
1420927, from ONR grant N00014-14-1-0486, and from ARO grant W911NF1410433.
8
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9
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5,651 | 6,113 | Learning to Poke by Poking: Experiential Learning of
Intuitive Physics
Pulkit Agrawal?
Ashvin Nair?
Pieter Abbeel
Jitendra Malik
Sergey Levine
Berkeley Artificial Intelligence Research Laboratory (BAIR)
University of California Berkeley
{pulkitag,anair17,pabbeel,malik,svlevine}@berkeley.edu
Abstract
We investigate an experiential learning paradigm for acquiring an internal model of
intuitive physics. Our model is evaluated on a real-world robotic manipulation task
that requires displacing objects to target locations by poking. The robot gathered
over 400 hours of experience by executing more than 100K pokes on different
objects. We propose a novel approach based on deep neural networks for modeling
the dynamics of robot?s interactions directly from images, by jointly estimating
forward and inverse models of dynamics. The inverse model objective provides
supervision to construct informative visual features, which the forward model can
then predict and in turn regularize the feature space for the inverse model. The
interplay between these two objectives creates useful, accurate models that can
then be used for multi-step decision making. This formulation has the additional
benefit that it is possible to learn forward models in an abstract feature space and
thus alleviate the need of predicting pixels. Our experiments show that this joint
modeling approach outperforms alternative methods.
1
Introduction
Humans can effortlessly manipulate previously unseen objects in novel ways. For example, if a
hammer is not available, a human might use a piece of rock or back of a screwdriver to hit a nail.
What enables humans to easily perform such tasks that machines struggle with? One possibility is that
humans possess an internal model of physics (i.e. ?intuitive physics? (Michotte, 1963; McCloskey,
1983)) that allows them to reason about physical properties of objects and forecast their dynamics
under the effect of applied forces. Such models can be used to transform a given task into a search
problem in a manner similar to how moves can be planned in a game of chess or tic-tac-toe by
searching through the game tree. Because the search algorithm is independent of task semantics,
solutions to different and possibly new tasks can be determined using the same mechanism.
In human development, it is well known that infants spend years worth of time playing with objects
in a seemingly random manner with no specific end goal (Smith & Gasser, 2005; Gopnik et al., 1999).
One hypothesis is that infants distill this experience into intuitive physics models that predict how
their actions effect the motion of objects. Once learnt, these models could be used for planning
actions for achieving novel goals later in life. Inspired by this hypothesis, in this work we investigate
whether a robot can use it?s own experience to learn an intuitive model of physics that is also effective
for planning actions. In our setup (see Figure 1), a Baxter robot interacts with objects kept on a table
in front of it by randomly poking them. The robot records the visual state of the world before and
after it executes a poke in order to learn a mapping between its actions and the accompanying change
in visual state caused by object motion. To date our robot has interacted with objects for more than
400 hours and in process collected more than 100K pokes on 16 distinct objects.
?
equal contribution, authors are listed in alphabetical order.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
CNN
CNN
Predict Poke
Figure 1: Infants spend years worth of time playing with objects in a seemingly random manner.
They might use this experience to learn a model of physics relating their actions with the resulting
motion of objects. Inspired by this hypothesis, we let a robot interact with objects by randomly
poking them. The robot pokes objects and records the visual state before (left) and after (right) the
poke. The triplet of before image, after image and the applied poke is used to train a neural network
(center) for learning the mapping between actions and the accompanying change in visual state. We
show that this learn model can be used to push objects into a desired configuration.
What kind of a model should the robot learn from it?s experience? One possibility is to build a model
that predicts the next visual state from the current visual state and the applied force (i.e forward
dynamics model). This is challenging because predicting the value of every pixel in the next image is
non-trivial in real world scenarios. Moreover, in most cases it is not the precise pixel values that are of
interest, but the occurrence of a more abstract event. For example, predicting that a glass jar will break
when pushed from the table onto the ground is of greater interest (and easier) than predicting exactly
how every piece of shattered glass will look. The difficulty, however, is that supervision for such
abstract concepts or events is not readily available in unsupervised settings such as ours. In this work,
we propose one solution to this problem by jointly training forward and inverse dynamics models. A
forward model predicts the next state from the current state and action, and an inverse model predicts
the action given the initial and target state. In joint training, the inverse model objective provides
supervision for transforming image pixels into an abstract feature space, which the forward model
can then predict. The inverse model alleviates the need for the forward model to make predictions in
the pixel space and the forward model in turn regularizes the feature space for the inverse model.
We empirically show that the joint model allows the robot to generalize and plan actions for achieving
tasks with significantly different visual statistics as compared to the data used in the learning phase.
Our model can be used for multi step decision making and displace objects with novel geometry
and texture into desired goal locations that are much farther apart as compared to position of objects
before and after a single poke. We probe the joint modeling approach further using simulation studies
and show that the forward model regularizes the inverse model.
2
Data
Figure 1 shows our experimental setup. The robot is equipped with a Kinect camera and a gripper for
poking objects kept on a table in front of it. At any given time there were 1-3 objects chosen from a
set of 16 distinct objects present on the table. The robot?s coordinate system was as following: X and
Y axis represented the horizontal and vertical axes, while the Z axis pointed away from the robot.
The robot poked objects by moving its finger along the XZ plane at a fixed height from the table.
Poke Representation: For collecting a sample of interaction data, the robot first selects a random
target point in its field of view to poke. One issue with random poking is that most pokes are executed
in free space which severely slows down collection of interesting interaction data. For speedy data
collection, a point cloud from the Kinect depth camera was used to only chose points that lie on any
object except the table. Point cloud information was only used during data collection and at test time
our system only requires RGB image data. After selecting a random point to poke (p) on the object,
2
Figure 2: These images depict the robot in the process of displacing the bottle away from the indicated
dotted line. In the middle of the poke, the object flips and ends up moving in the wrong direction.
Such occurrences are common because the real world objects have complex geometric and material
properties. This makes learning manipulation strategies without prior knowledge very challenging.
the robot randomly samples a poke direction (?) and length (l). Kinematically, the poke is defined
by points p1 , p2 that are 2l distance from p in the directions ?o , (180 + ?)o respectively. The robot
executes the poke by moving its finger from p1 to p2 .
Our robot can run autonomously 24x7 without any human intervention. Sometimes when objects are
poked they move as expected, but other times due to non-linear interaction between the robot?s finger
and the object they move in unexpected ways as shown in Figure 2. Any model of the poking data
must deal with such non-linear interactions (see project website for more examples). A small amount
of data in the early stages of the project was collected on a table with a green background, but most
of our data was collected in a wooden arena with walls for preventing objects from falling down. All
results in this paper are from data collected only from the wooden arena.
3
Method
The forward and inverse models can be formally described by equations 1 and 2, respectively. The
notation is as following: xt , ut are the world state and action applied time step t, x
?t+1 , u
?t+1 are the
predicted state and actions, and Wf wd and Winv are parameters of the functions F and G that are
used to construct the forward and inverse models.
x
?t+1 = F (xt , ut ; Wf wd )
(1)
u?t = G(xt , xt+1 ; Winv )
(2)
Given an initial and goal state, inverse models provide a direct mapping to actions required for
achieving the goal state in one step (if feasible). However, multiple possible actions can transform
the world from one visual state to another. For example, an object can appear in a certain part of the
visual field if the agent moves or if the agent uses its arms to move the object. This multi-modality
in the action space makes the learning hard. On the other hand, given xt and ut , there exists a next
state xt+1 that is unique up to dynamics noise. This suggests that forward models might be easier to
learn. However, learning forward models in image space is hard because predicting the value of each
pixel in the future frames is a non-trivial problem with no known good solution. However, in most
scenarios we are not interested in predicting every pixel, but predicting the occurrence of a more
abstract event such as object motion, change in object pose etc.
The ability to learn an abstract task relevant feature space should make it easier to learn a forward
dynamics model. One possible approach is to learn a dynamics model in the feature representation of
a higher layer of a deep neural network trained to perform image classification (say on ImageNet)
(Vondrick et al., 2016). However, this is not a general way of learning task relevant features and it is
unclear whether features adept at object recognition are also optimal for object manipulation. The
alternative of adapting higher layer features of a neural network while simultaneously optimizing
for the prediction loss leads to a degenerate solution of all the features reducing to zero, since the
prediction loss in this case is also zero. Our key observation is that this degenerate solution can be
avoided by imposing the constraint that it should be possible to infer the the executed action (ut )
from the feature representation of two images obtained before (xt ) and after (xt+1 ) the action (ut ) is
applied (i.e. optimizing the inverse model). This formulation provides a general mechanism for using
general purpose function approximators such as deep neural networks for simultaneously learning a
task relevant feature space and forecasting the future outcome of actions in this learned space.
A second challenge in using forward models is that inferring the optimal action inevitably leads to
finding a solution to non-convex problems that are subject to local optima. The inverse model does
not suffers from this drawback as it directly outputs the required action. These considerations suggest
that inverse and forward models have complementary strengths and therefore it is worthwhile to
investigate training a joint model of inverse and forward dynamics.
3
(c)
pt , ?t , lt
x
?t+1
(a)
It
p?t
xt
??t
(b)
It+1
?
lt
xt+1
Figure 3: (a) The collection of objects in the training set poked by the robot. (b) Example pairs
of before (It ) and after images (It+1 ) after a single poke was made by the robot. (c) A Siamese
convolutional neural network was trained to predict the poke location (pt ), angle (?t ) and length (lt )
required to transform objects in the image at the tth time step (It ) into their state in It+1 . Images It
and It+1 are transformed into their latent feature representations (xt , xt+1 ) by passing them through
a series of convolutional layers. For building the inverse model, xt , xt+1 are concatenated and passed
through fully connected layers to predict the discretized poke. For building the forward model, the
action ut = {pt , ?t , lt } and xt are passed through a series of fully connected layers to predict xt+1 .
3.1
Model
A deep neural network is used to simultaneously learn a model of forward and inverse dynamics (see
Figure 3). A tuple of before image (It ), after image (It+1 ) and the robot?s action (ut ) constitute one
training sample. Input images at consequent time steps (It , It+1 ) are transformed into their latent
feature representations (xt , xt+1 ) by passing them through a series of five convolutional layers with
the same architecture as the first five layers of AlexNet (Krizhevsky et al., 2012). For building the
inverse model, xt , xt+1 are concatenated and passed through fully connected layers to conditionally
predict the poke location (pt ), angle (?t ) and length (lt ) separately. For modeling multimodal poke
distributions, poke location, angle and length of poke are discretized into a 20 ? 20 grid, 36 bins and
11 bins respectively. The 11th bin of the poke length is used to denote no poke. For building the
forward model, the feature representation of the before image (xt ) and the action (ut ; real-valued
vector without discretization) are passed into a sequence of fully connected layer that predicts the
feature representation of the next image (xt+1 ). Training is performed to optimize the loss defined in
equation 3 below.
Ljoint = Linv (ut , u?t , W ) + ?Lf wd (xt+1 , x
?t+1 , W )
(3)
Linv is a sum of three cross entropy losses between the actual and predicted poke location, angle
and length. Lf wd is a L1 loss between the predicted (?
xt+1 ) and the ground truth (xt+1 ) feature
representation of the after image (It+1 ). W are the parameters of the neural network. We used
? = 0.1 in all our experiments. We call this the joint model and we compare its performance against
the inverse only model that was trained by setting ? = 0 in equation 3. More details about model
training are provided in the supplementary materials.
3.2
Evaluation Procedure
One way to test the learnt model is to provide the robot with an initial and goal image and task it to
apply pokes that would displace objects into the configuration shown in the goal image. If the robot
succeeds at achieving the goal configuration when the visual statistics of the pair of initial and goal
image is similar to before and after image in the training set, then this would not be a convincing
demonstration of generalization. However, if the robot is able to displace objects into goal positions
that are much farther apart as compared to position of objects before and after a single poke then it
might suggest that our model has not simply overfit but has learnt something about the underlying
physics of how objects move when poked. This suggestion would be further strengthened if the robot
is also able to push objects with novel geometry and texture in presence of multiple distractor objects.
If the objects in the initial and goal image are farther apart than the maximum distance that can be
pushed by a single poke, then the model would be required to output a sequence of pokes. We use a
4
(a) Greedy Planner
(b) Blob Model
Action Predictor
Current Image (It )
(c) Pose Error Evaluation
Goal Image (Ig )
Angle (?)
Next Image (It+1 )
Figure 4: (a) Greedy planner is used to output a sequence of pokes to displace the objects from their
configuration in initial to the goal image. (b) The blob model first detects the location of objects in
the current and goal image. Based on object positions, location and angle of the poke is computed
and then executed by the robot. The obtained next and goal image are used to compute the next poke
and this process is repeated iteratively. (c) The error of the models in poking objects to their correct
pose is measured as the angle between the major axis of the objects in the final and goal images.
greedy planning method (see Figure 4(a)) to output a sequence of pokes. First, images depicting the
initial and goal state are passed through the learnt model to predict the poke which is then executed
by the robot. Then, the image depicting the current world state (i.e. the current image) and the goal
image are fed again into the model to output a poke. This process is repeated iteratively unless the
robot predicts a no-poke (see section 3.1) or a maximum number of 10 pokes is reached.
Error Metrics: In all our experiments, the initial and goal images differ in the position of only a
single object. The location and pose of the object in the final image after the robot stops and the goal
image are compared for quantitative evaluation. The location error is the Euclidean distance between
the object locations. In order to account for different object distances in the initial and goal state, we
use relative instead of absolute location error. Pose error is defined as the angle (in degrees) between
the major axis of the objects in the final and goal images (see Figure 4(c)). Please see supplementary
materials for further details.
3.3
Blob Model
We compared the performance of the learnt model against a baseline blob model. This model first
estimates object locations in current and goal image using template based object detector. It then uses
the vector difference between these to compute the location, angle and length of poke executed by
the robot (see supplementary materials for details). In a manner similar to greedy planning with the
learnt model, this process is repeated iteratively until the object gets closer to the desired location in
the goal image by a pre-defined threshold or a maximum number of pokes is reached.
4
Results
The robot was tasked to displace objects in an initial image into their configuration depicted in a
goal image (see Figure 5). The three rows in the figure show the performance when the robot is
asked to displace an object (Nutella bottle) present in the training set, an object (red cup) whose
geometry is different from objects in the training set and when the task is to move an object around
an obstacle. These examples are representative of the robot?s performance and more examples can be
found on the project website. It can be seen that the robot is able to successfully poke objects present
in the training set and objects with novel geometry and texture into desired goal locations that are
significantly farther than pair of before and after images used in the training set.
Row 2 in Figure 5 also shows that the robot?s performance in unaffected by the presence of distractor
objects that occupy the same location in the current and goal images. These results indicate that the
learnt model allows the robot to perform tasks that show generalization beyond the training set (i.e.
poking object by small distances). Row 3 in Figure 5 depicts an example where the robots fails to
push the object around an obstacle (yellow object). The robot acts greedily and ends up pushing the
obstacle along with the object. One more side-effect of greedy planning is zig-zag instead of straight
trajectories taken by the object between its initial and goal locations. Investigating alternatives to
5
Initial State
Goal State
Training set
object
Unseen
object
End of Sequence (EoS)
Limitation
(EoS)
Figure 5: The robot is able to successfully displace objects in the training set (row 1; Nutella bottle)
and objects with previously unseen geometry (row 2; red cup) into goal locations that are significantly
farther than pair of before and after images used in the training set. The robot is unable to push
objects around obstacles (row 3; limitation of greedy planning).
greedy planning, such as using the learnt forward model for planning pokes is a very interesting
direction for future research.
What representation could the robot have learnt that allows it to generalize? One possibility is that
the robot ignores the geometry of the object and only infers the location of the object in the initial and
goal image and uses the difference vector between object locations to deduce what poke to execute.
This strategy is invariant to absolute distance between the object locations and is therefore capable
of explaining the observed generalization to large distances. While we cannot prove that the model
has learnt to detect object location, nearest neighbor visualizations of the learnt feature space clearly
suggest sensitivity to object location (see supplementary materials). This is interesting because the
robot received no direct supervision to locate objects.
Because different objects have different geometries, they need to be poked at different places to move
them in the same manner. For example, a Nutella bottle can be reliably moved forward without
rotating the bottle by poking it on the side along the direction toward its center of mass, whereas a
hammer is reliably moved by poking it where the hammer head meets the handle. Pushing an object to
a desired pose is harder and requires a more detailed understanding of object geometry in comparison
to pushing the object to a desired location. In order to test whether the learnt model represents any
information about object geometry, we compared its performance against the baseline blob model
(see section 3.3 and figure 4(b)) that ignores object geometry. For this comparison, the robot was
tasked to push objects to a nearby goal by making only a single poke (see supplementary materials
for more details). Results in Figure 6(a) show that both the inverse and joint model outperform the
blob model. This indicates that in addition to representing information about object location, the
learn models also represent some information about object geometry.
4.1
Forward model regularizes the inverse model
We tested the hypothesis whether the forward model regularizes the feature space learnt by the
inverse model in a 2-D simulation environment where the agent interacted with a red rectangular
object by poking it by small forces. The rectangle was allowed to freely translate and rotate (Figure
6(c)). Model training was performed using an architecture similar to the one described in section 3.1.
Additional details about the experimental setup, network architecture and training procedure for the
simulation experiments are provided in the supplementary materials. Figure 6(c) shows that when
less training data (10K, 20K examples) is available the joint model outperforms the inverse model
and reaches closer to the goal state in fewer steps (i.e. fewer actions). This shows that indeed the
forward model regularizes the inverse model and helps generalize better. However, when the number
of training examples is increased to 100K both models are at par. This is not surprising because
training with more data often results in better generalization and thus the inverse model is no longer
reliant on the forward model for the regularization.
Evaluation on the real robot supports the findings from the simulation experiments. Figure 6(b) shows
that in a test of generalization, when an object is required to be displaced by a long distance, the
joint model outperforms the inverse model. Similar performance of joint and blob model at this task
is not surprising because even if the pokes are somewhat inaccurate but generally in the direction
6
(a) Pose error for nearby goals
Initial
State
Goal
State
1.0
Inverse Model, #Train 10K
Joint Model, #Train 10K
Inverse Model, #Train 20K
Joint Model, #Train 20K
Inverse Model, #Train 100K
Joint Model, #Train 100K
0.9
20
Blob Model
40
60
Inverse Model
Relative Location Error
0
Joint Model
0.8
0.7
0.6
0.5
0.4
0.3
0.0
0.1
0.2
0.3
0.2
0
0.4
(b) Relative location error for far away goals
1
2
Number of Steps
3
4
(c) Simulation experiments
Figure 6: (a) Inverse and Joint model are more accurate than the blob model at pushing objects
towards the desired pose. (b) The joint model outperforms the inverse-only model when the robot
is tasked to push objects by distances that are significantly larger than object distance in before and
after images used in the training set (i.e. a test of generalization). (c) Simulation studies reveal that
when less number of training examples (10K, 20K) are available the joint model outperforms the
inverse model and the performance is comparable with larger amount of data (100K). This result
indicates that the forward model regularizes the inverse model.
from object?s current to goal location, the object might traverse a zig-zag path but it would eventually
reach the goal. The joint model is however more accurate at displacing objects into their correct pose
as compared to the blob model (Figure 6(a)).
5
Related Work
Learning visual control policies using reinforcement learning for tasks such as playing Atari
games (Mnih et al., 2015), controlling robots in simulation (Lillicrap et al., 2016) and in the real
world (Levine et al., 2016a) is of growing interest. However, these methods are model free and learn
goal specific policies, which makes it difficult to repurpose the learned policies for new tasks. In
contrast, the aim of this work is to learn intuitive physical models of object interaction which we show
allow the agent to generalize. Other works in visual control have relied on model free methods that
operate on a a low-dimensional state representation of images obtained using autoencoders (Lange
et al., 2012; Finn et al., 2016; Kietzmann & Riedmiller, 2009). It is unclear that features obtained by
optimizing pixelwise reconstruction are necessarily well suited for model based control.
Learning to grasp objects by trial and error from large amounts of interaction data has recently
been explored (Pinto & Gupta, 2016; Levine et al., 2016b). These methods aim to acquire a policy
for solving a single concrete task, while our work is concerned with learning a general predictive
model that could be used to achieve a variety of goals at test time. When an object is grasped, it is
possible to fully control the state of the grasped object. However, in non-prehensile manipulation
(i.e. manipulation without grasping (LaValle, 2006)) such as poking, the object state is not directly
controllable which makes manipulation by poking harder than grasping (Dogar & Srinivasa, 2012).
Learning a model of poking was considered by (Pinto et al., 2016), but their goal was to learn visual
representations and they did not consider using the learnt models to displace objects to goal locations.
A good review of model based control can be found in (Mayne, 2014) and (Jordan & Rumelhart,
1992; Wolpert et al., 1995) provide interesting perspectives. A model based deep learning method for
cutting vegetables was considered by (Lenz et al., 2015). However, as their system operated on the
robotic state space instead of vision and is thus limited in its generality. Model based control from
visual inputs was considered by (Fragkiadaki et al., 2016; Wahlstr?m et al., 2015; Watter et al., 2015;
Oh et al., 2015) in synthetic domains of manipulating two degree of freedom robotic arm, inverted
pendulum, billiards and Atari games. In contrast, we tackle manipulation of complex, compressible
real world objects. Instead of learning a model of physics, some recents works (Wu et al., 2015;
Mottaghi et al., 2016; Lerer et al., 2016) have proposed to use Newtonian physics in combination
with neural networks to forecast object dynamics.
7
In robotic manipulation, a number of prior methods have been proposed that use hand-designed visual
features and known object poses or key locations to plan and execute pushes and other non-prehensile
manipulations (Kopicki et al., 2011; Lau et al., 2011; Meri?li et al., 2015). Unlike these methods,
the goal in our work is to learn an intuitive physics model for pushing only from raw images, thus
allowing the robot to learn by exploring the environment on its own without human intervention.
6
Discussion and Future Work
In this work we propose to learn ?intuitive" model of physics using interaction data. An alternative is
to represent the world in terms of a fixed set of physical parameters such as mass, friction coefficient,
normal forces etc and use a physics simulator for computing object dynamics from this representation
(Kolev & Todorov, 2015; Mottaghi et al., 2016; Wu et al., 2015; Hamrick et al., 2011). This approach
is general because physics simulators inevitably use Newton?s laws that apply to a wide range of
physical phenomenon ranging from orbital motion of planets to a swinging pendulum. Estimating
parameters such as as mass, friction coefficient etc. from sensory data is subject to errors, and it is
possible that one parameterization is easier to estimate or more robust to sensory noise than another.
For example, the conclusion that objects with feather like appearance fall slower than objects with
stone like appearance can be reached by either correlating visual texture to the speed of falling objects,
or by computing the drag force after estimating the cross section area of the object. Depending on
whether estimation of visual texture or cross section area is more robust, one parameterization will
result in more accurate predictions than the other. Pre-defining a set of parameters for predicting
object dynamics, which is required by ?simulator-based" approach might therefore lead to suboptimal
solutions that are less robust.
For many practical object manipulation tasks of interest, such as re-arranging objects, cutting
vegetables, folding clothes, and so forth, small errors in execution are acceptable. The key challenge
is robust performance in the face of varying environmental conditions. This suggests that a more
robust but a somewhat imprecise model may in fact be desirable over a less robust and a more
precise model. While the arguments presented above suggest that intuitive physics models are likely
to be more robust than simulator based models, quantifying the robustness of these models is an
interesting direction for future work. Furthermore, it is non-trivial to use simulator based models
for manipulating deformable objects such as clothes and ropes because simulation of deformable
objects is hard and also also requires representing objects by heavily handcrafted features that are
unlikely to generalize across objects. The intuitive physics approach does not make any object
specific assumptions and can be easily extended to work with deformable objects. This approach is
in the spirit of recent successful deep learning techniques in computer vision and speech processing
that learn features directly from data, whereas the simulator based physics approach is more similar
to using hand-designed features. Current methods for learning intuitive physics models, such as ours
are data inefficient and it is possible that combining intuitive and simulator based approaches leads to
better models than either approach by itself.
In poking based interaction, the robot does not have full control of the object state which makes it
harder to predict and plan for the outcome of an action. The models proposed in this work generalize
and are able to push objects into their desired location. However, performance on setting objects
in the desired pose is not satisfactory, possibly because of the robot only executing pokes in large,
discrete time steps. An interesting area of future investigation is to use continuous time control with
smaller pokes that are likely to be more predictable than the large pokes used in this work. Further,
although our approach is evaluated on a specific robotic manipulation task, there are no task specific
assumptions, and the techniques are applicable to other tasks. In future, it would be interesting to
see how the proposed approach scales with more complex environments, diverse object collections,
different manipulation skills and to other non-manipulation based tasks, such as navigation. Other
directions for future investigation include the use of forward model for planning and developing
better strategies for data collection than random interaction.
Supplementary Materials: and videos can be found at http://ashvin.me/pokebot-website/.
Acknowledgement: We thank Alyosha Efros for inspiration and fruitful discussions throughout this
work. The title of this paper is partly influenced by the term ?pokebot" that Alyosha has been using
for several years. We thank Ruzena Bajcsy for access to Baxter robot and Shubham Tulsiani for
helpful comments. This work was supported in part by ONR MURI N00014-14-1-0671, ONR YIP
8
and by ARL through the MAST program. We are grateful to NVIDIA corporation for donating K40
GPUs and providing access to the NVIDIA PSG cluster.
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9
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5,652 | 6,114 | Weight Normalization: A Simple Reparameterization
to Accelerate Training of Deep Neural Networks
Tim Salimans
OpenAI
tim@openai.com
Diederik P. Kingma
OpenAI
dpkingma@openai.com
Abstract
We present weight normalization: a reparameterization of the weight vectors
in a neural network that decouples the length of those weight vectors from their
direction. By reparameterizing the weights in this way we improve the conditioning
of the optimization problem and we speed up convergence of stochastic gradient
descent. Our reparameterization is inspired by batch normalization but does not
introduce any dependencies between the examples in a minibatch. This means
that our method can also be applied successfully to recurrent models such as
LSTMs and to noise-sensitive applications such as deep reinforcement learning
or generative models, for which batch normalization is less well suited. Although
our method is much simpler, it still provides much of the speed-up of full batch
normalization. In addition, the computational overhead of our method is lower,
permitting more optimization steps to be taken in the same amount of time. We
demonstrate the usefulness of our method on applications in supervised image
recognition, generative modelling, and deep reinforcement learning.
1
Introduction
Recent successes in deep learning have shown that neural networks trained by first-order gradient
based optimization are capable of achieving amazing results in diverse domains like computer vision,
speech recognition, and language modelling [7]. However, it is also well known that the practical
success of first-order gradient based optimization is highly dependent on the curvature of the objective
that is optimized. If the condition number of the Hessian matrix of the objective at the optimum is
low, the problem is said to exhibit pathological curvature, and first-order gradient descent will have
trouble making progress [22, 32]. The amount of curvature, and thus the success of our optimization,
is not invariant to reparameterization [1]: there may be multiple equivalent ways of parameterizing
the same model, some of which are much easier to optimize than others. Finding good ways of
parameterizing neural networks is thus an important problem in deep learning.
While the architectures of neural networks differ widely across applications, they are typically mostly
composed of conceptually simple computational building blocks sometimes called neurons: each such
neuron computes a weighted sum over its inputs and adds a bias term, followed by the application of
an elementwise nonlinear transformation. Improving the general optimizability of deep networks is a
challenging task [6], but since many neural architectures share these basic building blocks, improving
these building blocks improves the performance of a very wide range of model architectures and
could thus be very useful.
Several authors have recently developed methods to improve the conditioning of the cost gradient for
general neural network architectures. One approach is to explicitly left multiply the cost gradient
with an approximate inverse of the Fisher information matrix, thereby obtaining an approximately
whitened natural gradient. Such an approximate inverse can for example be obtained by using a
Kronecker factored approximation to the Fisher matrix and inverting it (KFAC, [23]), by using an
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
approximate Cholesky factorization of the inverse Fisher matrix (FANG, [10]), or by whitening the
input of each layer in the neural network (PRONG, [5]).
Alternatively, we can use standard first order gradient descent without preconditioning, but change
the parameterization of our model to give gradients that are more like the whitened natural gradients
of these methods. For example, Raiko et al. [27] propose to transform the outputs of each neuron
to have zero output and zero slope on average. They show that this transformation approximately
diagonalizes the Fisher information matrix, thereby whitening the gradient, and that this leads to
improved optimization performance. Another approach in this direction is batch normalization [14],
a method where the output of each neuron (before application of the nonlinearity) is normalized by
the mean and standard deviation of the outputs calculated over the examples in the minibatch. This
reduces covariate shift of the neuron outputs and the authors suggest it also brings the Fisher matrix
closer to the identity matrix.
Following this second approach to approximate natural gradient optimization, we propose a simple
but general method, called weight normalization, for improving the optimizability of the weights
of neural network models. The method is inspired by batch normalization, but it is a deterministic
method that does not share batch normalization?s property of adding noise to the gradients. In
addition, the overhead imposed by our method is lower: no additional memory is required and the
additional computation is negligible. The method show encouraging results on a wide range of deep
learning applications.
2
Weight Normalization
We consider standard artificial neural networks where the computation of each neuron consists in
taking a weighted sum of input features, followed by an elementwise nonlinearity:
y = ?(w ? x + b),
(1)
where w is a k-dimensional weight vector, b is a scalar bias term, x is a k-dimensional vector of input
features, ?(.) denotes an elementwise nonlinearity such as the rectifier max(., 0), and y denotes the
scalar output of the neuron.
After associating a loss function to one or more neuron outputs, such a neural network is commonly
trained by stochastic gradient descent in the parameters w, b of each neuron. In an effort to speed up
the convergence of this optimization procedure, we propose to reparameterize each weight vector w
in terms of a parameter vector v and a scalar parameter g and to perform stochastic gradient descent
with respect to those parameters instead. We do so by expressing the weight vectors in terms of the
new parameters using
g
w=
v
(2)
||v||
where v is a k-dimensional vector, g is a scalar, and ||v|| denotes the Euclidean norm of v. This
reparameterization has the effect of fixing the Euclidean norm of the weight vector w: we now
have ||w|| = g, independent of the parameters v. We therefore call this reparameterizaton weight
normalization.
The idea of normalizing the weight vector has been proposed before (e.g. [31, 33]) but earlier work
typically still performed optimization in the w-parameterization, only applying the normalization
after each step of stochastic gradient descent. This is fundamentally different from our approach: we
propose to explicitly reparameterize the model and to perform stochastic gradient descent in the new
parameters v, g directly. Doing so improves the conditioning of the gradient and leads to improved
convergence of the optimization procedure: By decoupling the norm of the weight vector (g) from the
direction of the weight vector (v/||v||), we speed up convergence of our stochastic gradient descent
optimization, as we show experimentally in section 5.
Instead of working with g directly, we may also use an exponential parameterization for the scale, i.e.
g = es , where s is a log-scale parameter to learn by stochastic gradient descent. Parameterizing the g
parameter in the log-scale is more intuitive and more easily allows g to span a wide range of different
magnitudes. Empirically, however, we did not find this to be an advantage. In our experiments,
the eventual test-set performance was not significantly better or worse than the results with directly
learning g in its original parameterization, and optimization was slightly slower.
2
2.1
Gradients
Training a neural network in the new parameterization is done using standard stochastic gradient
descent methods. Here we differentiate through (2) to obtain the gradient of a loss function L with
respect to the new parameters v, g. Doing so gives
?g L =
?w L ? v
,
||v||
?v L =
g
g?g L
v,
?w L ?
||v||
||v||2
(3)
where ?w L is the gradient with respect to the weights w as used normally.
Backpropagation using weight normalization thus only requires a minor modification to the usual
backpropagation equations, and is easily implemented using standard neural network software, either
by directly specifying the network in terms of the v, g parameters and relying on auto-differentiation,
or by applying (3) in a post-processing step. We provide reference implementations using both
approaches for Theano, Tensorflow and Keras at https://github.com/openai/weightnorm.
Unlike with batch normalization, the expressions above are independent of the minibatch size and
thus cause only minimal computational overhead.
An alternative way to write the gradient is
?v L =
g
Mw ?w L,
||v||
with
Mw = I ?
ww0
,
||w||2
(4)
where Mw is a projection matrix that projects onto the complement of the w vector. This shows
that weight normalization accomplishes two things: it scales the weight gradient by g/||v||, and it
projects the gradient away from the current weight vector. Both effects help to bring the covariance
matrix of the gradient closer to identity and benefit optimization, as we explain below.
Due to projecting away from w, the norm of v grows monotonically with the number of weight
updates when learning a neural network with weight normalization using standard gradient descent
without momentum: Let v0 = v + ?v denote our parameter update, with ?v ? ?v L (steepest
ascent/descent), then ?v is necessarily orthogonal to the current weight vector w since we project
away from it when calculating ?v L (equation 4). Since v is proportional to w, the update is thus also
orthogonal to v and increases its norm by the Pythagorean
theorem. Specifically,
if ||?v||/||v|| = c
p
?
the new weight vector will have norm ||v0 || = ||v||2 + c2 ||v||2 = 1 + c2 ||v|| ? ||v||. The rate
of increase will depend on the the variance of the weight gradient. If our gradients are noisy, c will be
high and the norm of v will quickly increase,?which in turn will decrease the scaling factor g/||v||.
If the norm of the gradients is small, we get 1 + c2 ? 1, and the norm of v will stop increasing.
Using this mechanism, the scaled gradient self-stabilizes its norm. This property does not strictly
hold for optimizers that use separate learning rates for individual parameters, like Adam [15] which
we use in experiments, or when using momentum. However, qualitatively we still find the same effect
to hold.
Empirically, we find that the ability to grow the norm ||v|| makes optimization of neural networks
with weight normalization very robust to the value of the learning rate: If the learning rate is too
large, the norm of the unnormalized weights grows quickly until an appropriate effective learning rate
is reached. Once the norm of the weights has grown large with respect to the norm of the updates, the
effective learning rate stabilizes. Neural networks with weight normalization therefore work well
with a much wider range of learning rates than when using the normal parameterization. It has been
observed that neural networks with batch normalization also have this property [14], which can also
be explained by this analysis.
By projecting the gradient away from the weight vector w, we also eliminate the noise in that
direction. If the covariance matrix of the gradient with respect to w is given by C, the covariance
matrix of the gradient in v is given by D = (g 2 /||v||2 )Mw CMw . Empirically, we find that w is
often (close to) a dominant eigenvector of the covariance matrix C: removing that eigenvector then
gives a new covariance matrix D that is closer to the identity matrix, which may further speed up
learning.
3
2.2
Relation to batch normalization
An important source of inspiration for this reparameterization is batch normalization [14], which
normalizes the statistics of the pre-activation t for each minibatch as
t0 =
t ? ?[t]
,
?[t]
with ?[t], ?[t] the mean and standard deviation of the pre-activations t = v ? x. For the special
case where our network only has a single layer, and the input features x for that layer are whitened
(independently distributed with zero mean and unit variance), these statistics are given by ?[t] = 0
and ?[t] = ||v||. In that case, normalizing the pre-activations using batch normalization is equivalent
to normalizing the weights using weight normalization.
Convolutional neural networks usually have much fewer weights than pre-activations, so normalizing
the weights is often much cheaper computationally. In addition, the norm of v is non-stochastic, while
the minibatch mean ?[t] and variance ? 2 [t] can in general have high variance for small minibatch
size. Weight normalization can thus be viewed as a cheaper and less noisy approximation to batch
normalization. Although exact equivalence does not usually hold for deeper architectures, we still
find that our weight normalization method provides much of the speed-up of full batch normalization.
In addition, its deterministic nature and independence on the minibatch input also means that our
method can be applied more easily to models like RNNs and LSTMs, as well as noise-sensitive
applications like reinforcement learning.
3
Data-Dependent Initialization of Parameters
Besides a reparameterization effect, batch normalization also has the benefit of fixing the scale of the
features generated by each layer of the neural network. This makes the optimization robust against
parameter initializations for which these scales vary across layers. Since weight normalization lacks
this property, we find it is important to properly initialize our parameters. We propose to sample the
elements of v from a simple distribution with a fixed scale, which is in our experiments a normal
distribution with mean zero and standard deviation 0.05. Before starting training, we then initialize
the b and g parameters to fix the minibatch statistics of all pre-activations in our network, just like
in batch normalization, but only for a single minibatch of data and only during initialization. This
can be done efficiently by performing an initial feedforward pass through our network for a single
minibatch of data X, using the following computation at each neuron:
v?x
t ? ?[t]
t=
,
and y = ?
,
(5)
||v||
?[t]
where ?[t] and ?[t] are the mean and standard deviation of the pre-activation t over the examples in
the minibatch. We can then initialize the neuron?s biase b and scale g as
g?
1
,
?[t]
b?
??[t]
,
?[t]
(6)
so that y = ?(w ? x + b). Like batch normalization, this method ensures that all features initially have
zero mean and unit variance before application of the nonlinearity. With our method this only holds
for the minibatch we use for initialization, and subsequent minibatches may have slightly different
statistics, but experimentally we find this initialization method to work well. The method can also be
applied to networks without weight normalization, simply by doing stochastic gradient optimization
on the parameters w directly, after initialization in terms of v and g: this is what we compare to in
section 5. Independently from our work, this type of initialization was recently proposed by different
authors [24, 18] who found such data-based initialization to work well for use with the standard
parameterization in terms of w.
The downside of this initialization method is that it can only be applied in similar cases as where
batch normalization is applicable. For models with recursion, such as RNNs and LSTMs, we will
have to resort to standard initialization methods.
4
4
Mean-only Batch Normalization
Weight normalization, as introduced in section 2, makes the scale of neuron activations approximately
independent of the parameters v. Unlike with batch normalization, however, the means of the neuron
activations still depend on v. We therefore also explore the idea of combining weight normalization
with a special version of batch normalization, which we call mean-only batch normalization: With
this normalization method, we subtract out the minibatch means like with full batch normalization,
but we do not divide by the minibatch standard deviations. That is, we compute neuron activations
using
t = w ? x,
t? = t ? ?[t] + b,
y = ?(t?)
(7)
where w is the weight vector, parameterized using weight normalization, and ?[t] is the minibatch
mean of the pre-activation t. During training, we keep a running average of the minibatch mean
which we substitute in for ?[t] at test time.
The gradient of the loss with respect to the pre-activation t is calculated as
?t L = ?t?L ? ?[?t?L],
(8)
where ?[.] denotes once again the operation of taking the minibatch mean. Mean-only batch normalization thus has the effect of centering the gradients that are backpropagated. This is a comparatively
cheap operation, and the computational overhead of mean-only batch normalization is thus lower
than for full batch normalization. In addition, this method causes less noise during training, and
the noise that is caused is more gentle as the law of large numbers ensures that ?[t] and ?[?t?] are
approximately normally distributed. Thus, the added noise has much lighter tails than the highly
kurtotic noise caused by the minibatch estimate of the variance used in full batch normalization. As
we show in section 5.1, this leads to improved accuracy at test time.
5
Experiments
We experimentally validate the usefulness of our method using four different models for varied
applications in supervised image recognition, generative modelling, and deep reinforcement learning.
5.1
Supervised Classification: CIFAR-10
To test our reparameterization method for the application of supervised classification, we consider the
CIFAR-10 data set of natural images [19]. The model we are using is based on the ConvPool-CNN-C
architecture of [30], with some small modifications: we replace the first dropout layer by a layer
that adds Gaussian noise, we expand the last hidden layer from 10 units to 192 units, and we use
2 ? 2 max-pooling, rather than 3 ? 3. The only hyperparameter that we actively optimized (the
standard deviation of the Gaussian noise) was chosen to maximize the performance of the network
on a holdout set of 10000 examples, using the standard parameterization (no weight normalization
or batch normalization). A full description of the resulting architecture is given in table A in the
supplementary material.
We train our network for CIFAR-10 using Adam [15] for 200 epochs, with a fixed learning rate and
momentum of 0.9 for the first 100 epochs. For the last 100 epochs we set the momentum to 0.5 and
linearly decay the learning rate to zero. We use a minibatch size of 100. We evaluate 5 different
parameterizations of the network: 1) the standard parameterization, 2) using batch normalization,
3) using weight normalization, 4) using weight normalization combined with mean-only batch normalization, 5) using mean-only batch normalization with the normal parameterization. The network
parameters are initialized using the scheme of section 3 such that all four cases have identical parameters starting out. For each case we pick the optimal learning rate in {0.0003, 0.001, 0.003, 0.01}. The
resulting error curves during training can be found in figure 1: both weight normalization and batch
normalization provide a significant speed-up over the standard parameterization. Batch normalization
makes slightly more progress per epoch than weight normalization early on, although this is partly
offset by the higher computational cost: with our implementation, training with batch normalization
was about 16% slower compared to the standard parameterization. In contrast, weight normalization was not noticeably slower. During the later stage of training, weight normalization and batch
normalization seem to optimize at about the same speed, with the normal parameterization (with or
without mean-only batch normalization) still lagging behind. After optimizing the network for 200
5
training error
Model
Maxout [8]
0.1
Network in Network [21]
Deeply Supervised [20]
0.05
ConvPool-CNN-C [30]
ALL-CNN-C [30]
our CNN, mean-only B.N.
0
0
50
100
150
200
our CNN, weight norm.
training epochs
our CNN, normal param.
our CNN, batch norm.
Figure 1: Training error for CIFAR-10 using dif- ours, W.N. + mean-only B.N.
ferent parameterizations. For weight normaliza- DenseNet [13]
tion, batch normalization, and mean-only batch
normalization we show results using Adam with Figure 2: Classification results
a learning rate of 0.003. For the normal param- without data augmentation.
eterization we instead use 0.0003 which works
best in this case. For the last 100 epochs the
learning rate is linearly decayed to zero.
normal param.
weight norm.
batch norm.
WN + mean?only BN
mean?only BN
Test Error
11.68%
10.41%
9.6%
9.31%
9.08%
8.52%
8.46%
8.43%
8.05%
7.31%
5.77%
on CIFAR-10
epochs using the different parameterizations, we evaluate their performance on the CIFAR-10 test
set. The results are summarized in table 2: weight normalization, the normal parameterization, and
mean-only batch normalization have similar test accuracy (? 8.5% error). Batch normalization does
significantly better at 8.05% error. Mean-only batch normalization combined with weight normalization has the best performance at 7.31% test error, and interestingly does much better than mean-only
batch normalization combined with the normal parameterization: This suggests that the noise added
by batch normalization can be useful for regularizing the network, but that the reparameterization
provided by weight normalization or full batch normalization is also needed for optimal results. We
hypothesize that the substantial improvement by mean-only B.N. with weight normalization over
regular batch normalization is due to the distribution of the noise caused by the normalization method
during training: for mean-only batch normalization the minibatch mean has a distribution that is
approximately Gaussian, while the noise added by full batch normalization during training has much
higher kurtosis. The result with mean-only batch normalization combined with weight normalization
represented the state-of-the-art for CIFAR-10 among methods that do not use data augmentation,
until it was recently surpassed by DenseNets [13].
5.2
Generative Modelling: Convolutional VAE
Next, we test the effect of weight normalization applied to deep convolutional variational autoencoders (CVAEs) [16, 28, 29], trained on the MNIST data set of images of handwritten digits and
the CIFAR-10 data set of small natural images.
Variational auto-encoders are generative models that explain the data vector x as arising from a set of
latent variables z, through a joint distribution of the form p(z, x) = p(z)p(x|z), where the decoder
p(x|z) is specified using a neural network. A lower bound on the log marginal likelihood log p(x)
can be obtained by approximately inferring the latent variables z from the observed data x using
an encoder distribution q(z|x) that is also specified as a neural network. This lower bound is then
optimized to fit the model to the data.
We follow a similar implementation of the CVAE as in [29] with some modifications, mainly that the
encoder and decoder are parameterized with ResNet [11] blocks, and that the diagonal posterior is
replaced with a more flexible specification based on inverse autoregressive flow. A further developed
version of this model is presented in [17], where the architecture is explained in detail.
For MNIST, the encoder consists of 3 sequences of two ResNet blocks each, the first sequence acting
on 16 feature maps, the others on 32 feature maps. The first two sequences are followed by a 2-times
subsampling operation implemented using 2 ? 2 stride, while the third sequence is followed by
a fully connected layer with 450 units. The decoder has a similar architecture, but with reversed
direction. For CIFAR-10, we used a neural architecture with ResNet units and multiple intermediate
stochastic layers. We used Adamax [15] with ? = 0.002 for optimization, in combination with
6
Polyak averaging [26] in the form of an exponential moving average that averages parameters over
approximately 10 epochs.
In figure 3, we plot the test-set lower bound as a function of number of training epochs, including
error bars based on multiple different random seeds for initializing parameters. As can be seen, the
parameterization with weight normalization has lower variance and converges to a better optimum.
We observe similar results across different hyper-parameter settings.
Convolutional VAE on MNIST
84.0
Convolutional VAE on CIFAR-10
10000
bound on marginal likelihood
bound on marginal likelihood
84.5
85.0
85.5
86.0
86.5
87.0
normal parameterization
weight normalization
87.5
88.0
50
100
150
200
training epochs
250
9500
9000
8500
normal parameterization
weight normalization
8000
0
300
50
100
150
200 250 300
training epochs
350
400
450
Figure 3: Marginal log likelihood lower bound on the MNIST (top) and CIFAR-10 (bottom) test
sets for a convolutional VAE during training, for both the standard implementation as well as our
modification with weight normalization. For MNIST, we provide standard error bars to indicate
variance based on different initial random seeds.
5.3
Generative Modelling: DRAW
Next, we consider DRAW, a recurrent generative model by [9]. DRAW is a variational auto-encoder
with generative model p(z)p(x|z) and encoder q(z|x), similar to the model in section 5.2, but with
both the encoder and decoder consisting of a recurrent neural network comprised of Long Short-Term
Memory (LSTM) [12] units. LSTM units consist of a memory cell with additive dynamics, combined
with input, forget, and output gates that determine which information flows in and out of the memory.
The additive dynamics enables learning of long-range dependencies in the data.
At each time step of the model, DRAW uses the same set of weight vectors to update the cell states
of the LSTM units in its encoder and decoder. Because of the recurrent nature of this process it is not
trivial to apply batch normalization here: Normalizing the cell states diminishes their ability to pass
through information. Fortunately, weight normalization can easily be applied to the weight vectors
of each LSTM unit, and we find this to work well empirically. Some other potential solutions were
recently proposed in [4, 2].
bound on marginal log likelihood
?80
?85
?90
?95
?100
?105
?110
normal parameterization
weight normalization
?115
?120
0
10
20
30
40
50
60
70
80
90
100
training epochs
Figure 4: Marginal log likelihood lower bound on the MNIST test set for DRAW during training, for
both the standard implementation as well as our modification with weight normalization. 100 epochs
is not sufficient for convergence for this model, but the implementation using weight normalization
clearly makes progress much more quickly than with the standard parameterization.
7
We take the Theano implementation of DRAW provided at https://github.com/jbornschein/
draw and use it to model the MNIST data set of handwritten digits. We then make a single modification to the model: we apply weight normalization to all weight vectors. As can be seen in figure 4,
this significantly speeds up convergence of the optimization procedure, even without modifying the
initialization method and learning rate that were tuned for use with the normal parameterization.
5.4
Reinforcement Learning: DQN
Next we apply weight normalization to the problem of Reinforcement Learning for playing games on
the Atari Learning Environment [3]. The approach we use is the Deep Q-Network (DQN) proposed
by [25]. This is an application for which batch normalization is not well suited: the noise introduced
by estimating the minibatch statistics destabilizes the learning process. We were not able to get
batch normalization to work for DQN without using an impractically large minibatch size. In
contrast, weight normalization is easy to apply in this context, as is the initialization method of
section 3. Stochastic gradient learning is performed using Adamax [15] with momentum of 0.5. We
search for optimal learning rates in {0.0001, 0.0003, 0.001, 0.003}, generally finding 0.0003 to work
well with weight normalization and 0.0001 to work well for the normal parameterization. We also
use a larger minibatch size (64) which we found to be more efficient on our hardware (Amazon
Elastic Compute Cloud g2.2xlarge GPU instance). Apart from these changes we follow [25]
as closely as possible in terms of parameter settings and evaluation methods. However, we use a
Python/Theano/Lasagne reimplementation of their work, adapted from the implementation available
at https://github.com/spragunr/deep_q_rl, so there may be small additional differences in
implementation.
Figure 5 shows the training curves obtained using DQN with the standard parameterization and
with weight normalization on Space Invaders. Using weight normalization the algorithm progresses
more quickly and reaches a better final result. Table 6 shows the final evaluation scores obtained
by DQN with weight normalization for four games: on average weight normalization improves the
performance of DQN.
2500
Game
Breakout
Enduro
Seaquest
Space Invaders
test reward per episode
2000
1500
1000
500
0
0
normal parameterization
weight normalization
50
100
150
200
training epochs
Figure 5: Evaluation scores for Space Invaders obtained by DQN after each epoch of
training, for both the standard parameterization and using weight normalization. Learning rates for both cases were selected to maximize the highest achieved test score.
6
normal
410
1,250
7,188
1,779
weightnorm
403
1,448
7,375
2,179
Mnih
401
302
5,286
1,975
Figure 6: Maximum evaluation scores obtained by
DQN, using either the normal parameterization or
using weight normalization. The scores indicated by
Mnih et al. are those reported by [25]: Our normal
parameterization is approximately equivalent to their
method. Differences in scores may be caused by small
differences in our implementation. Specifically, the
difference in our score on Enduro and that reported
by [25] might be due to us not using a play-time limit
during evaluation.
Conclusion
We have presented weight normalization, a simple reparameterization of the weight vectors in a
neural network that accelerates the convergence of stochastic gradient descent optimization. Weight
normalization was applied to four different models in supervised image recognition, generative
modelling, and deep reinforcement learning, showing a consistent advantage across applications. The
reparameterization method is easy to apply, has low computational overhead, and does not introduce
dependencies between the examples in a minibatch, making it our default choice in the development
of new deep learning architectures.
8
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9
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5,653 | 6,115 | Linear-Memory and Decomposition-Invariant
Linearly Convergent Conditional Gradient Algorithm
for Structured Polytopes
Dan Garber
Toyota Technological Institute at Chicago
dgarber@ttic.edu
Ofer Meshi
Google
meshi@google.com
Abstract
Recently, several works have shown that natural modifications of the classical
conditional gradient method (aka Frank-Wolfe algorithm) for constrained convex
optimization, provably converge with a linear rate when: i) the feasible set is a
polytope, and ii) the objective is smooth and strongly-convex. However, all of these
results suffer from two significant shortcomings:
1. large memory requirement due to the need to store an explicit convex decomposition of the current iterate, and as a consequence, large running-time
overhead per iteration
2. the worst case convergence rate depends unfavorably on the dimension
In this work we present a new conditional gradient variant and a corresponding
analysis that improves on both of the above shortcomings. In particular:
1. both memory and computation overheads are only linear in the dimension
2. in case the optimal solution is sparse, the new convergence rate replaces a
factor which is at least linear in the dimension in previous work, with a linear
dependence on the number of non-zeros in the optimal solution
At the heart of our method and corresponding analysis, is a novel way to compute
decomposition-invariant away-steps. While our theoretical guarantees do not apply
to any polytope, they apply to several important structured polytopes that capture
central concepts such as paths in graphs, perfect matchings in bipartite graphs,
marginal distributions that arise in structured prediction tasks, and more. Our
theoretical findings are complemented by empirical evidence which shows that our
method delivers state-of-the-art performance.
1
Introduction
The efficient reduction of a constrained convex optimization problem to a constrained linear optimization problem is an appealing algorithmic concept, in particular for large-scale problems. The reason
is that for many feasible sets of interest, the problem of minimizing a linear function over the set
admits much more efficient methods than its non-linear convex counterpart. Prime examples for this
phenomenon include various structured polytopes that arise in combinatorial optimization, such as
the path polytope of a graph (aka the unit flow polytope), the perfect matching polytope of a bipartite
graph, and the base polyhedron of a matroid, for which we have highly efficient combinatorial
algorithms for linear minimization that rely heavily on the specific rich structure of the polytope
[21]. At the same time, minimizing a non-linear convex function over these sets usually requires the
use of generic interior point solvers that are oblivious to the specific combinatorial structure of the
underlying set, and as a result, are often much less efficient. Indeed, it is for this reason, that the
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
conditional gradient (CG) method (aka Frank-Wolfe algorithm), a method for constrained convex
optimization that is based on solving linear subproblems over the feasible domain, has regained much
interest in recent years in the machine learning, signal processing and optimization communities. It
has been recently shown that the method delivers state-of-the-art performance on many problems of
interest, see for instance [14, 17, 4, 10, 11, 22, 19, 25, 12, 15].
As part of the regained interest in the conditional gradient method, there is also a recent effort to
understand the convergence rates and associated complexities of conditional gradient-based methods,
which are in general far less understood than other first-order methods, e.g., the projected gradient
method. It is known, already from the first introduction of the method by Frank and Wolfe in the
1950?s [5] that the method converges with a rate of roughly O(1/t) for minimizing a smooth convex
function over a convex and compact set. However, it is not clear if this convergence rate improves
under an additional standard strong-convexity assumption. In fact, certain lower bounds, such as in
[18, 8], suggest that such improvement, even if possible, should come with a worse dependence on
the problem?s parameters (e.g., the dimension). Nevertheless, over the past years, various works tried
to design natural variants of the CG method that converge provably faster under the strong convexity
assumption, without dramatically increasing the per-iteration complexity of the method. For instance,
Gu?Lat and Marcotte [9] showed that a CG variant which uses the concept of away-steps converges
exponentially fast in case the objective function is strongly convex, the feasible set is a polytope,
and the optimal solution is located in the interior of the set. A similar result was presented by Beck
and Teboulle [3] who considered a specific problem they refer to as the convex feasibility problem
over an arbitrary convex set. They also obtained a linear convergence rate under the assumption that
an optimal solution that is far enough from the boundary of the set exists. In both of these works,
the exponent depends on the distance of the optimal solution from the boundary of the set, which
in general can be arbitrarily small. Later, Ahipasaoglu et al. [1] showed that in the specific case of
minimizing a smooth and strongly convex function over the unit simplex, a variant of the CG method
which also uses away-steps, converges with a linear rate. Unfortunately, it is not clear from their
analysis how this rate depends on natural parameters of the problem such as the dimension and the
condition number of the objective function.
Recently, Garber and Hazan presented a linearly-converging CG variant for polytopes without any
restrictions on the location of the optimum [8]. In a later work, Lacoste-Julien and Jaggi [16] gave
a refined affine-invariant analysis of an algorithm presented in [9] which also uses away steps, and
showed that it also converges exponentially fast in the same setting as the Garber-Hazan result. More
recently, Beck and Shtern [2] gave a different, duality-based, analysis for the algorithm of [9], and
showed that it can be applied to a wider class of functions than purely strongly convex functions.
However, the explicit dependency of their convergence rate on the dimension is suboptimal, compared
to [8, 16]. Aside from the polytope case, Garber and Hazan [7] have shown that in case the feasible
set is strongly-convex and the objective function satisfies certain strong convexity-like proprieties,
then the standard CG method converges with an accelerated rate of O(1/t2 ). Finally, in [6] Garber
showed a similar improvement (roughly quadratic) for the spectrahedron ? the set of unit trace
positive semidefinite matrices.
Despite the exponential improvement in convergence rate for polytopes obtained in recent results,
they all suffer from two major drawbacks. First, while in terms of the number of calls per-iteration to
the linear optimization oracle, these methods match the standard CG method, i.e., a single call per
iteration, the overhead of other operations both in terms of running time and memory requirements is
significantly worse. The reason is that in order to apply the so-called away-steps, which all methods
use, they require to maintain at all times an explicit decomposition of the current iterate into vertices
of the polytope. In the worst case, maintaining such a decomposition and computing the away-steps
require both memory and per-iteration runtime overheads that are at least quadratic in the dimension.
This is much worse than the standard CG method, whose memory and runtime overheads are only
linear in the dimension. Second, the convergence rate of all previous linearly convergent CG methods
depends explicitly on the dimension. While it is known that this dependency is unavoidable in certain
cases, e.g., when the optimal solution is, informally speaking, dense (see for instance the lower bound
in [8]), it is not clear that such an unfavorable dependence is mandatory when the optimum is sparse.
In this paper, we revisit the application of CG variants to smooth and strongly-convex optimization
over polytopes. We introduce a new variant which overcomes both of the above shortcomings from
which all previous linearly-converging variants suffer. The main novelty of our method, which is the
key to its improved performance, is that unlike previous variants, it is decomposition-invariant, i.e., it
2
Paper
Frank & Wolfe [5]
Garber & Hazan [8]
Lacoste-Julien & Jaggi [16]
Beck & Shtern [2]
This paper
#iterations to ? err.
D2 /?
?nD2 log(1/?)
?nD2 log(1/?)
?n2 D2 log(1/?)
?card(x? )D2 log(1/?)
#LOO calls
1
1
1
1
2
runtime
n
n min(n, t)
n min(n, t)
n min(n, t)
n
memory
n
n min(n, t)
n min(n, t)
n min(n, t)
n
Table 1: Comparison with previous work. We define ? := /?, we let n denote the dimension and D
denote the Euclidean diameter of the polytope. The third column gives the number of calls to the
linear optimization oracle per iteration, the fourth column gives the additional arithmetic complexity
at iteration t, and the fifth column gives the worst case memory requirement at iteration t. The
bounds for the algorithms of [8, 16, 2], which are independent of t, assume an algorithmic version of
Carath?odory?s theorem, as fully detailed in [2]. The bound on number of iterations of [16] depends
on the squared inverse pyramidal width of P, which is difficult to evaluate, however, this quantity is
at least proportional to n.
does not require to maintain an explicit convex decomposition of the current iterate. This principle
proves to be crucial both for eliminating the memory and runtime overheads, as well as to obtaining
shaper convergence rates for instances that admit a sparse optimal solution.
A detailed comparison of our method to previous art is shown in Table 1. We also provide in Section
5 empirical evidence that the proposed method delivers state-of-the-art performance on several tasks
of interest. While our method is less general than previous ones, i.e., our theoretical guarantees do not
hold for arbitrary polytopes, they readily apply to many structured polytopes that capture important
concepts such as paths in graphs, perfect matchings in bipartite graphs, Markov random fields, and
more.
2
Preliminaries
Throughout this work we let k ? k denote the standard Euclidean norm. Given a point x 2 Rn , we let
card(x) denote the number of non-zero entries in x.
Definition 1. We say that a function f (x) : Rn ! R is ?-strongly convex w.r.t. a norm k ? k, if for
all x, y 2 Rn it holds that f (y) f (x) + rf (x) ? (y x) + ?2 kx yk2 .
Definition 2. We say that a function f (x) : Rn ! R is -smooth w.r.t. a norm k ? k, if for all
x, y 2 Rn it holds that f (y) ? f (x) + rf (x) ? (y x) + 2 kx yk2 .
The first-order optimality condition implies that for a ?-strongly convex f , if x? is the unique
minimizer of f over a convex and compact set K ? Rn , then for all x 2 K it holds that
?
f (x) f (x? )
kx x? k2 .
(1)
2
2.1
Setting
In this work we consider the optimization problem minx2P f (x), where we assume that:
? f (x) is ?-strongly convex and -smooth with respect to the Euclidean norm.
? P is a polytope which satisfies the following two properties:
1. P can be described algebraically as P = {x 2 Rn | x 0, Ax = b} .
2. All vertices of P lie on the hypercube {0, 1}n .
We let x? denote the (unique) minimizer of f over P, and we let D denote the Euclidean diameter of
P, namely, D = maxx,y2P kx yk. We let V denote the set of vertices of P, where according to
our assumptions, it holds that V ? {0, 1}n .
While the polytopes that satisfy the above assumptions are not completely general, these assumptions
already capture several important concepts such as paths in graphs, perfect-matchings, Markov
3
random fields, and more. Indeed, a surprisingly large number of applications from machine learning,
signal processing and other domains are formulated as optimization problems in this category (e.g.,
[13, 15, 16]). We give detailed examples of such polytopes in Section A in the appendix. Importantly,
the above assumptions allow us to get rid of the dependency of the convergence rate on certain
geometric parameters (such as , ? in [8] or the pyramidal width in [16]), which can be polynomial
in the dimension, and hence result in an impractical convergence rate. Finally, for many of these
polytopes, the vertices are sparse, i.e., for any vertex v 2 V, card(v) << n. In this case, when the
optimum x? can be decomposed as a convex combination of only a few vertices (and thus, sparse by
itself), we get a sharper convergence rate that depends on the sparsity of x? and not explicitly on the
dimension, as in previous work. We believe that our theoretical guarantees could be well extended to
more general polytopes, as suggested in Section C in the appendix; we leave this extension for future
work.
3
Our Approach
In order to better communicate our ideas, we begin by first briefly introducing the standard conditional
gradient method and its accelerated away-steps-based variants. We discuss both the blessings and
shortcomings of these away-steps-based variants in Subsection 3.1. Then, in Subsection 3.2, we
present our new method, a decomposition-invariant away-steps-based conditional gradient algorithm,
and discuss how it addresses the shortcomings of previous variants.
3.1
The conditional gradient method and acceleration via away-steps
The standard conditional gradient algorithm is given below (Algorithm 1). It is well known that
when setting the step-size ?t in an appropriate way, the worst case convergence rate of the method is
O( D2 /t) [13]. This convergence rate is tight for the method in general, see for instance [18].
Algorithm 1 Conditional Gradient
1: let x1 be some vertex in V
2: for t = 1... do
3:
vt
arg minv2V v ? rf (xt )
4:
choose a step-size ?t 2 (0, 1]
5:
xt+1
xt + ?t (vt xt )
6: end for
Algorithm 2 Pairwise Conditional Gradient
1: let x1 be some vertex in V
2: for t =P
1... do
(i) (i)
kt
3:
let
be an explicitly maini=1 t vt
tained convex decomposition of xt
4:
vt+
arg minv2V v ? rf (xt )
(j)
5:
jt
arg minj2[kt ] vt ? ( rf (xt ))
(j )
6:
choose a step-size ?t 2 (0, t t ]
(jt )
+
7:
xt+1
xt + ?t (vt
vt )
8:
update the convex decomposition of xt+1
9: end for
Pk
Consider the iterate of Algorithm 1 on iteration t, and let xt = i=1 i vi be its convex decomposition
into vertices of the polytope P. Note that Algorithm 1 implicitly discounts each coefficient i by
a factor (1 ?t ), in favor of the new added vertex vt . A different approach is not to decrease all
vertices in the decomposition of xt uniformly, but to more-aggressively decrease vertices that are
worse than others with respect to some measure, such as their product with the gradient direction.
This key principle proves to be crucial to breaking the 1/t rate of the standard method, and to achieve
a linear convergence rate under certain strong-convexity assumptions, as described in the recent
works [8, 16, 2]. For instance, in [8] it has been shown, via the introduction of the concept of a Local
Linear Optimization Oracle, that using such a non-uniform reweighing rule, in fact approximates a
certain proximal problem, that together with the shrinking effect of strong convexity, as captured by
Eq. (1), yields a linear convergence rate. We refer to these methods as away-step-based CG methods.
As a concrete example, which will also serve as a basis for our new method, we describe the pairwise
variant recently studied in [16], which applies this principle in Algorithm 2.1 Note that Algorithm 2
decreases the weight of exactly one vertex in the decomposition: that with the largest product with
the gradient.
1
While the convergence rate of this pairwise variant, established in [16], is significantly worse than other
away-step-based variants, here we show that a proper analysis yields state-of-the-art performance guarantees.
4
It is important to note that since previous away-step-based CG variants do not decrease the coefficients
in the convex decomposition of the current iterate uniformly, they all require to explicitly store and
maintain a convex decomposition of the current iterate. This issue raises two main disadvantages:
Superlinear memory and running-time overheads Storing a decomposition of the current iterate
as a convex combination of vertices generally requires O(n2 ) memory in the worst case. While
the away-step-based variants increase the size of the decomposition by at most a single vertex per
iteration, they also typically exhibit linear convergence after performing at least ?(n) steps [8, 16, 2],
and thus, this O(n2 ) estimate still holds. Moreover, since these methods require i) to find the worst
vertex in the decomposition, in terms of dot-product with current gradient direction, and ii) to update
this decomposition at each iteration (even when using sophisticated update techniques such as in [2]),
then the worst case per-iteration overhead in terms of computation is also ?(n2 ).
Decomposition-specific performance The choice of away-step depends on the specific decomposition that is maintained by the algorithm. Since the feasible point xt may admit several different convex
decompositions, committing to one such decomposition, might result in sub-optimal away-steps. As
observable in Table 1, for certain problems in which the optimal solution is sparse, all analyses of
previous away-steps-based variants are significantly suboptimal, since they all depend explicitly on
the dimension. This seems to be an unavoidable side-effect of being decomposition-dependent. On
the other hand, the fact that our new approach is decomposition-invariant allows us to obtain sharper
convergence rates for such instances.
3.2
A new decomposition-invariant pairwise conditional gradient method
Our main observation is that in many cases of interest, given a feasible iterate xt , one can in fact
compute an optimal away-step from xt without relying on any single specific decomposition. This
observation allows us to overcome both of the main disadvantages of previous away-step-based CG
variants. Our algorithm, which we refer to as a decomposition-invariant pairwise conditional gradient
(DICG), is given below in Algorithm 3.
Algorithm 3 Decomposition-invariant Pairwise Conditional Gradient
1:
2:
3:
4:
5:
6:
input: sequence of step-sizes {?t }t
let x0 be an arbitrary point in P
x1
arg minv2V v ? rf (x0 )
for t = 1... do
vt+
arg minv2V v ? rf (xt )
1
? (xt ) 2 Rn as follows: [rf
? (xt )]i :=
define the vector rf
?
[rf (xt )]i
1
if xt (i) > 0
if xt (i) = 0
? (xt ))
vt
arg minv2V v ? ( rf
choose a new step-size ??t using one of the following two options:
Option 1: predefined step-size
let t be the smallest natural number such that 2 t ? ?t , and set a new step-size ??t
2
Option 2: line-search
max 2[0,1] {xt + (vt+ vt ) 0}, ??t
min?2(0, t ] f (xt + ?(vt+ vt ))
t
+
9:
xt+1
xt + ??t (vt
vt )
10: end for
7:
8:
t
The following observation shows the optimality of away-steps taken by Algorithm 3.
Observation 1 (optimal away-steps in Algorithm 3). Consider an iteration t of Algorithm 3 and
Pk
suppose that the iterate xt is feasible. Let xt = i=1 i vi for some integer k, be an irreducible
way of writing xt as a convex sum of vertices of P, i.e., i > 0 for all i 2 [k]. Then it holds that
8i 2 [k] : vi ? rf (xt ) ? vt ? rf (xt ), and t min{xt (i) | i 2 [n], xt (i) > 0}.
Pk
Proof. Let xt = i=1 i vi be a convex decomposition of xt into vertices of P, for some integer
k, where each i is positive. Note that it must hold that for any j 2 [n] and any i 2 [k], xt (j) =
0 ) vi (j) = 0, since by our assumption V ? Rn+ . The observation then follows directly from the
definition of vt .
We next state the main theorem of this paper, which bounds the convergence rate of Algorithm 3. The
proof is provided in Section B.3 in the appendix.
5
Theorem 1. Let M1 =
p
?/(8card(x? )) and M2 = D2 /2. Consider running Algorithm 3 with
t 1
p
Option 1 as the step-size, and suppose that 8t 1 : ?t = M1 /(2 M2 ) 1 M12 /(4M2 ) 2 .
Then, the iterates of Algorithm 3 are always feasible, and 8t 1:
?
?
D2
?
?
f (xt ) f (x ) ?
exp
t .
2
8 D2 card(x? )
We now turn to make several remarks regarding Algorithm 3 and Theorem 1:
The so-called dual gap, defined as gt := (xt vt+ ) ? rf (xt ), which serves as a certificate for the
sub-optimality of the iterates of Algorithm 3, also converges with a linear rate, as we prove in Section
B.4 in the appendix.
Note that despite the different parameters of the problem at hand (e.g., ?, , D, card(x? )), running
the algorithm with Option 1 for choosing the step-size,
p for which the guarantee of Theorem 1
holds, requires knowing a single parameter, i.e., M1 / M2 . p
In particular,
pit is an easy consequence
that running the algorithm with an estimate M 2 [0.5M1 / M2 , M1 / p
M2 ], will only affect the
leading constant in the convergence rate listed in the theorem. Hence, M1 / M2 could be efficiently
estimated via a logarithmic-scale search.
Theorem 1 improves significantly over the convergence rate established for the pairwise conditional
gradient variant in [16]. In particular, the number of iterations to reach an ? error in the analysis of
[16] depends linearly on |V|!, where |V| is the number of vertices of P.
4
Analysis
Throughout this section we let ht denote the approximation error of Algorithm 3 on iteration t, for
any t 1, i.e., ht = f (xt ) f (x? ).
4.1
Feasibility of the iterates generated by Algorithm 3
We start by proving that the iterates of Algorithm 3 are always feasible. While feasibility is straightforward when using the the line-search option to set the step-size (Option 2), it is less obvious when
using Option 1. We will make use of the following observation, which is a simple consequence of the
optimal choice of vt and our assumptions on P. A proof is given in Section B.1 in the appendix.
Observation 2. Suppose that on some iteration t of Algorithm 3, the iterate xt is feasible, and that
the step-size is chosen using Option 1. Then, if for all i 2 [n] for which xt (i) 6= 0 it holds that
xt (i) ??t , the following iterate xt+1 is also feasible.
Lemma 1 (feasibility of iterates under Option 1). Suppose that the sequence of step-sizes {?t }t 1 is
monotonically non-increasing, and contained in the interval [0, 1]. Then, the iterates generated by
Algorithm 3 using Option 1 for setting the step-size, are always feasible.
Proof. We are going to prove by induction that on each iteration t there exists a non-negative integervalued vector st 2 Nn , such that for any i 2 [n], it holds that xt (i) = 2 t st (i). The lemma
then follows since, by definition, ??t = 2 t , and by applying Observation 2. The base case t = 1
holds since x1 is a vertex of P and thus for any i 2 [n] we have that x1 (i) 2 {0, 1} (recall that
V ? {0, 1}n ). On the other hand, since ?1 ? 1, it follows that 1 0. Thus, there indeed exists a
non-negative integer-valued vector s1 , such that x1 = 2 1 s1 .
Suppose now that the induction holds for some t 1. Since by definition of vt , subtracting ??t vt
from xt can only decrease positive entries in xt (see proof of Observation 2), and both vt , vt+ are
vertices of P (and thus in {0, 1}n ), and ??t = 2 t , it follows that each entry i in xt+1 is given by:
xt+1 (i) = 2
8
< st (i)
t
st (i) 1
:
st (i) + 1
if st (i) 1 & vt (i) = vt+ (i) = 1 or vt (i) = vt+ (i) = 0
if st (i) 1 & vt (i) = 1 & vt+ (i) = 0
if vt (i) = 0 & vt+ (i) = 1
Thus, xt+1 can also be written in the form 2 t s?t+1 for some s?t+1 2 Nn . By definition of t and the
t
t
monotonicity of {?t }t 1 , we have that 22 t+1 is a positive integer. Thus, setting st+1 = 22 t+1 s?t+1 ,
the induction holds also for t + 1.
6
4.2
Bounding the per-iteration error-reduction of Algorithm 3
The following technical lemma is the key to deriving the linear convergence rate of our method, and
in particular, to deriving the improved dependence on the sparsity of x? , instead of the dimension. At
a high-level, the lemma translates the `2 distance between two feasible points into a `1 distance in a
simplex defined over the set of vertices of the polytope.
Lemma 2. Let x, y 2 P. There exists a way to write x as a convex combination of vertices of P,
Pk
Pk
Pk
x = i=1 i vi for some integer k, such that y can be written as y = i=1 ( i
i )vi +(
i )z
i=1
p
Pk
with i 2 [0, i ] 8i 2 [k],z 2 P, and i=1 i ? card(y)kx yk.
The proof is given in Section B.2 in the appendix. The next lemma bounds the per-iteration improvement of Algorithm 3 and is the key step to proving Theorem 1. We defer the rest of the proof of
Theorem 1 to Section B.3 in the appendix.
Lemmap3. Consider the iterates of Algorithm 3, when the step-sizes are chosen using Option 1. Let
1/2
M1 = ?/(8card(x? )) and M2 = D2 /2. For any t 1 it holds that ht+1 ? ht ?t M1 ht +
2
? t M2 .
p
Proof.
Define
=
2card(x? )ht /?, and note that from Eq. (1) we have that t
t
p
?
?
card(x )kxt x k. As a first step, we are going to show that the point yt := xt + t (vt+ vt )
satisfies: yt ? rf (xt ) ? x? ? rf (xt ). From Lemma 2 it follows that we can write x as a convex comPk
Pk
Pk
bination xt = i=1 i vi and write x? as x? = i=1 ( i
i )vi +
i z, where
i 2 [0, i ],
i=1
Pk
z 2 P, and i=1 i ? t . It holds that
Xk
+
(yt xt ) ? rf (xt ) = t (vt+ vt ) ? rf (xt ) ?
vt ) ? rf (xt )
i (vt
i=1
Xk
?
vi ) ? rf (xt ) = (x? xt ) ? rf (xt ),
i (z
i=1
where the first inequality follows since (vt+ vt ) ? rf (xt ) ? 0, and the second inequality follows
from the optimality of vt+ and vt (Observation 1). Rearranging, we have that indeed
xt +
+
t (vt
vt ) ? rf (xt ) ? x? ? rf (xt ).
Observe now that from the definition of ??t it follows for any t
smoothness of f (x) we have that
ht+1
=
f (xt + ??t (vt+
?
ht + ??t (vt+
=
ht +
?
?t
2 t
?t
ht +
2 t
vt ))
f (x? ) ? ht + ??t (vt+
1 that
(2)
?t
2
vt ) ? rf (xt ) +
? ??t ? ?t . Using the
??t2
kvt+
2
vt k 2
??t2 D2
?t
?2 D2
? ht + (vt+ vt ) ? rf (xt ) + t
2
2
2
?t2 D2
+
(xt + t (vt
vt ) xt ? rf (xt ) +
2
?t2 D2
?t
?2 D2
?
(x
xt ) ? rf (xt ) +
? ht
ht + t
,
2
2 t
2
vt ) ? rf (xt ) +
where the third inequality follows since (vt+ vt ) ? rf (xt ) ? 0, the forth inequality follows from
Eq. (2), and the last inequality follows from convexity of f (x). Finally, plugging the value of t
completes the proof.
5
Experiments
In this section we illustrate the performance of our algorithm in numerical experiments. We use
the two experimental settings from [16], which include a constrained Lasso problem and a video
co-localization problem. In addition, we test our algorithm on a learning problem related to an
optical character recognition (OCR) task from [23]. In each setting we compare the performance
of our algorithm (DICG) to standard conditional gradient (CG), as well as to the fast away (ACG)
and pairwise (PCG) variants [16]. For the baselines in the first two settings we use the publicly
available code from [16], to which we add our own implementation of Algorithm 3. Similarly, for the
OCR problem we extend code from [20], kindly provided by the authors. For all algorithms we use
line-search to set the step size.
7
Lasso
Video co-localization
5
OCR
?2
10
10
CG
ACG
PCG
DICG
?4
CG
ACG
PCG
DICG
?1
10
10
Gap
Gap
Gap
0
10
?2
10
?6
10
?5
10
0
CG
ACG
PCG
DICG
?3
10
?8
10
200
400
600
800
1000
0
500
1000
Iteration
1500
2000
0
500
Iteration
5
10
10
CG
ACG
PCG
DICG
1000
1500
2000
Effective passes
?2
10
CG
ACG
PCG
DICG
?4
-2
CG
ACG
PCG
DICG
10
Gap
Gap
Gap
0
10
?6
10
10
-3
?5
10
?8
10
0
2
4
6
Time (sec)
8
10
0
5
10
15
Time (sec)
20
25
30
0
1
2
3
4
5
6
Time (hr)
Figure 1: Duality gap gt vs. iterations (top) and time (bottom) in various settings.
?bk2 , where M is a
?
Lasso In the first example the goal is to solve the problem: minx2M kAx
scaled `1 ball. Notice that the constraints M do not match the required structure of P, however, with
a simple change of variables we can obtain an equivalent optimization problem over the simplex.
We generate the random matrix A? and vector ?b as in [16]. In Figure 1 (left, top) we observe that
our algorithm (DICG) converges similarly to the pairwise variant PCG and faster than the other
baselines. This is expected since the away direction v in DICG (Algorithm 3) is equivalent to the
away direction in PCG (Algorithm 2) in the case of simplex constraints.
Video co-localization The second example is a quadratic program over the flow polytope, originally
proposed in [15]. This is an instance of P that is mentioned in Section A in the appendix. As can
be seen in Figure 1 (middle, top), in this setting our proposed algorithm significantly outperforms
the baselines, as a result of finding a better away direction v . Figure 1 (middle, bottom) shows
convergence on a time scale, where the difference between the algorithms is even larger. One
reason for this difference is the costly search over the history of vertices maintained by the baseline
algorithms. Specifically, the number of stored vertices grows fast with the number of iterations and
reaches 1222 for away steps and 1438 for pairwise steps (out of 2000 iterations).
OCR We next conduct experiments on a structured SVM learning problem resulting from an OCR
task. The constraints in this setting are the marginal polytope corresponding to a chain graph over
the letters of a word (see [23]), and the objective function is quadratic. Notice that the marginal
polytope has a concise characterization in this case and also satisfies our assumptions (see Section A
in the appendix for more details). For this problem we actually run Algorithm 3 in a block-coordinate
fashion, where blocks correspond to training examples in the dual SVM formulation [17, 20]. In
Figure 1 (right, top) we see that our DICG algorithm is comparable to the PCG algorithm and faster
than the other baselines on the iteration scale. Figure 1 (right, bottom) demonstrates that in terms
of actual running time we get a noticeable speedup compared to all baselines. We point out that
for this OCR problem, both ACG and PCG each require about 5GB of memory to store the explicit
decomposition in the implementation of [20]. In comparison, our algorithm requires 220MB of
memory to store the current iterate, and the other variables in the code require 430MB (common to
all algorithms), so using DICG results in significant memory savings.
6
Extensions
Our results are readily extendable in two important directions. First, we can relax the strong convexity
requirement of f (x) and handle a broader class of functions, namely the class considered in [2].
Second, we extend the line-search variant of Algorithm 3 to handle arbitrary polytopes, but without
convergence guarantees, which is left as future work. Both extensions are brought in full detail in
Section C in the appendix.
8
References
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9
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5,654 | 6,116 | Proximal Stochastic Methods for Nonsmooth
Nonconvex Finite-Sum Optimization
Sashank J. Reddi
Carnegie Mellon University
sjakkamr@cs.cmu.edu
Suvrit Sra
Massachusetts Institute of Technology
suvrit@mit.edu
Barnab?s P?czos
Carnegie Mellon University
bapoczos@cs.cmu.edu
Alexander J. Smola
Carnegie Mellon University
alex@smola.org
Abstract
We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum
problems, where the nonsmooth part is convex. Surprisingly, unlike the smooth
case, our knowledge of this fundamental problem is very limited. For example,
it is not known whether the proximal stochastic gradient method with constant
minibatch converges to a stationary point. To tackle this issue, we develop fast
stochastic algorithms that provably converge to a stationary point for constant
minibatches. Furthermore, using a variant of these algorithms, we obtain provably
faster convergence than batch proximal gradient descent. Our results are based
on the recent variance reduction techniques for convex optimization but with a
novel analysis for handling nonconvex and nonsmooth functions. We also prove
global linear convergence rate for an interesting subclass of nonsmooth nonconvex
functions, which subsumes several recent works.
1
Introduction
We study nonconvex, nonsmooth, finite-sum optimization problems of the form
n
min
x2Rd
F (x) := f (x) + h(x), where f (x) :=
1X
fi (x),
n i=1
(1)
and each fi : Rd ! R is smooth (possibly nonconvex) for all i 2 {1, . . . , n} , [n], while
h : Rd ! R is nonsmooth but convex and relatively simple.
Such finite-sum optimization problems are fundamental to machine learning when performing regularized empirical risk minimization. While there has been extensive research in solving nonsmooth
convex finite-sum problems (i.e., each fi is convex for i 2 [n]) [4, 16, 31], our understanding of their
nonsmooth nonconvex counterparts is surprisingly limited. We hope to amend this situation (at least
partially), given the widespread importance of nonconvexity throughout machine learning.
A popular approach to handle nonsmoothness in convex problems is via proximal operators [14, 25],
but as we will soon see, this approach does not work so easily for the nonconvex problem (1).
Nevertheless, recall that proper closed convex function h, the proximal operator is defined as
?
?
1
prox?h (x) := argmin h(y) + 2?
ky xk2 ,
for ? > 0.
(2)
y2Rd
The power of proximal operators lies in how they generalize projections: e.g., if h is the indicator
function IC (x) of a closed convex set C, then proxIC (x) ? projC (x) ? argminy2C ky xk.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Throughout this paper, we assume that the proximal operator of h is easy to compute. This is true
for many applications in machine learning and statistics including `1 regularization, box-constraints,
simplex constraints, among others [2, 18].
Similar to other algorithms, we also assume access to a proximal oracle (PO) that takes a point
x 2 Rd and returns the output of (2). In addition to the number of PO calls, to describe our
P complexity
results we use the incremental first-order oracle (IFO) model.1 For a function f = n1 i fi , an IFO
takes an index i 2 [n] and a point x 2 Rd , and returns the pair (fi (x), rfi (x)).
A standard (batch) method for solving (1) is the proximal-gradient method (P ROX GD) [13], first
studied for (batch) nonconvex problems in [5]. This method performs the following iteration:
xt+1 = prox?h (xt ?rf (xt )),
t = 0, 1, . . . ,
(3)
where ? > 0 is a step size. The following convergence rate for P ROX GD was proved recently.
Theorem (Informal). [7]: The number of IFO and PO calls made by the proximal gradient method (3)
to reach ? close to a stationary point is O(n/?) and O(1/?), respectively.
We refer the reader to [7] for details. The key point to note here is that the IFO complexity of (3) is
O(n/?). This is due to the fact that a full gradient rf needs to be computed at each iteration (3),
which requires n IFO calls. When n is large, this high cost per iteration is prohibitive. A more
practical approach is offered by proximal stochastic gradient (P ROX S GD), which performs the
iteration
?
?
?t X
t+1
t
t
x
= prox?t h x
rfi (x ) , t = 0, 1, . . . ,
(4)
i2It
|It |
where It (referred to as minibatch) is a randomly chosen set (with replacement) from [n] and ?t is a
step size. Non-asymptotic convergence of P ROX S GD was also shown recently, as noted below.
Theorem (Informal). [7]: The number of IFO and PO calls made by P ROX S GD, i.e., iteration (4), to
reach ? close to a stationary point is O(1/?2 ) and O(1/?) respectively. For achieving this convergence,
we impose batch sizes |It | that increase and step sizes ?t that decrease with 1/?.
Notice that the PO complexity of P ROX S GD is similar to P ROX GD, but its IFO complexity is
independent of n; though, this benefit comes at the cost of an extra 1/? factor. Furthermore, the step
size must decrease with 1/? (or alternatively decay with the number of iterations of the algorithm).
The same two aspects are also seen for convex stochastic gradient, in both the smooth and proximal
versions. However, in the nonconvex setting there is a key third and more important aspect: the
minibatch size |It | increases with 1/?.
To understand this aspect, consider the case where |It | is a constant (independent of both n and ?),
typically the choice used in practice. In this case, the above convergence result no longer holds and
it is not clear if P ROX S GD even converges to a stationary point at all! To clarify, a decreasing step
size ?t trivially ensures convergence as t ! 1, but the limiting point is not necessarily stationary.
On the other hand, increasing |It | with 1/? can easily lead to |It | n for reasonably small ?, which
effectively reduces the algorithm to (batch) P ROX GD.
This dismal news does not apply to the convex setting, where P ROX S GD is known to converge (in
expectation) to an optimal solution using constant minibatch sizes |It |. Furthermore, this problem
does not afflict smooth nonconvex problems (h ? 0), where convergence with constant minibatches
is known [6, 21, 22]. Thus, there is a fundamental gap in our understanding of stochastic methods
for nonsmooth nonconvex problems. Given the ubiquity of nonconvex models in machine learning,
bridging this gap is important. We do so by analyzing stochastic proximal methods with guaranteed
convergence for constant minibatches, and faster convergence with minibatches independent of 1/?.
Main Contributions
We state our main contributions below and list the key complexity results in Table 1.
? We analyze nonconvex proximal versions of the recently proposed stochastic algorithms S VRG
and S AGA [4, 8, 31], hereafter referred to as P ROX S VRG and P ROX S AGA, respectively. We show
convergence of these algorithms with constant minibatches. To the best of our knowledge, this is
the first work to present non-asymptotic convergence rates for stochastic methods that apply to
nonsmooth nonconvex problems with constant (hence more realistic) minibatches.
1
Introduced in [1] to study lower bounds of deterministic algorithms for convex finite-sum problems.
2
? We show that by carefully choosing the minibatch size (to be sublinearly dependent on n but still
independent of 1/?), we can achieve provably faster convergence than both proximal gradient and
proximal stochastic gradient. We are not aware of any earlier results on stochastic methods for the
general nonsmooth nonconvex problem that have faster convergence than proximal gradient.
? We study a nonconvex subclass of (1) based on the proximal extension of Polyak-?ojasiewicz
inequality [9]. We show linear convergence of P ROX S VRG and P ROX S AGA to the optimal solution
for this subclass. This includes the recent results proved in [27, 32] as special cases. Ours is the
first stochastic method with provable global linear convergence for this subclass of problems.
1.1
Related Work
The literature on finite-sum problems is vast; so we summarize only a few closely related works.
Convex instances of (1) have been long studied [3, 15] and are fairly well-understood. Remarkable
recent progress for smooth convex instances of (1) is the creation of variance reduced (VR) stochastic
methods [4, 8, 26, 28]. Nonsmooth proximal VR stochastic algorithms are studied in [4, 31]
where faster convergence rates for both strongly convex and non-strongly convex cases are proved.
Asynchronous VR frameworks are developed in [20]; lower-bounds are studied in [1, 10].
In contrast, nonconvex instances of (1) are much less understood. Stochastic gradient for smooth
nonconvex problems is analyzed in [6], and only very recently, convergence results for VR stochastic
methods for smooth nonconvex problems were obtained in [21, 22]. In [11], the authors consider a VR
nonconvex setting different from ours, namely, where the loss is (essentially strongly) convex but hard
thresholding is used. We build upon [21, 22], and focus on handling nonsmooth convex regularizers
(h 6? 0 in (1)).2 Incremental proximal gradient methods for this class were also considered in [30]
but only asymptotic convergence was shown. The first analysis of a projection version of nonconvex
S VRG is due to [29], who considers the special problem of PCA. Perhaps, the closest to our work
is [7], where convergence of minibatch nonconvex P ROX S GD method is studied. However, typical
to the stochastic gradient method, the convergence is slow; moreover, no convergence for constant
minibatches is provided.
2
Preliminaries
We assume that the function h(x) in (1) is lower semi-continuous (lsc) and convex. Furthermore, we
also assume that its domain dom(h) = {x 2 Rd |h(x) < +1} is closed. We say f is L-smooth if
there is a constant L such that
krf (x)
rf (y)k ? Lkx
yk,
8 x, y 2 Rd .
Throughout, we assume that the functions fi in (1) are L-smooth, so that krfi (x) rfi (y)k ?
Lkx yk for all i 2 [n]. Such an assumption is typical in the analysis of first-order methods.
One crucial aspect of the analysis for nonsmooth nonconvex problems is the convergence criterion.
For convex problems, typically the optimality gap F (x) F (x? ) is used as a criterion. It is
unreasonable to use such a criterion for general nonconvex problems due to their intractability. For
smooth nonconvex problems (i.e., h ? 0), it is typical to measure stationarity, e.g., using krF k. This
cannot be used for nonsmooth problems, but a fitting alternative is the gradient mapping3 [17]:
G? (x) := ?1 [x
prox?h (x
?rf (x))].
(5)
When h ? 0 this mapping reduces to G? (x) = rf (x) = rF (x), the gradient of function F at x.
We analyze our algorithms using the gradient mapping (5) as described more precisely below.
Definition 1. A point x output by stochastic iterative algorithm for solving (1) is called an ?-accurate
solution, if E[kG? (x)k2 ] ? ? for some ? > 0.
Our goal is to obtain efficient algorithms for achieving an ?-accurate solution, where efficiency is
measured using IFO and PO complexity as functions of 1/? and n.
2
More recently, the authors have also developed VR Frank-Wolfe methods for handling constrained problems
that do not admit easy projection operators [24].
3
This mapping has also been used in the analysis of nonconvex proximal methods in [6, 7, 30].
3
Algorithm
IFO
PO
IFO (PL)
PO (PL)
Constant
minibatch?
P ROX S GD
O 1/?2
O (1/?)
O 1/?2
P ROX GD
O (n/?)
O (1/?)
?
O (1/?)
O (n? log(1/?))
O (? log(1/?))
P ROX S VRG
O(n + (n2/3 /?))
O(1/?)
O((n + ?n2/3 ) log(1/?))
O(? log(1/?))
p
P ROX S AGA
O(n + (n2/3 /?))
O(1/?)
O((n + ?n2/3 ) log(1/?))
O(? log(1/?))
p
Table 1: Table comparing the best IFO and PO complexity of different algorithms discussed in the paper.
The complexity is measured in terms of the number of oracle calls required to achieve an ?-accurate solution.
The IFO (PL) and PO (PL) represents the IFO and PO complexity of PL functions (see Section 4 for a formal
definition). The results marked in red are the contributions of this paper. In the table, ?constant minibatch?
indicates whether stochastic algorithm converges using a constant minibatch size. To the best of our knowledge,
it is not known if P ROX S GD converges on using constant minibatches for nonconvex nonsmooth optimization.
Also, we are not aware of any specific convergence results for P ROX S GD in the context of PL functions.
3
Algorithms
We focus on two algorithms: (a) proximal S VRG (P ROX S VRG) and (b) proximal S AGA (P ROX S AGA).
3.1
Nonconvex Proximal SVRG
We first consider a variant of P ROX S VRG [31]; pseudocode of this variant is stated in Algorithm 1.
When F is strongly convex, S VRG attains linear convergence rate as opposed to sublinear convergence
of S GD [8]. Note that, while S VRG is typically stated with b = 1, we use its minibatch variant with
batch size b. The specific reasons for using such a variant will become clear during the analysis.
While some other algorithms have been proposed for reducing the variance in the stochastic gradients,
S VRG is particularly attractive because of its low memory requirement; it requires just O(d) extra
memory in comparison to S GD for storing the average gradient (g s in Algorithm 1), while algorithms
like S AG and S AGA incur O(nd) storage cost. In addition to its strong theoretical results, S VRG is
known to outperform S GD empirically while being more robust to selection of step size. For convex
problems, P ROX S VRG is known to inherit these advantages of S VRG [31].
We now present our analysis of nonconvex P ROX S VRG, starting with a result for batch size b = 1.
Theorem 1. Let b = 1 in Algorithm 1. Let ? = 1/(3Ln), m = n and T be a multiple of m. Then the
output xa of Algorithm 1 satisfies the following bound:
?
?
18Ln2 F (x0 ) F (x? )
E[kG? (xa )k2 ] ?
,
3n 2
T
where x? is an optimal solution of (1).
Theorem 1 shows that P ROX S VRG converges for constant minibatches of size b = 1. This result
is in strong contrast to P ROX S GD whose convergence with constant minibatches is still unknown.
However, the result delivered by Theorem 1 is not stronger than that of P ROX GD. The following
corollary to Theorem 1 highlights this point.
Corollary 1. To obtain an ?-accurate solution, with b = 1 and parameters from Theorem 1, the IFO
and PO complexities of Algorithm 1 are O(n/?) and O(n/?), respectively.
Corollary 1 follows upon noting that each inner iteration (Step 7) of Algorithm 1 has an effective
IFO complexity of O(1) since m = n. This IFO complexity includes the IFO calls for calculating
the average gradient at the end of each epoch. Furthermore, each inner iteration also invokes the
proximal oracle, whereby the PO complexity is also O(n/?). While the IFO complexity of constant
minibatch P ROX S VRG is same as P ROX GD, we see that its PO complexity is much worse. This
is due to the fact that n IFO calls correspond to one PO call in P ROX GD, while one IFO call in
P ROX S VRG corresponds to one PO call. Consequently, we do not gain any theoretical advantage by
using constant minibatch P ROX S VRG over P ROX GD.
4
Algorithm 1: Nonconvex P ROX S VRG x0 , T, m, b, ?
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
Input: x
?0 = x0m = x0 2 Rd , epoch length m, step sizes ? > 0, S = dT /me, minibatch size b
for s = 0 to S 1 do
xs+1
= xsmP
0
g s+1 = n1 n
xs )
i=1 rfi (?
for t = 0 to m 1 do
UniformlyP
randomly pick It ? {1, . . . , n} (with replacement) such that |It | = b
vts+1 = 1b it 2It (rfit (xs+1
) rfit (?
xs )) + g s+1
t
s+1
s+1
s+1
xt+1 = prox?h (xt
?vt )
end for
x
?s+1 = xs+1
m
end for
1 S 1
Output: Iterate xa chosen uniformly at random from {{xs+1
}m
t
t=0 }s=0 .
The key question is therefore: can we modify the algorithm to obtain better theoretical guarantees?
To answer this question, we prove the following main convergence result. For ease of theoretical
exposition, we assume n2/3 to be an integer. This is only for convenience in stating our theoretical
results and all the results in the paper hold for the general case.
Theorem 2. Suppose b = n2/3 in Algorithm 1. Let ? = 1/(3L), m = bn1/3 c and T be a multiple of
m. Then for the output xa of Algorithm 1, we have:
E[kG? (xa )k2 ] ?
18L(F (x0 )
T
F (x? ))
,
where x? is an optimal solution to (1).
Rewriting Theorem 2 in terms of the IFO and PO complexity, we obtain the following corollary.
Corollary 2. Let b = n2/3 and set parameters as in Theorem 2. Then, to obtain an ?-accurate
solution the IFO and PO complexities of Algorithm 1 are O(n + n2/3 /?) and O(1/?), respectively.
The above corollary is due to the following observations. From Theorem 2, it can be seen that the
total number of inner iterations (across all epochs) of Algorithm 1 to obtain an ?-accurate solution
is O(1/?). Since each inner iteration of Algorithm 2 involves a call to the PO, we obtain a PO
complexity of O(1/?). Further, since b = n2/3 IFO calls are made at each inner iteration, we obtain
a net IFO complexity of O(n2/3 /?). Adding the IFO calls for the calculation of the average gradient
(and noting that T is a multiple of m), as well as noting that S
1, we obtain a total cost of
O(n + n2/3 /?). A noteworthy aspect of Corollary 2 is that its PO complexity matches P ROX GD, but
its IFO complexity is significantly decreased to O(n + n2/3 /?) as opposed to O(n/?) in P ROX GD.
3.2
Nonconvex Proximal SAGA
In the previous section, we investigated P ROX S VRG for solving (1). Note that P ROX S VRG is not a
fully ?incremental" algorithm since it requires calculation of the full gradient once per epoch. An
alternative to P ROX S VRG is the algorithm proposed in [4] (popularly referred to as SAGA). We build
upon the work of [4] to develop P ROX S AGA, a nonconvex proximal variant of S AGA.
The pseudocode for P ROX S AGA is presented in Algorithm 2. The key difference between Algorithm 1
and 2 is that P ROX S AGA, unlike P ROX S VRG, avoids computation of the full gradient. Instead, it
maintains an average gradient vector g t , which changes at each iteration (refer to [20]). However,
such a strategy entails additional storage costs. In particular, for implementing Algorithm 2, we must
store the gradients {rfi (?it )}ni=1 , which in general can cost O(nd) in storage. Nevertheless, in some
scenarios common to machine learning (see [4]), one can reduce the storage requirements to O(n).
Whenever such an implementation of P ROX S AGA is possible, it can perform similar to or even better
than P ROX S VRG [4]; hence, in addition to theoretical interest, it is of significant practical value.
We remark that P ROX S AGA in Algorithm 2 differs slightly from [4]. In particular, it uses minibatches
where two sets It , Jt are sampled at each iteration as opposed to one in [4]. This is mainly for the
ease of theoretical analysis.
5
Algorithm 2: Nonconvex P ROX S AGA x0 , T, b, ?
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
0
Input: xP
2 Rd , ?i0 = x0 for i 2 [n], step size ? > 0, minibatch size b
0
g 0 = n1 n
i=1 rfi (?i )
for t = 0 to T 1 do
Uniformly
P randomly pick sets It , Jt from [n] (with replacement) such that |It | = |Jt | = b
v t = 1b it 2It (rfit (xt ) rfit (?itt )) + g t
xt+1 = prox?h (xt ?v t )
?jt+1 = xt for j 2 Jt and ?jt+1 = ?jt for j 2
/ Jt
P
))
g t+1 = g t n1 jt 2Jt (rfjt (?jtt ) rfjt (?jt+1
t
end for
Output: Iterate xa chosen uniformly random from {xt }Tt=01 .
We prove that as in the convex case, nonconvex P ROX S VRG and P ROX S AGA share similar theoretical
guarantees. In particular, our first result for P ROX S AGA is a counterpart to Theorem 1 for P ROX S VRG.
Theorem 3. Suppose b = 1 in Algorithm 2. Let ? = 1/(5Ln). Then for the output xa of Algorithm 2
after T iterations, the following stationarity bound holds:
E[kG? (xa )k2 ] ?
50Ln2 F (x0 ) F (x? )
,
5n 2
T
where x? is an optimal solution of (1).
Theorem 3 immediately leads to the following corollary.
Corollary 3. The IFO and PO complexity of Algorithm 3 for b = 1 and parameters specified in
Theorem 3 to obtain an ?-accurate solution are O(n/?) and O(n/?) respectively.
Similar to Theorem 2 for P ROX S VRG, we obtain the following main result for P ROX S AGA.
Theorem 4. Suppose b = n2/3 in Algorithm 2. Let ? = 1/(5L). Then for the output xa of
Algorithm 2 after T iterations, the following holds:
50L(F (x0 ) F (x? ))
,
3T
where x? is an optimal solution of Problem (1).
E[kG? (xa )k2 ] ?
Rewriting this result in terms of IFO and PO access, we obtain the following important corollary.
Corollary 4. Let b = n2/3 and set parameters as in Theorem 4. Then, to obtain an ?-accurate
solution the IFO and PO complexities of Algorithm 2 are O(n + n2/3 /?) and O(1/?), respectively.
The above result is due to Theorem 4 and because each iteration of P ROX S AGA requires O(n2/3 )
IFO calls. The number of PO calls is only O(1/?), since make one PO call for every n2/3 IFO calls.
Discussion: It is important to note the role of minibatches in Corollaries 2 and 4. Minibatches are
typically used for reducing variance and promoting parallelism in stochastic methods. But unlike
previous works, we use minibatches as a theoretical tool to improve convergence rates of both
nonconvex P ROX S VRG and P ROX S AGA. In particular, by carefully selecting the minibatch size, we
can improve the IFO complexity of the algorithms described in the paper from O(n/?) (similar to
P ROX GD) to O(n2/3 /?) (matching the smooth nonconvex case). Furthermore, the PO complexity is
also improved in a similar manner by using the minibatch size mentioned in Theorems 2 and 4. 4
4
Extensions
We discuss some extensions of our approach in this section. Our first extension is to provide
convergence analysis for a subclass of nonconvex functions that satisfy a specific growth condition
popularly known as the Polyak-?ojasiewicz (PL) inequality. In the context of gradient descent,
4
We refer the readers to the full version [23] for a more general convergence analysis of the algorithms.
6
PL-SVRG:(x0 , K, T, m, ?)
for k = 1 to K do
xk = ProxSVRG(xk 1 , T, m, b, ?) ;
end
Output: xK
PL-SAGA:(x0 , K, T, m, ?)
for k = 1 to K do
xk = ProxSAGA(xk 1 , T, b, ?) ;
end
Output: xK
Figure 1: P ROX S VRG and P ROX S AGA variants for PL functions.
this inequality was proposed by Polyak in 1963 [19], who showed global linear convergence of
gradient descent for functions that satisfy the PL inequality. Recently, in [9] the PL inequality was
generalized to nonsmooth functions and used for proving linear convergence of proximal gradient.
The generalization presented in [9] considers functions F (x) = f (x)+h(x) that satisfy the following:
1
F (x? )) ? Dh (x, ?), where ? > 0
2
?
?
and Dh (x, ?) := 2? miny hrf (x), y xi + ky xk2 + h(y)
2
An F that satisfies (6) is called a ?-PL function.
?(F (x)
?
h(x) .
(6)
When h ? 0, condition (6) reduces to the usual PL inequality. The class of ?-PL functions includes
several other classes as special cases. It subsumes strongly convex functions, covers fi (x) = g(a>
i x)
with only g being strongly convex, and includes functions that satisfy a optimal strong convexity
property [12]. Note that the ?-PL functions also subsume the recently studied special case where fi ?s
are nonconvex but their sum f is strongly convex. Hence, it encapsulates the problems of [27, 32].
The algorithms in Figure 1 provide variants of P ROX S VRG and P ROX S AGA adapted to optimize
?-PL functions. We show the following global linear convergence result of P L -S VRG and P L -S AGA
in Figure 1 for PL functions. For simplicity, we assume ? = (L/?) > n1/3 . When f is strongly
convex, ? is referred to as the condition number, in which case ? > n1/3 corresponds to the high
condition number regime.
Theorem 5. Suppose F is a ?-PL function. Let b = n2/3 , ? = 1/5L, m = bn1/3 c and T = d30?e.
Then for the output xK of P L -S VRG and P L -S AGA (in Figure 1), the following holds:
[F (x0 ) F (x? )]
E[F (xK ) F (x? )] ?
,
2K
?
where x is an optimal solution of (1).
The following corollary on IFO and PO complexity of P L -S VRG and P L -S AGA is immediate.
Corollary 5. When F is a ?-PL function, then the IFO and PO complexities of P L -S VRG and
P L -S AGA with the parameters specified in Theorem 5 to obtain an ?-accurate solution are O((n +
?n2/3 ) log(1/?)) and O(? log(1/?)), respectively.
Note that proximal gradient also has global linear convergence for PL functions, as recently shown
in [9]. However, its IFO complexity is O(?n log(1/?)), which is much worse than that of P L -S VRG
and P L -S AGA (Corollary 5).
Other extensions: While we state our results for specific minibatch sizes, a more general convergence
analysis is provided for any minibatch size b ? n2/3 (Theorems 6 and 7 in the Appendix). Moreover,
our results can be easily generalized to the case where non-uniform sampling is used in Algorithm 1
and Algorithm 2. This is useful when the functions fi have different Lipschitz constants.
5
Experiments
We present our empirical results in this section. For our experiments, we study the problem of
non-negative principal component analysis (NN-PCA). More specifically, for a given set of samples
{zi }ni=1 , we solve the following optimization problem:
min
kxk?1, x 0
1 >
x
2
7
n
X
i=1
zi zi>
!
x.
(7)
10 -15
0
5
10
15
10 -10
10
-15
# grad/n
0
5
10
15
# grad/n
SGD
SAGA
SVRG
10 -5
10
f (x) ! f (^
x)
10 -10
SGD
SAGA
SVRG
10 -5
f (x) ! f (^
x)
f (x) ! f (^
x)
f (x) ! f (^
x)
SGD
SAGA
SVRG
10 -5
-10
10 -15
0
5
10
15
# grad/n
SGD
SAGA
SVRG
10 -5
10
-10
10 -15
0
5
10
15
# grad/n
Figure 2: Non-negative principal component analysis. Performance of P ROX S GD, P ROX S VRG and
P ROX S AGA on ?rcv1? (left), ?a9a?(left-center), ?mnist? (right-center) and ?aloi? (right) datasets. Here,
the y-axis is the function suboptimality i.e., f (x) f (?
x) where x
? represents the best solution obtained
by running gradient descent for long time and with multiple restarts.
The problem of NN-PCA is, in general, NP-hard. This variant of the standard PCA problem can
be written in the form (1) with fi (x) = (x> zi )2 for all i 2 [n] and h(x) = IC (x) where C is the
convex set {x 2 Rd |kxk ? 1, x 0}. In our experiments, we compare P ROX S GD with nonconvex
P ROX S VRG and P ROX S AGA. The choice of step size is important to P ROX S GD. The step size of
P ROX S GD is set using the popular t-inverse step size choice of ?t = ?0 (1 + ? 0 bt/nc) 1 where
?0 , ? 0 > 0. For P ROX S VRG and P ROX S AGA, motivated by the theoretical analysis, we use a fixed
step size. The parameters of the step size in each of these methods are chosen so that the method gives
the best performance on the objective value. In our experiments, we include the value ? 0 = 0, which
corresponds to P ROX S GD with fixed step size. For P ROX S VRG, we use the epoch length m = n.
We use standard machine learning datasets in LIBSVM for all our experiments 5 . The samples from
each of these datasets are normalized i.e. kzi k = 1 for all i 2 [n]. Each of these methods is initialized
by running P ROX S GD for n iterations. Such an initialization serves two purposes: (a) it provides a
reasonably good initial point, typically beneficial for variance reduction techniques [4, 26]. (b) it
provides a heuristic for calculating the initial average gradient g 0 [26]. In our experiments, we use
b = 1 in order to demonstrate the performance of the algorithms with constant minibatches.
We report the objective function value for the datasets. In particular, we report the suboptimality in
objective function i.e., f (xs+1
) f (?
x) (for P ROX S VRG) and f (xt ) f (?
x) (for P ROX S AGA). Here
t
x
? refers to the solution obtained by running proximal gradient descent for a large number of iterations
and multiple random initializations. For all the algorithms, we compare the aforementioned criteria
against for the number of effective passes through the dataset i.e., IFO complexity divided by n. For
P ROX S VRG, this includes the cost of calculating the full gradient at the end of each epoch.
Figure 2 shows the performance of the algorithms on NN-PCA problem (see Section D of the Appendix for more experiments). It can be seen that the objective value for P ROX S VRG and P ROX S AGA
is much lower compared to P ROX S GD, suggesting faster convergence for these algorithms. We
observed a significant gain consistently across all the datasets. Moreover, the selection of step size
was much simpler for P ROX S VRG and P ROX S AGA than that for P ROX S GD. We did not observe any
significant difference in the performance of P ROX S VRG and P ROX S AGA for this particular task.
6
Final Discussion
In this paper, we presented fast stochastic methods for nonsmooth nonconvex optimization. In
particular, by employing variance reduction techniques, we show that one can design methods that
can provably perform better than P ROX S GD and proximal gradient descent. Furthermore, in contrast
to P ROX S GD, the resulting approaches have provable convergence to a stationary point with constant
minibatches; thus, bridging a fundamental gap in our knowledge of nonsmooth nonconvex problems.
We proved that with a careful selection of minibatch size, it is possible to theoretically show superior
performance to proximal gradient descent. Our empirical results provide evidence for a similar
conclusion even with constant minibatches. Thus, we conclude with an important open problem of
developing stochastic methods with provably better performance than proximal gradient descent with
constant minibatch size.
Acknowledgment: SS acknowledges support of NSF grant: IIS-1409802.
5
The datasets can be
libsvmtools/datasets.
downloaded
from
8
https://www.csie.ntu.edu.tw/~cjlin/
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9
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alexander:5 handling:3 |
5,655 | 6,117 | Bayesian Optimization with
Robust Bayesian Neural Networks
Jost Tobias Springenberg Aaron Klein Stefan Falkner Frank Hutter
Department of Computer Science
University of Freiburg
{springj,kleinaa,sfalkner,fh}@cs.uni-freiburg.de
Abstract
Bayesian optimization is a prominent method for optimizing expensive-to-evaluate
black-box functions that is widely applied to tuning the hyperparameters of machine
learning algorithms. Despite its successes, the prototypical Bayesian optimization
approach ? using Gaussian process models ? does not scale well to either many
hyperparameters or many function evaluations. Attacking this lack of scalability
and flexibility is thus one of the key challenges of the field. We present a general
approach for using flexible parametric models (neural networks) for Bayesian
optimization, staying as close to a truly Bayesian treatment as possible. We obtain
scalability through stochastic gradient Hamiltonian Monte Carlo, whose robustness
we improve via a scale adaptation. Experiments including multi-task Bayesian
optimization with 21 tasks, parallel optimization of deep neural networks and deep
reinforcement learning show the power and flexibility of this approach.
1
Introduction
Hyperparameter optimization is crucial for obtaining good performance in many machine learning
algorithms, such as support vector machines, deep neural networks, and deep reinforcement learning.
The most prominent method for hyperparameter optimization is Bayesian optimization (BO) based
on Gaussian processes (GPs), as e.g., implemented in the Spearmint system [1].
While GPs are the natural probabilistic models for BO, unfortunately, their complexity is cubic in
the number of data points and they often do not gracefully scale to high dimensions [2]. Although
alternative methods based on tree models [3, 4] or Bayesian linear regression using features from
a neural network [5] exist, they obtain scalability by partially sacrificing a principled treatment of
model uncertainties.
Here, we propose to use neural networks as a powerful and scalable parametric model, while
staying as close to a truly Bayesian treatment as possible. Crucially, we aim to keep the wellcalibrated uncertainty estimates of GPs since BO relies on them to accurately determine promising
hyperparameters. To this end we derive a more robust variant of the recent stochastic gradient
Hamiltonian Monte Carlo (SGHMC) method [6].
After providing background (Section 2), we make the following contributions: We derive a general
formulation for both single-task and multi-task BO with Bayesian neural networks that leads to
a robust, scalable, and parallel optimizer (Section 3). We derive a scale adaptation technique to
substantially improve the robustness of stochastic gradient HMC (Section 4). Finally, using our
method ? which we dub Bayesian Optimization with Hamiltonian Monte Carlo Artificial Neural
Networks (BOHAMIANN) ? we demonstrate state-of-the-art performance for a wide range of
optimization tasks. This includes multi-task BO, parallel optimization of deep residual networks, and
deep reinforcement learning. An implementation of our method can be found at https://github.
com/automl/RoBO.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
2
Background
2.1
Bayesian optimization for single and multiple tasks
Let f : X ? R be an arbitrary function defined over a convex set X ? Rd that can be evaluated at
2
x ? X , yielding noisy observations y ? N (f (x), ?obs
). We aim to find x? ? arg minx?X f (x). To
solve this problem, BO (see, e.g., Brochu et al. [7]) typically starts by observing the function at an
initial design D = {(x1 , y1 ), . . . , (xI , yI )}. BO then repeatedly executes the following steps: (1) fit
a regression model p(f | D) to the current data D; (2) use p(f | D) to select an input xt+1 at which
to query f by maximizing an acquisition function (which trades off exploration and exploitation); (3)
2
observe yt+1 ? N (f (xt+1 ), ?obs
) and add the result to the dataset: D := D ? {xt+1 , yt+1 }.
In the generalized case of multi-task Bayesian optimization [8], there are K related black-box
functions, F = {f1 , . . . , fK }, each with the same domain X ; and, the goal is to find x? ?
arg minx?X ft (x) for a given t.1 In this case, the initial design is augmented with previous evaluations of the related functions. That is, D = D1 ? ? ? ? ? DK with Dk = {(xk1 , y1k ), . . . , (xknk , ynk k )},
2
where yik ? N (fk (xki ), ?obs
) and nk = |Dk | points have already been evaluated for function fk . BO
then requires a probabilistic model p(f | D) over the K functions, which can be used to transfer
knowledge from related tasks to the target task t (and thus reduce the required number of function
evaluations on t).
A concrete instantiation of BO is obtained by specifying the acquisition function and the probabilistic
model. As acquisition function, here, we will use the popular expected improvement (EI) criterion [9];
other commonly used options, such as UCB [10] could be directly applied. EI is defined as
?EI (x; D) = ?(f (x) | D) (?(x)?(?(x)) + ?(?(x))) , with ?(x)
=
y? ? ?(f (x) | D)
,
?(f (x) | D)
(1)
where ?(?) and ?(?) denote the cumulative distribution function and the probability density function
of a standard normal distribution, respectively, and ?(f (x) | D) and ?(f (x) | D) denote the posterior
mean and standard deviation of our probabilistic model based on data D. While the prototypical
probabilistic model in BO is a GP [1], we will use a Bayesian neural network (BNN).
2.2
Bayesian methods for neural networks
The ability to combine the flexibility and scalability of (deep) neural networks with well-calibrated
uncertainty estimates is highly desirable in many contexts. Not surprisingly, there thus exist many
approaches for this problem, including early work on (non-scalable) Hamiltonian Monte Carlo
[11], recent work on variational inference methods [12, 13] and expectation propagation [14], reinterpretations of dropout as approximate inference [15, 16], as well as stochastic gradient MCMC
methods based on Hamiltonian Monte Carlo [6] and stochastic gradient Langevin MCMC [17].
While any of these methods could, in principle, be used for BO, we found most of them to result in
suboptimal uncertainty estimates. Our preliminary experiments ? presented in the supplementary
material (Section B) ? suggest these methods often conservatively estimate the uncertainty for points
far away from the data, particularly when based on little training data. This is problematic for BO,
which crucially relies on well-calibrated uncertainty estimates based on few function evaluations.
One family of methods that consistently resulted in good uncertainty estimates in our tests were
Hamiltonian Monte Carlo (HMC) methods, which we will thus use throughout this paper. Concretely,
we will build on the scalable stochastic MCMC method from Chen et al. [6].
3
Bayesian optimization with Bayesian neural networks
We now formalize the Bayesian neural network regression model we use as the basis of our Bayesian
optimization approach. Formally, under the assumption that the observed function values (conditioned on x) are normally distributed (with unknown mean and variance), we start by defining our
probabilistic function model as
p(ft (x) | x, ?) = N (f?(x, t; ?? ), ??2 ),
1
The standard single-task case is recovered when K = t = 1.
2
(2)
where ? = [?? , ??2 ]T , f?(x, t; ?? ) is the output of a parametric model with parameters ?? , and where
we assume a homoscedastic noise model with zero mean and variance ??2 .2 A single-task model can
trivially be obtained from this definition:
Single-task model. In the single-task setting we simply model the function mean f?(x, t; ?? ) =
h(x; ?? ) using a neural network, with output h (i.e. h implements a forward-pass).
Multi-task model. For the multi-task model we use a slightly adapted network architecture. As
additional input,the networkis provided with a task-specific embedding vector. That is, we have
T
f?(x, t; ?? ) = h [x; ?t ] , ?h , where h(?), again, denotes the output of the neural network (here
with parameters ?h ) and ?t is the t-th row of an embedding matrix ? ? RK?L (we choose L = 5 for
our experiments). This embedding matrix is learned alongside all other parameters. Additionally,
if information about the dataset (such as data-set size etc.) is available it can be appended to this
embedding vector. The full vector of the network parameters then becomes ?? = [?h , vec(?)], where
vec(?) denotes vectorization. Instead of using a learned embedding we could have chosen to represent
the tasks through a one-out-of-K encoding vector, which functionally would be equivalent but would
induce a large number of additional parameters to be learned for large K. With these definitions, the
joint probability of the model parameters and the observed data is then
p(D, ?) = p(?? )p(??2 )
K |D
k|
Y
Y
N (yik |f?(xki , k; ?? ), ??2 ),
(3)
k=1 i=1
where p(?? ) and p(??2 ) are priors on the network parameters and on the variance, respectively.
For BO, we need to be able to compute the acquisition function at given candidate points x. For
this we require the predictive posterior p(ft (x)|x, D) (marginalized over the model parameters ?).
Unfortunately, for our choice of modeling ft with a neural network, evaluating this posterior exactly
is intractable. Let us, for now, assume that we can generate samples ?i ? p(? | D) from the posterior
for the model parameters given the data; we will show how to do this with stochastic gradient
Hamiltonian Monte Carlo (SGHMC) in Section 4. We can then use these samples to approximate the
predictive posterior p(ft (x)|x, D) as
Z
M
1 X
(4)
p(ft (x) | x, ?i ).
p(ft (x)|x, D) = p(ft (x) | x, ?)p(? | D)d? ?
M
?
i=1
Using the same samples ?i ? p(? | D), we make a Gaussian approximation to this predictive
distribution to obtain mean and variance to compute the EI value in Equation (1):
M
M
2
1 X?
1 X?
?(ft (x)|D) =
f (x, t; ??i ), ? 2 (f (x)|D) =
f (x, t; ??i ) ? ?(ft (x)|D) + ??i 2 .
M i=1
M i=1
(5)
Notably, we can compute partial derivatives of ?EI (with respect to x) via backpropagation through
all functions f?(x, t; ??i ) which allows gradient-based maximization of the acquisition function.
We also extended this formulation to parallel asynchronous BO by sampling possible outcomes for
currently-running function evaluations and using the acquisition function ?MCEI proposed by Snoek
et al. [1]. Details are given in the supplementary material (Section A).
4
Robust stochastic gradient HMC via scale adaptation
In this section, we show how stochastic gradient Hamiltonian Monte Carlo (SGHMC) can be used to
sample from the model defined by Equation (3). We first summarize the general formalism behind
SGHMC [6] and then derive a more robust variant suitable for BO.
4.1
Stochastic gradient HMC
HMC introduces a set of auxiliary variables, r, and then samples from the joint distribution
1
p(?, r | D) ? exp ?U (?) ? rT M?1 r ,
with
U (?) = ? log p(D, ?)
2
2
(6)
We note that, if required, we could model heteroscedastic functions by defining the observation noise
variance ??2 as a deterministic function of x (e.g. as the second output of the neural network).
3
by simulating a fictitious physical system described by a set of differential equations, called Hamilton?s
equations. In this system, the negative log-likelihood U (?) plays the role of a potential energy, r
corresponds to the momentum of the system, and M represents the (arbitrary) mass matrix [18].
Classically, the dynamics for ? and r depend on the gradient ?U (?) whose evaluation is too expensive
for our purposes, since it would involve evaluating the model on all data-points. By introducing a
user-defined friction matrix C, Chen et al. [6] showed how Hamiltonian dynamics can be modified to
? (?), e.g. computed from a mini-batch,
sample from the correct distribution if only a noisy estimate ?U
is available. In particular, their discretized system of equations reads
?? = M?1 r ,
?
? (?) ? CM?1 r + N (0, 2(C ? B))
?r = ??U
,
(7)
where, in a suggestive notation, we write N (0, ?) representing the addition of a sample from a
multivariate Gaussian with zero mean and covariance matrix ?. Besides the estimate for the noise
? and an undefined step length , all that is required for simulating the
of the gradient evaluation B,
dynamics in Equation (7) is a mechanism for computing gradients of the log likelihood (and thus
of our model) on small subsets (or batches) of the data. This makes SGHMC particularly appealing
when working with large models and data-sets. Furthermore, Equation (7) can be seen as an MCMC
analogue to stochastic gradient descent (with momentum) [6]. Following these update equations, the
distribution of (?, r) is the one in Equation (6), and ? is guaranteed to be distributed according to
p(? | D).
4.2
Scale adapted stochastic gradient HMC
Like many Monte Carlo methods, SGHMC does not come without caveats, namely the correct setting
? the mass
of the user-defined quantities: the friction term C, the estimate of the gradient noise B,
matrix M, the number of MCMC steps, and ? most importantly ? the step-size . We found the
friction term and the step-size to be highly model and data-set dependent3 , which is unacceptable for
BO, where robust estimates are required across many different functions F with as few parameter
choices as possible.
A closer look at Equation (7) shows why the step-size crucially impacts the robustness of SGHMC.
For the popular choice M = I, the change in the momentum is proportional to the gradient. If the
gradient elements are on vastly different scales (and potentially correlated), then the update effectively
assigns unequal importance to changes in different parameters of the model. This, in turn, can lead
to slow exploration of the target density. To correct for unequal parameter scales (and respect their
correlation), we would ideally like to use M as a pre-conditioner, reflecting the metric underlying
the model?s parameters. This would lead to a stochastic gradient analogue of Riemann Manifold
Hamiltonian Monte Carlo [19], which has been studied before by Ma et al. [20] and results in an
algorithm called generalized stochastic gradient Riemann Hamiltonian Monte Carlo (gSGRHMC).
Unfortunately, gSGRHMC requires computation (and storage) of the full Fisher information matrix
of U and its gradient, which is prohibitively expensive for our purposes.
As a pragmatic approach, we consider a pre-conditioning scheme increasing SGHMCs robustness
with respect to and C, while avoiding the costly computations of gSGRHMC. We want to note
that recently ? and directly related to our approach ? adaptive pre-conditioning using ideas from
SGD methods has been combined with stochastic gradient Langevin dynamics in Li et al. [21] and to
derive a hybrid between SGD optimization and HMC sampling in Chen et al. [22]. These approaches
however either come with additional hyperparameters that need to be set or do not guarantee unbiased
sampling. The rest of this section shows how all remaining SGHMC parameters in our method are
determined automatically.
Choosing M. For the mass matrix, we take inspiration from the connection between SGHMC
and SGD. Specifically, the literature [23, 24] shows how normalizing the gradient by its magnitude
(estimated over the whole dataset) improves the robustness of SGD. To perform the analogous
operation in SGHMC,
?1 we propose to adapt the mass matrix during the burn-in phase. We set
/2
?1
M = diag V??
, where V?? is an estimate of the (element-wise) uncentered variance of the
? (?))2 ]. We estimate V?? using an exponential moving average during the
gradient: V?? ? E[(?U
3
We refer to Section 5 for a quantitative evaluation of this claim.
4
burn-in phase yielding the update equation
? (?))2 ,
?V?? = ?? ?1 V?? + ? ?1 ?(U
(8)
where ? is a free parameter vector specifying the exponential averaging windows. Note that all
multiplications above are element-wise and ? is a vector with the same dimensionality as ?.
Automatically choosing ? . To avoid adding ? as a new hyperparameter ? that would have to be
tuned ? we automatically determine its value. For this purpose, we use an adaptive estimate previously
derived for adaptive learning rate procedures for SGD [25]. We maintain an additional smoothed
2
estimate of the gradient g? ? ?U (?) and consider the element-wise ratio g?/V?? between the squared
estimated gradient and the gradient variance. This ratio will be large if the estimated gradient is large
compared to the noise ? in which case we can use a small averaging window ? and it will be small
if the noise is large compared to the average gradient ? in which case we want a larger averaging
window. We formalize these desiderata by simultaneously updating Equation 8,
?? = ?g?2 V???1 ? + 1 ,
? (?) .
?g? = ?? ?1 g? + ? ?1 ?U
and
(9)
? While the above procedure removes the need to hand-tune M?1 (and will stabilize
Estimating B.
? Ideally, B
? should be
the method for different C and ), we have not yet defined an estimate for B.
the estimate of the empirical Fisher information matrix that, as discussed above, is too expensive
? = 1 V?? which is readily
to compute. We therefore resort to a diagonal approximation yielding B
2
available from Equation (8).
Scale adapted update equations. Finally, we can combine all parameter estimates to formulate
our automatically scale adapted SGHMC method. Following Chen et al. [6], we introduce the variable
?1/2
substitution v = M?1 r = V?? r which leads us to the dynamical equations
?1/2
? (?) ? V? ?1/2 Cv + N 0, 23 V? ?1/2 CV? ?1/2 ? 4 I , (10)
?? = v ,
?v = ?2 V?? ?U
?
?
?
using the quantities estimated in Equations (8)-(9) during the burn-in phase, and then fixing the
? cancels with the square of our estimate
choices for all parameters. Note that the approximation of B
?1
of M . In practice, we choose C = CI, i.e. the same independent noise for each element of ?. In
this case, Equation (10) constrains the choices of C and , as we need them to fulfill the relation
min(V??1 )C ? . For the remainder of the paper, we fix = 10?2 (a robust choice in our experience)
?1/2
and chose C such that we have V?? C = 0.05I (intuitively this corresponds to a constant decay in
momentum of 0.05 per time step) potentially increasing it to satisfy the mentioned constraint at the
end of the burn-in phase.
We want to emphasize that our estimation/adaptation of the parameters only changes the HMC
procedure during the burn-in phase. After it, when actual samples are recorded, all parameters stay
fixed. In particular, this entails that as long as our choice of and C satisfies min(V???1 )C ? , our
method samples from the correct distribution. Our choices are compatible with the constraints on the
free parameters of the original SGHMC [6]. Further, we note that the scale adaptation technique is
agnostic to the parametric form of the density we aim to sample from; and could therefore potentially
also simplify SGHMC sampling for models beyond those considered in this paper.
5
Experiments on the effects of scale adptation
First, to test the efficacy of the proposed scale adaptation technique, we performed an evaluation on
four common regression datasets following the protocol from Hern?ndez-Lobato and Adams [14],
presented in Table 1. The comparison shows that ? despite its guarantees for sampling from the
correct distribution ? SGHMC (without our adaptation) required tuning for each dataset to obtain
good uncertainty estimates. This effect can likely be attributed to the high dimensionality (and
non-uniformity) of the parameter space (for which the standard SGHMC procedure might just require
too many MCMC steps to sample from the target density). Our adaptation removed these problems.
Additionally we found our method to faithfully represent model uncertainty even in regimes were
only few data-points are available. This observation is qualitatively shown in Figure 1 (right) and
further explored in the supplementary material.
5
Table 1: Log likelihood for regression benchmarks from the UCI repository. For comparison, we
include results for VI (variational inference) and PBP (probabilistic backpropagation) taken from
Hern?ndez-Lobato and Adams [14]. We report mean ? standard deviation across 10 runs. The first
two SGHMC variants are the vanilla algorithm (without our modifications) optimized for best mean
performance (best average), and best performance on each dataset (tuned per dataset) via grid search.
Method/Dataset
Boston Housing Yacht Hydrodynamics
Concrete
Wine Quality Red
SGHMC (best average)
-3.474 ? 0.511
SGHMC (tuned per dataset) -2.489 ? 0.151
SGHMC (scale-adapted)
-2.536 ? 0.036
-13.579 ? 0.983
-1.753 ? 0.19
-1.107 ? 0.083
-4.871 ? 0.051
-4.165 ? 0.723
-3.384 ? 0.24
-1.825 ? 0.75
-1.287 ? 0.28
-1.041 ? 0.17
-2.903 ? 0.071
-2.574 ? 0.089
-3.439 ? 0.163
-1.634 ? 0.016
-3.391 ? 0.017
-3.161 ? 0.019
-0.980 ? 0.013
-0.968 ? 0.014
VI
PBP
6
Bayesian optimization experiments
We now show Bayesian optimization experiments for BOHAMIANN. Unless noted otherwise,
we used a three layer neural network with 50 tanh units for all experiments. For the priors we
let p(?? ) = N (0, ??2 ) be normally distributed and placed a Gamma hyperprior on ??2 , which is
periodically updated via Gibbs sampling. For p(??2 ) we chose a log-normal prior. To approximate EI
we used 50 samples acquired via SGHMC sampling. Maximization of the acquision function was
performed via gradient ascent. Due to space constraints, full details on the experimental setup as
well as the optimized hyperparameters for all experiments are given in the supplementary material
(Section C), which also contains additional plots and evaluations for all experiments.
Branin
103
GP
DNGO (10-10-10)
DNGO (50-50-50)
102
Immediate regret
101
2.0
BOHAMIANN (10-10-10)
BOHAMIANN (50-50-50)
Random Search
sinc(x)
BOHAMIANN
1.0
100
0.5
10-1
0.0
10-2
?0.5
10-3
?1.0
10-4
?1.5
10-5
10-6
Fit of the sinOne function after 20 BO steps using BOHAMIANN
1.5
?2.0
0
50
100
150
Number of function evaluations
200
?2.5
?2.0
?1.5
?1.0
?0.5
0.0
0.5
1.0
1.5
Figure 1: Evaluation on common benchmark problems. (Left) Immediate regret of various optimizers
averaged over 30 runs on the Branin function. For DNGO and BOHAMIANN, we denote the layer
sizes for the (3 layer) networks in parenthesis. (Right) A fit of the sinOne function after 20 steps of
BO using BOHAMIANN. We plot the mean of the predictive posterior and ? 2 standard deviations;
calculated based on 50 MCMC samples.
6.1
Common benchmark problems
As a first experiment, we compare BOHAMIANN to existing state-of-the-art BO on a set of synthetic
functions and hyperparameter optimization tasks devised by Eggensperger et al. [2]. All optimizers
achieved acceptable performance, but GP based methods were found to perform best on these lowdimensional benchmarks, which we thus take as a point of reference. Overall, on the 5 benchmarks
BOHAMIANN matched the performance of GP based BO on 4 and performed worse on one,
indicating that even in the low-data regime Bayesian neural networks (BNNs) are a feasible model
class for BO. A detailed listing of the results is given in the supplementary material.
We further compared to our re-implementation of the recently proposed DNGO method [5], which
uses features extracted from a maximum likelihood fit of a neural network as the basis for a Bayesian
linear regression fit (and was also proposed as a replacement of GPs for scalable BO). For the
benchmark tasks we found both DNGO and BOHAMIANN to perform well with BOHAMIANN
being slightly more robust to different architecture choices. This behavior is illustrated in Figure
1 (left) where we compare DNGO with two different network architectures to BOHAMIANN.
6
Additionally, DNGO performed well for some high-dimensional problems (cf. Section 6.3), but it got
stuck when we used it to optimize 13 hyperparameters of a Deep RL agent (cf. Section 6.4).
6.2
Multi-task hyperparameter optimization
Next, we evaluated BOHAMIANN for multi-task hyperparameter optimization of a support vector
machine (SVM) and a random forest (RF) over a range of different benchmarks. Concretely, we
considered a set of 21 different classification datasets downloaded from the OpenML repository [26].
These were grouped into four groups of related tasks (as determined by a distance based on metafeatures extracted from the datasets). Within each group (consisting of 3-6 datasets), we randomly
designated the optimization of the algorithms hyperparameters for one dataset as the target function
ft . The remaining datasets were used for collecting |Dk | = 30 additional training data points each,
which were used as the initial design for BO. To allow for fast evaluation of this benchmark, we
pre-computed the performance of different hyperparameter settings on all datasets following Feurer
et al. [27]. The task for the optimizer then is to find an optimal hyperparameter setting for the
target benchmark (for which it receives no initial data). We compared our method to the GP based
multi-task BO procedure from Swersky et al. [8], as well as to standard, single-task, GP based BO.
Overall, while all optimizers eventually found a solution close to the optimum the multi-task version
of BOHAMIANN was able to exploit the knowledge obtained from the related datasets, resulting in
quicker convergence. On average over all four benchmarks, MT-BOHAMIANN was 12 % faster
than GP based BO (to reach an immediate regret ? 0.25), whereas MTBO was only 5% faster. Plots
showing the optimizer behavior are included in the supplementary material.
Parallel hyperparameter optimization for deep residual networks
ResNet on CIFAR-10
30
Validation Error %
25
20
15
10
5
0
100000
200000
300000
Runtime in seconds
BO of deep RL algorithm (DDPG on CartPole)
1000
BOHAMIANN
DNGO
random search
Function value (episodes to success)
6.3
400000
BOHAMIANN
DNGO
900
800
700
600
500
400
0
50
100
150
200
250
Function evaluations
300
350
400
Figure 2: (Left) DNGO vs.BOHAMIANN for optimizing the 8 hyperparameters of a deep residual
network on CIFAR-10; we plotted each function evaluation performed over time, as well as the current
best; parallel random search is included as an additional baseline. (Right) DNGO vs. BOHAMIANN
for optimizing the 12 hyperparameters of an RL agent.
Next, we optimized the hyperparameters of the recently proposed residual network (ResNet) architecture [28] for classification of CIFAR-10. We adopted a general parameterization of this architecture,
tuning both the parameters of the stochastic gradient descent training as well as key architectural
choices (such as the dimensionality reduction strategy used between residual blocks). We kept the
maximum number of parameters fixed at the number used by the 32 layer ResNet [28]. Training a
single ResNet took up to 6 hours in our experiments and we therefore used the parallel BO procedure
described in Section 1 of the supplementary material (evaluating 8 ResNet configurations in parallel,
for all of DNGO, random search, and BOHAMIANN).
Interestingly, all methods quickly found good configurations of the hyperparameters as shown in
Figure 2(left), with BOHAMIANN reaching the validation performance of the manually-tuned
baseline ResNet after 104 function evaluations (or approximately 27 hours of total training time).
When re-training this model on the full dataset it obtained a classification error of 7.40 % ? 0.3,
matching the performance of the hand-tuned version from He et al. [28] (7.51 %). Perhaps surprisingly,
this result was reached with a different architecture than the one presented in He et al. [28]: (1)
it used max-pooling instead of strided convolutions for the spatial dimensionality reduction; (2)
approximately 50% of the weights in all residual blocks were shared (thus reducing the number of
parameters).
7
Table 2: Comparison between the original
DDPG algorithm and a version optimized using BOHAMIANN on two control tasks. We
show the number of episodes required to obtain
successful performance in 10 consecutive test
episodes (reward above -2 for CartPole, above -6
for reaching) and the maximum reward achieved
by the controller.
DDGP on Cartpole
0
?1
?2
Reward
?3
?4
?5
?6
?7
DDPG (optimized)
DDPG (original)
?8
?9
0
100
200
300
400
Collected episodes
500
600
Cartpole
700
Figure 3: Learning curve for DDPG on the Cartpole benchmark. We compare the original hyperparameter settings to an optimized version of DDPG.
The plot shows the cumulative reward (over 100
test episodes) obtained by the DDPG algorithm
after it obtained x episodes of data for training.
6.4
DDPG
DDPG + DNGO
DDPG + BOHAMIANN
2-link reaching task
DDPG
DDPG + DNGO
DDPG + BOHAMIANN
Reward
Episodes
-1.18
-1.39
-1.46
470
507
405
Reward
Episodes
-4.36
-4.39
-4.57
1512
1642
1102
Hyperparameter optimization for deep reinforcement learning
Finally, we optimized a neural reinforcement learning (RL) algorithm on two control tasks: the
Cartpole swing-up task and a two link robot arm reaching task. We used a re-implementation of the
DDPG algorithm by Lillicrap et al. [29] and aimed to minimize the interaction time with the simulated
system required to achieve stable performance (defined as: solving the task in 10 consecutive test
episodes). This is a critical performance metric for data-efficient RL.
The results of this experiment are given in Table 2 . While the original DDPG hyperparameters
were set to achieve robust performance on a large set of benchmarks (and out-of-the-box DDPG
performed remarkably well on the considered problems) our experiments indicate that the number
of samples required to achieve good performance can be substantially reduced for individual tasks
by hyperparameter optimization with BOHAMIANN. In contrast, DNGO did not perform as well
on this specific task, getting stuck during optimization, see Figure 2 (right). A comparison between
the learning curves of the original and the optimized DDPG, depicted in Figure 3, confirms this
observation. The parameters that had the most influence on this improved performance were (perhaps
unsurprisingly) the learning-rates of the Q-and policy networks and the number of SGD steps
performed between collected episodes. This observation was already used by domain experts in a
recent paper by Gu et al. [30] where they used 5 updates per sample (the hyperparameters found by
our method correspond to 10 updates per sample).
7
Conclusion
We proposed BOHAMIANN, a scalable and flexible Bayesian optimization method. It natively
supports multi-task optimization as well as parallel function evaluations, and scales to high dimensions
and many function evaluations. At its heart lies Bayesian inference for neural networks via stochastic
gradient Hamiltonian Monte Carlo, and we improved the robustness thereof by means of a scale
adaptation technique. In future work, we plan to implement Freeze-Thaw Bayesian optimization [31]
and Bayesian optimization across dataset sizes [32] in our framework, since both of these generate
many cheap function evaluations and thus reach the scalability limit of GPs. We thereby expect
substantial speedups in the practical hyperparameter optimization for ML algorithms on big datasets.
Acknowledgements
This work has partly been supported by the European Commission under Grant no. H2020-ICT645403-ROBDREAM, by the German Research Foundation (DFG), under Priority Programme
Autonomous Learning (SPP 1527, grant HU 1900/3-1), under Emmy Noether grant HU 1900/2-1,
and under the BrainLinks-BrainTools Cluster of Excellence (grant number EXC 1086).
8
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9
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5,656 | 6,118 | Gaussian Process Bandit Optimisation with Multi-fidelity Evaluations
Kirthevasan Kandasamy \ , Gautam Dasarathy ? , Junier Oliva \ ,
Jeff Schneider \ , Barnab?s P?czos \
\
Carnegie Mellon University, ? Rice University
{kandasamy, joliva, schneide, bapoczos}@cs.cmu.edu, gautamd@rice.edu
Abstract
In many scientific and engineering applications, we are tasked with the optimisation
of an expensive to evaluate black box function f . Traditional methods for this
problem assume just the availability of this single function. However, in many cases,
cheap approximations to f may be obtainable. For example, the expensive real
world behaviour of a robot can be approximated by a cheap computer simulation.
We can use these approximations to eliminate low function value regions cheaply
and use the expensive evaluations of f in a small but promising region and speedily
identify the optimum. We formalise this task as a multi-fidelity bandit problem
where the target function and its approximations are sampled from a Gaussian
process. We develop MF-GP-UCB, a novel method based on upper confidence
bound techniques. In our theoretical analysis we demonstrate that it exhibits
precisely the above behaviour, and achieves better regret than strategies which
ignore multi-fidelity information. MF-GP-UCB outperforms such naive strategies
and other multi-fidelity methods on several synthetic and real experiments.
1
Introduction
In stochastic bandit optimisation, we wish to optimise a payoff function f : X ? R by sequentially
querying it and obtaining bandit feedback, i.e. when we query at any x ? X , we observe a possibly
noisy evaluation of f (x). f is typically expensive and the goal is to identify its maximum while
keeping the number of queries as low as possible. Some applications are hyper-parameter tuning in
expensive machine learning algorithms, optimal policy search in complex systems, and scientific
experiments [20, 23, 27]. Historically, bandit problems were studied in settings where the goal is
to maximise the cumulative reward of all queries to the payoff instead of just finding the maximum.
Applications in this setting include clinical trials and online advertising.
Conventional methods in these settings assume access to only this single expensive function of
interest f . We will collectively refer to them as single fidelity methods. In many practical problems
however, cheap approximations to f might be available. For instance, when tuning hyper-parameters
of learning algorithms, the goal is to maximise a cross validation (CV) score on a training set, which
can be expensive if the training set is large. However CV curves tend to vary smoothly with training
set size; therefore, we can train and cross validate on small subsets to approximate the CV accuracies
of the entire dataset. For a concrete example, consider kernel density estimation (KDE), where we
need to tune the bandwidth h of a kernel. Figure 1 shows the CV likelihood against h for a dataset of
size n = 3000 and a smaller subset of size n = 300. The two maximisers are different, which is to
be expected since optimal hyper-parameters are functions of the training set size. That said, the curve
for n = 300 approximates the n = 3000 curve quite well. Since training/CV on small n is cheap,
we can use it to eliminate bad values of the hyper-parameters and reserve the expensive experiments
with the entire dataset for the promising candidates (e.g. boxed region in Fig. 1).
In online advertising, the goal is to maximise the cumulative number of clicks over a given period. In
the conventional bandit treatment, each query to f is the display of an ad for a specific time, say one
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
hour. However, we may display ads for shorter intervals, say a few minutes, to approximate its hourly
performance. The estimate is biased, as displaying an ad for a longer interval changes user behaviour,
but will nonetheless be useful in gauging its long run click through rate. In optimal policy search
in robotics and automated driving vastly cheaper computer simulations are used to approximate the
expensive real world performance of the system. Scientific experiments can be approximated to
varying degrees using less expensive data collection, analysis, and computational techniques.
In this paper, we cast these tasks as multi-fidelity bandit optimisation problems assuming the availability of cheap approximate functions (fidelities) to the payoff f . Our contributions are:
1. We present a formalism for multi-fidelity bandit optimisation using Gaussian Process (GP)
assumptions on f and its approximations. We develop a novel algorithm, Multi-Fidelity Gaussian
Process Upper Confidence Bound (MF-GP-UCB) for this setting.
2. Our theoretical analysis proves that MF-GP-UCB explores the space at lower fidelities and uses
the high fidelities in successively smaller regions to zero in on the optimum. As lower fidelity
queries are cheaper, MF-GP-UCB has better regret than single fidelity strategies.
3. Empirically, we demonstrate that MF-GP-UCB outperforms single fidelity methods on
a series of synthetic examples, three hyper-parameter tuning tasks and one inference
problem in Astrophysics. Our matlab implementation and experiments are available at
github.com/kirthevasank/mf-gp-ucb.
Related Work: Since the seminal work by Robbins [25], the multi-armed bandit problem has been
studied extensively in the K-armed setting. Recently, there has been a surge of interest in the
optimism under uncertainty principle for K armed bandits, typified by upper confidence bound
(UCB) methods [2, 4]. UCB strategies have also been used in bandit tasks with linear [6] and GP [28]
payoffs. There is a plethora of work on single fidelity methods for global optimisation both with
noisy and noiseless evaluations. Some examples are branch and bound techniques such as dividing
rectangles (DiRect) [12], simulated annealing, genetic algorithms and more [17, 18, 22]. A suite of
single fidelity methods in the GP framework closely related to our work is Bayesian Optimisation
(BO). While there are several techniques for BO [13, 21, 30], of particular interest to us is the
Gaussian process upper confidence bound (GP-UCB) algorithm of Srinivas et al. [28].
Many applied domains of research such as aerodynamics, industrial design and hyper-parameter
tuning have studied multi-fidelity methods [9, 11, 19, 29]; a plurality of them use BO techniques.
However none of these treatments neither formalise nor analyse any notion of regret in the multifidelity setting. In contrast, MF-GP-UCB is an intuitive UCB idea with good theoretical properties.
Some literature have analysed multi-fidelity methods in specific contexts such as hyper-parameter
tuning, active learning and reinforcement learning [1, 5, 26, 33]. Their settings and assumptions are
substantially different from ours. Critically, none of them are in the more difficult bandit setting where
there is a price for exploration. Due to space constraints we discuss them in detail in Appendix A.3.
The multi-fidelity poses substantially new theoretical and algorithmic challenges. We build on GPUCB and our recent work on multi-fidelity bandits in the K-armed setting [16]. Section 2 presents our
formalism including a notion of regret for multi-fidelity GP bandits. Section 3 presents our algorithm.
The theoretical analysis is in Appendix C with a synopsis for the 2-fidelity case in Section 4. Section 6
presents our experiments. Appendix A.1 tabulates the notation used in the manuscript.
2
Preliminaries
We wish to maximise a payoff function f : X ? R where X ? [0, r]d . We can interact with f only by
querying at some x ? X and obtaining a noisy observation y = f (x) + . Let x? ? argmaxx?X f (x)
and f? = f (x? ). Let xt ? X be P
the point queried at time t. The goal of a bandit strategy
n
is
to
maximise
the
sum
of
rewards
t=1 f (xt ) or equivalently minimise the cumulative regret
Pn
t=1 f? ? f (xt ) after n queries; i.e. we compete against an oracle which queries at x? at all t.
Our primary distinction from the classical setting is that we have access to M ?1 successively accurate
approximations f (1) , f (2) , . . . , f (M ?1) to the payoff f = f (M ) . We refer to these approximations as
fidelities. We encode the fact that fidelity m approximates fidelity M via the assumption, kf (M ) ?
f (m) k? ? ? (m) , where ? (1) > ? (2) > ? ? ? > ? (M ) = 0. Each query at fidelity m expends a cost
?(m) of a resource, e.g. computational effort or advertising time, where ?(1) < ?(2) < ? ? ? < ?(M ) .
A strategy for multi-fidelity bandits is a sequence of query-fidelity pairs {(xt , mt )}t?0 , where
2
n=300
n=3000
?t
f
x
Figure 1: Left: Average CV log likelihood on datasets of size 300, 3000 on a synthetic KDE task. The crosses
are the maxima. Right: Illustration of GP-UCB at time t. The figure shows f (x) (solid black line), the UCB
?t (x) (dashed blue line) and queries until t ? 1 (black crosses). We query at xt = argmaxx?X ?t (x) (red star).
(xn , mn ) could depend on the previous query-observation-fidelity tuples {(xt , yt , mt )}n?1
t=1 . Here
yt = f (mt ) (xt ) + . After n steps we will have queried any of the M fidelities multiple times.
Some smoothness assumptions on f (m) ?s are needed to make the problem tractable. A standard in the
Bayesian nonparametric literature is to use a Gaussian process (GP) prior [24] with covariance kernel
?. In this work we focus on the squared exponential (SE) ??,h and the Mat?rn ??,h kernels as they are
popularly used in practice and their theoretical properties are well studied. Writing
x0 k2 ,
? z ?= kx?
?
2?z
2?z
21??
0
2
2
0
they are defined as ??,h (x, x ) = ? exp ?z /(2h ) , ??,h (x, x ) = ?(?)
,
B?
h
h
where ?, B? are the Gamma and modified Bessel functions. A convenience the GP framework offers
is that posterior distributions are analytically tractable. If f ? GP(0, ?), and we have observations
Dn = {(xi , yi )}ni=1 , where yi = f (xi ) + and ? N (0, ? 2 ) is Gaussian noise, the posterior
distribution for f (x)|Dn is also Gaussian N (?n (x), ?n2 (x)) with
?n (x) = k> ??1 Y,
n
n
?n2 (x) = ?(x, x) ? k> ??1 k.
(1)
2
n?n
Here, Y ? R with Yi = yi , k ? R with ki = ?(x, xi ) and ? = K + ? I ? R
where
Ki,j = ?(xi , xj ). In keeping with the above, we make the following assumptions on our problem.
Assumption 1. A1: The functions at all fidelities are sampled from GPs, f (m) ? GP(0, ?) for all
m = 1, . . . , M . A2: kf (M ) ? f (m) k? ? ? (m) for all m = 1, . . . , M . A3: kf (M ) k? ? B.
The purpose of A3 is primarily to define the regret. In Remark 7, Appendix A.4 we argue that these
assumptions are probabilistically valid, i.e. the latter two events occur with nontrivial probability
when we sample the f (m) ?s from a GP. So a generative mechanism would keep sampling the functions
and deliver them when the conditions hold true. A point x ? X can be queried at any of the M
fidelities. When we query at fidelity m, we observe y = f (m) (x) + where ? N (0, ? 2 ).
We now present our notion of cumulative regret R(?) after spending capital ? of a resource in the
multi-fidelity setting. R(?) should reduce to the conventional definition of regret for any single
fidelity strategy that queries only at M th fidelity. As only the optimum of f = f (M ) is of interest
to us, queries at fidelities less than M should yield the lowest possible reward, (?B) according to
A3. Accordingly, we set the instantaneous reward qt at time to be ?B if mt 6= M and f (M ) (xt ) if
mt = M . If we let rt = f? ? qt denote the instantaneous regret, we have rt = f? + B if mt 6= M
and f? ? f (xt ) if mt = M . R(?) should also factor in the costs of the fidelity of each query. Finally,
we should also receive (?B) reward for any unused capital. Accordingly, we define R(?) as,
"N
#
N
N
X
X
X
(mt )
(mt )
R(?) = ?f? ?
?
qt + ? ?
?
(?B) ? 2B?res +
?(mt ) rt , (2)
t=1
(mt )
.
t=1 ?
PN
t=1
t=1
where ?res = ? ?
Here, N is the (random) number of queries at all fidelities within
Pn
capital ?, i.e. the largest n such that t=1 ?(mt ) ? ?. According to (2) above, we wish to compete
against an oracle that uses all its capital ? to query x? at the M th fidelity. R(?) is at best 0 when
we follow the oracle and at most 2?B. Our goal is a strategy that has small regret for all values of
(sufficiently large) ?, i.e. the equivalent of an anytime strategy, as opposed to a fixed time horizon
strategy in the usual bandit setting. For the purpose of optimisation, we also define the simple regret
as S(?) = mint rt = f? ? maxt qt . S(?) is the difference between f? and the best highest fidelity
query (and f? + B if we have never queried at fidelity M ). Since S(?) ? ?1 R(?), any strategy with
asymptotic sublinear regret lim??? ?1 R(?) = 0, also has vanishing simple regret.
Since, to our knowledge, this is the first attempt to formalise regret for multi-fidelity problems, the
definition for R(?) (2) necessitates justification. Consider a two fidelity robot gold mining problem
3
where the second fidelity is a real world robot trial, costing ?(2) dollars and the first fidelity is a
computer simulation costing ?(1) . A multi-fidelity algorithm queries the simulator to learn about
the real world. But it does not collect any actual gold during a simulation; hence no reward, which
according to our assumptions is ?B. Meantime the oracle is investing this capital on the best
experiment and collecting ? f? gold. Therefore, the regret at this time instant is f? + B. However
we weight this by the cost to account for the fact that the simulation costs only ?(1) . Note that lower
fidelities use up capital but yield the lowest reward. The goal however, is to leverage information
from these cheap queries to query prudently at the highest fidelity and obtain better regret.
That said, other multi-fidelity settings might require different definitions for R(?). In online advertising, the lower fidelities (displaying ads for shorter periods) would still yield rewards. In clinical trials,
the regret at the highest fidelity due to a bad treatment would be, say, a dead patient. However, a bad
treatment on a simulation may not warrant large penalty. We use the definition in (2) because it is
more aligned with our optimisation experiments: lower fidelities are useful to the extent that they
guide search on the expensive f (M ) , but there is no reward to finding the optimum of a cheap f (m) .
A crucial challenge for a multi-fidelity method is to not get stuck at the optimum of a lower fidelity,
which is typically suboptimal for f (M ) . While exploiting information from the lower fidelities, it is
also important to explore sufficiently at f (M ) . In our experiments we demonstrate that naive strategies
which do not do so would get stuck at the optimum of a lower fidelity.
A note on GP-UCB: Sequential optimisation methods adopting UCB principles maintain a high
probability upper bound ?t : X ? R for f (x) for all x ? X [2]. For GP-UCB, ?t takes the form
1/2
?t (x) = ?t?1 (x) + ?t ?t?1 (x) where ?t?1 , ?t?1 are the posterior mean and standard deviation of
the GP conditioned on the previous t ? 1 queries. The key intuition is that the mean ?t?1 encourages
an exploitative strategy ? in that we want to query where we know the function is high ? and the
1/2
confidence band ?t ?t?1 encourages an explorative strategy ? in that we want to query at regions
we are uncertain about f lest we miss out on high valued regions. We have illustrated GP-UCB in
Fig 1 and reviewed the algorithm and its theoretical properties in Appendix A.2.
MF-GP-UCB
3
The proposed algorithm, MF-GP-UCB, will also maintain a UCB for f (M ) obtained via the previous
queries at all fidelities. Denote the posterior GP mean and standard deviation of f (m) conditioned
(m)
(m)
only on the previous queries at fidelity m by ?t , ?t respectively (See (1)). Then define,
(m)
?t
(m)
1/2 (m)
?t?1 (x)
(x) = ?t?1 (x) + ?t
(m)
+ ? (m) , ? m,
?t (x) =
min
m=1,...,M
(m)
?t
(x).
(3)
1/2 (m)
For appropriately chosen ?t , ?t?1 (x) + ?t ?t?1 (x) will upper bound f (m) (x) with high probability.
(m)
By A2, ?t (x) upper bounds f (M ) (x) for all m. We have M such upper bounds, and their minimum
?t (x) gives the best bound. Our next query is at the maximiser of this UCB, xt = argmaxx?X ?t (x).
Next we need to decide which fidelity to query at. Consider any m < M . The ? (m) conditions
1/2 (m)
on f (m) constrain the value of f (M ) ? the confidence band ?t ?t?1 for f (m) is lengthened by
1/2
(m)
? (m) to obtain confidence on f (M ) . If ?t ?t?1 (xt ) for f (m) is large, it means that we have not
constrained f (m) sufficiently well at xt and should query at the mth fidelity. On the other hand,
1/2 (m)
querying indefinitely in the same region to reduce ?t ?t?1 in that region will not help us much as
the ? (m) elongation caps off how much we can learn about f (M ) from f (m) ; i.e. even if we knew
f (m) perfectly, we will only have constrained f (M ) to within a ?? (m) band. Our algorithm captures
1/2 (1)
this simple intuition. Having selected xt , we begin by checking at the first fidelity. If ?t ?t?1 (xt ) is
1/2 (m)
smaller than a threshold ? (1) , we proceed to the second fidelity. If at any stage ?t ?t?1 (xt ) ? ? (m)
we query at fidelity mt = m. If we proceed all the way to fidelity M , we query at mt = M . We will
discuss choices for ? (m) shortly. We summarise the resulting procedure in Algorithm 1.
Fig 2 illustrates MF-GP-UCB on a 2?fidelity problem. Initially, MF-GP-UCB is mostly exploring
1/2 (1)
X in the first fidelity. ?t ?t?1 is large and we are yet to constrain f (1) well to proceed to f (2) . By
t = 14, we have constrained f (1) around the optimum and have started querying at f (2) in this region.
4
Algorithm 1 MF-GP-UCB
(m)
D0
(m) M
Inputs: kernel ?, bounds {? (m) }M
}m=1 .
m=1 , thresholds {?
(m)
(m)
? For m = 1, . . . , M :
? ?, (?0 , ?0 ) ? (0, ?1/2 ).
? for t = 1, 2, . . .
1. xt ? argmaxx?X ?t (x).
(See Equation (3))
1/2 (m)
?t?1 (xt )
2. mt = minm { m |?t
3. yt ? Query f
(m )
Dt t
4. Update
(mt )
at xt .
? ? (m) or m = M }.
(m )
(mt )
? Dt?1t ? {(xt , yt )}. Obtain ?t
(1)
t=6
?t
(mt )
conditioned on Dt
(See (1)).
t = 14
(2)
?t
?t
f (2)
f
f (2)
(1)
xt
f (1)
x?
xt x?
mt = 1
mt = 2
1/2 (1)
? (1)
(mt )
, ?t
(See Appendix B, C for ?t )
?t ?t?1 (x)
? (1)
Figure 2: Illustration of MF-GP-UCB for a 2-fidelity problem initialised with 5 random points at the first
fidelity. In the top figures, the solid lines in brown and blue are f (1) , f (2) respectively, and the dashed lines are
(1)
(2)
(1)
(2)
?t , ?t . The solid green line is ?t = min(?t , ?t ). The small crosses are queries from 1 to t ? 1 and the
red star is the maximiser of ?t , i.e. the next query xt . x? , the optimum of f (2) is shown in magenta. In the
1/2 (1)
1/2 (1)
bottom figures, the solid orange line is ?t ?t?1 and the dashed black line is ? (1) . When ?t ?t?1 (xt ) ? ? (1)
we play at fidelity mt = 2 and otherwise at mt = 1. See Fig. 6 in Appendix B for an extended simulation.
(2)
Notice how ?t dips to change ?t in this region. MF-GP-UCB has identified the maximum with just
3 queries to f (2) . In Appendix B we provide an extended simulation and discuss further insights.
Finally, we make an essential observation. The posterior for any f (m) (x) conditioned on previous
queries at all fidelities is not Gaussian due to the ? (m) constraints (A2). However, |f (m) (x) ?
(m)
1/2 (m)
?t?1 (x)| < ?t ?t?1 (x) holds with high probability, since, by conditioning only on queries at the
mth fidelity we have Gaussianity for f (m) (x). Next we summarise our main theoretical contributions.
4
Summary of Theoretical Results
For pedagogical reasons we present our results for the M = 2 case. Appendix C contains statements
and proofs for general M . We also ignore constants and polylog terms when they are dominated
by other terms. ., denote inequality and equality ignoring constants. We begin by defining the
Maximum Information Gain (MIG) which characterises the statistical difficulty of GP bandits.
Definition 2. (Maximum Information Gain) Let f ? GP(0, ?). Consider any A ? Rd and let
e = {x1 , . . . , xn } ? A be a finite subset. Let f e, e ? Rn be such that (f e)i = f (xi ), ( e)i ?
A
A A
A
A
N (0, ? 2 ), and yAe = fAe + Ae. Let I denote the Shannon Mutual Information. The Maximum
Information Gain of A is ?n (A) = maxA?A,|
I(yAe; fAe).
e
e
A|=n
The MIG, which depends on the kernel ? and the set A, is an important quantity in our analysis. For a
given ?, it typically scales with the volume of A; i.e. if A = [0, r]d then ?n (A) ? O(rd ?n ([0, 1]d )).
d(d+1)
For the SE kernel, ?n ([0, 1]d ) ? O((log(n))d+1 ) and for Mat?rn, ?n ([0, 1]d ) ? O(n 2?+d(d+1) ) [28].
Recall, N is the (random) number of queries by a multi-fidelity strategy within capital ? at either
fidelity. Let n? = b?/?(2) c be the (non-random) number of queries by a single fidelity method
operating only at the second fidelity. As ?(1) < ?(2) , N could be large for an arbitrary multi-fidelity
method. However, our analysis reveals that for MF-GP-UCB, N is on the order of n? .
5
Fundamental to the 2-fidelity problem is the set Xg = {x ? X ; f? ? f (1) (x) ? ? (1) }. Xg is a
high valued region for f (2) (x): for all x ? Xg , f (2) (x) is at most 2? (1) away from the optimum.
More interestingly, when ? (1) is small, i.e. when f (1) is a good approximation to f (2) , Xg will
be much smaller than X . This is precisely the target domain for this research. For instance,
in the robot gold mining example, a cheap computer simulator can be used to eliminate several
bad policies and we could reserve the real world trials for the promising candidates. If a multifidelity strategy were to use the second fidelity queries only in Xg , then the regret will only have
?n (Xg ) dependence after n high fidelity queries. In contrast, a strategy that only operates at the
highest fidelity (e.g. GP-UCB) will have ?n (X ) dependence. In the scenario described above
?n (Xg ) ?n (X ), and the multi-fidelity strategy will have significantly better regret than a single
fidelity strategy. MF-GP-UCB roughly achieves this goal. In particular, we consider a slightly inflated
set Xeg,? = {x ? X ; f? ? f (1) (x) ? ? (1) + ?? (1) }, of Xg where ? > 0. The following result which
characterises the regret of MF-GP-UCB in terms of Xeg,? is the main theorem of this paper.
Theorem 3 (Regret of MF-GP-UCB ? Informal). Let X = [0, r]d and f (1) , f (2) ? GP(0, ?) satisfy
Assumption 1. Pick ? ? (0, 1) and run MF-GP-UCB with ?t d log(t/?). Then, with probability
> 1 ? ?, for sufficiently large ? and for all ? ? (0, 1), there exists ? depending on ? such that,
R(?) . ?(2)
q
q
q
(1)
?
n? ?n? ?n? (Xeg,? ) + ?(1) n? ?n? ?n? (X ) + ?(2) n?
?n,Xeg,? ,? (1)
? ?n? ?n? (X ) + ?
As we will explain shortly, the latter two terms are of lower order. It is instructive to compare the
above rates against that for GP-UCB (see Theorem 4, Appendix A.2). By dropping the common
1/2
1/2
and subdominant terms, the rate for MF-GP-UCB is ?(2) ?n? (Xeg,? ) + ?(1) ?n? (X ) whereas for
1/2
GP-UCB it is ?(2) ?n? (X ). When ?(1) ?(2) and vol(Xeg,? ) vol(X ) the rates for MF-GPUCB are very appealing. When the approximation worsens (Xg , Xeg,? become larger) and the costs
?(1) , ?(2) become comparable, the bound for MF-GP-UCB decays gracefully. In the worst case,
MF-GP-UCB is never worse than GP-UCB up to constant terms. Intuitively, the above result states
that MF-GP-UCB explores the entire X using f (1) but uses ?most? of its queries to f (2) inside Xeg,? .
Now let us turn to the latter two terms in the bound. The third term is the regret due to the second
fidelity queries outside Xeg,? . We are able to show that the number of such queries is O(n?
? ) for
all ? > 0 for an appropriate ?. This strong result is only possible in the multi-fidelity setting. For
example, in GP-UCB the best bound you can achieve on the number of plays on a suboptimal set is
1/2
O(n? ) for the SE kernel and worse for the Mat?rn kernel. The last term is due to the first fidelity
plays inside Xeg,? and it scales with vol(Xeg,? ) and polylogarithmically with n, both of which are small.
However, it has a 1/poly(? (1) ) dependence which could be bad if ? (1) is too small: intuitively, if
1/2 (1)
? (1) is too small then you will wait for a long time in step 2 of Algorithm 1 for ?t ?t?1 to decrease
(2)
without proceeding to f , incurring large regret (f? + B) in the process. Our analysis reveals that
an optimal choice for the SE kernel scales ? (1) (?(1) ? (1) /(t?(2) ))1/(d+2) at time t. However this
is of little practical use as the leading constant depends on several problem dependent quantities such
as ?n (Xg ). In Section 5 we describe a heuristic to set ? (m) which worked well in our experiments.
Theorem 3 can be generalised to cases where the kernels ?(m) and observation noises ? (m) are
different at each fidelity. The changes to the proofs are minimal. In fact, our practical implementation
uses different kernels. As with any nonparametric method, our algorithm has exponential dependence
on dimension. This can be alleviated by assuming additional structure in the problem [8, 15]. Finally,
we note that the above rates translate to bounds on the simple regret S(?) for optimisation.
5
Implementation Details
Our implementation uses some standard techniques in Bayesian optimisation to learn the kernel such
as initialisation with random queries and periodic marginal likelihood maximisation. The above
techniques might be already known to a reader familiar with the BO literature. We have elaborated
these in Appendix B but now focus on the ? (m) , ? (m) parameters of our method.
Algorithm 1 assumes that the ? (m) ?s are given with the problem description, which is hardly the
case in practice. In our implementation, instead of having to deal with M ? 1, ? (m) values we set
(? (1) , ? (2) , . . . , ? (M ?1) ) = ((M ? 1)?, (M ? 2)?, . . . , ?) so we only have one value ?. This for
6
BoreHole-8D, M = 2, Costs = [1; 10]
Query frequencies for Hartmann-3D
Hartmann-3D, M = 3, Costs = [1; 10; 100]
40
10 0
Number of Queries
S(?)
S(?)
10 -1
10 1
MF-GP-UCB
GP-UCB
EI
DiRect
MF-NAIVE
MF-SKO
200
10 -2
10 -3
400
600
800
1000
m=1
m=2
m=3
35
10 0
10 2
30
25
20
15
10
5
2000
4000
6000
?
?
8000
10000
0
0
0.5
1
1.5
2
2.5
3
3.5
f (3)(x)
Figure 3: The simple regret S(?) against the spent capital ? on synthetic functions. The title states the
function, its dimensionality, the number of fidelities and the costs we used for each fidelity in the experiment.
All curves barring DiRect (which is a deterministic), were produced by averaging over 20 experiments. The
error bars indicate one standard error. See Figures 8, 9 10 in Appendix D for more synthetic results. The last
panel shows the number of queries at different function values at each fidelity for the Hartmann-3D example.
instance, is satisfied if kf (m) ? f (m?1) k? ? ? which is stronger than Assumption A2. Initially, we
start with small ?. Whenever we query at any fidelity m > 1 we also check the posterior mean of
(m?1)
the (m ? 1)th fidelity. If |f (m) (xt ) ? ?t?1 (xt )| > ?, we query again at xt , but at the (m ? 1)th
fidelity. If |f (m) (xt ) ? f (m?1) (xt )| > ?, we update ? to twice the violation. To set ? (m) ?s we use
the following intuition: if the algorithm, is stuck at fidelity m for too long then ? (m) is probably too
small. We start with small values for ? (m) . If the algorithm does not query above the mth fidelity for
more than ?(m+1) /?(m) iterations, we double ? (m) . We found our implementation to be fairly robust
even recovering from fairly bad approximations at the lower fidelities (see Appendix D.3).
6
Experiments
We compare MF-GP-UCB to the following methods. Single fidelity methods: GP-UCB; EI: the
expected improvement criterion for BO [13]; DiRect: the dividing rectangles method [12]. Multifidelity methods: MF-NAIVE: a naive baseline where we use GP-UCB to query at the first fidelity a
large number of times and then query at the last fidelity at the points queried at f (1) in decreasing
order of f (1) -value; MF-SKO: the multi-fidelity sequential kriging method from [11]. Previous
works on multi-fidelity methods (including MF-SKO) had not made their code available and were
not straightforward to implement. Hence, we could not compare to all of them. We discuss this more
in Appendix D along with some other single and multi-fidelity baselines we tried but excluded in
the comparison to avoid clutter in the figures. In addition, we also detail the design choices and
hyper-parameters for all methods in Appendix D.
Synthetic Examples: We use the Currin exponential (d = 2), Park (d = 4) and Borehole (d = 8)
functions in M = 2 fidelity experiments and the Hartmann functions in d = 3 and 6 with M = 3
and 4 fidelities respectively. The first three are taken from previous multi-fidelity literature [32] while
we tweaked the Hartmann functions to obtain the lower fidelities for the latter two cases. We show
the simple regret S(?) against capital ? for the Borehole and Hartmann-3D functions in Fig. 3 with
the rest deferred to Appendix D due to space constraints. MF-GP-UCB outperforms other methods.
Appendix D also contains results for the cumulative regret R(?) and the formulae for these functions.
A common occurrence with MF-NAIVE was that once we started querying at fidelity M , the regret
barely decreased. The diagnosis in all cases was the same: it was stuck around the maximum of f (1)
which is suboptimal for f (M ) . This suggests that while we have cheap approximations, the problem
is by no means trivial. As explained previously, it is also important to ?explore? at the higher fidelities
to achieve good regret. The efficacy of MF-GP-UCB when compared to single fidelity methods is
that it confines this exploration to a small set containing the optimum. In our experiments we found
that MF-SKO did not consistently beat other single fidelity methods. Despite our best efforts to
reproduce this (and another) multi-fidelity method, we found them to be quite brittle (Appendix D.1).
The third panel of Fig. 3 shows a histogram of the number of queries at each fidelity after 184 queries
of MF-GP-UCB, for different ranges of f (3) (x) for the Hartmann-3D function. Many of the queries
at the low f (3) values are at fidelity 1, but as we progress they decrease and the second fidelity queries
increase. The third fidelity dominates very close to the optimum but is used sparingly elsewhere.
This corroborates the prediction in our analysis that MF-GP-UCB uses low fidelities to explore and
successively higher fidelities at promising regions to zero in on x? . (Also see Fig. 6, Appendix B.)
7
SVM-2D, M = 2, ntr = [500, 2000]
SALSA-6D, M = 3, ntr = [2000, 4000, 8000]
MF-GP-UCB
GP-UCB
EI
DiRect
MF-NAIVE
MF-SKO
0.13
0.125
0.12
0.115
CV (Classification) Error
0.135
V&J-22D, M = 2, ntr = [300, 3000]
0.35
1
CV (Least Squares) Error
CV (Classification) Error
0.14
0.8
0.6
0.4
0.2
0.3
0.25
0.2
0.15
0.1
0
0
2000
4000
6000
8000
0
1000
2000
3000
4000
5000
CPU Time (s)
CPU Time (s)
6000
7000
1000
2000
3000
4000
5000
6000
7000
8000
CPU Time (s)
Figure 4: Results on the hyper-parameter tuning experiments. The title states the experiment, dimensionality
(number of hyperparameters) and training set size at each fidelity. All curves were produced by averaging over
10 experiments. The error bars indicate one standard error. The lengths of the curves are different in time as we
ran each method for a pre-specified number of iterations and they concluded at different times.
Real Experiments: We present results on three hyper-parameter tuning tasks (results in Fig. 4), and
a maximum likelihood inference task in Astrophysics (Fig. 5). We compare methods on computation
time since that is the ?cost? in all experiments. We include the processing time for each method in
the comparison (i.e. the cost of determining the next query).
Classification using SVMs (SVM): We trained an SVM on the magic gamma dataset using the
SMO algorithm to an accuracy of 10?12 . The goal is to tune the kernel bandwidth and the soft margin
coefficient in the ranges (10?3 , 101 ) and (10?1 , 105 ) respectively on a dataset of size 2000. We set
this up as a M = 2 fidelity experiment with the entire training set at the second fidelity and 500
points at the first. Each query was 5-fold cross validation on these training sets.
Regression using Additive Kernels (SALSA): We used the regression method from [14] on the
4-dimensional coal power plant dataset. We tuned the 6 hyper-parameters ?the regularisation penalty,
the kernel scale and the kernel bandwidth for each dimension? each in the range (10?3 , 104 ) using
5-fold cross validation. This experiment used M = 3 and 2000, 4000, 8000 points at each fidelity.
Viola & Jones face detection (V&J): The V&J classifier [31], which uses a cascade of weak
classifiers, is a popular method for face detection. To classify an image, we pass it through each
classifier. If at any point the classifier score falls below a threshold, the image is classified negative. If
it passes through the cascade, then it is classified positive. One of the more popular implementations
comes with OpenCV and uses a cascade of 22 weak classifiers. The threshold values in OpenCV
are pre-set based on some heuristics and there is no reason to think they are optimal for a given face
detection task. The goal is to tune these 22 thresholds by optimising for them over a training set. We
modified the OpenCV implementation to take in the thresholds as parameters. As our domain X we
chose a neighbourhood around the configuration used in OpenCV. We set this up as a M = 2 fidelity
experiment where the second fidelity used 3000 images from the V&J face database and the first used
300. Interestingly, on an independent test set, the configurations found by MF-GP-UCB consistently
achieved over 90% accuracy while the OpenCV configuration achieved only 87.4% accuracy.
Type Ia Supernovae: We use Type Ia supernovae data [7]
for maximum likelihood inference on 3 cosmological parameters, the Hubble constant H0 ? (60, 80), the dark matter
and dark energy fractions ?M , ?? ? (0, 1). Unlike typical
parametric maximum likelihood problems, the likelihood
is only available as a black-box. It is computed using the
Robertson?Walker metric which requires a one dimensional
numerical integration for each sample in the dataset. We set
CPU Time (s)
this up as a M = 3 fidelity task. The goal is to maximise the
Figure 5: Results on the supernova infer- likelihood at the third fidelity where the integration was perence problem. The y-axis is the log likeli- formed using the trapezoidal rule on a grid of size 106 . For
hood so higher is better. MF-NAIVE is not the first and second fidelities, we used grids of size 102 , 104
visible as it performed very poorly.
respectively. The results are given in Fig. 5.
Supernova-3D, M = 3, Grid = [100, 10K, 1M ]
Log Likelihood
10
5
0
-5
-10
500
1000
1500
2000
2500
3000
3500
Conclusion: We introduced and studied the multi-fidelity bandit under Gaussian Process assumptions. We present, to our knowledge, the first formalism of regret and the first theoretical results
in this setting. They demonstrate that MF-GP-UCB explores the space via cheap lower fidelities,
and leverages the higher fidelities on successively smaller regions hence achieving better regret than
single fidelity strategies. Experimental results demonstrate the efficacy of our method.
8
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5,657 | 6,119 | Maximizing Influence in an Ising Network:
A Mean-Field Optimal Solution
Christopher W. Lynn
Department of Physics and Astronomy
University of Pennsylvania
chlynn@sas.upenn.edu
Daniel D. Lee
Department of Electrical and Systems Engineering
University of Pennsylvania
ddlee@seas.upenn.edu
Abstract
Influence maximization in social networks has typically been studied in the context
of contagion models and irreversible processes. In this paper, we consider an
alternate model that treats individual opinions as spins in an Ising system at dynamic
equilibrium. We formalize the Ising influence maximization problem, which has a
natural physical interpretation as maximizing the magnetization given a budget of
external magnetic field. Under the mean-field (MF) approximation, we present a
gradient ascent algorithm that uses the susceptibility to efficiently calculate local
maxima of the magnetization, and we develop a number of sufficient conditions
for when the MF magnetization is concave and our algorithm converges to a
global optimum. We apply our algorithm on random and real-world networks,
demonstrating, remarkably, that the MF optimal external fields (i.e., the external
fields which maximize the MF magnetization) shift from focusing on high-degree
individuals at high temperatures to focusing on low-degree individuals at low
temperatures. We also establish a number of novel results about the structure of
steady-states in the ferromagnetic MF Ising model on general graph topologies,
which are of independent interest.
1
Introduction
With the proliferation of online social networks, the problem of optimally influencing the opinions
of individuals in a population has garnered tremendous attention [1?3]. The prevailing paradigm
treats marketing as a viral process, whereby the advertiser is given a budget of seed infections and
chooses the subset of individuals to infect such that the spread of the ensuing contagion is maximized.
The development of algorithmic methods for influence maximization under the viral paradigm has
been the subject of vigorous study, resulting in a number of efficient techniques for identifying
meaningful marketing strategies in real-world settings [4?6]. While the viral paradigm accurately
describes out-of-equilibrium phenomena, such as the introduction of new ideas or products to a
system, these models fail to capture reverberant opinion dynamics wherein repeated interactions
between individuals in the network give rise to complex macroscopic opinion patterns, as, for example,
is the case in the formation of political opinions [7?10]. In this context, rather than maximizing
the spread of a viral advertisement, the marketer is interested in optimally shifting the equilibrium
opinions of individuals in the network.
To describe complex macroscopic opinion patterns resulting from repeated microscopic interactions,
we naturally employ the language of statistical mechanics, treating individual opinions as spins in an
Ising system at dynamic equilibrium and modeling marketing as the addition of an external magnetic
field. The resulting problem, which we call Ising influence maximization (IIM), has a natural physical
interpretation as maximizing the magnetization of an Ising system given a budget of external field.
While a number of models have been proposed for describing reverberant opinion dynamics [11], our
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
use of the Ising model follows a vibrant interdisciplinary literature [12, 13], and is closely related
to models in game theory [14, 15] and sociophysics [16, 17]. Furthermore, complex Ising models
have found widespread use in machine learning, and our model is formally equivalent to a pair-wise
Markov random field or a Boltzmann machine [18?20].
Our main contributions are as follows:
1. We formalize the influence maximization problem in the context of the Ising model, which
we call the Ising influence maximization (IIM) problem. We also propose the mean-field
Ising influence maximization (MF-IIM) problem as an approximation to IIM (Section 2).
2. We find sufficient conditions under which the MF-IIM objective is smooth and concave,
and we present a gradient ascent algorithm that guarantees an -approximation to MF-IIM
(Section 4).
3. We present numerical simulations that probe the structure and performance of MF optimal
marketing strategies. We find that at high temperatures, it is optimal to focus influence on
high-degree individuals, while at low temperatures, it is optimal to spread influence among
low-degree individuals (Sections 5 and 6).
4. Throughout the paper we present a number of novel results concerning the structure of
steady-states in the ferromagnetic MF Ising model on general (weighted, directed) stronglyconnected graphs, which are of independent interest. We name two highlights:
? The well-known pitchfork bifurcation structure for the ferromagnetic MF Ising model
on a lattice extends exactly to general strongly-connected graphs, and the critical
temperature is equal to the spectral radius of the adjacency matrix (Theorem 3).
? There can exist at most one stable steady-state with non-negative (non-positive) components, and it is smooth and concave (convex) in the external field (Theorem 4).
2
The Ising influence maximization problem
We consider a weighted, directed social network consisting of a set of individuals N = {1, . . . , n},
each of which is assigned an opinion ?i ? {?1} that captures its current state. By analogy with the
Ising model, we refer to ? = (?i ) as a spin configuration of the system. Individuals in the network
interact via a non-negative weighted coupling matrix J ? Rn?n
?0 , where Jij ? 0 represents the
amount of influence that individual j holds over the opinion of individual i, and the non-negativity of
J represents the assumption that opinions of neighboring individuals tend to align, known in physics
as a ferromagnetic interaction. Each individual also interacts with forces external to the network via
an external field h ? Rn . For example, if the spins represent the political opinions of individuals in a
social network, then Jij represents the influence that j holds over i?s opinion and hi represents the
political bias of node i due to external forces such as campaign advertisements and news articles.
The opinions of individuals in the network evolve according to asynchronous Glauber dynamics. At
each time t, an individual i is selected uniformly at random and her opinion is updated in response to
the external field h and the opinions of others in the network ?(t) by sampling from
P (?i (t + 1) = 1|?(t)) = P
e? (
?i0 =?1
P
j
Jij ?j (t)+hi )
e??i (
0
P
j
Jij ?j (t)+hi )
,
(1)
where ? is the inverse temperature, which we refer to as the interaction strength, and unless otherwise
specified, sums are assumed over N . Together,
the quadruple (N, J, h, ?) defines our system. We
P
refer to the total expected opinion, M = i h?i i, as the magnetization, where h?i denotes an average
over the dynamics in Eq. (1), and we often consider the magnetization as a function of the external
ii
field, denoted M (h). Another important concept is the susceptibility matrix, ?ij = ?h?
?hj , which
quantifies the response of individual i to a change in the external field on node j.
We study the problem of maximizing the magnetization of an Ising system with respect to the external
field. We assume
that an external field h can be added to the system, subject to the constraints
P
h ? 0 and i hi ? H, where H > 0 is theP
external field budget, and we denote the set of feasible
external fields by FH = {h ? Rn : h ? 0, i hi = H}. In general, we also assume that the system
experiences an initial external field b ? Rn , which cannot be controlled.
2
Definition 1. (Ising influence maximization (IIM)) Given a system (N, J, b, ?) and a budget H, find
a feasible external field h ? FH that maximizes the magnetization; that is, find an optimal external
field h? such that
h? = arg max M (b + h).
(2)
h?FH
Notation. Unless otherwise specified, bold symbols represent column vectors with the appropriate
number of components, while non-bold symbols with subscripts represent individual components.
We often abuse notation and write relations such as m ? 0 to mean mi ? 0 for all components i.
2.1
The mean-field approximation
In general, calculating expectations over the dynamics in Eq. (1) requires Monte-Carlo simulations
or other numerical approximation techniques. To make analytic progress, we employ the variational
mean-field approximation, which has roots in statistical physics and has long been used to tackle
inference problems in Boltzmann machines and Markov random fields [21?24]. The mean-field
approximation replaces the intractable task of calculating exact averages over Eq. (1) with the
problem of solving the following set of self-consistency equations:
? ?
??
X
mi = tanh ?? ?
Jij mj + hi ?? ,
(3)
j
for all i ? N , where mi approximates h?i i. We refer to the right-hand side of Eq. (3) as the
mean-field map, f (m) = tanh [?(Jm + h)], where tanh(?) is applied component-wise. In this
way, a fixed point of the mean-field map is a solution to Eq. (3), which we call a steady-state.
In general, there may be many solutions to Eq. (3), and we denote by Mh the set of steady-states for
a system (N, J, h, ?). We say that a steady-state m is stable if ?(f 0 (m)) < 1, where ?(?) denotes
the spectral radius and
?fi
0
f (m)ij =
= ? 1 ? m2i Jij ? f 0 (m) = ?D(m)J,
(4)
?mj m
where D(m)ij = (1 ? m2i )?ij . Furthermore, under the mean-field approximation, given a stable
steady-state m, the susceptibility has a particularly nice form:
!
X
?1
MF
2
?ij = ? 1 ? mi
Jik ?kj + ?ij
? ?M F = ? (I ? ?D(m)J) D(m),
(5)
k
where I is the n ? n identity matrix.
For the purpose of uniquely defining our objective, we optimistically choose to maximize the
maximum magnetization among the set of steady-states, defined by
X
M M F (h) = max
mi (h).
(6)
m?Mh
i
We note that the pessimistic framework of maximizing the minimum magnetization yields an equally
valid objective. We also note that simply choosing a steady-state to optimize does not yield a
well-defined objective since, as h increases, steady-states can pop in and out of existence.
Definition 2. (Mean-field Ising influence maximization (MF-IIM)) Given a system (N, J, b, ?) and a
budget H, find an optimal external field h? such that
h? = arg max M M F (b + h).
(7)
h?FH
3
The structure of steady-states in the MF Ising model
Before proceeding further, we must prove an important result concerning the existence and structure
of solutions to Eq. (3), for if there exists a system that does not admit a steady-state, then our objective
3
is ill-defined. Furthermore, if there exists a unique steady-state m, then M M F =
is no ambiguity in our choice of objective.
P
i
mi , and there
Theorem 3 establishes that every system admits a steady-state and that the well-known pitchfork
bifurcation structure for steady-states of the ferromagnetic MF Ising model on a lattice extends exactly
to general (weighted, directed) strongly-connected graphs. In particular, for any strongly-connected
graph described by J, there is a critical interaction strength ?c below which there exists a unique
and stable steady-state. For h = 0, as ? crosses ?c from below, two new stable steady-states appear,
one with all-positive components and one with all-negative components. Interestingly, the critical
interaction strength is equal to the inverse of the spectral radius of J, denoted ?c = 1/?(J).
Theorem 3. Any system (N, J, h, ?) exhibits a steady-state. Furthermore, if its network is stronglyconnected, then, for ? < ?c , there exists a unique and stable steady-state. For h = 0, as ? crosses
?c from below, the unique steady-state gives rise to two stable steady-states, one with all-positive
components and one with all-negative components.
Proof sketch. The existence of a steady-state follows directly by applying Brouwer?s fixed-point
theorem to f . For ? < ?c , it can be shown that f is a contraction mapping, and hence admits a unique
and stable steady-state by Banach?s fixed point theorem. For h = 0 and ? < ?c , m = 0 is the unique
steady-state and f 0 (m) = ?J. Because J is strongly-connected, the Perron-Frobenius theorem
guarantees a simple eigenvalue equal to ?(J) and a corresponding all-positive eigenvector. Thus,
when ? crosses 1/?(J) from below, the Perron-Frobenius eigenvalue of f 0 (m) crosses 1 from below,
giving rise to a supercritical pitchfork bifurcation with two new stable steady-states corresponding to
the Perron-Frobenius eigenvector.
Remark. Some of our results assume J is strongly-connected in order to use the Perron-Frobenius
theorem. We note that this assumption is not restrictive, since any graph can be efficiently decomposed
into strongly-connected components on which our results apply independently.
Theorem 3 shows that the objective M M F (b + h) is well-defined. Furthermore, for ? < ?c , Theorem
3 guarantees a unique P
and stable steady-state m for all b + h. In this case, MF-IIM reduces to
maximizing M M F = i mi , and because m is stable, M M F (b + h) is smooth for all h by the
implicit function theorem. Thus, for ? < ?c , we can use standard gradient ascent techniques to
efficiently calculate locally-optimal solutions to MF-IIM. In general, M M F is not necessarily smooth
in h since the topological structure of steady-states may change as h varies. However, in the next
section we show that if there exists a stable and entry-wise non-negative steady-state, and if J is
strongly-connected, then M M F (b + h) is both smooth and concave in h, regardless of the interaction
strength.
4
Sufficient conditions for when MF-IIM is concave
We consider conditions for which MF-IIM is smooth and concave, and hence exactly solvable by
efficient techniques. The case under consideration is when J is strongly-connected and there exists a
stable non-negative steady-state.
Theorem 4. Let (N, J, b, ?) describe a system with a strongly-connected graph for
P which there
exists a stable non-negative steady-state m(b). Then, for any H, M M F (b + h) = i mi (b + h),
M M F (b + h) is smooth in h, and M M F (b + h) is concave in h for all h ? FH .
Proof sketch. Our argument follows in three steps. We first show that m(b) is the unique stable
non-negative steady-state and
Pthat it attains the maximum total opinion among steady-states. This
guarantees that M M F (b) = i mi (b). Furthermore, m(b) gives rise to a unique and
P smooth branch
MF
of stable non-negative steady-states for additional h, and hence M
(b + h) = i mi (b + h) for
all h > 0. Finally, one can directly show that M M F (b + h) is concave in h.
Remark. By arguments similar to those in Theorem 4, it can be shown that any stable non-positive
steady-state is unique, attains the minimum total opinion among steady-states, and is smooth and
convex for decreasing h.
The above result paints a significantly simplified picture of the MF-IIM problem when J is stronglyconnected and there exists a stable non-negative steady-state m(b). Given a budget H, for any
feasible marketing strategy h ? FH , m(b + h) is the unique stable non-negative steady-state,
attains the maximum total opinion among steady-states, and is smooth in h. Thus, the objective
4
Algorithm 1: An -approximation to MF-IIM
Input: System (N, J, b, ?) for which there exists a stable non-negative steady-state, budget H,
accuracy parameter > 0
Output: External field h that approximates a MF optimal external field h?
t = 0; h(0) ? FH ; ? ? (0, L1 ) ;
repeat
P
?M M F (b+h(t))
F
= i ?M
ij (b + h(t));
?hj
h(t + 1) = PFH h(t) + ?Oh M M F (b + h(t)) ;
t++;
until M M F (b + h? ) ? M M F (b + h(t)) ? ;
h = h(t);
P
M M F (b + h) = i mi (b + h) is smooth, allowing us to write down a gradient ascent algorithm that
approximates a local maximum. Furthermore, since M M F (b+h) is concave in h, any local maximum
of M M F on FH is a global maximum, and we can apply efficient gradient ascent techniques to solve
MF-IIM.
Our algorithm, summarized in Algorithm 1, is initialized at a feasible external field. At each iteration,
P
MF
F
we calculate the susceptibility of the system, namely ?M
= i ?M
ij , and project this gradient
?hj
onto FH (the projection operator PFH is well-defined since FH is convex). Stepping along the
direction of the projected gradient with step size ? ? (0, L1 ), where L is a Lipschitz constant of
M M F , Algorithm 1 converges to an -approximation to MF-IIM in O(1/) iterations [25].
4.1
Sufficient conditions for the existence of a stable non-negative steady-state
In the previous section we found that MF-IIM is efficiently solvable if there exists a stable nonnegative steady-state. While this assumption may seem restrictive, we show, to the contrary, that the
appearance of a stable non-negative steady-state is a fairly general phenomenon. We first show, for J
strongly-connected, that the existence of a stable non-negative steady-state is robust to increases in h
and that the existence of a stable positive steady-state is robust to increases in ?.
Theorem 5. Let (N, J, h, ?) describe a system with a strongly-connected graph for which there
exists a stable non-negative steady-state m. If m ? 0, then as h increases, m gives rise to a unique
and smooth branch of stable non-negative steady-states. If m > 0, then as ? increases, m gives rise
to a unique and smooth branch of stable positive steady-states.
Proof sketch. By the implicit function theorem, any stable steady-state can be locally defined as a
function of both h and ?. Using the susceptibility, one can directly show that any stable non-negative
steady-state remains stable and non-negative as h increases and that any stable positive steady-state
remains stable and positive as ? increases.
The intuition behind Theorem 5 is that increasing the external field will never destroy a steady-state in
which all of the opinions are already non-positive. Furthermore, as the interaction strength increases,
each individual reacts more strongly to the positive influence of her neighbors, creating a positive
feedback loop that results in an even more positive magnetization. We conclude by showing for J
strongly-connected that if h ? 0, then there exists a stable non-negative steady-state.
Theorem 6. Let (N, J, h, ?) describe any system with a strongly-connected network. If h ? 0, then
there exists a stable non-negative steady-state.
Proof sketch. For h > 0 and ? < ?c , it can be shown that the unique steady-state is positive, and
hence Theorem 5 guarantees the result for all ? 0 > ?. For h = 0, Theorem 3 provides the result.
All together, the results of this section provide a number of sufficient conditions under which MF-IIM
is exactly and efficiently solvable by Algorithm 1.
5
5
A shift in the structure of solutions to MF-IIM
The structure of solutions to MF-IIM is of fundamental theoretical and practical interest. We
demonstrate, remarkably, that solutions to MF-IIM shift from focusing on nodes of high degree at
low interaction strengths to focusing on nodes of low degree at high interaction strengths.
Consider an Ising system described by (N, J, h, ?) in the limit ? ?c . To first-order in ?, the
self-consistency equations (3) take the form:
m = ? (Jm + h)
?
m = ?(I ? ?J)?1 h.
Since ? < ?c , we have ?(?J) < 1, allowing us to expand (I ? ?J)?1 in a geometric series:
X
X
3
m = ?h + ? 2 Jh + O(? 3 ) ? M M F (h) = ?
hi + ? 2
dout
i hi + O(? ),
i
(8)
(9)
i
P
where dout
=
i
j Jji is the out-degree of node i. Thus, for low interaction strengths, the MF
magnetization is maximized by focusing the external field on the nodes of highest out-degree in the
network, independent of b and H.
To study the structure of solutions to MF-IIM at high interaction strengths, we make the simplifying
assumptions that J is strongly-connected and b ? 0 so that Theorem 6 guarantees a stable nonnegative steady state m. For large ? and an additional external field h ? FH , m takes the form
? ?
??
X
in
mi ? tanh ?? ?
Jij + bi + hi ?? ? 1 ? 2e?2?(di +bi +hi ) ,
(10)
j
where
din
i
=
P
Jij is the in-degree of node i. Thus, in the high-? limit, we have:
X
(0)
in
in
M M F (b + h) ?
1 ? 2e?2?(di +bi +hi ) ? n ? 2e?2?(di? +hi? +hi? ) ,
j
(11)
i
where i? = arg mini (din
i + bi + hi ). Thus, for high interaction strengths, the solutions to MF-IIM
for an external field budget H are given by:
(0)
in
h? = arg max n ? 2e?2?(di? +hi? +hi? ) ? arg max min din
(12)
i + bi + hi .
h?FH
h?FH
i
Eq. (12) reveals that the high-? solutions to MF-IIM focus on the nodes for which din
i + bi + hi is
smallest. Thus, if b is uniform, the MF magnetization is maximized by focusing the external field on
the nodes of smallest in-degree in the network.
We emphasize the strength and novelty of the above results. In the context of reverberant opinion
dynamics, the optimal control strategy has a highly non-trivial dependence on the strength of
interactions in the system, a feature not captured by viral models. Thus, when controlling a social
system, accurately determining the strength of interactions is of critical importance.
6
Numerical simulations
We present numerical experiments to probe the structure and performance of MF optimal external
fields. We verify that the solutions to MF-IIM undergo a shift from focusing on high-degree nodes at
low interaction strengths to focusing on low-degree nodes at high interaction strengths. We also find
that for sufficiently high and low interaction strengths, the MF optimal external field achieves the
maximum exact magnetization, while admitting performance losses near ?c . However, even at ?c ,
we demonstrate that solutions to MF-IIM significantly outperform common node-selection heuristics
based on node degree and centrality.
We first consider an undirected hub-and-spoke network, shown in Figure 1, where Jij ? {0, 1} and
we set b = 0 for simplicity. Since b ? 0, Algorithm 1 is guaranteed to achieve a globally optimal MF
magnetization. Furthermore, because the network is small, we can calculate exact solutions to IIM
by brute force search. The left plot in Figure 1 compares the average degree of the MF and exact
optimal external fields over a range of temperatures for an external field budget H = 1, verifying
6
Figure 1: Left: A comparison of the structure of the MF and exact optimal external fields,
denoted h?M F and h? , in a hub-and-spoke network. Right: The relative performance of h?M F
compared to h? ; i.e., M (h?M F )/M (h?M F ), where M denotes the exact magnetization.
Figure 2: Left: A stochastic block network consisting of a highly-connected community
(Block 1) and a sparsely-connected community (Block2). Center: The solution to MF-IIM
shifts from focusing on Block 1 to Block 2 as ? increases. Right: Even at ?c , the MF solution
outperforms common node-selection heuristics.
that the solution to MF-IIM shifts from focusing on high-degree nodes at low interaction strengths
to low-degree nodes at high interaction strengths. Furthermore, we find that the shift in the MF
optimal external field occurs near the critical interaction strength ?c = .5. The performance of the
MF optimal strategy (measured as the ratio of the magnetization achieved by the MF solution to that
achieved by the exact solution) is shown in the right plot in Figure 1. For low and high interaction
strengths, the MF optimal external field achieves the maximum magnetization, while near ?c , it incurs
significant performance losses, a phenomenon well-studied in the literature [21].
We now consider a stochastic block network consisting of 100 nodes split into two blocks of 50
nodes each, shown in Figure 2. An undirected edge of weight 1 is placed between each pair of
nodes in Block 1 with probability .2, between each pair in Block 2 with probability .05, and between
nodes in different blocks with probability .05, resulting in a highly-connected community (Block
1) surrounded by a sparsely-connected community (Block 2). For b = 0 and H = 20, the center
plot in Figure 2 demonstrates that the solution to MF-IIM shifts from focusing on Block 1 at low ?
to focusing on Block 2 at high ? and that the shift occurs near ?c . The stochastic block network is
sufficiently large that exact calculation of the optimal external fields is infeasible. Thus, we resort
to comparing the MF solutions with three node-selection heuristics: one that distributes the budget
in amounts proportional to nodes? degrees, one that distributes the budget proportional to nodes?
centralities (the inverse of a node?s average shortest path length to all other nodes), and one that
distributes the budget randomly. The magnetizations are approximated via Monte Carlo simulations
of the Glauber dynamics, and we consider the system at ? = ?c to represent the worst-case scenario
for the MF optimal external fields. The right plot in Figure 2 shows that, even at ?c , the solutions to
MF-IIM outperform common node-selection heuristics.
We consider a real-world collaboration network (Figure 3) composed of 904 individuals, where each
edge is unweighted and represents the co-authorship of a paper on the arXiv [26]. We note that
co-authorship networks are known to capture many of the key structural features of social networks
7
Figure 3: Left: A collaboration network of 904 physicists where each edge represents the
co-authorship of a paper on the arXiv. Center: The solution to MF-IIM shifts from high- to lowdegree nodes as ? increases. Right: The MF solution out-performs common node-selection
heuristics, even at ?c .
[27]. For b = 0 and H = 40, the center plot in Figure 3 illustrates the sharp shift in the solution
to MF-IIM at ?c = 0.05 from high- to low-degree nodes. Furthermore, the right plot in Figure
3 compares the performance of the MF optimal external field with the node-selection heuristics
described above, where we again consider the system at ?c as a worst-case scenario, demonstrating
that Algorithm 1 is scalable and performs well on real-world networks.
7
Conclusions
We study influence maximization, one of the fundamental problems in network science, in the
context of the Ising model, wherein repeated interactions between individuals give rise to complex
macroscopic patterns. The resulting problem, which we call Ising influence maximization, has a
natural physical interpretation as maximizing the magnetization of an Ising system given a budget
of external magnetic field. Under the mean-field approximation, we develop a number of sufficient
conditions for when the problem is concave, and we provide a gradient ascent algorithm that uses the
susceptibility to efficiently calculate locally-optimal external fields. Furthermore, we demonstrate
that the MF optimal external fields shift from focusing on high-degree individuals at low interaction
strengths to focusing on low-degree individuals at high interaction strengths, a phenomenon not
observed in viral models. We apply our algorithm on random and real-world networks, numerically
demonstrating shifts in the solution structure and showing that our algorithm out-performs common
node-selection heuristics.
It would be interesting to study the exact Ising model on an undirected network, in which case the
spin statistics are governed by the Boltzmann distribution. Using this elegant steady-state description,
one might be able to derive analytic results for the exact IIM problem. Our work establishes a fruitful
connection between influence maximization and statistical physics, paving the way for exciting
cross-disciplinary research. For example, one could apply advanced mean-field techniques, such
as those in [21], to generate efficient algorithms of increasing accuracy. Furthermore, because our
model is equivalent to a Boltzmann machine, one could propose a framework for data-based influence
maximization based on well-known Boltzmann machine learning techniques.
Acknowledgements. We thank Michael Kearns and Eric Horsley for enlightening discussions, and
we acknowledge support from the U.S. National Science Foundation, the Air Force Office of Scientific
Research, and the Department of Transportation.
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9
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5,658 | 612 | Combining Neural and Symbolic
Learning to Revise Probabilistic Rule
Bases
J. Jeffrey Mahoney and Raymond J. Mooney
Dept. of Computer Sciences
University of Texas
Austin, TX 78712
mahoney@cs.utexas.edu, mooney@cs.utexas.edu
Abstract
This paper describes RAPTURE - a system for revising probabilistic knowledge bases that combines neural and symbolic learning
methods. RAPTURE uses a modified version of backpropagation
to refine the certainty factors of a MYCIN-style rule base and uses
ID3's information gain heuristic to add new rules. Results on refining two actual expert knowledge bases demonstrate that this
combined approach performs better than previous methods.
1
Introduction
In complex domains, learning needs to be biased with prior knowledge in order to
produce satisfactory results from limited training data. Recently, both connectionist
and symbolic methods have been developed for biasing learning with prior knowledge lFu, 1989; Towell et a/., 1990; Ourston and Mooney, 1990]. Most ofthese methods revise an imperfect knowledge base (usually obtained from a domain expert) to
fit a set of empirical data. Some of these methods have been successfully applied to
real-world tasks, such as recognizing promoter sequences in DNA [Towell et ai., 1990;
Ourston and Mooney, 1990]. The results demonstrate that revising an expert-given
knowledge base produces more accurate results than learning from training data
alone.
In this paper, we describe the RAPTURE system (Revising Approximate
107
108
Mahoney and Mooney
Probabilistic Theories Using Repositories of Examples), which combines connectionist and symbolic methods to revise both the parameters and structure of a
certainty-factor rule base.
2
The Rapture Algorithm
The RAPTURE algorithm breaks down into three main phases. First, an initial
rule-base (created by a human expert) is converted into a RAPTURE network. The
result is then trained using :ertainty-factor backpropagation (CFBP). The theory
is further revised through network architecture modification. Once the network is
fully trained, the solution is at hand-there is no need for retranslation. Each of
these steps is outlined in full below.
2.1
The Initial Rule-Base
RAPTURE uses propositional certainty factor rules to represent its theories. These
rules have the form A ~ D, which expresses the idea that belief in proposition A
gives a 0.8 measure of belief in proposition D [Shafer and Pearl, 1990]. Certainty
factors can range in value from -1 to +1, and indicate a degree of confidence in a
particular proposition. Certainty factor rules allow updating of these beliefs based
upon new observed evidence.
Rules combine evidence via probabilistic sum, which is defined as a EB b - a + b - abo
In general, all positive evidence is combined to determine the measure of belief(MB)
for a given proposition, and all negative evidence is combined to obtain a measure
of disbelief (MD). The certainty factor is then calculated using C F = M B + MD.
RAPTURE uses this formalism to represent its rule base for a variety of reasons.
First, it is perhaps the simplest method that retains the desired evidence-summing
aspect of uncertain reasoning. As each rule fires, additional evidence is contributed
towards belief in the rule's consequent. The use of probabilistic sum enables many
small pieces of evidence to add up to significant evidence. This is lacking in formalisms that use only MIN or MAX for combining evidence [Valtorta, 1988]. Second, probabilistic sum is a simple, differentiable, non-linear function. This is crucial for implementing gradient descent using backpropagation. Finally, and perhaps
most significantly, is the widespread use of certainty factors. Numerous knowledgebases have been implemented using this formalism, which immediately gives our
approach a large base of applicability.
2.2
Converting the Rule Base into a Network
Once the initial theory is obtained, it is converted into a RAPTURE -network. Building the network begins by mapping all identical propositions in the rule-base to the
same node in the network. Input features (those only appearing as rule-antecedents)
become input nodes, and output symbols (those only appearing as rule-consequents)
become output nodes. The certainty factors of the rules become the weights on the
links that connect nodes. Networks for classification problems contain one output
for each category. When an example is presented, the certainty factor for each of
the categories is computed and the example is assigned to the category with the
Combining Neural and Symbolic Learning to Revise Probabilistic Rule Bases
1.3
I
,
I
I
,,.2t'
Figure 1: A RAPTURE NETWORK
highest value. Figure 1 illustrates the following set of rules.
ABC~D
E~D
C~G
EF~G
HI~C
As shown in the network, conjuncts must first pass through a MIN node before
any activation reaches the consequent node. Note that each of the conjuncts is
connected to the corresponding MIN mode with a solid line. This represents the
fact that the link is non-adjustable, and simply passes its full activation value onto
the MIN node. The standard (certainty-factor) links are drawn as dotted lines
indicating that their values are adjustable.
This construction shows how easily a RAPTURE-network can model a MYCIN rulebase. Each representation can be converted into the other, without loss or corruption of information. They are two equivalent representations of the same set of
rules.
2.3
Certainty Factor Backpropagation
Using the constructed RAPTURE-network, we desire to maximize its predictive accuracy over a set of training examples. Cycling through the examples one at a time,
and slightly adjusting all relevant network weights in a direction that will minimize
the output error, results in hill-climbing to a local minimum. This is the idea behind gradient descent [Rumelhart et al., 1986), which RAPTURE accomplishes with
Certainty Factor Backpropagation (CFBP), using the following equations.
ApWji
= 7Jopj(1 ? LWjkOpk)
k#-i
If Uj is an output unit
(1)
109
110
Mahoney and Mooney
(2)
If Uj is not an output unit
hpj
=
L:
kmin
hpk w kj(1
?
EWjkOpk)
(3)
i;tk
The "Sigma with circle" notation is meant to represent probabilistic sum over the
index, and the ? notation is shorthand for two separate cases. If WjiOpi ~ 0, then
- is used, otherwise + is used. The kmin subscript refers to the fact that we do
not perform this summation for every unit k (as in standard backpropagation), but
only those units that received some contribution from unit j. Since a unit j may
be required to pass through a min or max-node before reaching the next layer (k),
it is possible that its value may not reach k.
RAPTURE deems a classification correct when the output value for the correct category is greater than that of any other category. No error propagation takes place in
this case (hpj
0). CFBP terminates when overall error reaches a minimal value.
=
2.4
Changing the Network Architecture
Whenever training accuracy fails to reach 100% through CFBP, it may be an indication that the network architecture is inappropriate for the current classification
task. To date, RAPTURE has been given two ways of changing network architecture.
First, whenever the weight of a link in the network approaches zero, it is removed
from the network along with all of the nodes and links that become detached due
to this removal. Further, whenever an intermediate node loses all of its input links
due to link deletion, it too is removed from the network, along with its output link.
This link/node deletion is performed immediately after CFBP, and before anything
new is introduced into the network.
also has a method for adding new nodes into the network. Specific nodes
are added in an attempt to maximize the number of training examples that are classified correctly. The simple solution employed by RAPTURE is to create new input
nodes that connect directly, either positively or negatively, to one or more output
nodes. These new nodes are created in a way that will best help the network distinguish among training examples that are being misclassified. Specifically, RAPTURE
attempts to distinguish for each output category, those examples of that category
that are being misclassified (Le. being classified into a different output category),
from those examples that do belong in these different output categories. Quinlan's
ID3 information gain metric [Quinlan, 1986] has been adopted by RAPTURE to select this new node, which becomes positive evidence for the correct category, and
negative evidence for mistaken categories.
RAPTURE
With these new nodes in place, we can now return to CFBP, where hopefully more
training examples will be successfully classified. This entire process (CFBP followed
by deleting links and adding new nodes) repeats until all training examples are
correctly classified. Once this has occurred, the network is considered trained, and
testing may begin.
Combining Neural and Symbolic Learning to Revise Probabilistic Rule Bases
Soybean Test Ac:eurac:y
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Figure 2: RAPTURE Testing Accuracy
3
Experimental Results
To date, RAPTURE has been tested on two real-world domains. The first of these
is a domain for recognizing promoter sequences in strings of DNA-nucleotides. The
second uses a theory for diagnosing soybean diseases. These datasets are discussed
in detail in the following sections.
3.1
Promoter Recognition Results
A prokaryotic promoter is a short DNA sequence that precedes the beginnings of
genes, and are locations where the protein RNA polymerase binds to the DNA
structure [Towell et al., 1990]. A set of propositional Horn-clause rules for recognizing promoters, along with 106 labelled examples (53 promoters, 53 non-promoters)
was provided as the initial theory.
In order for this theory to used by RAPTURE it had to be modified into a certainty
factor format. This was done by breaking up rules with multiple antecedents, into
several rules. In this fashion, each antecedent is able to contribute some evidence
towards belief in the consequent. Initial certainty factors were assigned in such a
way that if every antecedent (from the original rule) were true, a certainty factor of
0.9 would result for the consequent.
To test RAPTURE using this dataset, standard training and test runs were performed, which resulted in the learning curve of Figure 2a. This graph is a plot of
average performance in accuracy at classifying DNA strings over 25 independent
trials. A single trial consists of providing each system with increasing numbers of
examples to use for training, and then seeing how well it can classify unseen test
examples. This graph clearly demonstrates the advantages of an evidence summing
111
112
Mahoney and Mooney
system like RAPTURE over a pure Horn-clause system such as EITHER, a pure inductive system such as ID3, or a pure connectionist system, like backprop. Also plotted
in the graph, is KBANN [Towell et a/., 1990], a symbolic-connectionist system that
uses standard backpropagation, and RAPTURE-O, which is simply RAPTURE given
no initial theory, emphasizing the importance of the expert knowledge. For this
dataset, CFBP alone was all that was required in order to train the network. The
node addition module was never called.
3.2
Soybean Disease Diagnosis Results
The Soybean Data comes from [Michalski and Chilausky, 1980] and is a dataset
of 562 examples of diseased soybean plants. Examples were described by a string
of 35 features including the condition of the stem, the roots, the seeds, as well as
information such as the time of year, temperature, and features of the soil. An
expert classified each example into one of 15 soybean diseases. This dataset has
been used as a benchmark for a number of learning systems. Figure 2b is a learning
curve on this data comparing RAPTURE, RAPTURE-O, backpropagation, ID3, and
EITHER.
The headstart given to RAPTURE does not last throughout testing in this domain.
RAPTURE maintains a statistically significant lead over the other systems (except
RAPTURE-O) through 80 examples, but by 150 examples, all systems are performing
at statistically equivalent levels. A likely explanation for this is that the expert
provided theory is more helpful on the easier to diagnose diseases than on those that
are more difficult. But these easy ones are also easy to learn via pure induction,
and good rules can be created after seeing only a few examples. Trials have actually
been run out to 300 examples, though all systems are performing at equivalent levels
of accuracy.
4
Related Work
The SEEK system [Ginsberg et a/., 1988) revises rule bases containing M-of-N rules,
though can not modify real-valued weights and contains no means for adding new
rules. Valtorta [Valtorta, 1988) has examined the computational complexity of various refinement tasks for probabilistic knowledge bases, and shows that refining
the weights to fit a set of training data is an NP-Hard problem. Ma and Wilkins
[Ma and Wilkins, 1991) have developed methods for improving the accuracy of a
certainty-factor knowledge base by deleting rules, and they report modest improvements in the accuracy of a MYCIN rule base. Gallant [Gallant, 1988) designed and
implemented a system that combines expert domain knowledge with connectionist
learning, though is not suitable for multi-layer networks or for combination functions
like probabilistic sum. KBANN [Towell et a/., 1990] uses standard backpropagation
to refine a symbolic rule base, though the mapping between the symbolic rules and
the network is only an approximation. Fu [Fu, 1989) and Lacher [Lacher, 1992) have
also used backpropagation techniques to revise certainty factors on rules. However,
the current publications on these two projects do not address the problem of altering the network architecture (i.e. adding new rules) and do not present results on
revising actual expert knowledge bases.
Combining Neural and Symbolic Learning to Revise Probabilistic Rule Bases
5
Future Work
The current method for changing network architecture in RAPTURE is restricted
to adding new input units that directly feed the outputs. We hope to incorporate
newer techniques for creating and linking to hidden nodes, in order to improve the
range of architectural changes that it can make.
Another area requiring further research concerns the differences between certaintyfactor networks and traditional connectionist networks. Further comparison of the
RAPTURE and KBANN approaches to knowledge-base refinement are also indicated.
Finally, in recent years, certainty-factors have been the subject of considerable criticism from researchers in uncertain reasoning [Shafer and Pearl, 1990]. However,
the basic revision framework in RAPTURE should be applicable to other uncertain
reasoning formalisms such as Bayesian networks, Dempster-Shafer theory, or fuzzy
logic [Shafer and Pearl, 1990]. As long as the activation functions in the corresponding network implementations of these methods are differentiable, backpropagation
techniques should be employable.
6
Conclusions
Automatic refinement of probabilistic rule bases is an under-studied problem with
important applications to the development of intelligent systems. This paper has
described and evaluated an approach to refining certainty-factor rule bases that
integrates connectionist and symbolic learning. The approach is implemented in a
system called RAPTURE, which uses a revised backpropagation algorithm to modify
certainty factors and ID3's information gain criteria to determine new rules to add to
the network. In other words, connectionist methods are used to adjust parameters
and symbolic methods are used to make structural changes to the knowledge base.
In domains with limited training data or domains requiring meaningful explanations
for conclusions, refining existing expert knowledge has clear advantages. Results
on revising three real-world knowledge bases indicates that RAPTURE generally
performs better than purely inductive systems (ID3 and backpropagation), a purely
symbolic revision system (EITHER), and and purely connectionist revision system
(KBANN).
The certainty-factor networks used in RAPTURE blur the distinction between connectionist and symbolic representations. They can be viewed either as connectionist
networks or symbolic rule bases. RAPTURE demonstrates the utility of applying connectionist learning methods to "symbolic" knowledge bases and employing symbolic
methods to modify "connectionist" networks. Hopefully these results will encourage others to explore similar opportunities for cross-fertilization of ideas between
connectionist and symbolic learning.
Acknowledgements
This research was supported by the National Science Foundation under grant IRI9102926, the NASA Ames Research Center under grant NCC 2-629, and the Texas
Advanced Research Program under grant 003658114. We wish to thank R.S. Michal-
113
114
Mahoney and Mooney
ski for furnishing the soybean data, M. Noordewier, G.G. Towell, and J.W. Shavlik
for supplying the DNA data, and the KBANN results.
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5,659 | 6,120 | An urn model for majority voting
in classification ensembles
Victor Soto
Computer Science Department
Columbia University
New York, NY, USA
vsoto@cs.columbia.edu
Alberto Su?rez and Gonzalo Mart?nez-Mu?oz
Computer Science Department
Universidad Aut?noma de Madrid
Madrid, Spain
{gonzalo.martinez,alberto.suarez}@uam.es
Abstract
In this work we analyze the class prediction of parallel randomized ensembles by
majority voting as an urn model. For a given test instance, the ensemble can be
viewed as an urn of marbles of different colors. A marble represents an individual
classifier. Its color represents the class label prediction of the corresponding
classifier. The sequential querying of classifiers in the ensemble can be seen
as draws without replacement from the urn. An analysis of this classical urn
model based on the hypergeometric distribution makes it possible to estimate
the confidence on the outcome of majority voting when only a fraction of the
individual predictions is known. These estimates can be used to speed up the
prediction by the ensemble. Specifically, the aggregation of votes can be halted
when the confidence in the final prediction is sufficiently high. If one assumes
a uniform prior for the distribution of possible votes the analysis is shown to be
equivalent to a previous one based on Dirichlet distributions. The advantage of
the current approach is that prior knowledge on the possible vote outcomes can be
readily incorporated in a Bayesian framework. We show how incorporating this
type of problem-specific knowledge into the statistical analysis of majority voting
leads to faster classification by the ensemble and allows us to estimate the expected
average speed-up beforehand.
1
Introduction
Combining the outputs of multiple predictors is in many cases of interest a successful strategy to
improve the capabilities of artificial intelligence systems, ranging from agent architectures [19], to
committee learning [13, 15, 8, 9]. A common approach is to build a collection of individual subsystems and then integrate their outputs into a final decision by means of a voting process. Specifically,
in the machine learning literature, there is extensive empirical evidence on the improvements in
generalization capacity that can be obtained using ensembles of learners [7, 11]. However, one of the
drawbacks of these types of systems is the linear memory and time costs incurred in the computation
of the final ensemble prediction by combination of the individual predictions. There are various
strategies that alleviate these shortcomings. These techniques are grouped into static (or off-line)
and dynamic (or online). In static pruning techniques, only a subset of complementary predictors
from the original ensemble is kept [16, 21, 6]. By contrast, in dynamic pruning, the whole ensemble
is retained. The prediction of the class label of a particular instance is accelerated by halting the
sequential querying process when it is unlikely that the remaining (unknown) votes would change
the output prediction [10, 20, 14, 12, 2, 3, 17]. These techniques are online in the sense that, as new
individual predictions become known, the algorithm dynamically updated the estimated probability
of having a stable prediction; i.d. a prediction that coincides with that by the complete ensemble.
This is the basis of the Statistical Instance-Based Algorithm (SIBA) proposed in [14]. In a similar
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
approach, albeit with a different objective, Reyzin proposes to randomly sample hypotheses from
the original AdaBoost ensemble. The goal is to minimize the number of features that are used for
prediction, with a limited loss of accuracy [18]. This feature-efficient prediction is beneficial when
access to the features of a new instance at test time is costly (e.g., in some medical problems). A
different approach is followed in [3]. In this work, a policy is learned to decide which classifiers
should be queried and which discarded in the prediction of the class label of a given instance.
The dynamic ensemble pruning method proposed in this work is closely related to SIBA [14]. In
SIBA, the members of a committee are queried sequentially. At each step in the querying process, the
votes recorded are used to estimate the probability that the majority decision of the classifiers queried
up to that moment coincides with the complete ensemble. If this probability exceeds a specified
confidence level, ?, the voting process is halted. To compute this estimate, the probability that a
single predictor outputs a given decision for the particular instance considered is modeled as a random
variable. Starting from a uniform prior, Bayes? theorem is used to update the distribution of this
variable with the information provided by the actual votes, as they become known. In most of the
problems analyzed in [14], the assumption that the prior is uniform leads to conservative estimates
of the confidence on the stability of the predictions when only a fraction of the classifiers have
been queried. Analyzing the results of those experiments, it is apparent that the actual disagreement
percentages between the dynamic decision output and the decision made by the complete committee
are significantly lower than the specified target ?. As a consequence, more queries are made than the
ones that are actually needed.
The present work has two objectives. First, we propose an intuitive mathematical modeling of
the voting process in ensembles of classifiers based on the hypergeometric distribution. Under the
assumption that the distribution of possible vote outcomes is uniform, we prove that this derivation is
equivalent to the one presented in [14]. However, the vote distribution is, in general, not uniform.
Its shape depends on the classification task considered and on the base learning algorithm used to
generate the predictors. Second, to take into account this dependence, we propose to approximate
this distribution using a non-parametric prior. The use of this problem-specific prior knowledge
leads to more accurate estimations of the disagreement rates between the dynamic sub-committee
prediction and the complete committee, which are closer to the specified target ?. In this manner,
faster classification can be achieved with minimal loss of accuracy. In addition, the use of priors allow
us to estimate quite precisely the expected average number of trees that would be necessary to query.
2
Modeling ensemble voting processes as a classical urn problem
Consider the following process modeled as a classical urn model. Let us suppose we have marbles
of l different colors in an urn. The number of marbles of color yk in the urn is Tk , with k = 1 . . . l.
Pl
The total number of marbles in the urn is T = k=1 Tk . The contents of the urn can therefore be
described by vector T = hT1 , T2 . . . Tl i. Assume that t < T marbles are extracted from the urn
without replacement. This extraction process can be characterized by vector t = ht1 , t2 . . . tl i where
Pl
tk is the number of marbles of color yk extracted, with t = k=1 tk . The probability of extracting a
color distribution of marbles t, given the initial color distribution of the urn T is described by the
multivariate hypergeometric distribution
Ql
T1
Tl
Ti
i=1 ti
t1 . . . tl
P(t|T) =
=
.
(1)
T
T
t
t
Consider the case in which the total number of marbles in the urn, T , is known but that the color
distribution, T, is unknown. In this case, the color distribution of the extracted marbles, t, can be
used to estimate the content of the urn applying Bayes Theorem
T1
Tl
P(t|T)P(T)
P(t|T)P(T)
t1 . . . tl P(T)
P
P(T|t) =
=
=P
(2)
Tl?
T1?
?
?
?
P(t)
T? ??t P(t|T )P(T )
T? ??t t1 . . . tl P(T )
Pl
where ?t is the set of vectors T? , such that Ti? ? ti ?i and i=1 Ti? = T .
This problem is equivalent to the voting process in an ensemble of classifiers: Suppose we want
to predict the class label of an instance by combining the individual predictions of the ensemble
classifiers (marbles). Assuming that the individual predictions are deterministic, the class (color) that
2
each classifier (marble) would output if queried is fixed, but unknown before the query. Therefore,
for each instance considered we have a different "bag of colored marbles" with an unknown class
distribution. After a partial count of votes of the ensemble is known, Eq. 2 provides an estimate of the
distribution of votes for the complete ensemble. This estimate can be used to compute the probability
that the decision obtained using only a partial tally of votes, t, of size t < T and by the final decision
using all T votes, coincide
T1
Tl
X
t1 . . . tl P(T)
?
P (t, T ) =
,
(3)
P
Tl?
T1?
?
T? ??t t1 . . . tl P(T )
T?Tt
where Tt is the set of vectors of votes for the complete ensemble T = {T1 , T2 . . . Tl } such that
the class predicted by the subensemble of size t and the class predicted by the complete committee
Pl
coincide, with Ti ? ti , and i=1 Ti = T .
If P ? (t, T ) = 1, then the classification given by the partial ensemble and the full ensemble coincide.
This case happens when the difference between the number of votes for the first and second class
in t is greater than the remaining votes in the urn. In such case, the voting process can be halted
with full confidence that the decision of the partial ensemble will not change when the predictions of
the remaining classifiers are considered. In addition, if it is acceptable that, with a small probability
1 ? ?, the prediction of the partially polled ensemble and that of the complete ensemble disagree,
then the voting process can be stopped when the P ? (t, T ) exceeds the specified confidence level ?.
The final classification would be given as the combined decisions of the classifiers that have been
polled up to that point only.
2.1
Uniform prior
Assuming a uniform prior for the distribution of possible T vectors P(T) = 1/kTk, where kTk
stands for the number of possible T vectors, this derivation is equivalent to the one presented in
[14]. That formulation assumes that the base classifiers of the ensemble are independent realizations
from a pool of all possible classifiers given the training dataset. Assuming that an unlimited number
of realizations can be performed, the distribution of class votes in the ensemble converges to a
Dirichlet distribution in the limit of infinite ensemble size. Then, assuming a partial tally of t votes,
the probability that the ensemble?s decision will change if the precictions of the remaining T ? t
classifiers are considered, can be estimated.
In order to prove the equivalence between both formulations, we first need to introduce three results,
presented in the theorem and propositions below.
Theorem. Chu-Vandermonde Identity. Let s, t, r ? N then
X
r
s+t
s
t
=
r
k
r?k
(4)
k=0
Proposition 1. Upper negation. Let r ? C and k ? Z, then
r
k k?r?1
= (?1)
k
k
(5)
The previous theorem and proposition are used in the following proposition, which is the key to prove
the equivalence between the two formulations:
Proposition 2. Let n1 and n2 be positive integers such that n1 + n2 = n and n ? N . Then
NX
?n2
i
N ?i
N +1
=
n1
n2
N ?n
i=n
(6)
1
n
Proof. First the symmetry property of the binomial (i.e., nk = n?k
) is used to bring down the
indices
NX
NX
?n2
?n2
i
N ?i
i
N ?i
=
n1
n2
i ? n1
N ? i ? n2
i=n
i=n
1
1
3
The upper indices are removed by applying the upper negation property of proposition 1.
NX
?n2
NX
?n2
?n1 ? 1
?n2 ? 1
i
N ?i
(?1)i?n1 (?1)N ?i?n2
=
i
?
n
N
?
i
?
n
i
?
n
N
?
i
?
n
1
2
1
2
i=n
i=n
1
1
Now, the Chu-Vandermonde identity can be applied with r = N ? n1 ? n2 and k = i ? n1
NX
?n2
?n1 ? 1
?n2 ? 1
?n ? 2
i?n1
N ?i?n2
(?1)
(?1)
=
(?1)N ?n
i
?
n
N
?
i
?
n
N
?
n
1
2
i=n
1
Finally the upper negation is applied again
?n ? 2
N +1
(?1)N ?n =
N ?n
N ?n
Proposition 3 Following the hypergeometric reformulation given by Equation 2 and assuming that
P(T) follows a uniform distribution 1/kTk, where kTk stands for the number of possible T vectors
then
Ql
(T ? t)!
i=1 (ti + 1)Ti ?ti
P(T|t) = Ql
(t + l)T ?t
i=1 (Ti ? ti )!
where (x)n = x(x + 1) . . . (x + n ? 1) is the Pochhammer symbol. This formulation is equivalent
to the one proposed in [14].
Proof. Equation 2 can be simplified by taking into account the uniform prior P(T) = 1/kTk as
T1
Tl
P(t|T)P(T)
t1 . . . tl
=P
(7)
P(T|t) =
Tl?
T1?
P(t)
T? ??t t1 . . . tl
The indices of the summation, ?t , is the set of vectors T such that Ti ? ti ?i and
They can be changed for l classes to
T1
Tl
t1 . . . tl
P(T|t) = P ?
PT?2
PT?l?1
T1
Tl?
T1?
T ? =tl?1 t1 . . . tl
T ? =t1
T ? =t2 ? ? ?
1
2
Pl
i=1
Ti = T .
(8)
l?1
where T?k for k = 1, . . . , (l ? 1) are the maximum values for Tk? in the summations. Note that the
?
summation over Tl? is unnecessary since the value of Tl? becomes fixed once the values of T1? . . . Tl?1
Pl
are fixed since i=1 Ti? = T . In this sense, the values for T?k have a dependency on Ti? for i < k as
Pk?1
T?k = T ? t + tk ? i=1 (Ti? ? ti ), k = 1, . . . , (l ? 1). The summations in the denominator of
Eq. 8 can be rearranged
T?1
X
T?l?1
T?2
X
???
T1? =t1 T2? =t2
X
? =t
Tl?1
l?1
?
T?1 ? X
T?2 ?
X
T1?
Tl
T1
T2
...
=
???
t1
tl
t1
t2
?
?
T1 =t1
Proposition 2 (Eq. 6) can be used, together with N = T ?
?
Tl?1
in closed form
T2 =t2
Pl?2
i=1
T?l?1
X
? =t
Tl?1
l?1
?
Tl?1
tl?1
Tl?
.
tl
Ti? , to express the summation over
l?2
l?2
?
?
Xi=1 (Ti ?ti ) T ? T ? T ? i=1
XTi ?tl T ? T ? Pl?2 T ? ? T ?
l?1
l?1
l
l?1
i=1 i
=
=
t
t
t
t
l?1
l
l?1
l
?
?
T ?t+tl?1 ?
P
P
Tl?1 =tl?1
Tl?1 =tl?1
Pl?2
Pl?2
T ? i=1 Ti? + 1
T ? i=1 Ti? + 1
=
,
Pl?2
tl?1 + tl + 1
T ? i=1 Ti? ? tl?1 ? tl
4
where the symmetry property of the binomial has been used in the last step. The subsequent
summations are carried out in the same manner. The summation over Tk? requires the application of
Pk?1
Pl
Eq. 6 with N = T ? i=1 Ti? + (l ? k ? 1), n1 = tk and n2 = i=k+1 ti + (l ? k ? 1)
T?1 ?
X
T1
...
t1
?
T1 =t1
T?l?2
X
? =t
Tl?2
l?2
?
Tl?2
tl?2
Pl?2
T ? i=1 Ti? + 1
T +l?1
= ??? =
tl?1 + tl + 1
t+l?1
Employing this result in Eq. 8, one obtains
Ql
Tl !
T1 !
T1
Tl
(T ? t)!
t1 !(T1 ?t1 )! . . . tl !(Tl ?tl )!
t1 . . . tl
i=1 (ti + 1)Ti ?ti
= Ql
P(T|t) =
.
=
T +l?1
(T +l?1)!
(t + l)T ?t
t+l?1
i=1 (Ti ? ti )!
(t+l?1)!(T ?t)!
2.2
Non-uniform prior
The distribution P(T) can be modeled using a non-parametric non-uniform prior. The values of this
prior can be obtained from the training data by some form of validation; e.g., out-of-bag or cross
validation. Out-of-bag validation is faster because it does not require multiple generations of the
ensemble. Therefore, it will be the validation method used in our implementation of the method. To
compute the out-of-bag error, each training instance, xn , is classified by the ensemble predictors
that do not have that particular instance in their training set. Let T?n = T?1n + . . . + T?ln , be the
number of such classifiers, where T?in is the number of out-of-bag votes for class i, where i = 1, . . . , l,
assigned to instance xn . The number of votes for each class for an ensemble of size T is estimated as
Tin ? round(T T?in /T?n ). To mitigate the influence of the random fluctuations that appear because
of the finite size of the training set and to avoid spurious numeric artifacts, the prior is subsequently
smoothed using a sliding window of size 5 over the vote distribution.
As shown in Section 2, the response time of the ensemble can be reduced by using Eq. 3, if we allow
that a small fraction, 1 ? ?, of the predictions given by ensembles of size t and T do not coincide.
Assuming this tolerance, when P ? (t, T ) > ?, the voting process can be halted and the ensemble
will output the decision given by the t ? T queried classifiers. However, the computation of Eq. 3
is costly and should be performed off-line. In the SIBA formulation, a lookup table indexed by the
number of votes of the minority class (for binary problems) and whose values are the minimum
number of votes of the majority class such that P ? (t, T ) > ?, is used. Using a precomputed lookup
table to halt the voting process does not entail a significant overhead during classification: a single
lookup operation in the table is needed for each vote. The consequence of using a uniform prior is
that all classes are considered equivalent. Hence, it is sufficient to compute one lookup table and use
the minority class for indexing.
When prior knowledge is taken into account, the probability P ? (t1 = n, t2 = m, T ) is not necessarily
equal to P ? (t1 = m, t2 = n, T ) for n 6= m. Therefore, a different lookup table per class will be
necessary. In addition, it is necessary to compute a different set of tables for each dataset. In the
original formulation, the lookup table values depend only on T and ?. Therefore, they are independent
of the particular classification problem considered. In our case, the prior distribution is estimated
from the training data: Hence, it is problem dependent. However, the querying process is similar to
SIBA. For instance, if we have 1 vote for class 1 and 7 for class 2, one determines whether the value
in position 1 (minority class at this moment) of the lookup table for class 1 is greater or equal to 7. If
it is, the querying process stops. As a side effect, for the experimental comparison, it is necessary to
recompute the lookup tables for each realization of the data. Notwithstanding, in a real setting, these
tables need to be computed only once. This can be done offline. Therefore, the speed improvements
in the classification phase are independent of the size of the training set.
The lookup table and the estimated non-parametric prior can be used to estimate also the average
number of classifiers that are expected to be queried during test. This estimation can be made using
Monte Carlo simulation. To this end one would perform the following experiment repeatedly and
compute the average number of queries: extract a random vector T from the prior distribution;
generate a vector of votes of size T that contains exactly Ti votes for class i with i = 1 . . . l; finally,
query a random permutation of this vector of votes until the process can be halted as given by the
lookup table and keep the number of queries.
5
3
Experiments
In this section we present the results of an extensive empirical evaluation of the dynamical ensemble
pruning method described in the previous section. The experiments are performed in a series of
benchmark classification problems from the UCI Repository [1] and synthetic data [4] using Random
Forests [5]. The code is available at: https://github.com/vsoto/majority-ibp-prior.
The protocol for the experiments is as follows: for each problem, 100 partitions are created by
10 ? 10-fold cross-validation for real datasets and by random sampling in the synthetic datasets. All
the classification tasks considered are binary, except for New-thyroid, Waveform and Wine, which have
three classes. For each partition, the following steps are carried out: (i) a Random Forest ensemble
of size T = 101 is built; (ii) we compute the generalization error rate of the complete ensemble in
the test set and record the mean number of trees that are queried to determine the final prediction.
Note that this number need not be T : the voting process can be halted when the remaining votes (i.e.
the predictions of classifiers that have not been queried up to that point) cannot modify the partial
ensemble decision. This is the case when the number of remaining votes is below the difference
between the majority class and the second most voted class; (iii) The SIBA algorithm [14] is applied
to dynamically select the number of classifiers that are needed for each instance in the test set to
achieve a level of confidence in the prediction above ? = 0.99. We use SIBA as the benchmark for
comparison since in previous studies it has been shown to provide the best overall results, especially
for T < 500 [2]; (iv) The process is repeated using the proposed method with non-uniform priors
for the class vote distribution, with the same confidence threshold, ? = 0.99. The prior distribution
P(T) is estimated in the training set using out-of-bag data. This prior is also used to estimate the
expected number of trees to be queried in the testing phase. In addition, for steps (iii) and (iv) we
compute the test error rate, the average number of queried trees, and the disagreement rates between
the predictions of the partially queried ensembles and the complete ones.
Table 1: Error rates (left) and disagreement % (right). The statistical significant differences, using
paired t-tests at a significance level ? = 0.05, are highlighted in boldface.
Problem
Australian
Breast
Diabetes
Echocardiogram
German
Heart
Horse-colic
Ionosphere
Labor
Liver
Mushroom
New-thyroid
Ringnorm
Sonar
Spam
Threenorm
Tic-tac-toe
Twonorm
Votes
Waveform
Wine
RF
13.00?3.7
3.22?2.1
24.34?4.2
22.18?14.3
23.43?3.5
18.30?6.9
15.47?5.6
6.44?4.1
6.33?8.9
27.10?6.7
0.00?0.0
4.29?4.0
7.60?1.3
16.25?8.7
4.59?1.5
17.85?1.1
1.05?1.1
4.66?0.6
4.05?2.9
17.30?0.9
1.69?2.8
Error rates
SIBA
13.09?3.7
3.23?2.1
24.25?4.1
22.05?14.7
23.65?3.3
18.37?7.0
15.44?5.4
6.44?4.1
6.17?8.8
27.09?7.0
0.00?0.0
4.38?4.0
7.72?1.2
16.45?8.7
4.63?1.5
18.04?1.1
1.16?1.1
4.77?0.6
4.12?2.9
17.36?0.8
1.74?2.8
HYPER
13.25?3.8
3.76?2.3
24.23?4.0
22.18?14.1
23.62?3.3
18.37?7.2
15.44?5.4
6.52?3.9
6.43?9.1
27.01?6.9
0.08?0.2
4.66?4.2
7.82?1.2
16.45?8.8
4.86?1.4
17.97?1.1
1.72?1.5
4.90?0.6
4.30?2.9
17.45?0.8
2.30?3.5
Disagreement %
SIBA
HYPER
0.3?0.6 0.9?1.1
0.1?0.4 1.0?1.1
0.6?0.9 0.8?1.0
0.7?3.1 1.4?4.6
0.8?0.8 0.8?0.9
0.8?1.8 1.0?2.1
0.4?0.9 0.7?1.3
0.1?0.6 0.7?1.3
0.2?1.7 1.2?4.5
1.0?1.7 0.9?1.5
0.0?0.0 0.1?0.2
0.1?0.7 0.7?2.0
0.5?0.2 0.8?0.3
0.9?2.0 0.8?1.9
0.1?0.2 0.7?0.4
1.0?0.2 0.8?0.2
0.1?0.4 0.7?1.0
0.4?0.1 0.7?0.2
0.1?0.4 1.0?1.8
0.6?0.1 1.0?0.3
0.1?0.6 1.1?2.5
In Table 1, we compare the error rates of Random Forest (RF) and of the dynamically pruned
ensembles using the halting rule derived from assuming uniform priors (SIBA) and using nonuniform priors (HYPER), and the disagreement rates. The values displayed are averages over 100
realizations of the datasets The standard deviation is given after the ? symbol.
6
sonar
votes
0.12
0.025
0.06
HYPER
SIBA
0.1 FIXED
HYPER
SIBA
0.05 FIXED
0.3
0.25
disagreement
P(t1)
0.01
0.08
0.2
P(t1)
disagreement
0.02
0.015
0.06
0.15
0.04
0.04
0.03
0.02
0.1
0.02
0.005
0.01
0.05
0
0
0
10
20
30
40
50
t1
60
70
80
90
100
0
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
1-alpha
0
0
0
10
20
30
40
50
t1
60
70
80
90
100
0
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
1-alpha
Figure 1: Vote distribution, P(T), and disagreement rates for Sonar (left) and Votes (right)
From Table 1, one observes that the mean error rates of the pruned ensembles using SIBA and HYPER
are only slightly worse than the rates obtained by the complete ensemble (RF). These differences
should be expected since we are allowing a small disagreement of 1 ? ? = 1% between the decisions
of the partial and the complete ensemble. In any case, the differences in generalization error can
be made arbitrarily small by increasing ?. By design, the disagreement rates are expected to be
below, but close to 1%. From Table 1, one observes that the disagreement % of the proposed method
(HYPER) are closer to the specified threshold (1 ? ? = 1%) than those of SIBA, except for Liver,
Sonar and Threenorm, where the differences are small. In these problems (and in general in the
problems where SIBA obtains disagreement rates closer to 1 ? ?), the distribution of T is closer to a
uniform distribution (see Figure 1, left histogram). In consequence, the assumption of uniform prior
taken by SIBA is closer to the real one. However, when P(T) differs from the uniform distribution
(see for instance Votes in Figure 1 right histogram) the results of SIBA are rather different from the
expected disagreement rates.
Table 2: Number of queried trees and speed-up rate with respect to the full ensemble of 101 trees.
The statistical significant differences between SIBA and HYPER, using paired t-tests at a significance
level ? = 0.05, are highlighted in boldface.
Problem
Australian
Breast
Diabetes
Echocardiogram
German
Heart
Horse-colic
Ionosphere
Labor
Liver
Mushroom
New-thyroid
Ringnorm
Sonar
Spam
Threenorm
Tic-tac-toe
Twonorm
Votes
Waveform
Wine
RF?
62.2?1.4
54.2?0.9
68.8?1.8
68.0?4.6
71.8?1.3
67.2?2.5
66.2?2.1
57.9?1.5
61.6?4.0
74.5?2.3
51.0?0.0
55.2?1.8
68.6?0.8
73.9?3.0
57.1?0.3
76.6?0.5
60.7?0.9
67.2?0.2
54.5?1.2
72.3?0.7
57.3?2.1
# of trees
SIBA
HYPER
16.1?2.1 12.8?2.3
8.9?1.4
4.0?1.0
24.9?3.2 24.0?3.2
22.6?8.2 20.0?8.0
28.4?2.8 27.7?2.9
22.5?4.2 20.9?4.2
20.2?3.5 17.5?3.7
11.9?2.3
7.8?2.1
14.1?6.0
9.7?5.3
31.8?4.5 31.7?4.5
6.0?0.0
1.0?0.0
10.7?2.6
6.0?2.3
22.9?1.1 20.4?1.5
32.1?6.6 32.6?6.8
11.1?0.5
7.2?0.6
34.8?1.0 35.8?1.6
12.8?1.4
7.8?1.2
21.0?0.5 18.4?0.9
8.8?1.8
4.1?1.4
29.3?1.1 27.8?1.7
11.4?2.7
5.8?1.8
MC Estim
12.9?0.9
4.0?0.4
23.8?1.1
21.6?3.2
30.1?1.0
20.7?1.7
18.6?1.5
7.8?0.6
10.2?2.0
31.6?2.0
1.0?0.0
6.2?1.4
19.7?2.3
31.8?2.4
7.1?0.5
33.4?2.5
8.6?0.7
18.8?1.7
4.0?0.7
28.6?2.8
6.7?1.4
RF?
1.6
1.9
1.5
1.5
1.4
1.5
1.5
1.7
1.6
1.4
2.0
1.8
1.5
1.4
1.8
1.3
1.7
1.5
1.9
1.4
1.8
Speed-up rate
SIBA HYPER
6.3
7.9
11.3
25.3
4.1
4.2
4.5
5.1
3.6
3.6
4.5
4.8
5.0
5.8
8.5
12.9
7.2
10.4
3.2
3.2
16.8
101.0
9.4
16.8
4.4
5.0
3.1
3.1
9.1
14.0
2.9
2.8
7.9
12.9
4.8
5.5
11.5
24.6
3.4
3.6
8.9
17.5
In order to analyze this aspect in more detail, we have computed the disagreement rates for different
values of alpha (? = 0.999, 0.995, 0.99, 0.95). In Figure 1 the relation between the target 1 ? ? and
the actual disagreement rate is presented. A diagonal solid line marks the expected upper limit for
the disagreement. The results for SIBA, HYPER and for the case of using a fixed number of trees
for all instances (FIXED) (and equal to the average number of trees used by HYPER in those tasks)
7
are presented in these plots. This last case (FIXED) can be seen as a stochastic approximation to
the prediction of the whole ensemble. From these plots, we observe that the results for HYPER are
very close to the expected disagreement rates for cases in which the prior is approximately uniform
(Sonar), and for cases in which the prior is non-uniform (Votes). As expected, the results of SIBA are
close to the target only for the case of approximately uniform prior (Sonar). Finally, when a stochastic
approximation is used (FIXED) the disagreement rates are clearly above the target threshold given by
?. From these results we conclude that the proposed model provides a more accurate description
of the voting process used to compute the prediction of the ensemble. This means that taking into
account the prior distribution of possible vote outcomes, P(T), is important to obtain disagreement
rates that are closer to the threshold established.
Finally, in Table 2, we present the average number of trees used by Random Forest (RF? ), the SIBA
method,the proposed method using non-parametric priors (HYPER), and the expected average of
the number of trees to be queried in HYPER using Monte Carlo sampling (MC Estim). Note that
the number of trees used by RF? is not necessary T = 101: the voting process is halted when the
remaining (unknown) predictions cannot alter the decision of the ensemble. The number of trees
given in RF? is the same as the trees that SIBA or HYPER would use when ? = 100%. Finally, the
last three columns of Table 2 display the speed-up rate of the partial ensembles with respect to the
full ensemble of size T = 101. From this table it is clear that HYPER reduces the number of queried
classifiers with respect to SIBA in most of the tasks investigated. In addition, using only training data,
the Monte Carlo estimations of the average number of trees are very precise. The largest average
difference between this estimation and HYPER is 2.4 trees for German and Threenorm. The speed-up
rate of HYPER with respect to the full ensemble is remarkable: from 2.8 times faster for Threenorm
to 101 times faster in Mushroom. This dataset can be used to illustrate the benefits of using the prior
distribution. For this problem, most classifiers agree in their predictions. HYPER takes advantage of
this prior knowledge and queries only one classifier to cast the final decision. In this problem, the
chances that the prediction of a single classifier, and the prediction of the complete ensemble are
different, are below 1%. Similar behavior (but not as extreme) is observed in Breast and Votes.
4
Conclusions
In this work, we present an intuitive, rigorous mathematical description of the voting process in an
ensemble of classifiers: For a given an instance, the process is equivalent to extracting marbles (the
individual classifiers), without replacement, from a bag that contains a known number of marbles, but
whose color (class label prediction) distribution is unknown. In addition, we show that for the specific
case of a uniform prior distribution of class votes this process is equivalent to the one developed in
[14]. In the current description, which does not assume a uniform distribution prior for the class
votes, the hypergeometric distribution plays a central role.
The results of this statistical description are then used to design a dynamic ensemble pruning method,
with the goal of speeding up predictions in the test phase. For a given instance, it is possible to
compute the probability that the the partial decision made on the basis of the known votes (i.e., the
class label predictions of the subset of classifiers that have been queried) and the final ensemble
decision coincide. If this probability is above a specified threshold, sufficiently close to 1, a reliable
estimate of the class label that the complete ensemble would predict can be made on the basis of
the known votes. The effectiveness of this dynamic ensemble pruning method is illustrated using
random forests. The prior distribution of class votes is estimated using out-of-bag data. As a result of
incorporating this problem-specific knowledge in the statistical analysis of the voting process, the
differences between the predictions of the dynamically pruned ensemble and the complete ensemble
are closer to the specified threshold than when a uniform distribution is assumed, as in SIBA [14]. In
the empirical evaluation performed, this dynamic ensemble pruning algorithm consistently yields
improvements of classification speed over SIBA without a significant deterioration of accuracy.
Finally, the statistical model proposed is used to provide an accurate estimate of the average number
of individual classifier predictions that are needed to reach a stable ensemble prediction.
Acknowledgments
The authors acknowledge financial support from the Comunidad de Madrid (project CASI-CAMCM S2013/ICE-2845), and from the Spanish Ministerio de Econom?a y Competitividad (projects
TIN2013-42351-P and TIN2015-70308-REDT).
8
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5,660 | 6,121 | Dense Associative Memory for Pattern Recognition
Dmitry Krotov
Simons Center for Systems Biology
Institute for Advanced Study
Princeton, USA
krotov@ias.edu
John J. Hopfield
Princeton Neuroscience Institute
Princeton University
Princeton, USA
hopfield@princeton.edu
Abstract
A model of associative memory is studied, which stores and reliably retrieves many
more patterns than the number of neurons in the network. We propose a simple
duality between this dense associative memory and neural networks commonly used
in deep learning. On the associative memory side of this duality, a family of models
that smoothly interpolates between two limiting cases can be constructed. One limit
is referred to as the feature-matching mode of pattern recognition, and the other
one as the prototype regime. On the deep learning side of the duality, this family
corresponds to feedforward neural networks with one hidden layer and various
activation functions, which transmit the activities of the visible neurons to the
hidden layer. This family of activation functions includes logistics, rectified linear
units, and rectified polynomials of higher degrees. The proposed duality makes
it possible to apply energy-based intuition from associative memory to analyze
computational properties of neural networks with unusual activation functions ? the
higher rectified polynomials which until now have not been used in deep learning.
The utility of the dense memories is illustrated for two test cases: the logical gate
XOR and the recognition of handwritten digits from the MNIST data set.
1
Introduction
Pattern recognition and models of associative memory [1] are closely related. Consider image
classification as an example of pattern recognition. In this problem, the network is presented with an
image and the task is to label the image. In the case of associative memory the network stores a set of
memory vectors. In a typical query the network is presented with an incomplete pattern resembling,
but not identical to, one of the stored memories and the task is to recover the full memory. Pixel
intensities of the image can be combined together with the label of that image into one vector [2],
which will serve as a memory for the associative memory. Then the image itself can be thought of
as a partial memory cue. The task of identifying an appropriate label is a subpart of the associative
memory reconstruction. There is a limitation in using this idea to do pattern recognition. The standard
model of associative memory works well in the limit when the number of stored patterns is much
smaller than the number of neurons [1], or equivalently the number of pixels in an image. In order
to do pattern recognition with small error rate one would need to store many more memories than
the typical number of pixels in the presented images. This is a serious problem. It can be solved by
modifying the standard energy function of associative memory, quadratic in interactions between the
neurons, by including in it higher order interactions. By properly designing the energy function (or
Hamiltonian) for these models with higher order interactions one can store and reliably retrieve many
more memories than the number of neurons in the network.
Deep neural networks have proven to be useful for a broad range of problems in machine learning
including image classification, speech recognition, object detection, etc. These models are composed
of several layers of neurons, so that the output of one layer serves as the input to the next layer. Each
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
neuron calculates a weighted sum of the inputs and passes the result through a non-linear activation
function. Traditionally, deep neural networks used activation functions such as hyperbolic tangents or
logistics. Learning the weights in such networks, using a backpropagation algorithm, faced serious
problems in the 1980s and 1990s. These issues were largely resolved by introducing unsupervised
pre-training, which made it possible to initialize the weights in such a way that the subsequent
backpropagation could only gently move boundaries between the classes without destroying the
feature detectors [3, 4]. More recently, it was realized that the use of rectified linear units (ReLU)
instead of the logistic functions speeds up learning and improves generalization [5, 6, 7]. Rectified
linear functions are usually interpreted as firing rates of biological neurons. These rates are equal
to zero if the input is below a certain threshold and linearly grow with the input if it is above the
threshold. To mimic biology the output should be small or zero if the input is below the threshold, but
it is much less clear what the behavior of the activation function should be for inputs exceeding the
threshold. Should it grow linearly, sub-linearly, or faster than linearly? How does this choice affect
the computational properties of the neural network? Are there other functions that would work even
better than the rectified linear units? These questions to the best of our knowledge remain open.
This paper examines these questions through the lens of associative memory. We start by discussing
a family of models of associative memory with large capacity. These models use higher order (higher
than quadratic) interactions between the neurons in the energy function. The associative memory
description is then mapped onto a neural network with one hidden layer and an unusual activation
function, related to the Hamiltonian. We show that by varying the power of interaction vertex in
the energy function (or equivalently by changing the activation function of the neural network) one
can force the model to learn representations of the data either in terms of features or in terms of
prototypes.
2
Associative memory with large capacity
The standard model of associative memory [1] uses a system of N binary neurons, with values ?1. A
configuration of all the neurons is denoted by a vector i . The model stores K memories, denoted by
?i? , which for the moment are also assumed to be binary. The model is defined by an energy function,
which is given by
E=
N
1 X
2 i,j=1
i Tij j ,
Tij =
K
X
?i? ?j? ,
(1)
?=1
and a dynamical update rule that decreases the energy at every update. The basic problem is the
following: when presented with a new pattern the network should respond with a stored memory
which most closely resembles the input.
There has been a large amount of work in the community of statistical physicists investigating
the capacity of this model, which is the maximal number of memories that the network can store
and reliably retrieve. It has been demonstrated [1, 8, 9] that in case of random memories this
maximal value is of the order of K max ? 0.14N . If one tries to store more patterns, several
neighboring memories in the configuration space will merge together producing a ground state of
the Hamiltonian (1), which has nothing to do with any of the stored memories. By modifying the
Hamiltonian (1) in a way that removes second order correlations between the stored memories, it is
possible [10] to improve the capacity to K max = N .
The mathematical reason why the model (1) gets confused when many memories are stored is that
several memories produce contributions to the energy which are of the same order. In other words the
energy decreases too slowly as the pattern approaches a memory in the configuration space. In order
to take care of this problem, consider a modification of the standard energy
E=
K
X
F ?i?
i
(2)
?=1
In this formula F (x) is some smooth function (summation over index i is assumed). The computational capabilities of the model will be illustrated for two cases. First, when F (x) = xn (n is
an integer number), which is referred to as a polynomial energy function. Second, when F (x) is a
2
rectified polynomial energy function
?
xn , x 0
(3)
0, x < 0
In the case of the polynomial function with n = 2 the network reduces to the standard model of
associative memory [1]. If n > 2 each term in (2) becomes sharper compared to the n = 2 case, thus
more memories can be packed into the same configuration space before cross-talk intervenes.
F (x) =
Having defined the energy function one can derive an iterative update rule that leads to decrease of
the energy. We use asynchronous updates flipping one unit at a time. The update rule is:
?X
K ? ?
?
X ? (t) ?
X ? (t) ??
(t+1)
?
?
=
Sign
F
?
+
?
F
?
+
?j j
,
(4)
i
i
j j
i
?=1
j6=i
j6=i
The argument of the sign function is the difference of two energies. One, for the configuration with
all but the i-th units clumped to their current states and the i-th unit in the ?off? state. The other one
for a similar configuration, but with the i-th unit in the ?on? state. This rule means that the system
updates a unit, given the states of the rest of the network, in such a way that the energy of the entire
configuration decreases. For the case of polynomial energy function a very similar family of models
was considered in [11, 12, 13, 14, 15, 16]. The update rule in those models was based on the induced
magnetic fields, however, and not on the difference of energies. The two are slightly different due to
the presence of self-coupling terms. Throughout this paper we use energy-based update rules.
How many memories can model (4) store and reliably retrieve? Consider the case of random patterns,
so that each element of the memories is equal to ?1 with equal probability. Imagine that the system
is initialized in a state equal to one of the memories (pattern number ?). One can derive a stability
criterion, i.e. the upper bound on the number of memories such that the network stays in that initial
state. Define the energy difference between the initial state and the state with spin i flipped
K ?
K ?
?n X
?n
X
X
X
E=
?i? ?i? +
?j? ?j?
?i? ?i? +
?j? ?j? ,
?=1
?=1
j6=i
j6=i
where the polynomial energy function is used. This quantity has a mean h Ei = N n (N 2)n ?
2nN n 1 , which comes from the term with ? = ?, and a variance (in the limit of large N )
?2 = ?n (K 1)N n 1 ,
where ?n = 4n2 (2n 3)!!
The i-th bit becomes unstable when the magnitude of the fluctuation exceeds the energy gap h Ei
and the sign of the fluctuation is opposite to the sign of the energy gap. Thus the probability that the
state of a single neuron is unstable (in the limit when both N and K are large, so that the noise is
effectively gaussian) is equal to
r
Z1
Nn 1
x2
dx
(2n 3)!! K
p
Perror =
e 2?2 ?
e 2K(2n 3)!!
n
1
2?
N
2??2
h Ei
Requiring that this probability is less than a small value, say 0.5%, one can find the upper limit on
the number of patterns that the network can store
K max = ?n N n 1 ,
(5)
where ?n is a numerical constant, which depends on the (arbitrary) threshold 0.5%. The case
n = 2 corresponds to the standard model of associative memory and gives the well known result
K = 0.14N . For the perfect recovery of a memory (Perror < 1/N ) one obtains
1
Nn 1
max
Kno
(6)
errors ?
2(2n 3)!! ln(N )
For higher powers n the capacity rapidly grows with N in a non-linear way, allowing the network
to store and reliably retrieve many more patterns than the number of neurons that it has, in accord1
with [13, 14, 15, 16]. This non-linear scaling relationship between the capacity and the size of the
network is the phenomenon that we exploit.
1
The n-dependent coefficient in (6) depends on the exact form of the Hamiltonian and the update rule.
References [13, 14, 15] do not allow repeated indices in the products over neurons in the energy function,
therefore obtain a different coefficient. In [16] the Hamiltonian coincides with ours, but the update rule is
different, which, however, results in exactly the same coefficient as in (6).
3
We study a family of models of this kind as a function of n. At small n many terms contribute to the
sum over ? in (2) approximately equally. In the limit n ! 1 the dominant contribution to the sum
comes from a single memory, which has the largest overlap with the input. It turns out that optimal
computation occurs in the intermediate range.
3
The case of XOR
The case of XOR is elementary, yet instructive. It is presented here for three reasons. First, it illustrates
the construction (2) in this simplest case. Second, it shows that as n increases, the computational
capabilities of the network also increase. Third, it provides the simplest example of a situation in
which the number of memories is larger than the number of neurons, yet the network works reliably.
The problem is the following: given two inputs x and y produce an output z such that the truth table
x
-1
-1
1
1
y
-1
1
-1
1
z
-1
1
1
-1
is satisfied. We will treat this task as an associative memory problem and will simply embed the
four examples of the input-output triplets x, y, z in the memory. Therefore the network has N = 3
identical units: two of which will be used for the inputs and one for the output, and K = 4 memories
?i? , which are the four lines of the truth table. Thus, the energy (2) is equal to
En (x, y, z) =
x
y
z
n
x+y+z
n
x
y+z
n
x+y
z
n
,
(7)
where the energy function is chosen to be a polynomial of degree n. For odd n, energy (7) is an odd
function of each of its arguments, En (x, y, z) = En (x, y, z). For even n, it is an even function.
For n = 1 it is equal to zero. Thus, if evaluated on the corners of the cube x, y, z = ?1, it reduces to
8
n=1
<0,
En (x, y, z) = Cn ,
(8)
n = 2, 4, 6, ...
:
Cn xyz, n = 3, 5, 7, ...,
where coefficients Cn denote numerical constants.
In order to solve the XOR problem one can present to the network an ?incomplete pattern? of inputs
(x, y) and let the output z adjust to minimize the energy of the three-spin configuration, while holding
the inputs fixed. The network clearly cannot solve this problem for n = 1 and n = 2, since the energy
does not depend on the spin configuration. The case n = 2 is the standard model of associative
memory. It can also be thought of as a linear perceptron, and the inability to solve this problem
represents the well known statement [17] that linear perceptrons cannot compute XOR without hidden
neurons. The case of odd n 3 provides an interesting solution. Given two inputs, x and y, one can
choose the output z that minimizes the energy. This leads to the update rule
?
?
?
?
z = Sign En (x, y, 1) En (x, y, +1) = Sign
xy
Thus, in this simple case the network is capable of solving the problem for higher odd values of n,
while it cannot do so for n = 1 and n = 2. In case of rectified polynomials, a similar construction
solves the problem for any n 2. The network works well in spite of the fact that K > N .
4
An example of a pattern recognition problem, the case of MNIST
The MNIST data set is a collection of handwritten digits, which has 60000 training examples and
10000 test images. The goal is to classify the digits into 10 classes. The visible neurons, one for each
pixel, are combined together with 10 classification neurons in one vector that defines the state of
the network. The visible part of this vector is treated as an ?incomplete? pattern and the associative
memory is allowed to calculate a completion of that pattern, which is the label of the image.
Dense associative memory (2) is a recurrent network in which every neuron can be updated multiple
times. For the purposes of digit classification, however, this model will be used in a very limited
4
capacity, allowing it to perform only one update of the classification neurons. The network is
initialized in the state when the visible units vi are clamped to the intensities of a given image and the
classification neurons are in the off state x? = 1 (see Fig.1A). The network is allowed to make
one update of the classification neurons, while keeping the visible units clamped, to produce the
output c? . The update rule is similar to (4) except that the sign is replaced by the continuous function
g(x) = tanh(x)
? X
K ? ?
N
N
?
?
??
X
X
X
X
c? = g
F
??? x? +
??x +
?i? vi
F ??? x? +
??x +
?i? vi
, (9)
?=1
i=1
6=?
i=1
6=?
where parameter regulates the slope of g(x). The proposed digit class is given by the number
of a classification neuron producing the maximal output. Throughout this section the rectified
polynomials (3) are used as functions F . To learn effective memories for use in pattern classification,
an objective function is defined (see Appendix A in Supplemental), which penalizes the discrepancy
vi
vi
c?
x?
179-312 epochs
2
n=2
1.9
1.8
1.7
1.6
1.5
n=3
1.9
1.8
1.7
1.6
1.5
1.4
0
158-262 epochs
2
test
error,error,
test set %
B
test
error,error,
test set %
A
1.4
500
1000
1500
Epochs
2000
2500
number of epochs
3000
0
500
1000
1500
2000
2500
numberEpochs
of epochs
3000
Figure 1: (A) The network has N = 28 ? 28 = 784 visible neurons and Nc = 10 classification neurons.
The visible units are clamped to intensities of pixels (which is mapped on the segment [ 1, 1]), while the
classification neurons are initialized in the state x? and then updated once to the state c? . (B) Behavior of the
error on the test set as training progresses. Each curve corresponds to a different combination of hyperparameters
from the optimal window, which was determined on the validation set. The arrows show the first time when the
error falls below a 2% threshold. All models have K = 2000 memories (hidden units).
between the output c? and the target output. This objective function is then minimized using a
backpropagation algorithm. The learning starts with random memories drawn from a Gaussian
?
distribution. The backpropagation algorithm then finds a collection of K memories ?i,?
, which
minimize the classification error on the training set. The memories are normalized to stay within the
?
1 ? ?i,?
? 1 range, absorbing their overall scale into the definition of the parameter .
The performance of the proposed classification framework is studied as a function of the power n.
The next section shows that a rectified polynomial of power n in the energy function is equivalent
to the rectified polynomial of power n 1 used as an activation function in a feedforward neural
network with one hidden layer of neurons. Currently, the most common choice of activation functions
for training deep neural networks is the ReLU, which in our language corresponds to n = 2 for
the energy function. Although not currently used to train deep networks, the case n = 3 would
correspond to a rectified parabola as an activation function. We start by comparing the performances
of the dense memories in these two cases.
The performance of the network depends on n and on the remaining hyperparameters, thus the hyperparameters should be optimized for each value of n. In order to test the variability of performances
for various choices of hyperparameters at a given n, a window of hyperparameters for which the
network works well on the validation set (see the Appendix A in Supplemental) was determined.
Then many networks were trained for various choices of the hyperparameters from this window to
evaluate the performance on the test set. The test errors as training progresses are shown in Fig.1B.
While there is substantial variability among these samples, on average the cluster of trajectories for
n = 3 achieves better results on the test set than that for n = 2. These error rates should be compared
with error rates for backpropagation alone without the use of generative pretraining, various kinds
of regularizations (for example dropout) or adversarial training, all of which could be added to our
construction if necessary. In this class of models the best published results are all2 in the 1.6% range
[18], see also controls in [19, 20]. This agrees with our results for n = 2. The n = 3 case does
slightly better than that as is clear from Fig.1B, with all the samples performing better than 1.6%.
2
Although there are better results on pixel permutation invariant task, see for example [19, 20, 21, 22].
5
Higher rectified polynomials are also faster in training compared to ReLU. For the n = 2 case, the
error crosses the 2% threshold for the first time during training in the range of 179-312 epochs. For
the n = 3 case, this happens earlier on average, between 158-262 epochs. For higher powers n
this speed-up is larger. This is not a huge effect for a small dataset such as MNIST. However, this
speed-up might be very helpful for training large networks on large datasets, such as ImageNet. A
similar effect was reported earlier for the transition between saturating units, such as logistics or
hyperbolic tangents, to ReLU [7]. In our family of models that result corresponds to moving from
n = 1 to n = 2.
Feature to prototype transition
How does the computation performed by the neural network change as n varies? There are two
extreme classes of theories of pattern recognition: feature-matching and formation of a prototype.
According to the former, an input is decomposed into a set of features, which are compared with
those stored in the memory. The subset of the stored features activated by the presented input is then
interpreted as an object. One object has many features; features can also appear in more than one
object. The prototype theory provides an alternative approach, in which objects are recognized as a
whole. The prototypes do not necessarily match the object exactly, but rather are blurred abstract
n=3
n = 20
n = 30
n=2
1
256
256
256
256
256
192
192
192
192
192
128
128
128
128
128
64
64
64
64
64
0.5
0
0.5
1
n=2
n=3
n=20
n=30
40
30
20
10
0
0
1
2
3
4
5
6
7
8
9
number of strongly positively driven RU
number of RU with
???
10000
number
testimages
images
numberof
of test
percent of active memories
percent of memories, %
50
8000
6000
4000
2000
n=2
n=3
n=20
n=30
errortest = 1.51%
errortest = 1.44%
errortest = 1.61%
errortest = 1.80%
0
1 2 3 4 5 6 7 8 9 10 11 12
number of memories strongly contributing to the correct RU
10
number of memories making the decision
> 0.99
Figure 2: We show 25 randomly selected memories (feature detectors) for four networks, which use rectified polynomials of degrees n = 2, 3, 20, 30 as the energy function. The magnitude of a memory element
corresponding to each pixel is plotted in the location of that pixel, the color bar explains the color code. The
histograms at the bottom are explained in the text. The error rates refer to the particular four samples used in this
figure. RU stands for recognition unit.
representations which include all the features that an object has. We argue that the computational
models proposed here describe feature-matching mode of pattern recognition for small n and the
prototype regime for large n. This can be anticipated from the sharpness of contributions that each
memory makes to the total energy (2). For large n the function F (x) peaks much more sharply
around each memory compared to the case of small n. Thus, at large n all the information about a
digit must be written in only one memory, while at small n this information can be distributed among
several memories. In the case of intermediate n some learned memories behave like features while
others behave like prototypes. These two classes of memories work together to model the data in an
efficient way.
The feature to prototype transition is clearly seen in memories shown in Fig.2. For n = 2 or 3
each memory does not look like a digit, but resembles a pattern of activity that might be useful for
recognizing several different digits. For n = 20 many of the memories can be recognized as digits,
which are surrounded by white margins representing elements of memories having approximately
zero values. These margins describe the variability of thicknesses of lines of different training
examples and mathematically mean that the energy (2) does not depend on whether the corresponding
pixel is on or off. For n = 30 most of the memories represent prototypes of whole digits or large
portions of digits, with a small admixture of feature memories that do not resemble any digit.
6
The feature to prototype transition can be visualized by showing the feature detectors in situations
when there is a natural ordering of pixels. Such ordering exists in images, for example. In general
situations, however, there is no preferred permutation of visible neurons that would reveal this
structure (e.g. in the case of genomic data). It is therefore useful to develop a measure that permits a
distinction to be made between features and prototypes in the absence of such visual space. Towards
the end of training most of the recognition connections ??? are approximately equal to ?1. One
can choose an arbitrary cutoff, and count the number of recognition connections that are in the ?on?
state (??? = +1) for each memory. The distribution function of this number is shown on the left
histogram in Fig.2. Intuitively, this quantity corresponds to the number of different digit classes
that a particular memory votes for. At small n, most of the memories vote for three to five different
digit classes, a behavior characteristic of features. As n increases, each memory specializes and
votes for only a single class. In the case n = 30, for example, more than 40% of memories vote for
only one class, a behavior characteristic of prototypes. A second way to see the feature to prototype
transition is to look at the number of memories which make large contributions to the classification
decision (right histogram in Fig.2). For each test image one can find the memory that makes the
largest contribution to the energy gap, which is the sum over ? in (9). Then one can count the number
of memories that contribute to the gap by more than 0.9 of this largest contribution. For small n,
there are many memories that satisfy this criterion and the distribution function has a long tail. In
this regime several memories are cooperating with each other to make a classification decision. For
n = 30, however, more than 8000 of 10000 test images do not have a single other memory that would
make a contribution comparable with the largest one. This result is not sensitive to the arbitrary choice
(0.9) of the cutoff. Interestingly, the performance remains competitive even for very large n ? 20
(see Fig.2) in spite of the fact that these networks are doing a very different kind of computation
compared with that at small n.
5
Relationship to a neural network with one hidden layer
In this section we derive a simple duality between the dense associative memory and a feedforward
neural network with one layer of hidden neurons. In other words, we show that the same computational
model has two very different descriptions: one in terms of associative memory, the other one in terms
c?
vi
c?
vi
x? =
g
h?
f
"
vi
Figure 3: On the left a feedforward neural network with one layer of hidden neurons. The states of the visible
units are transformed to the hidden neurons using a non-linear function f , the states of the hidden units are
transformed to the output layer using a non-linear function g. On the right the model of dense associative
memory with one step update (9). The two models are equivalent.
of a network with one layer of hidden units. Using this correspondence one can transform the family
of dense memories, constructed for different values of power n, to the language of models used in
deep learning. The resulting neural networks are guaranteed to inherit computational properties of
the dense memories such as the feature to prototype transition.
The construction is very similar to (9), except that the classification neurons are initialized in the state
when all of them are equal to ", see Fig.3. In the limit " ! 0 one can expand the function F in (9)
so that the dominant contribution comes from the term linear in ". Then
K
N
K
K
h X
?X
?
i
hX
i
hX
i
c? ? g
F0
?i? vi ( 2??? x? ) = g
??? F 0 ?i? vi = g
??? f ?i? vi , (10)
?=1
?=1
i=1
?=1
where the parameter is set to = 1/(2") (summation over the visible index i is assumed). Thus,
the model of associative memory with one step update is equivalent to a conventional feedforward
neural network with one hidden layer provided that the activation function from the visible layer to
the hidden layer is equal to the derivative of the energy function
f (x) = F 0 (x)
(11)
7
The visible part of each memory serves as an incoming weight to the hidden layer, and the recognition
part of the memory serves as an outgoing weight from the hidden layer. The expansion used in (10)
Nc
N
P
P
is justified by a condition
?i? vi
??? x? , which is satisfied for most common problems, and
i=1
?=1
is simply a statement that labels contain far less information than the data itself3 .
From the point of view of associative memory, the dominant contribution shaping the basins of
attraction comes from the low energy states. Therefore mathematically it is determined by the
asymptotics of the activation function f (x), or the energy function F (x), at x ! 1. Thus different
activation functions having similar asymptotics at x ! 1 should fall into the same universality class
and should have similar computational properties. In the table below we list some common activation
activation function
energy function
n
f (x) = tanh(x)
F (x) = ln cosh(x) ? x, at x ! 1 1
f (x) = logistic function
F (x) = ln 1 + ex ? x, at x ! 1
1
f (x) =ReLU
F (x) ? x2 , at x ! 1
2
f (x) = RePn 1
F (x) = RePn
n
functions used in models of deep learning, their associative memory counterparts and the power n
which determines the asymptotic behavior of the energy function at x ! 1.The results of section 4
suggest that for not too large n the speed of learning should improve as n increases. This is consistent
with the previous observation that ReLU are faster in training than hyperbolic tangents and logistics
[5, 6, 7]. The last row of the table corresponds to rectified polynomials of higher degrees. To the
best of our knowledge these activation functions have not been used in neural networks. Our results
suggest that for some problems these higher power activation functions should have even better
computational properties than the rectified liner units.
6
Discussion and conclusions
What is the relationship between the capacity of the dense associative memory, calculated in section 2,
and the neural network with one step update that is used for digit classification? Consider the limit of
very large in (9), so that the hyperbolic tangent is approximately equal to the sign function, as in (4).
In the limit of sufficiently large n the network is operating in the prototype regime. The presented
image places the initial state of the network close to a local minimum of energy, which corresponds
to one of the prototypes. In most cases the one step update of the classification neurons is sufficient
to bring this initial state to the nearest local minimum, thus completing the memory recovery. This is
true, however, only if the stored patterns are stable and have basins of attraction around them of at
least the size of one neuron flip, which is exactly (in the case of random patterns) the condition given
by (6). For correlated patterns the maximal number of stored memories might be different from (6),
however it still rapidly increases with increase of n. The associative memory with one step update (or
the feedforward neural network) is exactly equivalent to the full associative memory with multiple
updates in this limit. The calculation with random patterns thus theoretically justifies the expectation
of a good performance in the prototype regime.
To summarize, this paper contains three main results. First, it is shown how to use the general
framework of associative memory for pattern recognition. Second, a family of models is constructed
that can learn representations of the data in terms of features or in terms of prototypes, and that
smoothly interpolates between these two extreme regimes by varying the power of interaction vertex.
Third, there exists a simple duality between a one step update version of the associative memory
model and a feedforward neural network with one layer of hidden units and an unusual activation
function. This duality makes it possible to propose a class of activation functions that encourages
the network to learn representations of the data with various proportions of features and prototypes.
These activation functions can be used in models of deep learning and should be more effective than
the standard choices. They allow the networks to train faster. We have also observed an improvement
of generalization ability in networks trained with the rectified parabola activation function compared
to the ReLU for the case of MNIST. While these ideas were illustrated using the simplest architecture
of the neural network with one layer of hidden units, the proposed activation functions can also be
used in multilayer architectures. We did not study various regularizations (weight decay, dropout,
etc), which can be added to our construction. The performance of the model supplemented with these
regularizations, as well as performance on other common benchmarks, will be reported elsewhere.
3
A relationshp similar to (11) was discussed in [23, 24] in the context of autoencoders.
8
References
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9
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5,661 | 6,122 | Cooperative Graphical Models
Josip Djolonga
Dept. of Computer Science, ETH Z?urich
josipd@inf.ethz.ch
Stefanie Jegelka
CSAIL, MIT
stefje@mit.edu
Sebastian Tschiatschek
Dept. of Computer Science, ETH Z?urich
stschia@inf.ethz.ch
Andreas Krause
Dept. of Computer Science, ETH Z?urich
krausea@inf.ethz.ch
Abstract
We study a rich family of distributions that capture variable interactions significantly more expressive than those representable with low-treewidth or pairwise
graphical models, or log-supermodular models. We call these cooperative graphical models. Yet, this family retains structure, which we carefully exploit for
efficient inference techniques. Our algorithms combine the polyhedral structure of
submodular functions in new ways with variational inference methods to obtain
both lower and upper bounds on the partition function. While our fully convex upper
bound is minimized as an SDP or via tree-reweighted belief propagation, our lower
bound is tightened via belief propagation or mean-field algorithms. The resulting
algorithms are easy to implement and, as our experiments show, effectively obtain
good bounds and marginals for synthetic and real-world examples.
1
Introduction
Probabilistic inference in high-order discrete graphical models has
been an ongoing computational challenge, and all existing methods
rely on exploiting specific structure: either low-treewidth or pairwise
graphical models, or functional properties of the distribution such as
log-submodularity. Here, we aim to compute approximate marginal
probabilities in complex models with long-range variable interactions
that do not possess any of these properties. Instead, we exploit a
combination of structural and functional properties in new ways.
X1,1
X1,2
X1,3
X1,4
X2,1
X2,2
X2,3
X2,4
X3,1
X3,2
X3,3
X3,4
The classical example of image segmentation may serve to motivate
our family of models: we would like to estimate a posterior marginal Figure 1: Example cooperadistribution over k labels for each pixel in an image. A common tive model. Edge colors inapproach uses Conditional Random Fields on a pixel neighborhood dicate the edge cluster. Dotgraph with pairwise potentials that encourage neighboring pixels to ted edges are cut under the
take on the same label. From the perspective of the graph, this model current assignment.
prefers configurations with few edges cut, where an edge is said to
be cut if its endpoints have different labels. Such cut-based models,
however, short-cut elongated structures (e.g. tree branches), a problem known as shrinking bias.
Jegelka and Bilmes [1] hence replace the bias towards short cuts (boundaries) by a bias towards
configurations with certain higher-order structure: the cut edges occur at similar-looking pixel pairs.
They group the graph edges into clusters (based on, say, color gradients across the endpoints),
observing that the true object boundary is captured by few of these clusters. To encourage cutting
edges from few clusters, the cost of cutting an edge decreases as more edges in its cluster are cut.
In short, the edges ?cooperate?. In Figure 1, each pixel takes on one of two labels (colors), and cut
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
edges are indicated by dotted lines. The current configuration cuts three red edges and one blue edge,
and has lower probability than the configuration that swaps X3,1 to gray, cutting only red edges. Such
a model can?
be implemented by an energy (cost) h(#red edges cut) + h(#blue edges cut), where
e.g. h(u) = u. Similar cooperative models can express a preference for shapes [2].
While being expressive, such models are computationally very challenging: the nonlinear function
on pairs of variables (edges) is equivalent to a graphical model of extremely high order (up to the
number of variables). Previous work hence addressed only MAP inference [3, 4]; the computation of
marginals and partition functions was left as an open problem. In this paper, we close this gap, even
for a larger family of models.
We address models, which we call cooperative graphical models, that are specified by an undirected
graph G = (V, E): each node i ? V is associated with a random variable Xi that takes values in
X = {1, 2, . . . , k}. To each vertex i ? V and edge {i, j}, we attach a potential function ?i : X ? R
and ?i,j : X 2 ? R, respectively. Our distribution is then
?
?
X
X
1
P (x) = exp ??
?i (xi ) +
?i,j (xi , xj ) + f (y(x)) ? ?(x),
(1)
Z
i?V
{i,j}?E
where we call y : X n ? {0, 1}E the disagreement variable1 , defined as yi,j = Jxi 6= xj K. The term
? : X n ? R?0 is the base-measure and allows to encode constraints, e.g., conditioning on some
variables. With f ? 0 we obtain a Markov random field.
Probabilistic inference in our model class (1) is very challenging, since we make no factorization
assumption about f . One solution would be to encode P (x) as a log-linear model via a new
variable z ? {0, 1}E and constraints ?(x, z) = Jy(x) = zK, but this in general requires computing
exponential-sized sufficient statistics from z. In contrast, we make one additional key assumption
that will enable the development of efficiently computable variational lower and upper bounds: we
henceforth assume that f : {0, 1}E ? R is submodular, i.e., it satisfies
f (min(y, y0 )) + f (max(y, y0 )) ? f (y) + f (y0 )
for all y, y0 ? {0, 1}E ,
where the min and max operations are taken element-wise. For example, the pairwise potentials ?i,j
are submodular if ?i,j (0, 0) + ?i,j (1, 1) ? ?i,j (0, 1) + ?i,j (1, 0). In our introductory example, f is
submodular if h is concave. As opposed to [3], we do not assume that f is monotone increasing.
Importantly, even if f is submodular, P (x) neither has low treewdith, nor is its logarithm sub- or
supermodular in x, properties that have commonly been exploited for inference.
Contributions. We make the following contributions: (1) We introduce a new family of probabilistic models that can capture rich non-submodular interactions, while still admitting efficient
inference. This family includes pairwise and certain higher-order graphical models, cooperative cuts
[1], and other, new models. We develop new inference methods for these models; in particular, (2)
upper bounds that are amenable to convex optimization, and (3) lower bounds that we optimize with
traditional variational methods. Finally, we demonstrate the efficacy of our methods empirically.
1.1
Related work
Maximum-a-posteriori (MAP). Computing the mode of (1) for binary models is also known as
the cooperative cut problem, and has been analyzed for the case when both the pairwise interactions
?i,j are submodular and f is monotone [1]. While the general problem is NP-hard, it can be solved if
f is defined by a piecewise linear concave function [4].
Variational inference. Since computing marginal probabilities for (1) is #P-hard even for pairwise
models (when f ? 0) [5, 6], we revert to approximate inference. Variational inference methods for
discrete pairwise models have been studied extensively; a comprehensive overview may be found
in [7]. We will build on a selection of techniques that we discuss in the next section. Most existing
methods focus on pairwise models (f ? 0), and many scale exponentially with the size of the largest
factor, which is infeasible for our cooperative models. Some specialized tractable inference methods
exist for higher-order models [8, 9], but they do not apply to our family of models (1).
1
The results presented in this paper can be easily extended to arbitrary binary-valued functions y(x).
2
Log-supermodular models. A related class of relatively tractable models are distributions P (x) =
1
Z exp(?g(x)) for some submodular function g; Djolonga and Krause [10] showed variational
inference methods for those models. However, our models are not log-supermodular. While [10]
also obtain upper and lower bounds, we need different optimization techniques, and also different
polytopes. In fact, submodular and multi-class submodular [11] settings are a strict subset of ours:
the function g(x) can be expressed via an auxiliary variable z ? {0, 1} that is fixed to zero using
?(x, z) = Jz = 0K. We then set f (y(x, z)) = g(x1 6= z, x2 6= z, . . . , xn 6= z).
2
Notation and Background
Throughout this paper, we have n variables in a graph of m edges, and the potentials ?i and ?i,j are
stored in a vector ?. The characteristic vector (or indicator vector) 1A of a set A is the binary vector
which contains 1 in the positions corresponding to elements in A, and zeros elsewhere. Moreover,
the vector of all ones is 1, and the neighbours of i ? V are denoted by ?(i) ? V .
Submodularity. We assume that f in Eqn. (1) is submodular. Occasionally (in Sec. 4 and 5, where
stated), we assume that f is monotone: for any y and y0 in {0, 1}E such that y ? y0 coordinate-wise,
it holds that f (y) ? f (y0 ). When defining the inference schemes, we make use of two polytopes
associated with f . First, the base polytope of a submodular function f is
B(f ) = {g ? Rm | ?y ? {0, 1}E : gT y ? f (y)} ? {g ? Rm | gT 1 = f (1)}.
Although B(f ) is defined by exponentially many inequalities, an influential result [12] states that
it is tractable: we can optimize linear functions over B(f ) in time O(m log m + mF ), where F is
the time complexity of evaluating f . This algorithm is part of our scheme in Figure 2. Moreover,
as a result of this (linear) tractability, it is possible to compute orthogonal projections onto B(f ).
Projection is equivalent to the minimum norm point problem [13]. While the general projection
problem has a high degree polynomial time complexity, there are many very commonly used models
that admit practically fast projections [14, 15, 16].
The second polytope is the upper submodular polyhedron of f [17], defined as
U(f ) = {(g, c) ? Rm+1 | ?y ? {0, 1}E : gT y + c ? f (y)}.
Unfortunately, U(f ) is not as tractable as B(f ): even checking membership in U(f ) is hard [17].
However, we can still succinctly describe specific elements of U(f ). In ?4, we show how to efficiently
optimize over those elements.
Variational inference. We briefly summarize key results for variational inference for pairwise
models, following Wainwright and Jordan [7]. We write pairwise models as2
?
?
X
X
P (x) = exp ??
?i (xi ) +
(gi,j Jxi 6= xj K + ?i,j (xi , xj ) ? A(g)? ?(x),
i?V
{i,j}?E
where g ? RE is an arbitrary vector and A(g) is the log-partition function. For any choice of
2
parameters (?, g), there is a resulting vector of marginals ? ? [0, 1]k|V |+k |E| . Specifically, for every
i ? V , ? has k elements ?i,xi = P (Xi = xi ), one for each xi ? X . Similarly, for each {i, j} ? E,
there are k 2 elements ?ij,xi xj so that ?ij,xi xj = P (Xi = xi , Xj = xj ). The marginal polytope M is
now the set of all such vectors ? that are realizable under some distribution P (x), and the partition
function can equally be expressed in terms of the marginals [7]:
?
?
X
X X
A(g) = sup ??
?i,xi ?i (xi ) ?
?ij,xi xj ?i,j (xi , xj ) ? ?(?)T g? + H(?), (2)
??M
i?V,xi
|
{i,j}?E xi ,xj
{z
hstack(?,g),?i
}
where H(?) is the entropy
of the distribution, ?(?) is the vector of disagreement probabilities with
P
entries ?(?)i,j = xi 6=xj ?ij,xi xj , and stack(?, g) adds the elements of ? and g into a single
2
This formulation is slightly nonstandard, but will be very useful for the subsequent discussion in ?3.
3
vector so that the sum can be written as an inner product. Alas, neither M nor H(?) have succinct
descriptions and we will have to approximate them. Because the vectors in the approximation of
M are in general not correct marginals, they are called pseudo-marginals and will be denoted by
? instead of ?. Different approximations of M and H yield various methods, e.g. mean-field [7],
the semidefinite programming (SDP) relaxation of Wainwright and Jordan [18], tree-reweighted
belief propagation (TRWBP) [19], or the family of weighted entropies [20, 21]. Due to the space
constraints, we only discuss the latter. They approximate M with the local polytope
X
X
L = {? ? 0 | (?i ? V )
?i,xi = 1 and (?j ? ?(i)) ?i,xi =
?ij,xi xj }.
xi
xj
The approximations H to the entropy H are parametrized by one weight ?i,j per edge and one ?i per
vertex i, all collected in a vector ? ? R|V |+|E| . Then, they take the following form
P
X
X
Hi (? i ) = ?
?i,x log ?i,xi , and
H(? , ?) =
?i Hi (? i )+
?i,j Hi,j (? i,j ), where H (? ) = ? Pxi ?i
i,j
i,j
xi ,xj ij,xi jxj log ?ij,xi xj .
i?V
{i,j}?E
The most prominent example is traditional belief propagation, i.e., using the Bethe entropy, which
sets ?e = 1 for all e ? E, and assigns to each vertex i ? V a weight of ?i = 1 ? |?(i)|.
3
Convex upper bounds
The above variational methods do not directly generalize to our cooperative models: the vectors of
marginals could be exponentially large. Hence, we derive a different approach that relies on the
submodularity of f . Our first step is to approximate f (y(x)) by a linear lower bound, f (y(x)) ?
gT y(x), so that the resulting (pairwise) linearized model will have a partition function upper
bounding that of the original model. Ensuring that g indeed remains a lower bound means to satisfy
an exponential number of constraints f (y(x)) ? gT y(x), one for each x ? {0, 1}n . While this is
hard in general, the submodularity of f implies that these constraints are easily satisfied if g ? B(f ),
a very tractable constraint. For g ? B(f ), we have
X
X
XX
log Z = log
exp ? (
?i (xi ) +
?i,j (xi , xj ) + f (y(x)))
? log
x?{0,1}V
i?V xi
X
XX
x?{0,1}V
exp ? (
{i,j}?E
X
?i (xi ) +
i?V xi
{i,j}?E
(?i,j (xi , xj ) + gi,j Jxi 6= xj K)) ? A(g).
Unfortunately, A(g) is still very hard to compute and we need to approximate it. If we use an
approximation A(g) that upper bounds A(g), then the above inequality will still hold when we
replace A by A. Such approximations can be obtained by relaxing the marginal polytope M to an
outer bound M ? M, and using a concave entropy surrogate H that upper bounds the true entropy
H. TRWBP [19] or the SDP formulation [18] implement this approach. Our central optimization
problem is now to find the tightest upper bound, an optimization problem3 in g:
minimize sup hstack(?, g), ? i + H(? ).
(3)
g?B(f ) ? ?M
Because the inner problem is linear in g, this is a convex optimization problem over the base polytope.
To obtain the gradient with respect to g (equal to the negative disagreement probabilities ??(? )), we
have to solve the inner problem. This subproblem corresponds to performing variational inference in
a pairwise model, e.g. via TRWBP or an SDP. The optimization properties of the problem (3) depend
on its Lipschitz continuity of the gradients (smoothness). Informally, the inferred pseudomarginals
should not drastically change if we perturb the linearization g. The formal condition is that there
exists some ? > 0 so that k?(? ) ? ?(? 0 )k ? ?k? ? ? 0 k for all ? , ? 0 ? M. We discuss below
when this condition holds. Before that, we discuss two different algorithms for solving problem (3),
and how their convergence depends on ?.
3
If we compute the Fenchel dual, we obtain a special case of the problem considered in [22] with the Lov?asz
extension acting as a non-smooth non-local energy function (in the terminology introduced therein).
4
Frank-Wolfe. Given that we can efficiently solve linear programs over B(f ), the Frank-Wolfe [23]
algorithm is a natural candidate for solving the problem. We present it in Figure 2. It iteratively
moves towards the minimizer of a linearization of the objective around the current iterate. The method
has a convergence rate of O(?/t) [24], where ? is the assumed smoothness parameter. One can
either use a fixed step size ? = 2/(t + 2), or determine it using line search. In each iteration, the
algorithm calls the procedure L INEAR -O RACLE, which finds the vector s ? B(f ) that minimizes the
linearization of the objective function in (3) over the base polytope B(f ). The linearization is given
by the (approximate) gradient ?(? ), determined by the computed approximate marginals ? .
When taking a step towards s, the weight of edge ei is changed by sei = f ({e1 , e2 , . . . , ei }) ?
f ({e1 , e2 , . . . , ei?1 }). Due to the submodularity4 of f , an edge will obtain a higher weight if it
appears earlier in the order determined by the disagreement probabilities ?(? ). Hence, in every
iteration, the algorithm will re-adjusts the pairwise potentials, by encouraging the variables to agree
more as a function of their (approximate) disagreement probability.
1: procedure FW-I NFERENCE(f, ?)
2:
g ? L INEAR -O RACLE(f, 0)
3:
for t = 0, 1, . . . , max steps do
4:
? ? VAR -I NFERENCE(?, g)
5:
s ? L INEAR -O RACLE(f, ? )
6:
? ? C OMPUTE -S TEP -S IZE(g, s)
7:
g ? (1 ? ?)g + ?s
8:
return ? , A?
1: procedure L INEAR -O RACLE(f, ? )
Let e1 , e2 , . . . , e|E| be the edges E sorted so
2:
that ?(? )e1 ? ?(? )e2 ? . . . ? ?(? )e|E|
3:
for i = 0, 1, . . . , |E| do
4:
f?i ? f ({e1 , e2 , . . . , ei?1 })
5:
f+i ? f ({e1 , e2 , . . . , ei })
6:
sei ? f+i ? f?i
7:
return s
Figure 2: Inference with Frank-Wolfe, assuming that VAR -I NFERENCE guarantees an upper bound.
Projected gradient descent (PGD). Since it is possible to compute projections onto B(f ), and
practically so for many submodular functions f , we can alternatively use projected ?
gradient or
subgradient descent (PGD). Without smoothness, PGD converges at a rate of O(1/ t). If the
objective is smooth, we can use an accelerated methods like FISTA [25], which has both a much
better O(?/t2 ) rate and seems to converge faster than many Frank-Wolfe variants in our experiments.
Smoothness and convergence. The final question that remains to be answered is under which
conditions problem (3) is smooth (the proof can be found in the appendix).
Theorem 1 Problem (3) is k 2 ?-smooth over B(f ) if the entropy surrogate ?H is ?1 -strongly convex.
This result follows from the duality between smoothness and strong convexity for convex conjugates,
see e.g. [26]. It implies that the convergence rates of the proposed algorithms depend on the strong
convexity of the entropy approximation ?H. The benefits of strongly convex entropy approximations
are known. For instance, the tree-reweighted entropy approximation is strongly convex with a
modulus ? depending on the size of the graph; similarly, the SDP relaxation is strongly convex [27].
London et al. [28] provide an even sharper bound for the tree reweighted entropy, and show how one
can strong-convexify any weighted entropy by solving a QP over the weights ?.
In practice, because the inner problem is typically solved using an iterative algorithm and because the
problem is smooth, we obtain speedups by warm-starting the solver with the solution at the previous
iterate. We can moreover easily obtain duality certificates using the results in [24].
Joint optimization. When using weighted entropy approximations, it makes sense to optimize over
both the linearization g and the weights ? jointly. Specifically, let T be some set of weights that yield
an entropy approximation H that upper bounds H. Then, if we expand H in problem (3), we obtain
X
X
minimize sup hstack(?, g), ? i +
?i Hi (? i ) +
?i,j Hi,j (? i,j ).
g?B(f ),??T ? ?L
i?V
{i,j}?E
Note that inside the supremum, both g and ? appear only linearly, and there is no summand that has
terms from both of them. Thus, the problem is convex in (g, ?), and we can optimize jointly over
4
This is also known as the diminishing returns property.
5
both variables. As a final remark, if we already perform inference in a pairwise model and repeatedly
tighten the approximation by optimizing over ? via Frank-Wolfe (as suggested in [19]), then the
complexity per iteration remains the same even if we use the higher-order term f .
4
Submodular lower bounds
While we just derived variational upper bounds, we next develop lower bounds on the partition
function. Specifically, analogously to the linearization for the upper bound, if we pick an element
(g, c) of U(f ), the partition function of the resulting pairwise approximation always lower bounds
the partition function of (1). Formally,
X
X
X
log Z ? log
exp ? (aT x +
?ij,xi xj +
gi,j Jxi 6= xj K + c) = A(g) ? c.
x?{0,1}V
{i,j}?E
{i,j}?E
As before, after plugging in a lower bound estimate of A, we obtain a variational lower bound over
the partition function, which takes the form
log Z ?
sup
?c + hstack(?, g), ? i + H(? ),
(4)
(g,c)?U (f ),? ?M
for any pair of approximations of M and H that guarantee a lower bound of the pairwise model.
We propose to optimize this lower bound in a block-coordinate-wise manner: first with respect to
the pseudo-marginals ? (which amounts to approximate inference in the linearized model), and
then with respect to the supergradient (g, c) ? U(f ). As already noted, this step is in general
intractable. However, it is well-known [29] that for any Y ? E we can construct a point (so
called bar supergradient) in U(f ) as follows. First, define the vectors ai,j = f (1{i,j} ) and bi,j =
f (1)?f (1?1{i,j} ). Then, the vector (g, c) with g = b 1Y +(1?1Y ) a and c = f (Y )?bT 1Y
belongs to U(f ), where denotes element-wise multiplication.
Theorem 2 Optimizing problem (4) for a fixed ? over all bar supergradients is equal to the following
T
submodular minimization problem minY ?E f (Y ) + ?(? ) (b ? a) ? b 1Y .
In contrast to computing the MAP, the above problem has no constraints and can be easily solved
using existing algorithms. As the approximation algorithm for the linearized pairwise model, one
can always use mean-field [7]. Moreover, if (i) the problem is binary with submodular pairwise
potentials ?i,j and (ii) f is monotone, we can also use belief propagation. This is an implication
of the result of Ruozzi [30], who shows that traditional belief-propagation yields a lower bound on
the partition function for binary pairwise log-supermodular models. It is easy to see that the above
conditions are sufficient for the log-supermodularity of the linearized model, as g ? 0 when f is
monotone (because both a and b have non-negative components). Moreover, in this setting both
the mean-field and belief propagation objectives (i.e. computing ? ) can be cast as an instance of
continuous submodular minimization (see e.g. [31]), which means that they can be solved to arbitrary
precision in polynomial time. Unfortunately, problem (4) will not be jointly submodular, so we still
need to use the block-coordinate ascent method we have just outlined.
5
Approximate inference via MAP perturbations
For binary models with submodular pairwise potentials and monotone f we can (approximately)
solve the MAP problem using the techniques in [1, 4]. Hence, this opens as an alternative approach
the perturb-and-MAP method of Papandreou and Yuille [32]. This method relies on a set of tractable
first order perturbations: For any i ? V define ?i0 (xi ) = ?i (xi ) ? ?i,xi , where ? = (?i,xi )i?V,xi ?X
are a set of independently drawn Gumbel
randomP
variables. The optimizer argminx G? (x) of the
P
perturbed model energy G? (x) = i?V ?i0 (xi ) + {i,j}?E ?i,j (xi , xj ) + f (y(x)) is then a sample
from (an approximation to) the true distribution. If this MAP problem can be solved exactly (which is
not always the case here), then it is possible to obtain an upper bound on the partition function [33].
6
Experiments
Synthetic experiments. Our first set of experiments uses a complete graph on n variables. The
unary potentials were sampled as ?i (xi ) ? Uniform(??, ?). The edges E were randomly split
6
P5
into five disjoint buckets E1 , E2 , . . . , E5 , and we used f (y) = j=1 hj (yEj ), where yEi are the
coordinates of y corresponding to that group, and the functions {hj } will be defined below. To
perform inference in the linearized pairwise models, we used: trwbp, jtree+ (exact inference, upper
bound), jtree- (same, lower bound), sdp (SDP), mf (mean-field), bp (belief propagation), pmap
(perturb-and-MAP with approximate MAP) and epmap (perturb-and-MAP with exact MAP). We
used libDAI [34] and implemented sdp using cvxpy [35] and SCS [36]. As a maxflow solver we
used [37]. Errors bars denote three standard errors.
qP
p
Figure 3 shows the results for hi (yEi ) = wi
e?Ei ye / |Ei |, with weights wi ? Uniform(0, ?).
In panel (c) we use mixed (attractive and repulsive) pairwise potentials, chosen as ?i,j (xi , xj ) =
wi,j Jxi 6= xj K, where wi,j ? Uniform(??, ?). First, the results imply that the methods optimizing
the fully convex upper bound yield very good marginal probabilities over a large set of parameter
configurations. The estimate of the log-partition function from trwbp is also very good, while sdp is
much worse, which we believe can be attributed to the very loose entropy bound used in the relaxation.
The lower bounds (bp and mf) work well for settings when the pairwise strength ? is small compared
to the unary strength ?. Otherwise, both the bound and the marginals become worse, while jtreestill performs very well. This could be explained by the hardness of the pairwise models obtained
after linearizing f . Finally, pmap (when applicable) seems very promising for small ?.
To better understand the regimes when one should use trwbp or pmap, we compare their marginal
errors in Figure 5. We see that for most parameter configurations, trwbp performs better, and
significantly so when the edge interactions are strong.
Finally, we evaluate the effects of the approximate MAP solver for pmap
4. To be able
P in Figure P
to solve the MAP problem exactly (see [4]), we used h(yEj ) = max{ e?Ej ye ve , e?Ej ve /2},
where ve ? Uniform(0, ?). As evident from the figure, the gains from the exact solver seem minimal,
and it seems that solving the MAP problem approximately does not strongly affect the results.
An example from computer vision. To demonstrate the scalability of our method and obtain a
better qualitative understanding of the resulting marginals, we ran trwbp and pmap on a real world
image segmentation task. We use the same setting, data and models as [1], as implemented in
the pycoop5 package. Because libDAI was too slow, we wrote our own TRWBP implementation.
Figure 6 shows the results for two specific images (size 305 ? 398 and 214 ? 320). The example
in the first row is particularly difficult for pairwise models, but the rich higher-order model has no
problem capturing the details even in the challenging shaded regions of the image. The second row
shows results for two different model parameters. The second model uses a function f that is closer to
being linear, while the first one is more curved (see the appendix for details). We observe that trwbp
requires lower temperature parameters (i.e. relatively larger functions ?i , ?i,j and f ) than pmap, and
that the bottleneck of the complete inference procedure is running the trwbp updates. In other words,
the added complexity from our method is minimal and the runtime is dominated by the message
passing updates of TRWBP. Hence, any algorithms that speed up TRWBP (e.g., by parallelization or
better message scheduling) will result in a direct improvement on the proposed inference procedure.
7
Conclusion
We developed new inference techniques for a new broad family of discrete probabilistic models by
exploiting the (indirect) submodularity in the model, and carefully combining it with ideas from
classical variational inference in graphical models. The result are inference schemes that optimize
rigorous bounds on the partition function. For example, our upper bounds lead to convex variational
inference problems. Our experiments indicate the scalability, efficacy and quality of these schemes.
Acknowledgements. This research was supported in part by SNSF grant CRSII2 147633, ERC StG 307036,
a Microsoft Research Faculty Fellowship, a Google European Doctoral Fellowship, and NSF CAREER 1553284.
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(a) Original image
(b) trwbp, pairwise
(c) pmap, pairwise
(d) trwbp, coop.
(e) pmap, coop.
(f) Original image
(g) trwbp, model 1
(h) pmap, model 1
(i) trwbp, model 2
(j) pmap, model 2
Figure 6: Inferred marginals on an image segmentation task. The first row showcases an example
that is particularly hard for pairwise models. In the second row we show the results for two different
models (the cooperative function f is more curved for model 1).
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9
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5,662 | 6,123 | Finite-Sample Analysis of Fixed-k Nearest Neighbor
Density Functional Estimators
Shashank Singh
Statistics & Machine Learning Departments
Carnegie Mellon University
sss1@andrew.cmu.edu
Barnab?s P?czos
Machine Learning Departments
Carnegie Mellon University
bapoczos@cs.cmu.edu
Abstract
We provide finite-sample analysis of a general framework for using k-nearest neighbor statistics to estimate functionals of a nonparametric continuous probability
density, including entropies and divergences. Rather than plugging a consistent
density estimate (which requires k ? ? as the sample size n ? ?) into the
functional of interest, the estimators we consider fix k and perform a bias correction. This is more efficient computationally, and, as we show in certain cases,
statistically, leading to faster convergence rates. Our framework unifies several
previous estimators, for most of which ours are the first finite sample guarantees.
1
Introduction
Estimating entropies and divergences of probability distributions in a consistent manner is of importance in a number of problems in machine learning. Entropy estimators have applications in
goodness-of-fit testing [13], parameter estimation in semi-parametric models [51], studying fractal
random walks [3], and texture classification [14, 15]. Divergence estimators have been used to
generalize machine learning algorithms for regression, classification, and clustering from inputs in
RD to sets and distributions [40, 33].
Divergences also include mutual informations as a special case; mutual information estimators have
applications in feature selection [35], clustering [2], causality detection [16], optimal experimental
design [26, 38], fMRI data analysis [7], prediction of protein structures [1], and boosting and facial
expression recognition [41]. Both entropy estimators and mutual information estimators have been
used for independent component and subspace analysis [23, 47, 37, 17], as well as for image
registration [14, 15]. Further applications can be found in [25].
This paper considers the more general problem of estimating functionals of the form
F (P ) := E [f (p(X))] ,
(1)
X?P
using n IID samples from P , where P is an unknown probability measure with smooth density
function p and f is a known smooth function. We are interested in analyzing a class of nonparametric
estimators based on k-nearest neighbor (k-NN) distance statistics. Rather than plugging a consistent
estimator of p into (1), which requires k ? ? as n ? ?, these estimators derive a bias correction
for the plug-in estimator with fixed k; hence, we refer to this type of estimator as a fixed-k estimator.
Compared to plug-in estimators, fixed-k estimators are faster to compute. As we show, fixed-k
estimators can also exhibit superior rates of convergence.
As shown in Table 1, several authors have derived bias corrections necessary for fixed-k estimators of
entropies and divergences, including, most famously, the Shannon entropy estimator of [20]. 1 The
estimators in Table 1 estimators are known to be weakly consistent, 2 but, except for Shannon entropy,
1
MATLAB code for these estimators is in the ITE toolbox https://bitbucket.org/szzoli/ite/ [48].
Several of these proofs contain errors regarding the use of integral convergence theorems when their
conditions do not hold, as described in [39].
2
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Functional Name
Shannon Entropy
R?nyi-? Entropy
KL Divergence
?-Divergence
Functional Form
E [log p(X)]
E p??1 (X)
i
h
p(X)
E log q(X)
??1
p(X)
E q(X)
Bias Correction
Additive constant: ?(n) ? ?(k) + log(k/n)
?(k)
Multiplicative constant: ?(k+1??)
Ref.
[20][13]
[25, 24]
None?
[50]
2
Multiplicative constant:
? (k)
?(k??+1)?(k+??1)
[39]
Table 1: Functionals with known bias-corrected
k-NN estimators, their bias corrections, and references. All
R
?
d
expectations are over X ? P . ?(t) = 0 xt?1 e?x dx is the gamma function, and ?(x) = dx
log (?(x)) is
the digamma function. ? ? R\{1} is a free parameter. ? For KL divergence, bias corrections for p and q cancel.
no finite sample bounds are known. The main goal of this paper is to provide finite-sample analysis
of these estimators, via unified analysis of the estimator after bias correction. Specifically, we show
conditions under which, for ?-H?lder continuous
space, the
(? ? (0, 2]) densities on D dimensional
bias of fixed-k estimators decays as O n??/D and the variance decays as O n?1 , giving a mean
squared error of O n?2?/D + n?1 . Hence, the estimators converge at the parametric O(n?1 ) rate
when ? ? D/2, and at the slower rate O(n?2?/D ) otherwise. A modification of the estimators would
be necessary to leverage additional smoothness for ? > 2, but we do not pursue this here. Along the
way, we prove a finite-sample version of the useful fact [25] that (normalized) k-NN distances have
an Erlang asymptotic distribution, which may be of independent interest.
We present our results for distributions P supported on the unit cube in RD because this significantly
simplifies the statements of our results, but, as we discuss in the supplement, our results generalize
fairly naturally, for example to distributions supported on smooth compact manifolds. In this context,
it is worth noting that our results scale with the intrinsic dimension of the manifold. As we discuss
later, we believe deriving finite sample rates for distributions with unbounded support may require a
truncated modification of the estimators we study (as in [49]), but we do not pursue this here.
2
Problem statement and notation
Let X := [0, 1]D denote the unit cube in RD , and let ? denote the Lebesgue measure. Suppose P is an
unknown ?-absolutely continuous Borel probability measure supported on X , and let p : X ? [0, ?)
denote the density of P . Consider a (known) differentiable function f : (0, ?) ? R. Given n
samples X1 , ..., Xn drawn IID from P , we are interested in estimating the functional
F (P ) := E [f (p(X))] .
X?P
Somewhat more generally (as in divergence estimation), we may have a function f : (0, ?)2 ? R
of two variables and a second unknown probability measure Q, with density q and n IID samples
Y1 , ..., Yn . Then, we are interested in estimating
F (P, Q) := E [f (p(X), q(X))] .
X?P
Fix r ? [1, ?] and a positive integer k. We will work with distances induced by the r-norm
!1/r
D
D
X
(2?(1 + 1/r))
r
kxkr :=
xi
and define cD,r :=
= ?(B(0, 1)),
?(1 + D/r)
i=1
where B(x, ?) := {y ? RD : kx ? ykr < ?} denotes the open radius-? ball centered at x. Our
estimators use k-nearest neighbor (k-NN) distances:
Definition 1. (k-NN distance): Given n IID samples X1 , ..., Xn from P , for x ? RD , we define the
k-NN distance ?k (x) by ?k (x) = kx ? Xi kr , where Xi is the k th -nearest element (in k ? kr ) of the set
{X1 , ..., Xn } to x. For divergence estimation, given n samples Y1 , ..., Yn from Q, then we similarly
define ?k (x) by ?k (x) = kx ? Yi kr , where Yi is the k th -nearest element of {Y1 , ..., Yn } to x.
?-absolute continuity of P precludes the existence of atoms (i.e., ?x ? RD , P ({x}) = ?({x}) = 0).
Hence, each ?k (x) > 0 a.s. We will require this to study quantities such as log ?k (x) and 1/?k (x).
2
3
Estimator
3.1 k-NN density estimation and plug-in functional estimators
The k-NN density estimator
p?k (x) =
k/n
k/n
=
?(B(x, ?k (x))
cD ?D
k (x)
is well-studied nonparametric density estimator [28], motivated by noting that, for small ? > 0,
p(x) ?
P (B(x, ?))
,
?(B(x, ?))
and that, P (B(x, ?k (x))) ? k/n. One can show that, for x ? RD at which p is continuous, if
k ? ? and k/n ? 0 as n ? ?, then p?k (x) ? p(x) in probability ([28], Theorem 3.1). Thus, a
natural approach for estimating F (P ) is the plug-in estimator
n
1X
F?P I :=
f (?
pk (Xi )) .
n i=1
(2)
Since p?k ? p in probability pointwise as k, n ? ? and f is smooth, one can show F?P I is consistent,
and in fact derive finite sampleconvergence rates
(depending on how k ? ?). For example, [44]
2?
show a convergence rate of O n? min{ ?+D ,1} for ?-H?lder continuous densities (after sample
?
splitting and boundary correction) by setting k n ?+d .
Unfortunately, while necessary to ensure V [?
pk (x)] ? 0, the requirement k ? ? is computationally
burdensome. Furthermore, increasing k can increase the bias of p?k due to over-smoothing (see (5)
below), suggesting that this may be sub-optimal for estimating F (P ). Indeed, similar work based on
kernel density estimation [42] suggests that, for plug-in functional estimators, under-smoothing may
be preferable, since the empirical mean results in additional smoothing.
3.2 Fixed-k functional estimators
Anhalternative
approach is to fix k as n ? ?. Since F?P I is itself an empirical mean, unlike V [?
pk (x)],
i
V F?P I ? 0 as n ? ?. The more critical complication of fixing k is bias. Since f is typically
non-linear, the non-vanishing variance of p?k translates into asymptotic bias. A solution adopted by
several papers is to derive a bias correction function B (depending only on known factors) such that
P (B(x, ?k (x)))
k/n
f
B f
=
.
(3)
E
E
?(B(x, ?k (x))
?(B(x, ?k (x))
X1 ,...,Xn
X1 ,...,Xn
For continuous p, the quantity
p?k (x) (x) :=
P (B(x, ?k (x)))
?(B(x, ?k (x))
(4)
is a consistent estimate of p(x) with k fixed, but it is not computable, since P is unknown. The bias
correction B gives us an asymptotically unbiased estimator
n
n
1X
1X
k/n
?
FB (P ) :=
B (f (?
pk (Xi ))) =
B f
.
n i=1
n i=1
?(B(Xi , ?k (Xi ))
that uses k/n in place of P (B(x, ?k (x))). This estimate extends naturally to divergences:
n
1X
F?B (P, Q) :=
B (f (?
pk (Xi ), q?k (Xi ))) .
n i=1
As an example, if f = log (as in Shannon entropy), then it can be shown that, for any continuous p,
E [log P (B(x, ?k (x)))] = ?(k) ? ?(n).
3
Hence, for Bn,k := ?(k) ? ?(n) + log(n) ? log(k),
k/n
P (B(x, ?k (x)))
+ Bn,k =
.
f
f
E
E
?(B(x, ?k (x))
?(B(x, ?k (x))
X1 ,...,Xn
X1 ,...,Xn
giving the estimator of [20]. Other examples of functionals for which the bias correction is known
are given in Table 1.
In general, deriving an appropriate bias correction can be quite a difficult problem specific to the
functional of interest, and it is not our goal presently to study this problem; rather, we are interested in
bounding the error of F?B (P ), assuming the bias correction is known. Hence, our results apply to all
of the estimators in Table 1, as well as any estimators of this form that may be derived in the future.
4
Related work
4.1
Estimating information theoretic functionals
Recently, there has been much work on analyzing estimators for entropy, mutual information,
divergences, and other functionals of densities. Besides bias-corrected fixed-k estimators, most of
this work has taken one of three approaches. One series of papers [27, 42, 43] studied a boundarycorrected plug-in approach based on under-smoothed kernel density estimation. This approach has
strong finite sample guarantees, but requires prior knowledge of the support of the density, and can
have a slow rate of convergence. A second approach [18, 22] uses von Mises expansion to partially
correct the bias of optimally smoothed density estimates. This is statistically more efficient, but can
require computationally demanding numerical integration over the support of the density. A final line
of work [30, 31, 44, 46] studied plug-in estimators based on consistent, boundary corrected k-NN
density estimates (i.e., with k ? ? as n ? ?). [32] study a divergence estimator based on convex
risk minimization, but this relies of the context of an RKHS, making results are difficult to compare.
Rates of Convergence: For densities over RD satisfying a H?lder smoothness condition parametrized
by ? ? (0, ?), the minimax mean squared error
functionals of the form
rate for8?estimating
R
f (p(x)) dx has been known since [6] to be O n? min{ 4?+D ,1} . [22] recently derived identical minimax rates for divergence estimation.
2?
Most of the above estimators have been shown to converge at the rate O n? min{ ?+D ,1} . Only
the von Mises approach [22] is known to achieve the minimax rate for general ? and D, but due to
its computational demand (O(2D n3 )), 3 the authors suggest using other statistically less efficient
estimators for moderate sample size. Here, we show
that, for ? ? (0, 2], bias-corrected fixed-k
? min{ 2?
D ,1} . For ? > 2, modifications are
estimators converge at the relatively fast rate O n
needed for the estimator to leverage the additional smoothness of the density. Notably, this rate is
adaptive; that is, it does not require selecting a smoothing parameter depending on the unknown ?;
our results (Theorem 5) imply the above rate is achieved for any fixed choice of k. On the other hand,
since no empirical error metric is available for cross-validation, parameter selection is an obstacle for
competing estimators.
4.2
Prior analysis of fixed-k estimators
As of writing this paper, the only finite-sample results for F?B (P ) were those of [5] for the KozachenkoLeonenko (KL) 4 Shannon entropy estimator. [20] Theorem 7.1 of [5] shows that, if the density p has
compact support, then the variance of the KL estimator decays as O(n?1 ). They also claim (Theorem
7.2) to bound the bias of the KL estimator by O(n?? ), under the assumptions that p is ?-H?lder
continuous (? ? (0, 1]), bounded away from 0, and supported on the interval [0, 1]. However, in
their proof, [5] neglect to bound the additional bias incurred near the boundaries of [0, 1], where the
density cannot simultaneously be bounded away from 0 and continuous. In fact, because the KL
estimator does not attempt to correct for boundary bias, it is not clear that the bias should decay as
O(n?? ) under these conditions; we require additional conditions at the boundary of X .
Fixed-k estimators can be computed in O Dn2 time, or O 2D n log n using k-d trees for small D.
4
Not to be confused with Kullback-Leibler (KL) divergence, for which we also analyze an estimator.
3
4
?
[49] studied a closely related entropy estimator for which they prove n-consistency.
Their estimator
?
is identical to?the KL estimator, except that it truncates k-NN distances at n, replacing ?k (x) with
min{?k (x), n}. This sort of truncation may be necessary for certain fixed-k estimators to satisfy
finite-sample bounds for densities of unbounded support, though consistency can be shown regardless.
Finally, two very recent papers [12, 4] have analyzed the KL estimator. In this case, [12] generalize
the results of [5] to D > 1, and [4] weaken the regularity and boundary assumptions required by our
bias bound, while deriving the same rate of convergence. Moreover, they show that, if k increases
with n at the rate k log5 n, the KL estimator is asymptotically efficient (i.e., asymptotically normal,
with optimal asymptotic variance). As explained in Section 8, together with our results this elucidates
the role of k in the KL estimator: fixing k optimizes the convergence rate of the estimator, but
increasing k slowly can further improve error by constant factors.
5
Discussion of assumptions
The lack of finite-sample results for fixed-k estimators is due to several technical challenges. Here,
we discuss some of these challenges, motivating the assumptions we make to overcome them.
First, these estimators are sensitive to regions of low probability (i.e., p(x) small), for two reasons:
1. Many functions f of interest (e.g., f = log or f (z) = z ? , ? < 0) have singularities at 0.
2. The k-NN estimate p?k (x) of p(x) is highly biased when p(x) is small. For example, for p
?-H?lder continuous (? ? (0, 2]), one has ([29], Theorem 2)
?/D
k
Bias(?
pk (x))
.
(5)
np(x)
For these reasons, it is common in analysis of k-NN estimators to assume the following [5, 39]:
(A1) p is bounded away from zero on its support. That is, p? := inf x?X p(x) > 0.
Second, unlike many functional estimators (see e.g., [34, 45, 42]), the fixed-k estimators we consider
do not attempt correct for boundary bias (i.e., bias incurred due to discontinuity of p on the boundary
?X of X ). 5 The boundary bias of the density estimate p?k (x) does vanish at x in the interior X ?
of X as n ? ?, but additional assumptions are needed to obtain finite-sample rates. Either of the
following assumptions would suffice:
(A2) p is continuous not only on X ? but also on ?X (i.e., p(x) ? 0 as dist(x, ?X ) ? 0).
(A3) p is supported on all of RD . That is, the support of p has no boundary. This is the approach
of [49], but we reiterate that, to handle an unbounded domain, they require truncating ?k (x).
Unfortunately, both assumptions (A2) and (A3) are inconsistent with (A1). Our approach is to assume
(A2) and replace assumption (A1) with a much milder assumption that p is locally lower bounded on
its support in the following sense:
(A4) There exist ? > 0 and a function p? : X ? (0, ?) such that, for all x ? X , r ? (0, ?],
(B(x,r))
p? (x) ? P?(B(x,r))
.
We show in Lemma 2 that assumption (A4) is in fact very mild; in a metric measure space of positive
dimension D, as ?
long as p is continuous on X , such a p? exists for any desired ? > 0. For simplicity,
we will use ? = D = diam(X ).
As hinted by (5) and the fact that F (P ) is an expectation, our bounds will contain terms of the form
"
# Z
1
p(x)
=
d?(x)
E
?/D
?/D
X?P (p? (X))
X (p? (x))
(with an additional f 0 (p? (x)) factor if f has a singularity at zero). Hence, our key assumption is that
these quantities are finite. This depends primarily on how quickly p approaches zero near ?X . For
many functionals, Lemma 6 gives a simple sufficient condition.
5
This complication was omitted in the bias bound (Theorem 7.2) of [5] for entropy estimation.
5
6
Preliminary lemmas
Here, we present some lemmas, both as a means of summarizing our proof techniques and also
because they may be of independent interest for proving finite-sample bounds for other k-NN methods.
Due to space constraints, all proofs are given in the appendix. Our first lemma states that, if p is
continuous, then it is locally lower bounded as described in the previous section.
Lemma 2. (Existence of?Local Bounds) If p is continuous on X and strictly positive on the interior
X ? of X , then, for ? := D = diam(X ), there exists a continuous function p? : X ? ? (0, ?) and
a constant p? ? (0, ?) such that
P (B(x, r))
0 < p? (x) ?
? p? < ?, ?x ? X , r ? (0, ?].
?(B(x, r))
We now use these local lower and upper bounds to prove that k-NN distances concentrate around a
1/D
term of order (k/(np(x)))
. Related lemmas, also based on multiplicative Chernoff bounds, are
used by [21, 9] and [8, 19] to prove finite-sample bounds on k-NN methods for cluster tree pruning
and classification, respectively. For cluster tree pruning, the relevant inequalities bound the error of
the k-NN density estimate, and, for classification, they lower bound the probability of nearby samples
of the same class. Unlike in cluster tree pruning, we are not using a consistent density estimate, and,
unlike in classification, our estimator is a function of k-NN distances themselves (rather than their
ordering). Thus, our statement is somewhat different, bounding the k-NN distances themselves:
Lemma 3. (Concentration of k-NN Distances) Suppose p is continuous on X and strictly positive
on X ? . Let p? and p? be as in Lemma 2. Then, for any x ? X ? ,
k
1/D
p? (x)rD n
?p? (x)r D n
k
1. if r > p? (x)n
.
, then P [?k (x) > r] ? e
e
k
? D kp? (x)/p?
1/D
ep r n
?p? (x)r D n
k
2. if r ? 0, p? n
.
, then P [?k (x) < r] ? e
k
It is worth noting an asymmetry in the above bounds: counter-intuitively, the lower bound depends on
p? . This asymmetry is related to the large bias of k-NN density estimators when p is small (as in (5)).
The next lemma uses Lemma 3 to bound expectations of monotone functions of the ratio p?k /p? . As
suggested by the form of integrals (6) and (7), this is essentially a finite-sample statement of the fact
that (appropriately normalized) k-NN distances have Erlang asymptotic distributions; this asymptotic
statement is key to consistency proofs of [25] and [39] for ?-entropy and divergence estimators.
Lemma 4. Let p be continuous on X and strictly positive on X ? . Define p? and p? as in Lemma 2.
Suppose f : (0, ?) ? R is continuously differentiable and f 0 > 0. Then, we have the upper bound 6
y
? Z ? e?y y k
p? (x)
sup E f+
? f+ (1) + e k
f+
dy,
(6)
p?k (x)
?(k + 1)
k
x?X ?
k
and, for all x ? X ? , for ?(x) := kp? (x)/p? , the lower bound
s
Z ?(x) ?y ?(x)
y
p? (x)
k
e y
? f? (1) + e
f?
dy
E f?
p?k (x)
?(x) 0
?(?(x) + 1)
k
Note that plugging the function z 7? f
kz
cD,r np? (x)
D1
(7)
into Lemma 4 gives bounds on
E [f (?k (x))]. As one might guess from Lemma 3 and the assumption that f is smooth, this bound is
D1
k
roughly of the order np(x)
. For example, for any ? > 0, a simple calculation from (6) gives
D?
k
?
?
.
(8)
[?
(x)]
?
1
+
E k
D
cD,r np? (x)
(8) is used for our bias bound, and more direct applications of Lemma 4 are used in variance bound.
6
f+ (x) = max{0, f (x)} and f? (x) = ? min{0, f (x)} denote the positive and negative parts of f . Recall
that E [f (X)] = E [f+ (X)] ? E [f? (X)].
6
7
Main results
Here, we present our main results on the bias and variance of F?B (P ). Again, due to space constraints,
all proofs are given in the appendix. We begin with bounding the bias:
Theorem 5. (Bias Bound) Suppose that, for some ? ? (0, 2], p is ?-H?lder continuous with constant
L > 0 on X , and p is strictly positive on X ? . Let p? and p? be as in Lemma 2. Let f : (0, ?) ? R
be differentiable, and define Mf,p : X ? [0, ?) by
d
Mf,p (x) :=
sup
f (z)
dz
?
z?[p? (x),p ]
Assume
"
Cf := E
X?p
Mf,p (X)
#
?
< ?.
Then,
(p? (X)) D
D?
k
?
.
E FB (P ) ? F (P ) ? Cf L
n
The statement for divergences is similar, assuming that q is also ?-H?lder continuous with constant L
and strictly positive on X ? . Specifically, we get the same bound if we replace Mf,o with
?
f
(w,
z)
Mf,p (x) :=
sup
?w
?
?
(w,z)?[p? (x),p ]?[q? (x),q ]
and define Mf,q similarly (i.e., with
"
Cf := E
X?p
?
?z )
and we assume that
"
#
#
Mf,p (X)
Mf,q (X)
+ E
< ?.
?
?
X?p (q (X)) D
(p? (X)) D
?
As an example of the applicability of Theorem 5, consider estimating the Shannon entropy. Then,
R
??/D
f (z) = log(x), and so we need Cf = X (p? (x))
d?(x) < ?.
The assumption Cf < ? is not immediately transparent. For the functionals in Table 1, Cf has the
R
?c
form X (p(x)) dx, for some c > 0, and hence Cf < ? intuitively means p(x) cannot approach
zero too quickly as dist(x, ?X ) ? 0. The following lemma gives a formal sufficient condition:
Lemma 6. (Boundary Condition) Let c > 0. Suppose there exist b? ? (0, 1c ), c? , ?? > 0 such that,
R
?c
for all x ? X with ?(x) := dist(x, ?X ) < ?? , p(x) ? c? ?b? (x). Then, X (p? (x)) d?(x) < ?.
In the supplement, we give examples showing that this condition is fairly general, satisfied by
densities proportional to xb? near ?X (i.e., those with at least b? nonzero one-sided derivatives on
the boundary).
We now bound the variance. The main obstacle here is that the fixed-k estimator is an empirical
mean of dependent terms (functions of k-NN distances). We generalize the approach used by [5] to
bound the variance of the KL estimator of Shannon entropy. The key insight is the geometric fact
that, in (RD , k ? kp ), there exists a constant Nk,D (independent of n) such that any sample Xi can be
amongst the k-nearest neighbors of at most Nk,D other samples. Hence, at most Nk,D + 1 of the
terms in (2) can change when a single Xi is added, suggesting a variance bound via the Efron-Stein
inequality [10], which bounds the variance of a function of random variables in terms of its expected
change when its arguments are resampled. [11] originally used this approach to prove a general Law
of Large Numbers (LLN) for nearest-neighbors statistics. Unfortunately, this LLN relies on bounded
kurtosis assumptions that are difficult to justify for the log or negative power statistics we study.
Theorem 7. (Variance Bound) Suppose
B ? f is continuously
R ? differentiable and strictly monotone.
Assume Cf,p := EX?P B 2 (f (p? (X))) < ?, and Cf := 0 e?y y k f (y) < ?. Then, for
h
i C
V
CV := 2 (1 + Nk,D ) (3 + 4k) (Cf,p + Cf ) , we have V F?B (P ) ?
.
n
As an example,
if f = log (as in Shannon entropy), then, since B is an additive constant, we simply
R
cD
require X p(x) log2 (p? (x)) < ?. In general,
h Nk,D iis of the order k2 , for some c > 0. Our
bound is likely quite loose in k; in practice, V F?B (P ) typically decreases somewhat with k.
7
8
Conclusions and discussion
In this paper, we gave finite-sample bias and variance error bounds for a class of fixed-k estimators
of functionals of probability density functions, including the entropy and divergence estimators in
Table 1. The bias and variance bounds in turn imply a bound on the mean squared error (MSE) of the
bias-corrected estimator via the usual decomposition into squared bias and variance:
Corollary 8. (MSE Bound) Under the conditions of Theorems 5 and 7,
2?/D
2
k
CV
2 2
?
? Cf L
+
.
(9)
E Hk (X) ? H(X)
n
n
Choosing k: Contrary to the name, fixing k is not required for ?fixed-k? estimators. [36] empirically
studied the effect of changing k with n and found that fixing k = 1 gave best results for estimating
F (P ). However, there has been no theoretical justification for fixing k. Assuming tightness of our
bias bound in k, we provide this in a worst-case sense: since our bias bound is nondecreasing in k and
our variance bound is no larger than the minimax MSE rate for these estimation problems, reducing
variance (i.e., increasing k) does not improve the (worst-case) convergence rate. On the other hand,
[4] recently showed that slowly increasing k can improves the asymptotic variance of the estimator,
with the rate k log5 n leading to asymptotic efficiency. In view of these results, we suggest that
increasing k can improve error by constant factors, but cannot improve the convergence rate.
Finally, we note that [36] found increasing k quickly (e.g., k = n/2) was best for certain hypothesis
tests based on these estimators. Intuitively, this is because, in testing problems, bias is less problematic
than variance (e.g., an asymptotically biased estimator can still lead to a consistent test).
Acknowledgments
This material is based upon work supported by a National Science Foundation Graduate Research
Fellowship to the first author under Grant No. DGE-1252522.
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5,663 | 6,124 | Achieving Budget-optimality with Adaptive Schemes
in Crowdsourcing
Ashish Khetan and Sewoong Oh
Department of ISE, University of Illinois at Urbana-Champaign
Email: {khetan2,swoh}@illinois.edu
Abstract
Adaptive schemes, where tasks are assigned based on the data collected thus far, are
widely used in practical crowdsourcing systems to efficiently allocate the budget.
However, existing theoretical analyses of crowdsourcing systems suggest that the
gain of adaptive task assignments is minimal. To bridge this gap, we investigate this
question under a strictly more general probabilistic model, which has been recently
introduced to model practical crowdsourcing datasets. Under this generalized
Dawid-Skene model, we characterize the fundamental trade-off between budget
and accuracy. We introduce a novel adaptive scheme that matches this fundamental
limit. A given budget is allocated over multiple rounds. In each round, a subset of
tasks with high enough confidence are classified, and increasing budget is allocated
on remaining ones that are potentially more difficult. On each round, decisions are
made based on the leading eigenvector of (weighted) non-backtracking operator
corresponding to the bipartite assignment graph. We further quantify the gain of
adaptivity, by comparing the tradeoff with the one for non-adaptive schemes, and
confirm that the gain is significant and can be made arbitrarily large depending on
the distribution of the difficulty level of the tasks at hand.
1
Introduction
Crowdsourcing platforms provide labor markets in which pieces of micro-tasks are electronically
distributed to a pool of workers. In typical crowdsourcing scenarios, such as those on Amazon?s
Mechanical Turk, a requester posts a collection of tasks, and a batch is picked up by any worker
who is willing to complete it. The worker is subsequently rewarded for each task he/she completes.
However, some workers are spammers trying to make easy money. Moreover, since the reward is
small and tasks are tedious, errors are common even among those who try. To correct for the errors,
a common approach is to introduce redundancy by assigning each task to multiple workers and
aggregating their responses using some schemes such as majority voting.
A fundamental problem of interest is how to maximize the accuracy of thus inferred solutions, while
using as small number of repetitions as possible. There are two challenges in achieving such an
optimal tradeoff between accuracy and the budget: (a) we need a scheme for deciding which tasks to
assign to which workers; and (b) at the same time infer the true solutions from their responses.
Since the workers are fleeting, the requester has no control over who gets to work on which tasks. It
is impossible to make a trust relationship with the workers. In particular, it does not make sense to
explore reliable workers, and exploit them in subsequent steps. Each arriving worker is completely
new and you may never get him back. Nevertheless, by comparing responses from multiple workers,
we can estimate the true answer to the task, and use it in subsequent steps to learn the reliability of the
workers. Our beliefs on the true answers as well as the difficulty of the tasks and the reliability of the
workers can be iteratively refined, and one can potentially choose to assign more workers to the more
difficult tasks. We would like to understand such intricate interplay of task assignment and inference.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Setup. We have m binary classification tasks to be completed by workers. We assume a recent generalization of the Dawid-Skene model introduced in [22] to model the responses, which captures the
heterogeneity in the tasks as well as the workers. Precisely, each new arriving worker is parametrized
by a quality parameter pj ? [0, 1] (for the j-th arriving worker), which is i.i.d. according to some
prior distribution F . Each task is parametrized by a difficulty parameter qi ? [0, 1] (for the i-th task),
which is drawn i.i.d. according to some prior distribution G . When a worker j is assigned a task i,
the task is perceived as a positive task with probability qi , and as a negative task otherwise. Hence, if
qi is close to a half then it is confusing and difficult to correctly classify, and easy if close to one or
zero. When task i is assigned to worker j, the response is a noisy perception of the task:
1, w.p. qi pj + q?i p?j ,
Aij =
,
(1)
?1, w.p. q?i pj + qi p?j .
where q?i = 1 ? qi and p?j = 1 ? pj . With probability pj , the worker answers truthfully as he perceives
the task, and otherwise gives the opposite answer. Hence, if pj is close to one then he tells the truth
(in his opinion) and if it is close to half he gives random answers. If it is zero, he is also reliable, in
the sense that a requester who can correctly decode his reliability can extract the truths exactly.
We define the ground truth of a task as what the majority of the workers agree on, had we asked
all the workers. Accordingly, we assume that EF [pj ] > 1/2 and the true labels are defined as
ti = I{qi >(1/2)} ? I{qi <(1/2)} . Otherwise, we do not impose any condition on the distribution of
pj ?s. However, we assume qi ?s are discrete random variables with support at K points. Our results
do not directly depend on this support size K, and therefore K can be made arbitrarily large. Note
that we focus on only binary tasks with two types of classes, and also the workers are assumed to be
symmetric, i.e. the error probability does not depend on the perceived label of the task. The original
Dawid-Skene model introduced in [3] and analyzed in [9] is a special case, when all tasks are equally
easy, i.e. qi ?s are either one or zero. This makes inference easier as all tasks are perceived their true
class; the only source of error is in workers? noisy responses.
We assume the following task assignment scenario to model practical crowdsourcing systems. It is a
discrete time system, where at the beginning of each time step the requester can create a batch of
tasks. This batch is picked up by a new arriving worker, and his/her responses are collected. To model
real-world constraints we assume there is a limit on how many tasks a single worker can complete,
which we denote by r. The requester (also called task master) has no control over who is arriving
next, but he has control over which of the m tasks are to be solved by the next arriving worker. This
allows for adaptive task assignment schemes, where the requester can choose to include those tasks
that he is most uncertain about based on all the history of responses collected thus far.
We consider all randomized task assignment schemes, whose expected number of assignment per task
is `, and all inference algorithms. We study the minimax rate when the nature chooses the worst case
priors F and G (from a family of priors parametrized by average worker reliability ? and average task
difficulty ? defined in (2)), and we choose the best possible adaptive task assignment together with
the best possible inference algorithm. We further propose a novel adaptive approach that achieves
this minimax rate up to a constant factor. Our approach is different from existing adaptive schemes in
[5], where there are multiple types of tasks and the main source of uncertainty is which type the next
arriving worker is expert on. Golden tasks with known answers are used to explore expertise and
tasks are assigned accordingly.
Related work. Existing work on crowdsourcing systems study the standard Dawid-Skene (DS)
model [3], where all tasks are equally difficult and hence qi ? {0, 1} for all tasks. Several inference
algorithms have been proposed [3, 17, 6, 16, 4, 7, 11, 23, 10, 21, 2, 8, 14], and the question of task
assignment is addressed in [9], where the minimax rate on the probability of error is characterized and
a matching task assignment scheme and an inference algorithm are proposed. Perhaps surprisingly,
for the standard DS model, a non-adaptive task assignment scheme achieves the fundamental limit.
Namely, given m tasks and a total budget for m` responses, the requester first constructs a bipartite
task-assignment graph with m task nodes, n = m`/r worker nodes, and edges drawn uniformly at
random with degree ` for the task nodes and r for the worker nodes. Then, j-th arriving worker is
assigned a batch of r tasks that are adjacent to the j-th worker node. Together with an inference
algorithm explained in detail in Section 2, this achieves a near-optimal performance. Namely, to
achieve an average probability of error ?, it is sufficient to have total budget O((m/?) log(1/?)),
where ? = EF [(2pj ? 1)2 ] is the quality of the workers defined in (2). Perhaps surprisingly, no
adaptive assignment can improve upon it. Even the best adaptive scheme and the best inference
2
algorithm still requires ?((m/?) log(1/?)) total budget. Hence, there is no gain in adaptivity. This
negative result relies crucially in the fact that under the standard DS model, all tasks are inherently
equally difficult. Hence, adaptively assigning more workers to relatively more ambiguous tasks
has only a marginal gain. However, simple adaptive schemes are widely used in practice, where
significant gains are achieved; in real-world systems, tasks are widely heterogeneous. To capture such
varying difficulties in the tasks, generalizations of the DS model were proposed in [19, 18, 22, 15]
and significant improvement has been reported on inference problems for real datasets.
The generalized DS model serves as the missing piece in bridging the gap between practical gains of
adaptivity and theoretical limitations of adaptivity. We investigate the fundamental question of ?do
adaptive task assignments improve accuracy?? under this generalized Dawid-Skene model of Eq. (1).
Contributions. To investigate the gain of adaptivity, we first characterize the fundamental lower
bound on the budget required to achieve a target accuracy. To match this fundamental limit, we
introduce a novel adaptive task assignment scheme. Our approach consists of multiple rounds of
non-adaptive schemes, and we provide sharp analyses on the performance at each round, which guides
the design of the task assignment in each round adaptively using the data from previous rounds. The
proposed adaptive task assignment is simple to apply in practice, and numerical simulations confirm
the superiority compared to state-of-the-art non-adaptive schemes. Under a certain assumption on
the choice of parameters in the algorithm, which requires a moderate access to an oracle, we can
prove that the performance of the proposed adaptive scheme matches that of the fundamental limit
up to a constant factor. Finally, we quantify the gain of adaptivity by proving a strictly larger lower
bound on the budget required for any non-adaptive schemes. Precisely, we show that the minimax
rate on the budget required to achieve a target average error rate of ? scales as ?((m/??) log(1/?)).
The dependence on the prior F and G are solely captured in ? (the quality of the crowd as a whole)
and ? (the quality of the tasks as a whole). We show that the fundamental tradeoff for non-adaptive
schemes is ?((m/?min ?) log(1/?)), requiring a factor of ?/?min larger budget for non-adaptive
schemes. This factor of ?/?min is precisely how much we gain by adaptivity, and this gain can be
made arbitrarily large in the worst case distribution G .
2
Main Results
The following quantities are fundamental in capturing the dependence of the minimax rate on the
distribution of task difficulties and worker reliabilities:
?1
1
? ? EG
, ? ? EG [(2qi ? 1)2 ], and ? ? EF [(2pj ? 1)2 ] .
(2)
(2qi ? 1)2
Let n denote the total number of workers used, and Tj denote the set of all tasks assigned to worker
j ? [n] and Wi denote the set of all workers assigned to task i ? [m] until the adaptive task
assignment scheme has terminated. We consider discrete distribution G with K types of tasks of
varying difficulty levels. Define effective difficulty level of each task i to be ?i ? (2qi ? 1)2 , and
?min = mini?[m] ?i . A task with a small ?i is more difficult, since qi close to 1/2 means the task is
more
P ambiguous. Let ?a denote fraction of total tasks having difficulty level ?a for a ? [K] such that
a?[K] ?a = 1, and ?max ? maxa?[K] ?a , ?min ? mina?[K] ?a .
2.1
Fundamental limit under the adaptive scenario
We prove a lower bound on the minimax error rate: the error that is achieved by the best inference
algorithm t? using the best adaptive task assignment scheme ? under a worst case worker distribution
F and the worst-case?true answers t for the given distribution of?
difficulty level ?i ?s. Note that given
?i , either qi = (1 + ?i )/2 in which case ti = 1 or qi = (1 ? ?i )/2 and ti = ?1. Let T` be the
set of all task assignment schemes that use at most m` queries in total, and let F? be the set of all the
worker distributions such that expectation of worker quality is ?, i.e. F? ? {F |EF [(2pj ?1)2 ] = ?}.
Then we can show the following lower bound on the minimax rate on the probability of error. A
proof of this theorem is provided in Section 4 in the supplementary material.
Theorem 2.1. When ? < 1, there exists a positive constant C 0 such that for each task i ? [m],
min
max
m
? ?T` ,t? t?{?1} ,F ?F?
P[ti 6= t?i |?i ] ?
3
1 ?C 0 ?i ? E[|Wi | | ?i ]
e
.
2
This proves a lower bound on per task probability of error that decays exponentially with exponent
scaling as ?i ?E[|Wi | | ?i ]. The easier the task (?i large), the more reliable the workers are (? large),
and the more workers assigned to that task (|Wi | large), the smaller the achievable error. To get a
lower bound on the average probability of error, suppose we know the difficulties
of the tasks and
P
assign `a workers to tasks of difficulty ?a . With average budget constraint a?[K] `a ?a ? `,
m
min
max
m
? ?T` ,t? t?{?1} ,F ?F?
1 X
P[ti 6= t?i ] ?
m i=1
=
min
P
`a : a?[K] ?a `a =`
K
X
1
2
a=1
?a e?C
0
`a ?a ?
(3)
K
P
1 ?C 0 `?? X
?? a6=a0 (?a0 /?a0 ) log(?a /?a0 )
?a e
,
e
2
a=1
where the equality follows from solving the optimization problem. Note that the summand in the
bound does not depend upon the budget `, and it is lower bounded by ?min > 0. The error scales as
0
e?C `?? , where ? = 1/(E[1/?i ]) as defined in (2), and captures how difficult the set of tasks are
collectively. This gives a lower bound on the budget ? required to achieve error ?; there exists a
constant C 00 such that if
?min
00 m
log
,
(4)
?? ? C
??
?
then no task assignment scheme (adaptive or not) with any inference algorithm can achieve error less
than . Intuitively, ? captures the (collective) quality of the workers as specified by F and ? captures
the (collective) difficulty of the tasks as specified by G . This recovers the known fundamental limit
1
for standard DS model where all tasks have ?i = 1 and hence ? = 1 in [9]: ?? > C 000 m
? log .
2.2
Upper bound on the achievable error rate
We present an adaptive task assignment scheme and an iterative inference algorithm that asymptotically achieve an error rate of C1 e?(C? /4)`?? , when m grows large and ` = ?(log m) where
C1 = log2 (2?max /?min ) log2 (?1 /?K ). This matches the lower bound in (3) and the expected number of queries (or task-worker assignments) is bounded by m`. Comparing it to a fundamental lower
bound in Theorem 2.1 establishes the near-optimality of our approach, and the sufficient condition to
achieve average error ? is for the average total budget to be larger than,
C
m
1
?? ? C 0
log
.
(5)
??
?
2.2.1
Adaptive algorithm
Since difficulty level is varying across the tasks, it is intuitive to assign fewer workers to easy tasks
and more workers to hard tasks. Suppose we know the difficulty levels, then optimizing the lower
bound (3) over `?i ?s, it suggests to assign `?i ' `(?/?i ) workers to the task i with difficulty ?i , when
given a fixed budget of ` workers per task on average. However, the difficulty levels are not known.
Non-adaptive schemes can be arbitrarily worse (see Theorem 2.4). We propose a novel approach of
adaptively assigning workers in multiple rounds, refining our belief on ?i , and making decisions on
the tasks with higher confidence.
The main algorithmic component is the sub-routine in line 8-13 of Algorithm 1. For a choice of the
(per task) budget `t , we collect responses according to a (`t , rt = `t ) regular random graph on |M |
tasks and |M | workers. The leading eigen-vector of the non-backtracking operator on this bipartite
graph, weighted by the ?1 responses reveals a noisy observation of the true class and the difficulty
levels of the tasks. Let x ? R|M | denote the top left eigenvector, computed as per Algorithm 2.
Then the i-th entry xi asymptotically converges in the large number of tasks m limit to a Gaussian
random variable with mean proportional to the difficulty level (2qi ? 1), with mean and variance
specified in Lemma 5.1 in the the supplementary material. This non-backtracking operator approach
to crowdsourcing was first introduced in [7] for the standard DS model, is a single-round non-adaptive
scheme, and uses a threshold of zero to classify tasks based on the sign of xi ?s. We generalize their
analysis to this generalized DS model in Theorem 2.3 for finite sample regime, and further give a
sharper characterization based on central limit theorem in the asymptotic regime (Lemma 5.1 in the
supplementary material).
4
This provides us a sub-routine that reveals (2qi ? 1)?s we want, corrupted by additive Gaussian noise.
This resembles the setting in racing algorithms introduced in [12] where the goal is to choose the
variable (i.e. task) with largest mean (i.e. easiest) with minimal budget. However, our goal is to
identify the sign of the mean of the variables (i.e. classes) with sufficient accuracy. The key idea is
to classify the easier tasks first with minimal budget, and then classify the remaining more difficult
tasks with more budget allocated per task. We can set a threshold Xt,u at each round, and make a
permanent decision on a subset of tasks that have large xi ?s in absolute value, since those are the
tasks we are most confident about in its class, i.e. sign(2qi ? 1). We are now left to choose the budget
`t and the threshold Xt,u for each round.
We prescribe a choice using following notations. Assume that ?a ?s are indexed such that ?1 >
?2 > . . . > ?K . For simplicity, assume that ?K = ?1 2?(T ?1) for some T ? Z+ \ {1}. Given the
? a , ??a }a?[T ] which is supported
distribution {?a , ?a }a?[K] , we first bin it to get another distribution {?
? 1 = ?1 and ?
? a+1 = ?
? a 2?1 for each a ? [T ? 1]. ??a is the total
at most at T points. We take ?
fraction
?i is smaller than ? 1 2?(a?2) and larger than ?1 2?(a?1) . Precisely,
difficulty
Pof tasks whose
(a?1)
?
?a = a0 ?[K] ?a0 I ?1 /2
? ?a0 < ?1 /2(a?2) , for a ? [T ] . The choice of 2 for the ratio of
? a ?s is arbitrary and can be further optimized for a given distribution of ?i ?s. For ease of notations in
?
? a , ??a } ? , for T? ? T , such that
writing the algorithm, we re-index the binned distribution to get {?
a?[T ]
??a 6= 0 for all a ? T?. Note that T? ? dlog (?1 /?K )e.
2
We start with a set of all tasks M = [m]. A fraction of tasks are classified in each round and the
un-classified ones are taken to the next round. At round t ? {1, . . . , T?}, our goal is to classify
sufficient fraction of those tasks in the same difficulty group {i ? M : ?i = ?t } to be classified
with desired level of accuracy. If `t is too low and/or threshold Xt,u too small, then misclassification
rate will be too large. If `t is too large, we are wasting our budget unnecessarily. If Xt,u is too
? ?
? t and an appropriate Xt,u to
large, not enough tasks will be classified. We choose `t = `C? ?/
?(C? /4)??`
ensure that the misclassification probability is at most C1 e
based on the central limit
theorem on the leading eigen vector (see (21) in the supplementary material). We run this sub-routine
st = max{0, dlog2 (??t (1 + ?t )/??t+1 ?t+1 )e} times to ensure that enough fraction from t-th group is
classified. We make sure that the expected number of unclassified tasks is at most equal to the number
of tasks in the next group, i.e., difficulty level ?i = ?t+1 . We provide a near-optimal performance
guarantee for ?t = 1 for all t ? [T?], and ?t provides an extra degree of freedom for practitioners to
further optimize the efficiency.
? t ) that get classified
Note that statistically, the fraction of the t-th group (i.e. tasks with difficulty ?
before the t-th round is very small as the threshold set in these rounds is more than their absolute
? t will get classified in round t. Further, the binning of the original
mean message. Most tasks with ?
?
?
given distribution to get {?a , ?a } ensures that `t+1 ? 2`t . It ensures that the total extraneous budget
? t tasks is not more than a constant times the allocated budget of those tasks, and the
spent on ?
constant can be made one, by changing the initial choice of `1 by a constant factor.
2.2.2
Performance Guarantee
Since we are not wasting any budget on any of the tasks, with the right choice of the constant C? ,
we are guaranteed that this algorithm uses at most m`
P assignments in expectation. One caveat is
that, the threshold Xt,u depends on ?t,u = (1/|M |) i?[M ] ?i , which is the average difficulty of
the remaining tasks. As the remaining tasks are changing over the course of the algorithm, we
need to estimate this value in each sub-routine. We provide an estimator of ?t,u in Algorithm 3 (in
the supplementary material) that only uses the sampled responses that are already collected. All
numerical results are based on this estimator. However, analyzing the sensitivity of the performance
with respect to the estimation error in ?t,u is quite challenging, and for a theoretical analysis, we
assume we have access to an oracle that provides the exact value of ?t,u , replacing Algorithm 3.
P
Theorem 2.2. Suppose Algorithm 3 returns the exact value of ?t,u = (1/|M |) i?[M ] ?i . With
the choice of ?a = 1 for all a ? [T?] and C? = (4 + dlog(2?max /?min )e)?1 for any given distribution of task difficulty {?a , ?a }a?[K] of m tasks and an average number of workers per task
` = ?(logP
m), the expected number of queries made by Algorithm 1 is asymptotically bounded by
limm?? t?[T?],u?st `t E[|Mt,u |]/(m`) ? 1, where Mt,u is the number of tasks remaining at round
5
(t, u). Further, Algorithm 1 returns estimates {t?i }i?[m] that asymptotically achieves,
m
1 X
P[ti 6= t?i ] ? C1 e?(C? /4)`?? ,
m?? m
i=1
lim
(6)
where C1 = log2 (2?max /?min ) log2 (?1 /?K ) for ?? scaling as 1/` such that `?? = ?(1).
A proof of this theorem is provided in Section 5 in the supplementary material. This shows the nearoptimal sufficient condition of our approach in (5). The constant C? can be improved by optimizing
over the choice of ?a ?s by minimizing the expected number of queries that the algorithm makes.
Algorithm 1 Adaptive Task Assignment and Inference Algorithm
? a , ??a } ? , `, C? , {?a } ? , ?, ?, ? = E[2pj ? 1]
Require: m, {?
a?[T ]
Ensure: Estimate {t?i }i?[m]
?=
1: M ? {1, 2, ? ? ? , m}, ?
a?[T ]
P
2: for all t = 1, 2, ? ? ? , T? do
? ?
?
3:
`t ? (`C? ?)/
n lt , rt ? `t
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
? ?
a?[T? ] (?a /?a )
?1
m o
?
t)
I{t < T?} + 1 I{t = T?}
st ? max 0, log ???t (1+?
t+1 ?t+1
for all u = 1, 2, ? ? ? , st do
if M 6= ? then p
n ? |M | , k ? log |M |
Draw E ? {0, 1}|M |?n ? (`t , rt )-regular random graph
Collect answers {Ai,j ? {1,
?1}}(i,j)?E
{xi }i?M ? Algorithm 2 E, {Ai,j }(i,j)?E , k
?t,u ? Algorithm 3 [E, {Ai,j }(i,j)?E , `t , rt ]
p
? t ?`t (`t ? 1)(rt ? 1)?t,u ? k?1 I{t < T?} + 0 I{t = T?}
Xt,u ? ?
t?i = I{xi > Xt,u } ? I{xi < ?Xt,u } i?M , M ? {i ? M : |xi | ? Xt,u }
end if
end for
end for
Algorithm 2 Message-Passing Algorithm
Require: E ? {0, 1}|M |?n , {Aij ? {1, ?1}}(i,j)?E , kmax
Ensure: {xi ? R}i?[|M |]
1: for all (i, j) ? E do
(0)
2:
Initialize yj?i with random Zj?i ? N (1, 1)
3: end for
4: for all k = 1, 2, ? ? ? , kmax do
5:
for all (i, j) ? E do
P
(k)
6:
xi?j ? j 0 ?Wi \j Aij 0 yjk?1
0 ?i
7:
end for
8:
for all (i, j) ? E do
P
(k)
9:
yj?i ? i0 ?Tj \i Ai0 j xki0 ?j
10:
end for
11: end for
12: for all i ? [m] do
P
kmax ?1
13:
xi ? j?Wi Aij yj?i
14: end for
In Figure 1, we compare performance of our algorithm with majority voting and also non-adaptive
version of our Algorithm 1, where we assign to each task ` (the given budget) number of workers in
6
one round and set classification threshold Xt,u = 0 so as to classify all the tasks. This non-adaptive
special case has been introduced for the standard DS model in [9].
We make a slight modification to Algorithm 1. In the final round, when the classification threshold
is set to zero, we include all the responses collected thus far when running the message passing
Algorithm 2, and not just the fresh samples collected in that round. This creates dependencies between
rounds, which makes the analysis challenging. However, in practice we see improved performance
and it allows us to use the given fixed budget efficiently.
We run synthetic experiments with m = 1800 and fix n = 1800 for the non-adaptive version. The
crowds are generated from the spammer-hammer model with hammer probability equal to 0.3. In
the left panel, we take difficulty level ?a to be uniformly distributed over {1, 1/4, 1/16}, that gives
? = 1/7. In the right panel, we take ?a = 1 with probability 3/4, otherwise we take it to be 1/4 or
1/16 with equal probability, that gives ? = 4/13. As predicted from the theoretical analysis, our
adaptive algorithm improves significantly over its non-adaptive version. In particular, for the left
panel, the non-adaptive algorithm?s error scaling depends on smallest ?i that is 1/16 while for the
adaptive algorithm it scales with ? = 1/7. In the figure, it can be seen that the adaptive algorithm
requires approximately (7/16)` queries to acheive the same error as achieved by the non-adaptive
one using ` queries. This gap widens in the right panel to approximately (13/64) as predicted, and
the adaptive algorithm achieves zero error as the number of queries increase. For a fair comparison
with the non-adaptive version, we fix total budget to be m` and assign workers in each round until
the budget is exhausted. C? is 1 and st = 1 for t ? {1, 2, 3}.
probability of error
probability of error
0.1
0.1
0.01
0.01
0.001
0.001
0.0001
0.0001
1e-005
1e-006
1e-005
Majority voting
Non-adaptive
Adaptive
50
100
150
Majority voting
Non-adaptive
Adaptive
1e-006
200
250
300
350
50
number of queries per task `
100
150
200
250
300
number of queries per task `
Figure 1: Algorithm 1 improves significantly over its non-adaptive version and majority voting.
2.3
Achievable error rate under the non-adaptive scenario
Consider a non-adaptive version of our approach where we apply it for one round using an (`, r)
random regular graph, where ` is the given budget. Naturally, the classification threshold is set to
Xt,u = 0 so as to classify all the tasks. We provide a sharp upper bound on the achieved error, that
holds for all (non-asymptotic) regimes of m. Define ?k2 as
?r(??)2 k?1
1 ? 1/ `?
2?
1
2
+3 1+
(7)
?k ?
.
?r(??)2
r???
?r(??)2 k?1
1 ? 1/ `?
?2 `?
This captures the effective variance in the sub-Gaussian tail of the messages xi ?s after k iterations
of the inference algorithm (Algorithm 2), as shown in the proof of the following theorem (see the
supplementary material in Section 6).
Theorem 2.3. For any ` > 1 and r > 1, suppose m tasks are assigned according to a random
?r?2 ? 2 > 1, and r?? > 1, then
(`, r)-regular graph drawn from the configuration model. If ? > 0, `?
m
?
for any t ? {?1} , the estimate ti = sign(xi ) after k iterations of Algorithm 2 achieves
2
3`r ? 2k?2
(k)
P ti 6= t?i ?i ? e?`??i /(2?k ) +
(`?
r)
.
(8)
m
Therefore, the average error rate is bounded by
m
1 X
(k)
P[ti 6= t?i ] ?
m i=1
?`??i
3`r ? 2k?2
2
EG e 2?k +
(`?
r)
.
m
7
(9)
The second term, which is the probability that the resulting (`,
? r) regular random graph is not locally
tree-like, can be made small for large m as long as k = O( log m) (which is the choice we make
in Algorithm 1). Hence, the dominant term in the error bound is the first term. Further, when we
run our algorithm for large enough numbers of iterations, ?k2 converges linearly to a finite limit
2
2
?r??)2 /((`?
?r??)2 ? 1), which for large enough
??
? limk?? ?k2 such that ??
= 3 1 + 1/(?
r??) (`?
?
r??? and `?
r is upper bounded by
Hence, for a wide range of parameters, the average
a constant.
P
2
error in (9) is dominated by EG e?`??i /2?k = a ?a e?C`??a . When all ??s are strictly positive,
the error is dominated by the difficult tasks with ?min = mina ?a , as illustrated in Figure 2. Hence, it
is sufficient to have budget ?? ? C 00 m/(?min ?) log(1/?) to achieve an average error of ? > 0. Such
a scaling is also necessary as we show in the next section.
This is further illustrated in Figure 2. The error decays exponentially in ` and ? as predicted,
but the rate of decay crucially hinges on the difficulty level. We run synthetic experiments with
m = n = 1000 and the crowds are generated from the spammer-hammer model where pj = 1 with
probability ? and 1/2 otherwise. We fix ? = 0.3 and vary ` in the left figure and fix ` = 30 and
vary ? in the right figure. We let qi ?s take values in {0.6, 0.8, 1} with equal probability such that
? = 1.4/3. The error rate of each task grouped by their difficulty is plotted in the dashed lines,
2
matching predicted e??(`?(2qi ?1) ) . The average error rates in solid lines are dominated by those of
the difficult tasks, which is a universal drawback for all non-adaptive schemes.
probability of error
probability of error
1
1
0.1
0.1
0.01
0.01
0.001
0.001
q = 1.0
q = 0.8
q = 0.6
Mean Error
0.0001
1e-005
0
5
q = 1.0
q = 0.8
q = 0.6
Mean Error
0.0001
1e-005
10
15
20
25
30
0
number of queries per task `
0.1
0.2
0.3
0.4
crowd quality ?
Figure 2: Non-adaptive schemes suffer as average error is dominated by difficult tasks.
2.4
Fundamental limit under the non-adaptive scenario
Theorem 2.3 implies that it suffices to assign ` ? (c/(??i )) log(1/?) to achieve an error smaller than
? for a task i. We show in the following theorem that this scaling is also necessary. Hence, applying
one round of Algorithm 1 is near-optimal in the non-adaptive scenario compared to a minimax rate
where the nature chooses the worst distribution of worker pj ?s among the set of distributions with the
same ?. We provide a proof of the theorem in Section 7 in the supplementary material.
Theorem 2.4. There exists a positive constant C 0 and a distribution F of workers with average
reliability E[(2pj ? 1)2 ] = ? s.t. when ?i < 1, if the number of workers assigned to task i by
any non-adaptive task assignment scheme is less than (C 0 /(??i )) log(1/), then no algorithm can
achieve conditional probability of error on task i less than for any m and r.
Since in this non-adaptive scheme, task assignments are done a priori, there are on average ` workers
assigned to any set of tasks of the same difficulty. Hence, if the total budget is less than
m
?min
?? ? C 0
log
,
(10)
?min ?
?
then no algorithm can achieve average error less than ?, where ?min = mina ?a . Compared to the
adaptive case in (4) (nearly achieved in (5)), the gain of adaptivity is a factor
Pa of ?/?min . The RHS
is negative when ?min < P
?, and can be tightened to C 0 (m/?a ?) log( b=1 ?b /?) where a is the
a
smallest integer such that b=1 ?b > ?.
Acknowledgements
This work is supported by NSF SaTC award CNS-1527754, and NSF CISE award CCF-1553452.
8
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9
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5,664 | 6,125 | Improved Techniques for Training GANs
Tim Salimans
tim@openai.com
Ian Goodfellow
ian@openai.com
Wojciech Zaremba
woj@openai.com
Alec Radford
alec@openai.com
Vicki Cheung
vicki@openai.com
Xi Chen
peter@openai.com
Abstract
We present a variety of new architectural features and training procedures that we
apply to the generative adversarial networks (GANs) framework. Using our new
techniques, we achieve state-of-the-art results in semi-supervised classification on
MNIST, CIFAR-10 and SVHN. The generated images are of high quality as confirmed by a visual Turing test: our model generates MNIST samples that humans
cannot distinguish from real data, and CIFAR-10 samples that yield a human error
rate of 21.3%. We also present ImageNet samples with unprecedented resolution
and show that our methods enable the model to learn recognizable features of
ImageNet classes.
1
Introduction
Generative adversarial networks [1] (GANs) are a class of methods for learning generative models
based on game theory. The goal of GANs is to train a generator network G(z; ? (G) ) that produces
samples from the data distribution, pdata (x), by transforming vectors of noise z as x = G(z; ? (G) ).
The training signal for G is provided by a discriminator network D(x) that is trained to distinguish
samples from the generator distribution pmodel (x) from real data. The generator network G in turn
is then trained to fool the discriminator into accepting its outputs as being real.
Recent applications of GANs have shown that they can produce excellent samples [2, 3]. However,
training GANs requires finding a Nash equilibrium of a non-convex game with continuous, highdimensional parameters. GANs are typically trained using gradient descent techniques that are
designed to find a low value of a cost function, rather than to find the Nash equilibrium of a game.
When used to seek for a Nash equilibrium, these algorithms may fail to converge [4].
In this work, we introduce several techniques intended to encourage convergence of the GANs game.
These techniques are motivated by a heuristic understanding of the non-convergence problem. They
lead to improved semi-supervised learning peformance and improved sample generation. We hope
that some of them may form the basis for future work, providing formal guarantees of convergence.
All code and hyperparameters may be found at https://github.com/openai/improved-gan.
2
Related work
Several recent papers focus on improving the stability of training and the resulting perceptual quality
of GAN samples [2, 3, 5, 6]. We build on some of these techniques in this work. For instance, we
use some of the ?DCGAN? architectural innovations proposed in Radford et al. [3], as discussed
below.
One of our proposed techniques, feature matching, discussed in Sec. 3.1, is similar in spirit to
approaches that use maximum mean discrepancy [7, 8, 9] to train generator networks [10, 11].
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Another of our proposed techniques, minibatch features, is based in part on ideas used for batch
normalization [12], while our proposed virtual batch normalization is a direct extension of batch
normalization.
One of the primary goals of this work is to improve the effectiveness of generative adversarial
networks for semi-supervised learning (improving the performance of a supervised task, in this case,
classification, by learning on additional unlabeled examples). Like many deep generative models,
GANs have previously been applied to semi-supervised learning [13, 14], and our work can be seen
as a continuation and refinement of this effort. In concurrent work, Odena [15] proposes to extend
GANs to predict image labels like we do in Section 5, but without our feature matching extension
(Section 3.1) which we found to be critical for obtaining state-of-the-art performance.
3
Toward Convergent GAN Training
Training GANs consists in finding a Nash equilibrium to a two-player non-cooperative game.
Each player wishes to minimize its own cost function, J (D) (? (D) , ? (G) ) for the discriminator and
J (G) (? (D) , ? (G) ) for the generator. A Nash equilibirum is a point (? (D) , ? (G) ) such that J (D) is at a
minimum with respect to ? (D) and J (G) is at a minimum with respect to ? (G) . Unfortunately, finding Nash equilibria is a very difficult problem. Algorithms exist for specialized cases, but we are not
aware of any that are feasible to apply to the GAN game, where the cost functions are non-convex,
the parameters are continuous, and the parameter space is extremely high-dimensional.
The idea that a Nash equilibrium occurs when each player has minimal cost seems to intuitively motivate the idea of using traditional gradient-based minimization techniques to minimize each player?s
cost simultaneously. Unfortunately, a modification to ? (D) that reduces J (D) can increase J (G) , and
a modification to ? (G) that reduces J (G) can increase J (D) . Gradient descent thus fails to converge
for many games. For example, when one player minimizes xy with respect to x and another player
minimizes ?xy with respect to y, gradient descent enters a stable orbit, rather than converging to
x = y = 0, the desired equilibrium point [16]. Previous approaches to GAN training have thus
applied gradient descent on each player?s cost simultaneously, despite the lack of guarantee that this
procedure will converge. We introduce the following techniques that are heuristically motivated to
encourage convergence:
3.1
Feature matching
Feature matching addresses the instability of GANs by specifying a new objective for the generator
that prevents it from overtraining on the current discriminator. Instead of directly maximizing the
output of the discriminator, the new objective requires the generator to generate data that matches
the statistics of the real data, where we use the discriminator only to specify the statistics that we
think are worth matching. Specifically, we train the generator to match the expected value of the
features on an intermediate layer of the discriminator. This is a natural choice of statistics for the
generator to match, since by training the discriminator we ask it to find those features that are most
discriminative of real data versus data generated by the current model.
Letting f (x) denote activations on an intermediate layer of the discriminator, our new objective for
the generator is defined as: ||Ex?pdata f (x) ? Ez?pz (z) f (G(z))||22 . The discriminator, and hence
f (x), are trained in the usual way. As with regular GAN training, the objective has a fixed point
where G exactly matches the distribution of training data. We have no guarantee of reaching this
fixed point in practice, but our empirical results indicate that feature matching is indeed effective in
situations where regular GAN becomes unstable.
3.2
Minibatch discrimination
One of the main failure modes for GAN is for the generator to collapse to a parameter setting where
it always emits the same point. When collapse to a single mode is imminent, the gradient of the
discriminator may point in similar directions for many similar points. Because the discriminator
processes each example independently, there is no coordination between its gradients, and thus no
mechanism to tell the outputs of the generator to become more dissimilar to each other. Instead,
all outputs race toward a single point that the discriminator currently believes is highly realistic.
After collapse has occurred, the discriminator learns that this single point comes from the generator,
but gradient descent is unable to separate the identical outputs. The gradients of the discriminator
2
then push the single point produced by the generator around space forever, and the algorithm cannot
converge to a distribution with the correct amount of entropy. An obvious strategy to avoid this type
of failure is to allow the discriminator to look at multiple data examples in combination, and perform
what we call minibatch discrimination.
The concept of minibatch discrimination is quite general: any discriminator model that looks
at multiple examples in combination, rather than in isolation, could potentially help avoid collapse of the generator. In fact, the successful application of batch normalization in the discriminator by Radford et al. [3] is well explained from this perspective. So far, however, we
have restricted our experiments to models that explicitly aim to identify generator samples that
are particularly close together. One successful specification for modelling the closeness between
examples in a minibatch is as follows: Let f (xi ) ? RA denote a vector of features for input xi , produced by some intermediate layer in the discriminator. We then multiply the vector
f (xi ) by a tensor T ? RA?B?C , which results in a matrix Mi ? RB?C . We then compute
the L1 -distance between the rows of the resulting matrix Mi across samples i ? {1, 2, . . . , n}
and apply a negative exponential (Fig. 1): cb (xi , xj ) = exp(?||Mi,b ? Mj,b ||L1 ) ? R.
The output o(xi ) for this minibatch layer for a sample xi
is then defined as the sum of the cb (xi , xj )?s to all other
samples:
n
X
o(xi )b =
cb (xi , xj ) ? R
j=1
h
i
o(xi ) = o(xi )1 , o(xi )2 , . . . , o(xi )B ? RB
o(X) ? Rn?B
Next, we concatenate the output o(xi ) of the minibatch
layer with the intermediate features f (xi ) that were its Figure 1: Figure sketches how miniinput, and we feed the result into the next layer of the batch discrimination works. Features
discriminator. We compute these minibatch features sep- f (xi ) from sample xi are multiplied
arately for samples from the generator and from the train- through a tensor T , and cross-sample
ing data. As before, the discriminator is still required to distance is computed.
output a single number for each example indicating how
likely it is to come from the training data: The task of the discriminator is thus effectively still to
classify single examples as real data or generated data, but it is now able to use the other examples in
the minibatch as side information. Minibatch discrimination allows us to generate visually appealing
samples very quickly, and in this regard it is superior to feature matching (Section 6). Interestingly,
however, feature matching was found to work much better if the goal is to obtain a strong classifier
using the approach to semi-supervised learning described in Section 5.
3.3
Historical averaging
Pt
When applying this technique, we modify each player?s cost to include a term ||? ? 1t i=1 ?[i]||2 ,
where ?[i] is the value of the parameters at past time i. The historical average of the parameters can
be updated in an online fashion so this learning rule scales well to long time series. This approach is
loosely inspired by the fictitious play [17] algorithm that can find equilibria in other kinds of games.
We found that our approach was able to find equilibria of low-dimensional, continuous non-convex
games, such as the minimax game with one player controlling x, the other player controlling y, and
value function (f (x) ? 1)(y ? 1), where f (x) = x for x < 0 and f (x) = x2 otherwise. For
these same toy games, gradient descent fails by going into extended orbits that do not approach the
equilibrium point.
3.4
One-sided label smoothing
Label smoothing, a technique from the 1980s recently independently re-discovered by Szegedy et.
al [18], replaces the 0 and 1 targets for a classifier with smoothed values, like .9 or .1, and was
recently shown to reduce the vulnerability of neural networks to adversarial examples [19].
Replacing positive classification targets with ? and negative targets with ?, the optimal discriminator
data (x)+?pmodel (x)
becomes D(x) = ?ppdata
(x)+pmodel (x) . The presence of pmodel in the numerator is problematic
because, in areas where pdata is approximately zero and pmodel is large, erroneous samples from
3
pmodel have no incentive to move nearer to the data. We therefore smooth only the positive labels to
?, leaving negative labels set to 0.
3.5
Virtual batch normalization
Batch normalization greatly improves optimization of neural networks, and was shown to be highly
effective for DCGANs [3]. However, it causes the output of a neural network for an input example
x to be highly dependent on several other inputs x0 in the same minibatch. To avoid this problem
we introduce virtual batch normalization (VBN), in which each example x is normalized based on
the statistics collected on a reference batch of examples that are chosen once and fixed at the start
of training, and on x itself. The reference batch is normalized using only its own statistics. VBN is
computationally expensive because it requires running forward propagation on two minibatches of
data, so we use it only in the generator network.
4
Assessment of image quality
Generative adversarial networks lack an objective function, which makes it difficult to
compare performance of different models.
One intuitive metric of performance can be
obtained by having human annotators judge the visual quality of samples [2].
We
automate this process using Amazon Mechanical Turk (MTurk), using the web interface in figure Fig. 2 (live at http://infinite-chamber-35121.herokuapp.com/
cifar-minibatch/), which we use to ask annotators to distinguish between generated data
and real data. The resulting quality assessments of our models are described in Section 6.
Figure 2: Web interface given to annotators. Annotators are asked to distinguish computer generated images from
real ones.
A downside of using human annotators is that the metric
varies depending on the setup of the task and the motivation of the annotators. We also find that results change
drastically when we give annotators feedback about their
mistakes: By learning from such feedback, annotators are
better able to point out the flaws in generated images, giving a more pessimistic quality assessment. The left column of Fig. 2 presents a screen from the annotation process, while the right column shows how we inform annotators about their mistakes.
As an alternative to human annotators, we propose an automatic method to evaluate samples, which
we find to correlate well with human evaluation: We apply the Inception model1 [20] to every
generated image to get the conditional label distribution p(y|x). Images that contain meaningful
objects should have a conditional label distribution p(y|x)
with low entropy. Moreover, we expect
R
the model to generate varied images, so the marginal p(y|x = G(z))dz should have high entropy.
Combining these two requirements, the metric that we propose is: exp(Ex KL(p(y|x)||p(y))), where
we exponentiate results so the values are easier to compare. Our Inception score is closely related
to the objective used for training generative models in CatGAN [14]: Although we had less success
using such an objective for training, we find it is a good metric for evaluation that correlates very
well with human judgment. We find that it?s important to evaluate the metric on a large enough
number of samples (i.e. 50k) as part of this metric measures diversity.
5
Semi-supervised learning
Consider a standard classifier for classifying a data point x into one of K possible classes. Such
a model takes in x as input and outputs a K-dimensional vector of logits {l1 , . . . , lK }, that can
exp(lj )
be turned into class probabilities by applying the softmax: pmodel (y = j|x) = PK exp(l
. In
k)
k=1
supervised learning, such a model is then trained by minimizing the cross-entropy between the
observed labels and the model predictive distribution pmodel (y|x).
1
We use the pretrained Inception model from http://download.tensorflow.org/models/image/
imagenet/inception-2015-12-05.tgz. Code to compute the Inception score with this model will be made
available by the time of publication.
4
We can do semi-supervised learning with any standard classifier by simply adding samples from
the GAN generator G to our data set, labeling them with a new ?generated? class y = K + 1, and
correspondingly increasing the dimension of our classifier output from K to K + 1. We may then
use pmodel (y = K + 1 | x) to supply the probability that x is fake, corresponding to 1 ? D(x) in
the original GAN framework. We can now also learn from unlabeled data, as long as we know that
it corresponds to one of the K classes of real data by maximizing log pmodel (y ? {1, . . . , K}|x).
Assuming half of our data set consists of real data and half of it is generated (this is arbitrary), our
loss function for training the classifier then becomes
L = ?Ex,y?pdata (x,y) [log pmodel (y|x)] ? Ex?G [log pmodel (y = K + 1|x)]
= Lsupervised + Lunsupervised , where
Lsupervised = ?Ex,y?pdata (x,y) log pmodel (y|x, y < K + 1)
Lunsupervised = ?{Ex?pdata (x) log[1 ? pmodel (y = K + 1|x)] + Ex?G log[pmodel (y = K + 1|x)]},
where we have decomposed the total cross-entropy loss into our standard supervised loss function
Lsupervised (the negative log probability of the label, given that the data is real) and an unsupervised
loss Lunsupervised which is in fact the standard GAN game-value as becomes evident when we substitute D(x) = 1 ? pmodel (y = K + 1|x) into the expression:
Lunsupervised = ?{Ex?pdata (x) log D(x) + Ez?noise log(1 ? D(G(z)))}.
The optimal solution for minimizing both Lsupervised and Lunsupervised is to have
exp[lj (x)] = c(x)p(y=j, x)?j<K+1 and exp[lK+1 (x)] = c(x)pG (x) for some undetermined scaling function c(x). The unsupervised loss is thus consistent with the supervised loss in
the sense of Sutskever et al. [13], and we can hope to better estimate this optimal solution from
the data by minimizing these two loss functions jointly. In practice, Lunsupervised will only help if
it is not trivial to minimize for our classifier and we thus need to train G to approximate the data
distribution. One way to do this is by training G to minimize the GAN game-value, using the
discriminator D defined by our classifier. This approach introduces an interaction between G and
our classifier that we do not fully understand yet, but empirically we find that optimizing G using
feature matching GAN works very well for semi-supervised learning, while training G using GAN
with minibatch discrimination does not work at all. Here we present our empirical results using this
approach; developing a full theoretical understanding of the interaction between D and G using this
approach is left for future work.
Finally, note that our classifier with K + 1 outputs is over-parameterized: subtracting a general
function f (x) from each output logit, i.e. setting lj (x) ? lj (x) ? f (x)?j, does not change the
output of the softmax. This means we may equivalently fix lK+1 (x) = 0?x, in which case Lsupervised
becomes the standard supervised loss function of our original
classifier with K classes, and our
PK
Z(x)
discriminator D is given by D(x) = Z(x)+1
, where Z(x) = k=1 exp[lk (x)].
5.1
Importance of labels for image quality
Besides achieving state-of-the-art results in semi-supervised learning, the approach described above
also has the surprising effect of improving the quality of generated images as judged by human
annotators. The reason appears to be that the human visual system is strongly attuned to image
statistics that can help infer what class of object an image represents, while it is presumably less
sensitive to local statistics that are less important for interpretation of the image. This is supported
by the high correlation we find between the quality reported by human annotators and the Inception
score we developed in Section 4, which is explicitly constructed to measure the ?objectness? of a
generated image. By having the discriminator D classify the object shown in the image, we bias it to
develop an internal representation that puts emphasis on the same features humans emphasize. This
effect can be understood as a method for transfer learning, and could potentially be applied much
more broadly. We leave further exploration of this possibility for future work.
5
6
Experiments
We performed semi-supervised experiments on MNIST, CIFAR-10 and SVHN, and sample generation experiments on MNIST, CIFAR-10, SVHN and ImageNet. We provide code to reproduce the
majority of our experiments.
6.1
MNIST
The MNIST dataset contains 60, 000 labeled
images of digits. We perform semi-supervised
training with a small randomly picked fraction
of these, considering setups with 20, 50, 100,
and 200 labeled examples. Results are averaged
over 10 random subsets of labeled data, each
chosen to have a balanced number of examples
from each class. The remaining training images
are provided without labels. Our networks have
5 hidden layers each. We use weight normalization [21] and add Gaussian noise to the output Figure 3: (Left) samples generated by model durof each layer of the discriminator. Table 1 sum- ing semi-supervised training. Samples can be
marizes our results.
clearly distinguished from images coming from
Samples generated by the generator during MNIST dataset. (Right) Samples generated with
semi-supervised learning using feature match- minibatch discrimination. Samples are coming (Section 3.1) do not look visually appealing pletely indistinguishable from dataset images.
(left Fig. 3). By using minibatch discrimination
instead (Section 3.2) we can improve their visual quality. On MTurk, annotators were able to distinguish samples in 52.4% of cases (2000 votes total), where 50% would be obtained by random
guessing. Similarly, researchers in our institution were not able to find any artifacts that would allow them to distinguish samples. However, semi-supervised learning with minibatch discrimination
does not produce as good a classifier as does feature matching.
Model
20
DGN [22]
Virtual Adversarial [23]
CatGAN [14]
Skip Deep Generative Model [24]
Ladder network [25]
Auxiliary Deep Generative Model [24]
Our model
Ensemble of 10 of our models
Number of incorrectly predicted test examples
for a given number of labeled samples
50
100
1677 ? 452
1134 ? 445
221 ? 136
142 ? 96
333 ? 14
212
191 ? 10
132 ? 7
106 ? 37
96 ? 2
93 ? 6.5
86 ? 5.6
200
90 ? 4.2
81 ? 4.3
Table 1: Number of incorrectly classified test examples for the semi-supervised setting on permutation invariant MNIST. Results are averaged over 10 seeds.
6.2
CIFAR-10
Model
1000
Ladder network [25]
CatGAN [14]
Our model
Ensemble of 10 of our models
21.83?2.01
19.22?0.54
Test error rate for
a given number of labeled samples
2000
4000
19.61?2.09
17.25?0.66
20.40?0.47
19.58?0.46
18.63?2.32
15.59?0.47
8000
17.72?1.82
14.87?0.89
Table 2: Test error on semi-supervised CIFAR-10. Results are averaged over 10 splits of data.
CIFAR-10 is a small, well studied dataset of 32 ? 32 natural images. We use this data set to study
semi-supervised learning, as well as to examine the visual quality of samples that can be achieved.
For the discriminator in our GAN we use a 9 layer deep convolutional network with dropout and
weight normalization. The generator is a 4 layer deep CNN with batch normalization. Table 2
summarizes our results on the semi-supervised learning task.
6
Figure 4: Samples generated during semi-supervised training on CIFAR-10 with feature matching
(Section 3.1, left) and minibatch discrimination (Section 3.2, right).
When presented with 50% real and 50% fake data generated by our best CIFAR-10 model, MTurk
users correctly categorized 78.7% of images correctly. However, MTurk users may not be sufficiently familiar with CIFAR-10 images or sufficiently motivated; we ourselves were able to categorize images with > 95% accuracy. We validated the Inception score described above by observing
that MTurk accuracy drops to 71.4% when the data is filtered by using only the top 1% of samples
according to the Inception score. We performed a series of ablation experiments to demonstrate that
our proposed techniques improve the Inception score, presented in Table 3. We also present images
for these ablation experiments?in our opinion, the Inception score correlates well with our subjective judgment of image quality. Samples from the dataset achieve the highest value. All the models
that even partially collapse have relatively low scores. We caution that the Inception score should be
used as a rough guide to evaluate models that were trained via some independent criterion; directly
optimizing Inception score will lead to the generation of adversarial examples [26].
Samples
Model
Score ? std.
Real data
11.24 ? .12
Our methods
8.09 ? .07
-VBN+BN
7.54 ? .07
-L+HA
6.86 ? .06
-LS
6.83 ? .06
-L
4.36 ? .04
-MBF
3.87 ? .03
Table 3: Table of Inception scores for samples generated by various models for 50, 000 images.
Score highly correlates with human judgment, and the best score is achieved for natural images.
Models that generate collapsed samples have relatively low score. This metric allows us to avoid
relying on human evaluations. ?Our methods? includes all the techniques described in this work,
except for feature matching and historical averaging. The remaining experiments are ablation experiments showing that our techniques are effective. ?-VBN+BN? replaces the VBN in the generator
with BN, as in DCGANs. This causes a small decrease in sample quality on CIFAR. VBN is more
important for ImageNet. ?-L+HA? removes the labels from the training process, and adds historical
averaging to compensate. HA makes it possible to still generate some recognizable objects. Without
HA, sample quality is considerably reduced (see ?-L?). ?-LS? removes label smoothing and incurs a
noticeable drop in performance relative to ?our methods.? ?-MBF? removes the minibatch features
and incurs a very large drop in performance, greater even than the drop resulting from removing the
labels. Adding HA cannot prevent this problem.
6.3
SVHN
For the SVHN data set, we used the same architecture and experimental setup as for CIFAR-10.
Figure 5 compares against the previous state-of-the-art, where it should be noted that the model
7
of [24] is not convolutional, but does use an additional data set of 531131 unlabeled examples. The
other methods, including ours, are convolutional and do not use this data.
Model
Percentage of incorrectly predicted test examples
for a given number of labeled samples
500
1000
2000
Virtual Adversarial [23]
Stacked What-Where Auto-Encoder [27]
DCGAN [3]
Skip Deep Generative Model [24]
Our model
Ensemble of 10 of our models
24.63
23.56
22.48
16.61?0.24
8.11 ? 1.3
5.88 ? 1.0
18.44 ? 4.8
6.16 ? 0.58
Figure 5: (Left) Error rate on SVHN. (Right) Samples from the generator for SVHN.
6.4
ImageNet
We tested our techniques on a dataset of unprecedented scale: 128 ? 128 images from the
ILSVRC2012 dataset with 1,000 categories. To our knowledge, no previous publication has applied a generative model to a dataset with both this large of a resolution and this large a number
of object classes. The large number of object classes is particularly challenging for GANs due to
their tendency to underestimate the entropy in the distribution. We extensively modified a publicly
available implementation of DCGANs2 using TensorFlow [28] to achieve high performance, using
a multi-GPU implementation. DCGANs without modification learn some basic image statistics and
generate contiguous shapes with somewhat natural color and texture but do not learn any objects.
Using the techniques described in this paper, GANs learn to generate objects that resemble animals,
but with incorrect anatomy. Results are shown in Fig. 6.
Figure 6: Samples generated from the ImageNet dataset. (Left) Samples generated by a DCGAN.
(Right) Samples generated using the techniques proposed in this work. The new techniques enable
GANs to learn recognizable features of animals, such as fur, eyes, and noses, but these features are
not correctly combined to form an animal with realistic anatomical structure.
7
Conclusion
Generative adversarial networks are a promising class of generative models that has so far been
held back by unstable training and by the lack of a proper evaluation metric. This work presents
partial solutions to both of these problems. We propose several techniques to stabilize training
that allow us to train models that were previously untrainable. Moreover, our proposed evaluation
metric (the Inception score) gives us a basis for comparing the quality of these models. We apply
our techniques to the problem of semi-supervised learning, achieving state-of-the-art results on a
number of different data sets in computer vision. The contributions made in this work are of a
practical nature; we hope to develop a more rigorous theoretical understanding in future work.
2
https://github.com/carpedm20/DCGAN-tensorflow
8
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5,665 | 6,126 | Robust k-means: a Theoretical Revisit
Alexandros Georgogiannis
School of Electrical and Computer Engineering
Technical University of Crete, Greece
alexandrosgeorgogiannis at gmail.com
Abstract
Over the last years, many variations of the quadratic k-means clustering procedure
have been proposed, all aiming to robustify the performance of the algorithm in
the presence of outliers. In general terms, two main approaches have been developed: one based on penalized regularization methods, and one based on trimming
functions. In this work, we present a theoretical analysis of the robustness and
consistency properties of a variant of the classical quadratic k-means algorithm,
the robust k-means, which borrows ideas from outlier detection in regression. We
show that two outliers in a dataset are enough to breakdown this clustering procedure. However, if we focus on ?well-structured? datasets, then robust k-means
can recover the underlying cluster structure in spite of the outliers. Finally, we
show that, with slight modifications, the most general non-asymptotic results for
consistency of quadratic k-means remain valid for this robust variant.
1
Introduction
Let ? : R ? R+ be a lower semi-continuous (lsc) and symmetric function with minimum value
?(0). Given a set of points X n = {x1 , . . . , xn } ? Rp , consider the generalized k-means problem
(GKM) [7]
n
!
min Rn (c1 , . . . , ck ) =
min ?(||xi ? cl ||2 )
c1 ,...,ck
1?l?k
(GKM)
i=1
subject to cl ? Rp , l ? {1, . . . , k}.
Our aim is to find a set of k centers {c1 , . . . , ck } that minimize the clustering risk Rn . These centers
define a partition of X n into k clusters A = {A1 , . . . , Ak }, defined as
"
#
Al = x ? X n : l = argmin1?j?k ?(||x ? cj ||2 ) ,
(1)
where ties are broken randomly. Varying ? beyond the usual quadratic function (?(t) = t2 ) we
expect to gain some robustness against the outliers [9]. When ? is upper bounded by ?, the clusters
are defined as follows. For l ? k, let
"
#
Al = x ? X n : l = argmin1?j?k ?(||x ? cj ||2 ) and ?(||x ? cl ||2 ) ? ? ,
(2)
and define the extra cluster
"
#
Ak+1 = x ? X n : min ?(||x ? cj ||2 ) > ? .
1?j?k
(3)
This extra cluster contains points whose distance from their closest center, when measured according
to ?(||x?cl ||2 ), is larger than ? and, as will become clear later, it represents the set of outliers. From
now on, given a set of centers {c1 , . . . , ck }, we write just A = {A1 , . . . , Ak } and implicitly mean
A ? Ak+1 when ? is bounded.1
1
For a similar definition for the set of clusters induced by a bounded ? see also Section 4 in [2].
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Now, consider the following instance of (GKM), for the same set of points X n ,
n
"
#
!
1
min Rn? (c1 , . . . , ck ) =
min min ||xi ? cl ? oi ||22 + f? (||oi ||2 )
c1 ,...,ck
oi
1?l?k
2
i=1
$
%&
'
?(||xi ?cl ||2 )
p
subject to cl ? R , l = 1, . . . , k,
oi ? Rp , i = 1, . . . , n,
(RKM)
where f? : R ? R+ is a symmetric, lsc, proper2 and bounded from below function, with minimum
value f? (0), and ? a non-negative parameter. This problem is called robust k-means (RKM) and,
as we show later, it takes the form of (GKM) when ? equals the Moreau envelope of f? . The
problem (RKM) [5, 24] describes the following simple model: we allow each observation xi to take
on an ?error? term oi and we penalize the errors, using a group penalty, in order to encourage most
of the observations? errors to be equal to zero. We consider functions f? where the parameter ? ? 0
has the following effect: for ? = 0, all oi ?s may become arbitrary large (all observations are outliers),
while, for ? ? ?, all oi ?s become zero (no outliers); non-trivial cases occur for intermediate values
0 < ? < ?. Our interest is in understanding the robustness and consistency properties of (RKM).
Robustness: Although robustness is an important notion, it has not been given a standard technical
definition in the literature. Here, we focus on the finite sample breakdown point [18], which counts
how many outliers a dataset may contain without causing significant damage in the estimates of
the centers. Such damage is reflected to an arbitrarily large magnitude of at least one center. In
Section 3, we show that two outliers in a dataset are enough to breakdown some centers. On the other
hand, if we restrict our focus on some ?well structured? datasets, then (RKM) has some remarkable
robustness properties even if there is a considerable amount of contamination.
Consistency: Much is known about the consistency of (GKM) when the function ? is lsc and
increasing [11, 15]. It turns out that this case also includes the case of (RKM) when f? is convex
(see Section 3.1 for details). In Section 4, we show that the known non-asymptotic results about
consistency of quadratic k-means may remain valid even when f? is non-convex.
2
Preliminaries and some technical remarks
We start our analysis with a few technical tools from variational analysis [19]. Here, we introduce
the necessary notation and a lemma (the proofs are in the appendix). The Moreau envelope e?f (x)
with parameter ? > 0 (Definition 1.22 in [19]) of an lsc, proper, and bounded from below function
f : Rp ? R and the associated (possibly multivalued) proximal map Pf? : Rp ?
? Rp are
e?f (x) = minp
z?R
1
1
||x ? z||22 + f (z) and Pf? (x) = argminz?Rp ||x ? z||22 + f (z),
2?
2?
(4)
respectively. In order to simplify the notation, in the following, we fix ? to 1 and suppress the
superscript. The Moreau envelope is a continuous approximation from below of f having the same
set of minimizers while the proximal map gives the (possibly non-unique) minimizing arguments
in (4). For (GKM), we define ? : Rp ? R as ?(x) := ?(||x||2 ). Accordingly, for (RKM), we
define F? : Rp ? R as F? (x) := f? (||x||2 ). Thus, we obtain the following pairs:
1
(x ? o)2 + f? (o), Pf? (x) := argmino?R ef? (x), x ? R
(5a)
2
1
eF? (x) := minp ||x ? o||22 + F? (o), PF? (x) := argmino?Rp eF? (x), x ? Rp .
(5b)
o?R 2
Obviously, (RKM) is equivalent to (GKM) when ?(x) = eF? (x). Every map P : R ?
? R throughout
the text is assumed to be i) odd, i.e., P(?x) = ?P(x), ii) compact-valued, iii) non-decreasing, and
iv) have a closed graph. We know that for any such map there exists at least one function f? such
that P = Pf? (Proposition 3 in [26]).3 Finally, for our purposes (outlier detection), it is natural
ef? (x) := min
o?R
2
We call f proper if f (x) < ? for at least one x ? Rn , and f (x) > ?? for all x ? Rn ; in words, if the
domain of f is a nonempty set on which f is finite (see page 5 in [19]).
3
Accordingly, for a general function ? : R ? [0, ?) to be a Moreau envelope, i.e., ?(?) = ef? (?) as
defined in (5a) for some function f? , we require that ?(?) ? 12 | ? |2 is a concave function (Proposition 1 in [26]).
2
to require that v) P is a shrinkage rule, i.e., P(x) ? x, ?x ? 0. The following corollary is quite
straightforward and useful in the sequel.
Corollary 1. Using the notation in definitions (5a) and (5b), we have
x
PF? (x) =
Pf (||x||2 ) and eF? (x) = ef? (||x||2 ).
(6)
||x||2 ?
Passing from a model of minimization in terms of a single problem, like (GKM), to a model in which
a problem is expressed in a particular parametric form, like (RKM) with the Moreau envelope, the
description of optimality conditions is opened to the incorporation of the multivalued map PF? . The
next lemma describes the necessary conditions for a center cl to be (local) optimal for (RKM). Since
we deal with the general case, well known results, such as smoothness of the Moreau envelope or
convexity of its subgradients, can no longer be taken for granted.
?
Remark 1. Let ?(?) = eF? (?). The usual subgradient, denoted as ??(x),
is not sufficient to
characterize the differentiability properties of Rn? in (RKM). Instead, we use the (generalized) subd?
ifferential ??(x) (Definition 8.3 in [19]). For all x, we have ??(x)
? ??(x). Usually, the previous
two sets coincide at a point x. In this case, ? is called regular at x. However, it is common in
?
practice that the sets ??(x)
and ??(x) are different (for a detailed exposition on subgradients see
Chapter 8 in [19]; see also Example 1 in Appendix A.9).
Lemma 1. Let PF? : Rp ?
? Rp be a proximal map and set ?(?) = eF? (?). The necessary (generalized) first order conditions for the centers {c1 , . . . , ck } ? Rp to be optimal for (RKM) are
"!
# !
!
0??
?(xi ? cl ) ?
??(xi ? cl ) ?
(cl ? xi + PF? (xi ? cl )) , l ? {1, . . . , k}.
i?Al
i?Al
i?Al
(7)
The interpretation of the set inclusion above is the following: for any center cl ? Rp , every subgradient vector in ??(xi ? cl ) must be a vector associated with a vector in PF? (xi ? cl ) (Theorem
10.13 in [19]). However, in general, the converse does not hold true. We note that when the proximal
map is single-valued and continuous, which happens for example not only when f? is convex, but
also for many popular non-convex penalties, both set inclusions become equalities and the converse
holds, i.e., every vector in PF? (xi ? cl ) is a vector associated with a subgradient in ??(xi ? cl )
(Theorem 10.13 in [19] and Proposition 7 in [26]).
We close this section with some useful remarks on the properties of the Moreau envelope as a map
between two spaces of functions. There exist cases where two different functions, f? ?= f?? , have
equal Moreau envelopes, ef? = ef?? (Proposition 1 in [26]), implying that two different forms
of (RKM) correspond to the same ? in (GKM). For example, the proximal hull of f? , defined as
h?f? (x) := ?e?(?e? ) (x), is a function different from f? but has the same Moreau envelope as f?
f?
(see also Example 1.44 in [19], Proposition 2 and Example 3 in [26]). This is the main reason we
preferred the proximal map instead of the penalty function point of view for the analysis of (RKM).
3
On the breakdown properties of robust k-means
In this section, we study the finite sample breakdown point of (RKM) and, more specifically, its
universal breakdown point. Loosely speaking, the breakdown point measures the minimum fraction
of outliers that can cause excessive damage in the estimates of the centers. Here, it will become clear
how the interplay between the two forms, (GKM) and (RKM), helps the analysis. Given a dataset
n
X n = {x1 , . . . , xn } and a nonnegative integer m ? n, we say that Xm
is an m-modification if it
arises from X n after replacing m of its elements by arbitrary elements x?i ? Rp [6]. Denote as r(?)
the non-outlier samples, as counted after solving (RKM), for a dataset X n and some ? ? 0, i.e., 4
(
(
(
(
r(?) := ({xi ? X n : ||oi ||2 = 0, i = 1, . . . , n}(.
(8)
Then, the number of estimated outliers is q(?) = n ? r(?). In order to simplify notation, we drop
the dependence of r and q on ?. With this notation, we proceed to the following definition.
4
More than one ? can yield the same r, but this does not affect our analysis.
3
Definition 1 (universal breakdown point for the centers [6]). Let n, r, k be such that n ? r ? k + 1.
n
Given a dataset Xm
in Rp , let {c1 , . . . , ck } denote the (global) optimal set of centers for (RKM).
The universal breakdown value of (RKM) is
"m
#
?(n, r, k) := min
min
:
sup
max
||c
||
=
?
.
(9)
l
2
n 1?l?k
X n 1?m?n
n Xm
n
Here, X n = {x1 , . . . , xn } ? Rp while Xm
? Rp runs over all m-modifications of X n .
According to the concept of universal breakdown point, (RKM) breaks down at the first integer m
for which there exists a set X n such that the estimates of the cluster centers become arbitrarily bad
n
for a suitable modification Xm
. Our analysis is based on Pf? and considers two cases: those of
biased and unbiased proximal maps. The former corresponds to the class of convex functions f? ,
while the latter corresponds to a class of non-convex f? .
3.1
Biased proximal maps: the case of convex f?
If f? is convex, then ? = eF? is also convex while PF? is continuous, single-valued, and satisfies [19]
||x ? PF? (x)||2 ? ? as ||x||2 ? ?.
(10)
Proximal maps with this property are called biased since, as the l2 -norm of x increases, so does the
norm of the difference in (10). In this case, for each xi ? Al , from Lemma 1 and expression (10),
we have
||??(xi ?cl )||2 = ||?eF? (xi ?cl )||2 = ||cl ?xi +PF? (xi ?cl )||2 ? ? as ||xi ?cl ||2 ? ?. (11)
The supremum value of ||??(x ? cl )||2 is closely related to the gross error sensitivity of an estimator [9]. It is interpreted as the worst possible influence which a sample x can have on cl [7]. In view
of (11) and the definition of the clusters in (1), (RKM) is extremely sensitive. Although it can detect
an outlier, i.e., a sample xi with a nonzero estimate for ||oi ||2 , it does not reject it since the influence
of xi on its closest center never vanishes.5 The l1 -norm, f? (x) = ?|x|, which has Moreau envelope
equal to the Huber loss-function [24], is the limiting case for the class of convex penalty functions
that, although it keeps the difference ||x ? PF? (x)||2 in (10) constant and equal to ?, introduces a
bias term proportional to ? in the estimate cl . The following proposition shows that (RKM) with a
biased PF? has breakdown point equal to n1 , i.e., one outlier suffices to breakdown a center.
Proposition 1. Assume k ? 2, k + 1 < r ? n. Given a biased proximal map, there exist a dataset
X n and a modification X1n such that (RKM) breaks down.
3.2
Unbiased proximal maps: the case of non-convex f?
2
Consider now the l0 -(pseudo)norm on R, f? (z) := ?|z|0 = ?2 {z?=0} ,
thresholding proximal operator P?|?|0 : R ?
? R,
?
?0,
P?|?|0 (x) = arg minz?R 12 (x ? z)2 + f? (z) = {0, x},
?
x,
According to Lemma 1, for p = 1 (scalar case), we have
and the associated hard|x| < ?,
|x| = ?,
|x| > ?.
(12)
??(xi ? cl ) ? cl ? xi + P?|?|0 (xi ? cl ) = {0} for |xi ? cl | > ?, xi ? Al ,
(12)
(13)
implying that ?(xi ? cl ), as a function of,
cl , remains constant for |xi ? cl | > ?. As a consequence
of (13), if cl is local optimal, then 0 ? ?{ i?Al ?(xi ? cl )} and
!
! .
0?
(cl ? xi ) +
cl ? xi + P?|?| (xi ? cl ) .
(14)
i?Al ,
i?Al ,
|xi ?cl |<?
|xi ?cl |=?
Depending on the value of ?, (RKM) with the l0 -norm is able to ignore samples with distance from
their closest center larger than ?. This is done since P?|?|0 (xi ? cl ) = xi ? cl whenever |xi ? cl | > ?
5
See the analysis in [7] about the influence function of (GKM) when ? is convex.
4
and the influence of xi vanishes. In fact, there is a whole family of non-convex f? ?s whose proximal
map Pf? satisfies
Pf? (x) = x, for all |x| > ?,
(15)
for some ? > 0. These are called unbiased proximal maps [13, 20] and have the useful property
that, as one observation is arbitrarily modified, all estimated cluster centers remain bounded by a
constant that depends only on the remaining unmodified samples. Under certain circumstances, the
proof of the following proposition reveals that, if there exists one outlier in the dataset, then robust
k-means will reject it.
Proposition 2. Assume k ? 2, k + 1 < r ? n, and consider the dataset X n = {x1 , . . . , xn }
along with its modification by one replacement y, X1n = {x1 , . . . , xn?1 , y}. If we solve (RKM) with
X1n and an unbiased proximal map satisfying (15), then all estimates for the cluster centers remain
bounded by a constant that depends only on the unmodified samples of X n .
Next, we show that, even for this class of maps, there always exists a dataset that causes one of the
estimated centers to breakdown as two particular observations are suitably replaced.
Theorem 1 (Universal breakdown point for (RKM)). Assume k ? 2 and n ? r ? k + 2. Given an
unbiased proximal map satisfying (15), there exist a dataset X n and a modification X2n , such that
(RKM) breaks down.
Hence, the universal breakdown point of
(RKM) with an unbiased proximal map is n2 . In
Figure 1, we give a visual interpretation of Theorem 1. The top subfigure depicts the unmodified initial dataset X 9 = {x1 , . . . , x9 } (black
circles) with a clear two-cluster structure; the
bottom subfigure shows the modification X29
(dashed line arrows). Theorem 1 states that
(RKM) on X29 fails to be robust since, every
subset of X29 with r = 8 points has a cluster
containing an outlier.
3.3
Figure 1: The top subfigure is the unmodified
dataset X 9 . Theorem 1 states that every subset of
the modification X29 (bottom subfigure) with size
8 contains an outlier.
Restricted robustness of robust k-means for well-clustered data
The result of Theorem 1 is disappointing but it is not (RKM) to be blamed for the poor performance but the tight notion of the definition about the breakdown point [6, 7]; allowing any kind of
contamination in a dataset is a very general assumption.
In this section, we place two restrictions: i) we consider datasets where inlier samples can be covered
by unions of balls with centers that are ?far apart? each other, and ii) we ask a question different
from the finite sample breakdown point. We want to exploit as much as possible the results of [2]
concerning a new quantitative measure of noise robustness which compares the output of (RKM) on
a contaminated dataset to its output on the uncontaminated version of the dataset. Our aim is to show
that (RKM), with a certain class of proximal maps and datasets that are well-structured ignores the
influence of outliers when grouping the inliers.
First, we introduce Corollary 2 which states the form that Pf? should have in order the results
of [2] to apply to (RKM) and, second, we give details about the datasets which we consider as wellstructured. Using this corollary we are able to design proximal maps for which Theorems 3, 4, and
5 in [2] apply; otherwise, it is not clear how the analysis of [2] is valid for (RKM).
Let h : R ? R be a continuous function with the following properties:
1. h is odd and non-decreasing (h+ (?) is used to denote its restriction on [0, ?));
2. h is a shrinkage rule: 0 ? h+ (x) ? x, ?x ? [0, ?);
3. the difference x ? h+ (x) is non-decreasing, i.e., for 0 ? x1 ? x2 we have x1 ? h+ (x1 ) ?
x2 ? h+ (x2 ).
5
Define the map
?
?h(x),
Pf? (x) := {h(x), x},
?
x,
|x| < ?,
|x| = ?,
|x| > ?.
(16)
Multivaluedness of Pf? at |x| = ? signals that ef? is non-smooth at these points. An immediate
consequence for the Moreau envelope associated with the previous map is the following.
Corollary 2. Let the function g : [0, ?) ? [0, ?) be defined as
/ x
g(x) :=
(u ? h(u))du, x ? [0, ?).
(17)
0
Then, the Moreau envelope associated with Pf? in (16) is
ef? (x) = min{g(|x|), g(?)} = g(min{|x|, ?}).
(18)
Next, we define what it means for a dataset to be (?1 , ?2 )-balanced; this is the class of datasets
which we consider to be well-structured.
Definition 2 ((?1 , ?2 ) balanced dataset [2]). Assume that a set X n ? Rp has a subset I (inliers),
with at least n2 samples, and the following properties:
1. I =
0k
l=1
Bl , where Bl = B(bl , r) is a ball in Rp with bounded radius r and center bl ;
2. ?1 |I| ? |Bl | ? ?2 |I| for every l, where |Bl | is the number of samples in Bl and ?1 ,?2 > 0;
3. ||bl ? bl? ||2 > v for every l ?= l? , i.e., the centers of the balls are at least v > 0 apart.
Then, X n is a (?1 , ?2 )-balanced dataset.
We now state the form that Theorem 3 in [2] takes for (RKM).
Theorem 2 (Restricted robustness of (RKM)). If i) ef? is as in Corollary 2, i.e., ef? (||x||2 ) =
g(min{||x||2 , ?}), ii) X n has a (?1 , ?2 )-balanced subset of samples I with k balls, and
2
iii) the centers of the balls are at least v > 4r + 2g ?1 ( ?1?+?
g(r)) apart, then for ? ?
1
1
2
33
|I|
v
v
?1
the set of outliers X n \I has no effect on the
2, g
|X n \I| (?1 g( 2 ? 2r) ? (?1 + ?2 )g(r))
grouping of inliers I. In other words, if {x, y} ? Bl and {c1 , . . . , ck } are the optimal centers when
solving (RKM) for a ? as described before, then
l = argmin1?j?k ef? (||x ? cj ||2 ) = argmin1?j?k ef? (||y ? cj ||2 ).
For the sake of completeness, we give a proof of this theorem in the appendix. In a similar way, we
can recast the results of Theorems 4 and 5 in [2] to be valid for (RKM).
4
On the consistency of robust k-means
Let X n be a set with n independent and identically distributed random samples xi from a fixed but
unknown probability distribution ?. Let C? be the empirical optimal set of centers, i.e.,
C? := argminc1 ...,ck ?Rp Rn? (c1 , . . . , ck ).
(19)
C ? := argminc1 ...,ck ?Rp R? (c1 , . . . , ck ),
(20)
The population optimal set of centers is the set
where R? is the population clustering risk, defined as
/
"
#
1
R? (c1 , . . . , ck ) :=
min minp ||x ? cl ? o||22 + f? (||o||2 ) ?(dx).
1?l?k o?R 2
$
%&
'
(21)
?(||x?cl ||2 )=ef? (||x?cl ||2 )
Loss consistency and (simply) consistency for (RKM) require, respectively, that
n??
? n??
Rn? (C)
?? R? (C ? ) and C? ?? C ? .
6
(22)
? converges
In words, as the size n of the dataset X n increases, the empirical clustering risk Rn? (C)
? ?
almost surely to the minimum population risk R (C ) and (for n large enough) C? can effectively
replace the optimal set C ? in quantizing the unknown probability measure ?.
For the case of convex f? , non-asymptotic results describing the rate of convergence of Rn? to R
in (22) are already known ([11], Theorem 3). Noting that the Moreau envelope of a non-convex f?
belongs to a class of functions with polynomial discrimination [16] (the shatter coefficient of this
class is bounded by a polynomial) we give a sketch proof of the following result.
Theorem 3 (Consistency of (RKM)). Let the samples xi ? X n , i ? {1, . . . , n}, come from a fixed
but unknown probability measure ?. For any k ? 1 and any unbiased proximal map, we have
? ? R? (C ? ) and
lim ER? (C)
n??
lim C? ? C ? (convergence in probability).
n??
(23)
Theorem 3 reads like an asymptotic convergence result. However, its proof (given in the appendix)
uses combinatorial tools from Vapnik-Chervonenkis
4theory, revealing that the non-asymptotic rate
? to R? (C ? ) is of order O( log n/n) (see Corollary 12.1 in [4]).
of convergence of ER? (C)
5
Relating (RKM) to trimmed k-means
As the effectiveness of robust k-means on real world and synthetic data has already been evaluated [5,
24], the purpose of this section is to relate (RKM) to trimmed k-means (TKM) [7]. Trimmed kmeans is based on the methodology of ?impartial trimming?, which is a combinatorial problem
fundamentally different from (RKM). Despite their differences, the experiments show that, both
(RKM) and (TKM) perform remarkably similar in practice. The solution of (TKM) (which is also
a set of k centers) is the solution of quadratic k-means on the subsample containing ?n(1 ? ?)?
points with the smallest mean deviation (0 < ? < 1). The only common characteristic of (RKM)
and (TKM) is that they both have the same universal breakdown point, i.e., n2 , for arbitrary datasets.
Trimmed k-means takes as input a dataset X n , the number of clusters k, and a proportion of outliers
a ? (0, 1) to remove.6 A popular heuristic algorithm for (TKM) is the following. After the initialization, each iteration of (TKM) consists of the following steps: i) the distance of each observation
from its closest center is computed, ii) the top ?an? observations with larger distance from its closest center are removed, iii) the remaining points are used to update the centers. The previous three
steps are repeated untill the centers converge.7 As for robust k-means, we solve the (RKM) problem
with a coordinate optimization procedure (see Appendix A.9 for details).
The synthetic data for the experiments come from a mixture of Gaussians with 10 components and
without any overlap between them.8 The number of inlier samples is 500 and each inlier xi ?
[?1, 1]10 for i ? {1, . . . , 500}. On top of the inliers lie 150 outliers in R10 distributed uniformly
in general positions over the entire space. We consider two scenarios: in the first, the outliers lie
in [?3, 3]10 (call it mild-contamination), while, in the second, the outliers lie in [?6, 6]10 (call it
heavy-contamination). The parameter a in trimmed k-means (the percentage of outliers) is set to
a = 0.3, while the value of the parameter ? for which (RKM) yields 150 outliers is found through
a search over a grid on the set ? ? (0, ?max ) (we set ?max as the maximum distance between two
points in a dataset). Both algorithms, as they are designed, require as input an initial set of k points;
these points form the initial set of centers. In all experiments, both (RKM) and (TKM) take the same
k vectors as initial centers, i.e., k points sampled randomly from the dataset.
The statistics we use for the comparison are: i) the rand-index for clustering accuracy [17] ii) the
cluster estimation error, i.e., the root mean square error between the estimated cluster centers and
the sample mean of each cluster, iii) the true positive outlier detection rate, and finally, iv) the false
positive outlier detection rate. In Figures 2-3, we plot the results for a proximal map Pf like the one
in (16) with h(x) = ?x and ? = 0.005; with this choice for h, we mimic the hard-thresholding
operator. The results for each scenario (accuracy, cluster estimation error, etc) are averages over 150
runs of the experiment. As seen, both algorithms share almost the same statistics in all cases.
6
We use the implementation of trimmed k-means in the R package trimcluster [10].
The previous three steps are performed also by another robust variant of k-means, the k-means? (see [3]).
8
We use the R toolbox MixSim [14] that guarantees no overlap among the 10 mixtures.
7
7
0.5
?
?
?
?
?
?
?
7.5
5.0
?
?
0.950
0.925
?
?
robust k?means trimmed k?means
robust k?means trimmed k?means
0.015
0.010
?
0.005
?
?
0.000
robust k?meanstrimmed k?means
Cluster Radius Estimation Error
?
?
?
?
?
?
10.0
0.975
False Positive Error Rate
?
?
?
True Positive Error Rate
Accuracy
0.7
12.5
?
Center Estimation Error
?
?
0.9
?
?
?
9
?
?
?
6
3
?
?
?
?
0
robust k?meanstrimmed k?means
robust k?means trimmed k?means
0.4
?
?
?
?
?
?
?
?
?
?
0.2
?
?
15.0
12.5
10.0
?
robust k?means trimmed k?means
?
0.75
?
?
0.50
?
?
0.25
?
?
?
?
0.00
?
0.3
?
?
?
?
False Positive Error Rate
0.6
1.00
?
True Positive Error Rate
Accuracy
0.8
Center Estimation Error
17.5
robust k?means trimmed k?means
?
?
0.2
Cluster Radius Estimation Error
Figure 2: Performance of robust and trimmed k-means on a mixture of 10 Gaussians without overlap.
On top of the 500 samples from the mixture there are 150 outliers uniformly distributed in [?1, 1]10 .
?
?
?
?
?
?
?
?
?
?
0.1
?
0.0
robust k?means trimmed k?means
?
?
?
4
3
2
1
robust k?means trimmed k?means
robust k?means trimmed k?means
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
20
1.000
?
?
?
10
?
robust k?means trimmed k?means
?
?
?
?
?
0.975
?
?
0.950
0.925
0.900
robust k?means trimmed k?means
?
?
?
?
?
?
?
?
?
?
robust k?meanstrimmed k?means
False Positive Error Rate
0.6
Center Estimation Error
Accuracy
0.8
?
True Positive Error Rate
30
1.0
0.04
?
?
?
0.03
?
?
0.02
0.01
?
?
?
?
?
?
?
?
?
0.00
robust k?means trimmed k?means
Figure 4: Results on two spherical clusters with equal radius r, each one with 150 samples, and
centers are at least 4r apart. On top of the samples lie 150 outliers uniformly distributed in [?6, 6]10 .
In Figure 4, we plot the results for the case of two spherical clusters in R10 with equal radius r, each
one with 150 samples, and centers that are at least 4r apart from each other. The inlier samples are
in [?3, 3]10 . The outliers are 150 (half of the dataset is contaminated) and are uniformly distributed
in [?6, 6]10 . The results (accuracy, cluster estimation error, etc) are averages over 150 runs of the
experiment. This configuration is a heavy contamination scenario but, due to the structure of the
dataset, as expected from Theorem 2, (RKM) performs remarkably well; the same holds for (TKM).
6
Conclusions
We provided a theoretical analysis for the robustness and consistency properties of a variation of
the classical quadratic k-means called robust k-means (RKM). As a by-product of the analysis, we
derived a detailed description of the optimality conditions for the associated minimization problem.
In most cases, (RKM) shares the computational simplicity of quadratic k-means, making it a ?computationally cheap? candidate for robust nearest neighbor clustering. We show that (RKM) cannot
be robust against any type of contamination and any type of datasets, no matter the form of the
proximal map we use. If we restrict our attention to ?well-structured? datasets, then the algorithm
exhibits some desirable noise robustness. As for the consistency properties, we showed that most
general results for consistency of quadratic k-means still remain valid for this robust variant.
Acknowledgments
The author would like to thank Athanasios P. Liavas for useful comments and suggestions that
improved the quality of the article.
8
Cluster Radius Estimation Error
Figure 3: The same setup as in Figure 2 except that the coordinates of each outlier lie in [?3, 3]10 .
7.5
5.0
2.5
0.0
robust k?means trimmed k?means
References
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9
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5,666 | 6,127 | Stochastic Three-Composite Convex Minimization
? and Volkan Cevher
Alp Yurtsever, B`a? ng C?ng Vu,
Laboratory for Information and Inference Systems (LIONS)
?cole Polytechnique F?d?rale de Lausanne, Switzerland
alp.yurtsever@epfl.ch, bang.vu@epfl.ch, volkan.cevher@epfl.ch
Abstract
We propose a stochastic optimization method for the minimization of the sum of
three convex functions, one of which has Lipschitz continuous gradient as well
as restricted strong convexity. Our approach is most suitable in the setting where
it is computationally advantageous to process smooth term in the decomposition
with its stochastic gradient estimate and the other two functions separately with
their proximal operators, such as doubly regularized empirical risk minimization
problems. We prove the convergence characterization of the proposed algorithm in
expectation under the standard assumptions for the stochastic gradient estimate of
the smooth term. Our method operates in the primal space and can be considered as
a stochastic extension of the three-operator splitting method. Numerical evidence
supports the effectiveness of our method in real-world problems.
1
Introduction
We propose a stochastic optimization method for the three-composite minimization problem:
minimize f (x) + g(x) + h(x),
x?Rd
(1)
where f : Rd ? R and g : Rd ? R are proper, lower semicontinuous convex functions that admit
tractable proximal operators, and h : Rd ? R is a smooth function with restricted strong convexity.
We assume that we have access to unbiased, stochastic estimates of the gradient of h in the sequel,
which is key to scale up optimization and to address streaming settings where data arrive in time.
Template (1) covers a large number of applications in machine learning, statistics, and signal processing by appropriately choosing the individual terms. Operator splitting methods are powerful in this
setting, since they reduce the complex problem (1) into smaller subproblems. These algorithms are
easy to implement, and they typically exhibit state-of-the-art performance.
To our knowledge, there is no operator splitting framework that can currently tackle template (1)
using stochastic gradient of h and the proximal operators of f and g separately, which is critical to
the scalability of the methods. This paper specifically bridges this gap.
Our basic framework is closely related to the deterministic three operator splitting method proposed
in [11], but we avoid the computation of the gradient ?h and instead work with its unbiased estimates.
We provide rigorous convergence guarantees for our approach and provide guidance in selecting the
learning rate under different scenarios.
Road map. Section 2 introduces the basic optimization background. Section 3 then presents the
main algorithm and provides its convergence characterization. Section 4 places our contributions in
light of the existing work. Numerical evidence that illustrates our theory appears in Section 5. We
relegate the technical proofs to the supplementary material.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
2
Notation and background
This section recalls a few basic notions from the convex analysis and the probability theory, and
presents the notation used in the rest of the paper. Throughout, ?0 (Rd ) denotes the set of all proper,
lower semicontinuous convex functions from Rd to [??, +?], and h? | ?i is the standard scalar
product on Rd with its associated norm k ? k.
Subdifferential. The subdifferential of f ? ?0 (Rd ) at a point x ? Rd is defined as
?f (x) = {u ? Rd | f (y) ? f (x) ? hy ? x | ui , ?y ? Rd }.
We denote the domain of ?f as
dom(?f ) = {x ? Rd | ?f (x) 6= ?}.
If ?f (x) is a singleton, then f is a differentiable function, and ?f (x) = {?f (x)}.
Indicator function. Given a nonempty subset C in Rd , the indicator function of C is given by
0
if x ? C,
?C (x) =
+? if x 6? C.
Proximal operator. The proximal operator of a function f ? ?0 (Rd ) is defined as follows
1
2
proxf (x) = arg min f (z) + kz ? xk .
2
z?Rd
(2)
(3)
Roughly speaking, the proximal operator is tractable when the computation of (3) is cheap. If f is
the indicator function of a nonempty, closed convex subset C, its proximity operator is the projection
operator on C.
Lipschitz continuos gradient. A function f ? ?0 (Rd ) has Lipschitz continuous gradient with
Lipschitz constant L > 0 (or simply L-Lipschitz), if
k?f (x) ? ?f (y)k ? Lkx ? yk,
?x, y ? Rd .
Strong convexity. A function f ? ?0 (Rd ) is called strongly convex with some parameter ? > 0 (or
simply ?-strongly convex), if
hp ? q | x ? yi ? ?kx ? yk2 ,
?x, y ? dom(?f ), ?p ? ?f (x), ?q ? ?f (y).
Solution set. We denote optimum points of (1) by x? , and the solution set by X ? :
x? ? X ? = {x ? Rd | 0 ? ?h(x) + ?g(x) + ?f (x)}.
Throughout this paper, we assume that X ? is not empty.
Restricted strong convexity. A function f ? ?0 (Rd ) has restricted strong convexity with respect to
a point x? in a set M ? dom(?f ), with parameter ? > 0, if
hp ? q | x ? x? i ? ?kx ? x? k2 ,
?x ? M, ?p ? ?f (x), ?q ? ?f (x? ).
Let (?, F , P) be a probability space. An Rd -valued random variable is a measurable function
x : ? ? Rd , where Rd is endowed with the Borel ?-algebra. We denote by ?(x) the ?-field
generated by x. The expectation of a random variable x is denoted by E[x]. The conditional
expectation of x given a ?-field A ? F is denoted by E[x|A]. Given a random variable y : ? ? Rd ,
the conditional expectation of x given y is denoted by E[x|y]. See [17] for more details on probability
theory. An Rd -valued random process is a sequence (xn )n?N of Rd -valued random variables.
3
Stochastic three-composite minimization algorithm and its analysis
We present stochastic three-composite minimization method (S3CM) in Algorithm 1, for solving the
three-composite template (1). Our approach combines the stochastic gradient of h, denoted as r, and
the proximal operators of f and g in essentially the same structrure as the three-operator splitting
method [11, Algorithm 2]. Our technique is a nontrivial combination of the algorithmic framework
of [11] with stochastic analysis.
2
Algorithm 1 Stochastic three-composite minimization algorithm (S3CM)
Input: An initial point xf,0 , a sequence of learning rates (?n )n?N , and a sequence of squared
integrable Rd -valued stochastic gradient estimates (rn )n?N .
Initialization:
xg,0 = prox?0 g (xf,0 )
ug,0 = ?0?1 (xf,0 ? xg,0 )
Main loop:
for n = 0, 1, 2, . . . do
xg,n+1 = prox?n g (xf,n + ?n ug,n )
ug,n+1 = ?n?1 (xf,n ? xg,n+1 ) + ug,n
xf,n+1 = prox?n+1 f (xg,n+1 ? ?n+1 ug,n+1 ? ?n+1 rn+1 )
end for
Output: xg,n as an approximation of an optimal solution x? .
Theorem 1 Assume that h is ?h -strongly convex and has L-Lipschitz continuous gradient. Further
assume that g is ?g -strongly convex, where we allow ?g = 0. Consider the following update rule for
the learning rate:
p
??n2 ?h ? + (?n2 ?h ?)2 + (1 + 2?n ?g )?n2
?n+1 =
,
for some ?0 > 0 and ? ?]0, 1[.
1 + 2?n ?g
Define F n = ?(xf,k )0?k?n , and suppose that the following conditions hold for every n ? N:
1. E[rn+1 |F n ] = ?h(xg,n+1 ) almost surely,
2. There exists c ? [0, +?[ and t ? R, that satisfies
Pn
k=0
E[krk ? ?h(xg,k )k2 ] ? cnt .
Then, the iterates of S3CM satisfy
E[kxg,n ? x? k2 ] = O(1/n2 ) + O(1/n2?t ).
(4)
Remark 1 The variance condition of the stochastic gradient estimates in the theorems above is
satisfied when E[krn ? ?h(xg,n )k2 ] ? c for all n ? N and for some constant c ? [0, +?[. See
[15, 22, 26] for details.
Remark 2 When rn = ?h(xn ), S3CM reduces to the deterministic three-operator splitting scheme
[11, Algorithm 2] and we recover the convergence rate O(1/n2 ) as in [11]. When g is zero, S3CM
reduces to the standard stochastic proximal point algorithm [2, 13, 26].
Remark 3 Learning rate sequence (?n )n?N in Theorem 1 depends on the strong convexity parameter
?h , which may not be available a priori. Our next result avoids the explicit reliance on the strong
convexity parameter, while providing essentially the same convergence rate.
Theorem 2 Assume that h is ?h -strongly convex and has L-Lipschitz continuous gradient. Consider a positive decreasing learning rate sequence ?n = ?(1/n? ) for some ? ?]0, 1], and denote
? = limn?? 2?h n? ?n .
Define F n = ?(xf,k )0?k?n , and suppose that the following conditions hold for every n ? N:
1. E[rn+1 |F n ] = ?h(xg,n+1 ) almost surely,
2. E[krn ? ?h(xg,n )k2 ] is uniformly bounded by some positive constant.
3. E[kug,n ? x? k2 ] is uniformly bounded by some positive constant.
Then, the iterates of S3CM satisfy
?
O
?
?
?
O
E[kxg,n ? x? k2 ] =
?
O
?
?
O
1/n?
1/n?
(log n)/n
1/n
3
if
if
if
if
0<?<1
? = 1, and ? < 1
? = 1, and ? = 1,
? = 1, and ? > 1.
Proof outline. We consider the proof of three-operator splitting method as a baseline, and we use
the stochastic fixed point theory to derive the convergence of the iterates via the stochastic Fej?r
monotone sequence. See the supplement for the complete proof.
Remark 4 Note that ug,n ? ?g(xg,n ). Hence, we can replace condition 3 in Theorem 2 with the
bounded subgradient assumption: kpk ? c, ?p ? ?g(xg,n ), for some positive constant c.
Remark 5 (Restricted strong convexity) Let M be a subset of Rd that contains (xg,n )n?N and x? .
Suppose that h has restricted strong convexity on M with parameter ?h . Then, Theorems 1 and 2
still hold. An example role of the restricted strong convexity assumption on algorithmic convergence
can be found in [1, 21].
Remark 6 (Extension to arbitrary number of non-smooth terms.) Using the product space technique [5, Section 6.1], S3CM can be applied to composite problems with arbitrary number of
non-smooth terms:
m
X
minimize
fi (x) + h(x),
x?Rd
i=1
d
where fi : R ? R are proper, lower semicontinuous convex functions, and h : Rd ? R is a smooth
function with restricted strong convexity. We present this variant in Algorithm 2. Theorems 1 and 2
hold for this variant, replacing xg,n by xn , and ug,n by ui,n for i = 1, 2, . . . , m.
Algorithm 2 Stochastic m(ulti)-composite minimization algorithm (SmCM)
Input: Initial points {xf1 ,0 , xf2 ,0 , . . . , xfm ,0 }, a sequence of learning rates (?n )n?N , and a sequence of squared integrable Rd -valued stochastic gradient estimates (rn )n?N
Initialization:
Pm
x0 = m?1 i=1 xfi ,0
for i=1,2,. . . ,m do
ui,0 = ?0?1 (xfi ,0 ? x0 )
end for
Main loop:
for n = 0, 1, 2, . P
. . do
m
xn+1 = m?1 i=1 (xfi ,n + ?n ui,n )
for i=1,2,. . . ,m do
ui,n+1 = ?n?1 (xfi ,n ? xn+1 ) + ui,n
xfi ,n+1 = prox?n+1 mfi (xn+1 ? ?n+1 ui,n+1 ? ?n+1 rn+1 )
end for
end for
Output: xn as an approximation of an optimal solution x? .
Remark 7 With a proper learning rate, S3CM still converges even if h is not (restricted) strongly
convex under mild assumptions. Suppose that h has L-Lipschitz continuous gradient. Set the learning
rate such that ? ? ?n ? ? ? ?(2L?1 ? ?), for some ? and ? in ]0, 1[. Define F n = ?(xf,k )0?k?n ,
and suppose that the following conditions hold for every n ? N:
1. E[rn+1 |F n ] = ?h(xg,n+1 ) almost surely.
P
2
2.
n?N E[krn+1 ? ?h(xg,n+1 )k |F n ] < +? almost surely.
Then, (xg,n )n?N converges to a X ?-valued random vector almost surely. See [7] for details.
Remark 8 All the results above hold for any separable Hilbert space, except that the strong convergence in Remark 7 is replaced by weak convergence. Note however that extending Remark 7 to
variable metric setting as in [10, 27] is an open problem.
4
4
Contributions in the light of prior work
Recent algorithms in the operator splitting, such as generalized forward-backward splitting [24],
forward-Douglas-Rachford splitting [5], and the three-operator splitting [11], apply to our problem
template (1). These key results, however, are in the deterministic setting.
Our basic framework can be viewed as a combination of the three-operator splitting method in [11]
with the stochastic analysis.
The idea of using unbiased estimates of the gradient dates back to [25]. Recent developments
of this idea can be viewed as proximal based methods for solving the generic composite convex
minimization template with a single non-smooth term [2, 9, 12, 13, 15, 16, 19, 26, 23]. This generic
form arises naturally in regularized or constrained composite problems [3, 13, 20], where the smooth
term typically encodes the data fidelity. These methods require the evaluation of the joint prox of f
and g when applied to the three-composite template (1).
Unfortunately, evaluation of the joint prox is arguably more expensive compared to the individual
prox operators. To make comparison stark, consider the simple example where f and g are indicator
functions for two convex sets. Even if the projection onto the individual sets are easy to compute,
projection onto the intersection of these sets can be challenging.
Related literature also contains algorithms that solve some specific instances of template (1). To point
out a few, random averaging projection method [28] handles multiple constraints simultaneously
but cannot deal with regularizers. On the other hand, accelerated stochastic gradient descent with
proximal average [29] can handle multiple regularizers simultaneously, but the algorithm imposes a
Lipschitz condition on regularizers, and hence, it cannot deal with constraints.
To our knowledge, our method is the first operator splitting framework that can tackle optimization
template (1) using the stochastic gradient estimate of h and the proximal operators of f and g
separately, without any restriction on the non-smooth parts except that their subdifferentials are
maximally monotone. When h is strongly convex, under mild assumptions, and with a proper learning
rate, our algorithm converges with O(1/n) rate, which is optimal for the stochastic methods under
strong convexity assumption for this problem class.
5
Numerical experiments
We present numerical evidence to assess the theoretical convergence guarantees of the proposed
algorithm. We provide two numerical examples from Markowitz portfolio optimization and support
vector machines.
As a baseline, we use the deterministic three-operator splitting method [11]. Even though the random
averaging projection method proposed in [28] does not apply to our template (1) with its all generality,
it does for the specific applications that we present below. In our numerical tests, however, we
observed that this method exhibits essentially the same convergence behavior as ours when used
with the same learning rate sequence. For the clarity of the presentation, we omit this method in our
results.
5.1
Portfolio optimization
Traditional Markowitz portfolio optimization aims to reduce risk by minimizing the variance for a
given expected return. Mathematically, we can formulate this as a convex optimization problem [6]:
minimize E |aTi x ? b|2 subject to x ? ?, aTav x ? b,
x?Rd
where ? is the standard simplex for portfolios with no-short positions or a simple sum constraint,
aav = E [ai ] is the average returns for each asset that is assumed to be known (or estimated), and b
encodes a minimum desired return.
This problem has a streaming nature where new data points arrive in time. Hence, we typically do not
have access to the whole dataset, and the stochastic setting is more favorable. For implementation,
5
we replace the expectation with the empirical sample average:
p
minimize
x?Rd
1X T
(a x ? b)2
p i=1 i
subject to x ? ?, aTav x ? b.
(5)
This problem fits into our optimization template (1) by setting
p
h(x) =
1X T
(a x ? b)2 ,
p i=1 i
g(x) = ?? (x),
and
f (x) = ?{x | aTav x?b} (x).
We compute the unbiased estimates of the gradient by rn = 2(aTin x ? b)ain , where index in is
chosen uniformly random.
We use 5 different real portfolio datasets: Dow Jones industrial average (DJIA, with 30 stocks for
507 days), New York stock exchange (NYSE, with 36 stocks for 5651 days), Standard & Poor?s 500
(SP500, with 25 stocks for 1276 days), Toronto stock exchange (TSE, with 88 stocks for 1258 days)
that are also considered in [4]; and one dataset by Fama and French (FF100, 100 portfolios formed
on size and book-to-market, 23,647 days) that is commonly used in financial literature, e.g., [6, 14].
We impute the missing data in FF100 using nearest-neighbor method with Euclidean distance.
Figure 1: Comparison of the deterministic three-operators splitting method [11, Algorithm 2] and
our stochastic three-composite minimization method (S3CM) for Markowitz portfolio optimization
(5). Results are averaged over 100 Monte-Carlo simulations, and the boundaries of the shaded area
are the best and worst instances.
For the deterministic algorithm, we set ? = 0.1. We evaluate the Lipschitz constant L and the strong
convexity parameter ?h to determine the step-size. For the stochastic algorithm, we do not have
access to the whole data, so we cannot compute these parameter. Hence, we adopt the learning
rate sequence defined in Theorem 2. We simply use ?n = ?0 /(n + 1) with ?0 = 1 for FF100, and
?0 = 103 for others.1 We start both algorithms from the zero vector.
1
Note that a fine-tuned learning rate with a more complex definition can improve the empirical performance,
e.g., ?n = ?0 /(n + ?) for some positive constants ?0 and ?.
6
We split all the datasets into test (10%) and train (90%) partitions randomly. We set the desired
return as the average return over all assets in the training set, b = mean(aav ). Other b values exhibit
qualitatively similar behavior.
The results of this experiment are compiled in Figure 1. We compute the objective function over
the datapoints in the test partition, htest . We compare our algorithm against the deterministic threeoperator splitting method [11, Algorithm 2]. Since we seek statistical solutions, we compare the
algorithms to achieve low to medium accuracy. [11] provides other variants of the deterministic algorithm, including two ergodic averaging schemes that feature improved theoretical rate of convergence.
However, these variants performed worse in practice than the original method, and are omitted.
Solid lines in Figure 1 present the average results over 100 Monte-Carlo simulations, and the
boundaries of the shaded area are the best and worst instances. We also assess empirical evidence of
the O(1/n) convergence rate guaranteed in Theorem 2, by presenting squared relative distance to the
optimum solution for FF100 dataset. Here, we approximate the ground truth by solving the problem
to high accuracy with the deterministic algorithm for 105 iterations.
5.2
Nonlinear support vector machines classification
This section demonstrates S3CM on a support vector machines (SVM) for binary classification
problem. We are given a training set A = {a1 , a2 , . . . , ad } and the corresponding class labels
{b1 , b2 , . . . , bd }, where ai ? Rp and bi ? {?1, 1}. The goal is to build a model that assigns new
examples into one class or the other correctly.
As common in practice, we solve the dual soft-margin SVM formulation:
d
d
d
X
1 XX
K(ai , aj )bi bj xi xj ?
xi
2 i=1 j=1
i=1
minimize
x?Rd
subject to x ? [0, C]d , bT x = 0,
where C ? [0, +?[ is the penalty parameter and K : Rp ? Rp ? R is a kernel function. In our
example we use the Gaussian kernel given by K? (ai , aj ) = exp(??kai ? aj k2 ) for some ? > 0.
Define symmetric positive semidefinite matrix M ? Rd?d with entries Mij = K? (ai , aj )bi bj .
Then the problem takes the form
minimize
x?Rd
d
X
1 T
x Mx ?
xi
2
i=1
subject to x ? [0, C]d , bT x = 0.
(6)
This problem fits into three-composite optimization template (1) with
h(x) =
d
X
1 T
x Mx ?
xi ,
2
i=1
g(x) = ?[0,C]d (x),
and
f (x) = ?{x | bT x=0} (x).
One can solve this problem using three-operator splitting method [11, Algorithm 1]. Note that proxf
and proxg , which are projections onto the corresponding constraint sets, incur O(d) computational
cost, whereas the cost of computing the gradient is O(d2 ).
To compute an unbiased gradient estimate, we choose an index in uniformly random, and we form
rn = dM in xin ? 1. Here M in denotes ith
n column of matrix M , and 1 represents the vector of ones.
We can compute rn in O(d) computations, hence each iteration of S3CM costs an order cheaper
compared to deterministic algorithm.
We use UCI machine learning dataset ?a1a?, with d = 1605 datapoints and p = 123 features [8, 18].
Note that our goal here is to demonstrate the optimization performance of our algorithm for a real
world problem, rather than competing the prediction quality of the best engineered solvers. Hence,
to keep experiments simple, we fix problem parameters C = 1 and ? = 2?2 , and we focus on the
effects of algorithmic parameters on the convergence behavior.
Since p < d, M is rank deficient and h is not strongly convex. Nevertheless we use S3CM with the
learning rate ?n = ?0 /(n + 1) for various values of ?0 . We observe O(1/n) empirical convergence
rate on the squared relative error for large enough ?0 , which is guaranteed under restricted strong
convexity assumption. See Figure 2 for the results.
7
Figure 2: [Left] Convergence of S3CM in the squared relative error with learning rate
?n = ?0 /(n + 1). [Right] Comparison of the deterministic three-operators splitting method [11,
Algorithm 1] and S3CM with ?0 = 1 for SVM classification problem. Results are averaged over 100
Monte-Carlo simulations. Boundaries of the shaded area are the best and worst instances.
Acknowledgments
This work was supported in part by ERC Future Proof, SNF 200021-146750, SNF CRSII2-147633,
and NCCR-Marvel.
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9
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5,667 | 6,128 | Normalized Spectral Map Synchronization
Yanyao Shen
UT Austin
Austin, TX 78712
shenyanyao@utexas.edu
Qixing Huang
TTI Chicago and UT Austin
Austin, TX 78712
huangqx@cs.utexas.edu
Nathan Srebro
TTI Chicago
Chicago, IL 60637
nati@ttic.edu
Sujay Sanghavi
UT Austin
Austin, TX 78712
sanghavi@mail.utexas.edu
Abstract
Estimating maps among large collections of objects (e.g., dense correspondences
across images and 3D shapes) is a fundamental problem across a wide range of
domains. In this paper, we provide theoretical justifications of spectral techniques
for the map synchronization problem, i.e., it takes as input a collection of objects
and noisy maps estimated between pairs of objects along a connected object graph,
and outputs clean maps between all pairs of objects. We show that a simple
normalized spectral method (or NormSpecSync) that projects the blocks of the top
eigenvectors of a data matrix to the map space, exhibits surprisingly good behavior
? NormSpecSync is much more efficient than state-of-the-art convex optimization
techniques, yet still admitting similar exact recovery conditions. We demonstrate
the usefulness of NormSpecSync on both synthetic and real datasets.
1
Introduction
The problem of establishing maps (e.g., point correspondences or transformations) among a collection
of objects is connected with a wide range of scientific problems, including fusing partially overlapped
range scans [1], multi-view structure from motion [2], re-assembling fractured objects [3], analyzing
and organizing geometric data collections [4] as well as DNA sequencing and modeling [5]. A
fundamental problem in this domain is the so-called map synchronization, which takes as input noisy
maps computed between pairs of objects, and utilizes the natural constraint that composite maps
along cycles are identity maps to obtain improved maps.
Despite the importance of map synchronization, the algorithmic advancements on this problem remain
limited. Earlier works formulate map synchronization as solving combinatorial optimizations [1, 6, 7,
8]. These formulations are restricted to small-scale problems and are susceptible to local minimums.
Recent works establish the connection between the cycle-consistency constraint and the low-rank
property of the matrix that stores pairwise maps in blocks; they cast map synchronization as low-rank
matrix inference [9, 10, 11]. These techniques exhibit improvements on both the theoretical and
practical sides. In particular, they admit exact recovery conditions (i.e., on the underlying maps can
be recovered from noisy input maps). Yet due to the limitations of convex optimization, all of these
methods do not scale well to large-scale datasets.
In contrast to convex optimizations, we demonstrate that spectral techniques work remarkably well
for map synchronization. We focus on the problem of synchronizing permutations and introduce
a robust and efficient algorithm that consists of two simple steps. The first step computes the top
eigenvectors of a data matrix that encodes the input maps, and the second step rounds each block of
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
the top-eigenvector matrix into a permutation matrix. We show that such a simple algorithm possesses
a remarkable denoising ability. In particular, its exact recovery conditions match the state-of-the-art
convex optimization techniques. Yet computation-wise, it is much more efficient, and such a property
enables us to apply the proposed algorithm on large-scale dataset (e.g., many thousands of objects).
Spectral map synchronization has been considered in [12, 13] for input observations between all
pairs of objects. In contrast to these techniques, we consider incomplete pair-wise observations, and
provide theoretical justifications on a much more practical noise model.
2
Algorithm
In this section, we describe the proposed algorithm for permutation synchronization. We begin with
the problem setup in Section 2.1. Then we introduce the algorithmic details in Section 2.2.
2.1
Problem Setup
Suppose we have n objects S1 , ? ? ? , Sn . Each object is represented by m points (e.g., feature points
on images and shapes). We consider bijective maps ?ij : Si ? Sj , 1 ? i, j ? n between pairs of
objects. Following the convention, we encode each such map ?ij as a permutation matrix Xij ? Pm ,
where Pm is the space of permutation matrices of dimension m:
Pm := {X|X ? [0, 1]m?m , X1m = 1m , X T 1m = 1m },
where 1m = (1, ? ? ? , 1)T ? Rm is the vector whose elements are 1.
in
The input permutation synchronization consists of noisy permutations Xij
? G along a connected
object graph G. As described in [4, 9], a widely used pipeline to generate such input is to 1) establish
the object graph G by connecting each object and similar objects using object descriptors (e.g.,
HOG [14] for images) , and 2) apply off-the-shelf pair-wise object matching methods to compute the
input pair-wise maps (e.g., SIFTFlow [15] for images and BIM [16] for 3D shapes).
The output consists of improved maps between all of objects
Xij , 1 ? i, j ? n.
2.2
Algorithm
We begin with defining a data matrix X obs ? Rnm?nm that encodes the initial pairwise maps in
blocks:
(
in
? 1 Xij
, (i, j) ? G
obs
di dj
(1)
Xij =
0,
otherwise
where di := |{Sj |(Si , Sj ) ? G}| is the degree of object Si in graph G.
Remark 1. Note that the way we encode the data matrix is different from [12, 13] in the sense that
we follow the common strategy for handling irregular graphs and use a normalized data matrix.
The proposed algorithm is motivated from the fact that when the input pair-wise maps are correct, the
correct maps between all pairs of objects can be recovered from the leading eigenvectors of X obs :
Proposition 2.1. Suppose there exist latent maps (e.g., the ground-truth maps to one object) Xi , 1 ?
in
i ? n so that Xij
= XjT Xi , (i, j) ? G. Denote W ? Rnm?m as the matrix that collects the first m
obs
eigenvectors of X in its columns. Then the underlying pair-wise maps can be computed from the
corresponding matrix blocks of matrix W W T :
Pn
di
T
Xj Xi = pi=1 (W W T )ij , 1 ? i, j ? n.
(2)
di dj
The key insight of the proposed approach is that even when the input maps are noisy (i.e., the blocks
of X obs are corrupted), the leading eigenvectors of X obs are still stable under these perturbations
(we will analyze this stability property in Section 3). This motivates us to design a simple two-step
permutation synchronization approach called NormSpecSync. The first step of NormSpecSync
computes the leading eigenvectors of W ; the second step of NormSpecSync rounds the induced
2
Algorithm 1 NormSpecSync
Input: Xobs based on (1), ?max
Initialize W0 : set W0 as an initial guess for the top-m orthonormal eigenvectors, k ? 0
while kW (k) ? W (k?1) k > ?max do
+
W (k+1) = X obs ? W (k) ,
+
W (k+1) R(k+1) = W (k+1) ,
(QR factorization),
k ? k + 1.
end while
spec
Set W = W (k) and X i1 = (W W T )i1 .
spec
Round each X i1 into the corresponding Xi1 by solving (3).
T
Xi1 , 1 ? i, j ? n.
Output: Xij = Xj1
matrix blocks (2) into permutations. In the following, we elaborate these two steps and analyze the
complexity. Algorithm 1 provides the pseudo-code.
Leading eigenvector computation. Since we only need to compute the leading m eigenvectors of
X obs , we propose to use generalized power method. This is justified by the observation that usually
there exists a gap between ?m and ?m+1 . In fact, when the input pair-wise maps are correct, it is
easy to derive that the leading eigenvectors of X obs are given by:
?1 (X obs ) = ? ? ? = ?m (X obs ) = 1, ?m+1 (X obs ) = ?n?1 (G),
where ?n?1 (G) is the second largest eigenvalue of the normalized adjacency matrix of G. As we
will see later, the eigen-gap ?m (X obs ) ? ?m+1 (X obs ) is still persistent in the presence of corrupted
pair-wise maps, due to the stability of eigenvalues under perturbation.
Pn
d
spec
Projection onto Pm . Denote Xij
:= ?i=1 i (W W T )ij . Since the underlying ground-truth maps
di dj
spec
T
Xij , 1 ? i, j ? n obey Xij = Xjk
Xik , 1 ? i, j ? n for any fixed k, we only need to round Xik
into Xik . Without losing generality, we set k = 1 in this paper.
The rounding is done by solving the following constrained optimization problem, which projects
obs
Xi1
onto the space of permutations via the Frobenius norm:
2
obs
Xi1 = arg min kX ? Xi1
kF = arg min
X?Pm
X?Pm
2
2
obs
obs
kXkF + kXi1
kF ? 2hX, Xi1
i
obs
= arg max hX, Xi1
i.
(3)
X?Pm
The optimization problem described in (3) is the so-called linear assignment problem, which can be
solved exactly using the Hungarian algorithm whose complexity is O(m3 ) (c.f. [17]). Note
that the
Pn
d
obs
optimal solution of (3) is invariant under global scaling and shifting of Xi1
, so we omit ?i=1 i and
di dj
T
1
m 11
when generating
obs
Xij
(See Algorithm 1).
Time complexity of NormSpecSync. Each step of the generalized power method consists of a matrixvector multiplication and a QR factorization. The complexity of the matrix-vector multiplication,
which leverages of the sparsity in X obs , is O(nE ? m2 ), where nE is the number of edges in G.
The complexity of each QR factorization is O(nm3 ). As we will analyze laser, generalized power
method converges linearly, and setting ?max = 1/n provides a sufficiently accurate estimation
of the leading eigenvectors.
So the total time complexity of the Generalized power method is
O (nE m2 + nm3 log(n)). The time complexity of therounding step is O(nm3 ). In summary, the
total complexity of NormSpecSync is O (nE m2 + nm3 log(n)). In comparison, the complexity of
the SDP formulation [9], even when it is solved using the fast ADMM method (alternating direction
of multiplier method), is at least O(n3 m3 nadmm . So NormSpecSync exhibits significant speedups
when compared to SDP formulations.
3
3
Analysis
In this section, we provide an analysis of NormSpecSync under a generalized Erd?os-R?nyi noise
model.
3.1
Noise Model
The noise model we consider is given by two parameters m and p. Specifically, we assume the
observation graph G is fixed. Then independently for each edge (i, j) ? E,
Im with probability p
in
Xij =
(4)
Pij with probability 1 ? p
where Pij ? Pm is a random permutation.
Remark 2. The noise model described above assumes the underlying permutations are identity maps.
In fact, one can assume a generalized noise model
T
Xj1 Xi1 with probability p
in
Xij
=
Pij
with probability 1 ? p
where Xi1 , 1 ? i ? n are pre-defined underlying permutations from object Si to the first object S1 .
However, since Pij are independent of Xi1 . It turns out the model described above is equivalent to
Im with probability p
in T
Xj1 Xij Xi1 =
Pij with probability 1 ? p
Where Pij are independent random permutations. This means it is sufficient to consider the model
described in (4).
Remark 3. The fundamental difference between our model and the one proposed in [11] or the ones
used in low-rank matrix recovery [18] is that the observation pattern (i.e., G) is fixed, while in other
models it also follows a random model. We argue that our assumption is more practical because
the observation graph is constructed by comparing object descriptors and it is dependent on the
distribution of the input objects. On the other hand, fixing G significantly complicates the analysis of
NormSpecSync, which is the main contribution of this paper.
3.2
Main Theorem
Now we state the main result of the paper.
P
Theorem 3.1. Let dmin := min1?i?n di , davg := i di /n, and denote ? as the second top eigen?
value of normalized adjacency matrix of G. Assume dmin = ?( n ln3 n), davg = O(dmin ),
? < min{p, 1/2}. Then under the noise model described above, NormSpecSync recovers the
underlying pair-wise maps with high probability if
p>C?
ln3 n
? ,
dmin / n
(5)
for some constant C.
Proof Roadmap. The proof of Theorem 3.1 combines two stability bounds. The first one considers
the projection step:
Proposition 3.1. Consider a permutation matrix X = (xij ) ? Pm and another matrix X = (xij ) ?
Rm?m . If kX ? Xk < 12 , then
2
X = arg minkY ? XkF .
Y ?Pm
Proof. The proof is quite straight-forward. In fact,
kX ? Xk? ? kX ? Xk <
4
1
.
2
Varying graph density
Varying graph density
0.1
0.2
0.2
0.2
0.3
0.4
p-true
0.1
p-true
p-true
Varying graph density
0.1
0.3
0.4
0.4
0.5
0.5
0.5
0.1
0.3
0.5
0.7
0.9
0.3
0.1
0.3
0.5
0.7
0.1
0.9
0.3
0.5
0.7
0.9
Graph density
Graph density
Graph density
(a) NormSpecSync (2.25 seconds)
(b) SDP (203.12 seconds)
(c) DiffSync(1.07 seconds)
Figure 1: Comparisons between NormSpecSync, SDP[9], DiffSync[13] on the noise model described in Sec. 2.
This means the corresponding element xij of each non-zero element in xij is dominant in its row and
column, i.e.,
xij 6= 0 ? xij > max(max xik , max xkj ),
k6=j
k6=i
which ends the proof.
The second bound concerns the block-wise stability of the leading eigenvectors of X
Lemma 3.1. Under the assumption of Theorem 3.1, then w.h.p.,
Pn
i=1 di
1
T
?
d d (W W )i1 ? Im
< 3 , 1 ? i ? n.
i 1
obs
:
(6)
It is easy to see that we can prove Theorem 3.1 by combing Lemma 3.1 and Prop. 3.1. Yet unlike
Prop. 3.1, the proof of Lemma 3.1 is much harder. The major difficulty is that (6) requires controlling
each block of the leading eigenvectors, namely, it requires a L? bound, whereas most stability results
on eigenvectors are based on the L2 -norm. Due to space constraint, we defer the proof of Lemma 3.1
to Appendix A and the supplemental material.
Near-optimality of NormSpecSync. Theorem 3.1 implies that NormSpecSync is near-optimal with
respect to the information theoretical bound described in [19]. In fact, when G is a clique, (5) becomes
3
p > C ? ln?(n)
, which aligns with the lower bound in [19] up to a polylogarithmic factor. Following
n
the model described in [19], we can also assume that the observation graph G is sampled with a
density factor q, namely, two objects are connected independently with probability q. In this case,
4
? n . This bound also
it is easy to see that dmin > O(nq/ ln n) w.h.p., and (5) becomes p > C ? ln
nq
stays within a polylogarithmic factor from the lower bound in [19], indicating the near-optimality of
NormSpecSync.
4
Experiments
In this section, we perform quantitative evaluations of NormSpecSync on both synthetic and real
examples. Experimental results show that NormSpecSync is superior to state-of-the-art map synchronization methods in the literature. We organize the remainder of this section as follows. In
Section 4.1, we evaluate NormSpecSync on synthetic examples. Then in section 4.2, we evaluate
NormSpecSync on real examples.
4.1
Quantitative Evaluations on Synthetic Examples
We generate synthetic data by following the same procedure described in Section 2. Specifically,
each synthetic example is controlled by three parameters G, m, and p. Here G specifies the input
graph; m describes the size of each permutation matrix; p controls the noise level of the input maps.
The input maps follow a generalized Erdos-Renyi model, i.e., independently for each edge (i, j) ? G
in
in
in the input graph, with probability p the input map Xij
= Im , and otherwise Xij
is a random
permutation. To simplify the discussion, we fix m = 10, n = 200 and vary the observation graph G
and p to evaluate NormSpecSync and existing algorithms.
5
Varying vertex degrees
0.1
0.2
0.2
p-true
p-true
Varying vertex degrees
0.1
0.3
0.3
0.4
0.4
0.5
0.5
0.0
0.1
0.2
0.3
0.0
0.4
0.1
0.2
0.3
Irregularty
Irregularty
(a) NormSpecSync
(b) SpecSync
0.4
Figure 2: Comparison between NorSpecSync and SpecSync on irregular observation graphs.
Dense graph versus sparse graph. We first study the performance of NormSpecSync with respect
to the density of the graph. In this experiment, we control the density of G by following a standard
Erd?os-R?nyi model with parameter q, namely independently, each edge is connected with probability
q. For each pair of fixed p and q, we generate 10 examples. We then apply NormSpecSync and count
the ratio that the underlying permutations are recovered. Figure 1(a) illustrates the success rate of
NormSpecSync on a grid of samples for p and q. Blue and yellow colors indicate it succeeded and
failed on all the examples, respectively, and the colors in between indicate a mixture of success and
failure. We can see that NormSpecSync tolerates more noise when the graph becomes denser. This
aligns with our theoretical analysis result.
NormSpecSync versus SpecSync. We also compare NormSpecSync with SpecSync [12], and
show the advantage of NormSpecSync on irregular observation graphs. To this end, we generate
G using a different model. Specifically, we let the degree of the vertex to be uniformly distribute
between ( 12 ? q)n and ( 12 + q)n. As illustrated in Figure 2, when q is small, i.e., all the vertices have
similar degrees, the performance of NormSpecSync and SpecSync are similar. When q is large, i.e.,
G is irregular, NormSpecSync tend to tolerate more noise than SpecSync. This shows the advantage
of utilizing a normalized data matrix.
NormSpecSync versus DiffSync. We proceed to compare NormSpecSync with DiffSync [13],
which is a permutation synchronization method based on diffusion distances. NormSpecSync and
DiffSync exhibit similar computation efficiency. However, NormSpecSync can tolerate significantly
more noise than DiffSync, as illustrated in Figure 1(c).
NormSpecSync versus SDP. Finally, we compare NormSpecSync with SDP [9], which formulates
permutation synchronization as solving a semidefinite program. As illustrated in Figure 1(b), the
exact recovery ability of NormSpecSync and SDP are similar. This aligns with our theoretical analysis
result, which shows the near-optimality of NormSpecSync under the noise model of consideration.
Yet computationally, NormSpecSync is much more efficient than SDP. The averaged running time for
SpecSync is 2.25 second. In contrast, SDP takes 203.12 seconds in average.
4.2
Quantitative Evaluations on Real Examples
In this section, we present quantitative evaluation of NormSpecSync on real datasets.
CMU Hotel/House.
We first evaluate NormSpecSync on CMU Hotel and CMU House
datasets [20]. The CMU Hotel dataset contains 110 images, where each image has 30 marked
feature points. In our experiment, we estimate the initial map between a pair of images using
RANSAC [21]. We consider two observation graphs: a clique observation graph Gf ull , where we
have initial maps computed between all pairs of images, and a sparse observation graph Gsparse .
Gsparse is constructed to only connect similar images. In this experiment, we connect an edge between
two images if the difference in their HOG descriptors [22] is smaller than 12 of the average descriptor
differences among all pairs of images. Note that Gsparse shows high variance in terms of vertex
6
CMU-G-Sparse
CMU-G-Full
90
90
80
80
% correspondences
100
% correspondences
100
70
60
50
RANSAC
DiffSync
SpecSync
NonSpecSync
SDP
40
30
20
10
70
60
50
RANSAC
DiffSync
SpecSync
NonSpecSync
SDP
40
30
20
10
0
0
0
1
2
3
4
5
0
6
1
2
3
4
5
6
Euclidean distance (pixels)
Euclidean distance (pixels)
SCAPE-G-Sparse
SCAPE-G-Full
100
100
90
% correspondences
% correspondences
90
80
70
60
50
RANSAC
DiffSync
SpecSync
NonSpecSync
SDP
40
30
20
10
80
70
60
50
RANSAC
DiffSync
SpecSync
NonSpecSync
SDP
40
30
20
10
0
0
0
0.05
0.1
0.15
0
0.2
0.05
0.1
0.15
0.2
Geodesic distance (diameter)
Geodesic distance (diameter)
Figure 3: Comparison between NorSpecSync, SpecSync, DiffSync and SDP on CMU Hotel/House and SCAPE.
In each dataset, we consider a full observation graph and a sparse observation graph that only connects potentially
similar objects.
degree. The CMU House dataset is similar to CMU Hotel, containing 100 images and exhibiting
slightly bigger intra-cluster variability than CMU Hotel. We construct the observation graphs and the
initial maps in a similar fashion. For quantitative evaluation, we measure the cumulative distribution
of distances between the predicted target points and the ground-truth target points.
Figure 3(Left) compares NormSpecSync with the SDP formulation, SpecSync, and DiffSync. On
both full and sparse observation graphs, we can see that NormSpecSync, SDP and SpecSync are
superior to DiffSync. The performance of NormSpecSync and SpecSync on Gf ull is similar, while
on Gsparse , NormSpecSync shows a slight advantage, due to its ability to handle irregular graphs.
Moreover, although the performance of NormSpecSync and SDP are similar, SDP is much slower
than NormSpecSync. For example, on Gsparse , SDP took 1002.4 seconds, while NormSpecSync
only took 3.4 seconds.
SCAPE. Next we evaluate NormSpecSync on the SCAPE dataset. SCAPE consists of 71 different
poses of a human subject. We uniformly sample 128 points on each model. Again we consider a
full observation graph Gf ull and a sparse observation graph Gsparse . Gsparse is constructed in the
same way as above, except we use the shape context descriptor [4] for measuring the similarity
between 3D models. In addition, the initial maps are computed from blended-intrinsic-map [16],
which is the state-of-the-art technique for computing dense correspondences between organic shapes.
For quantitative evaluation, we measure the cumulative distribution of geodesic distances between
the predicted target points and the ground-truth target points. As illustrated in Figure 3(Right), the
relative performance between NormSpecSync and the other three algorithms is similar to CMU Hotel
and CMU House. In particular, NormSpecSync shows an advantage over SpecSync on Gsparse . Yet
in terms of computational efficiency, NormSpecSync is far better than SDP.
5
Conclusions
In this paper, we propose an efficient algorithm named NormSpecSync towards solving the permutation synchronization problem. The algorithm adopts a spectral view of the mapping problem and
exhibits surprising behavior both in terms of computation complexity and exact recovery conditions.
The theoretical result improves upon existing methods from several aspects, including a fixed obser7
vation graph and a practical noise method. Experimental results demonstrate the usefulness of the
proposed approach.
There are multiple opportunities for future research. For example, we would like to extend NormSpecSync to handle the case where input objects only partially overlap with each other. In this scenario,
developing and analyzing suitable rounding procedures become subtle. Another example is to extend
NormSpecSync for rotation synchronization, e.g., by applying Spectral decomposition and rounding
in an iterative manner.
Acknowledgement. We would like to thank the anonymous reviewers for detailed comments on how
to improve the paper. The authors would like to thank the support of DMS-1700234, CCF-1302435,
CCF-1320175, CCF-1564000, CNS-0954059, IIS-1302662, and IIS-1546500.
A
Proof Architecture of Lemma 3.1
In this section, we provide a roadmap for the proof of Lemma 3.1. The detailed proofs are deferred to
the supplemental material.
1
1
Reformulate the observation matrix. The normalized adjacency matrix A? = D? 2 AD? 2 can
be decomposed as A? = ssT + V ?V T , where the dominant eigenvalue is 1 and corresponding
? , and it is clear
eigenvector is s. We reformulate the observation matrix as p1 M = A? ? Im + N
to see that the ground truth result relates to the term (ssT ) ? Im , while the noise comes from two
? . More specifically, the noise not only comes from the randomness of
terms: (V ?V T ) ? Im and N
uncertainty of the measurements, but also from the graph structure, and we use ? to represent the
spectral norm of ?. When the graph is disconnected or near disconnected, ? is close to 1 and it is
impossible to recover the ground truth.
? . The noise term N
? consists of random matrices with mean zero
Bound the spectral norm of N
in each block. In a complete graph, the spectral norm is bounded by O( p?1 n ), however, when
considering the graph structure, we give a O( p?d1 ) bound.
min
Measure the block-wise distance between U and s ? Im . Let M = U ?U T + U2 ?2 U2T , we
want to show the distance between U and s ? 1m is small, where the distance function dist(?) is
defined as:
dist(U, V ) = min
U ? V R
,
(7)
R:RRT =I
B
and this B?norm for any matrix X represented in the form X = [X1T , ? ? ? , XnT ]T ? Rmn?m is
defined as
kXkB = maxkXi kF .
(8)
i
More specifically, we bound the distance between U and s ? Im by constructing a series of matrix
{Ak }, and we can show for some k = O(log n), the distances from s ? Ak to both U and s ? Im
are small. Therefore, by using the triangle inequality, we can show that U and s ? Im is close.
Sketch proof of Lemma 3.1. Once we are able to show that there exists some rotation matrix R,
such that dist(U, s ? Im ) is in the order of o( ?1n ), then it is straightforward to prove Lemma 3.1.
Intuitively, this is because the measurements from the eigenvectors is close enough to the ground
truth, hence their second moment will still be close. Formally speaking,
Ui UjT ? (si ? Im )(sj ? Im )
(9)
=
Ui RRT UjT ? (si ? Im )(sj ? Im )
(10)
T T
T
T
= Ui R(R Uj ? (sj ? Im ) ) + (Ui R ? si ? Im )(sj ? Im )
(11)
? Ui ? dist(U, s ? Im ) + dist(U, s ? Im ) ? sj ? Im
(12)
On the other hand, notice that
Pn d
Pn d
i=1 i
i
T
Ui Uj ? Im
= pi=1
Ui UjT ? (si ? Im )(sj ? Im )
,
(13)
p
di dj
di dj
and we only need to show that (13) is in the order of o(1). The details are included in the supplemental
material.
8
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9
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5,668 | 6,129 | Reconstructing Parameters of Spreading Models
from Partial Observations
Andrey Y. Lokhov
Center for Nonlinear Studies and Theoretical Division T-4
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
lokhov@lanl.gov
Abstract
Spreading processes are often modelled as a stochastic dynamics occurring on top
of a given network with edge weights corresponding to the transmission probabilities. Knowledge of veracious transmission probabilities is essential for prediction,
optimization, and control of diffusion dynamics. Unfortunately, in most cases the
transmission rates are unknown and need to be reconstructed from the spreading
data. Moreover, in realistic settings it is impossible to monitor the state of each
node at every time, and thus the data is highly incomplete. We introduce an efficient
dynamic message-passing algorithm, which is able to reconstruct parameters of the
spreading model given only partial information on the activation times of nodes in
the network. The method is generalizable to a large class of dynamic models, as
well to the case of temporal graphs.
1
Introduction
Knowledge of the underlying parameters of the spreading model is crucial for understanding the
global properties of the dynamics and for development of effective control strategies for an optimal dissemination or mitigation of diffusion [1, 2]. However, in many realistic settings effective
transmission probabilities are not known a priori and need to be recovered from a limited number of
realizations of the process. Examples of such situations include spreading of a disease [3], propagation
of information and opinions in a social network [4], correlated infrastructure failures [5], or activation
cascades in biological and neural networks [6]: precise model and parameters, as well as propagation
paths are often unknown, and one is left at most with several observed diffusion traces. It can be
argued that for many interesting systems, even the functional form of the dynamic model is uncertain.
Nevertheless, the reconstruction problem still makes sense in this case: the common approach is to
assume some simple and reasonable form of the dynamics, and recover the parameters of the model
which explain the data in the most accurate and minimalistic way; this is crucial for understanding
the basic mechanisms of the spreading process, as well as for making further predictions without
overfitting. For example, if only a small number of samples is available, a few-parameter model
should be used.
In practice, it is very costly or even impossible to record the state of each node at every time step of
the dynamics: we might only have access to a subset of nodes, or monitor the state of the system
at particular times. For instance, surveys may give some information on the health or awareness of
certain individuals, but there is no way to get a detailed account for the whole population; neural
avalanches are usually recorded in cortical slices, representing only a small part of the brain; it is
costly to deploy measurement devices on each unit of a complex infrastructure system; finally, hidden
nodes play an important role in the artificial learning architectures. This is precisely the setting
that we address in this article: reconstruction of parameters of a propagation model in the presence
of nodes with hidden information, and/or partial information in time. It is not surprising that this
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
challenging problem turns out to be notably harder then its detailed counterpart and requires new
algorithms which would be robust with respect to missing observations.
Related work. The inverse problem of network and couplings reconstruction in the dynamic setting
has attracted a considerable attention in the past several years. However, most of the existing works
are focused on learning the propagation networks under the assumption of availability of full diffusion
information. The papers [7, 8, 9, 10] developed inference methods based on the maximization of
the likelihood of the observed cascades, leading to distributed and convex optimization algorithms
in the case of continuous and discrete dynamics, principally for the variants of the independent
cascade (IC) model [11]. These algorithms have been further improved under the sparse recovery
framework [12, 13], particularly efficient for structure learning of treelike networks. A careful
rigorous analysis of these likelihood-based and alternative [14, 15] reconstruction algorithms give
an estimation of the number of observed cascades required for an exact network recovery with high
probability. Precise conditions for the parameters recovery at a given accuracy are still lacking. The
fact that the aforementioned algorithms rely on the fully observed spreading history represents an
important limitation in the case of incomplete information. The case of missing time information has
been addressed in two recent papers: focusing primarily on tree graphs, [16] studied the structure
learning problem in which only initial and final spreading states are observed; [17] addressed the
network reconstruction problem in the case of partial time snapshots of the network, using relaxation
optimization techniques and assuming that full probabilistic trace for each node in the network is
available. A standard technique for dealing with incomplete data involves maximizing the likelihood
marginalized over the hidden information; for example, this approach has been used in [18] for
identifying the diffusion source. In what follows, we use this method for benchmarking our results.
Overview of results. In this article, we propose a different algorithm, based on recently introduced
dynamic message-passing (DMP) equations for cascading processes [19, 20], which will be referred
to as DMP REC (DMP-reconstruction) throughout the text. Making use of all available information,
it yields significantly more accurate reconstruction results, outperforming the likelihood method
and having a substantially lower algorithmic complexity, independent on the number of nodes with
unobserved information. More generally, the DMP REC framework can be easily adapted to allow
reconstruction of heterogeneous transmission probabilities in a large class of cascading processes,
including the IC and threshold models, SIR and other epidemiological models, rumor spreading
dynamics, etc., as well as for the processes occurring on dynamically-changing networks.
2
Problem formulation
Model. For the sake of simplicity and definiteness, we assume that cascades follow the dynamics of
stochastic susceptible-infected (SI) model in discrete time, defined on a network G = (V, E) with set
of nodes denoted by V and set of directed edges E [3]. Each node i ? V at times t = 1, 2, . . . , T
can be in either of two states: susceptible (S) or infected (I). At each time step, node i in the I state
can activate one of its susceptible neighbors j with probability ?ij 1 . The dynamics is non-recurrent:
once the node is activated (infected), it can never change its state back to susceptible. In what follows,
the network G is supposed to be known.
Incomplete observations and inference problem. We assume that the input is formed from M
independent cascades, where a cascade ?c is defined as a collection of activation times of nodes in
the network {?ic }i?V . Each cascade is observed up to the final observation time T . Notice that T is
an important parameter: intuitively, the larger is T , the more information is contained in cascades,
and the less samples are needed. We assume that T is given and fixed, being related to the availability
of the finite-time observation window. If node i in cascade c does not get activated at a certain time
prior to the horizon T , we put by definition ?ic = T ; hence, ?ic = T means that node i changes its
state at time T or later. The full information on the cascades ? = ?c ?c is divided into the observed
part, ?O , and the hidden part ?H . Thus, in general ?O contains only a subset of activation times
in T ? [0, T ] for a part of observed nodes in the network O ? V . The task is to reconstruct the
1
We chose this two-state model since it has slightly more general dynamic rules compared to the popular IC
model [11] with an additional restriction: a node infected at time t has a single chance to activate its susceptible
neighbors at time step t+1, while further infection attempts in subsequent rounds are not allowed. The DMP REC
method presented below can be easily applied to the case of IC model by noticing that it corresponds to the SIR
model with a recovery probability equal to one, for which the DMP equations are known [20].
2
?
couplings {?ij
}(ij)?E ? G?? , where G?? with a star denotes the original transmission probabilities
that have been used to generate the data.
Maximization of the likelihood. Similarly to the formulations considered in [7, 8, 10], it is possible
to explicitly write the expression for the likelihood of the discrete-time SI model in the case of fully
available information ?O = ? under the assumption that the data has been generated using the
couplings G? :
Y Y
P (? | G? ) =
Pi (?ic | ?c , G? ),
(1)
i?V 1?c?M
with
?
Pi (?ic
c
| ? , G? ) = ?
?"
?ic ?2
Y Y
(1 ? ?ki 1?kc ?t0 )? 1 ?
t0 =0 k??i
!
Y
#
(1 ? ?ki 1?kc ??ic ?1 ) 1?ic <T , (2)
k??i
where ?i denotes the set of neighbors of node i in the graph G, and 1 is the indicator function. The
expression (2) has the following meaning: the probability that node i has been activated at time ?i
given the activation times of its neighbors is equal to the probability that the activation signal has
not been transmitted by any infected neighbor of i until the time ?i ? 2 (first term in the product),
and that at least one of the active neighbors actually transmitted the infection at time ?i ? 1 (second
term). A straightforward adaptation of the N ET R ATE algorithm, suggested in [8], to the present
b ?? is obtained as a solution of
setting implies that the estimation of the transmission probabilities G
the convex optimization problem
b ?? = arg min (? ln P (? | G? )) ,
G
(3)
which can be solved locally for each node i and its neighborhood due to the factorization of the
likelihood (1) under assumption of asymmetry of the couplings. In the case of partial observations,
the optimization problem (3) is not well defined since it requires the full knowledge of activation
times for each node. A simple and natural extension of this scheme, which we will refer to as the
maximum likelihood estimator (MLE), is to consider the likelihood function marginalized over
unknown activation times:
X
P (?O | G? ) =
P (? | G? ).
(4)
{?hc },h?H
An exact evaluation of (4) is a computationally hard high-dimensional integration problem with
complexity proportional to T H in the presence of H nodes with hidden information. In order to
correct for this fact, we propose a heuristic scheme which we denote as the heuristic two-stage
(HTS) algorithm. The idea of HTS consists of completing the missing part {?hc }h?H of the cascades
at each step of the optimization process with the most probable values according to the current
b? , ?
b H = arg max P (? | G
b ? ), and solving the optimization problem
estimation of the couplings G
b H ; these two alternating steps are iterated
(3) using the full information on the cascades ? = ?O ? ?
b H requires an
until the global convergence of the algorithm. An exact (brute-force) estimation of ?
H
exponential number of operations T , as the original MLE formulation. However, we found that
in practice the computational time can be significantly reduced with the use of the Monte Carlo
sampling. The corresponding approximation is based on the observation that the likelihood (1) is
non-zero only for {?ic }i?V forming possible (realizable) cascades. Hence, for each c, we sample
LH,T auxiliary cascades, and choose the set of {?hc }h?H maximizing (1). LH,T is typically a large
sampling parameter, growing with T and H to ensure a proper convergence. This procedure leads
to an algorithm with a complexity O(N M |E|2 LH,T ) at each step of the optimization, where |E|
denotes the number of edges; see the journal version of the paper [21] for a more in-depth discussion.
Hence, both MLE and HTS algorithms are practically intractable; the remaining part of the paper
is devoted to the development of an accurate algorithm with a polynomial-time computational
complexity for this hard problem. The next section introduces dynamic message-passing equations
which serve as a basis for such algorithm.
3
Dynamic message-passing equations.
The dynamic message-passing equations for the SI model in continuous [19] and discrete [20] settings
allow to compute marginal probabilities that node i is in the state S at time t:
Y
?k?i (t)
(5)
PSi (t) = PSi (0)
k??i
3
for t > 0 and a given initial condition PSi (0). The variables ?k?i (t) represent the probability that
node k did not pass the activation signal to the node i until time t. The intuition behind the key
Equation (5) is that the probability of node i to be susceptible at time t is equal to the probability of
being in the S state at initial time times the probability that neither of its neighbors infected it until
time t. The quantities ?k?i (t) can be computed iteratively using the following expressions:
?k?i (t) = ?k?i (t ? 1) ? ?ki ?k?i (t ? 1),
(6)
?
?k?i (t) = (1 ? ?ki )?k?i (t ? 1) + PSk (0) ?
?
Y
?l?k (t ? 1) ?
l??k\i
Y
?l?k (t)? ,
(7)
l??k\i
where ?k\i denotes the set of neighbors of k excluding i. The Equation (6) translates the fact that
?k?i (t) can only decrease if the infection is actually transmitted along the directed link (ki) ? E;
this happens with probability ?ki times ?k?i (t ? 1) which denotes the probability that node k is in
the state I at time t, but has not transmitted the infection to node i until time t ? 1. The Equation
(7), which allows to close the system of dynamic equations, describes the evolution of probability
?k?i (t): at time t ? 1, it decreases if the infection is transmitted (first term in the sum), and increases
if node k goes from the state S to the state I (difference of terms 2 and 3). Note that node i is
excluded from the corresponding products over ?-variables because this equation is conditioned on
the fact that i is in the state S, and therefore can not infect k. The Equations (6) and (7) are iterated
in time starting from initial conditions ?i?j (0) = 1 and ?i?j (0) = 1 ? PSi (0) which are consistent
with the definitions above. The name ?DMP equations? comes from the fact the whole scheme can
be interpreted as the procedure of passing ?messages? along the edges of the network.
Theorem 1. DMP equations for the SI model, defined by Equations (5)-(7), yield exact marginal
probabilities on tree networks. On general networks, the quantities PSi (t) give lower bound on values
of marginal probabilities.
Proof Sketch. The exactness of solution on tree graphs immediately follows from the fact that the
DMP equations can be derived from belief propagation equations on time trajectories [20], which
provide exact marginals on trees. The fact that PSi (t) computed according to (5) represent a lower
bound on marginal probabilities in general networks can be derived from a counting argument,
considering multiple infection paths on a loopy graph which contribute to the computation of PSi (t),
effectively lowering its value through the Equation (5); the proof technique is borrowed from [19],
where similar dynamic equations in the continuous-time case have been considered.
Using the definition (5) of PSi (t), it is convenient to define the marginal probability mi (t) of activation
"
#
of node i at time t:
Y
Y
mi (t) = PSi (0)
?k?i (t ? 1) ?
?k?i (t) .
(8)
k??i
k??i
As it often happens with message-passing algorithms, although being exact only on tree networks,
DMP equations provide accurate results even on loopy networks. An example is provided in the
Figure 1, where the DMP-predicted marginals are compared with the values obtained from extensive
simulations of the dynamics on a network of retweets with N = 96 nodes [22]. This observation will
allow us to use DMP equations as a suitable approximation tool on general networks. In the next
section we describe an efficient reconstruction algorithm, DMP REC, which is based on the resolution
of the dynamics given by DMP equations and makes use of all available information.
4
Proposed algorithm: DMP REC
Probability of cascades and free energy. The marginalization over hidden nodes in (4) creates a
complex relation between couplings in the whole graph, resulting in a non-explicit expression. The
main idea behind the DMP REC algorithm is to approximate the likelihood of observed cascades (4)
through the marginal probability distributions (5) and (8):
P (?O | G? ) ?
M Y
Y
i c
m (?i | G? )1?ic ?T + PSi (?ic | G? )1?ic =T .
(9)
c=1 i?O
The expression (9) is at the core of the suggested algorithm. As there is no tractable way to compute
exactly the joint probability of partial observations, we approximate it using a mean-field-type
4
MC-predicted PSi (t)
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
DMP-predicted PSi (t)
(a)
(b)
Figure 1: Illustration of the accuracy of DMP equations on a network of retweets with N = 96 nodes [22].
(a) Comparison of DMP-predicted PSi (t) with PSi (t) estimated from 106 runs of Monte Carlo simulations with
t = 10 and one infected node at initial time. The couplings {?ij } have been generated uniformly at random in
the range [0, 1]. (b) Visualization of the network topology created with the Gephi software.
approach as a product of marginal probabilities provided by the dynamic message-passing equations.
The reasoning behind this approach is that each marginal is expressed through an average of all
possible realizations of dynamics with a given initial condition; this is in contrast with the likelihood
function which considers only particular instance realized in the given cascade. Therefore, equation
(9) summarizes the effect of different propagation paths, and the maximization of this probability
function will yield the most likely consensus between the ensemble of couplings in the network.
Precisely this key property makes the reconstruction possible in the case involving nodes with
hidden information via maximization of the objective (9) which can be interpreted as a cost function
representing the product of individual probabilities of activation taken precisely at the value of the
observed infection times. Starting from this expression, one can define the associated ?free energy?:
X
i
fDMP = ? ln P (?O | G? ) =
fDMP
,
(10)
i?O
i
i
where fDMP
= ? c ln mi (?ic )1?ic ?T ?1 + PSi (T ? 1)1?ic =T . In the last expression for fDMP
we
i
i
i
used the fact that m (T ) + PS (T ) = PS (T ? 1). Our goal is to minimize the free energy (10)
with respect to {?ij }(ij)?E . A similar approach has been previously outlined by [23] as a way to
learn homogeneous couplings in the spreading source inference algorithm. In order to carry out this
optimization task, we need to develop an efficient way of gradient evaluation.
P
Computation of the gradient. The gradient of the free energy reads (note that the indicator functions
point to disjoint events):
i
i
X h ?mi (? c | G? )/??rs
?fDMP
?PSi (T ? 1 | G? )/??rs
i
c ?T ?1 +
c =T ,
=?
1
1
(11)
?
?
i
i
??rs
mi (?ic | G? )
PSi (T ? 1 | G? )
c
where the derivatives of the marginal probabilities can be computed explicitly by taking the derivative
k?i
of the DMP equations (5)-(8). Let us denote ??k?i (t)/??rs ? pk?i
(t)/??rs ?
rs (t) and ??
k?i
i?j
i?j
qrs (t). Since the dynamic messages at initial time {?
(0)} and {?
(0)} are independent
k?i
on the couplings, we have pk?i
rs (0) = qrs (0) = 0 for all k, i, r, s, and these quantities can be
computed iteratively using the analogues of the Equations (6) and (7):
k?i
k?i
k?i
pk?i
(t ? 1)1k=r,i=s ,
rs (t) = prs (t ? 1) ? ?ki qrs (t ? 1) ? ?
q k?i
rs (t)
k?i
?ki )qrs
(t
(12)
k?i
= (1 ?
? 1) ? ?
(t ? 1)1k=r,i=s
X
Y
X
Y
+ PSk (0)
pl?k
?n?k (t ? 1) ? PSk (0)
pl?k
rs (t ? 1)
rs (t)
l??k\i
n??k\{i,l}
l?k\i
?n?k (t). (13)
n??k\{i,l}
Using these quantities, the derivatives of the marginals entering in Equation (11) can be written as
X
Y
?PSi (t)
= PSi (0)
pk?i
?l?i (t),
rs (t)
??rs
k??i
l??i\k
5
?PSi (t ? 1) ?PSi (t)
?mi (t)
=
?
.
??rs
??rs
??rs
(14)
The following observation shows that at least on tree networks, corresponding to the regime in
which DMP equations have been derived, the values of the original transmission probabilities G??
correspond to the point in which the gradient of the free energy takes zero value.
Claim 1. On a tree network, in the limit of large number of samples M ? ?, the derivative of the
free energy is equal to zero at the values of couplings G?? used for generating cascades.
Proof. Let us first look at samples originating from the same initial condition. According to Theorem
1, the DMP equations are exact on tree graphs, and hence it is easy to see that
X
i
lim fDMP
=?
mi (t | G?? ) ln mi (t | G? ) ? PSi (T ? 1 | G?? ) ln PSi (T ? 1 | G? ). (15)
M ??
t?T ?1
Therefore,
i
?fDMP
?
lim
|G?? = ?
M ?? ??rs
??rs
"
#
X
i
m (t | G?? ) +
PSi (T
? 1 | G?? ) = 0,
t?T ?1
since the expression inside the brackets sums exactly to one. This result trivially holds by summing up
samples with different initial conditions. Combination of this result with the definition (10) completes
the proof.
The DMP REC algorithm consists of running the message-passing equations for the derivatives of the
dynamic variables (12), (13) in parallel with DMP equations (5)-(7), allowing for the computation
of the gradient of the free energy (11) through (14), which is used afterwards in the optimization
procedure. Let us analyse the computational complexity of each step of parameters update. The
number of runs is equal to the number of distinct initial conditions in the ensemble of observed
cascades, so if all M cascades start with distinct initial conditions, the complexity of the DMP REC
algorithm is equal to O(|E|2 T M ) for each step of the update of {?rs }(rs)?E . Hence, in a typical
situation where each cascade is initiated at one particular node, the number of runs will be limited by
N , and the overall update-step complexity of DMP REC will be O(|E|2 T N ).
Missing information in time. On top of inaccessible nodes, the state of the network can be monitored
at a lower frequency compared to the natural time scale of the dynamics. It is easy to adapt the
algorithm to the case of observations at K time steps T ? {tk }k?[1,K] . Since the activation time
c
c
?ic of node i in cascade c is now known only up to the interval [tki + 1, tki +1 ] ? ?kic , where
P
c
c
c
c
tki < ?ic ? tki +1 , one should maximize t??kc mi (t) = PSi (tki ) ? PSi (tki +1 ) ? ?kic PSi (t | G? )
i
instead of mi (?ic ) in this case. This leads to obvious modifications to the expressions (10) and (11),
using the differences of derivatives at corresponding times instead of one-step differences as in (14).
For instance, if the final time is not included in the observations, we have
"
#
i
X
X ??kc PSi (t | G? )/??rs
?f
DMP
i
i
fDMP
=?
ln ?kic PSi (t | G? ) ,
=?
.
i
??
?
rs
kic PS (t | G? )
c
c
5
Numerical results
We evaluate the performance of the DMP REC algorithm on synthetic and real-world networks under
assumption of partial observations. In numerical experiments, we focus primarily on the presence of
inaccessible nodes, which is a more computationally difficult case compared to the setting of missing
information in time. An example involving partial time observations is shown in section 5.1.
5.1
Tests with synthetic data
Experimental setup. In the tests described in this section, the couplings {?ij } are sampled uniformly
in the range [0, 1], the final observation time is set to T = 10. Each cascade is generated using a
discrete-time SI model defined in section 2 from randomly selected sources. In the case of inaccessible
nodes, the activation times data is hidden in all the samples for H randomly selected nodes. We use the
likelihood methods for benchmarking the accuracy of our approach. The MLE algorithm introduced
above is not tractable even on small graphs, therefore we compare the results of DMP REC with
6
the HTS algorithm outlined in the section 2. Still, HTS has a very high computational complexity,
and therefore we are bound to run comparative tests on small graphs: a connected component of an
artificially-generated network with N = 20, sampled using a power-law degree distribution, and a
real directed network of relationships in a New England monastery with N = 18 nodes [24]. Both
algorithms are initialized with ?ij = 0.5 for all (ij) ? E. The accuracy of reconstruction is assessed
using the `1 norm of the difference between reconstructed and original couplings, normalized over
the number of directed edges in the graph2 . Intuitively, this measure gives an average expected error
for each parameter ?ij .
HTS
DMPrec
0.1
0.05
0.05
0
(a)
0.1
?*ij
???ij - ?*ij??
0.15
102 103 104 105 106
M(?0.64)
1
0.8
0.6
0.4
0.2
0
0
2
5
H
(b)
7
0.2 0.4 0.6 0.8
10
1
?ij
(c)
Figure 2: Tests for DMP REC and HTS on a small power-law network: (a) for fixed number of nodes with
???ij - ?*ij??
unobserved information H = 5, (b) for fixed number of samples M = 6400. (c) Scatter plot of {?ij } obtained
?
with DMP REC versus original parameters {?ij
} in the case of missing information in time with M = 6400,
T = 10; the state of the network is observed every other time step.
0.15
0.1
0.1
0.05
0.05
0
(a)
HTS
DMPrec
102 103 104 105 106
M(?0.64)
0
0
(b)
2
4
H
6
8
(c)
Figure 3: Numerical results for the real-world Monastery network of [24]: (a) for fixed number of nodes with
unobserved information H = 4, (b) for fixed number of samples M = 6400. (c) The topology of the network
?
(thickness of edges proportional to {?ij
} used for generating cascades).
Results. In the Figure 2 we present results for a small power-law network with short loops, which
is not a favorable situation for DMP equations derived in the treelike approximation of the graph.
Figures 2 (a) and 2 (b) show the dependence of an average reconstruction error as a function of M
(for fixed H/N = 0.25) and H (for fixed M = 6400), respectively. DMP REC clearly outperforms
the HTS algorithm, yielding surprisingly accurate reconstruction of transmission probabilities even
in the case where a half of network nodes do not report any information. Most importantly, DMP REC
achieves reconstruction with a significantly lower computational time: for example, while it took
more than 24 hours to compute the point corresponding to H = 4 and M = 6400 with HTS (MLE
at this test point took several weeks to converge), the computation involving DMP REC converged to
the presented level of accuracy in less than 10 minutes on a standard laptop. These times illustrate
the hardness of the learning problem involving incomplete information.
We have also used this case study network to test the estimation of transmission probabilities with the
DMP REC algorithm when the state of the network is recorded only at a subset of times T ? [0, T ].
Results for the case where every other time stamp is missing are given in the Figure 2 (c): couplings
?
estimated with DMP REC are compared to the original values {?ij
}; despite the fact that only 50% of
time stamps are available, the inferred couplings show an excellent agreement with the ground truth.
Equivalent results for the real-world relationship network extracted from the study [24] and containing
both directed and undirected links, are shown in the Figure 3; an ability of DMP REC to capture the
mutual dependencies of different couplings through dynamic correlations is even more pronounced in
this case, with almost perfect reconstruction of couplings for large M and a rather weak dependence
2
Note that this measure excludes those few parameters which are impossible to reconstruct: e.g. no algorithm
can learn the coupling associated with the ingoing edge of the hidden node located at the leaf of a network.
7
on the number of nodes with removed observations. We have run tests on larger synthetic networks
which show similar reconstruction results for DMP REC, but where comparisons with the likelihood
method could not be carried out. In the next section we focus on an application involving real-world
data which represents a more interesting and important case for the validation of the algorithm.
5.2
Test with a real-world data
As a proxy for the real statistics, we used the data provided by the Bureau of Transportation Statistics
[25], from which we reconstructed a part of the U.S. air transportation network, where airports are
the nodes, and directed links correspond to traffic between them. The reason behind this choice is
based on the fact that the majority of large-scale influenza pandemics over the past several decades
represented the air-traffic mediated epidemics. For illustration purposes, we selected top N = 30
airports ranked according to the total number of passenger enplanements and commonly classified as
large hubs, and extracted a sub-network of flights between them. The weight of each edge is defined
by the annual number of transported passengers, aggregated over multiple routes; we have pruned
links with a relatively low traffic ? below 10% of the traffic level on the busiest routes, so that the
total number of remaining directed links is |E| = 210. The final weights are based on the assumption
that the probability of infection transmission is proportional to the flux; the weights have been
renormalized accordingly so that the busiest route received the coupling ?ij = 0.5. The resulting
network is depicted in the Figure 4 . We have generated M = 10, 000 independent cascades in this
network, and have hidden the information at H = 15 nodes (50% of airports) selected at random.
We observe that even with a significantly large portion of missing information, the reconstructed
parameters show a good agreement with the original ones.
0.5
?
?
???ij - ?*ij??=0.0400
0.5
???ij - ?*ij??=0.0473
?
?
?
?
0.4
0.4
?
?
?
?
?
?
?
?
?
?
*
?
?ij
? ??
?
?
?ij
?
?
?
*
?
??
?
?
?
?
0.3
0.3
?
?
?
?
?
0.2
?
?
?
?
?
?
?
0.2
?
?
?
?
0.1
?
?
?
?
H=0
0.1
H=15
?
?
0.1 0.2 0.3 0.4 0.5
?ij
?
?
?
?
0.1 0.2 0.3 0.4 0.5
?ij
Figure 4: Left: Sub-network of flights between major U.S. hubs, where the thickness of edges is proportional to
the aggregated traffic between them; nodes which do not report information are indicated in red. Right: Scatter
?
plots of reconstructed {?ij } versus original {?ij
} couplings for H = 0 and H = 15 and M = 10, 000.
6
Conclusions and path forward
From the algorithmic point of view, inference of spreading parameters in the presence of nodes with
incomplete information considerably complicates the problem because the reconstruction can no
longer be performed independently for each neighborhood. In this paper, it is shown how the dynamic
interdependence of parameters can be exploited in order to be able to recover the couplings in the
setting involving hidden information. Let us discuss several directions for future work. DMP REC
can be straightforwardly generalized to more complicated spreading models using a generic form
of DMP equations [20] and the key approximation ingredient (9), as well as adapted to the case
of temporal graphs by encoding network dynamics via time-dependent coefficients ?ij (t), which
might be more appropriate in certain real situations. It would also be useful to extend the present
framework to the case of continuous dynamics using the continuous-time version of DMP equations
of [19]. An important direction would be to generalize the learning problem beyond the assumption
of a known network, and formulate precise conditions for detection of hidden nodes and for a perfect
network recovery in this case. Finally, in the spirit of active learning, we anticipate that DMP REC
could be helpful for the problems involving an optimal placement of observes in the situations where
collection of full measurements is costly.
Acknowledgements. The author is grateful to M. Chertkov and T. Misiakiewicz for discussions and comments,
and acknowledges support from the LDRD Program at Los Alamos National Laboratory by the National Nuclear
Security Administration of the U.S. Department of Energy under Contract No. DE-AC52-06NA25396.
8
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9
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5,669 | 613 | Bayesian Learning
via Stochastic Dynamics
Radford M. Neal
Department of Computer Science
University of Toronto
Toronto, Ontario, Canada M5S lA4
Abstract
The attempt to find a single "optimal" weight vector in conventional network training can lead to overfitting and poor generalization. Bayesian methods avoid this, without the need for a validation set, by averaging the outputs of many networks with weights
sampled from the posterior distribution given the training data.
This sample can be obtained by simulating a stochastic dynamical
system that has the posterior as its stationary distribution.
1
CONVENTIONAL AND BAYESIAN LEARNING
I view neural networks as probabilistic models, and learning as statistical inference.
Conventional network learning finds a single "optimal" set of network parameter
values, corresponding to maximum likelihood or maximum penalized likelihood inference. Bayesian inference instead integrates the predictions of the network over
all possible values of the network parameters, weighting each parameter set by its
posterior probability in light of the training data.
1.1
NEURAL NETWORKS AS PROBABILISTIC MODELS
Consider a network taking a vector of real-valued inputs, x, and producing a vector
of real-valued outputs, y, perhaps computed using hidden units. Such a network
architecture corresponds to a function, I, with y
I(x, w), where w is a vector of
connection weights. If we assume the observed outputs, y, are equal to y plus Gaussian noise of standard deviation (j, the network defines the conditional probability
=
475
476
Neal
for an observed output vector given an input vector as follows:
P(y I x, 0")
exp( -IY - !(x, w)12 /20"2)
ex:
(1)
The probability of the outputs in a training set (Xl, yt), ... , (Xn, Yn) given this fixed
noise level is therefore
P(Yl, ... , Yn IXl,?.?, Xn , 0")
ex:
exp( -
E lYe -
!(Xe, w)12 /20"2)
(2)
e
Often 0" is unknown. A Bayesian approach to handling this is to assign 0" a vague
prior distribution and then ?.ntcgrating it away, giving the following probability for
the training set (see (Buntine and Weigend, 1991) or (Neal, 1992) for details):
P(Yl,"" Yn IXl, ... , Xn)
ex:
(so + E lYe - !(Xe, w)12) -
mp?nD
2
(3)
e
where So and mo are parameters of the prior for 0".
1.2
CONVENTIONAL LEARNING
Conventional backpropagation learning tries to find the weight vector that assigns
the highest probability to the training data, or equivalently, that minimizes minus
the log probability of the training data. When 0" is assumed known, we can use (2)
to obtain the following objective function to minimize:
M(w)
= E
lYe - !(Xe, w)12 / 20"2
(4)
e
When 0" is unknown, we can instead minimize the following, derived from (3):
M(w)
(5)
e
Conventional learning often leads to the network overfitting the training data modeling the noise, rather than the true regularities. This can be alleviated by
stopping learning when the the performance of the network on a separate validation
set begins to worsen, rather than improve. Another way to avoid overfitting is to
include a weight decay term in the objective function, as follows:
M'(w)
=
Alwl 2 +
M(w)
(6)
Here, the data fit term, M(w), may come from either (4) or (5). We must somehow
find an appropriate value for A, perhaps, again, using a separate validation set.
1.3
BAYESIAN LEARNING AND PREDICTION
Unlike conventional training, Bayesian learning does not look for a single "optimal"
set of network weights. Instead, the training data is used to find the posterior
probability distribution over weight vectors. Predictions for future cases are made
by averaging the outputs obtained with all possible weight vectors, with each contributing in proportion to its posterior probability.
To obtain the posterior, we must first define a prior distribution for weight vectors.
We might, for example, give each weight a Gaussian prior of standard deviation w:
(7)
Bayesian Learning via Stochastic Dynamics
We can then obtain the posterior distribution over weight vectors given the training
cases (Xl, yt), ... , (Xn, Yn) using Bayes' Theorem:
P(w I (Xl, yt}, ... , (Xn, Yn))
oc
P(w) P(YI, ... , Yn I Xl, ... , Xn , w)
(8)
Based on the training data, the best prediction for the output vector in a test case
with input vector X., assuming squared-error loss, is
Y.
=
J
/(x.,w)P(w
I (xI,yd,? .. ,(xn,Yn))dw
(9)
A full predictive distribution for the outputs in the test case can also be obtained,
quantifying the uncertainty in the above prediction.
2
INTEGRATION BY MONTE CARLO METHODS
Integrals such as that of (9) are difficult to evaluate. Buntine and Weigend (1991)
and MacKay (1992) approach this problem by approximating the posterior distribution by a Gaussian. Instead, I evaluate such integrals using Monte Carlo methods.
If we randomly select weight vectors, wo, ... , WN-I, each distributed according to
the posterior, the prediction for a test case can be found by approximating the
integral of (9) by the average output of networks with these weights:
y. ~ ~
L/(x.,Wt)
(10)
t
This formula is valid even if the Wt are dependent, though a larger sample may
then be needed to achieve a given error bound. Such a sample can be obtained
by simulating an ergodic Markov chain that has the posterior as its stationary
distribution. The early part of the chain, before the stationary distribution has
been reached, is discarded. Subsequent vectors are used to estimate the integral.
2.1
FORMULATING THE PROBLEM IN TERMS OF ENERGY
Consider the general problem of obtaining a sample of (dependent) vectors, qt,
with probabilities given by P( q). For Bayesian network learning, q will be the
weight vector, or other parameters from which the weights can be obtained, and
the distribution of interest will be the posterior.
It will be convenient to express this probability distribution in terms of a potential
energy function, E( q), chosen so that
P(q)
oc
exp(-E(q))
(11)
A momentum vector, p, of the same dimensions as q, is also introduced, and defined
to have a kinetic energy of ~ \pI2. The sum of the potential and kinetic energies is
the Hamiltonian:
(12)
H(q,p) = E(q) + ~lpl2
From the Hamiltonian, we define ajoint probability distribution over q and p (phase
space) as follows:
P(q,p) oc exp(-H(q,p))
(13)
The marginal distribution for q in (13) is that of (11), from which we wish to sample.
477
478
Neal
We can therefore proceed by sampling from this joint distribution for q and p, and
then just ignoring the values obtained for p.
2.2
HAMILTONIAN DYNAMICS
Sampling from the distribution (13) can be split into two subproblems - first,
to sample uniformly from a surface where H, and hence the probability, is constant, and second, to visit points of differing H with the correct probabilities. The
solutions to these subproblems can then be interleaved to give an overall solution.
The first subproblem can be solved by simulating the Hamiltonian dynamics of
the system, in which q and p evolve through a fictitious time, r, according to the
following equations:
dq
dr
8H
8p
= p,
8H
-dp
=
- = -VE(q)
dr
8q
(14)
This dynamics leaves H constant, and preserves the volumes of regions of phase
space. It therefore visits points on a surface of constant H with uniform probability.
When simulating this dynamics, some discrete approximation must be used. The
leapfrog method exactly maintains the preservation of phase space volume. Given
a size for the time step, E, an iteration of the leapfrog method goes as follows:
p(r+ E/2)
2.3
per) - (E/2)VE(q(r?)
q(r+ E)
q(r)+Ep
p(r + E)
p(r + E) - (E/2)V E(q(r + E?
(15)
THE STOCHASTIC DYNAMICS METHOD
To create a Markov chain that converges to the distribution of (13), we must interleave leapfrog iterations, which keep H (approximately) constant, with steps that
can change H. It is convenient for the latter to affect only p, since it enters into H
in a simple way. This general approach is due to Anderson (1980).
I use stochastic steps of the following form to change H:
p'
(16)
where 0 < (l' < 1, and n is a random vector with components picked independently
from Gaussian distributions of mean zero and standard deviation one. One can
show that these steps leave the distribution of (13) invariant. Alternating these
stochastic steps with dynamical leapfrog steps will therefore sample values for q
and p with close to the desired probabilities. In so far as the discretized dynamics
does not keep H exactly constant, however, there will be some degree of bias, which
will be eliminated only in the limit as E goes to zero.
It is best to use a value of (l' close to one, as this reduces the random walk aspect
of the dynamics. If the random term in (16) is omitted, the procedure is equivalent
to ordinary batch mode backpropagation learning with momentum.
Bayesian Learning via Stochastic Dynamics
2.4
THE HYBRID MONTE CARLO METHOD
The bias introduced into the stochastic dynamics method by using an approximation to the dynamics is eliminated in the Hybrid Monte Carlo method of Duane,
Kennedy, Pendleton, and Roweth (1987).
This method is a variation on the algorithm of Metropolis, et al (1953), which
generates a Markov chain by considering randomly-selected changes to the state.
A change is always accepted if it lowers the energy (H), or leaves it unchanged. If
it increases the energy, it is accepted with probability exp( -LlH), and is rejected
otherwise, with the old state then being repeated.
In the Hybrid Monte Carlo method, candidate changes are produced by picking a
random value for p from its distribution given by (13) and then performing some predetermined number of leapfrog steps. If the leapfrog method were exact, H would
be unchanged, and these changes would always be accepted. Since the method
is actually only approximate, H sometimes increases, and changes are sometimes
rejected, exactly cancelling the bias introduced by the approximation.
Of course, if the errors are very large, the acceptance probability will be very low,
and it will take a long time to reach and explore the stationary distribution. To
avoid this, we need to choose a step size (f) that is small enough.
3
RESULTS ON A TEST PROBLEM
I use the "robot arm" problem of MacKay (1992) for testing. The task is to learn
the mapping from two real-valued inputs, Xl and X2, to two real-valued outputs, YI
and Y2, given by
(17)
ih
2.0 cos(xI) + 1.3 COS(XI + X2)
(18)
Y2
2.0 sin(xI) + 1.3 sin(xi + X2)
=
=
Gaussian noise of mean zero and standard deviation 0.05 is added to (YI' Y2) to give
the observed position, (YI, Y2). The training and test sets each consist of 200 cases,
with Xl picked randomly from the ranges [-1.932, -0.453] and [+0.453, +1.932],
and X2 from the range [0.534,3.142].
A network with 16 sigmoidal hidden units was used. The output units were linear.
Like MacKay, I group weights into three categories - input to hidden, bias to
hidden, and hidden/bias to output. MacKay gives separate priors to weights in
each category, finding an appropriate value of w for each. I fix w to one, but
multiply each weight by a scale factor associated with its category before using it,
giving an equivalent effect. For conventional training with weight decay, I use an
analogous scheme with three weight decay constants (.\ in (6?.
In all cases, I assume that the true value of u is not known. I therefore use (3) for
the training set probability, and (5) for the data fit term in conventional training.
I set 80 rno 0.1, which corresponds to a very vague prior for u.
= =
3.1
PERFORMANCE OF CONVENTIONAL LEARNING
Conventional backpropagation learning was tested on the robot arm problem to
gauge how difficult it is to obtain good generalization with standard methods.
479
480
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Herations
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X
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Iterations
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Figure 1: Conventional backpropagation learning - (a) with no weight decay, (b) with
carefully-chosen weight decay constants. The solid lines give the squared error on the
training data, the dotted lines the squared error on the test data.
Fig. l(a) shows results obtained without using weight decay. Error on the test set
declined initially, but then increased with further training. To achieve good results,
the point where the test error reaches its minimum would have to be identified using
a separate validation set.
Fig. l(b) shows results using good weight decay constants, one for each category
of weights, taken from the Bayesian runs described below. In this case there is no
need to stop learning early, but finding the proper weight decay constants by nonBayesian methods would be a problem. Again, a validation set seems necessary, as
well as considerable computation.
Use of a validation set is wasteful, since data that could otherwise be included in
the training set must be excluded. Standard techniques for avoiding this, such as
"N-fold" cross-validation, are difficult to apply to neural networks.
3.2
PERFORMANCE OF BAYESIAN LEARNING
Bayesian learning was first tested using the unbiased Hybrid Monte Carlo method.
The parameter vector in the simulations (q) consisted of the unsealed network
weights together with the scale factors for the three weight categories. The actual
weight vector (w) was obtained by multiplying each unsealed weight by the scale
factor for its category.
Each Hybrid Monte Carlo run consisted of 500 Metropolis steps. For each step, a
trajectory consisting of 1000 leapfrog iterations with f = 0.00012 was computed,
and accepted or rejected based on the change in H at its end-point. Each run
therefore required 500,000 batch gradient evaluations, and took approximately four
hours on a machine rated at about 25 MIPS.
Fig. 2(a) shows the training and test error for the early portion of one Hybrid Monte
Carlo run. After initially declining, these values fluctuate about an average. Though
not apparent in the figure, some quantities (notably the scale factors) require a
hundred or more steps to reach their final distribution. The first 250 steps of each
run were therefore discarded as not being from the stationary distribution.
Fig. 2(b) shows the training and test set errors produced by networks with weight
vectors taken from the last 250 steps of the same run. Also shown is the error on
the test set using the average of the outputs of all these networks - that is, the
estimate given by (10) for the Bayesian prediction of (9). For the run shown, this
Bayesian Learning via Stochastic Dynamics
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350
300
I&era&ions X 1000
Figure 2: Bayesian learning using Hybrid
400
450
I&era&ions x 1000
Mon~e
Carlo - (a) early portion of run, (b) last
250 iterations. The solid lines give the squared error on the training set, the dotted lines
the squared error on the test set, for individual networks. The dashed line in (b) is the
test error when using the average of the outputs of all 250 networks.
+3? -
+2? -
Figure 3: Predictive distribution for
outputs. The two regions from which
training data was drawn are outlined.
Circles indicate the true, noise-free outputs for a grid of cases in the input
space. The dots in the vicinity of each
circle (often piled on top of it) are the
outputs of every fifth network from the
last 250 iterations of a Hybrid Monte
Carlo run.
' ??
",.
?
+1? -
?
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:.:~:.'.
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?10
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?1.0
0?
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+3.0
test set error using averaged outputs is 0.00559, which is (slightly) better than any
results obtained using conventional training. Note that with Bayesian training no
validation set is necessary. The analogues of the weight decay constants - the
weight scale factors - are found during the course of the simulation.
Another advantage of the Bayesian approach is that it can provide an indication
of how uncertain the predictions for test cases are. Fig. 3 demonstrates this. As
one would expect, the uncertainty is greater for test cases with inputs outside the
region where training data was supplied.
3.3
STOCHASTIC DYNAMICS VS. HYBRID MONTE CARLO
The uncorrected stochastic dynamics method will have some degree of systematic
bias, due to inexact simulation of the dynamics. Is the amount of bias introduced
of any practical importance, however?
481
482
Neal
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~------+---~-++-----~----~-----4-(b)
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Iterations X 1000
.. ?
~
~
\.
21
\
?
?
~
II
Iterations X 1000
Figure 4: Bayesian learning using uncorrected stochastic dynamics - (a) Training and
test error for the last 250 iterations of a run with c = 0.00012, (b) potential energy (E)
for a run with c = 0.00030. Note the two peaks where the dynamics became unstable.
To help answer this question, the stochastic dynamics method was run with parameters analogous to those used in the Hybrid Monte Carlo runs. The step size of
0.00012 used in those runs was chosen to be as large as possible while keeping
the number of trajectories rejected low (about 10%). A smaller step size would not
give competitive results, so this value was used for the stochastic dynamics runs as
well. A value of 0.999 for 0' in (16) was chosen as being (loosely) equivalent to the
use of trajectories 1000 iterations long in the Hybrid Monte Carlo runs.
(=
The results shown in Fig. 4(a) are comparable to those obtained using Hybrid Monte
Carlo in Fig. 2(b). Fig. 4(b) shows that with a larger step size the uncorrected
stochastic dynamics method becomes unstable. Large step sizes also cause problems
for the Hybrid Monte Carlo method, however, as they lead to high rejection rates.
The Hybrid Monte Carlo method may be the more robust choice in some circumstances, but uncorrected stochastic dynamics can also give good results. As it is
simpler, the stochastic dynamics method may be better for hardware implementation, and is a more plausible starting point for any attempt to relate Bayesian
methods to biology. Numerous other variations on these methods are possible as
well, some of which are discussed in (Neal, 1992).
References
Andersen, H. C. (1980) "Molecular dynamics simulations at constant pressure and/or
temperature", Journal of Chemical Physics, vol. 72, pp. 2384-2393.
Buntine, W. L. and Weigend, A. S. (1991) "Bayesian back-propagation", Complex Systems,
vol. 5, pp. 603-643.
Duane, S., Kennedy, A. D., Pendleton, B. J., and Roweth, D. (1987) "Hybrid Monte
Carlo", Physics Letters B, vol. 195, pp. 216-222.
MacKay, D. J. C. (1992) "A practical Bayesian framework for backpropagation networks",
Neural Computation, vol. 4, pp. 448-472.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. (1953)
"Equation of state calculations by fast computing machines", Journal of Chemical
Physics, vol. 21, pp. 1087-1092.
Neal, R. M. (1992) "Bayesian training of backpropagation networks by the hybrid Monte
Carlo method", CRG-TR-92-1, Dept. of Computer Science, University of Toronto.
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5,670 | 6,130 | Probing the Compositionality of Intuitive Functions
Eric Schulz
University College London
e.schulz@cs.ucl.ac.uk
Joshua B. Tenenbaum
MIT
jbt@mit.edu
Maarten Speekenbrink
University College London
m.speekenbrink@ucl.ac.uk
David Duvenaud
University of Toronto
duvenaud@cs.toronto.edu
Samuel J. Gershman
Harvard University
gershman@fas.harvard.edu
Abstract
How do people learn about complex functional structure? Taking inspiration from
other areas of cognitive science, we propose that this is accomplished by harnessing
compositionality: complex structure is decomposed into simpler building blocks.
We formalize this idea within the framework of Bayesian regression using a grammar over Gaussian process kernels. We show that participants prefer compositional
over non-compositional function extrapolations, that samples from the human prior
over functions are best described by a compositional model, and that people perceive compositional functions as more predictable than their non-compositional but
otherwise similar counterparts. We argue that the compositional nature of intuitive
functions is consistent with broad principles of human cognition.
1
Introduction
Function learning underlies many intuitive judgments, such as the perception of time, space and
number. All of these tasks require the construction of mental representations that map inputs to
outputs. Since the space of such mappings is infinite, inductive biases are necessary to constrain the
plausible inferences. What is the nature of human inductive biases over functions?
It has been suggested that Gaussian processes (GPs) provide a good characterization of these inductive
biases [15]. As we describe more formally below, GPs are distributions over functions that can encode
properties such as smoothness, linearity, periodicity, and other inductive biases indicated by research
on human function learning [5, 3]. Lucas et al. [15] showed how Bayesian inference with GP priors
can unify previous rule-based and exemplar-based theories of function learning [18].
A major unresolved question is how people deal with complex functions that are not easily captured
by any simple GP. Insight into this question is provided by the observation that many complex
functions encountered in the real world can be broken down into compositions of simpler functions
[6, 11]. We pursue this idea theoretically and experimentally, by first defining a hypothetical
compositional grammar for intuitive functions (based on [6]) and then investigating whether this
grammar quantitatively predicts human function learning performance. We compare the compositional
model to a flexible non-compositional model (the spectral mixture representation proposed by [21]).
Both models use Bayesian inference to reason about functions, but differ in their inductive biases.
We show that (a) participants prefer compositional pattern extrapolations in both forced choice
and manual drawing tasks; (b) samples elicited from participants? priors over functions are more
consistent with the compositional grammar; and (c) participants perceive compositional functions as
more predictable than non-compositional ones. Taken together, these findings provide support for the
compositional nature of intuitive functions.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
2
Gaussian process regression as a theory of intuitive function learning
A GP is a collection of random variables, any finite subset of which are jointly Gaussian-distributed
(see [18] for an introduction). A GP can be expressed as a distribution over functions: f ? GP(m, k),
where m(x) = E[f (x)] is a mean function modeling the expected output of the function given input
x, and k(x, x0 ) = E [(f (x) ? m(x))(f (x0 ) ? m(x0 ))] is a kernel function modeling the covariance
between points. Intuitively, the kernel encodes an inductive bias about the expected smoothness of
functions drawn from the GP. To simplify exposition, we follow standard convention in assuming a
constant mean of 0.
Conditional on data D = {X, y}, where yn ? N (f (xn ), ? 2 ), the posterior predictive distribution
for a new input x? is Gaussian with mean and variance given by:
2 ?1
E[f (x? )|D] = k>
y
(1)
? (K + ? I)
2 ?1
V[f (x? )|D] = k(x? , x? ) ? k>
k? ,
(2)
? (K + ? I)
where K is the N ? N matrix of covariances evaluated at each input in X and k? =
[k(x1 , x? ), . . . , k(xN , x? )].
As pointed out by Griffiths et al. [10] (see also [15]), the predictive distribution can be viewed as
an exemplar (similarity-based) model of function learning [5, 16], since it can be written as a linear
combination of the covariance between past and current inputs:
N
X
f (x? ) =
?n k(xn , x? )
(3)
n=1
with ? = (K + ? 2 I)?1 y. Equivalently, by Mercer?s theorem any positive definite kernel can be
expressed as an outer product of feature vectors:
?
X
k(x, x0 ) =
?d ?d (x)?d (x0 ),
(4)
d=1
where {?d (x)} are the eigenfunctions of the kernel and {?d } are the eigenvalues. The posterior
predictive mean is a linear combination of the features, which from a psychological perspective can
be thought of as encoding ?rules? mapping inputs to outputs [4, 14]. Thus, a GP can be expressed
as both an exemplar (similarity-based) model and a feature (rule-based) model, unifying the two
dominant classes of function learning theories in cognitive science [15].
3
Structure learning with Gaussian processes
So far we have assumed a fixed kernel function. However, humans can adapt to a wide variety of
structural forms [13, 8], suggesting that they have the flexibility to learn the kernel function from
experience. The key question addressed in this paper is what space of kernels humans are optimizing
over?how rich is their representational vocabulary? This vocabulary will in turn act as an inductive
bias, making some functions easier to learn, and other functions harder to learn.
Broadly speaking, there are two approaches to parameterizing the kernel space: a fixed functional
form with continuous parameters, or a combinatorial space of functional forms. These approaches
are not mutually exclusive; indeed, the success of the combinatorial approach depends on optimizing
the continuous parameters for each form. Nonetheless, this distinction is useful because it allows us
to separate different forms of functional complexity. A function might have internal structure such
that when this structure is revealed, the apparent functional complexity is significantly reduced. For
example, a function composed of many piecewise linear segments might have a long description
length under a typical continuous parametrization (e.g., the radial basis kernel described below),
because it violates the smoothness assumptions of the prior. However, conditional on the changepoints between segments, the function can be decomposed into independent parts each of which is
well-described by a simple continuous parametrization. If internally structured functions are ?natural
kinds,? then the combinatorial approach may be a good model of human intuitive functions.
In the rest of this section, we describe three kernel parameterizations. The first two are continuous,
differing in their expressiveness. The third one is combinatorial, allowing it to capture complex
patterns by composing simpler kernels. For all kernels, we take the standard approach of choosing
the parameter values that optimize the log marginal likelihood.
2
3.1
Radial basis kernel
The radial basis kernel is a commonly used kernel in machine learning applications, embodying the
assumption that the covariance between function values decays exponentially with input distance:
|x ? x0 |2
0
2
,
(5)
k(x, x ) = ? exp ?
2l2
where ? is a scaling parameter and l is a length-scale parameter. This kernel assumes that the same
smoothness properties apply globally for all inputs. It provides a standard baseline to compare with
more expressive kernels.
3.2
Spectral mixture kernel
The second approach is based on the fact that any stationary kernel can be expressed as an integral
using Bochner?s theorem. Letting ? = |x ? x0 | ? RP , then
Z
>
k(? ) =
e2?is ? ?(ds).
(6)
RP
If ? has a density S(s), then S is the spectral density of k; S and k are thus Fourier duals [18]. This
means that a spectral density fully defines the kernel and that furthermore every stationary kernel
can be expressed as a spectral density. Wilson & Adams [21] showed that the spectral density can be
approximated by a mixture of Q Gaussians, such that
k(? ) =
Q
X
wq
q=1
P
Y
exp ?2? 2 ?p2 ?qp cos 2??p ?(p)
q
(7)
p=1
(1)
(P )
Here, the qth component has mean vector ?q = ?q , . . . , ?q
and a covariance matrix
(1)
(P )
Mq = diag ?q , . . . , ?q
. The result is a non-parametric approach to Gaussian process regression, in which complex kernels are approximated by mixtures of simpler ones. This approach is
appealing when simpler kernels fail to capture functional structure. Its main drawback is that because
structure is captured implicitly via the spectral density, the building blocks are psychologically less
intuitive: humans appear to have preferences for linear [12] and periodic [1] functions, which are not
straightforwardly encoded in the spectral mixture (though of course the mixture can approximate
these functions). Since the spectral kernel has been successfully applied to reverse engineer human
kernels [22], it is a useful reference of comparison to more structured compositional approaches.
3.3
Compositional kernel
As positive semidefinite kernels are closed under addition and multiplication, we can create richly
structured and interpretable kernels from well understood base components. For example, by
summing kernels, we can model the data as a superposition of independent functions. Figure 1
shows an example of how different kernels (radial basis, linear, periodic) can be combined. Table 1
summarizes the kernels used in our grammar.
LIN
PER
PER+LIN
RBFxPER
f(x)
RBF
x
Figure 1: Examples of base and compositional kernels.
Many other compositional grammars are possible. For example, we could have included a more
diverse set of kernels, and other composition operators (e.g., convolution, scaling) that generate valid
kernels. However, we believe that our simple grammar is a useful starting point, since the components
are intuitive and likely to be psychologically plausible. For tractability, we fix the maximum number
of combined kernels to 3. Additionally, we do not allow for repetition of kernels in order to restrict
the complexity of the kernel space.
3
Linear
k(x, x0 ) = (x ? ?1 )(x0 ? ?1 )
Radial basis function
? )2
k(? ) = ?22 exp ? (2?
2
3
Periodic
2
k(? ) = ?42 exp ? 2 sin ?(?2 ? ?5 )
6
Table 1: Utilized base kernels in our compositional grammar. ? = |x ? x0 |
.
4
Experiment 1: Extrapolation
The first experiment assessed whether people prefer compositional over non-compositional extrapolations. In experiment 1a, functions were sampled from a compositional GP and different extrapolations
(mean predictions) were produced using each of the aforementioned kernels. Participants were then
asked to choose among the 3 different extrapolations for a given function (see Figure 2). In detail, the
outputs for xlearn = [0, 0.1, ? ? ? , 7] were used as a training set to which all three kernels were fitted
and then used to generate predictions for the test set xtest = [7.1, 7.2, ? ? ? , 10]. Their mean predictions
were then used to generate one plot for every approach that showed the learned input as a blue line and
the extrapolation as a red line. The procedure was repeated for 20 different compositional functions.
Figure 2: Screen shot of first choice experiment. Predictions in this example (from left to right) were
generated by a spectral mixture, a radial basis, and a compositional kernel.
52 participants (mean age=36.15, SD = 9.11) were recruited via Amazon Mechanical Turk and
received $0.5 for their participation. Participants were asked to select one of 3 extrapolations
(displayed as red lines) they thought best completed a given blue line. Results showed that participants
chose compositional predictions 69%, spectral mixture predictions 17%, and radial basis predictions
14% of the time. Overall, the compositional predictions were chosen significantly more often than
the other two (?2 = 591.2, p < 0.01) as shown in Figure 3a.
Compositional
Spectral mixture
1.00
Proportion chosen
Proportion chosen
1.00
0.75
?
0.50
0.25
?
0.75
?
0.50
?
0.25
?
0.00
0.00
Compositional
RBF
Spectral
Compositional
Kernel
Spectral
Kernel
(a) Choice proportion for compositional ground truth. (b) Choice proportion for spectral mixture ground truth.
Figure 3: Results of extrapolation experiments. Error bars represent the standard error of the mean.
In experiment 1b, again 20 functions were sampled but this time from a spectral mixture kernel
and 65 participants (mean age=30, SD = 9.84) were asked to choose among either compositional
or spectral mixture extrapolations and received $0.5 as before. Results (displayed in Figure 3b)
showed that participants again chose compositional extrapolations more frequently (68% vs. 32%,
?2 = 172.8, p < 0.01), even if the ground truth happened to be generated by a spectral mixture
kernel. Thus, people seem to prefer compositional over non-compositional extrapolations in forced
choice extrapolation tasks.
4
5
Markov chain Monte Carlo with people
In a second set of experiments, we assessed participants? inductive biases directly using a Markov
chain Monte Carlo with People (MCMCP) approach [19]. Participants accept or reject proposed
extrapolations, effectively simulating a Markov chain whose stationary distribution is in this case the
posterior predictive. Extrapolations from all possible kernel combinations (up to 3 combined kernels)
were generated and stored a priori. These were then used to generate plots of different proposal
extrapolations (as in the previous experiment). On each trial, participants chose between their most
recently accepted extrapolation and a new proposal.
5.1
Experiment 2a: Compositional ground truth
In the first MCMCP experiment, we sampled functions from compositional kernels. Eight different
functions were sampled from various compositional kernels, the input space was split into training
and test sets, and then all kernel combinations were used to generate extrapolations. Proposals were
sampled uniformly from this set. 51 participants with an average age of 32.55 (SD = 8.21) were
recruited via Amazon?s Mechanical Turk and paid $1. There were 8 blocks of 30 trials, where each
block corresponded to a single training set. We calculated the average proportion of accepted kernels
over the last 5 trials, as shown in Figure 4.
Proportion of accepted kernels
LIN + PER 1
0.4
LIN + PER 2
?
?
0.3
0.4
?
0.2
0.2
0.1
?
?
?
0.0
?
?
l
p
?
?
r
?
?
?
?
?
?
?
0.0
l+p l+r p+r lxp lxr pxr p+r+lpxr+lpxl+rrxl+ppxlxr
?
?
l
?
p
r
LIN + PER 3
?
?
?
?
?
LIN + PER 4
0.25
?
?
0.3
0.20
?
?
0.15
0.2
?
?
0.10
?
0.1
0.0
?
?
?
l+p l+r p+r lxp lxr pxr p+r+lpxr+lpxl+rrxl+ppxlxr
?
?
?
l
?
?
p
?
r
?
?
0.05
?
?
?
?
0.00
?
l+p l+r p+r lxp lxr pxr p+r+lpxr+lpxl+rrxl+ppxlxr
?
l
?
?
p
r
?
?
l+p l+r p+r lxp lxr pxr p+r+lpxr+lpxl+rrxl+ppxlxr
LIN x PER
PER x RBF + LIN
?
0.3
?
?
?
?
0.2
?
?
0.2
?
?
?
?
0.1
0.1
?
?
?
?
?
?
0.0
?
?
l
p
r
?
?
?
?
?
?
?
?
?
?
0.0
?
l+p l+r p+r lxp lxr pxr p+r+lpxr+lpxl+rrxl+ppxlxr
l
p
r
?
l+p l+r p+r lxp lxr pxr p+r+lpxr+lpxl+rrxl+ppxlxr
PER
LIN + PER + RBF
?
?
?
0.3
0.2
?
0.2
?
0.1
0.1
?
?
?
?
0.0
l
?
?
?
?
?
p
r
?
?
?
?
?
?
?
?
?
0.0
?
l+p l+r p+r lxp lxr pxr p+r+lpxr+lpxl+rrxl+ppxlxr
l
p
r
?
?
?
?
l+p l+r p+r lxp lxr pxr p+r+lpxr+lpxl+rrxl+ppxlxr
Figure 4: Proportions of chosen predictions over last 5 trials. Generating kernel marked in red.
In all cases participants? subjective probability distribution over kernels corresponded well with the
data-generating kernels. Moreover, the inverse marginal likelihood, standardized over all kernels,
correlated highly with the subjective beliefs assessed by MCMCP (? = 0.91, p < .01). Thus, participants seemed to converge to sensible structures when the functions were generated by compositional
kernels.
5
5.2
Experiment 2b: Naturalistic functions
The second MCMCP experiment assessed what structures people converged to when faced with real
world data. 51 participants with an average age of 32.55 (SD = 12.14) were recruited via Amazon
Mechanical Turk and received $1 for their participation. The functions were an airline passenger data
set, volcano CO2 emission data, the number of gym memberships over 5 years, and the number of
times people googled the band ?Wham!? over the last 8 years; all shown in Figure 5a. Participants
were not told any information about the data set (including input and output descriptions) beyond
the input-output pairs. As periodicity in the real world is rarely ever purely periodic, we adapted
the periodic component of the grammar by multiplying a periodic kernel with a radial basis kernel,
thereby locally smoothing the periodic part of the function.1 Apart from the different training sets,
the procedure was identical to the last experiment.
Real world data
Proportion of accepted kernels
Airline passengers
Airline passengers
300
0.4
200
0.3
100
?
?
0.2
0
?
0.1
?100
0.0
?200
0
50
100
150
?
l
p
r
?
?
?
?
?
?
l+p
l+r
p+r
lxp
lxr
pxr
Gym memberships
?
p+r+l
pxr+l
?
?
pxl+r
rxl+p
?
pxlxr
Gym memberships
100
?
0.6
80
0.4
60
0.2
40
0.0
200
y
0
400
600
?
?
l
?
?
p
r
l+p
?
?
?
?
?
l+r
p+r
lxp
lxr
pxr
Volcano
0.3
375
?
?
?
pxl+r
rxl+p
pxlxr
?
?
0.2
350
?
?
0.1
?
325
0.0
1980
?
pxr+l
Volcano
400
1960
p+r+l
2000
?
?
?
l
p
r
l+p
l+r
p+r
Wham!
lxp
?
?
?
?
lxr
pxr
?
p+r+l
pxr+l
?
pxl+r
rxl+p
pxlxr
Wham!
?
20
0.6
15
0.4
10
0.2
5
0.0
?
?
0
100
200
300
400
?
?
l
p
r
l+p
?
?
?
?
?
l+r
p+r
lxp
lxr
pxr
p+r+l
?
?
?
?
pxr+l
pxl+r
rxl+p
pxlxr
x
(a) Data.
(b) Proportions of chosen predictions over last 5 trials.
Figure 5: Real world data and MCMCP results. Error bars represent the standard error of the mean.
Results are shown in Figure 5b, demonstrating that participants converged to intuitively plausible
patterns. In particular, for both the volcano and the airline passenger data, participants converged to
compositions resembling those found in previous analyses [6]. The correlation between the mean
proportion of accepted predictions and the inverse standardized marginal likelihoods of the different
kernels was again significantly positive (? = 0.83, p < .01).
6
Experiment 3: Manual function completion
In the next experiment, we let participants draw the functions underlying observed data manually.
As all of the prior experiments asked participants to judge between ?pre-generated? predictions of
functions, we wanted to compare this to how participants generate predictions themselves. On each
round of the experiment, functions were sampled from the compositional grammar, the number of
points to be presented on each trial was sampled uniformly between 100 and 200, and the noise
variance was sampled uniformly between 0 and 25. Finally, the size of an unobserved region of the
1
See the following page for an example: http://learning.eng.cam.ac.uk/carl/mauna.
6
function was sampled to lie between 5 and 50. Participants were asked to manually draw the function
best describing observed data and to inter- and extrapolate this function in two unobserved regions. A
screen shot of the experiment is shown in Figure 6.
Figure 6: Manual pattern completion experiment. Extrapolation region is delimited by vertical lines.
36 participants with a mean age of 30.5 (SD = 7.15) were recruited from Amazon Mechanical Turk
and received $2 for their participation. Participants were asked to draw lines in a cloud of dots that
they thought best described the given data. To facilitate this process, participants placed black dots
into the cloud, which were then automatically connected by a black line based on a cubic Bezier
smoothing curve. They were asked to place the first dot on the left boundary and the final dot on the
right boundary of the graph. In between, participants were allowed to place as many dots as they
liked (from left to right) and could remove previously placed dots. There were 50 trials in total. We
assessed the average root mean squared distance between participants? predictions (the line they drew)
and the mean predictions of each kernel given the data participants had seen, for both interpolation
and extrapolation areas. Results are shown in Figure 7.
Interpolation
Extrapolation
?
?
?
75
90
?
?
RMSE
RMSE
?
50
25
60
30
0
0
Compositional
RBF
Spectral
Compositional
Kernel
RBF
Spectral
Kernel
(a) Distance for interpolation drawings.
(b) Distance for extrapolation drawings.
Figure 7: Root mean squared distances. Error bars represent the standard error of the mean.
The mean distance from participants? drawings was significantly higher for the spectral mixture
kernel than for the compositional kernel in both interpolation (86.96 vs. 58.33, t(1291.1) = ?6.3,
p < .001) and extrapolation areas (110.45 vs 83.91, t(1475.7) = 6.39, p < 0.001). The radial basis
kernel produced similar distances as the compositional kernel in interpolation (55.8), but predicted
participants? drawings significantly worse in extrapolation areas (97.9, t(1459.9) = 3.26, p < 0.01).
7
Experiment 4: Assessing predictability
Compositional patterns might also affect the way in which participants perceive functions a priori
[20]. To assess this, we asked participants to judge how well they thought they could predict 40
different functions that were similar on many measures such as their spectral entropy and their average
wavelet distance to each other, but 20 of which were sampled from a compositional and 20 from a
spectral mixture kernel. Figure 8 shows a screenshot of the experiment.
50 participants with a mean age of 32 (SD = 7.82) were recruited via Amazon Mechanical Turk and
received $0.5 for their participation. Participants were asked to rate the predictability of different
functions. On each trial participants were shown a total of nj ? {50, 60, . . . , 100} randomly sampled
input-output points of a given function and asked to judge how well they thought they could predict the
output for a randomly sampled input point on a scale of 0 (not at all) to 100 (very well). Afterwards,
they had to rate which of two functions was easier to predict (Figure 8) on a scale from -100 (left
graph is definitely easier to predict) to 100 (right graph is definitely easier predict).
As shown in Figure 9, compositional functions were perceived as more predictable than spectral
functions in isolation (t(948) = 11.422, p < 0.01) and in paired comparisons (t(499) = 13.502,
p < 0.01). Perceived predictability increases with the number of observed outputs (r = 0.23,
7
(b) Comparative judgements.
(a) Predictability judgements.
Figure 8: Screenshot of the predictablity experiment.
Predictability
Group
?
Compositional
?
Direct Comparison
Spectral
50
70
?
?
?
Mean Judgement
Mean Judgement
40
60
?
?
50
?
?
?
?
40
?
?
70
80
?
?
30
?
?
?
20
10
?
?
0
30
50
60
90
100
50
Sample Size
60
70
80
90
100
Sample Size
(a) Predictability judgements.
(b) Comparative judgements.
Figure 9: Results of the predictablity experiment. Error bars represent the standard error of the mean.
p < 0.01) and the larger the number of observations, the larger the difference between compositional
and spectral mixture functions (r = 0.14, p < 0.01).
8
Discussion
In this paper, we probed human intuitions about functions and found that these intuitions are best
described as compositional. We operationalized compositionality using a grammar over kernels within
a GP regression framework and found that people prefer extrapolations based on compositional kernels
over other alternatives, such as a spectral mixture or the standard radial basis kernel. Two Markov
chain Monte Carlo with people experiments revealed that participants converge to extrapolations
consistent with the compositional kernels. These findings were replicated when people manually drew
the functions underlying observed data. Moreover, participants perceived compositional functions as
more predictable than non-compositional ? but otherwise similar ? ones.
The work presented here is connected to several lines of previous research, most importantly that
of Lucas et al. [15], which introduced GP regression as a model of human function learning, and
Wilson et al. [22], which attempted to reverse-engineer the human kernel using a spectral mixture.
We see our work as complementary; we need both a theory to describe how people make sense of
structure as well as a method to indicate what the final structure might look like when represented
as a kernel. Our approach also ties together neatly with past attempts to model structure in other
cognitive domains such as motion perception [9] and decision making [7].
Our work can be extended in a number of ways. First, it is desirable to more thoroughly explore
the space of base kernels and composition operators, since we used an elementary grammar in
our analyses that is probably too simple. Second, the compositional approach could be used in
traditional function learning paradigms (e.g., [5, 14]) as well as in active input selection paradigms
[17]. Another interesting avenue for future research would be to explore the broader implications of
compositional function representations. For example, evidence suggests that statistical regularities
reduce perceived numerosity [23] and increase memory capacity [2]; these tasks can therefore provide
clues about the underlying representations. If compositional functions alter number perception or
memory performance to a greater extent than alternative functions, that suggests that our theory
extends beyond simple function learning.
8
References
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Neurobiology, 20:251?256, 2010.
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[14] K. Koh and D. E. Meyer. Function learning: Induction of continuous stimulus-response relations. Journal
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Comparison of rule-based and associative-based models. Psychonomic Bulletin & Review, 12:24?42, 2005.
[17] P. Parpart, E. Schulz, M. Speekenbrink, and B. C. Love. Active learning as a means to distinguish among
prominent decision strategies. In Proceedings of the 37th Annual Meeting of the Cognitive Science Society,
pages 1829?1834, 2015.
[18] C. Rasmussen and C. Williams. Gaussian Processes for Machine Learning. MIT Press, 2006.
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Monte Carlo. Cognitive Psychology, 60(2):63?106, 2010.
[20] E. Schulz, J. B. Tenenbaum, D. N. Reshef, M. Speekenbrink, and S. J. Gershman. Assessing the perceived
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9
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5,671 | 6,131 | A Bayesian method for reducing bias in neural
representational similarity analysis
Ming Bo Cai
Princeton Neuroscience Institute
Princeton University
Princeton, NJ 08544
mcai@princeton.edu
Nicolas W. Schuck
Princeton Neuroscience Institute
Princeton University
Princeton, NJ 08544
nschuck@princeton.edu
Jonathan W. Pillow
Princeton Neuroscience Institute
Princeton University
Princeton, NJ 08544
pillow@princeton.edu
Yael Niv
Princeton Neuroscience Institute
Princeton University
Princeton, NJ 08544
yael@princeton.edu
Abstract
In neuroscience, the similarity matrix of neural activity patterns in response to
different sensory stimuli or under different cognitive states reflects the structure
of neural representational space. Existing methods derive point estimations of
neural activity patterns from noisy neural imaging data, and the similarity is
calculated from these point estimations. We show that this approach translates
structured noise from estimated patterns into spurious bias structure in the resulting
similarity matrix, which is especially severe when signal-to-noise ratio is low and
experimental conditions cannot be fully randomized in a cognitive task. We propose
an alternative Bayesian framework for computing representational similarity in
which we treat the covariance structure of neural activity patterns as a hyperparameter in a generative model of the neural data, and directly estimate this
covariance structure from imaging data while marginalizing over the unknown
activity patterns. Converting the estimated covariance structure into a correlation
matrix offers a much less biased estimate of neural representational similarity. Our
method can also simultaneously estimate a signal-to-noise map that informs where
the learned representational structure is supported more strongly, and the learned
covariance matrix can be used as a structured prior to constrain Bayesian estimation
of neural activity patterns. Our code is freely available in Brain Imaging Analysis
Kit (Brainiak) (https://github.com/IntelPNI/brainiak).
1
Neural pattern similarity as a way to understand neural representations
Understanding how patterns of neural activity relate to internal representations of the environment
is one of the central themes of both system neuroscience and human neural imaging [20, 5, 7, 15].
One can record neural responses (e.g. by functional magnetic resonance imaging; fMRI) while
participants observe sensory stimuli, and in parallel, build different computational models to mimic
the brain?s encoding of these stimuli. Neural activity pattern corresponding to each feature of an
encoding model can then be estimated from the imaging data. Such activity patterns can be used
to decode the perceived content with respect to the encoding features from new imaging data. The
degree to which stimuli can be decoded from one brain area based on different encoding models
informs us of the type of information represented in that area. For example, an encoding model based
on motion energy in visual stimuli captured activity fluctuations from visual cortical areas V1 to V3,
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
and was used to successfully decode natural movie watched during an fMRI scan [14]. In contrast,
encoding models based on semantic categories can more successfully decode information from higher
level visual cortex [7].
While the decoding performance of different encoding models informs us of the type of information
represented in a brain region, it does not directly reveal the structure of the representational space in
that area. Such structure is indexed by how distinctively different contents are represented in that
region [21, 4]. Therefore, one way to directly quantify the structure of the representational space
in the neural population activity is to estimate the neural activity pattern elicited by each sensory
stimulus, and calculate the similarity between the patterns corresponding to each pair of stimuli.
This analysis of pair-wise similarity between neural activity patterns to different stimuli was named
Representational Similarity Analysis (RSA) [11]. In fact, one of the earliest demonstrations of
decoding from fMRI data was based on pattern similarity [7]. RSA revealed that the representational
structures in the inferotemporal (IT) cortex of natural objects are highly similar between human and
monkey [12] and a continuum in the abstract representation of biological classes exist in human
ventral object visual cortex [2]. Because the similarity structure can be estimated from imaging data
even without building an encoding model, RSA allows not only for model testing (by comparing the
similarity matrix of neural data with the similarity matrix of the feature vectors when stimuli are
represented with an encoding model) but also for exploratory study (e.g., by projecting the similarity
structure to a low-dimensional space to visualize its structure, [11]). Therefore, originally as a tool
for studying visual representations [2, 16, 10], RSA has recently attracted neuroscientists to explore
the neural representational structure in many higher level cognitive areas [23, 18].
2
Structured noise in pattern estimation translates into bias in RSA
Although RSA is gaining popularity, a few recent studies revealed that in certain circumstances
the similarity structure estimated by standard RSA might include a significant bias. For example,
the estimated similarity between fMRI patterns of two stimuli is much higher when the stimuli are
displayed closer in time [8]. This dependence of pattern similarity on inter-stimulus interval was
hypothesized to reflect "temporal drift of pattern"[1], but we believe it may also be due to temporal
autocorrelation in fMRI noise. Furthermore, we applied RSA to a dataset from a structured cognitive
task (Fig 1A) [19] and found that the highly structured representational similarity matrix obtained
from the neural data (Fig 1B,C) is very similar to the matrix obtained when RSA is applied to pure
white noise (Fig 1D). Since no task-related similarity structure should exist in white noise while the
result in Fig 1D is replicable from noise, this shows that the standard RSA approach can introduce
similarity structure not present in the data.
We now provide an analytical derivation to explain the source of both types of bias (patterns closer
in time are more similar and spurious similarity emerges from analyzing pure noise). It is notable
that almost all applications of RSA explicitly or implicitly assume that fMRI responses are related to
task-related events through a general linear model (GLM):
Y = X ? ? + .
(1)
Here, Y ? RnT ?nS is the fMRI time series from an experiment with nT time points from nS brain
voxels. The experiment involves nC different conditions (e.g., different sensory stimuli, task states,
or mental states), each of which comprises events whose onset time and duration is either controlled
by the experimenter, or can be measured experimentally (e.g., reaction times). In fMRI, the measured
blood oxygen-level dependent (BOLD) response is protracted, such that the response to condition
c is modelled as the time course of events in the experimental condition sc (t) convolved with a
typical hemodynamic response function (HRF) h(t). Importantly, each voxel can respond to different
conditions with different amplitudes ? ? RnC ?nS , and the responses to all conditions are assumed
to contribute linearly to the measured signal. Thus, denoting the matrix of HRF-convolved event time
courses for each task condition with X ? RnT ?nC , often called the design matrix, the measured Y is
assumed to be a linear sum of X weighted by response amplitude ? plus zero-mean noise.
Each row of ? is the spatial response pattern (i.e., the response across voxels) to an experimental
condition. The goal of RSA is therefore to estimate the similarity between the rows of ?. Because
? is unknown, pattern similarity is usually calculated based on ordinary least square estimation of
? = (XT X)?1 XT Y, and then using Pearson correlation of ?
? to measure similarity. Because
?: ?
2
A Markovian state
C
transition
Low-dimensional
projection
Enter
Internal
Exit
D ?Similarity? from noise
Similarity in brain
Exit Internal
Enter
B
Figure 1: Standard RSA introduces bias structure to the similarity matrix. (A) A cognitive task
that includes 16 different experimental conditions. Transitions between conditions follow a Markov
process. Arrows indicate possible transitions, each with p = 0.5. The task conditions can be grouped
to 3 categories (color coded) according to the semantics, or mental operations, required in each
condition (the exact meaning of these conditions is not relevant to this paper). (B) Standard RSA of
activity patterns corresponding to each condition estimated from a region of interest (ROI) reveal
a highly structured similarity matrix. (C) Converting the similarity matrix C to a distance matrix
1 ? C and projecting it to a low-dimensional space using multi-dimensional scaling [13] reveals a
highly regular structure. Seeing such a result, one may infer that representational structure in the
ROI is strongly related to the semantic meanings of the task conditions. (D) However, a very similar
similarity matrix can also be obtained if one applies standard RSA to pure white noise, with a similar
low-dimensional projection (not shown). This indicates that standard RSA can introduce spurious
structure in the resulting similarity matrix that does not exist in the data.
calculating sample correlation implies the belief that there exists an underlying covariance structure
? compared to that of true ?.
of ?, we examine the source of bias by focusing on the covariance of ?
We assume ? of all voxels in the ROI are indeed random vectors generated from a multivariate
Gaussian distribution N(0, U) (the size of U being nC ? nC ). If one knew the true U, similarity
measures such as correlation could be derived from it. Substituting the expression Y from equation 1
? = ? + (XT X)?1 XT . We assume that the signal ? is independent from the noise ,
we have ?
and therefore also independent from its linear transformation (XT X)?1 XT . Thus the covariance of
? is the sum of the true covariance of ? and the covariance of (XT X)?1 XT :
?
? ? N(0, U + (XT X)?1 XT ? X(XT X)?1 )
?
(2)
Where ? ? RnT ?nT is the temporal covariance of the noise (for illustration purposes, in this
section we assume that all voxels have the same noise covariance).
The term
(XT X)?1 XT ? X(XT X)?1
? has this bias term adding to U which we are
is the source of the bias. Since the covariance of ?
interested in, their sample correlation is also biased. So are many other similarity measures based on
? such as Eucledian distance.
?,
3
The bias term (XT X)?1 XT ? X(XT X)?1 depends on both the design matrix and the properties of
the noise. It is well known that autocorrelation exists in fMRI noise [24, 22]. Even if we assume that
the noise is temporally independent (i.e., ? is a diagonal matrix, which may be a valid assumption
if one "pre-whitens" the data before further analysis [22]), the bias structure still exists but reduces to
(XT X)?1 ? 2 , where ? 2 is the variance of the noise. Diedrichsen et al. [6] realized that the noise in
? could contribute to a bias in the correlation matrix but assumed the bias is only in the diagonal
?
of the matrix. However, the bias is a diagonal matrix only if the columns of X (hypothetical fMRI
response time courses to different conditions) are orthogonal to each other and if the noise has no
autocorrelation. This is rarely the case for most cognitive tasks. In the example in Figure 1A, the
transitions between experimental conditions follow a Markov process such that some conditions are
always temporally closer than others. Due to the long-lasting HRF, conditions of temporal proximity
will have higher correlation in their corresponding columns in X. Such correlation structure in X is
the major determinant of the bias structure in this case. On the other hand, if each single stimulus is
?
modelled as a condition in X and regularization is used during regression, the correlation between ?
of temporally adjacent stimuli is higher primarily because of the autocorrelation property of the noise.
This can be the major determinant of the bias structure in cases such as [8].
It is worth noting that the magnitude of bias is larger relative to the true covariance structure U when
the signal-to-noise ratio (SNR) is lower, or when X has less power (i.e., there are few repetitions of
each condition, thus few measurements of the related neural activity), as illustrated later in Figure 2B.
The bias in RSA was not noticed until recently [1, 8], probably because RSA was initially applied
to visual tasks in which stimuli are presented many times in a well randomized order. Such designs
made the bias structure close to a diagonal matrix and researchers typically only focus on off-diagonal
elements of a similarity matrix. In contrast, the neural signals in higher-level cognitive tasks are
typically weaker than those in visual tasks [9]. Moreover, in many decision-making and memory
studies the orders of different task conditions cannot be fully counter-balanced. Therefore, we expect
the bias in RSA to be much stronger and highly structured in these cases, misleading researchers and
hiding the true (but weaker) representational structure in the data.
? using regression as above, is to perform RSA on the raw conditionOne alternative to estimating ?
averaged fMRI data (for instance, taking the average signal ? 6 sec after the onset of an event as a
? This is equivalent to using a design matrix that assumes a 6-sec delayed single-pulse
proxy for ?).
? is still biased, so is its
HRF. Although here columns of X are orthogonal by definition, the estimate ?
T
T
T
T
?1 T
?1
?1 T
covariance (X X) X Xtrue U Xtrue X(X X) + (X X) X ? X(XT X)?1 (where Xtrue
is the design matrix reflecting the true HRF in fMRI). See supplementary material for illustration of
this bias.
3
Maximum likelihood estimation of similarity structure directly from data
As shown in equation 2, the bias in RSA stems from treating the noisy estimate of ? as the true ? and
performing a secondary analysis (correlation) on this noisy estimate. The similarly-structured noise
? translates into bias in
(in terms of the covariance of their generating distribution) in each voxel?s ?
the secondary analysis. Since the bias comes from inferring U indirectly from point estimation of ?,
a good way to avoid such bias is by not relying analysis on this point estimation. With a generative
model relating U to the measured fMRI data Y, we can avoid the point estimation of unknown ? by
marginalizing it in the likelihood of observing the data. In this section, we propose a method which
performs maximum-likelihood estimation of the shared covariance structure U of activity patterns
directly from the data.
Our generative model of fMRI data follows most of the assumptions above, but also allows the noise
property and the SNR to vary across voxels. We use an AR(1) process to model the autocorrelation
of noise in each voxel: for the ith voxel, we denote the noise at time t(> 0) as t,i , and assume
t,i = ?i ? t?1,i + ?t,i ,
where
?i2
?t,i ? N(0, ?i2 )
is the variance of the "new" noise and ?i is the autoregressive coefficient for the i
(3)
th
voxel.
We assume that the covariance of the Gaussian distribution from which the activity amplitudes ? i of
the ith voxel are generated has a scaling factor that depends on its SNR si :
? i ? N(0, (si ?i )2 U).
(4)
4
This is to reflect the fact that not all voxels in an ROI respond to tasks (voxels covering partially or
entirely white matter might have little or no response). Because the magnitude of the BOLD response
to a task is determined by the product of the magnitude of X and ?, but s is a hyper-parameter only
of ?, we hereforth refer to s as pseudo-SNR.
We further use the Cholesky decomposition to parametrize the shared covariance structure across
voxels: U = LLT , where L is a lower triangular matrix. Thus, ? i can be written as ? i = si ?i L?i ,
where ?i ? N (0, I) (this change of parameter allows for estimating U of less than full rank
by setting L as lower-triangular matrix with a few rightmost-columns truncated). And we have
Yi ? si ?i XL?i ? N (0, ?i (?i , ?i )). Therefore, for the ith voxel, the likelihood of observing
data Yi given the parameters is:
Z
p(Yi |L, ?i , ?i , si ) =
p(Yi |L, ?i , ?i , si , ?i )p(?i )d?i
Z
nT
1
1
T ?1
2
2
= (2?)? 2 |??1
i | exp[? (Yi ? si ?i XL?i ) ?i (Yi ? si ?i XL?i )]
2
nC
1
? (2?)? 2 exp[? ?T
?i ]d?i
2 i
nT
1
1
1
2 T ?1
T
T ?1
T ?1
2
2
=(2?)? 2 |??1
i | |?i | exp[ ((si ?i ) Yi ?i XL?i L X ?i Yi ? Yi ?i Yi )]
2
(5)
?1
where ?i = (s2i ?i2 LT X T ??1
. ??1
i XL + I)
i is the inverse of the noise covariance matrix of
th
the i voxel, which is a function of ?i and ?i (see supplementary material).
For simplicity, we assume that the noise for different voxels is independent, which is the common
assumption of standard RSA (although see [21]). The likelihood of the whole dataset, including all
voxels in an ROI, is then
Y
p(Y |L, ?, ?, s) =
p(Yi |L, ?i , ?i , si ).
(6)
i
We can use gradient-based methods to optimize the model, that is, to search for the values of
parameters that maximize the log likelihood of the data. Note that s are determined only up to a scale,
because L can be scaled down by a factor and all si can be scaled up by the same factor without
influencing the likelihood. Therefore, we set the geometric mean of s to be 1 to circumvent this
indeterminacy, and fit s and L iteratively. The spatial pattern of s thus only reflects the relative SNR
of different voxels.
? the estimate of L, we can convert the covariance matrix U
? = L
?L
? T into a
Once we obtain L,
correlation matrix, which is our estimation of neural representational similarity. Because U is a
hyper-parameter of the activity pattern in our generative model and we estimate it directly from data,
this is an empirical Bayesian approach. We therefore refer to our method as ?Bayesian RSA? now.
4
4.1
Performance of the method
Reduced bias in recovering the latent covariance structure from simulated data
To test if the proposed method indeed reduces bias, we simulated fMRI data with a predefined
covariance structure and compared the structure recovered by our method with that recovered by
standard RSA. Fig 2A shows the hypothetical covariance structure from which we drew ?i for each
voxel. The bias structure in Fig 1D is the average structure induced by the design matrices of all
participants. To simplify the comparison, we use the design matrices of the experiment experienced
by one participant. As a result, the bias structure induced by the design matrix deviates slightly from
that in Fig 1D.
? depends on both the level of noise
As mentioned, the contribution of the bias to the covariance of ?
and the power in the design matrix X. The more each experimental condition is measured during an
? and
experiment (roughly speaking, the longer the experiment), the less noisy the estimation of ?,
the less biased the standard RSA is. To evaluate the improvement of our method over standard RSA
5
A Covariance
structure
of simulated ?
B
standard
individual
Recovered covariance
structure
standard
average
Bayesian
individual
Bayesian
average
C % of recovered
structure not explained
by true structure
individual
average
SNR
0.16
0.31
0.63
Figure 2: Bayesian RSA reduces bias in the recovered shared covariance structure of activity
patterns. (A) The covariance structure from which we sampled neural activity amplitudes ? for
each voxel. fMRI data were synthesized by weighting the design matrix of the task from Fig 1A
with the simulated ? and adding AR(1) noise. (B) The recovered covariance structure for different
? as is done in
simulated pseudo-SNR. Standard individual: covariance calculated directly from ?
standard RSA, for one simulated participant. Standard average: average of covariance matrices of
? from 20 simulated participants. Bayesian individual: covariance estimated directly from data by
?
our method for one simulated participant. Bayesian average: average of the covariance matrices
estimated by Bayesian RSA from 20 simulated participants. (C) The ratio of the variation in the
recovered covariance structure which cannot be explained by the true covariance structure in Fig 2A.
Left: the ratio for covariance matrix from individual simulation (panel 1 and 3 of Fig 2B). Right: the
ratio for average covariance matrix (panel 2 and 4 of Fig 2B). Number of runs: the design matrices of
1, 2, or 4 runs of a participant in the experiment of Fig 1A were used in each simulation, to test the
effect of experiment duration. Error bar: standard deviation.
in different scenarios, we therefore varied two factors: the average SNR of voxels and the duration
of the experiment. 500 voxels were simulated. For each voxel, ?i was sampled uniformly from
[1.0, 3.0], ?i was sampled uniformly from [?0.2, 0.6] (our empirical investigation of example
fMRI data shows that small negative autoregressive coefficient can occur in white matter), si was
sampled uniformly from f ? [0.5, 2.0]. The average SNR was manipulated by choosing f from one
of three levels {1, 2, 4} in different simulations. The duration of the experiment was manipulated by
using the design matrices of run 1, runs 1-2, and runs 1-4 from one participant.
Fig 2B displays the covariance matrix recovered by standard RSA (first two columns) and Bayesian
RSA (last two columns), with an experiment duration of approximately 10 minutes (one run, measurement resolution: TR = 2.4 sec). The rows correspond to different levels of average SNR (calculated
std(X ?i )
post-hoc by averaging the ratio
across voxels). Covariance matrices recovered from one
?i
simulated participant and the average of covariance matrices recovered from 20 simulated participants
(?average?) are displayed. Comparing the shapes of the matrix and the magnitudes of values (color
bars) across rows, one can see that the bias structure in standard RSA is most severe when SNR is
low. Averaging the estimated covariance matrices across simulated participants can reduce noise, but
not bias. Comparing between columns, one can see that strong residual structure exists in standard
RSA even after averaging, but almost disappears for Bayesian RSA. This is especially apparent for
low SNR ? the block structure of the true covariance matrix from Figure 2A is almost undetectable
for standard RSA even after averaging (column 2, row 1 of Fig 2B), but emerges after averaging
for Bayesian RSA (column 4, row 1 of Fig 2B). Fig 2C compares the proportion of variation in the
recovered covariance structure that cannot be explained by the true structure in Fig 2A, for different
levels of SNR and different experiment durations, for individual simulated participants and for
average results. This comparison confirms that the covariance recovered by Bayesian RSA deviates
much less from the true covariance matrix than that by standard RSA, and that the deviation observed
in an individual participant can be reduced considerably by averaging over multiple participants
(comparing the left with right panels of Fig 2C for Bayesian RSA).
6
4.2
Application to real data: simultaneous estimation of neural representational similarity
and spatial location supporting the representation
In addition to reducing bias in estimation of representational similarity, our method also has an
advantage over standard RSA: it estimates the pseudo-SNR map s. This map reveals the locations
within the ROI that support the identified representational structure. When a researcher looks into
an anatomically defined ROI, it is often the case that only some of the voxels respond to the task
? in voxels with little or no response to tasks is dominated by structured
conditions. In standard RSA, ?
noise following the bias covariance structure (XT X)?1 XT ? X(XT X)?1 , but all voxels are taken
into account equally in the analysis. In contrast, si in our model is a hyper-parameter learned directly
from data ? if a voxel does not respond to any condition of the task, si would be small and the
contribution of the voxel to the total log likelihood is small. The fitting of the shared covariance
structure is thus less influenced by this voxel.
From our simulated data, we found that parameters of the noise (? and ?) can be recovered reliably with small variance. However, the estimation of s had large variance from the true values
used in the simulation. One approach to reduce variance of estimation is by harnessing prior
knowledge about data. Voxels supporting similar representation of sensory input or tasks tend to
spatially cluster together. Therefore, we used a Gaussian Process to impose a smooth prior on
log(s) [17]. Specifically, for any two voxels i and j, we assumed cov(log(si ), log(sj )) =
(x ?x )T (x ?x )
(I ?I )2
b2 exp(? i 2lj 2 i j ? 2li2 j ), where xi and xj are the spatial coordinates of the two
space
inten
voxels and Ii and Ij are the average intensities of fMRI signals of the two voxels. Intuitively, this
means that if two voxels are close together and have similar signal intensity (that is, they are of the
same tissue type), then they should have similar SNR. Such a kernel of a Gaussian Process imposes
spatial smoothness but also allows the pseudo-SNR to change quickly at tissue boundaries. The
variance of the Gaussian process b2 , the length scale lspace and linten were fitted together with the
other parameters by maximizing the joint log likelihood of all parameters (here again, we restrict the
geometric mean of s to be 1).
B
lemur
lemur
C Map of
pseudo-SNR
lunamoth
monkey
ladybug
monkey
warbler
mallard
mallard
mallard
lunamoth
warbler
ladybug
warbler
warbler
ladybug
mallard
ladybug
monkey
lunamoth
lemur
lunamoth
Similarity in IT by
Bayesian RSA
monkey
Subjectively judged
similarity
lemur
A
Figure 3: Bayesian RSA estimates both the representational similarity structure from fMRI
data and the spatial map supporting the learned representation. (A) Similarity between 6 animal
categories, as judged behaviorally (reproduced from [2]). (B) Average representational similarity
estimated from IT cortex from all participants of [2], using our approach. The estimated structure
resembles the subjectively-reported structure. (C) Pseudo-SNR map in IT cortex corresponding to
one participant. Red: high pseudo-SNR, green: low pseudo-SNR. Only small clusters of voxels show
high pseudo-SNR.
We applied our method to the dataset of Connolly et al. (2012) [2]. In their experiment, participants
viewed images of animals from 6 different categories during an fMRI scan and rated the similarity
between animals outside the scanner. fMRI time series were pre-processed in the same way as in their
work [2]. Inferior temporal (IT) cortex is generally considered as the late stage of ventral pathway of
the visual system, in which object identity is represented. Fig 3 shows the similarity judged by the
participants and the average similarity matrix estimated from IT cortex, which shows similar structure
but higher correlations between animal classes. Interestingly, the pseudo-SNR map shows that only
part of the anatomically-defined ROI supports the representational structure.
7
5
Discussion
In this paper, we demonstrated that representational similarity analysis, a popular method in many
recent fMRI studies, suffers from a bias. We showed analytically that such bias is contributed by both
the structure of the experiment design and the covariance structure of measurement and neural noise.
The bias is induced because standard RSA analyzes noisy estimates of neural activation level, and
the structured noise in the estimates turns into bias. Such bias is especially severe when SNR is low
and when the order of task conditions cannot be fully counterbalanced. To overcome this bias, we
proposed a Bayesian framework of the fMRI data, incorporating the representational structure as the
shared covariance structure of activity levels across voxels. Our Bayesian RSA method estimates this
covariance structure directly from data, avoiding the structured noise in point estimation of activity
levels. Our method can be applied to neural recordings from other modalities as well.
Using simulated data, we showed that, as compared to standard RSA, the covariance structure
estimated by our method deviates much less from the true covariance structure, especially for low
SNR and short experiments. Furthermore, our method has the advantage of taking into account the
variation in SNR across voxels. In future work, we will use the pseudo-SNR map and the covariance
structure learned from the data jointly as an empirical prior to constrain the estimation of activation
levels ?. We believe that such structured priors learned directly from data can potentially provide
more accurate estimation of neural activation patterns?the bread and butter of fMRI analyses.
A number of approaches have recently been proposed to deal with the bias structure in RSA, such
as using the correlation or Mahalanobis distance between neural activity patterns estimated from
separate fMRI scans instead of from the same fMRI scan, or modeling the bias structure as a diagonal
matrix or by a Taylor expansion of an unknown function of inter-events intervals [1, 21, 6]. Such
approaches have different limitations. The correlation between patterns estimated from different scans
[1] is severely underestimated if SNR is low (for example, unless there is zero noise, the correlation
between the neural patterns corresponding to the same conditions estimated from different fMRI
scans is always smaller than 1, while the true patterns should presumably be the same across scans in
order for such an analysis to be justified). Similar problems exists for using Mahalanobis distance
between patterns estimated from different scans [21]: with noise in the data, it is not guaranteed that
the distance between patterns of the same condition estimated from separate scans is smaller than the
distance between patterns of different conditions. Such a result cannot be interpreted as a measure
of ?similarity? because, theoretically, neural patterns should be more similar if they belong to the
same condition than if they belong to different conditions. Our approach does not suffer from such
limitations, because we are directly estimating a covariance structure, which can always be converted
to a correlation matrix. Modeling the bias as a diagonal matrix [6] is not sufficient, as the bias can
be far from diagonal, as shown in Fig 1D. Taylor expansion of the bias covariance structure as a
function of inter-event intervals can potentially account for off-diagonal elements of the bias structure,
but it has the risk of removing structure in the true covariance matrix if it happens to co-vary with
inter-event intervals, and becomes complicated to set up if conditions repeat multiple times [1].
One limitation of our model is the assumption that noise is spatially independent. Henriksson et
al. [8] suggested that global fluctuations of fMRI time series over large areas (which is reflected as
spatial correlation) might contribute largely to their RSA pattern. This might also be the reason that
the overall correlation in Fig 1B is higher than the bias obtained from standard RSA on independent
Gaussian noise (Fig 1D). Our future work will explicitly incorporate such global fluctuations of noise.
Acknowledgement
This publication was made possible through the support of grants from the John Templeton Foundation and the
Intel Corporation. The opinions expressed in this publication are those of the authors and do not necessarily reflect
the views of the John Templeton Foundation. JWP was supported by grants from the McKnight Foundation,
Simons Collaboration on the Global Brain (SCGB AWD1004351) and the NSF CAREER Award (IIS-1150186).
We thank Andrew C. Connolly etc. for sharing of the data used in 4.2. Data used in the supplementary material
were obtained from the MGH-USC Human Connectome Project (HCP) database.
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5,672 | 6,132 | Average-case hardness of RIP certification
Tengyao Wang
Centre for Mathematical Sciences
Cambridge, CB3 0WB, United Kingdom
t.wang@statslab.cam.ac.uk
Quentin Berthet
Centre for Mathematical Sciences
Cambridge, CB3 0WB, United Kingdom
q.berthet@statslab.cam.ac.uk
Yaniv Plan
1986 Mathematics Road
Vancouver BC V6T 1Z2, Canada
yaniv@math.ubc.ca
Abstract
The restricted isometry property (RIP) for design matrices gives guarantees for
optimal recovery in sparse linear models. It is of high interest in compressed
sensing and statistical learning. This property is particularly important for computationally efficient recovery methods. As a consequence, even though it is in
general NP-hard to check that RIP holds, there have been substantial efforts to
find tractable proxies for it. These would allow the construction of RIP matrices
and the polynomial-time verification of RIP given an arbitrary matrix. We consider the framework of average-case certifiers, that never wrongly declare that a
matrix is RIP, while being often correct for random instances. While there are
such functions which are tractable in a suboptimal parameter regime, we show
that this is a computationally hard task in any better regime. Our results are based
on a new, weaker assumption on the problem of detecting dense subgraphs.
Introduction
In many areas of data science, high-dimensional signals contain rich structure. It is of great interest to leverage this structure to improve our ability to describe characteristics of the signal and
to make future predictions. Sparsity is a structure of wide applicability (see, e.g. Mallat, 1999;
Rauhut and Foucart, 2013; Eldar and Kutyniok, 2012), with a broad literature dedicated to its study
in various scientific fields.
The sparse linear model takes the form y = X? + ?, where y ? Rn is a vector of observations,
X ? Rn?p is a design matrix, ? ? Rn is noise, and the vector ? ? Rp is assumed to have a
small number k of non-zero entries. Estimating ? or the mean response, X?, are among the most
widely studied problems in signal processing, as well as in statistical learning. In high-dimensional
problems, one would wish to recover ? with as few observations as possible. For an incoherent
design matrix, it is known that an order of k 2 observations suffice (Donoho, Elad and Temlyakov,
2006; Donoho and Elad, 2003). However, this appears to require a number of observations far
exceeding the information content of ?, which has only k variables, albeit with unknown locations.
This dependence in k can be greatly improved by using design matrices that are almost isometries
on some low dimensional subspaces, i.e., matrices that satisfy the restricted isometry property with
parameters k and ?, or RIP(k, ?) (see Definition 1.1). It is a highly robust property, and in fact
implies that many different polynomial time methods, such as greedy methods (Blumensath and
Davies, 2009; Needell and Tropp, 2009; Dai and Milenkovic, 2009) and convex optimization (Cand?s, 2008; Cand?s, Romberg and Tao, 2006b; Cand?s and Tao, 2005), are stable in recovering ?.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Random matrices are known to satisfy the RIP when the number n of observation is more than about
k log(p)/?2 . These results were developed in the field of compressed sensing (Cand?s, Romberg and
Tao, 2006a; Donoho, 2006; Rauhut and Foucart, 2013; Eldar and Kutyniok, 2012) where the use of
randomness still remains pivotal for near-optimal results. Properties related to the conditioning of
design matrices have also been shown to play a key role in the statistical properties of computationally efficient estimators of ? (Zhang, Wainwright and Jordan, 2014). While the assumption of
randomness allows great theoretical leaps, it leaves open questions for practitioners.
Scientists working on data closely following this model cannot always choose their design matrix
X, or at least choose one that is completely random. Moreover, it is in general practically impossible to check that a given matrix satisfies these desired properties, as RIP certification is NP-hard
(Bandeira et al., 2012). Having access to a function, or statistic, of X that could be easily computed,
which determines how well ? may be estimated, would therefore be of a great help. The search
for such statistics has been of great importance for over a decade now, and several have been proposed (d?Aspremont and El Ghaoui, 2011; Lee and Bresler, 2008; Juditsky and Nemirovski, 2011;
d?Aspremont, Bach and El Ghaoui, 2008). Perhaps the simplest and most popular is the incoherence
parameter, which measures the maximum inner product between distinct, normalized, columns of
X. However, all of these are known to necessarily fail to guarantee good recovery when p ? 2n
unless n is of order k 2 (d?Aspremont and El Ghaoui, 2011). Given a specific problem instance, the
strong recovery guarantees of compressed sensing cannot be verified based on these statistics.
In this article, we study the problem of average-case certification of the Restricted Isometry Property
(RIP). A certifier takes as input a design matrix X, always outputs ?false? when X does not satisfy
the property, and outputs ?true? for a large proportion of matrices (see Definition 2.1). Indeed, worstcase hardness does not preclude a problem from being solvable for most instances. The link between
restricted isometry and incoherence implies that polynomial time certifiers exists in a regime where
n is of order k 2 log(p)/?2 . It is natural to ask whether the RIP can be certified for sample size
n k log(p)/?2 , where most matrices (with respect to, say, the Gaussian measure) are RIP. If it
does, it would also provide a Las Vegas algorithm to construct RIP design matrices of optimal sizes.
This should be compared with the currently existing limitations for the deterministic construction of
RIP matrices.
Our main result is that certification in this sense is hard even in a near-optimal regime, assuming a
new, weaker assumption on detecting dense subgraphs, related to the Planted Clique hypothesis.
Theorem (Informal). For any ? < 1, there is no computationally efficient, average-case certifier
for the class RIPn,p (k, ?) uniformly over an asymptotic regime where n k 1+? /?2 .
This suggests that even in the average case, RIP certification requires almost k 2 log(p)/?2 observations. This contrasts highly with the fact that a random matrix satisfies RIP with high probability
when n exceeds about k log(p)/?2 . Thus, there appears to be a large gap between what a practitioner
may be able to certify given a specific problem instance, and what holds for a random matrix.On the
other hand, if a certifier is found which fills this gap, the result would not only have huge practical
implications in compressed sensing and statistical learning, but would also disprove a long-standing
conjecture from computational complexity theory.
We focus solely on the restricted isometry property, but other conditions under which compressed
sensing is possible are also known. Extending our results to the restricted eigenvalue condition
Bickel, Ritov and Tsybakov (2009) or other conditions (see, van de Geer and Buhlmann, 2009, and
references therein) is an interesting path for future research.
Our result shares many characteristics with a hypothesis by Feige (2002) on the hardness of refuting
random satisfiability formulas. Indeed, our statement is also about the hardness of verifying that
a property holds for a particular instance (RIP for design matrices, instead of unsatisfiability for
boolean formulas). It concerns a regime where such a property should hold with high probability (n
of order k 1+? /?2 , linear regime for satisfiability), cautiously allowing only one type of errors, false
negatives, for a problem that is hard in the worst case. In these two examples, such certifiers exist in
a sub-optimal regime. Our problem is conceptually different from results regarding the worst-case
hardness of certifying this property (see, e.g. Bandeira et al., 2012; Koiran and Zouzias, 2012; Tillmann and Pfetsch, 2014). It is closer to another line of work concerned with computational lower
bounds for statistical learning problems based on average-case assumptions. The planted clique
assumption has been used to prove computational hardness results for statistical problems such as
estimation and testing of sparse principal components (Berthet and Rigollet, 2013a,b; Wang, Berthet
2
and Samworth, 2016), testing and localization of submatrix signals (Ma and Wu, 2013; Chen and
Xu, 2014), community detection (Hajek, Wu and Xu, 2015) and sparse canonical correlation analysis (Gao, Ma and Zhou, 2014). The intractability of noisy parity recovery problem (Blum, Kalai
and Wasserman, 2003) has also been used recently as an average-case assumption to deduce computational hardness of detection of satisfiability formulas with lightly planted solutions (Berthet and
Ellenberg, 2015). Additionally, several unconditional computational hardness results are shown for
statistical problems under constraints of learning models (Feldman et al., 2013). The present work
has two main differences compared to previous computational lower bound results. First, in a detection setting, these lower bounds concern two specific distributions (for the null and alternative
hypothesis), while ours is valid for all sub-Gaussian distributions, and there is no alternative distribution. Secondly, our result is not based on the usual assumption for the Planted Clique problem.
Instead, we use a weaker assumption on a problem of detecting planted dense graphs. This does
not mean that the planted graph is a random graph with edge probability q > 1/2 as considered
in (Arias-Castro and Verzelen, 2013; Bhaskara et al., 2010; Awasthi et al., 2015), but that it can be
any graph with an unexpectedly high number of edges (see section 3.1). This choice is made to
strengthen our result: it would ?survive? the discovery of an algorithm that would use very specific
properties of cliques (or even of random dense graphs) to detect their presence. As a consequence,
the analysis of our reduction is more technically complicated.
Our work is organized in the following manner: We recall in Section 1 the definition of the restricted
isometry property, and some of its known properties. In Section 2, we define the notion of certifier,
and prove the existence of a computationally efficient certifier in a sub-optimal regime. Our main
result is developed in Section 3, focused on the hardness of average-case certification. The proofs
of the main results are in Appendix A of the supplementary material and those of auxiliary results
in Appendix B of the the supplementary material.
1
1.1
Restricted Isometric Property
Formulation
We use the definition of Cand?s and Tao (2005), who introduced this notion. Below, for a vector
u ? Rp , we define kuk0 is the number of its non-zero entries.
Definition (RIP). A matrix X ? Rn?p satisfies the restricted isometry property with sparsity k ?
{1, . . . , p} and distortion ? ? (0, 1), denoted by X ? RIPn,p (k, ?), if it holds that
1 ? ? ? kXuk22 ? 1 + ?,
for every u ? Sp?1 (k) := {u ? Rp : kuk2 = 1, kuk0 ? k}.
This can be equivalently defined by a property on submatrices of the design matrix: X is in
RIPn,p (k, ?) if and only if for any set S of k columns of X, the submatrix, X?S , formed by taking
any these columns is almost an isometry, i.e. if the spectrum of its Gram matrix is contained in the
interval [1 ? ?, 1 + ?]:
>
kX?S
X?S ? Ik kop ? ? .
Denote by k ? kop,k the k-sparse operator norm, defined for a matrix A as kAkop,k =
supx?Sp?1 (k) kAxk2 . This yields another equivalent formulation of the RIP property: X ?
RIPn,p (k, ?) if and only if
kX > X ? Ip kop,k ? ? .
We assume in the following discussion that the distortion parameter ? is upper-bounded by 1. For
v ? Rp and T ? {1, . . . , p}, we write vT for the #T -dimensional vector obtained by restricting
v to coordinates indexed by T . Similarly, for an n ? p matrix A and subsets S ? {1, . . . , n} and
T ? {1, . . . , p}, we write AS? for the submatrix obtained by restricting A to rows indexed by S,
A?T for the submatrix obtained by restricting A to columns indexed by T .
1.2
Generation via random design
Matrices that satisfy the restricted isometry property have many interesting applications in highdimensional statistics and compressed sensing. However, there is no known way to generate them
3
deterministically in general. It is even NP-hard to check whether a given matrix X belongs to
RIPn,p (k, ?) (see, e.g Bandeira
? et al., 2012). Several deterministic constructions of RIP matrices
tight frames and Gershgorin?s
exist for sparsity level k . ? n. For example, using equitriangular ?
circle theorem, one can construct RIP matrices with sparsity k ? ?
n and distortion ? bounded
away from 0 (see, e.g. Bandeira et al., 2012). The limitation k ? ? n is known as the ?square
root bottleneck?. To date, the only constructions that break the ?square root bottleneck? are due to
Bourgain et al. (2011) and Bandeira, Mixon and Moreira (2014), both of which give RIP guarantee
for k of order n1/2+ for some small > 0 and fixed ? (the latter construction is conditional on a
number-theoretic conjecture being true).
Interestingly though, it is easy to generate large matrices satisfying the restricted isometry property
through random design, and compared to the fixed design matrices mentioned in the previous paragraph, these random design constructions are much less restrictive on the sparsity level, typically
allowing k up to the order n/ log(p) (assuming ? is bounded away from zero). They can be constructed easily from any centred sub-Gaussian distribution. We recall that Ra distribution Q (and its
2 2
associated random variable) is said to be sub-Gaussian with parameter ? if R e?x dQ(x) ? e? ? /2
for all ? ? R.
Definition. Define Q = Q? to be the set of sub-Gaussian distributions Q over R with zero mean,
unit variance, and sub-Gaussian parameter at most ?.
The most common choice for a Q ? Q is the standard normalR distribution N (0, 1). Note that by
Taylor expansion, for any Q ? Q, we necessarily have ? 2 ? R x2 dQ(x) = 1. In the rest of the
? to be the distribution of Z/?n for
paper, we treat ? as fixed. Define the normalized distribution Q
Z ? Q. The following well-known result states that by concentration of measure, random matrices
? ?(n?p) satisfy restricted isometries (see, e.g. Cand?s and Tao (2005)
generated with distribution Q
and Baraniuk et al. (2008)). For completeness, we include a proof that establishes these particular
constants stated here. All proofs are deferred to Appendix A or Appendix B of the supplementary
material.
? ?(n?p) , where Q ? Q. It holds
Proposition 1. Suppose X is a random matrix with distribution Q
that
9ep
n?2
P X ? RIPn,p (k, ?) ? 1 ? 2 exp k log
?
.
(1)
k
256? 4
In order to clarify the notion of asymptotic regimes used in this paper, we introduce the following
definition.
Definition. For 0 ? ? ? 1, define the asymptotic regime
kn1+? log(pn )
R? := (pn , kn , ?n )n : p, k ? ? and n
.
?n2
We note that in this notation, Proposition 1 implies that for (p, k, ?) = (pn , kn , ?n ) ? R0 we have,
? ?(n?p) (X ? RIPn,p (k, ?)) = 1, and this convergence is uniform over Q ? Q.
limn?? Q
2
2.1
Certification of Restricted Isometry
Objectives and definition
In practice, it is useful to know with certainty whether a particular realization of a random design
matrix satisfies the RIP condition. It is known that the problem of deciding if a given matrix is RIP
is NP-hard (Bandeira et al., 2012). However, NP-hardness is a only a statement about worst-case
instances. It would still be of great use to have an algorithm that can correctly decide RIP property
for an average instance of a design matrix, with some accuracy. Such an algorithm should identify a
high proportion of RIP matrices generated through random design and make no false positive claims.
We call such an algorithm an average-case certifier, or a certifier for short.
Definition (Certifier). Given a parameter sequence (p, k, ?) = (pn , kn , ?n ), we define a certifier for
? ?(n?p) -random matrices to be a sequence (?n )n of measurable functions ?n : Rn?p ? {0, 1},
Q
such that
? ?(n?p) ?n?1 (0) ? 1/3.
?n?1 (1) ? RIPn,p (k, ?)
and
lim sup Q
(2)
n??
4
Note the definition of a certifier depends on both the asymptotic parameter sequence (pn , kn , ?n ) and
the sub-Gaussian distribution Q. However, when it is clear from the context, we will suppress the
? ?(n?p) -random matrices simply
dependence and refer to certifiers for RIPn,p (k, ?) properties of Q
as ?certifiers?.
The two defining properties in (2) can be understood as follows. The first condition means that if a
certifier outputs 1, we know with certainty that the matrix is RIP. The second condition means that
the certifier is not overly conservative; it is allowed to output 0 for at most one third (with respect
? ?(n?p) measure) of the matrices. The choice of 1/3 in the definition of a certifier is made to
to Q
simplify proofs. However, all subsequent results will still hold if we replace 1/3 by any constant in
(0, 1). In view of Proposition 1, the second condition in (2) can be equivalently stated as
? ?(n?p) ?n (X) = 1 X ? RIPn,p (k, ?) ? 2/3.
lim inf Q
n??
With such a certifier, given an arbitrary problem fitting the sparse linear model, the matrix X could
be tested for the restricted isometry property, with some expectation of a positive result. This would
be particularly interesting given a certifier in the parameter regime n ?n2 kn2 , in which presently
known polynomial-time certifiers cannot give positive results.
Even though it is not the main focus of our paper, we also note that a certifier ? with the above
properties for some distribution Q ? Q would form a certifier/distribution couple (?, Q), that yields
in the usual manner a Las Vegas algorithm to generate RIP matrices. The (random) algorithm keeps
? ?(n?p) until ?n (X) = 1. The number of times that the certifier
generating random matrices X ? Q
? ?(n?p) ?n?1 (1) . Hence, the
is invoked has a geometric distribution with success probability Q
Las Vegas algorithm runs in randomized polynomial time if and only if ?n runs in randomized
polynomial time.
2.2
Certifier properties
Although our focus is on algorithmically efficient certifiers, we establish first the properties of a
certifier that is computationally intractable. This certifier serves as a benchmark for the performance
of other candidates. Indeed, we exhibit in the following proposition a certifier, based on the k-sparse
operator norm, that works uniformly well in the same asymptotic parameter regime R0 , where
? ?(n?p) -random matrices are RIP with asymptotic probability 1. For clarity, we stress that our
Q
criterion when judging a certifier will always be its uniform performance over asymptotic regimes
R? for some ? ? [0, 1].
Proposition 2. Suppose (p, k, ?) = (pn , kn , ?n ) ? R0 . Furthermore, Let Q ? Q and X ?
? ?(n?p) . Then the sequence of tests (?op,k )n based on sparse operator norms, defined by
Q
?op,k (X) := 1 kX > X ? Ip kop,k ? ? .
? ?(n?p) -random matrices.
is a certifier for Q
By a direct reduction from the clique problem, one can show that it is NP-hard to compute the ksparse operator norm of a matrix. Hence the certifier ?op,k is computationally intractable. The next
proposition concerns the certifier property of a test based on the maximum incoherence between
columns of the design matrix. It follows directly from a well-known result on the incoherence
parameter of a random matrix (see, e.g. Rauhut and Foucart (2013, Proposition 6.2)) and allows the
construction of a polynomial-time certifier that works uniformly well in the asymptotic parameter
regime R1 .
Proposition 3. Suppose (p, k, ?) = (pn , kn , ?n ) satisfies n ? 196? 4 k 2 log(p)/?2 . Let Q ? Q and
? ?(n?p) , then the tests ?? defined by
X?Q
r
log(p)
>
2
?? (X) := 1 kX X ? Ip k? ? 14?
n
? ?(n?p) -random matrices.
is a certifier for Q
5
Proposition 3 shows that, when the sample size n is above k 2 log(p)/?2 in magnitude (in particular,
this is satisfied asymptotically when (p, k, ?) = (pn , kn , ?n ) ? R1 ), there is a polynomial time
certifier. In other words, in this high-signal regime, the average-case decision problem for RIP
property is much more tractable than indicated by the worst-case result. On the other hand, the
certifier in Proposition 3 works in a much smaller parameter range when compared to ?op,k in
Proposition 2. Combining Proposition 2 and 3, we have the following schematic diagram (Figure 1).
When the sample size is lower than specified in R0 , the property does not hold, with high probability,
and no certifier exists. A computationally intractable certifier works uniformly over R0 . On the other
end of the spectrum, when the sample size is large enough to be in R1 , a simple certifier based on
the maximum incoherence of the design matrix is known to work in polynomial time. This leaves
open the question of whether (randomized) polynomial time certifiers can work uniformly well in
R0 , or R? for any ? ? [0, 1). We will see in the next section that, assuming a weaker variant of
the Planted Clique hypothesis from computational complexity theory, R1 is essentially the largest
asymptotic regime where a randomized polynomial time certifier can exist.
Figure 1: Schematic digram for existence of certifiers in different asymptotic regimes.
3
3.1
Hardness of Certification
Planted dense subgraph assumptions
We show in this section that certification of RIP property is an average-case hard problem in the
parameter regime R? for any ? < 1. This is precisely the regime not covered by Proposition 3. The
average-case hardness result is proved via reduction to the planted dense subgraph assumption.
For any integer m ? 0, denote Gm the collection of all graphs on m vertices. We write V (G)
and E(G) for the set of vertices and edges of a graph G. For H ? G? where ? ? {0, . . . , m}, let
G(m, 1/2, H) be the random graph model that generates a random graph G on m vertices as follows.
It first picks ? random vertices K ? V (G) and plants an isomorphic copy of H on these ? vertices,
then every pair of vertices not in K ? K is connected by an edge independently with probability
1/2. We write PH for the probability measure on Gm associated with G(m, 1/2, H). Note that if H
is the empty graph, then G(m, 1/2, ?) describes the Erd?os?R?nyi random graph. With a slight abuse
of notation, we write P0 in place of P? . On the other hand, for ? (0, 1/2], if H belongs to the set
?(? ? 1)
H = H?, := H ? G? : #E(H) ? (1/2 + )
,
2
then G(m, 1/2, H) generates random graphs that contain elevated local edge density. The planted
dense graph problem concerns testing apart the following two hypotheses:
H0 : G ? G(m, 1/2, ?)
and
H1 : G ? G(m, 1/2, H) for some H ? H?, .
1/2??
(3)
It is widely believed that for ? = O(m
), there does not exist randomized polynomial time
tests to distinguish between H0 and H1 (see, e.g. Jerrum (1992); Feige and Krauthgamer (2003);
Feldman et al. (2013)). More precisely, we have the following assumption.
Assumption (A1) 1. Fix ? (0, 1/2] and
? ? (0, 1/2). let (?m )m be any sequence of integers
such that ?m ? ? and ?m = O m1/2?? . For any sequence of randomized polynomial time tests
(?m : Gm ? {0, 1})m , we have
n
o
lim inf P0 ?(G) = 1 + max PH ?(G) = 0) > 1/3 .
m
H?H?,
6
We remark that if = 1/2, then H?, contains only the ?-complete graph and the testing problem
becomes the well-known planted clique problem (cf. Jerrum (1992) and references in Berthet and
Rigollet (2013a,b)).
The difficulty of this problem has been used as a primitive for the hardness of other tasks, such
as cryptographic applications, in Juels and Peinado (2000), testing for k-wise dependence in Alon
et al. (2007), approximating Nash equilibria in Hazan and Krauthgamer (2011). In this case, Assumption (A1) is a version of the planted clique hypothesis (see, e.g. Berthet and Rigollet (2013b,
Assumption APC )). We emphasize that Assumption A1 is significantly milder than the planted
clique hypothesis (since it allows any
? ? (0, 1/2]), or that a hypothesis on planted random graphs.
We also note that when ? ? C m, spectral methods can be used to detect such graphs with
high probability. Indeed, when G contains a graph of H, denoting AG its adjacency
? matrix, then
AG ? 11> /2 has a leading eigenvalue greater than (? ? 1), whereas it is of order m for a usual
Erd?os?R?nyi random graph.
The following theorem relates the hardness of the planted dense subgraph testing problem to the
hardness of certifying restricted isometry of random matrices. We recall that the distribution of X is
?
d
?=
that of an n?p random matrix with entries independently and identically sampled from Q
Q/ n,
for some Q ? Q. We also write ?rp for the class of randomized polynomial time certifiers.
Theorem 4. Assume (A1) and fix any ? ? [0, 1). Then there exists a sequence (p, k, ?) =
(pn , kn , ?n ) ? R? , such that there is no certifier/distribution couple (?, Q) ? ?rp ? Q with respect
to this sequence of parameters.
Our proof of Theorem 4 relies on the following ideas: Given a graph G, an instance of the planted
clique problem in the assumed hard regime, we construct n random vectors based on the adjacency
matrix of a bipartite subgraph of G, between two random sets of vertices. Each coefficient of these
vectors is then randomly drawn from one of two carefully chosen distributions, conditionally on the
presence or absence of a particular edge. This construction ensures that if the graph is an Erd?os?
R?nyi random graph (i.e. with no planted graph), the vectors are independent with independent
? Otherwise, we show that with high probability, the presence of an
coefficients, with distribution Q.
unusually dense subgraph will make it very likely that the matrix does not satisfy the restricted isometry property, for a set of parameters in R? . As a consequence, if there existed a certifier/distribution
couple (?, Q) ? ?rp ? Q in this range of parameters, it could be used - by using as input in the
certifier the newly constructed matrix - to determine with high probability the distribution of G,
violating our assumption (A1).
We remark that this result holds for any distribution in Q, in contrast to computational lower bounds
in statistical learning problems, that apply to a specific distribution. For the sake of simplicity, we
have kept the coefficients of X identically distributed, but our analysis is not dependent on that
fact, and our result can be directly extended to the case where the coefficients are independent, with
different distributions in Q.
Theorem 4 may be viewed as providing an asymptotic lower bound of the sample size n for the
existence of a computationally feasible certifier. It establishes this computational lower bound by
exhibiting some specific ?hard? sequences of parameters inside R? , and show that any algorithm
violating the computational lower bound could be exploited to solve the planted dense subgraph
problem. All hardness results, whether in a worst-case (NP-hardness, or other) or the average-case
(by reduction from a hard problem), are by nature statements on the impossibility of accomplishing a
task in a computationally efficient manner, uniformly over a range of parameters. They are therefore
always based on the construction of a ?hard? sequence of parameters used in the reduction, for
which a contradiction is shown. Here, the ?hard? sequence is explicitly constructed in the proof
to be some (p, k, ?) = (pn , kn , ?n ) satisfying p ? n and n1/(3???4?) k n1/(2??)?? , for
? ? [0, (1 ? ?)/3) and any small ? > 0. The tuning parameter ? is to allow additional flexibility
in choosing these ?hard? sequences. More precisely, using an averaging trick first seen in Ma and
Wu (2013), we are able to show that the existence of such ?hard? sequences is not confined only in
the sparsity regime k n1/2 . We note that in all our ?hard? sequences, ?n must depend on n. An
interesting extension is to see if similar computational lower bounds hold when restricted to a subset
of R? where ? is constant.
7
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9
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5,673 | 6,133 | Learning in Games: Robustness of Fast Convergence
Dylan J. Foster?
Zhiyuan Li?
Thodoris Lykouris?
Karthik Sridharan?
?va Tardos?
Abstract
We show that learning algorithms satisfying a low approximate regret property
experience fast convergence to approximate optimality in a large class of repeated
games. Our property, which simply requires that each learner has small regret
compared to a (1 + ?)-multiplicative approximation to the best action in hindsight,
is ubiquitous among learning algorithms; it is satisfied even by the vanilla Hedge
forecaster. Our results improve upon recent work of Syrgkanis et al. [28] in a
number of ways. We require only that players observe payoffs under other players?
realized actions, as opposed to expected payoffs. We further show that convergence
occurs with high probability, and show convergence under bandit feedback. Finally,
we improve upon the speed of convergence by a factor of n, the number of players.
Both the scope of settings and the class of algorithms for which our analysis
provides fast convergence are considerably broader than in previous work.
Our framework applies to dynamic population games via a low approximate regret
property for shifting experts. Here we strengthen the results of Lykouris et al. [19]
in two ways: We allow players to select learning algorithms from a larger class,
which includes a minor variant of the basic Hedge algorithm, and we increase the
maximum churn in players for which approximate optimality is achieved.
In the bandit setting we present a new algorithm which provides a ?small loss?-type
bound with improved dependence on the number of actions in utility settings, and
is both simple and efficient. This result may be of independent interest.
1
Introduction
Consider players repeatedly playing a game, all acting independently to minimize their cost or
maximize their utility. It is natural in this setting for each player to use a learning algorithm that
guarantees small regret to decide on their strategy, as the environment is constantly changing due
to each player?s choice of strategy. It is well known that such decentralized no-regret dynamics
are guaranteed to converge to a form of equilibrium for the game. Furthermore, in a large class of
games known as smooth games [23] they converge to outcomes with approximately optimal social
welfare matching the worst-case efficiency loss of Nash equilibria (the price of anarchy). In smooth
cost minimization games the overall cost is /(1 ?) times the minimum cost, while in smooth
mechanisms [29] such as auctions it is / max(1, ?) times the maximum total utility (where and ?
are parameters of the smoothness condition). Examples of smooth games and mechanisms include
routing games and many forms of auction games (see e.g. [23, 29, 24]).
The speed at which the game outcome converges to this approximately optimal welfare is governed by
individual players? regret bounds. There are a large number of simple regret minimization algorithms
(Hedge/Multiplicative Weights, Mirror Decent, Follow the Regularized Leader; see e.g. [12]) that
?
Cornell University {djfoster,teddlyk,sridharan,eva}@cs.cornell.edu. Work supported in
part under NSF grants CDS&E-MSS 1521544, CCF-1563714, ONR grant N00014-08-1-0031, a Google faculty
research award, and an NDSEG fellowship.
?
Tsinghua University, lizhiyuan13@mails.tsinghua.edu.cn. Research performed while author was
visiting Cornell University.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
p
guarantee that the average regret goes down as O(1/ T ) with time T , which is tight in adversarial
settings.
Taking advantage of the fact that playing a game against opponents who themselves are also using
regret minimization is not a truly adversarial setting, a sequence of papers [9, 22, 28] showed that by
using specific learning algorithms, the dependence on T of the convergence rate can be improved to
O(1/T ) (?fast convergence?). Concretely, Syrgkanis et al. [28] show that all algorithms satisfying the
so-called RVU property (Regret by Variation in Utilities), which include Optimistic Mirror Descent
[22], converge at a O(1/T ) rate with a fixed number of players.
One issue with the works of [9, 22, 28] is that they use expected cost as their feedback model for the
players. In each round every player receives the expected cost for each of their available actions, in
expectation over the current action distributions of all other players. This clearly represents more
information than is realistically available to players in games ? at most each player sees the cost
of each of their actions given the actions taken by the other players (realized feedback). In fact,
even if each player had access to the action distributions of the other players, simply computing this
expectation is generally intractable when n, the number of players, is large.
We improve the result of [28] on the convergence to approximate optimality in smooth games in a
number of different aspects. To achieve this, we relax the quality of approximation from the bound
guaranteed by smoothness. Typical smoothness bounds on the price of anarchy in auctions are small
constants, such a factor of 1.58 or 2 in item auctions. Increasing the approximation factor by an
arbitrarily small constant ? > 0 enables the following results:
? We show that learning algorithms obtaining fast convergence are ubiquitous.
? We improve the speed of convergence by a factor of n, the number of players.
? For all our results, players only need feedback based on realized outcomes, instead of
expected outcomes.
? We show that convergence occurs with high probability in most settings.
? We extend the results to show that it is enough for the players to observe realized bandit
feedback, only seeing the outcome of the action they play.
? Our results apply to settings where the set of players in the game changes over time [19].
We strengthen previous results by showing that a broader class of algorithms achieve
approximate efficiency under significant churn.
We achieve these results using a property we term Low Approximate Regret, which simply states
that an online learning algorithm achieves good regret against a multiplicative approximation of the
best action in hindsight. This property is satisfied by many known algorithms including even the
vanilla Hedge algorithm, as well as Optimistic Hedge [21, 28] (via a new analysis). The crux of our
analysis technique is the simple observation that for many types of data-dependent regret bounds
we can fold part of the regret bound into the comparator term, allowing us to explore the trade-off
between additive and multiplicative approximation.
In Section 3, we show that Low Approximate Regret implies fast convergence to the social welfare
guaranteed by the price of anarchy via the smoothness property. This convergence only requires
feedback from the realized actions played by other players, not their action distribution or the
expectation over their actions. We further show that this convergence occurs with high probability in
most settings. For games with a large number of players we also improve the speed of convergence.
[28] shows that players using Optimistic Hedge in a repeated game with n players converge to the
approximately optimal outcome guaranteed by smoothness at a rate of O(n2 /T ). They also offer an
analysis guaranteeing convergence of O(n/T ) , at the expense of a constant factor decrease in the
quality of approximation (e.g., a factor of 4 in atomic congestion games with affine congestion). We
achieve the convergence bound of O(n/T ) with only an arbitrarily small loss in the approximation.
Algorithms that satisfy the Low Approximate Regret property are ubiquitous and include simple,
efficient algorithms such as Hedge and variants. The observation that this broad class of algorithms
enjoys fast convergence in realistic settings suggests that fast convergence occurs in practice.
Comparing our work to [28] with regard to feedback, Low Approximate Regret algorithms require
only realized feedback, while the analysis of the RVU property in [28] requires expected feedback.
To see the contrast, consider the load balancing game introduced in [17] with two players and two
bins, where each player selects a bin and observes cost given by the number of players in that bin.
Initialized at the uniform distribution, any learning algorithm with expectation feedback (e.g. those
in [28]) will stay at the uniform distribution forever, because the expected cost vector distributes
2
cost equally across the two bins. This gives low regret under expected costs, but suppose we were
interested in realized costs: The only ?black box? way to lift [28] to this case would be to simply
p
evaluate the regret bound above under realized costs, but here players will experience ?(1/ T )
variation because they select bins uniformly at random, ruining the fast convergence. Our analysis
sidesteps this issue because players achieve Low Approximate Regret with high probability.
In Section 4 we consider games where players can only observe the cost of the action they played
given the actions taken by the other players, and receive no feedback for actions not played (bandit
feedback). [22] analyzed zero-sum games with bandit feedback, but assumed that players receive
expected cost over the strategies of all other players. In contrast, the Low Approximate Regret
property can be satisfied by just observing realizations, even with bandit feedback. We propose a
new bandit algorithm based on log-barrier regularization with importance sampling that guarantees
fast convergence of O(d log T /?) where d is the number of actions. Known techniques would either
result in a convergence rate of O(d3 log T ) (e.g. adaptations of SCRiBLe [21]) or would not extend
to utility maximization settings (e.g. GREEN [2]). Our technique is of independent interest since it
improves the dependence of approximate regret bounds on the number of experts while applying to
both cost minimization and utility maximization settings.
Finally, in Section 5, we consider the dynamic population game setting of [19], where players enter
and leave the game over time. [19] showed that regret bounds for shifting experts directly influence
the rate at which players can turn over and still guarantee close to optimal solutions on average. We
show that a number of learning algorithms have the Low Approximate Regret property in the shifting
experts setting, allowing us to extend the fast convergence result to dynamic games. Such learning
algorithms include a noisy version of Hedge as well as AdaNormalHedge [18], which was previously
studied in the dynamic setting in [19]. Low Approximate Regret allows us to increase the turnover
rate from the one in [19], while also widening and simplifying the class of learning algorithms that
players can use to guarantee the close to optimal average welfare.
2
Repeated Games and Learning Dynamics
We consider a game G among a set of n players. Each player i has an action space Si and a cost
function costi : S1 ? ? ? ? ? Sn ! [0, 1] that maps an action profile s = (s1 , . . . , sn ) to the cost
costi (s) that player experiences1 . We assume that the action space of each player has cardinality
d, i.e. |Si | = d. We let w = (w1 , . . . , wn ) denote a list of probability distributions over all players?
actions, where wi 2 (Si ) and wi,x is the probability of action x 2 Si .
The game is repeated for T rounds. At each round t each player i picks a probability distribution
wit 2 (Si ) over actions and draws their action sti from this distribution. Depending on the game
playing environment under consideration, players will receive different types of feedback after each
round. In Sections 3 and 5 we consider feedback where at the end of the round each player i observes
the utility they would have received had they played any possible action x 2 Si given the actions
taken by the other players. More formally let cti,x = costi (x, st i ), where st i is the set of strategies
of all but the ith player at round t, and let cti = (cti,x )x2Si . Note that the expected cost of player i at
round t (conditioned on the other players? actions) is simply the inner product hwit , cti i.
We refer to this form of feedback as realized feedback since it only depends on the realized actions st i sampled by the opponents; it does not directly depend on their distributions wt i . This
should be contrasted with the expectation feedback used by [28, 9, 22], where player i observes
Est i ?wt i [costi (x, st i )] for each x.
Sections 4 and 5 consider extensions of our repeated game model. In Section 4 we examine partial
information (?bandit?) feedback, where players observe only the cost of their own realized actions. In
Section 5 we consider a setting where the player set is evolving over time. Here we use the dynamic
population model of [19], where at each round t each player i is replaced (?turns over?) with some
probability p. The new player has cost function costti (?) and action space Sit which may change
arbitrarily subject to certain constraints. We will formalize this notion later on.
Learning Dynamics We assume that players select their actions using learning algorithms satisfying a property we call Low Approximate Regret, which simply requires that the cumulative cost of the
learner multiplicatively approximates the cost of the best action they could have chosen in hindsight.
1
See Appendix D for analogous definitions for utility maximization games.
3
We will see in subsequent sections that this property is ubiquitous and leads to fast convergence in a
robust range of settings.
Definition 1. (Low Approximate Regret) A learning algorithm for player i satisfies the Low Approximate Regret property for parameter ? > 0 and function A(d, T ) if for all action distributions
f 2 (Si ),
(1
?)
T
X
t=1
hwit , cti i ?
T
X
t=1
hf, cti i +
A(d, T )
.
?
(1)
A learning algorithm satisfies Low Approximate Regret against shifting experts if for all sequences
f 1 , . . . , f T 2 (Si ), letting K = |{i > 2 : f t 1 6= f t }| be the number of shifts,
(1
?)
T
X
t=1
hwit , cti i ?
T
X
t=1
hf t , cti i + (1 + K)
A(d, T )
.
?
(2)
In the bandit feedback setting, we require (1) to hold in expectation over the realized strategies of
player i for any f 2 (Si ) fixed before the game begins.
We use the version of the Low Approximate Regret property with shifting experts when considering
players in dynamic population games in Section 5. In this case, the game environment is constantly
changing due to churn in the population, and we need the players to have low approximate regret
with shifting experts to guarantee high social welfare despite the churn.
We emphasize that all algorithms we are aware of that satisfy Low Approximate Regret can be made
to do so for any fixed choice of the approximation factor ? via an appropriate selection of parameters.
Many algorithms have an even stronger property: They satisfy (1) or (2) for all ? > 0 simultaneously.
We say that such algorithms satisfy the Strong Low Approximate Regret property. This property has
favorable consequences in the context of repeated games.
The Low Approximate Regret property differs from previous properties such as RVU in that it only
requires that the learner?s cost be close to a multiplicative approximation to the cost of the best action
in hindsight. Consequently, it is always smaller than the regret. For instance, if we consider
p only
uniform (i.e. not data-dependent) regret bounds the Hedge algorithm can only achieve O( T log d)
exact regret, but can achieve Low Approximate Regret with parameters ? and A(d, T ) = O(log d)
for any ? > 0. Low Approximate Regret is analogous to the notion of ?-regret from [15], with
? = (1 + ?).
In Appendix D we show that the Low Approximate Regret property and our subsequent results
naturally extend to utility maximization games.
Smooth Games It is well-known that in a large class of games, termed smooth games by Roughgarden [23], traditional learning dynamics converge to approximately optimal social welfare. In
subsequent sections we analyze the convergence of Low Approximate Regret learning dynamics
in such smooth games. We will see that Low Approximate Regret (for sufficiently small A(d, T ))
coupled with smoothness of the game implies fast convergence of learning dynamics to desirable
social welfare under a variety of conditions. Before proving this result we review social welfare and
smooth games.
Pn
For a given action profile s, the social cost is C(s) = i=1 costi (s). To bound the efficiency loss
due to the selfish behavior of the players we define
O PT = min
o
s
n
X
costi (so ).
i=1
Definition 2. (Smooth
P game [23]) A cost minimization game is called ( , ?)-smooth if for all strategy
profiles s and s? : i costi (s?i , s i ) ? ? costi (s? ) + ? ? costi (s).
This property is typically applied using a (close to) optimal action profile s? = so . For this case the
property implies that if s is an action profile with very high cost, then some player deviating to her
share of the optimal profile s?i will improve her cost.
For smooth games, the price of anarchy is at most /(1 ?), meaning that Nash equilibria of
the game, as well as no-regret learning outcomes in the limit, have social cost at most a factor of
/(1 ?) above the optimum. Smooth cost minimization games include congestion games such
4
as routing or load balancing. For example, atomic congestion games with affine cost functions are
( 53 , 13 )-smooth [8], non-atomic games are (1, 0.25) smooth [25], implying a price of anarchy of 2.5
and 1.33 respectively. While we focus on cost-minimization games for simplicity of exposition, an
analogous definition also applies for utility maximization, including smooth mechanisms [29], which
we elaborate on in Appendix D. Smooth mechanisms include most simple auctions. For example, the
first price item auction is (1 1/e, 1)-smooth and all-pay actions are (1/2, 1)-smooth, implying a
price of anarchy of 1.58 and 2 respectively. All of our results extend to such mechanisms.
3
Learning in Games with Full Information Feedback
We now analyze the efficiency of algorithms with the Low Approximate Regret property in the
full information setting. Our first proposition shows that, for smooth games with full information
feedback, learners with the Low Approximate Regret property converge to efficient outcomes.
Proposition 1. In any ( , ?)-smooth game, if all players use Low Approximate Regret algorithms
satisfying Eq. (1) with parameters ? and A(d, T ), then for the action profiles st drawn on round t
from the corresponding mixed actions of the players,
?
1 X ?
E C(st ) ?
T t
1
?
?
O PT +
n
?
T 1
1
?
?
?
A(d, T )
.
?
Proof. This proof is a straightforward modification
P of the usualPprice
P of anarchy proof for smooth
games. We obtain the claimed bound by writing t E[C(st )] = i t E[costi (st )], using the Low
?
Approximate Regret property with f = s?i for each
P player i?fort the optimal solution s , then using the
smoothness property for each time t to bound i costi (si , s i ), and finally rearranging terms.
For ? << (1 ?) the approximation factor of /(1 ? ?) is very close to the price of anarchy
/(1 ?). This shows that Low Approximate Regret learning dynamics quickly converge to outcomes
with social welfare arbitrarily close to the welfare guaranteed for exact Nash equilibria by the price
of anarchy. A simple corollary of this proposition is that, when players use learning algorithms that
satisfy the Strong Low Approximate Regret property, the bound above can be taken to depend on
O PT even though this value is unknown to the players.
Whenever the Low Approximate Regret property is satisfied, a high probability version of the property
with similar dependence on ? and A(d, T ) is also satisfied. This implies that in addition to quickly
converging to efficient outcomes in expectation, Low Approximate Regret learners experience fast
convergence with high probability.
Proposition 2. In any ( , ?)-smooth game, if all players use Low Approximate Regret algorithms
satisfying Eq. (1) for parameters ? and A(d, T ), then for the action profile st drawn on round t from
the players? mixed actions and = 2?/(1 + ?), we have that 8 > 0, with probability at least 1
,
1 X
C(st ) ?
T t
1
?
O PT +
n
?
T 1
1
?
?
?
4A(d, T )
+
12 log(n log2 (T )/ ))
,
Examples of Simple Low Approximate Regret Algorithms Propositions 1 and 2 are informative
when applied with algorithms for which A(d, T ) is sufficiently small. One would hope that such
algorithms are relatively simple and easy to find. We show now that the well-known Hedge algorithm
as well as basic variants such as Optimistic Hedge and Hedge with online learning rate tuning satisfy
the property with A(d, T ) = O(log d), which will lead to fast convergence both in terms of n and
T . For these algorithms and indeed all that we consider in this paper, we can achieve the Low
Approximate Regret property for any fixed ? > 0 via an appropriate parameter setting. In Appendix
A.2, we provide full descriptions and proofs for these algorithms.
Example 1. Hedge satisfies the Low Approximate Regret property with A(d, T ) = log(d). In
particular one can achieve the property for any fixed ? > 0 by using ? as the learning rate.
Example 2. Hedge with online learning rate tuning satisfies the Strong Low Approximate Regret
property with A(d, T ) = O(log d).
Example 3. Optimistic Hedge satisfies the Low Approximate Regret property with A(d, T ) =
8 log(d). As with vanilla Hedge, we can choose the learning rate to achieve the property with any ?.
5
p
Example
4. Any algorithm satisfying a ?small loss? regret bound of the form (Learner?s cost) ? A
p
or p(Cost of best action) ? A satisfies Strong Low Approximate Regret via the AM-GM inequality,
i.e. (Learner?s cost) ? A / inf ?>0 [? ? (Learner?s cost) + A/?]. In particular, this implies that the
following algorithms have Strong Low Approximate Regret: Canonical small loss and self-confident
algorithms, e.g. [11, 4, 30], Algorithm of [7], Variation MW [13], AEG-Path [26], AdaNormalHedge
[18], Squint [16], and Optimistic PAC-Bayes [10].
Example 4 shows that the Strong Low Approximate Regret property in fact is ubiquitous, as it is
satisfied by any algorithm that provides small loss regret bounds or one of many variants on this type
of bound. Moreover, all algorithms that satisfy the Low Approximate Regret property for all fixed ?
can be made to satisfy the strong property using the doubling trick.
Main Result for Full Information Games:
Theorem 3. In any ( , ?)-smooth game, if all players use Low Approximate Regret algorithms
satisfying (1) for parameter ? 2 and A(d, T ) = O(log d), then
?
1 X ?
E C(st ) ?
T t
1
?
?
O PT +
n
?
T 1
and furthermore, 8 > 0, with probability at least 1
?
1 X ?
E C(st ) ?
T t
1
?
?
O PT +
n
?
T 1
1
?
,
?
?
?
1
?
?
?
O(log d)
,
?
O(log d)
O(log(n log2 (T )/ ))
+
.
?
?
Corollary 4. If all players use Strong Low Approximate Regret algorithms then: 1. The above results
hold for all ? > 0 simultaneously. 2. Individual players have regret bounded by O(T 1/2 ), even in
adversarial settings. 3. The players approach a coarse correlated equilibrium asymptotically.
Comparison with Syrgkanis et al. [28]. By relaxing the standard /(1 ?) price of anarchy
bound, Theorem 3 substantially broadens the class of algorithms that experience fast convergence to
include even the common Hedge algorithm. The main result of [28] shows that learning algorithms
that satisfy their RVU property converge to the price of anarchy bound /(1 ?) at rate n2 log d/T .
They further achieve a worse approximation of (1 + ?)/(?(1 ?)) at the improved (in terms of
n) rate of n log d/T . We converge to an approximation arbitrarily close to /(1 + ?) at a rate of
n log d/T . Note that in atomic congestion games with affine congestion function ? = 1/3, so their
bound of (1 + ?)/?(1 ?) loses a factor of 4 compared to the price of anarchy.
Strong Low Approximate Regret algorithms such as Hedge with online learning
rate tuning simultap
neously experience both fast O(n/T ) convergence in games and
an
O(1/
T
)
bound
on individual
p
regret in adversarial settings. In contrast, [28] only shows O(n/ T ) individual regret and O(n3 /T )
convergence to price of anarchy simultaneously.
Low Approximate Regret algorithms only need realized feedback, whereas [28] require expectation
feedback. Having players receive expectation feedback is unrealistic in terms of both information
and computation. Indeed, even if the necessary information was available, computing expectations
over discrete probability distributions is not tractable unless n is taken to be constant.
Our results imply that Optimistic Hedge enjoys the best of two worlds: It enjoys fast convergence
to the exact /(1 ?) price of anarchy using expectation feedback as well as fast convergence to
the ?-approximate price of anarchy using realized feedback. Our new analysis of Optimistic Hedge
(Appendix A.2.2) sheds light on another desirable property of this algorithm: Its regret is bounded in
terms of the net cost incurred by Hedge. Figure 1 summarizes the differences between our results.
RVU property [28]
LAR property (section 2)
Feedback
Expected costs
Realized costs
POA
exact
?-approx
Rate
O(n2 log d/T )
O(n log d/(?T ))
Time comp.
dO(n) per round
O(d) per round
Figure 1: Comparison of Low Approximate Regret and RVU properties.
2
We can also show that the theorem holds if players satisfy the property for different values of ?, but with a
dependence on the worst case value of ? across all players.
6
4
Bandit Feedback
In many realistic scenarios, the players of a game might not even know what they would have lost
or gained if they had deviated from the action they played. We model this lack of information with
bandit feedback, in which each player observes a single scalar, costi (st ) = hsti , cti i, per round.3
When the game considered is smooth, one can use the Low Approximate Regret property as in the
full information setting to show that players quickly converge to efficient outcomes. Our results here
hold with the same generality as in the full information setting: As long as learners satisfy the Low
Approximate Regret property (1), an efficiency result analogous to Proposition 1 holds.
Proposition 5. Consider a ( , ?)-smooth game. If all players use bandit learning algorithms with
Low Approximate Regret A(d, T ) then
"
#
X
1
t
E
C(s ) ?
T
1
t
?
?
O PT +
n
?
T 1
1
?
?
?
A(d, T )
.
?
Bandit Algorithms with Low Approximate Regret The bandit Low Approximate Regret property
requires that (1) holds in expectation against any sequence of adaptive and potentially adversarially
chosen costs, but only for an obliviously chosen comparator f .4 This is weaker than requiring that an
algorithm achieve a true expected regret bound; it is closer to pseudo-regret.
The Exp3Light algorithm [27] satisfies Low Approximate Regret with A(d, T ) = d2 log T . The
SCRiBLe algorithm introduced in [1] (via the analysis in [21]) enjoys the Low Approximate Regret
property with A(d, T ) = d3 log(dT ). The GREEN algorithm [2] achieves the Low Approximate
Regret property with A(d, T ) = d log(T ), but only works with costs and not gains. This prevents it
from being used in utility settings such as auctions, as in Appendix D.
We present a new bandit algorithm (Algorithm 3) that achieves Low Approximate Regret with
A(d, T ) = d log(T /d) and thus matches the performance of GREEN, but works in both cost
minimization and utility maximization settings. This method is based on Online Mirror Descent
with a logarithmic barrier for the positive orthant, but differs from earlier algorithms based on the
logarithmic barrier (e.g. [21]) in that it uses the classical importance-weighted estimator for costs
?
instead of sampling based on the Dikin elipsoid. It can be implemented in O(d)
time per round, using
line search to find . We provide proofs and further discussion of Algorithm 3 in Appendix B.
Algorithm 3: Initialize w1 to the uniform distribution. On each round t, perform update:
Algorithm 3 update:
wstt
1
=
wstt
1 + ?ctst
1
1
1
+ wstt
1
1
and 8j 6= st
1
wjt =
wjt
1
1 + wjt
1,
(3)
where ? 0 is chosen so that wt is a valid probability distribution.
Lemma 6. Algorithm 3 with ? = ?/(1 + ?) has Low Approximate Regret with A(d, T ) = O(d log T ).
Comparison to Other Algorithms In contrast to the full information setting where the most
common algorithm, Hedge, achieves Low Approximate Regret with competitive parameters, the most
common adversarial bandit algorithm Exp3 does not seem to satisfy Low Approximate Regret. [3]
provide a small loss bound for bandits which would be sufficient for Low Approximate Regret, but
their algorithm requires prior knowledge on the loss of the best action (or a bound on it), which is
not appropriate in our game setting. Similarly, the small loss bound in [20] is not applicable in our
setting as the work assumes an oblivious adversary and so does not apply to the games we consider.
5
Dynamic Population Games
In this section we consider the dynamic population repeated game setting introduced in [19]. Detailed
discussion and proofs are deferred to Appendix C. Given a game G as described in Section 2, a
dynamic population game with stage game G is a repeated game where at each round t game G is
played and every player i is replaced by a new player with a turnover probability p. Concretely, when
a player turns over, their strategy set and cost function are changed arbitrarily subject to the rules
3
4
With slight abuse of notation, sti denotes the identity vector associated to the strategy player i used at time t.
This is because we only need to evaluate (1) with the game?s optimal solution s? to prove efficiency results.
7
of the game. This models a repeated game setting where players have to adapt to an adversarially
changing environment. We denote the cost function of player i at round t as costti (?). As in Section 3,
we assume that the players receive full information feedback. At the end of each round they observe
the entire cost vector cti = costti (?, st i ), but are not aware of the costs of other players in the game.
Learning in Dynamic Population Games and the Price of Anarchy To guarantee small overall
cost using the smoothness analysis from Section 2, players need to exhibitP
low regret against a shifting
t
t ?t
benchmark s?t
i of socially optimal strategies achieving O PT = mins?t
i costi (s ). Even with a
small probability p of change, the sequence of optimal solutions can have too many changes to be
able to achieve low regret. In spite of this apparent difficulty, [19] prove that at least a ? /(1 ? ?)
fraction of the optimal welfare is guaranteed if 1. players are using low adaptive regret algorithms
(see [14, 18]) and 2. for the underlying optimization problem there exists a relatively stable sequence
of solutions which at each step approximate the optimal solution by a factor of ?. This holds as long
as the turnover probability p is upper bounded by a function of ? (and of certain other properties of
the game, such as the stability of the close to optimal solution).
We consider dynamic population games where each player uses a learning algorithm satisfying Low
Approximate Regret for shifting experts (2). This shifting version of Low Approximate Regret
implies a dynamic game analog of our main efficiency result, Proposition 1.
Algorithms with Low Approximate Regret for Shifting Experts A simple variant of Hedge we
term Noisy Hedge, which mixes the Hedge update at each round with a small amount of uniform noise,
satisfies the Low Approximate Regret property for shifting experts with A(d, T ) = O(log(dT )).
Moreover, algorithms that satisfy a small loss version of the adaptive regret property [14] used in
[19] satisfy the Strong Low Approximate Regret property.
Proposition 7. Noisy Hedge with learning rate ? = ? satisfies the Low Approximate Regret property
for shifting experts with A(d, T ) = 2 log(dT ).
Extending Proposition 1 to the DynamicP
Population Game Setting Let s?1:T denote a stable
?t
sequence of near-optimal solutions s with i costti (s?t ) ? ? ? O PTt for all rounds t. As discussed
in [19], such stable sequences can come from simple greedy algorithms (where each change in the
input of one player affects the output of few other players) or via differentially private algorithms
(where each change in the input of one player affects the output of all other players with small
probability); in the latter case the sequence is randomized. For a deterministic sequence s?1:T
of
i
player i?s actions, we let the random variable Ki denote the number of changes in the sequence. For
a randomized sequence s?1:T
, we let Ki be the sum of total variation distances between
subsequent
i
P
1
?t
pairs s?t
and
s
.
The
stability
of
a
sequence
of
solutions
is
determined
by
E[
K
].
i
i
i
i
Proposition 8. (PoA with Dynamic Population) If all players use Low Approximate Regret algorithms
satisfying (2) in a dynamic population game, where the stage game is ( , ?)-smooth, and Ki as
defined above then
?
1 X ?
1
E C(st ) ?
T t
T 1
?P
?
? n+E
?? X ?
i Ki
E O PTt +
?
? ? t
T
1
1
?
?
?
A(d, T )
.
?
(4)
Here the expectation is taken over the random turnover in the population playing the game, as well
as the random choices of the players on the left hand side.
To claim a price of anarchy bound, we need to ensure that the additive term in (4) is a small fraction
of P
the optimal cost. The challenge is that high turnover probability reduces stability,
increasing
P
E[ i Ki ]. By using algorithms with smaller A(d, T ), we can allow for higher E[ i Ki ] and hence
higher turnover probability. Combining Noisy Hedge with Proposition 8 strengthens the results in
[19] by both weakening the behavioral assumption on the players, allowing them to use simpler
learning algorithms, and allowing a higher turnover probability.
Comparison to Previous Results [19] use the more complex AdaNormalHedge algorithm of [18],
which satisfies the adaptive regret property of [14], but has O(dT ) space complexity. In contrast,
Noisy Hedge only requires space complexity of just O(d). Moreover, a broader class of algorithms
satisfy the Low Approximate Regret property which makes the efficiency guarantees more prescriptive
since this property serves as a behavioral assumption. Finally, the our guarantees we provide improve
on the turnover probability that can be accommodated as discussed in Appendix C.1.
Acknowledgements We thank Vasilis Syrgkanis for sharing his simulation software and the NIPS
reviewers for pointing out the GREEN algorithm [2].
8
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9
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5,674 | 6,134 | Stochastic Structured Prediction
under Bandit Feedback
Artem Sokolov,?, Julia Kreutzer? , Christopher Lo?,? , Stefan Riezler?,?
?
Computational Linguistics & ? IWR, Heidelberg University, Germany
{sokolov,kreutzer,riezler}@cl.uni-heidelberg.de
?
Department of Mathematics, Tufts University, Boston, MA, USA
chris.aa.lo@gmail.com
Amazon Development Center, Berlin, Germany
Abstract
Stochastic structured prediction under bandit feedback follows a learning protocol
where on each of a sequence of iterations, the learner receives an input, predicts
an output structure, and receives partial feedback in form of a task loss evaluation
of the predicted structure. We present applications of this learning scenario to
convex and non-convex objectives for structured prediction and analyze them as
stochastic first-order methods. We present an experimental evaluation on problems
of natural language processing over exponential output spaces, and compare convergence speed across different objectives under the practical criterion of optimal
task performance on development data and the optimization-theoretic criterion of
minimal squared gradient norm. Best results under both criteria are obtained for a
non-convex objective for pairwise preference learning under bandit feedback.
1
Introduction
We present algorithms for stochastic structured prediction under bandit feedback that obey the
following learning protocol: On each of a sequence of iterations, the learner receives an input,
predicts an output structure, and receives partial feedback in form of a task loss evaluation of the
predicted structure. In contrast to the full-information batch learning scenario, the gradient cannot
be averaged over the complete input set. Furthermore, in contrast to standard stochastic learning,
the correct output structure is not revealed to the learner. We present algorithms that use this
feedback to ?banditize? expected loss minimization approaches to structured prediction [18, 25]. The
algorithms follow the structure of performing simultaneous exploration/exploitation by sampling
output structures from a log-linear probability model, receiving feedback to the sampled structure, and
conducting an update in the negative direction of an unbiased estimate of the gradient of the respective
full information objective. The algorithms apply to situations where learning proceeds online on a
sequence of inputs for which gold standard structures are not available, but feedback to predicted
structures can be elicited from users. A practical example is interactive machine translation where
instead of human generated reference translations only translation quality judgments on predicted
translations are used for learning [20]. The example of machine translation showcases the complexity
of the problem: For each input sentence, we receive feedback for only a single predicted translation
out of a space that is exponential in sentence length, while the goal is to learn to predict the translation
with the smallest loss under a complex evaluation metric.
[19] showed that partial feedback is indeed sufficient for optimization of feature-rich linear structured
prediction over large output spaces in various natural language processing (NLP) tasks. Their
experiments follow the standard online-to-batch conversion practice in NLP applications where the
?
The work for this paper was done while the authors were at Heidelberg University.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
model with optimal task performance on development data is selected for final evaluation on a test set.
The contribution of our paper is to analyze these algorithms as stochastic first-order (SFO) methods
in the framework of [7] and investigate the connection of optimization for task performance with
optimization-theoretic concepts of convergence.
Our analysis starts with revisiting the approach to stochastic optimization of a non-convex expected
loss criterion presented by [20]. The iteration complexity of stochastic optimization of a non-convex
objective J(wt ) can be analyzed in the framework of [7] as O(1/2 ) in terms of the number of
iterations needed to reach an accuracy of for the criterion E[k?J(wt )k2 ] ? . [19] attempt to
improve convergence speed by introducing a cross-entropy objective that can be seen as a (strong)
convexification of the expected loss objective. The known best iteration complexity for strongly
convex stochastic optimization is O(1/) for the suboptimality criterion E[J(wt )] ? J(w? ) ? .
Lastly, we analyze the pairwise preference learning algorithm introduced by [19]. This algorithm can
also be analyzed as an SFO method for non-convex optimization. To our knowledge, this is the first
SFO approach to stochastic learning form pairwise comparison feedback, while related approaches
fall into the area of gradient-free stochastic zeroth-order (SZO) approaches [24, 1, 7, 4]. Convergence
rate for SZO methods depends on the dimensionality d of the function to be evaluated, for example,
the non-convex SZO algorithm of [7] has an iteration complexity of O(d/2 ). SFO algorithms do
not depend on d which is crucial if the dimensionality of the feature space is large as is common in
structured prediction.
Furthermore, we present a comparison of empirical and theoretical convergence criteria for the NLP
tasks of machine translation and noun-phrase chunking. We compare the empirical convergence
criterion of optimal task performance on development data with the theoretically motivated criterion
of minimal squared gradient norm. We find a correspondence of fastest convergence of pairwise
preference learning on both tasks. Given the standard analysis of asymptotic complexity bounds,
this result is surprising. An explanation can be given by measuring variance and Lipschitz constant
of the stochastic gradient, which is smallest for pairwise preference learning and largest for crossentropy minimization by several orders of magnitude. This offsets the possible gains in asymptotic
convergence rates for strongly convex stochastic optimization, and makes pairwise preference learning
an attractive method for fast optimization in practical interactive scenarios.
2
Related Work
The methods presented in this paper are related to various other machine learning problems where
predictions over large output spaces have to be learned from partial information.
Reinforcement learning has the goal of maximizing the expected reward for choosing an action at
a given state in a Markov Decision Process (MDP) model, where unknown rewards are received at
each state, or once at the final state. The algorithms in this paper can be seen as one-state MDPs with
context where choosing an action corresponds to predicting a structured output. Most closely related
are recent applications of policy gradient methods to exponential output spaces in NLP problems
[22, 3, 15]. Similar to our expected loss minimization approaches, these approaches are based on
non-convex models, however, convergence rates are rarely a focus in the reinforcement learning
literature. One focus of our paper is to present an analysis of asymptotic convergence and convergence
rates of non-convex stochastic first-order methods.
Contextual one-state MDPs are also known as contextual bandits [11, 13] which operate in a scenario
of maximizing the expected reward for selecting an arm of a multi-armed slot machine. Similar to
our case, the feedback is partial, and the models consist of a single state. While bandit learning
is mostly formalized as online regret minimization with respect to the best fixed arm in hindsight,
we characterize our approach in an asymptotic convergence framework. Furthermore, our highdimensional models predict structures over exponential output spaces. Since we aim to train these
models in interaction with real users, we focus on the ease of elicitability of the feedback and on
speed of convergence. In the spectrum of stochastic versus adversarial bandits, our approach is
semi-adversarial in making stochastic assumptions on inputs, but not on rewards [12].
Pairwise preference learning has been studied in the full information supervised setting [8, 10, 6]
where given preference pairs are assumed. Work on stochastic pairwise learning has been formalized
as derivative-free stochastic zeroth-order optimization [24, 1, 7, 4]. To our knowledge, our approach
2
Algorithm 1 Bandit Structured Prediction
1: Input: sequence of learning rates ?t
2: Initialize w0
3: for t = 0, . . . , T do
4:
Observe xt
5:
Sample y?t ? pwt (y|xt )
6:
Obtain feedback ?(?
yt )
7:
wt+1 = wt ? ?t st
8: Choose a solution w
? from the list {w0 , . . . , wT }
to pairwise preference learning from partial feedback is the first SFO approach to learning from
pairwise preferences in form of relative task loss evaluations.
3
Expected Loss Minimization for Structured Prediction
[18, 25] introduce the expected loss criterion for structured prediction as the minimization of the
expectation of a given task loss function with respect to the conditional distribution over structured
outputs. Let X be a structured input space, let Y(x) be the set of possible output structures for input
x, and let ?y : Y ? [0, 1] quantify the loss ?y (y 0 ) suffered for predicting y 0 instead of the gold
standard structure y. In the full information setting, for a given (empirical) data distribution p(x, y),
the learning problem is defined as
X
X
min Ep(x,y)pw (y0 |x) [?y (y 0 )] = min
p(x, y)
?y (y 0 )pw (y 0 |x),
(1)
w?Rd
where
w?Rd
x,y
y 0 ?Y(x)
pw (y|x) = exp(w> ?(x, y))/Zw (x)
(2)
d
d
is a Gibbs distribution with joint feature representation ? : X ? Y ? R , weight vector w ? R , and
normalization constant Zw (x). Despite being a highly non-convex optimization problem, positive
results have been obtained by gradient-based optimization with respect to
h
i
?Ep(x,y)pw (y0 |x) [?y (y 0 )] = Ep(x,y)pw (y0 |x) ?y (y 0 ) ?(x, y 0 ) ? Epw (y0 |x) [?(x, y 0 )] . (3)
Unlike in the full information scenario, in structured learning under bandit feedback the gold standard
output structure y with respect to which the objective function is evaluated is not revealed to the
learner. Thus we can neither evaluate the task loss ? nor calculate the gradient (3) as in the full
information case. A solution to this problem is to pass the evaluation of the loss function to the user,
i.e, we access the loss directly through user feedback without assuming existence of a fixed reference
y. In the following, we will drop the subscript referring to the gold standard structure in the definition
of ? to indicate that the feedback is in general independent of gold standard outputs. In particular,
we allow ? to be equal to 0 for several outputs.
4
Stochastic Structured Prediction under Partial Feedback
Algorithm Structure. Algorithm 1 shows the structure of the methods analyzed in this paper. It
assumes a sequence of input structures xt , t = 0, . . . , T that are generated by a fixed, unknown
distribution p(x) (line 4). For each randomly chosen input, an output y?t is sampled from a Gibbs
model to perform simultaneous exploitation (use the current best estimate) / exploration (get new
information) on output structures (line 5). Then, feedback ?(?
yt ) to the predicted structure is obtained
(line 6). An update is performed by taking a step in the negative direction of the stochastic gradient
st , at a rate ?t (line 7). As a post-optimization step, a solution w
? is chosen from the list of vectors
wt ? {w0 , . . . , wT } (line 8).
Given Algorithm 1, we can formalize the notion of ?banditization? of objective functions by presenting
different instantiations of the vector st , and showing them to be unbiased estimates of the gradients
of corresponding full information objectives.
3
Expected Loss Minimization. [20] presented an algorithm that minimizes the following expected
loss objective. It is non-convex for the specific instantiations in this paper:
X
X
Ep(x)pw (y|x) [?(y)] =
p(x)
?(y)pw (y|x).
(4)
x
y?Y(x)
The vector st used in their algorithm can be seen as a stochastic gradient of this objective, i.e., an
evaluation of the full gradient at a randomly chosen input xt and output y?t :
st = ?(?
yt ) ?(xt , y?t ) ? Epwt (y|xt ) [?(xt , y)] .
(5)
Instantiating st in Algorithm 1 to the stochastic gradient in equation (5) yields an update that compares
the sampled feature vector to the average feature vector, and performs a step into the opposite direction
of this difference, the more so the higher the loss of the sampled structure is. In the following, we
refer to the algorithm for expected loss minimization defined by the update (5) as Algorithm EL.
Pairwise Preference Learning. Decomposing complex problems into a series of pairwise comparisons has been shown to be advantageous for human decision making [23]. For the example
of machine translation, this means that instead of requiring numerical assessments of translation
quality from human users, only a relative preference judgement on a pair of translations needs to be
elicited. This idea is formalized in [19] as an expected loss objective with respect to a conditional
distribution of pairs of structured outputs. Let P(x) = {hyi , yj i |yi , yj ? Y(x)} denote the set of
output pairs for an input x, and let ?(hyi , yj i) : P(x) ? [0, 1] denote a task loss function that
specifies a dispreference of yi compared to yj . In the experiments reported in this paper, we simulate
two types of pairwise feedback. Firstly, continuous pairwise feedback is computed as
?(yi ) ? ?(yj ) if ?(yi ) > ?(yj ),
?(hyi , yj i) =
(6)
0 otherwise.
A binary feedback function is computed as
?(hyi , yj i) =
1
0
if ?(yi ) > ?(yj ),
otherwise.
(7)
Furthermore, we assume a feature representation ?(x, hyi , yj i) = ?(x, yi ) ? ?(x, yj ) and a Gibbs
model on pairs of output structures
pw (hyi , yj i |x) =
ew
P
>
(?(x,yi )??(x,yj ))
ew
> (?(x,y
i )??(x,yj ))
= pw (yi |x)p?w (yj |x).
(8)
hyi ,yj i?P(x)
The factorization of this model into the product pw (yi |x)p?w (yj |x) allows efficient sampling and
calculation of expectations. Instantiating objective (4) to the case of pairs of output structures defines
the following objective that is again non-convex in the use cases in this paper:
X
X
Ep(x)pw (hyi ,yj i|x) [?(hyi , yj i)] =
p(x)
?(hyi , yj i) pw (hyi , yj i |x).
(9)
x
hyi ,yj i?P(x)
Learning from partial feedback on pairwise preferences will ensure that the model finds a ranking
function that assigns low probabilities to discordant pairs with respect the the observed preference
pairs. Stronger assumptions on the learned ranking can be made if asymmetry and transitivity of the
observed ordering of pairs is required.2 An algorithm for pairwise preference learning can be defined
by instantiating Algorithm 1 to sampling output pairs h?
yi , y?j it , receiving feedback ?(h?
yi , y?j it ), and
performing a stochastic gradient update using
st = ?(h?
yi , y?j it ) ?(xt , h?
yi , y?j it ) ? Epwt (hyi ,yj i|xt ) [?(xt , hyi , yj i)] .
(10)
The algorithms for pairwise preference ranking defined by update (10) are referred to as Algorithms
PR(bin) and PR(cont), depending on the use of binary or continuous feedback.
2
See [2] for an overview of bandit learning from consistent and inconsistent pairwise comparisons.
4
Cross-Entropy Minimization. The standard theory of stochastic optimization predicts considerable improvements in convergence speed depending on the functional form of the objective. This
motivated the formalization of a convex upper bounds on expected normalized loss in [19]. If a
P
normalized gain function g?(y) = Zg(y)
is used where Zg (x) = y?Y(x) g(y), and g = 1 ? ?, the
g (x)
objective can be seen as the cross-entropy of model pw (y|x) with respect to g?(y):
X
X
Ep(x)?g(y) [? log pw (y|x)] = ?
p(x)
g?(y) log pw (y|x).
(11)
x
y?Y(x)
For a proper probability distribution g?(y), an application of Jensen?s inequality to the convex negative
logarithm function shows that objective (11) is a convex upper bound on objective (4). However,
normalizing the gain function is prohibitive in a partial feedback setting since it would require to
elicit user feedback for each structure in the output space. [19] thus work with an unnormalized gain
function g(y) that preserves convexity. This can be seen by rewriting the objective as the sum of a
linear and a convex function in w:
X
X
Ep(x)g(y) [? log pw (y|x)] = ?
p(x)
g(y)w> ?(x, y)
(12)
x
+
X
y?Y(x)
p(x)(log
x
X
exp(w> ?(x, y)))?(x),
y?Y(x)
P
where ?(x) = y?Y(x) g(y) is a constant factor not depending on w. Instantiating Algorithm 1 to the
following stochastic gradient st of this objective yields an algorithm for cross-entropy minimization:
st =
g(?
yt )
pwt (?
yt |xt )
? ?(xt , y?t ) + Epwt [?(xt , yt )] .
(13)
Note that the ability to sample structures from pwt (?
yt |xt ) comes at the price of having to normalize
yt |xt ). While minimization of this objective will assign high probabilities to structures
st by 1/pwt (?
with high gain, as desired, each update is affected by a probability that changes over time and is
unreliable when training is started. This further increases the variance already present in stochastic
optimization. We deal with this problem by clipping too small sampling probabilities to p?wt (?
yt |xt ) =
max{pwt (?
yt |xt ), k} for a constant k [9]. The algorithm for cross-entropy minimization based on the
stochastic gradient (13) is referred to as Algorithm CE in the following.
5
Convergence Analysis
To analyze convergence, we describe Algorithms EL, PR, and CE as stochastic first-order (SFO)
methods in the framework of [7]. We assume lower bounded, differentiable objective functions J(w)
with Lipschitz continuous gradient ?J(w) satisfying
k?J(w + w0 ) ? ?J(w)k ? Lkw0 k ?w, w0 , ?L ? 0.
(14)
For an iterative process of the form wt+1 = wt ? ?t st , the conditions to be met concern unbiasedness
of the gradient estimate
E[st ] = ?J(wt ), ?t ? 0,
(15)
and boundedness of the variance of the stochastic gradient
E[||st ? ?J(wt )||2 ] ? ? 2 , ?t ? 0.
(16)
Condition (15) is met for all three Algorithms by taking expectations over all sources of randomness,
i.e., over random inputs and output structures. Assuming k?(x, y)k ? R, ?(y) ? [0, 1] and
yt )
g(y) ? [0, 1] for all x, y, and since the ratio p?w g(?
yt |xt ) is bounded, the variance in condition (16) is
t (?
bounded. Note that the analysis of [7] justifies the use of constant learning rates ?t = ?, t = 0, . . . , T .
Convergence speed can be quantified in terms of the number of iterations needed to reach an accuracy
of for a gradient-based criterion E[k?J(wt )k2 ] ? . For stochastic optimization of non-convex
objectives, the iteration complexity with respect to this criterion is analyzed as O(1/2 ) in [7]. This
complexity result applies to our Algorithms EL and PR.
5
The iteration complexity of stochastic optimization of (strongly) convex objectives has been analyzed
as at best O(1/) for the suboptimality criterion E[J(wt )] ? J(w? ) ? for decreasing learning rates
[14].3 Strong convexity of objective (12) can be achieved easily by adding an `2 regularizer ?2 kwk2
with constant ? > 0. Algorithm CE is then modified to use the following regularized update rule
wt+1 = wt ? ?t (st + T? wt ).
This standard analysis shows two interesting points: First, Algorithms EL and PR can be analyzed as
SFO methods where the latter only requires relative preference feedback for learning, while enjoying
an iteration complexity that does not depend on the dimensionality of the function as in gradient-free
stochastic zeroth-order (SZO) approaches. Second, the standard asymptotic complexity bound of
O(1/2 ) for non-convex stochastic optimization hides the constants L and ? 2 in which iteration
complexity increases linearly. As we will show, these constants have a substantial influence, possibly
offsetting the advantages in asymptotic convergence speed of Algorithm CE.
6
Experiments
Measuring Numerical Convergence and Task Loss Performance. In the following, we will
present an experimental evaluation for two complex structured prediction tasks from the area of
NLP, namely statistical machine translation and noun phrase chunking. Both tasks involve dynamic
programming over exponential output spaces, large sparse feature spaces, and non-linear nondecomposable task loss metrics. Training for both tasks was done by simulating bandit feedback by
evaluating ? against gold standard structures which are never revealed to the learner. We compare
the empirical convergence criterion of optimal task performance on development data with numerical
results on theoretically motivated convergence criteria.
For the purpose of measuring convergence with respect to optimal task performance, we report an
evaluation of convergence speed on a fixed set of unseen data as performed in [19]. This instantiates
the selection criterion in line (8) in Algorithm 1 to an evaluation of the respective task loss function
?(?
ywt (x)) under MAP prediction y?w (x) = arg maxy?Y(x) pw (y|x) on the development data. This
corresponds to the standard practice of online-to-batch conversion where the model selected on the
development data is used for final evaluation on a further unseen test set. For bandit structured
prediction algorithms, final results are averaged over three runs with different random seeds.
For the purpose of obtaining numerical results on convergence speed, we compute estimates of the
expected squared gradient norm E[k?J(wt )k2 ], the Lipschitz constant L and the variance ? 2 in
which the asymptotic bounds on iteration complexity grow linearly.4 We estimate the squared gradient
norm by the squared norm of the stochastic gradient ksT k2 at a fixed time horizon T . The Lipschitz
ks ?s k
constant L in equation (14) is estimated by maxi,j kwii ?wjj k for 500 pairs wi and wj randomly drawn
from the weights produced during training. The variance ? 2 in equation (16) is computed as the
empirical variance of the stochastic gradient, taken at regular intervals after each epoch of size D,
PK
PK
1
1
T
2
yielding the estimate K
k=1 kskD ? K
k=1 skD k where K = b D c. All estimates include
multiplication of the stochastic gradient with the learning rate. For comparability of results across
different algorithms, we use the same T and the same constant learning rates for all algorithms.5
Statistical Machine Translation. In this experiment, an interactive machine translation scenario
is simulated where a given machine translation system is adapted to user style and domain based on
feedback to predicted translations. Domain adaptation from Europarl to NewsCommentary domains
using the data provided at the WMT 2007 shared task is performed for French-to-English translation.6
The MT experiments are based on the synchronous context-free grammar decoder cdec [5]. The
models use a standard set of dense and lexicalized sparse features, including an out-of and an in3
For constant learning rates, [21] show even faster convergence in the search phase of strongly-convex
stochastic optimization.
2
4
For example, these constants appear as O( L + L?
) in the complexity bound for non-convex stochastic
2
optimization of [7].
5
Note that the squared gradient norm upper bounds the suboptimality criterion s.t. k?J(wt )k2 ?
2?J(wt )] ? J(w? ) for strongly convex functions. Together with the use of constant learning rates this means
that we measure convergence to a point near an optimum for strongly convex objectives.
6
http://www.statmt.org/wmt07/shared-task.html
6
Task
SMT
Chunking
Algorithm
CE
EL
PR(bin)
Iterations
281k
370k
115k
Score
0.271?0.001
0.267?8e?6
0.273?0.0005
?
1e-6
1e-5
1e-4
?
1e-6
k
5e-3
CE
EL
PR(cont)
5.9M
7.5M
4.7M
0.891?0.005
0.923?0.002
0.914?0.002
1e-6
1e-4
1e-4
1e-6
1e-2
Table 1: Test set evaluation for stochastic learning under bandit feedback from [19], for chunking
under F1-score, and for machine translation under BLEU. Higher is better for both scores. Results
for stochastic learners are averaged over three runs of each algorithm, with standard deviation shown
in subscripts. The meta-parameter settings were determined on dev sets for constant learning rate ?,
clipping constant k, `2 regularization constant ?.
domain language model. The out-of-domain baseline model has around 200k active features. The
pre-processing, data splits, feature sets and tuning strategies are described in detail in [19]. The
difference in the task loss evaluation between out-of-domain (BLEU: 0.2651) and in-domain (BLEU:
0.2831) models gives the range of possible improvements (1.8 BLEU points) for bandit learning.
Learning under bandit feedback starts at the learned weights of the out-of-domain median models.
It uses parallel in-domain data (news-commentary, 40,444 sentences) to simulate bandit feedback,
by evaluating the sampled translation against the reference using as loss function ? a smoothed
per-sentence 1 ? BLEU (zero n-gram counts being replaced with 0.01). After each update, the
hypergraph is re-decoded and all hypotheses are re-ranked. Training is distributed across 38 shards
using a multitask-based feature selection algorithm [17].
Noun-phrase Chunking. The experimental setting for chunking is the same as in [19]. Following
[16], conditional random fields (CRF) are applied to the noun phrase chunking task on the CoNLL2000 dataset7 . The implemented set of feature templates is a simplified version of [16] and leads to
around 2M active features. Training under full information with a log-likelihood objective yields
0.935 F1. In difference to machine translation, training with bandit feedback starts from w0 = 0, not
from a pre-trained model.
Task Loss Evaluation. Table 1 lists the results of the task loss evaluation for machine translation
and chunking as performed in [19], together with the optimal meta-parameters and the number of
iterations needed to find an optimal result on the development set. Note that the pairwise feedback
type (cont or bin) is treated as a meta-parameter for Algorithm PR in our simulation experiment.
We found that bin is preferable for machine translation and cont for chunking in order to obtain the
highest task scores.
For machine translation, all bandit learning runs show significant improvements in BLEU score
over the out-of-domain baseline. Early stopping by task performance on the development led to the
selection of algorithm PR(bin) at a number of iterations that is by a factor of 2-4 smaller compared to
Algorithms EL and CE.
For the chunking experiment, the F1-score results obtained for bandit learning are close to the fullinformation baseline. The number of iterations needed to find an optimal result on the development
set is smallest for Algorithm PR(cont), compared to Algorithms EL and CE. However, the best
F1-score is obtained by Algorithm EL.
Numerical Convergence Results. Estimates of E[k?J(wt )k2 ], L and ? 2 for three runs of each
algorithm and task with different random seeds are listed in Table 2.
For machine translation, at time horizon T , the estimated squared gradient norm for Algorithm PR
is several orders of magnitude smaller than for Algorithms EL and CE. Furthermore, the estimated
Lipschitz constant L and the estimated variance ? 2 are smallest for Algorithm PR. Since the iteration
complexity increases linearly with respect to these terms, smaller constants L and ? 2 and a smaller
7
http://www.cnts.ua.ac.be/conll2000/chunking/
7
Task
SMT
Chunking
Algorithm
CE
EL
PR(bin)
PR(cont)
T
767,000
767,000
767,000
767,000
ksT k2
3.04?0.02
0.02?0.03
2.88e-4?3.40e?6
1.03e-8?2.91e?10
L
0.54?0.3
1.63?0.67
0.08?0.01
0.10?5.70e?3
?2
35 ?6
3.13e-4?3.60e?6
3.79e-5?9.50e?8
1.78e-7?1.45e?10
CE
EL
PR(bin)
PR(cont)
3,174,400
3,174,400
3,174,400
3,174,400
4.20?0.71
1.21e-3?1.1e?4
7.71e-4?2.53e?4
5.99e-3?7.24e?4
1.60?0.11
1.16?0.31
1.33?0.24
1.11?0.30
4.88?0.07
0.01?9.51e?5
4.44e-3?2.66e?5
0.03?4.68e?4
Table 2: Estimates of squared gradient norm ksT k2 , Lipschitz constant L and variance ? 2 of stochastic
gradient (including multiplication with learning rate) for fixed time horizon T and constant learning
rates ? = 1e ? 6 for SMT and for chunking. The clipping and regularization parameters for CE are
set as in Table 1 for machine translation, except for chunking CE ? = 1e ? 5. Results are averaged
over three runs of each algorithm, with standard deviation shown in subscripts.
value of the estimate E[k?J(wt )k2 ] at the same number of iterations indicates fastest convergence
for Algorithm PR. This theoretically motivated result is consistent with the practical convergence
criterion of early stopping on development data: Algorithm PR which yields the smallest squared
gradient norm at time horizon T also needs the smallest number of iterations to achieve optimal
performance on the development set. In the case of machine translation, Algorithm PR even achieves
the nominally best BLEU score on test data.
For the chunking experiment, after T iterations, the estimated squared gradient norm and either of
the constants L and ? 2 for Algorithm PR are several orders of magnitude smaller than for Algorithm
CE, but similar to the results for Algorithm EL. The corresponding iteration counts determined by
early stopping on development data show an improvement of Algorithm PR over Algorithms CE and
EL, however, by a smaller factor than in the machine translation experiment.
Note that for comparability across algorithms, the same constant learning rates were used in all runs.
However, we obtained similar relations between algorithms by using the meta-parameter settings
chosen on development data as shown in Table 1. Furthermore, the above tendendencies hold for
both settings of the meta-parameter bin or cont of Algorithm PR.
7
Conclusion
We presented learning objectives and algorithms for stochastic structured prediction under bandit
feedback. The presented methods ?banditize? well-known approaches to probabilistic structured
prediction such as expected loss minimization, pairwise preference ranking, and cross-entropy minimization. We presented a comparison of practical convergence criteria based on early stopping with
theoretically motivated convergence criteria based on the squared gradient norm. Our experimental
results showed fastest convergence speed under both criteria for pairwise preference learning. Our
numerical evaluation showed smallest variance for pairwise preference learning, which possibly
explains fastest convergence despite the underlying non-convex objective. Furthermore, since this
algorithm requires only easily obtainable relative preference feedback for learning, it is an attractive
choice for practical interactive learning scenarios.
Acknowledgments.
This research was supported in part by the German research foundation (DFG), and in part by a
research cooperation grant with the Amazon Development Center Germany.
8
References
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9
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5,675 | 6,135 | The Multiscale Laplacian Graph Kernel
Risi Kondor
Department of Computer Science
Department of Statistics
University of Chicago
Chicago, IL 60637
risi@cs.uchicago.edu
Horace Pan
Department of Computer Science
University of Chicago
Chicago, IL 60637
hopan@uchicago.edu
Abstract
Many real world graphs, such as the graphs of molecules, exhibit structure at multiple different scales, but most existing kernels between graphs are either purely
local or purely global in character. In contrast, by building a hierarchy of nested
subgraphs, the Multiscale Laplacian Graph kernels (MLG kernels) that we define
in this paper can account for structure at a range of different scales. At the heart
of the MLG construction is another new graph kernel, called the Feature Space
Laplacian Graph kernel (FLG kernel), which has the property that it can lift a
base kernel defined on the vertices of two graphs to a kernel between the graphs.
The MLG kernel applies such FLG kernels to subgraphs recursively. To make the
MLG kernel computationally feasible, we also introduce a randomized projection
procedure, similar to the Nystr?om method, but for RKHS operators.
1
Introduction
There is a wide range of problems in applied machine learning from web data mining [1] to protein function prediction [2] where the input space is a space of graphs. A particularly important
application domain is chemoinformatics, where the graphs capture the structure of molecules. In
the pharmaceutical industry, for example, machine learning algorithms are regularly used to screen
candidate drug compounds for safety and efficacy against specific diseases [3].
Because kernel methods neatly separate the issue of data representation from the statistical learning
component, it is natural to formulate graph learning problems in the kernel paradigm. Starting with
[4], a number of different graph kernels have appeared in the literature (for an overview, see [5]). In
general, a graph kernel k(G1 , G2 ) must satisfy the following requirements:
1. The kernel should capture the right notion of similarity between G1 and G2 . For example, if G1
and G2 are social networks, then k might capture to what extent their clustering structure, degree
distribution, etc. match. If, on the other hand, G1 and G2 are molecules, then we are probably
more interested in what functional groups are present, and how they are arranged relative to each
other.
2. The kernel is usually computed from the adjacency matrices A1 and A2 of the two graphs, but it
must be invariant to the ordering of the vertices. In other words, writing the kernel explicitly in
terms of A1 and A2 , we must have k(A1 , A2 ) = k(A1 ,P A2 P >) for any permutation matrix P .
Permutation invariance has proved to be the central constraint around which much of the graph
kernels literature is organized, effectively stipulating that graph kernels must be built out of graph
invariants. Efficiently computable graph invariants offered by the mathematics literature tend to fall
in one of two categories:
1. Local invariants, which can often be reduced to simply counting some local properties, such as
the number of triangles, squares, etc. that appear in G as subgraphs.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
2. Spectral invariants, which can be expressed as functions of the eigenvalues of the adjacency
matrix or the graph Laplacian.
Correspondingly, while different graph kernels are motivated in very different ways from random
walks [4] through shortest paths [6, 7] to Fourier transforms on the symmetric group [8], most graph
kernels in the literature ultimately reduce to computing a function of the two graphs that is either
purely local or purely spectral. Any of the kernels based on the ?subgraph counting? idea (e.g., [9])
are local. On the other hand, most of the random walk based kernels are reducible to a spectral form
involving the eigenvalues of either the two graphs individually, or their Kronecker product [5] and
therefore are really only sensitive to the large scale structure of graphs.
In practice, it would be desirable to have a kernel that can take structure into account at multiple
different scales. A kernel between molecules, for example, should not only be sensitive to the overall
large-scale shape of the graphs (whether they are more like a chain, a ring, a chain that branches,
etc.), but also to what smaller structures (e.g., functional groups) are present in the graphs, and how
they are related to the global structure (e.g., whether a particular functional group is towards the
middle or one of the ends of a chain).
For the most part, such a multiscale graph kernel has been missing from the literature. Two notable
exceptions are the Weisfeiler?Lehman kernel [10] and Propagation Kernel [11]. The WL kernel
uses a combination of message passing and hashing to build summaries of the local neighborhoods
of vertices at different scales. While shown to be effective, the Weisfeiler?Lehman kernel?s hashing
step is somewhat ad-hoc; perturbing the edges by a small amount leads to completely different hash
features. Similarly, the propagation kernel monitors how the distribution of node/edge labels spreads
through the graph and then uses locality sensitivity hashing to efficiently bin the label distributions
into feature vectors.
Most recently, structure2vec[12] attempts to represent each graph with a latent variable model and
then embeds them into a feature space, using the inner product as a kernel. This approach compares
favorably to the standard kernel methods in both accuracy and computational efficiency.
In this paper we present a new graph kernel, the Multiscale Laplacian Graph Kernel (MLG kernel),
which, we believe, is the first kernel in the literature that can truly compare structure in graphs
simultaneously at multiple different scales. We begin by introducing the Feature Space Laplacian
Graph Kernel(FLG kernel) in Section 2. The FLG kernel operates at a single scale, while combining
information from the nodes?s vertex features with topological information through its Laplacian.
An important property of the FLG kernel is that it can work with vertex labels provided as a ?base
kernel? on the vertices, which allows us to apply the FLG kernel recursively.
The MLG kernel, defined in Section 3, uses the FLG kernel?s recursive property to build a hierarchy
of subgraph kernels that are sensitive to both the topological relationships between individual vertices, and between subgraphs of increasing sizes. Each kernel is defined in terms of the preceding
kernel in the hierarchy. Efficient computability is a major concern in our paper, and recursively
defined kernels on combinatorial data structures, can be very expensive. Therefore, in Section 4
we describe a strategy based on a combination of linearizing each level of the kernel (relative to a
given dataset) and a randomized low rank projection step, which reduces every stage of the kernel
computation to simple operations involving small matrices, leading to a very fast algorithm. Finally,
section 5 presents experimental comparisons of our kernel with competing methods.
2
Laplacian Graph Kernels
Let G be a weighted undirected graph with vertex set V = {v1 , . . . , vn } and edge set E. Recall that
the graph Laplacian of G is an n ? n matrix LG , with
LGi,j =
?
?
??w
P i,j
j : {vi ,vj }?E wi,j
?
?0
if {vi , vj } ? E
if i = j
otherwise,
where wi,j is the weight of edge {vi , vj }. The graph Laplacian is positive semi-definite, and in terms
of the adjacency matrix A and the weighted degree matrix D it can be expressed as L = D ? A.
2
Spectral graph theory tells us that the low eigenvalue eigenvectors of LG are informative about the
overall shape of G. One way of seeing this is to note that for any vector z ? Rn
X
z>LG z =
wi,j (zi ? zj )2 ,
{i,j}?E
so the low eigenvalue eigenvectors are the smoothest functions on G, in the sense that they vary the
least between adjacent vertices. An alternative interpretation emerges if we use G to construct a
Gaussian graphical model (Markov Random Field or MRF) over n variables x1 , . . . , xn with clique
2
2
potentials ?(xi , xj ) = e?wi,j (xi ?xj ) /2 for each edge and ?(xi ) = e??xi /2 for each vertex. The
>
joint distribution of x = (x1 , . . . , xn ) is then
Y
Y
2
2
>
p(x) ?
e?wi,j (xi?xj ) /2
e??xi /2 = e?x (L+?I) x/2 ,
(1)
vi ?V
{vi ,vj }?E
G
showing that the covariance matrix of x is (L + ?I)?1 . Note that the ? factors were added to
ensure that the distribution is normalizable, and ? is typically just a small constant ?regularizer?:
LG actually has a zero eigenvalue eigenvector (namely the constant vector n?1/2 (1, 1, . . . , 1)> ), so
without adding ?I we would not be able to invert it. In the following we will call LG + ?I the
regularized Laplacian, and denote it simply by L.
Both the above views suggest that if we want define a kernel between graphs that is sensitive to
their overall shape, comparing the low eigenvalue eigenvectors of their Laplacians is a good place
to start. Previous work by [13] also used the graph Laplacian for constructing a similarity function
on graphs. Following the MRF route, given two graphs G1 and G2 of n vertices, we can define the
kernel between them to be a kernel between the corresponding distributions p1 = N (0, L?1
1 ) and
p2 = N (0, L?1
).
Specifically,
we
will
use
the
Bhattacharyya
kernel
[14]
2
Z p
p
p1 (x) p2 (x) dx,
(2)
k(p1 , p2 ) =
because for Gaussian distributions it can be computed in closed form, giving
1
L1 + 1 L2 ?1 1/2
2
2
k(p1 , p2 ) =
.
L?1 1/4 L?1 1/4
1
L?1
1
2
L?1
2
If some of the eigenvalues of
or
are zero or very close to zero, along certain directions
in space the two distributions in (2) become very flat, leading to vanishingly small kernel values
(unless the ?flat? directions of the two Gaussians are perfectly aligned). To remedy this problem,
similarly to [15], we ?soften? (or regularize) the kernel by adding some small constant ? times the
?1
identity to L?1
1 and L2 . This leads to what we call the Laplacian Graph Kernel.
Definition 1. Let G1 and G2 be two graphs with n vertices with (regularized) Laplacians L1 and
L2 , respectively. We define the Laplacian graph kernel (LG kernel) with parameter ? between G1
1 ?1 1 ?1 ?1 1/2
and G2 as
S + S
kLG (G1 , G2 ) = 2 1 1/42 2 1/4
,
(3)
|S1 |
|S2 |
?1
where S1 = L?1
1 + ?I and S2 = L2 + ?I.
By virtue of (2), the LG kernel is positive semi-definite, and because the value of the overlap integral
is largely determined by the extent to which the subspaces spanned by the largest eigenvalue eigen?1
vectors of L?1
1 and L2 are aligned, it effectively captures similarity between the overall shapes
of G1 and G2 . However, the LG kernel does suffer from three major limitations: it assumes that
both graphs have the same number of vertices, it is only sensitive to the overall structure of the two
graphs, and it is not invariant to permuting the vertices. Our goal for the rest of this paper is to
overcome each of these limitations, while retaining the LG kernel?s attractive spectral interpretation.
2.1
The feature space Laplacian graph kernel (FLG kernel)
>
In the probabilistic view of the LG kernel, every graph generates random vectors x = (x1 , . . . , xn )
according to (1), and the kernel between two graphs is determined by comparing the corresponding
3
distributions. The invariance problem arises because the ordering of the variables x1 , . . . , xn is
arbitrary: even if G1 and G2 are topologically the same, kLG (G1 , G2 ) might be low if their vertices
happen to be numbered differently.
One of the central ideas of this paper is to address this issue by transforming from
P the ?vertex space
variables? x1 , . . . , xn to ?feature space variables? y1 , . . . , ym , where yi = j ti,j (xj ), and each
ti,j function may only depend on j through local and reordering invariant properties of vertex vj . If
we then compute an analogous kernel to the LG kernel, but now between the distributions of the y?s
rather than the x?s, the resulting kernel will be permutation invariant.
In the simplest case, the ti,j functions are linear, i.e., ti,j (xj ) = ?i (vj ) ? xj , where (?1 , . . . , ?m ) is
a collection of m local (and permutation invariant) vertex features. For example, ?i (vj ) may be the
degree of vertex vj , or the value of h? (vj , vj ), where h is the diffusion kernel on G with length scale
parameter ? (c.f., [16]). In the chemoinformatics setting, the ?i ?s might be some way of encoding
what type of atom is located at vertex vj .
The linear transform of a multivariate normal random variable is multivariate normal. In our case,
defining Ui,j = ?i (vj )i,j and y = U x, we have E(y) = 0 and Cov(y, y) = U Cov(x, x)U > =
U L?1 U > , leading to the following kernel, which is the workhorse of the present paper.
Definition 2. Let G1 and G2 be two graphs with regularized Laplacians L1 and L2 , respectively, ? ?
0 a parameter, and (?1 , . . . , ?m ) a collection of m local vertex features. Define the corresponding
feature mapping matrices
[U2 ]i,j = ?i (vj0 ),
[U1 ]i,j = ?i (vj )
where vj is the j?th vertex of G1 and vj0 is the j?th vertex of G2 . The corresponding Feature space
Laplacian graph kernel (FLG kernel) is defined
1 ?1 1 ?1 ?1 1/2
S + S
,
(4)
kFLG (G1 , G2 ) = 2 1 1/42 2 1/4
|S1 |
|S2 |
?1 >
>
where S1 = U1 L?1
1 U1 + ?I and S2 = U2 L2 U2 + ?I.
Since the ?1 , . . . , ?m vertex features, by definition, are local and invariant to vertex renumbering,
the FLG kernel is permutation invariant. Moreover, the distributions now live in the space of features
rather than the space defined by the vertices, so we can apply the kernel to two graphs with different
numbers of vertices. The major remaining shortcoming of the FLG kernel is that it cannot take into
account structure at multiple different scales.
2.2
The ?kernelized? FLG kernel
The key to boosting kFLG to a multiscale kernel is that it itself can be ?kernelized?, i.e., it can be
computed from just the inner products between the feature vectors of the vertices (which we call the
base kernel) without having to know the actual ?i (vj ) features values.
Definition 3. Given a collection ? = (?1 , . . . , ?m )> of local vertex features, we define the corresponding base kernel ? between two vertices v and v 0 as the dot product of their feature vectors:
?(v, v 0 ) = ?(v)> ?(v 0 ).
Note that in this definition v and v 0 may be two vertices of the same graph, or of two different graphs.
We first show that, similarly to other kernel methods [17], to compute kFLG (G1 , G2 ) one only needs
to consider the subspace of Rm spanned by the feature vectors of their vertices.
Proposition 1. Let G1 and G2 be two graphs with vertex sets V1 = {v1 . . . vn1 } and V2 =
{v10 . . . vn0 2 }, and let {?1 , . . . , ?p }
be an orthonormal basis for the subspace
W = span ?(v1 ), . . . , ?(vn1 ), ?(v10 ), . . . , ?(vn0 2 ) .
dim(W ) = p. Then, (4) can be rewritten as
1 ?1 1 ?1 ?1 1/2
S + S
kFLG (G1 , G2 ) = 2 1 1/42 2 1/4
,
(5)
|S 1 |
|S 2 |
where [S 1 ]i,j = ?i>S1 ?j and [S 2 ]i,j = ?i>S2 ?j . In other words, S 1 and S 2 are the projections of S1
and S2 to W .
4
Similarly to kernel PCA [18] or the Bhattacharyya kernel [15], the easiest way to construct the basis
{?1 , . . . , ?p } required by (5) is to compute the eigendecomposition of the joint Gram matrix of the
vertices of the two graphs.
Proposition 2. Let G1 and G be as in Proposition 1, V = {v 1 , . . . , v n1 +n2 } be the union of their
vertex
sets (where it is assumed that the first n1 vertices are {v1 , . . . , vn1 } and the second n2 vertices
are v10 , . . . , vn0 2 ), and define the joint Gram matrix K ? R(n1 +n2 )?(n1 +n2 ) as
Ki,j = ?(v i , v j ) = ?(v i )> ?(v j ).
Let u1 , . . . , up be a maximal orthonormal set of the non-zero eigenvalue eigenvectors of K with
corresponding eigenvalues. Then the vectors
n1 +n2
1 X
?i = ?
[ui ]` ?(v ` )
?i `=1
(6)
1/2
1/2
form an orthonormal basis for W . Moreover, defining Q = [?1 u1 , . . . , ?p up ] ? Rp?p and
setting Q1 = Q1:n1 , : and Q2 = Qn1 +1:n2 , : (the first n1 and remaining n2 rows of Q, respectively),
the matrices S 1 and S 2 appearing in (5) can be computed as
?1
S 1 = Q>
1 L1 Q1 + ?I,
?1
S 2 = Q>
2 L2 Q2 + ?I.
(7)
Proofs of these two propositions are given in the Supplemental Material. As in other kernel methods,
the significance of Propositions 1 and 2 is not just that they show how kFLG (G1 , G2 ) can be efficiently
computed when ? is very high dimensional, but that they make it clear that the FLG kernel may
be induced from any base kernel. For completeness, we close this section with the generalized
definition of the FLG kernel.
Definition 4. Let G1 and G2 be two graphs. Assume that each of their vertices comes from an
abstract vertex space V and that ? : V ? V ? R is a symmetric positive semi-definite kernel on V.
The generalized FLG kernel induced from ? is then defined as
1 ?1 1 ?1 ?1 1/2
S + S
?
,
(8)
kFLG (G1 , G2 ) = 2 1 1/42 2 1/4
|S 1 |
|S 2 |
where S 1 and S 2 are as defined in Proposition 2.
3
The multiscale Laplacian graph kernel (MLG kernel)
By multiscale graph kernel we mean a kernel that is able to capture similarity between graphs not
just based on the topological relationships between their individual vertices, but also the topological
relationships between subgraphs. The key property of the FLG kernel that allows us to build such a
kernel is that it can be applied recursively. In broad terms, the construction goes as follows:
1. Given a graph G, associate each vertex with a subgraph centered around it and compute the FLG
kernel between every pair of these subgraphs.
2. Reinterpret the FLG kernel between these subgraphs as a new base kernel between the center
vertices of the subgraphs.
3. Consider larger subgraphs centered at each vertex, compute the FLG kernel between them induced from the new base kernel constructed in the previous step, and recurse.
To compute the actual multiscale graph kernel K between G and another graph G 0 , we follow the
same process for G 0 and then set K(G, G 0 ) equal to the FLG kernel induced from their top level base
kernels. The following definitions formalize this construction.
Definition 5. Let G be a graph with vertex set V , and ? a positive semi-definite kernel on V . Assume
that for each v ? V we have a nested sequence of L neighborhoods
v ? N1 (v) ? N2 (v) ? . . . ? NL (v) ? V,
(9)
and for each N` (v), let G` (v) be the corresponding induced subgraph of G. We define the Multiscale
Laplacian Subgraph Kernels (MLS kernels), K1 , . . . , KL : V ? V ? R as follows:
?
1. K1 is just the FLG kernel kFLG
induced from the base kernel ? between the lowest level subgraphs:
?
K1 (v, v 0 ) = kFLG
(G1 (v), G1 (v 0 )).
5
2. For ` = 2, 3, . . . , L, K` is the FLG kernel induced from K`?1 between G` (v) and G` (v 0 ):
K
`?1
K` (v, v 0 ) = kFLG
(G` (v), G` (v 0 )).
Definition 5 defines the MLS kernel as a kernel between different subgraphs of the same graph G.
However, if two graphs G1 and G2 share the same base kernel, the MLS kernel can also be used to
compare any subgraph of G1 with any subgraph of G2 . This is what allows us to define an L + 1?th
FLG kernel, which compares the two full graphs.
Definition 6. Let G be a collection of graphs such that all their vertices are members of an abstract
vertex space V endowed with a symmetric positive semi-definite kernel ? : V ? V ? R. Assume that
the MLS kernels K1 , . . . , KL are defined as in Definition 5, both for pairs of subgraphs within the
same graph and across pairs of different graphs. We define the Multiscale Laplacian Graph Kernel
(MLG kernel) between any two graphs G1 , G2 ? G as
KL
K(G1 , G2 ) = kFLG
(G1 , G2 ).
Definition 5 leaves open the question of how the neighborhoods N1 (v), . . . , NL (v) are to be defined.
In the simplest case, we set N` (v) to be the ball Br (v) (i.e., the set of vertices at a distance at most
r from v), where r = r0 d`?1 for some d > 1.
3.1
Computational complexity
Definitions 5 and 6 suggest a recursive approach to computing
the MLG kernel: computing
2
K(G1 , G2 ) first requires computing KL (v, v 0 ) between all n1 +n
pairs of top level subgraphs across
2
G1 and G2 ; each of these kernel evaluations requires computing KL?1 (v, v 0 ) between up to O(n2 )
level L ? 1 subgraphs, and so on. Following this recursion blindly would require up to O(n2L+2 )
kernel evaluations, which is clearly infeasible.
The recursive strategy is wasteful because it involves evaluating the same kernel entries over and
over again in different parts of the recursion tree. An alternative solution that requires only O(Ln2 )
kernel evaluations would be to first compute K1 (v, v 0 ) for all (v, v 0 ) pairs, then compute K2 (v, v 0 )
for all (v, v 0 ) pairs and so on.
4
Linearized Kernels and Low Rank Approximation
Computing the MLG kernel between two graphs, as described in the previous section, may involve O(Ln2 ) kernel evaluations. At the top levels of the hierarchy each G` (v) might have ?(n)
vertices, so the cost of a single FLG kernel evaluation can be as high as O(n3 ). Somewhat pessimistically, this means that the overall cost of computing kFLG (G1 , G2 ) is O(Ln5 ). Given a dataset
of M graphs, computing their Gram matrix requires repeating this for all {G1 , G2 } pairs, giving
O(LM 2 n5 ), which is even more problematic. The solution that we propose in this section is to
compute for each level ` = 1, 2, . . . , L + 1 a single joint basis for all subgraphs at the given level
across all graphs G1 , . . . , GM . For concreteness, we go back to the definition of the FLG kernel.
Definition 7. Let G = {G1 , . . . , GM } be a collection of graphs, V1 , . . . , VM their vertex sets, and
assume that V1 , . . . , VM ? V for some general vertex space V. Further, assume that ? : V ? V ? R
is a positive semi-definite kernel on V, H? is its Reproducing Kernel Hilbert Space, and ? : V ? H?
is the corresponding feature map satisfying ?(v, v 0 ) = h?(v), ?(v 0 )i for any v, v 0 ? V. The joint
SM S
vertex feature space of {G1 , . . . , GM } is then WG = span
i=1 v?Vi {?(v)} .
WG is just the generalization of the W space defined in Proposition 1 from two graphs to M . The
following generalization of Propositions 1 and 2 is then immediate.
PM
Proposition 3. Let N = i=1 | Vi |, V = (v 1 , . . . , v N ) be the concatenation of the vertex sets
V1 , . . . , VM , and K the corresponding joint Gram matrix Ki,j = ?(v i , v j ) = h?(v i ), ?(v j )i . Let
u1 , . . . , uP be a maximal orthonormal set of non-zero eigenvalue eigenvectors of K with corresponding eigenvalues ?1 , . . . , ?P , and P = dim(WG ). Then the vectors
N
1 X
?
[ui ]` ?(v ` )
?i =
?i `=1
6
i = 1, . . . , P
1/2
1/2
form an orthonormal basis for WG . Moreover, defining Q = [?1 u1 , . . . , ?p uP ] ? RP ?P , and
setting Q1 to be the submatrix of Q composed of its first |V1 | rows; Q2 be the submatrix composed
of the next |V2 | rows, and so on, for any Gi , Gj ? G, the generalized FLG kernel induced from ?
(Definition 4) can be expressed as
1 ?1 1 ?1 ?1 1/2
Si + Sj
2
,
(10)
kFLG (Gi , Gj ) = 2
|S i |1/4 |S j |1/4
?1
> ?1
where S i = Q>
i Li Qi + ?I and S j = Qj Lj Qj + ?I.
The significance of Proposition 3 is that S1 , . . . , SM are now fixed matrices that do not need to be
recomputed for each kernel evaluation. Once we have constructed the joint basis {?1 , . . . , ?P }, the
Si matrix of each graph Gi can be computed independently, as a precomputation step, and individual
kernel evaluations reduce to just plugging them into (10). At a conceptual level, Proposition 3
linearizes the kernel ? by projecting everything down to WG . In particular, it replaces the {?(v i )}
RKHS vectors with explicit finite dimensional feature vectors given by the corresponding rows of
Q, just like we had in the ?unkernelized? FLG kernel of Definition 2.
?
For our multiscale kernels this is particularly important, because linearizing not just kFLG
, but also
K1
K2
kFLG , kFLG , . . ., allows us to compute the MLG kernel level by level, without recursion. After linearizing the base kernel ?, we attach explicit, finite dimensional vectors to each vertex of each graph.
K1
Then we compute compute kFLG
between all pairs of lowest level subgraphs, and linearizing this kernel as well, each vertex effectively just gets an updated feature vector. Then we repeat the process
KL
K2
for kFLG
. . . kFLG
, and finally we compute the MLG kernel K(G1 , G2 ).
4.1
Randomized low rank approximation
The difficulty in the above approach of course is that at each level (3) is a Gram matrix between all
vertices of all graphs, so storing it is already very costly, let along computing its eigendecomposition.
Morever, P = dim(WG ) is also very large, so managing the S 1 , . . . , S M matrices (each of which is
of size P?P ) becomes infeasible. The natural alternative is to replace WG by a smaller, approximate
joint features space, defined as follows.
? N
Definition 8. Let G, ?, H? and ? be defined as in Definition 7. Let V? = (?
v1 , . . . , v? ? ) be N
N
vertices sampled from the joint vertex set V = (v 1 , . . . , v N ). Then the corresponding subsampled
vertex feature space is
? G = span{ ?(?
W
v ) | v? ? V? }.
? G ). Similarly to before, we construct an orthonormal basis {?1 , . . . , ? ? } for W
?G
Let P? = dim(W
P
?
by forming the (now much smaller) Gram matrix Ki,j = ?(?
vi , v?j ), computing its eigenvalues and
PN?
1
?
v` ). The resulting approximate FLG kernel is
eigenvectors, and setting ?i = ?
`=1 [ui ]` ?(?
i
kFLG (Gi , Gj ) =
1 ??1
1 ??1 ?1 1/2
2 Si + 2 Sj
,
| S?i |1/4 | S?j |1/4
(11)
? > L?1 Q
? i + ?I and S?j = Q
? > L?1 Q
? j + ?I are the projections of S i and S j to W
? G.
where S?i = Q
i
j j
i
?
We introduce a further layer of approximation by restricting WG to be the space spanned by the
first P? < P? basis vectors (ordered by descending eigenvalue), effectively doing kernel PCA on
?
{?(?
v )}v??V? , equivalently, a low rank approximation of K.
?s
Assuming that vjg is the j?th vertex of Gg , in contrast to Proposition 2, now the j?th row of Q
g
?
consists of the coordinates of the projection of ?(vj ) onto WG , i.e.,
?
?
N
N
1 X
1 X
g
g
?
[Q ]j,i = ?
[ui ]` ?(vj ), ?(?
v` ) = ?
[ui ]` ?(vjg , v?` ).
?i `=1
?i `=1
The above procedure is similar to the popular Nystr?om approximation for kernel matrices [19, 20],
except that in our case the ultimate goal is not to approximate the Gram matrix (3), but the
7
Table 1: Classification Results (Mean Accuracy ? Standard Deviation)
Method
WL
WL-Edge
SP
Graphlet
p?RW
MLG
MUTAG[22]
84.50(?2.16)
82.94(?2.33)
85.50(?2.50)
82.44(?1.29)
80.33(?1.35)
84.21(?2.61)
PTC[23]
59.97(?1.60)
60.18(?2.19)
59.53(?1.71)
55.88(?0.31)
59.85(?0.95)
63.62(?4.69)
ENZYMES[2]
53.75(?1.37)
52.00(?0.72)
42.31(?1.37)
30.95(?0.73)
28.17(?0.76)
57.92(?5.39)
PROTEINS[2]
75.43(?1.95)
73.63(?2.12)
75.61(?0.45)
71.63(?0.33)
71.67(?0.78)
76.14(?1.95)
NCI1[24]
84.76(?0.32)
84.65(?0.25)
73.61(?0.36)
62.40(?0.27)
TIMED OUT
80.83(?1.29)
NCI109[24]
85.12(?0.29)
85.32(?0.34)
73.23(?0.26)
62.35(?0.28)
TIMED OUT
81.30(?0.80)
S1 , . . . , SM matrices used to form the FLG kernel. In practice, we found that the eigenvalues of
K usually drop off very rapidly, suggesting that W can be safely approximated by a surprisingly
? can be kept quite
small dimensional subspace (P? ? 10), and correspondingly the sample size N
small as well (on the order of 100). The combination of these two factors makes computing the
entire stack of kernels feasible, reducing the complexity of computing the Gram matrix for a dataset
? 2 P? 3 + M LN
? 3 + M 2 P? 3 ). It is also important to
of M graphs of ?(n) vertices each to O(M LN
note that this linearization step requires the graphs(not the labels) in the test set to be known during
? G.
training in order to project the features of the test graphs onto the low rank approximation of W
5
Experiments
We tested the efficacy of the MLG kernel by performing classification on benchmark bioinformatics
datasets using a binary C-SVM solver [21], and compared our classification results against those
from other representative graph kernels from the literature: the Weisfeiler?Lehman Kernel, the
Weisfeiler?Lehman Edge Kernel [9], the Shortest Path Kernel [6], the Graphlet Kernel [9], and
the p-random Walk Kernel [5].
We randomly selected 20% of each dataset to be used as a test set. On the other 80% we did 10 fold
cross validation to select the parameters for each kernel method to be used on the test set and repeated
this setup 10 times. For the Weisfeiler?Lehman kernels, the height parameter h is chosen from
{1, 2, ..., 5}, the random walk size p for the p-random walk kernel was chosen from {1, 2, ..., 5},
for the Graphlets kernel the graphlet size n was chosen from {3, 4, 5}. For the parameters of the
MLG kernel: we chose ? from {0.01, 0.1, 1}, radius size n from {1, 2, 3}, number of levels l from
{1, 2, 3}, and fixed gamma to be 0.01. For the MLG kernel, we used the given discrete node labels to
create a one-hot binary feature vector for each node and used the dot product between nodes? binary
feature vector labels as the base kernel. All experiments were done on a 16 core Intel E5-2670 @
2.6GHz processor with 32 GB of memory.
We are fairly competitive in accuracy for all datasets except NCI1, and NCI109, where it performs
better than all non-Weisfeiler Lehman kernels. The Supplementary Materials give a more detailed
discussion of the experiments and datasets.
6
Conclusions
In this paper we have proposed two new graph kernels: (1) The FLG kernel, which is a very simple
single level kernel that combines information attached to the vertices with the graph Laplacian; (2)
The MLG kernel, which is a multilevel, recursively defined kernel that captures topological relationships between not just individual vertices, but also subgraphs. Clearly, designing kernels that can
optimally take into account the multiscale structure of actual chemical compounds is a challenging
task that will require further work and domain knowledge. However, it is encouraging that even just
?straight out of the box?, tuning only two or three parameters, the MLG kernel is competitive with
other well known kernels in the literature. Beyond just graphs, the general idea of multiscale kernels
is of interest for other types of data as well (such as images) that have multiresolution structure, and
the way that the MLG kernel chains together local spectral analysis at multiple scales is potentially
applicable to these domains as well, which will be the subject of further research.
Acknowledgements
This work was completed in part with computing resources provided by the University of Chicago
Research Computing Center and with the support of DARPA-D16AP00112 and NSF-1320344.
8
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9
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5,676 | 6,136 | Learning Bound for Parameter Transfer Learning
Wataru Kumagai
Faculty of Engineering
Kanagawa University
kumagai@kanagawa-u.ac.jp
Abstract
We consider a transfer-learning problem by using the parameter transfer approach,
where a suitable parameter of feature mapping is learned through one task and applied to another objective task. Then, we introduce the notion of the local stability
and parameter transfer learnability of parametric feature mapping, and thereby
derive a learning bound for parameter transfer algorithms. As an application of
parameter transfer learning, we discuss the performance of sparse coding in selftaught learning. Although self-taught learning algorithms with plentiful unlabeled
data often show excellent empirical performance, their theoretical analysis has not
been studied. In this paper, we also provide the first theoretical learning bound for
self-taught learning.
1
Introduction
In traditional machine learning, it is assumed that data are identically drawn from a single distribution. However, this assumption does not always hold in real-world applications. Therefore, it would
be significant to develop methods capable of incorporating samples drawn from different distributions. In this case, transfer learning provides a general way to accommodate these situations. In
transfer learning, besides the availability of relatively few samples related with an objective task,
abundant samples in other domains that are not necessarily drawn from an identical distribution, are
available. Then, transfer learning aims at extracting some useful knowledge from data in other domains and applying the knowledge to improve the performance of the objective task. In accordance
with the kind of knowledge that is transferred, approaches to solving transfer-learning problems can
be classified into cases such as instance transfer, feature representation transfer, and parameter transfer (Pan and Yang (2010)). In this paper, we consider the parameter transfer approach, where some
kind of parametric model is supposed and the transferred knowledge is encoded into parameters.
Since the parameter transfer approach typically requires many samples to accurately learn a suitable
parameter, unsupervised methods are often utilized for the learning process. In particular, transfer learning from unlabeled data for predictive tasks is known as self-taught learning (Raina et al.
(2007)), where a joint generative model is not assumed to underlie unlabeled samples even though
the unlabeled samples should be indicative of a structure that would subsequently be helpful in predicting tasks. In recent years, self-taught learning has been intensively studied, encouraged by the
development of strong unsupervised methods. Furthermore, sparsity-based methods such as sparse
coding or sparse neural networks have often been used in empirical studies of self-taught learning.
Although many algorithms based on the parameter transfer approach have empirically demonstrated
impressive performance in self-taught learning, some fundamental problems remain. First, the theoretical aspects of the parameter transfer approach have not been studied, and in particular, no learning
bound was obtained. Second, although it is believed that a large amount of unlabeled data help to
improve the performance of the objective task in self-taught learning, it has not been sufficiently
clarified how many samples are required. Third, although sparsity-based methods are typically employed in self-taught learning, it is unknown how the sparsity works to guarantee the performance
of self-taught learning.
The aim of the research presented in this paper is to shed light on the above problems. We first
consider a general model of parametric feature mapping in the parameter transfer approach. Then,
we newly formulate the local stability of parametric feature mapping and the parameter transfer
learnability for this mapping, and provide a theoretical learning bound for parameter transfer learning algorithms based on the notions. Next, we consider the stability of sparse coding. Then we
discuss the parameter transfer learnability by dictionary learning under the sparse model. Applying
the learning bound for parameter transfer learning algorithms, we provide a learning bound of the
sparse coding algorithm in self-taught learning.
This paper is organized as follows. In the remainder of this section, we refer to some related studies.
In Section 2, we formulate the stability and the parameter transfer learnability of the parametric
feature mapping. Then, we present a learning bound for parameter transfer learning. In Section 3,
we show the stability of the sparse coding under perturbation of the dictionaries. Then, by imposing
sparsity assumptions on samples and by considering dictionary learning, we derive the parameter
transfer learnability for sparse coding. In particular, a learning bound is obtained for sparse coding
in the setting of self-taught learning. In Section 4, we conclude the paper.
1.1
Related Works
Approaches to transfer learning can be classified into some cases based on the kind of knowledge
being transferred (Pan and Yang (2010)). In this paper, we consider the parameter transfer approach.
This approach can be applied to various notable algorithms such as sparse coding, multiple kernel
learning, and deep learning since the dictionary, weights on kernels, and weights on the neural network are regarded as parameters, respectively. Then, those parameters are typically trained or tuned
on samples that are not necessarily drawn from a target region. In the parameter transfer setting, a
number of samples in the source region are often needed to accurately estimate the parameter to be
transferred. Thus, it is desirable to be able to use unlabeled samples in the source region.
Self-taught learning corresponds to the case where only unlabeled samples are given in the source
region while labeled samples are available in the target domain. In this sense, self-taught learning is
compatible with the parameter transfer approach. Actually, in Raina et al. (2007) where self-taught
learning was first introduced, the sparse coding-based method is employed and the parameter transfer approach is already used regarding the dictionary learnt from images as the parameter to be
transferred. Although self-taught learning has been studied in various contexts (Dai et al. (2008);
Lee et al. (2009); Wang et al. (2013); Zhu et al. (2013)), its theoretical aspects have not been sufficiently analyzed. One of the main results in this paper is to provide a first theoretical learning bound
in self-taught learning with the parameter transfer approach. We note that our setting differs from
the environment-based setting (Baxter (2000), Maurer (2009)), where a distribution on distributions
on labeled samples, known as an environment, is assumed. In our formulation, the existence of the
environment is not assumed and labeled data in the source region are not required.
Self-taught learning algorithms are often based on sparse coding. In the seminal paper by Raina et al.
(2007), they already proposed an algorithm that learns a dictionary in the source region and transfers it to the target region. They also showed the effectiveness of the sparse coding-based method.
Moreover, since remarkable progress has been made in unsupervised learning based on sparse neural
networks (Coates et al. (2011), Le (2013)), unlabeled samples of the source domain in self-taught
learning are often preprocessed by sparsity-based methods. Recently, a sparse coding-based generalization bound was studied (Mehta and Gray (2013); Maurer et al. (2012)) and the analysis in
Section 3.1 is based on (Mehta and Gray (2013)).
2
2.1
Learning Bound for Parameter Transfer Learning
Problem Setting of Parameter Transfer Learning
We formulate parameter transfer learning in this subsection. We first briefly introduce notations and
terminology in transfer learning (Pan and Yang (2010)). Let X and Y be a sample space and a label
space, respectively. We refer to a pair of Z := X ? Y and a joint distribution P (x, y) on Z as a
region. Then, a domain comprises a pair consisting of a sample space X and a marginal probability
of P (x) on X and a task consists of a pair containing a label set Y and a conditional distribution
P (y|x). In addition, let H = {h : X ? Y} be a hypothesis space and ? : Y ? Y ? R?0
2
represent a loss function. Then, the expected risk and the empirical risk are defined by R(h) :=
b n (h) := 1 ?n ?(yj , h(xj )), respectively. In the setting of transfer
E(x,y)?P [?(y, h(x))] and R
j=1
n
learning, besides samples from a region of interest known as a target region, it is assumed that
samples from another region known as a source region are also available. We distinguish between
the target and source regions by adding a subscript T or S to each notation introduced above, (e.g.
PT , RS ). Then, the homogeneous setting (i.e., XS = XT ) is not assumed in general, and thus, the
heterogeneous setting (i.e., XS ?= XT ) can be treated. We note that self-taught learning, which is
treated in Section 3, corresponds to the case when the label space YS in the source region is the set
of a single element.
We consider the parameter transfer approach, where the knowledge to be transferred is encoded into
a parameter. The parameter transfer approach aims to learn a hypothesis with low expected risk for
the target task by obtaining some knowledge about an effective parameter in the source region and
transfer it to the target region. In this paper, we suppose that there are parametric models on both
the source and target regions and that their parameter spaces are partly shared. Then, our strategy
is to learn an effective parameter in the source region and then transfer a part of the parameter to
the target region. We describe the formulation in the following. In the target region, we assume that
YT ? R and there is a parametric feature mapping ?? : XT ? Rm on the target domain such that
each hypothesis hT ,?,w : XT ? YT is represented by
hT ,?,w (x) := ?w, ?? (x)?
(1)
with parameters ? ? ? and w ? WT , where ? is a subset of a normed space with a norm ? ? ? and
WT is a subset of Rm . Then the hypothesis set in the target region is parameterized as
HT = {hT ,?,w |? ? ?, w ? WT }.
b T (hT ,?,w ) by RT (?, w) and R
b T (?, w),
In the following, we simply denote RT (hT ,?,w ) and R
respectively. In the source region, we suppose that there exists some kind of parametric model such
as a sample distribution PS,?,w or a hypothesis hS,?,w with parameters ? ? ? and w ? WS , and
a part ? of the parameter space is shared with the target region. Then, let ? ?S ? ? and wS? ? WS
be parameters that are supposed to be effective in the source region (e.g., the true parameter of the
sample distribution, the parameter of the optimal hypothesis with respect to the expected risk RS );
however, explicit assumptions are not imposed on the parameters. Then, the parameter transfer
algorithm treated in this paper is described as follows. Let N - and n-samples be available in the
source and target regions, respectively. First, a parameter transfer algorithm outputs the estimator
bN ? ? of ? ? by using N -samples. Next, for the parameter
?
S
wT?
:= argmin RT (? ?S , w)
w?WT
in the target region, the algorithm outputs its estimator
b N,n
w
bN , w) + ?r(w)
bT ,n (?
:= argmin R
w?WT
by using n-samples, where r(w) is a 1-strongly convex function with respect to ? ? ?2 and ? > 0.
If the source region relates to the target region in some sense, the effective parameter ? ?S in the
source region is expected to also be useful for the target task. In the next subsection, we regard
RT (? ?S , wT? ) as the baseline of predictive performance and derive a learning bound.
2.2
Learning Bound Based on Stability and Learnability
We newly introduce the local stability and the parameter transfer learnability as below. These notions
are essential to derive a learning bound in Theorem 1.
Definition 1 (Local Stability). A parametric feature mapping ?? is said to be locally stable if there
exist ?? : X ? R>0 for each ? ? ? and L? > 0 such that for ? ? ? ?
?? ? ? ? ? ? ?? (x) ? ??? (x) ? ??? (x)?2 ? L? ?? ? ? ? ?.
We term ?? (x) the permissible radius of perturbation for ? at x. For samples Xn = {x1 , . . . xn },
we denote as ?? (Xn ) := minj?[n] ?? (xj ), where [n] := {1, . . . , n} for a positive integer n. Next,
we formulate the parameter transfer learnability based on the local stability.
3
Definition 2 (Parameter Transfer Learnability). Suppose that N -samples in the source domain and
n-samples Xn in the target domain are available. Let a parametric feature mapping {?? }??? be
locally stable. For ?? ? [0, 1), {?? }??? is said to be parameter transfer learnable with probability
1 ? ?? if there exists an algorithm that depends only on N -samples in the source domain such that,
bN of the algorithm satisfies
the output ?
[
]
bN ? ? ? ? ? ??? (Xn ) ? 1 ? ?.
?
Pr ??
S
S
In the following, we assume that parametric feature mapping is bounded as ??? (x)?2 ? R? for
arbitrary x ? X and ? ? ? and linear predictors are also bounded as ?w?2 ? RW for any w ? W.
In addition, we suppose that a loss function ?(?, ?) is L? -Lipschitz and convex with respect to the
second variable. We denote as Rr := supw?W |r(w)|. Then, the following learning bound is
obtained, where the strong convexity of the regularization term ?r(w) is essential.
Theorem 1 (Learning Bound). Suppose that the parametric feature mapping ?? is locally stable
bN learned in the source region satisfies the parameter transfer learnability with
and an estimator ?
?
? When ? = L? R? 8(32+log(2/?)) , the following inequality holds with probability
probability 1 ? ?.
Rr n
?
1 ? (? + 2?):
(
)
bN , w
b N,n ? RT (? ?S , wT? )
RT ?
(
) 1
?
?
b
?
? L? R? RW 2 log(2/?) + 2 2Rr (32 + log(2/?)) ? + L? L? R?
?
N ? ?S
n
?
(
) 14
?
1
Rr
b
?
n 4
?
+L? L? RW R?
(2)
N ? ? S
.
2(32 + log(2/?))
bN ? ? ? ? can be evaluated in terms of the number N of samples, Theorem
If the estimation error ??
S
1 clarifies which term is dominant, and in particular, the number of samples required in the source
domain such that this number is sufficiently large compared to the samples in the target domain.
2.3
Proof of Learning Bound
We prove Theorem 1 in this subsection. In this proof, we omit the subscript T for simplicity. In
addition, we denote ? ?S simply by ? ? . We set as
1?
?(yj , ?w, ??? (xj )?) + ?r(w).
n j=1
n
b n?
w
:= argmin
w?W
Then, we have
(
)
bN , w
b N,n ? RT (? ? , w? )
RT ?
[
]
b N,n , ??bN (x)?) ? E(x,y)?P [?(y, ?w
b N,n , ??? (x)?)]
= E(x,y)?P ?(y, ?w
b N,n , ??? (x)?)] ? E(x,y)?P [?(y, ?w
b n? , ??? (x)?)]
+E(x,y)?P [?(y, ?w
b n? , ??? (x)?)] ? E(x,y)?P [?(y, ?w? , ??? (x)?)] .
+E(x,y)?P [?(y, ?w
(3)
In the following, we bound three parts of (3). First, we have the following inequality with probability
?
1 ? (?/2 + ?):
[
]
b N,n , ??bN (x)?) ? E(x,y)?P [?(y, ?w
b N,n , ??? (x)?)]
E(x,y)?P ?(y, ?w
]
[
? L? RW E(x,y)?P
??bN (x) ? ??? (x)
?
n
1 ?
2 log(2/?)
? L? RW
??bN (xj ) ? ??? (xj )
+ L? RW R?
n j=1
n
?
2 log(2/?)
b
?
? L? L? RW
?
,
N ? ?
+ L? RW R?
n
4
where we used Hoeffding?s inequality as the third inequality, and the local stability and parameter
transfer learnability in the last inequality. Second, we have the following inequality with probability
?
1 ? ?:
?
?
?
b N,n , ??? (x)?)] ? E(x,y)?P [?(y, ?w
b n? , ??? (x)?)]
E(x,y)?P [?(y, ?w
b N,n , ??? (x)? ? ?w
b n? , ??? (x)?|]
L? E(x,y)?P [|?w
b N,n ? w
b n? ?2
L? R? ?w
?
2L? L? RW
b
L? R?
? N ? ? ?
,
?
(4)
where the last inequality is derived by the strong convexity of the regularizer ?r(w) in the Appendix.
Third, the following holds by Theorem 1 of Sridharan et al. (2009) with probability 1 ? ?/2:
b n? , ??? (x)?)] ? E(x,y)?P [?(y, ?w? , ??? (x)?)]
E(x,y)?P [?(y, ?w
b n? )]
b n? , ??? (x)?) + ?r(w
= E(x,y)?P [?(y, ?w
b n? ))
?E(x,y)?P [?(y, ?w? , ??? (x)?) + ?r(w? )] + ?(r(w? ) ? r(w
(
)
2
8L2? R?
(32 + log(2/?))
?
+ ?Rr .
?n
?
Thus, when ? = L? R?
3
8(32+log(2/?))
,
Rr n
?
we have (2) with probability 1 ? (? + 2?).
Stability and Learnability in Sparse Coding
In this section, we consider the sparse coding in self-taught learning, where the source region essentially consists of the sample space XS without the label space YS . We assume that the sample
spaces in both regions are Rd . Then, the sparse coding method treated here consists of a two-stage
procedure, where a dictionary is learnt on the source region, and then a sparse coding with the learnt
dictionary is used for a predictive task in the target region.
First, we show that sparse coding satisfies the local stability in Section 3.1 and next explain that
appropriate dictionary learning algorithms satisfy the parameter transfer learnability in Section 3.4.
As a consequence of Theorem 1, we obtain the learning bound of self-taught learning algorithms
based on sparse coding. We note that the results in this section are useful independent of transfer
learning.
We here summarize the notations used in this section. Let ? ? ?p be the p-norm on Rd . We define as
supp(a) := {i ? [m]|ai ?= 0} for a ? Rm . We denote the number of elements of a set S by |S|.
When a vector a satisfies ?a?0 = |supp(a)| ? k, a is said to be k-sparse. We denote the ball with
radius R centered at 0 by BRd (R) := {x ? Rd |?x?2 ? R}. We set as D := {D = [d1 , . . . , dm ] ?
BRd (1)m | ?dj ?2 = 1 (i = 1, . . . , m)} and each D ? D a dictionary with size m.
Definition 3 (Induced matrix norm). For an arbitrary matrix E = [e1 , . . . , em ] ? Rd?m ,
induced matrix norm is defined by ?E?1,2 := maxi?[m] ?ei ?2 .
1)
the
We adopt ? ? ?1,2 to measure the difference of dictionaries since it is typically used in the framework
? 1,2 ? 2 holds for arbitrary dictionaries D, D
? ? D.
of dictionary learning. We note that ?D ? D?
3.1
Local Stability of Sparse Representation
We show the local stability of sparse representation under a sparse model. A sparse representation
with dictionary parameter D of a sample x ? Rd is expressed as follows:
1
?D (x) := argmin ?x ? Dz?22 + ??z?1 ,
z?Rm 2
In general, the (p, q)-induced norm for p, q ? 1 is defined by ?E?p,q := supv?Rm ,?v?p =1 ?Ev?q . Then,
? ? ?1,2 in this general definition coincides with that in Definition 3 by Lemma 17 of Vainsencher et al. (2011).
1)
5
where ? > 0 is a regularization parameter. This situation corresponds to the case where ? = D
and ?? = ?D in the setting of Section 2.1. We prepare some notions to the stability of the sparse
representation. The following margin and incoherence were introduced by Mehta and Gray (2013).
Definition 4 (k-margin). Given a dictionary D = [d1 , . . . , dm ] ? D and a point x ? Rd , the
k-margin of D on x is
Mk (D, x) :=
max
min {? ? |?dj , x ? D?D (x)?|} .
I?[m],|I|=m?k j?I
Definition 5 (?-incoherence).
A dictionary matrix D = [d1 , . . . , dm ] ? D is termed ?-incoherent
?
if |?di , dj ?| ? ?/ d for all i ?= j.
Then, the following theorem is obtained.
? 1,2 ? ?. When
Theorem 2 (Sparse Coding Stability). Let D ? D be ?-incoherent and ?D ? D?
? 1,2 ? ?k,D (x) :=
?D ? D?
Mk,D (x)2 ?
,
64 max{1, ?x?}4
(5)
the following stability bound holds:
?
4?x?2 k
? 1,2 .
?
?D ? D?
??D (x) ? ?D
? (x)?2 ?
(1 ? ?k/ d)?
From Theorem 2, ?k,D (x) becomes the permissible radius of perturbation in Definition 1.
Here, we refer to the relation with the sparse coding stability (Theorem 4) of Mehta and Gray (2013),
who measured the difference of dictionaries by ? ? ?2,2 instead of ? ? ?1,2 and the permissible radius
of perturbation is given by Mk,D (x)2 ? except for a constant factor. Applying the simple inequality
?
?E?2,2 ? m?E?1,2 for E ? Rd?m , we can obtain a variant of the sparse coding stability with the
norm ? ? ?1,2 . However, then the dictionary size m affects the permissible radius of perturbation and
the stability bound of the sparse coding stability. On the other hand, the factor of m does not appear
in Theorem 2, and thus, the result is effective even for a large m. In addition, whereas ?x? ? 1
is assumed in Mehta and Gray (2013), Theorem 2 does not assume that ?x? ? 1 and clarifies the
dependency for the norm ?x?.
In existing studies related to sparse coding, the sparse representation ?D (x) is modified as ?D (x) ?
x (Mairal et al. (2009)) or ?D (x) ? (x ? D?D (x)) (Raina et al. (2007)) where ? is the tensor
product. By the stability of sparse representation (Theorem 2), it can be shown that such modified
representations also have local stability.
3.2
Sparse Modeling and Margin Bound
In this subsection, we assume a sparse structure for samples x ? Rd and specify a lower bound
for the k-margin used in (5). The result obtained in this section plays an essential role to show the
parameter transfer learnability in Section 3.4.
Assumption 1 (Model). There exists a dictionary matrix D? such that every sample x is independently generated by a representation a and noise ? as
x = D? a + ?.
Moreover, we impose the following three assumptions on the above model.
Assumption 2 (Dictionary). The dictionary matrix D? = [d1 , . . . , dm ] ? D is ?-incoherent.
Assumption 3 (Representation). The representation a is a random variable that is k-sparse (i.e.,
?a?0 ? k) and the non-zero entries are lower bounded by C > 0 (i.e., ai ?= 0 satisfy |ai | ? C).
Assumption
? 4 (Noise). The noise ? is independent across coordinates and sub-Gaussian with parameter ?/ d on each component.
We note that the assumptions do not require the representation a or noise ? to be identically distributed while those components are independent. This is essential because samples in the source
and target domains cannot be assumed to be identically distributed in transfer learning.
6
Theorem 3 (Margin Bound). Let 0 < t < 1. We set as
(
)
(
)
2?
(1 ? t)2 d?2
2?m
d?2
? exp ?
?t,? :=
+ ? exp ? 2
8? 2
8?
(1 ? t) d?
d?
(
? )
)
(
2
4?k
C d(1 ? ?k/ d)
8?(d ? k)
d?2
+ ?
+ ?
exp ?
. (6)
? exp ?
8? 2
32? 2
d?
C d(1 ? ?k/ d)
{(
) }2
6
We suppose that d ?
1 + (1?t)
?k and ? = d?? for arbitrary 1/4 ? ? ? 1/2. Under
Assumptions 1-4, the following inequality holds with probability 1 ? ?t,? at least:
Mk,D? (x) ? t?.
(7)
We refer to the regularization parameter ?. An appropriate reflection of the sparsity of samples
requires the regularization parameter ? to be set suitably. According to Theorem 4 of Zhao and Yu
(2006)2) , when samples follow the sparse model as in Assumptions 1-4 and ? ?
= d?? for 1/4 ? ? ?
1/2, the representation ?D (x) reconstructs the true sparse representation a of sample x with a small
error.
In particular, when ? = 1/4 (i.e., ? ?
= d?1/4 ) in Theorem 3, the failure probability ?t,? ?
=
?
? d
e
on the margin is guaranteed to become sub-exponentially small with respect to dimension d
and is negligible for the high-dimensional case. On the other hand, the typical choice ? = 1/2 (i.e.,
??
= d?1/2 ) does not provide a useful result because ?t,? is not small at all.
3.3
Proof of Margin Bound
We give a sketch of proof of Theorem 3. We denote the first term, the second term and the sum of
the third and fourth terms of (6) by ?1 , ?2 and ?3 , respectively From Assumptions 1 and 3, a sample
is represented as x = D? a + ? and ?a?0 ? k. Without loss of generality, we assume that the first
m ? k components of a are 0 and the last k components are not 0. Since
Mk,D? (x) ?
min
1?j?m?k
? ? ?dj , x ? D? ?D (x)? =
min
1?j?m?k
? ? ?dj , ?? ? ?D?? dj , a ? ?D (x)?,
it is enough to show that the following holds an arbitrary 1 ? j ? m ? k to prove Theorem 3:
Pr[?dj , ?? + ?D?? dj , a ? ?D (x)? > (1 ? t)?] ? ?t,? .
Then, (8) follows from the following inequalities:
]
[
1?t
?
? ?1 ,
Pr ?dj , ?? >
2
[
]
1?t
Pr ?D?? dj , a ? ?D (x)? >
?
? ?2 + ?3 .
2
(8)
(9)
(10)
The inequality (9) holds since ?dj ? = 1 by the definition and Assumption 4. Thus, all we have to
do is to show (10). We have
?D?? dj , a ? ?D (x)? = ?[?d1 , dj ?, . . . , ?dm , dj ?]? , a ? ?D (x)?
= ?(1supp(a??D (x)) ? [?d1 , dj ?, . . . , ?dm , dj ?])? , a ? ?D (x)?
?
?1supp(a??D (x)) ? [?d1 , dj ?, . . . , ?dm , dj ?]?2 ?a ? ?D (x)?2 ,(11)
where u ? v is the Hadamard product (i.e. component-wise product) between u and v, and 1A for a
set A ? [m] is a vector whose i-th component is 1 if i ? A and 0 otherwise.
Applying Theorem 4 of Zhao and Yu (2006) and using the condition for ?, the following holds with
probability 1 ? ?3 :
supp(a) = supp(?D (x)).
2)
(12)
Theorem 4 of Zhao and Yu (2006) is stated for Gaussian noise. However, it can be easily generalized to
sub-Gaussian noise as in Assumption 4. Our setting corresponds to the case in which c1 = 1/2, c2 = 1, c3 =
c3
(log ? + log log d)/ log d for some ? > 1 (i.e., ed ?
= d? ) and c4 = c in Theorem 4 of Zhao and Yu (2006).
Note that our regularization parameter ? corresponds to ?d /d in (Zhao and Yu (2006)).
7
Moreover, under (12), the following holds with probability 1 ? ?2 by modifying Corollary 1 of
Negahban et al. (2009) and using the condition for ?:
?
6 k?
?a ? ?D (x)?2 ?
.
(13)
?k
1? ?
d
Thus, if both of (12) and (13) hold, the right hand side of (11) is bounded as follows:
?1supp(a??D (x)) ? [?d1 , dj ?, . . . , ?dm , dj ?]?2 ?a ? ?D (x)?2
?
?
? 6 k?
1?t
6?k
?
|supp(a ? ?D (x))| ?
? ?
?,
=?
?k
2
?
d1? d
d ? ?k
where we used Assumption 2 in the first inequality, (12) and Assumption 3 in the equality and the
condition for d in the last inequality. From the above discussion, the left hand side of (10) is bounded
by the sum of the probability ?3 that (12) does not hold and the probability ?2 that (12) holds but
(13) does not hold.
3.4
Transfer Learnability for Dictionary Learning
b N of a suitable
When the true dictionary D? exists as in Assumption 1, we show that the output D
dictionary learning algorithm from N -unlabeled samples satisfies the parameter transfer learnability
for the sparse coding ?D . Then, Theorem 1 guarantees the learning bound in self-taught learning
since the discussion in this section does not assume the label space in the source region. This
bN = D
b N and ? ? ? = ? ? ?1,2 in Section 2.1.
situation corresponds to the case where ? ?S = D? , ?
We show that an appropriate dictionary learning algorithm satisfies the parameter transfer learnability for the sparse coding ?D by focusing on the permissible radius of perturbation in (5) under some
assumptions. When Assumptions 1-4 hold and ? = d?? for 1/4 ? ? ? 1/2, the margin bound (7)
for x ? X holds with probability 1 ? ?t,? , and thus, we have
t2 ?3
= ?(d?3? ).
64 max{1, ?x?}4
b N such that
Thus, if a dictionary learning algorithm outputs the estimator D
?k,D? (x) ?
b N ? D? ?1,2 ? O(d?3? )
?D
(14)
?
b N of D satisfies the parameter transfer learnability for the
with probability 1 ? ?N , the estimator D
sparse coding ?D with probability ?? = ?N + n?t,? . Then, by the local stability of the sparse representation and the parameter transfer learnability of such a dictionary learning, Theorem 1 guarantees
that sparse coding in self-taught learning satisfies the learning bound in (2).
We note that Theorem 1 can apply to any dictionary learning
? algorithm as long as (14) is satisfied.
For example, Arora et al. (2015) show that, when k = O( d/ log d), m = O(d), Assumptions 1-4
b N which
and some additional conditions are assumed, their dictionary learning algorithm outputs D
satisfies
b N ? D? ?1,2 = O(d?M )
?D
?
with probability 1 ? d?M for arbitrarily large M, M ? as long as N is sufficiently large.
4
Conclusion
We derived a learning bound (Theorem 1) for a parameter transfer learning problem based on the
local stability and parameter transfer learnability, which are newly introduced in this paper. Then,
applying it to a sparse coding-based algorithm under a sparse model (Assumptions 1-4), we obtained
the first theoretical guarantee of a learning bound in self-taught learning. Although we only consider
sparse coding, the framework of parameter transfer learning includes other promising algorithms
such as multiple kernel learning and deep neural networks, and thus, our results are expected to
be effective to analyze the theoretical performance of these algorithms. Finally, we note that our
learning bound can be applied to different settings from self-taught learning because Theorem 1
includes the case in which labeled samples are available in the source region.
8
References
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9
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5,677 | 6,137 | Combinatorial semi-bandit with known covariance
R?my Degenne
LMPA, Universit? Paris Diderot
CMLA, ENS Paris-Saclay
degenne@cmla.ens-cachan.fr
Vianney Perchet
CMLA, ENS Paris-Saclay
CRITEO Research, Paris
perchet@normalesup.org
Abstract
The combinatorial stochastic semi-bandit problem is an extension of the classical
multi-armed bandit problem in which an algorithm pulls more than one arm at
each stage and the rewards of all pulled arms are revealed. One difference with the
single arm variant is that the dependency structure of the arms is crucial. Previous
works on this setting either used a worst-case approach or imposed independence
of the arms. We introduce a way to quantify the dependency structure of the
problem and design an algorithm that adapts to it. The algorithm is based on linear
regression and the analysis develops techniques from the linear bandit literature.
By comparing its performance to a new lower bound, we prove that it is optimal,
up to a poly-logarithmic factor in the number of pulled arms.
1
Introduction and setting
The multi-armed bandit problem (MAB) is a sequential learning task in which an algorithm takes at
each stage a decision (or, ?pulls an arm?). It then gets a reward from this choice, with the goal of
maximizing the cumulative reward [Robbins, 1985]. We consider here its stochastic combinatorial
extension, in which the algorithm chooses at each stage a subset of arms [Audibert et al., 2013,
Cesa-Bianchi and Lugosi, 2012, Chen et al., 2013, Gai et al., 2012]. These arms could form, for
example, the path from an origin to a destination in a network. In the combinatorial setting, contrary
to the the classical MAB, the inter-dependencies between the arms can play a role (we consider that
the distribution of rewards is invariant with time). We investigate here how the covariance structure
of the arms affects the difficulty of the learning task and whether it is possible to design a unique
algorithm capable of performing optimally in all cases from the simple scenario with independent
rewards to the more challenging scenario of general correlated rewards.
Formally, at each stage t ? N, t ? 1, an algorithm pulls m ? 1 arms among d ? m. Such a set of m
arms is called an ?action? and will be denoted by At ? {0, 1}d , a vector with exactly m non-zero
entries. The possible actions are restricted to an arbitrary fixed subset A ? {0, 1}d . After choosing
d
action At , the algorithm receives the reward A>
t Xt , where Xt ? R is the vector encapsulating the
reward of the d arms at stage t. The successive reward vectors (Xt )t?1 are i.i.d with unknown mean
? ? Rd . We consider a semi-bandit feedback system: after choosing the action At , the algorithm
observes the reward of each of the arms in that action, but not the other rewards. Other possible
feedbacks previously studied include bandit (only A>
t Xt is revealed) and full information (Xt is
revealed). The goal of the algorithm is to maximize the cumulated reward up to stage T ? 1 or
equivalently to minimize the expected regret, which is the difference of the reward that would have
been gained by choosing the best action in hindsight A? and what was actually gained:
ERT = E
T
X
(A?> ? ? A>
t ?) .
t=1
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
For an action A ? A, the difference ?A = (A?> ? ? A> ?) is called gap of A. We denote by ?t the
PT
gap of At , so that regret rewrites as ERT = E t=1 ?t . We also define the minimal gap of an arm,
?i,min = min{A?A:i?A} ?A .
This setting was already studied Cesa-Bianchi and Lugosi [2012], most recently in Combes et al.
[2015], Kveton et al. [2015], where two different algorithms are used to tackle on one hand the case
where the arms have independent rewards and on the other hand the general bounded case. The regret
guaranties of the two algorithms are different and reflect that the independent case is easier. Another
algorithm for the independent arms case based on Thompson Sampling was introduced in Komiyama
et al. [2015]. One of the main objectives of this paper is to design a unique algorithm that can adapt
to the covariance structure of the problem when prior information is available.
The following notations will be used throughout the paper: given a matrix M (resp. vector v), its
(i, j)th (resp. ith ) coefficient is denoted by M (ij) (resp. v (i) ). For a matrix M , the diagonal matrix
with same diagonal as M is denoted by ?M .
We denote by ?t the noise in the reward, i.e. ?t := Xt ? ?. We consider a subgaussian setting, in
which we suppose that there is a positive semi-definite matrix C such that for all t ? 1,
?u ? Rd , E[eu
>
?t
1
>
] ? e2u
Cu
.
This is equivalent to the usual setting for bandits where we suppose
? that the individual arms are
(i)
subgaussian. Indeed if we have such a matrix C then each ?t is C (ii) -subgaussian. And under
a subgaussian arms assumption, such a matrix always exists. This setting encompasses the case of
bounded rewards.
We call C a subgaussian covariance matrix of the noise (see appendix A of the supplementary
material). A good knowledge of C can simplify the problem greatly, as we will show. In the case
of 1-subgaussian independent rewards, in which C can be chosen diagonal, a known lower bound
d
on the regret appearing in Combes et al. [2015] is ?
log T , while Kveton et al. [2015] proves a
dm
log
T
lower
bound
in
general.
Our
goal
here
is
to
investigate
the spectrum of intermediate cases
?
between these two settings, from the uninformed general case to the independent case in which one
has much information on the relations between the arm rewards. We characterize the difficulty of the
problem as a function of the subgaussian covariance matrix C. We suppose that we know a positive
semi-definite matrix ? such that for all vectors v with positive coordinates, v > Cv ? v > ?v, property
that we denote by C + ?. ? reflects the prior information available about the possible degree of
independence of the arms. We will study algorithms that enjoy regret bounds as functions of ?.
The matrix
? ? can be chosen such that all its coefficients are non-negative and verify for all i, j,
?(ij) ? ?(ii) ?(jj) . From now on, we suppose that it is the case. In the following, we will use t
such that ?t = C 1/2 t and write for the reward: Xt = ? + C 1/2 t .
2
Lower bound
We first prove a lower bound on the regret of any algorithm, demonstrating the link between the subgaussian covariance matrix and the difficulty of the problem. It depends on the maximal off-diagonal
(ij)
correlation coefficient of the covariance matrix. This coefficient is ? = max{(i,j)?[d],i6=j} ? C
.
C (ii) C (jj)
The bound is valid for consistent algorithms [Lai and Robbins, 1985], for which the regret on any
problem verifies ERt = o(ta ) as t ? +? for all a > 0.
Theorem 1. Suppose to simplify that d is a multiple of m. Then, for any ? > 0, for any consistent
algorithm, there is a problem with gaps ?, ?-subgaussian arms and correlation coefficients smaller
than ? ? [0, 1] on which the regret is such that
lim inf
t?+?
ERt
2? 2 (d ? m)
? (1 + ?(m ? 1))
log t
?
This bound is a consequence of the classical result of Lai and Robbins [1985] for multi-armed bandits,
applied to the problem of choosing one among d/m paths, each of which has m different successive
edges (Figure 1). The rewards in the same path are correlated but the paths are independent. A
complete proof can be found in appendix B.1 of the supplementary material.
2
Figure 1: Left: parallel paths problem. Right: regret of OLS-UCB as a function of m and ? in the
parallel paths problem with 5 paths (average over 1000 runs).
3
OLS-UCB Algorithm and analysis
Faced with the combinatorial semi-bandit at stage t ? 1, the observations from t ? 1 stages form as
many linear equations and the goal of an algorithm is to choose the best action. To find the action with
the highest mean, we estimate the mean of all arms. This can be viewed as a regression problem. The
design of our algorithm stems from this observation and is inspired by linear regression in the fixed
design setting, similarly to what was done in the stochastic linear bandit literature [Rusmevichientong
and Tsitsiklis, 2010, Filippi et al., 2010]. There are many estimators for linear regression and we
focus on the one that is simple enough and adaptive: Ordinary Least Squares (OLS).
3.1
Fixed design linear regression and OLS-UCB algorithm
For an action A ? A, let IA be the diagonal matrix with a 1 at line i if A(i) = 1 and 0 otherwise. For
a matrix M , we also denote by MA the matrix IA M IA . At stage t, if all actions A1 , . . . , At were
independent of the rewards, we would have observed a set of linear equations
IA1 X1 = IA1 ? + IA1 ?1
..
.
IAt?1 Xt?1 = IAt?1 ? + IAt?1 ?t?1
and we could use the OLS estimator to estimate ?, which is unbiased and has a known subgaussian
constant controlling its variance. This is however not true in our online setting since the successive
actions are not independent. At stage t, we define
(i)
nt =
t?1
X
(ij)
I{i ? As }, nt
s=1
=
t?1
X
I{i ? As }I{j ? As } and Dt =
s=1
t?1
X
IAs ,
s=1
(i)
where nt is the number of times arm i has been pulled before stage t and Dt is a diagonal matrix of
these numbers. The OLS estimator is, for an arm i ? [d],
(i)
?
?t =
1
X
(i)
nt s<t:i?As
Xs(i) = ?(i) + (Dt?1
t?1
X
IAs C
1/2
s )(i) .
s=1
Then for all A ? A, A> (?
?t ? ?) in the fixed design setting has a subgaussian matrix equal to
Pt?1
Dt?1 ( s=1 CAs )Dt?1 . We get confidence intervals for the estimates and can use an upper confidence
bound strategy [Lai and Robbins, 1985, Auer et al., 2002]. In the online learning setting the actions
are not independent but we will show that using this estimator still leads to estimates that are
well concentrated around ?, with confidence intervals given by the same subgaussian matrix. The
algorithm OLS-UCB (Algorithm 1) results from an application of an upper confidence bound strategy
with this estimator.
We now turn to an analysis of the regret of OLS-UCB. At any stage t ? 1 of the algorithm, let
?(ij)
?t = max{(i,j)?At ,i6=j} ? (ii)
be the maximal off-diagonal correlation coefficient of ?At and
?
?(jj)
let ? = max{t?[T ]} ?t be the maximum up to stage T .
3
Algorithm 1 OLS-UCB.
Require: Positive semi-definite matrix ?, real parameter ? > 0.
1: Choose actions such that each arm is pulled at least one time.
2: loop: at stage t,
3:
At = arg maxA A> ?
?t +qEt (A)
p
Pt?1
with Et (A) = 2f (t) A> Dt?1 (??? Dt + s=1 ?As )Dt?1 A.
4:
Choose action At , observe IAt Xt .
5:
Update ?
?t , Dt .
6: end loop
Theorem 2. The OLS-UCB algorithm with parameter ? > 0 and f (t) = log t + (m + 2) log log t +
m
e
2 log(1 + ? ) enjoys for all times T ? 1 the regret bound
!
2
X ?(ii)
log m
E[RT ] ?16f (T )
5(? + 1 ? ?)
+ 45?m
?i,min
1.6
i?[d]
8dm2 maxi {C (ii) }?max
+
+ 4?max ,
?2min
where dxe stands for the smallest positive integer bigger than or equal to x. In particular, d0e = 1.
This bound shows the transition between a general case with a
2
dm log T
?
regime and an independent
d log m log T
?
T
case with a
upper bound (we recall that the lower bound is of the order of d log
? ). The
weight of each case is given by the maximum correlation parameter ?. The parameter ? seems to be
an artefact of the analysis and can in practice be taken very small or even equal to 0.
Figure 1 illustrates the regret of OLS-UCB on the parallel paths problem used to derive the lower
bound. It shows a linear dependency in ? and supports the hypothesis that the true upper bound
matches the lower bound with a dependency in m and ? of the form (1 + ?(m ? 1)).
Corollary 1. The OLS-UCB algorithm with matrix ? and parameter ? > 0 has a regret bounded as
v
!
u
2
u
log m
t
(ii)
+ 45?m ) .
E[RT ] ? O( dT log T max{? } 5(? + 1 ? ?)
1.6
i?[d]
Proof. We write that the regret up to stage T is bounded by ?T for actions with gap smaller than
some ? and bounded using theorem 2 for other actions (with ?min ? ?). Maximizing over ? then
gives the result.
3.2
Comparison with other algorithms
Previous works supposed that the rewards of the individual arms are in [0, 1], which gives them a
property. Hence we suppose (?i ? [d], C (ii) = 1/2) for our comparison.
1/2-subgaussian
In the independent case, our algorithm is the same as ESCB-2
from Combes et al. [2015], up to the
?
d m log T
parameter ?. That paper shows that ESCB-2 enjoys an O(
) upper bound but our analysis
?
tighten it to O( d log
2
m log T
?
).
log T
In the general (worst) case, Kveton et al. [2015] prove an O( dm ?
) upper bound (which is tight)
using CombUCB1, a UCB based algorithm
introduced
in
Chen
et
al.
[2013] which at stage t uses
q
?
P
(i)
the exploration term 1.5 log t i?A 1/ nt . Our exploration term always verifies Et (A) ?
q
p
P
(i)
f (t) i?A 1/ nt with f (t) ? log t (see section 3.6). Their exploration term is a worst-case
confidence interval for the means. Their broader confidence intervals however have the desirable
property that one can find the action that realizes the maximum index by solving a linear optimization
problem, making their algorithm computationally efficient, quality that both ESCB and OLS-UCB
are lacking.
4
None of the two former algorithms benefits from guaranties in the other regime. The regret of ESCB
in the general possibly correlated case is unknown and the regret bound for CombUCB1 is not
improved in the independent case. In contrast, OLS-UCB is adaptive in the sense that its performance
gets better when more information is available on the independence of the arms.
3.3
Regret Decomposition
(i)
?t
Let Hi,t = {|?
?t ? ?(i) | ? 2m
} and Ht = ?di=1 Hi,t . Ht is the event that at least one coordinate of
?
?t is far from the true mean. Let Gt = {A?> ? ? A?> ?
?t + Et (A? )} be the event that the estimate
of the optimal action is below its true mean by a big margin. We decompose the regret according to
these events:
RT ?
T
X
?t I{Gt , Ht } +
t=1
T
X
?t I{Gt } +
t=1
T
X
?t I{Ht }
t=1
Events Gt and Ht are rare and lead to a finite regret (see below). We first simplify the regret due to
Gt ? Ht and show that it is bounded by the "variance" term of the algorithm.
Lemma 1. With the algorithm choosing at stage t the action At = arg maxA (A> ?
?t + Et (A)), we
have ?t I{Gt , Ht } ? 2Et (At )I{?t ? Et (At )}.
Proof in appendix B.2 of the supplementary material. Then the regret is cut into three terms,
RT ? 2
T
X
Et (At )I{?t ? 2Et (At )} +
T
X
t=1
t=1
?t I{Gt } +
T
X
?t I{Ht } .
t=1
The three terms will be bounded as follows:
? The Ht term leads to a finite regret from a simple application of Hoeffding?s inequality.
? The Gt term leads to a finite regret for a good choice of f (t). This is where we need to show
that the exploration term of the algorithm gives a high probability upper confidence bound
of the reward.
? The Et (At ) term, or variance term, is the main source of the regret and is bounded using
ideas similar to the ones used in existing works on semi-bandits.
3.4
Expected regret from Ht
PT
Lemma 2. The expected regret due to the event Ht is E[ t=1 ?t I{Ht }] ?
8dm2 maxi {C (ii) }?max
?2min
.
The proof uses Hoeffding?s inequality on the arm mean estimates and can be found in appendix B.2
of the supplementary material.
3.5
Expected regret from Gt
We want to bound the probability that the estimated reward for the optimal action is far from its mean.
We show that it is sufficient to control a self-normalized sum and do it using arguments from Pe?a
et al. [2008], or Abbasi-Yadkori et al. [2011] who applied them to linear bandits. The analysis also
involves a peeling argument, as was done in one dimension by Garivier [2013] to bound a similar
quantity.
e
Lemma 3. Let ?t > 0. With f?(?t ) = log(1/?t ) + m log log t + m
2 log(1 + ? ) and an algorithm
q
q
P
t?1
given by the exploration term Et (A) = 2f?(?t ) A> Dt?1 (??? Dt + s=1 ?As )Dt?1 A , then the
event Gt = {A?> ? ? A?> ?
?t + Et (A? )} verifies P{Gt } ? ?t .
1
?
With ?1 = 1 and ?t = t log
2 t for t ? 2, such that f (?t ) = f (t), the regret due to Gt is finite in
expectation, bounded by 4?max .
5
Proof. We use a peeling argument: let ? > 0 and for a = (a1 , . . . , am ) ? Nm , let Da ? [T ] be a
(i)
subset of indices defined by (t ? Da ? ?i ? A? , (1 + ?)ai ? nt < (1 + ?)ai +1 ). For any Bt ? R,
X ?>
P A?> (? ? ?
?t ) ? Bt ?
P A (? ? ?
?t ) ? Bt |t ? Da .
a
The number of possible sets Da for t is bounded by (log t/ log(1 + ?))m , since each number of pulls
(i)
nt for i ? A? is bounded by t. We now search a bound of the form P A?> (? ? ?
?t ) ? Bt |t ? Da .
Suppose t ? Da and let D be a positive definite diagonal matrix (that depends on a).
Pt?1
Pt?1
2
Let St = s=1 IAs ?A? C 1/2 s , Vt = s=1 CAs ?A? and IVt +D () = 12 kSt k(Vt +D)?1 .
Lemma 4. Let ?t > 0 and let f?(?t ) be a function of ?t . With a choice of D such that IA? D
?IA? ?C Dt for all t in Da ,
q
n
o
P A?> (???
?t )? 2f?(?t )A?> Dt?1 (??C Dt +Vt )Dt?1 A? t?Da ? P IVt +D ()?f?(?t )|t?Da .
Proof in appendix B.2 of the supplementary material.
The self-normalized sum IVt () is an interesting quantity for the following reason: exp( 12 IVt ()) =
Qt?1
maxu?Rd s=1 exp(u> IAs ?A? C 1/2 s ? u> CAs ?A? u). For a given u, the exponential is smaller
that 1 in expectation, from the subgaussian hypothesis. The maximum of the expectation is then
smaller than 1. To control IVt (), we are however interested in the expectation of this maximum and
cannot interchange max and E. The method of mixtures circumvents this difficulty: it provides an
approximation of the maximum by integrating the exponential against a multivariate normal centered
at the point Vt?1 St , where the maximum is attained. The integrals over u and can then be swapped
by Fubini?s theorem to get an approximation of the expectation of the maximum using an integral of
the expectations. Doing so leads to the following lemma, extracted from the proof of Theorem 1 of
Abbasi-Yadkori et al. [2011].
Lemma 5.q
Let D be a positive definite matrix that does not depend on t and
det D
Mt (D) = det(V
exp(IVt +D ()). Then E[Mt (D)] ? 1.
n t +D)
o
We rewrite P IVt +D () ? f?(?t ) to introduce Mt (D),
(
n
o
P IVt +D () ? f?(?t )|t?Da = P Mt (D) ? p
1
det(Id + D?1/2 Vt D?1/2 )
exp(f?(?t )) t?Da
)
.
The peeling lets us bound Vt . Let Da be the diagonal matrix with entry (i, i) equal to (1 + ?)ai for
i ? A? and 0 elsewhere.
Lemma 6. With D = ??C Da + I[d]\A? , det(Id + D?1/2 Vt D?1/2 ) ? (1 +
1+? m
? )
.
The union bound on the sets Da and Markov?s inequality give
q
q
P A?> (? ? ?
?t ) ? 2f?(?t ) ?A?> ?C Dt?1 A? + A?> Dt?1 Vt Dt?1 A?
X
1 + ? ?m/2
)
exp(f?(?t ))|t ? Da
?
P Mt (D) ? (1 +
?
Da
m
log t
1 + ? m/2
?
(1 +
)
exp(?f?(?t ))
log(1 + ?)
?
For ? = e ? 1 and f?(?t ) as in lemma 3, this is bounded by ?t . The result with ? instead of C is a
consequence of C + ?. With ?1 = 1 and ?t = 1/(t log2 t) for t ? 2, the regret due to Gt is
E[
T
X
?t I{Gt }] ? ?max (1 +
t=1
T
X
t=2
6
1
) ? 4?max .
t log2 t
3.6
Bounding the variance term
The goal of this section is to bound Et (At ) under
? the event {?t ? Et (At )}. Let ?t ? [0, 1] such
that for q
all i, j ? At with i 6= j, ?(ij) ? ?t ?(ii) ?(jj) . From the Cauchy-Schwartz inequality,
(ij)
nt
(i) (j)
?
nt nt . Using these two inequalities,
t?1
X
?1
A>
t Dt (
?As )Dt?1 At
X n(ij) ?(ij)
t
=
s=1
i,j?At
(i) (j)
nt nt
? (1 ? ?t )
X ?(ii)
i?At
(i)
nt
s
X
?(ii)
i?At
(i)
nt
+ ?t (
)2 .
We recognize here the forms of the indexes used in Combes et al. [2015] for independent arms (left
term) and Kveton et al. [2015] for general arms (right term). Using ?t ? Et (At ) we get
s
X ?(ii)
X ?(ii)
?2t
? (? + 1 ? ?t )
+ ?t (
)2 .
(1)
(i)
(i)
8f (t)
nt
nt
i?At
i?At
The strategy from here is to find events that must happen when (1) holds and to show that these events
cannot happen very often. For positive integers j and t and for e ? {1, 2}, we define the set of arms
(ii)
(i)
j
in At that were pulled less than a given threshold: St,e
= {i ? At , nt ? ?j,e 8f (t)?
ge (m,?t )
},
?2t
j
At . (St,e )j?0
0
with ge (m, ?t ) to be stated later and (?i,e )i?1 a decreasing sequence. Let also St,e
=
is decreasing for the inclusion of sets and we impose limj?+? ?j,e = 0, such that there is an index
j?
j? with St,e
= ?. We introduce another positive sequence (?j,e )j?0 and consider the events that
j
at least m?j,e arms in At are in the set St,e
and that the same is false for k < j, i.e. for t ? 1,
j
j
k
At,e = {|St,e | ? m?j,e ; ?k < j, |St,e | < m?k,e }. To avoid having some of these events being
0
impossible we choose (?j,e )j?0 decreasing. We also impose ?0,e = 1, such that |St,e
| = m?0,e .
j
Let then At,e = ?+?
j=1 At,e and At = At,1 ? At,2 . We will show that At must happen for (1) to be
true. First, remark that under a condition on (?j,e )j?0 , At is a finite union of events,
0
Lemma 7. For e ? {1, 2}, if there exists j0,e such that ?j0,e ,e ? 1/m, then At,e = ?jj=1
Ajt,e .
We now show that At is impossible by proving a contradiction in (1).
Lemma 8. Under the event At,1 , if there exists j0 such that ?j0 ,1 ? 1/m, then
?
?
j0
X ?(ii)
X
m?2t
?
?
?
?
j?1,1
j,1
j ,1
?
<
+ 0 ?.
(i)
8f (t)g1 (m, ?t ) j=1
?j,1
?j0 ,1
nt
i?At
P+?
?
Under the event At,2 , if limj?+? ?j,2 / ?j,2 = 0 and j=1
s
X
?(ii)
i?At
nt
(i)
?p
?j?1,2 ??j,2
?
?j,2
exists, then
+?
X
?j?1,2 ? ?j,2
.
?
?j,2
8f (t)g2 (m, ?t ) j=1
m?t
A proof can be found in appendix B.2 of the supplementary material. To ensure that the conditions of these lemmas are fulfilled, we impose that (?i,1 )i?0 and (?i,2 )i?0 have limit 0 and
?
that limj?+? ?j,2 / ?j,2 = 0. Let j0,1 be the smallest integer such that ?j0,1 ,1 ? 1/m. Let
Pj0,1 ?j?1,1 ??j,1
P+? ?
?j ,1
??j,2
?
l1 = ?j0,1 ,1 + j=1
and l2 = j=1 j?1,2
. Using the two last lemmas with (1),
?j,1
?j,2
0,1
we get that if At is true,
?2t
?2t
<
8f (t)
8f (t)
(? + 1 ? ?t )
ml1
m2 l22
+ ?t
g1 (m, ?t )
g2 (m, ?t )
.
Taking g1 (m, ?t ) = 2(? + 1 ? ?t )ml1 and g2 (m, ?t ) = 2?t m2 l22 , we get a contradiction. Hence
with these choices At must happen. The regret bound will be obtained by a union bound on the events
that form At . First suppose that all gaps are equal to the same ?.
7
Lemma 9. Let ?
(ii)
d?j,e 8f (T ) maxi {?
m?j,e ?2
=
maxt?1 ?t .
}ge (m,?)
For j
N? , the event Ajt,e happens at most
?
times.
(i)
Proof. Each time that Ajt,e happens, the counter of plays nt of at least m?je arms is incremented.
(ii)
After
?j,e 8f (T ) maxi {?
?2
}ge (m,?)
(i)
increments, an arm cannot verify the condition on nt any more.
There are d arms, so the event can happen at most d m?1j
e
?j,e 8f (T ) maxi {?(ii) }ge (m,?)
?2
times.
If all gaps are equal to ?, an union bound for At gives
?
?
j0,1
+?
T
X
X
X
?j,1
?j,2 ?
f (T ) ?
.
E[
d (? + 1 ? ?)l1
+ ?ml22
?I{Ht ? Gt }] ? 16 max{?(ii) }
?
?
?
i?[d]
j,1
t=1
j=1
j=1 j,2
The general case requires more involved manipulations but the result is similar and no new important
idea is used. The following lemma is proved in appendix B.2 of the supplementary material:
Lemma 10. Let ? (i) = max{t,i?At } ?t . The regret from the event Ht ? Gt is such that
?
?
j0
T
+?
X
X ?(ii)
X
X
?j,1
?j,2 ?
?(? + 1 ? ?)l1
?t I{Ht ? Gt }] ? 16f (T )
E[
.
+ ?ml22
?
?
?
i,min
j,1
t=1
j=1
j=1 j,2
i?[d]
Finally we can find sequences (?j,1 )j?1 , (?j,2 )j?1 , (?j,1 )j?0 and (?j,2 )j?0 such that
!
2
T
X
X ?(ii)
log m
(i)
(i)
5(? + 1 ? ? )
+ 45? m
?I{Ht ? Gt }] ? 16f (T )
E[
?i,min
1.6
t=1
i?[d]
See appendix C of the supplementary material. In Combes et al. [2015], ?i,1 and ?i,1 were such
?
that the log2 m term was replaced by m. Our choice is also applicable to their ESCB algorithm.
Our use of geometric sequences is only optimal among sequences such that ?i,1 = ?i,1 for all i ? 1.
It is unknown to us if one can do better. With this control of the variance term, we finally proved
Theorem 2.
4
Conclusion
We defined a continuum of settings from the general to the independent arms cases which is suitable
for the analysis of semi-bandit algorithms. We exhibited a lower bound scaling with a parameter that
quantifies the particular setting in this continuum and proposed an algorithm inspired from linear
regression with an upper bound that matches the lower bound up to a log2 m term. Finally we showed
how to use tools from the linear bandits literature to analyse algorithms for the combinatorial bandit
case that are based on linear regression.
It would be interesting to estimate the subgaussian covariance matrix online to attain good regret
bounds without prior knowledge. Also, our algorithm is not computationally efficient since it requires
the computation of an argmax over the actions at each stage. It may be possible to compute this
argmax less often and still keep the regret guaranty, as was done in Abbasi-Yadkori et al. [2011] and
Combes et al. [2015].
On a broader scope, the inspiration from linear regression could lead to algorithms using different
estimators, adapted to the structure of the problem. For example, the weighted least-square estimator
is also unbiased and has smaller variance than OLS. Or one could take advantage of a sparse
covariance matrix by using sparse estimators, as was done in the linear bandit case in Carpentier and
Munos [2012].
Acknowledgements
The authors would like to acknowledge funding from the ANR under grant number ANR-13-JS010004 as well as the Fondation Math?matiques Jacques Hadamard and EDF through the Program
Gaspard Monge for Optimization and the Irsdi project Tecolere.
8
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9
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5,678 | 6,138 | Automatic Neuron Detection in Calcium Imaging
Data Using Convolutional Networks
Noah J. Apthorpe1? Alexander J. Riordan2? Rob E. Aguilar1 Jan Homann2
Yi Gu2 David W. Tank2 H. Sebastian Seung12
1
Computer Science Department 2 Princeton Neuroscience Institute
Princeton University
{apthorpe, ariordan, dwtank, sseung}@princeton.edu
?
These authors contributed equally to this work
Abstract
Calcium imaging is an important technique for monitoring the activity of thousands
of neurons simultaneously. As calcium imaging datasets grow in size, automated
detection of individual neurons is becoming important. Here we apply a supervised
learning approach to this problem and show that convolutional networks can achieve
near-human accuracy and superhuman speed. Accuracy is superior to the popular
PCA/ICA method based on precision and recall relative to ground truth annotation
by a human expert. These results suggest that convolutional networks are an
efficient and flexible tool for the analysis of large-scale calcium imaging data.
1
Introduction
Two-photon calcium imaging is a powerful technique for monitoring the activity of thousands of
individual neurons simultaneously in awake, behaving animals [1, 2]. Action potentials cause
transient changes in the intracellular concentration of calcium ions. Such changes are detected by
observing the fluorescence of calcium indicator molecules, typically using two-photon microscopy
in the mammalian brain [3]. Repeatedly scanning a single image plane yields a time series of 2D
images. This is effectively a video in which neurons blink whenever they are active [4, 5].
In the traditional workflow for extracting neural activities from the video, a human expert manually
annotates regions of interest (ROIs) corresponding to individual neurons [5, 1, 2]. Within each
ROI, pixel values are summed for each frame of the video, which yields the calcium signal of the
corresponding neuron versus time. A subsequent step may deconvolve the temporal filtering of
the intracellular calcium dynamics for an estimate of neural activity with better time resolution.
The traditional workflow has the deficiency that manual annotation becomes laborious and timeconsuming for very large datasets. Furthermore, manual annotation does not de-mix the signals from
spatially overlapping neurons.
Unsupervised basis learning methods (PCA/ICA [6], CNMF [7], dictionary learning [8], and sparse
space-time deconvolution [9]) express the video as a time-varying superposition of basis images.
The basis images play a similar role as ROIs in the traditional workflow, and their time-varying
coefficients are intended to correspond to neural activities. While basis learning methods are useful
for finding active neurons, they do not detect low-activity cells?making these methods inappropriate
for studies involving neurons that may be temporarily inactive depending on context or learning [10].
Such subtle difficulties may explain the lasting popularity of manual annotation. At first glance, the
videos produced by calcium imaging seem simple (neurons blinking on and off). Yet automating
image analysis has not been trivial. One difficulty is that images are corrupted by noise and artifacts
due to brain motion. Another difficulty is variability in the appearance of cell bodies, which vary
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
in shape, size, spacing, and resting-level fluorescence. Additionally, different neuroscience studies
may require differing ROI selection criteria. Some may require only cell bodies [5, 11], while others
involve dendrites [6]. Some may require only active cells, while others necessitate both active and
inactive cells [10]. Some neuroscientists may wish to reject slightly out-of-focus neurons. For all of
these reasons, a neuroscientist may spend hours or days tuning the parameters of nominally automated
methods, or may never succeed in finding a set of parameters that produces satisfactory results.
As a way of dealing with these difficulties, we focus here on a supervised learning approach to
automated ROI detection. An automated ROI detector could be used to replace manual ROI detection
by a human expert in the traditional workflow, or could be used to make the basis learning algorithms
more reliable by providing good initial conditions for basis images. However, the usability of an
automated algorithm strongly depends on it attaining high accuracy. A supervised learning method can
adapt to different ROI selection criteria and generalize them to new datasets. Supervised learning has
become the dominant approach for attaining high accuracy in many computer vision problems [12].
We assemble ground truth datasets consisting of calcium imaging videos along with ROIs drawn by
human experts and employ a precision-recall formalism for quantifying accuracy. We train a sliding
window convolutional network (ConvNet) to take a calcium video as input and output a 2D image
that matches the human-drawn ROIs as well as possible. The ConvNet achieves near-human accuracy
and exceeds that of PCA/ICA [6].
The prior work most similar to ours used supervised learning based on boosting with hand-designed
features [13]. Other previous attempts to automate ROI detection did not employ supervised machine
learning. For example, hand-designed filtering operations [14] and normalized cuts [15] were applied
to image pixel correlations.
The major cost of supervised learning is the human effort required to create the training set. As a
rough guide, our results suggest that on the order of 10 hours of effort or 1000 annotated cells are
sufficient to yield a ConvNet with usable accuracy. This initial time investment, however, is more than
repaid by the speed of a ConvNet at classifying new data. Furthermore, the marginal effort required
to create a training set is essentially zero for those neuroscientists who already have annotated data.
Neuroscientists can also agree to use the same trained ConvNets for uniformity of ROI selection
across labs.
From the deep learning perspective, an interesting aspect of our work is that a ConvNet that processes
a spatiotemporal (2+1)D image is trained using only spatial (2D) annotations. Full spatiotemporal
annotations (spatial locations and times of activation) would have been more laborious to collect. The
use of purely spatial annotations is possible because the neurons in our videos are stationary (apart
from motion artifacts). This makes our task simpler than other applications of ConvNets to video
processing [16].
2
Neuron detection benchmark
We use a precision-recall framework to quantify accuracy of neuron detection. Predicted ROIs are
classified as false positives (FP), false negatives (FN), and true positives (TP) relative to ground truth
ROIs. Precision and recall are defined by
TP
TP
precision =
recall =
(1)
TP + FP
TP + FN
Both measures would be equal to 1 if predictions were perfectly accurate, i.e. higher numbers are
better. If a single measure of accuracy is required, we use the harmonic mean of precision and recall,
1/F1 = (1/precision + 1/recall) /2. The F1 score favors neither precision nor recall, but in practice
a neuroscientist may care more about one measure than the other. For example, some neuroscientists
may be satisfied if the algorithm fails to detect many neurons (low recall) so long as it produces
few false positives (high precision). Other neuroscientists may want the algorithm to find as many
neurons as possible (high recall) even if there are many false positives (low precision).
For computing precision and recall, it is helpful to define the overlap between two ROIs R1 and R2
as the Jaccard similarity coefficient |R1 ? R2 |/|R1 ? R2 | where |R| denotes the number of pixels in
R. For each predicted ROI, we find the ground truth ROI with maximal overlap. The ground truth
ROIs with overlap greater than 0.5 are assigned to the predicted ROIs with which they overlap the
most. These assignments are true positives. Leftover ROIs are the false positives and false negatives.
2
We prefer the precision-recall framework over the receiver operating characteristic (ROC), which
was previously used as a quantitative measure of neuron detection accuracy [13]. This is because
precision and recall do not depend on true negatives, which are less well-defined. (The ROC depends
on true negatives through the false positive rate.)
Ground truth generation by human annotation The quantitative measures of accuracy proposed
above depend on the existence of ground truth. For the vast majority of calcium imaging datasets,
no objectively defined ground truth exists, and we must rely on subjective evaluation by human
experts. For a dataset with low noise in which the desired ROIs are cell bodies, human experts
are typically confident about most of their ROIs, though some are borderline cases that may be
ambiguous. Therefore our measures of accuracy should be able to distinguish between algorithms
that differ widely in their performance but may not be adequate to distinguish between algorithms
that are very similar.
Two-photon calcium imaging data were gathered from both the primary visual cortex (V1) and medial
entorhinal cortex (MEC) from awake-behaving mice (Supplementary Methods). All experiments
were performed according to the Guide for the Care and Use of Laboratory Animals, and procedures
were approved by Princeton University?s Animal Care and Use Committee.
Each time series of calcium images was corrected for motion artifacts (Supplementary Methods),
average-pooled over time with stride 167, and then max-pooled over time with stride 6. This
downsampling in time was arbitrarily chosen to reduce noise and make the dataset into a more
manageable size. Human experts then annotated ROIs using the ImageJ Cell Magic Wand Tool [17],
which automatically generates a region of interest (ROI) based on a single mouse click. The human
experts found 4006 neurons in the V1 dataset with an average of 148 neurons per image series and
538 neurons in the MEC dataset with an average of 54 neurons per image series.
Human experts used the following criteria to select neurons: 1. the soma was in the focal plane of
the image?apparent as a light doughnut-like ring (the soma cytosol) surrounding a dark area (the
nucleus), or 2. the area showed significantly changing brightness distinguishable from background
and had the same general size and shape expected from a neuron in the given brain region.
After motion correction, downsampling, and human labeling, the V1 dataset consisted of 27 16-bit
grayscale multi-page TIFF image series ranging from 28 to 142 frames per series with 512 ? 512
pixels per frame. The MEC dataset consisted of 10 image series ranging from 5 to 28 frames in the
same format. Human annotation time was estimated at one hour per image series for the V1 dataset
and 40 minutes per images series for the MEC dataset. Each human-labeled ROI was represented as
a 512 ? 512 pixel binary mask.
3
Convolutional network
Preprocessing of images and ground truth ROIs. Microscopy image series from the V1 and MEC
datasets were preprocessed prior to network training (Figure 1). Image contrast was enhanced by
clipping all pixel values above the 99th percentile and below the 3rd percentile. Pixel values were
then normalized to [0, 1]. We divided the V1 series into 60% training, 20% validation, and 20% test
sets and the MEC series into 50% training, 20% validation, and 30% test sets.
Neighboring ground truth ROIs often touched or even overlapped with each other. For the purpose
of ConvNet training, we shrank the ground truth ROIs by replacing each one with a 4-pixel radius
disk located at the centroid of the ROI. The shrinkage was intended to encourage the ConvNets to
separate neighboring neurons.
Convolutional network architecture and training. The architecture of the (2+1)D ConvNet is depicted in Figure 2. The input is an image stack containing T time slices. There are four convolutional
layers, a max pooling over all time slices, and then two pixelwise fully connected layers. This yields
two 2D grayscale images as output, which together represent the softmax probability of each pixel
being inside an ROI centroid.
The convolutional layers were chosen to contain only 2D kernels, because the temporal downsampling
used in the preprocessing (?2) caused most neural activity to last for only a single time frame. Each
output pixel depended on a 37 ? 37 ? T pixel field of view in the input, where T is the number of
frames in the input image stack?governed by the length of the imaging experiment and the imaging
3
Initial#Image
Contrast#Enhancement
Human#Labeled#ROIs
ROI#Centroids
V1#Dataset
20?m
MEC#Dataset
Figure 1: Preprocessing steps for calcium images and human-labeled ROIs. Col 1) Calcium imaging
stacks were motion-corrected and downsampled in time. Col 2) Image contrast was enhanced by
clipping pixel intensities below the 3rd and above the 99th percentile then linearly rescaling pixel
intensities between these new bounds. Col 3) Human-labeled ROIs were converted into binary
masks. Col 4) Networks were trained to detect 4-pixel radius circular centroids of human-labeled
ROIs. Primary visual cortex (V1, Row 1) and medial entorhinal cortex (MEC, Row 2) datasets were
preprocessed identically.
sampling rate. T was equalized to 50 for all image stacks in the V1 dataset and 5 for all image stacks
in the MEC dataset using averaging and bicubic interpolation. In the future, we will consider less
temporal downsampling and the use of 3D kernels in the convolutional layers.
The ConvNet was applied in a 37 ? 37 ? T window, sliding in two dimensions over the input image
stack to produce an output pixel for every location of the window fully contained within the image
bounds. For comparison, we also trained a 2D ConvNet that took as input the time-averaged image
stack and did no temporal computation (Figure 2).
We used ZNN, an open-source sliding window ConvNet package with multi-core CPU parallelism
and FFT-based convolution [18]. ZNN automatically augmented training sets by random rotations
(multiples of 90 degrees) and reflections of image patches to facilitate ConvNet learning of invariances.
The training sets were also rebalanced by the fraction of pixels in human-labeled ROIs to the total
number of pixels. See Supplementary Methods for further details.
The (2+1)D network was trained with softmax loss and output patches of size 120 ? 120. The
learning rate parameter was annealed by hand from 0.01 to 0.002, and the momentum parameter was
annealed by hand from 0.9 to 0.5. The network was trained for 16800 stochastic gradient descent
(SGD) updates for the V1 dataset, which took approximately 1.2 seconds/update (? 5.5hrs) on an
Amazon EC2 c4.8xlarge instance (Supplementary Figure 1). The network was trained for 200000
SGD updates for the MEC dataset, which took approximately 0.1 seconds/update (? 5.5hrs).
The 2D network training omitted annealing of the learning rate and momentum parameters. The
2D network was trained for 14000 SGD updates for the V1 dataset, which took approximately 0.9
seconds/update (? 3.75hrs) on an Amazon EC2 c4.8xlarge instance (Supplementary Figure 1). We
performed early stopping on the network after 10200 SGD updates based on the validation loss.
Network output postprocessing. Network outputs were converted into individual ROIs by: 1.
Thresholding out pixels with low probability values, 2. Removing small connected components,
3. Weighting resulting pixels with a normalized distance transform, 4. Performing marker-based
watershed labeling with local max markers, 5. Merging small watershed regions, and 6. Automatically applying the ImageJ Cell Magic Wand tool to the original images at the centroids of the
watershed regions. Thresholding and minimum size values were optimized using the validation sets
(Supplementary Methods).
Source code. A ready-to-use pipeline, including pre- and postprocessing, ConvNet training, and
precision-recall scoring, will be publicly available for community use (https://github.com/
NoahApthorpe/ConvnetCellDetection).
4
A.
conv
10x10x1
conv
10x10x1
conv
conv max'filter' conv
1x1x1
10x10x1 10x10x1 1x1xT
2D'output'
image
3D'image'
input
20'unit'FC'
layer
10'units/conv layer
B.
conv
3x3
conv max'filter' conv
3x3
2x2
3x3
conv max'filter' conv
3x3
2x2
3x3
conv
1x1
2D'image'
output
2D'image'
input
24 units
48
48
72
96
96
120
50'unit'FC'
layer
Figure 2: A) Schematic of the (2+1)D network architecture. The (2+1)D network transforms 3D
calcium imaging stacks ? stacks of 2D calcium images changing over time ? into 2D images of
predicted neuron locations. All convolutional filters are 2D except for the 1x1xT max filter layer,
where T is the number of frames in the image stack. B) The 2D network architecture. The 2D network
takes as input calcium imaging stacks that are mean projected over time down to two dimensions.
4
Results
ConvNets successfully detect cells in calcium images. A sample image from the V1 test set and
ConvNet output is shown in Figure 4. Postprocessing of the ConvNet output yielded predicted ROIs,
many of which are the same as the human ROIs (Figure 4c). As described in Section 2, we quantified
agreement between ConvNet and human using the precision-recall formalism. Both (2+1)D and 2D
networks attained the same F1 score (0.71). Full precision-recall curves are given in Supplementary
Figure 1.
Inspection of the ConvNet-human disagreements suggested that some were not actually ConvNet
errors. To investigate this hypothesis, the original human expert reevaluated all disagreements
with the (2+1)D network. After reevaluation, 131 false positives became true positives, and 30
false negatives became true negatives (Figure 4D). Some of these reversals appeared to involve
unambiguous human errors in the original annotation, while others were ambiguous cases (Figure 4E?
G). After reevaluation, the F1 score of the (2+1)D network increased to 0.82. The F1 score of the
human expert?s reevaluation relative to his original annotation was 0.89. These results indicate that
the ConvNet is nearing human performance.
(2+1)D versus 2D network. The (2+1)D and 2D networks achieved similar precision, recall, and
F1 scores on the V1 dataset; however, the (2+1)D network produced raw output with less noise
than the 2D network (Figure 3). Qualitative inspection also indicates that the (2+1)D network finds
transiently active and transiently in focus neurons missed by the 2D network (Figure 3). Although
such neurons occurred infrequently in the V1 dataset and did not noticeably affect network scores,
these results suggest that datasets with larger populations of transiently active or variably focused
cells will particularly benefit from (2+1)D network architectures.
ConvNet segmentation outperforms PCA/ICA. The (2+1)D network was also able to successfully
locate neurons in the MEC dataset (Figure 5). For comparison, we also implemented and applied
PCA/ICA as described by Ref. [6]. The (2+1)D network achieved an F1 score of 0.51, while
PCA/ICA achieved 0.27. Precision and recall numbers are given in Figure 5.
5
A.
Transiently'active'neuron
(2+1)D network
2D'network
Overlay
20?m
Neuron'falls'in'and'out'of'focus
20?m
(2+1)D network
2D'network
2
number pixels in image
C.
B.
#10
4
(2+1)D network
2D network
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
pixel intensity
Figure 3: A) The (2+1)D network detected neurons that the 2D network failed to locate. The sequence
of greyscale images shows a patch of V1 neurons over time. Both transiently active neurons and
neurons that wane in and out of the focal plane are visible. The color image shows the output of both
networks. The (2+1)D network detects these transiently visible neurons, whereas the 2D network is
unable to find these cells using only the mean-flattened image. B) The raw outputs of the (2+1)D and
2D networks. C) Representative histogram of output pixel intensities. The (2+1)D network output
has more values clustered around 0 and 1 compared to the 2D network. This suggests that (2+1)D
network output has a higher signal to noise ratio than 2D network output.
ConvNet accuracy was lower on the MEC dataset than the V1 dataset, probably because the former
has more noise and larger motion artifacts. The amount of training data for the MEC dataset was also
much smaller.
PCA/ICA accuracy was numerically worse, but this result should be interpreted cautiously. PCA/ICA
is intended to identify active neurons, while the ground truth included both active and inactive
neurons. Furthermore, the ground truth depends on the human expert?s selection criteria, which are
not accessible to PCA/ICA.
Training and post-processing optimization for ConvNet segmentation took ?6 hours with a forward
pass taking ?1.2 seconds per image series. Parameter optimization for PCA/ICA performed by a
human expert took ?2.5 hours with a forward pass taking ?40 minutes. This amounted to ?6 hours
total computation time for the ConvNet and ?9 hours for the PCA/ICA algorithm. This suggests that
ConvNet segmentation is faster than PCA/ICA for all but the smallest datasets.
5
Discussion
The lack of quantitative difference between (2+1)D and 2D ConvNet accuracy (same F1 score on
the V1 dataset) may be due to limitations of our study, such as imperfect ground truth and temporal
downsampling in preprocessing. It may also be because the vast majority of neurons in the V1 dataset
are clearly visible in the time-averaged image. We do have qualitative evidence that the (2+1)D
architecture may turn out to be superior for other datasets, because its output looks cleaner, and it is
able to detect transiently active or transiently in-focus cells (Figure 3).
The (2+1)D ConvNet outperformed PCA/ICA in the precision-recall metrics. We are presently
working to compare against recently released basis learning methods [7]. ConvNets readily locate
inactive neurons and process new images rapidly once trained. ConvNets adapt to the selection
criteria of the neuroscientist if they are implicitly contained in the training set. They do not depend on
hand-designed features and so require little expertise in computer vision. ConvNet speed could enable
novel applications involving online ROI detection, such as computer-guided single-cell optogenetics
[11] or real-time neural feedback experiments.
6
B.
A.
20?m
C.
D.
Human
(2+1)D&network
Added&by&human&relabeling&
Removed&by&human&relabeling&
Human&&&(2+1)D&network&overlay
B.
F.
C.
G.
H.
F1&Score
A.
E.
2D
Original
Labels
20?m
(2+1)D
Temporal
Original
Labels
2D
Relabeled
(2+1)D
Temporal
Relabeled
Human
Original to
Relabeled
Figure 4: The (2+1)D network successfully detected neurons in the V1 test set with near-human
accuracy. A) Slice from preprocessed calcium imaging stack input to network. B) Network softmax
probability output. Brighter regions are considered by the network to have higher probability of
being a neuron. C) ROIs found by the (2+1)D network after post-processing, overlaid with human
labels. Network output is shown by green outlines, whereas human labels are red. Regions of
agreement are indicated by yellow overlays. D) ROI labels added by human reevaluation are shown
in blue. ROI labels removed by reevaluation are shown in magenta. Post hoc assessment of network
output revealed a sizable portion of ROIs that were initially missed by human labeling. E) Examples
of formerly negative ROIs that were reevaluated as positive. F) Initial positive labels that were
reevaluated to be false. G) Examples of ROIs that remained negative even after reevaluation. H) F1
scores for (2+1)D and 2D networks before and after ROI reevaluation. Human labels before and after
reevaluation were also compared to assess human labeling variability. Boxplots depict the variability
of F1 scores around the median score across test images.
7
A.
B.
20?m
C.
D.
Human
(2+1)D)network
PCA/ICA
Human)&)(2+1)D)network)overlay
E.
(2+1)D
Temporal
network
PCA/
PCA
ICA
(2+1)D
Temporal
network
PCA/
PCA
ICA
(2+1)D
Temporal
network
PCA/
PCA
ICA
Figure 5: The (2+1)D network successfully detected neurons in the MEC test set with higher
precision and recall than PCA/ICA. A) Slice from preprocessed calcium imaging stack that was input
to network. B) Network output, normalized by softmax. C) ROIs found by the (2+1)D network after
postprocessing, overlaid with ROIs previously labeled by a human. Network output is shown by red
outlines, whereas human labels are green. Regions of agreement are indicated by yellow overlays. D)
The ROIs found by PCA/ICA are overlaid in blue. E) Quantitative comparison of F1 score, precision,
and recall for (2+1)D network and PCA/ICA on human-labeled MEC data.
8
Acknowledgments
We thank Kisuk Lee, Jingpeng Wu, Nicholas Turner, and Jeffrey Gauthier for technical assistance.
We also thank Sue Ann Koay, Niranjani Prasad, Cyril Zhang, and Hussein Nagree for discussions.
This work was supported by IARPA D16PC00005 (HSS), the Mathers Foundation (HSS), NIH
R01 MH083686 (DWT), NIH U01 NS090541 (DWT, HSS), NIH U01 NS090562 (HSS), Simons
Foundation SCGB (DWT), and U.S. Army Research Office W911NF-12-1-0594 (HSS).
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9
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5,679 | 6,139 | Supervised Word Mover?s Distance
Gao Huang? , Chuan Guo?
Cornell University
{gh349,cg563}@cornell.edu
Yu Sun, Kilian Q. Weinberger
Cornell University
{ys646,kqw4}@cornell.edu
Matt J. Kusner?
Alan Turing Institute, University of Warwick
mkusner@turing.ac.uk
Fei Sha
University of California, Los Angeles
feisha@cs.ucla.edu
Abstract
Recently, a new document metric called the word mover?s distance (WMD) has
been proposed with unprecedented results on kNN-based document classification.
The WMD elevates high-quality word embeddings to a document metric by formulating the distance between two documents as an optimal transport problem
between the embedded words. However, the document distances are entirely unsupervised and lack a mechanism to incorporate supervision when available. In
this paper we propose an efficient technique to learn a supervised metric, which
we call the Supervised-WMD (S-WMD) metric. The supervised training minimizes the stochastic leave-one-out nearest neighbor classification error on a perdocument level by updating an affine transformation of the underlying word embedding space and a word-imporance weight vector. As the gradient of the original WMD distance would result in an inefficient nested optimization problem, we
provide an arbitrarily close approximation that results in a practical and efficient
update rule. We evaluate S-WMD on eight real-world text classification tasks on
which it consistently outperforms almost all of our 26 competitive baselines.
1
Introduction
Document distances are a key component of many text retrieval tasks such as web-search ranking
[24], book recommendation [16], and news categorization [25]. Because of the variety of potential applications, there has been a wealth of work towards developing accurate document distances
[2, 4, 11, 27]. In large part, prior work focused on extracting meaningful document representations,
starting with the classical bag of words (BOW) and term frequency-inverse document frequency
(TF-IDF) representations [30]. These sparse, high-dimensional representations are frequently nearly
orthogonal [17] and a pair of similar documents may therefore have nearly the same distance as a
pair that are very different. It is possible to design more meaningful representations through eigendecomposing the BOW space with Latent Semantic Indexing (LSI) [11], or learning a probabilistic
clustering of BOW vectors with Latent Dirichlet Allocation (LDA) [2]. Other work generalizes LDA
[27] or uses denoising autoencoders [4] to learn a suitable document representation.
Recently, Kusner et al. [19] proposed the Word Mover?s Distance (WMD), a new distance for text
documents that leverages word embeddings [22]. Given these high-quality embeddings, the WMD
defines the distances between two documents as the optimal transport cost of moving all words from
one document to another within the word embedding space. This approach was shown to lead to
state-of-the-art error rates in k-nearest neighbor (kNN) document classification.
?
?
Authors contributing equally
This work was done while the author was a student at Washington University in St. Louis
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Importantly, these prior works are entirely unsupervised and not learned explicitly for any particular
task. For example, text documents could be classified by topic or by author, which would lead to
very different measures of dissimilarity. Lately, there has been a vast amount of work on metric
learning [10, 15, 36, 37], most of which focuses on learning a generalized linear Euclidean metric.
These methods often scale quadratically with the input dimensionality, and can only be applied to
high-dimensional text documents after dimensionality reduction techniques such as PCA [36].
In this paper we propose an algorithm for learning a metric to improve the Word Mover?s Distance.
WMD stands out from prior work in that it computes distances between documents without ever
learning a new document representation. Instead, it leverages low-dimensional word representations,
for example word2vec, to compute distances. This allows us to transform the word embedding
instead of the documents, and remain in a low-dimensional space throughout. At the same time we
propose to learn word-specific ?importance? weights, to emphasize the usefulness of certain words
for distinguishing the document class.
At first glance, incorporating supervision into the WMD appears computationally prohibitive, as
each individual WMD computation scales cubically with respect to the (sparse) dimensionality of
the documents. However, we devise an efficient technique that exploits a relaxed version of the
underlying optimal transport problem, called the Sinkhorn distance [6]. This, combined with a
probabilistic filtering of the training set, reduces the computation time significantly.
Our metric learning algorithm, Supervised Word Mover?s Distance (S-WMD), directly minimizes a
stochastic version of the leave-one-out classification error under the WMD metric. Different from
classic metric learning, we learn a linear transformation of the word representations while also learning re-weighted word frequencies. These transformations are learned to make the WMD distances
match the semantic meaning of similarity encoded in the labels. We show across 8 datasets and 26
baseline methods the superiority of our method.
2
Background
Here we describe the word embedding technique we use (word2vec) and the recently introduced
Word Mover?s Distance. We then detail the setting of linear metric learning and the solution proposed by Neighborhood Components Analysis (NCA) [15], which inspires our method.
word2vec may be the most popular technique for learning a word embedding over billions of words
and was introduced by Mikolov et al. [22]. Each word in the training corpus is associated with
an initial word vector, which is then optimized so that if two words w1 and w2 frequently occur
together, they have high conditional probability p(w2 |w1 ). This probability is the hierarchical softmax of the word vectors vw1 and vw2 [22], an easily-computed quantity which allows a simplified
neural language model (the word2vec model) to be trained efficiently on desktop computers. Training an embedding over billions of words allows word2vec to capture surprisingly accurate word
relationships [23]. Word embeddings can learn hundreds of millions of parameters and are typically
by design unsupervised, allowing them to be trained on large unlabeled text corpora ahead of time.
Throughout this paper we use word2vec, although many word embeddings could be used [5, 21? ].
Word Mover?s Distance. Leveraging the compelling word vector relationships of word embeddings, Kusner et al. [19] introduced the Word Mover?s Distance (WMD) as a distance between text
documents. At a high level, the WMD is the minimum distance required to transport the words
from one document to another. We assume that we are given a word embedding matrix X ? Rd?n
for a vocabulary of n words. Let xi ? Rd be the representation of the ith word, as defined by this
embedding. Additionally, let da , db be the n-dimensional normalized bag-of-words (BOW) vectors
for two documents, where dai is the number of times word i occurs in da (normalized over all words
in da ). The WMD introduces an auxiliary ?transport? matrix T ? Rn?n , such that Tij describes
how much of dai should be transported to dbj . Formally, the WMD learns T to minimize
n
n
n
X
X
X
D(xi , xj ) = min
Tij kxi ? xj kp2 , subject to,
Tij = dai ,
Tij = dbj ?i, j, (1)
T?0
j=1
i,j=1
i=1
where p is usually set to 1 or 2. In this way, documents that share many words (or even related ones)
should have smaller distances than documents with very dissimilar words. It was noted in Kusner
et al. [19] that the WMD is a special case of the Earth Mover?s Distance (EMD) [29], also known
more generally as the Wasserstein distance [20]. The authors also introduce the word centroid distance (WCD), which uses a fast approximation first described by Rubner et al. [29]: kXd ? Xd0 k2 .
2
It can be shown that the WCD always lower bounds the WMD. Intuitively the WCD represents each
document by the weighted average word vector, where the weights are the normalized BOW counts.
The time complexity of solving the WMD optimization problem is O(q 3 log q) [26], where q is the
maximum number of unique words in either d or d0 . The WCD scales asymptotically by O(dq).
Regularized Transport Problem. To alleviate the cubic time complexity of the Wasserstein distance computation, Cuturi [6] formulated a smoothed version of the underlying transport problem by
adding an entropy regularizer to the transport objective. This makes the objective function strictly
convex, and efficient
Pn algorithms can be adopted to solve it. In particular, given a transport matrix
T, let h(T) = ? i,j=1 Tij log(Tij ) be the entropy of T. For any ? > 0, the regularized (primal)
transport problem is defined as
n
n
n
X
X
X
1
min
Tij kxi ? xj kp2 ? h(T) subject to,
Tij = dai ,
Tij = dbj ?i, j.
(2)
T?0
?
i,j=1
j=1
i=1
The larger ? is, the closer this relaxation is to the original Wasserstein distance. Cuturi [6] propose
an efficient algorithm to solve for the optimal transport T?? using a clever matrix-scaling algorithm.
Specifically, we may define the matrix Kij = exp(??kxi ? xj k2 ) and solve for the scaling vectors
u, v to a fixed-point by computing u = da ./(Kv), v = db ./(K> u) in an alternating fashion.
These yield the relaxed transport T?? = diag(u)K diag(v). This algorithm can be shown to have
empirical time complexity O(q 2 ) [6], which is significantly faster than solving the WMD problem
exactly.
Linear Metric Learning. Assume that we have access to a training set {x1 , . . . , xn } ? Rd , arranged as columns in matrix X ? Rd?n , and corresponding labels {y1 , . . . , yn } ? Y n , where Y
contains some finite number of classes C = |Y|. Linear metric learning learns a matrix A ? Rr?d ,
where r ? d, and defines the generalized Euclidean distance between two documents xi and xj as
dA (xi , xj ) = kA(xi ?xj )k2 . Popular linear metric learning algorithms are NCA [15], LMNN [36],
and ITML [10] amongst others [37]. These methods learn a matrix A to minimize a loss function
that is often an approximation of the leave-one-out (LOO) classification error of the kNN classifier.
Neighborhood Components Analysis (NCA) was introduced by Goldberger et al. [15] to learn
a generalized Euclidean metric. Here, the authors approximate the non-continuous leave-one-out
kNN error by defining a stochastic neighborhood process. An input xi is assigned input xj as its
nearest neighbor with probability
exp(?d2A (xi , xj ))
,
(3)
pij = P
2
k6=i exp (?dA (xi , xk ))
where we define pii = 0. Under this stochastic neighborhood assignment, an input xi with label
yi is classified correctly if its nearest neighbor isP
any xj 6= xi from the same class (yj = yi ). The
probability of this event can be stated as pi =
learns A by maximizing the
j:yj =yi pij . NCA
P
P
expected LOO accuracy i pi , or equivalently by minimizing ? i log(pi ), the KL-divergence
from a perfect classification distribution (pi = 1 for all xi ).
3
Learning a Word Embedding Metric
In this section we propose a method for learning a supervised document distance, by way of learning a generalized Euclidean metric within the word embedding space and a word importance vector. We will refer to the learned document distance as the Supervised Word Mover?s Distance (SWMD). To learn such a metric we assume we have a training dataset consisting of m documents
{d1 , . . . , dm } ? ?n , where ?n is the (n?1)-dimensional simplex (thus each document is represented as a normalized histogram over the words in the vocabulary, of size n). For each document
we are given a label out of C possible classes, i.e. {y1 , . . . , ym } ? {1, . . . , C}m . Additionally,
we are given a word embedding matrix X ? Rd?n (e.g., the word2vec embedding) which defines a
d-dimensional word vector for each of the words in the vocabulary.
Supervised WMD. As described in the previous section, it is possible to define a distance between
any two documents da and db as the minimum cumulative word distance of moving da to db in
word embedding space, as is done in the WMD. Given a labeled training set we would like to
improve the distance so that documents that share the same labels are close, and those with different
labels are far apart. We capture this notion of similarity in two ways: First we transform the word
embedding, which captures a latent representation of words. We adapt this representation with a
3
linear transformation xi ? Axi , where xi represents the embedding of the ith word. Second, as
different classification tasks and data sets may value words differently, we also introduce a histogram
importance vector w that re-weighs the word histogram values to reflect the importance of words
for distinguishing the classes:
? a = (w ? da )/(w> da ),
d
(4)
where ??? denotes the element-wise Hadamard product. After applying the vector w and the linear
mapping A, the WMD distance between documents da and db becomes
DA,w (da , db ) , min
T?0
n
X
Tij kA(xi ? xj )k22 s.t.
i,j=1
n
X
j=1
Tij = d?ai and
n
X
Tij = d?bj ?i, j. (5)
i=1
Loss Function. Our goal is to learn the matrix A and vector w to make the distance DA,w reflect
the semantic definition of similarity encoded in the labeled data. Similar to prior work on metric
learning [10, 15, 36] we achieve this by minimizing the kNN-LOO error with the distance DA,w
in the document space. As the LOO error is non-differentiable, we use the stochastic neighborhood
relaxation proposed by Hinton & Roweis [18], which is also used for NCA. Similar to prior work
we use the squared Euclidean word distance in Eq. (5). We use the KL-divergence loss proposed in
NCA alongside the definition of neighborhood probability in (3) which yields,
?
?
m
m
X
X
exp(?D
(d
,
d
))
A,w
a
b
?.
P
(6)
`(A, w) = ?
log ?
c6=a exp (?DA,w (da , dc ))
a=1
b:yb =ya
Gradient. We can compute the gradient of the loss `(A, w) with respect to A and w as follows,
m X
X
?
pab
?
`(A, w) =
DA,w (da , db ),
(?ab ? pa )
?(A, w)
p
?(A,
w)
a
a=1
(7)
b6=a
where ?ab = 1 if and only if ya = yb , and ?ab = 0 otherwise.
3.1
Fast computation of ?DA,w (da , db )/?(A, w)
Notice that the remaining gradient term above ?DA,w (da , db )/?(A, w) contains the nested linear
program defined in (5). In fact, computing this gradient just for a single pair of documents will
require time complexity O(q 3 log q), where q is the largest set of unique words in either document
[8]. This quickly becomes prohibitively slow as the document size becomes large and the number
of documents increase. Further, the gradient is not always guaranteed to exist [1, 7] (instead we
must resort to subgradient descent). Motivated by the recent works on fast Wasserstein distance
computation [6, 8, 12], we propose to relax the modified linear program in eq. (5) using the entropy
as in eq. (2). As described in Section 2, this allows us to approximately solve eq. (5) in O(q 2 ) time
via T?? = diag(u)K diag(v). We will use this approximate solution in the following gradients.
Gradient w.r.t. A. It can be shown that,
n
X
?
>
DA,w (da , db ) = 2A
Tab
ij (xi ? xj )(xi ? xj ) ,
?A
i,j=1
(8)
where Tab is the optimizer of (5), so long as it is unique (otherwise it is a subgradient) [1]. We
replace Tab by T?? which is always unique as the relaxed transport is strongly convex [9].
Gradient w.r.t. w. To obtain the gradient with respect to w, we need the optimal solution to the
dual transport problem:
?
? a + ?> d
? b ; s.t. ?i + ?j ? kA(xi ? xj )k2 ?i, j.
DA,w
(da , db ) , max ?> d
2
(?,?)
(9)
? a and d
? b are functions of w, we have
Given that both d
?
?
? a ?DA,w
? b ?? ?da ?(?? > d
? a)da ? ? ?db ?(? ? > d
? b)db
?DA,w
?
?d
?d
DA,w (da , db ) =
+
=
+
.
>
a
>
b
? a ?w
? b ?w
?w
w d
w d
?d
?d
(10)
4
Instead of solving the dual directly, we obtain the relaxed optimal dual variables ??? , ? ?? via the
vectors u, v that were used to derive our relaxed transport T?? . Specifically, we can solve for the
>
>
1
1
? log(u)
1 and ? ?? = log(v)
? log(v)
1, where 1 is the
dual variables as such: ??? = log(u)
?
p
?
p
p-dimensional all ones vector. In general, we can observe from eq. (2) that the above approximation
process becomes more accurate as ? grows. However, setting ? too large can make the algorithm
converges slower. In our experiments, we use ? = 10, which leads to a nice trade-off between speed
and approximation accuracy.
3.2
Optimization
Alongside the fast gradient computation process in- Algorithm 1 S-WMD
troduced above, we can further speed up the training with a clever initialization and batch gradient de- 1: Input: word1 embedding: X,
2: dataset: {(d , y1 ), . . . , (dm , ym )}
scent.
3: ca = Xda , ?a ? {1, . . . , m}
Initialization. The loss function in eq. (6) is non4: A = NCA((c1 , y1 ), . . . , (cm , ym ))
convex and is thus highly dependent on the initial 5: w = 1
setting of A and w. A good initialization also dras- 6: while loop until convergence do
tically reduces the number of gradient steps required. 7:
Randomly select B ? {1, . . . , m}
For w, we initialize all its entries to 1, i.e., all words 8:
Compute gradients using eq. (11)
are assigned with the same weights at the begin- 9:
A ? A ? ?A gA
ning. For A, we propose to learn an initial projection 10:
w ? w ? ?w gw
within the word centroid distance (WCD), defined 11: end while
as D0 (da , db ) = kXda ? Xdb k2 , described in Section 2. The WCD should be a reasonable approximation to the WMD. Kusner et al. [19] point out
that the WCD is a lower bound on the WMD, which holds true after the transformation with A.
We obtain our initialization by applying NCA in word embedding space using the WCD distance
between documents. This is to say that we can construct the WCD dataset: {c1 , . . . , cm } ? Rd ,
representing each text document as its word centroid, and apply NCA in the usual way as described
in Section 2. We call this learned word distance Supervised Word Centroid Distance (S-WCD).
Batch Gradient Descent. Once the initial matrix A is obtained, we minimize the loss `(A, w) in
(6) with batch gradient descent. At each iteration, instead of optimizing over the full training set,
we randomly pick a batch of documents B from the training set, and compute the gradient for these
documents. We can further speed up training by observing that the vast majority of NCA probabilities pab near zero. This is because most documents are far away from any given document. Thus,
for a document da we can use the WCD to get a cheap neighbor ordering and only compute the
NCA probabilities for the closest set of documents Na , based on the WCD. When we compute the
gradient for each of the selected documents, we only use the document?s M nearest neighbor documents (defined by WCD distance) to compute the NCA neighborhood probabilities. In particular,
the gradient is computed as follows,
X X
?
gA,w =
(pab /pa )(?ab ? pa )
D(A,w) (da , db ),
(11)
?(A, w)
a?B b?Na
where again Na is the set of nearest neighbors of document a. With the gradient, we update A and
w with learning rates ?A and ?w , respectively. Algorithm 1 summarizes S-WMD in pseudo code.
Complexity. The empirical time complexity of solving the dual transport problem scales quadratically with p [26]. Therefore, the complexity of our algorithm is O(T BN [p2 + d2 (p + r)]), where
T denotes the number of batch gradient descent iterations, B = |B| the batch size, N = |Na | the
size of the nearest neighbor set, and p the maximum number of unique words in a document. This
is because computing T?ij , ?? and ? ? using the alternating fixed point algorithm in Section 3.1
requires O(p2 ) time, while constructing the gradients from eqs. (8) and (10) takes O(d2 (p + r))
time. The approximated gradient eq. (11) requires this computation to be repeated BN times. In
our experiments, we set B = 32 and N = 200, and computing the gradient at each iteration can be
done in seconds.
4
Results
We evaluate S-WMD on 8 different document corpora and compare the kNN error with unsupervised
WCD, WMD, and 6 document representations. In addition, all 6 document representation baselines
5
Table 1: The document datasets (and their descriptions) used for visualization and evaluation.
name
BBCSPORT
TWITTER
RECIPE
OHSUMED
CLASSIC
REUTERS
AMAZON
20 NEWS
twitter
recipe
ohsumed
C
5
3
15
10
4
8
4
20
classic
n
517
2176
3059
3999
4965
5485
5600
11293
ne
220
932
1311
5153
2128
2189
2400
7528
reuters
BOW
dim.
13243
6344
5708
31789
24277
22425
42063
29671
avg
words
117
9.9
48.5
59.2
38.6
37.1
45.0
72
amazon
20news
S-WMD
WMD
bbcsport
description
BBC sports articles labeled by sport
tweets categorized by sentiment [31]
recipe procedures labeled by origin
medical abstracts (class subsampled)
academic papers labeled by publisher
news dataset (train/test split [3])
reviews labeled by product
canonical news article dataset [3]
Figure 1: t-SNE plots of WMD and S-WMD on all datasets.
are used with and without 3 leading supervised metric learning algorithms?resulting in an overall
total of 26 competitive baselines. Our code is implemented in Matlab and is freely available at
https://github.com/gaohuang/S-WMD.
Datasets and Baselines. We evaluate all approaches on 8 document datasets in the settings of
news categorization, sentiment analysis, and product identification, among others. Table 1 describes
the classification tasks as well as the size and number of classes C of each of the datasets. We
evaluate against the following document representation/distance methods: 1. bag-of-words (BOW):
a count of the number of word occurrences in a document, the length of the vector is the number
of unique words in the corpus; 2. term frequency-inverse document frequency (TF-IDF): the BOW
vector normalized by the document frequency of each word across the corpus; 3. Okapi BM25 [28]:
a TF-IDF-like ranking function, first used in search engines; 4. Latent Semantic Indexing (LSI)
[11]: projects the BOW vectors onto an orthogonal basis via singular value decomposition; 5. Latent Dirichlet Allocation (LDA) [2]: a generative probabilistic method that models documents as
mixtures of word ?topics?. We train LDA transductively (i.e., on the combined collection of training
& testing words) and use the topic probabilities as the document representation ; 6. Marginalized
Stacked Denoising Autoencoders (mSDA) [4]: a fast method for training stacked denoising autoencoders, which have state-of-the-art error rates on sentiment analysis tasks [14]. For datasets larger
than RECIPE we use either a high-dimensional variant of mSDA or take 20% of the features that
occur most often, whichever has better performance.; 7. Word Centroid Distance (WCD), described
in Section 2; 8. Word Mover?s Distance (WMD), described in Section 2. For completeness, we
also show results for the Supervised Word Centroid Distance (S-WCD) and the initialization of SWMD (S-WMD init.), described in Section 3. For methods that propose a document representation
(as opposed to a distance), we use the Euclidean distance between these vector representations for
visualization and kNN classification. For the supervised metric learning results we first reduce the
dimensionality of each representation to 200 dimensions (if necessary) with PCA and then run either NCA, ITML, or LMNN on the projected data. We tune all free hyperparameters in all compared
methods with Bayesian optimization (BO), using the implementation of Gardner et al. [13]3 .
kNN classification. We show the kNN test error of all document representation and distance methods in Table 2. For datasets that do not have a predefined train/test split: BBCSPORT, TWITTER,
RECIPE , CLASSIC , and AMAZON we average results over five 70/30 train/test splits and report standard errors. For each dataset we highlight the best results in bold (and those whose standard error
3
http://tinyurl.com/bayesopt
6
Table 2: The kNN test error for all datasets and distances.
DATASET
BBCSPORT
TWITTER
BOW
TF-IDF
O KAPI BM25 [28]
LSI [11]
LDA [2]
M SDA [4]
20.6 ? 1.2
21.5 ? 2.8
16.9 ? 1.5
4.3 ? 0.6
6.4 ? 0.7
8.4 ? 0.8
43.6 ? 0.4
33.2 ? 0.9
42.7 ? 7.8
31.7 ? 0.7
33.8 ? 0.3
32.3 ? 0.7
BOW
TF-IDF
O KAPI BM25 [28]
LSI [11]
LDA [2]
M SDA [4]
7.4 ? 1.4
1.8 ? 0.2
3.7 ? 0.5
5.0 ? 0.7
6.5 ? 0.7
25.5 ? 9.4
32.0 ? 0.4
31.1 ? 0.3
31.9 ? 0.3
32.3 ? 0.4
33.9 ? 0.9
43.7 ? 7.4
BOW
TF-IDF
O KAPI BM25 [28]
LSI [11]
LDA [2]
M SDA [4]
2.4 ? 0.4
4.0 ? 0.6
1.9 ? 0.7
2.4 ? 0.5
4.5 ? 0.4
22.7 ? 10.0
31.8 ? 0.3
30.8 ? 0.3
30.5 ? 0.4
31.6 ? 0.2
31.9 ? 0.6
50.3 ? 8.6
BOW
TF-IDF
O KAPI BM25 [28]
LSI [11]
LDA [2]
M SDA [4]
9.6 ? 0.6
0.6 ? 0.3
4.5 ? 0.5
2.4 ? 0.7
7.1 ? 0.9
21.8 ? 7.4
31.1 ? 0.5
30.6 ? 0.5
31.8 ? 0.4
31.1 ? 0.8
32.7 ? 0.3
37.9 ? 2.8
WCD [19]
WMD [19]
S-WCD
S-WMD INIT.
S-WMD
11.3 ? 1.1
4.6 ? 0.7
4.6 ? 0.5
2.8 ? 0.3
2.1 ? 0.5
30.7 ? 0.9
28.7 ? 0.6
30.4 ? 0.5
28.2 ? 0.4
27.5 ? 0.5
RECIPE
OHSUMED
CLASSIC
REUTERS
U NSUPERVISED
61.1
36.0 ? 0.5
13.9
62.7
35.0 ? 1.8
29.1
66.2
40.6 ? 2.7
32.8
44.2
6.7 ? 0.4
6.3
51.0
5.0 ? 0.3
6.9
49.3
6.9 ? 0.4
8.1
ITML [10]
63.1 ? 0.9
70.1
7.5 ? 0.5
7.3
51.0 ? 1.4
55.1
9.9 ? 1.0
6.6
53.8 ? 1.8
77.0
18.3 ? 4.5
20.7
55.7 ? 0.8
54.7
5.5 ? 0.7
6.9
59.3 ? 0.8
59.6
6.6 ? 0.5
9.2
54.5 ? 1.3
61.8
14.9 ? 2.2
5.9
LMNN [36]
48.4 ? 0.4
49.1
4.7 ? 0.3
3.9
43.7 ? 0.3
40.0
4.9 ? 0.3
5.8
41.7 ? 0.7
59.4
19.0 ? 9.3
9.2
44.8 ? 0.4
40.8
3.0 ? 0.1
3.2
51.4 ? 0.4
49.9
4.9 ? 0.4
5.6
46.3 ? 1.2
41.6
11.1 ? 1.9
5.3
NCA [15]
55.2 ? 0.6
57.4
4.0 ? 0.1
6.2
41.4 ? 0.4
35.8
5.5 ? 0.2
3.8
45.8 ? 0.5
56.6
20.6 ? 4.8
10.5
41.6 ? 0.5
37.5
3.1 ? 0.2
3.3
50.9 ? 0.4
50.7
5.0 ? 0.2
7.9
48.0 ? 1.6
40.4
11.2 ? 1.8
5.2
D ISTANCES IN THE W ORD M OVER ? S FAMILY
49.4 ? 0.3
48.9
6.6 ? 0.2
4.7
42.6 ? 0.3
44.5
2.8 ? 0.1
3.5
51.3 ? 0.2
43.3
5.8 ? 0.2
3.9
39.8 ? 0.4
38.0
3.3 ? 0.3
3.5
39.2 ? 0.3
34.3
3.2 ? 0.2
3.2
59.3 ? 1.0
53.4 ? 1.0
53.4 ? 1.9
45.4 ? 0.5
51.3 ? 0.6
48.0 ? 1.4
AMAZON
20 NEWS
AVERAGE - RANK
28.5 ? 0.5
41.5 ? 1.2
58.8 ? 2.6
9.3 ? 0.4
11.8 ? 0.6
17.1 ? 0.4
57.8
54.4
55.9
28.9
31.5
39.5
26.1
25.0
26.1
12.0
16.6
18.0
20.5 ? 2.1
11.1 ? 1.9
11.4 ? 2.9
10.6 ? 2.2
15.7 ? 2.0
37.4 ? 4.0
60.6
45.3
81.5
39.6
87.8
47.7
23.0
14.8
21.5
17.6
22.5
23.9
10.7 ? 0.3
6.8 ? 0.3
6.9 ? 0.2
6.6 ? 0.2
12.1 ? 0.6
24.0 ? 3.6
40.7
28.1
57.4
25.1
32.0
27.1
11.5
7.8
14.4
5.1
14.6
17.3
16.8 ? 0.3
6.5 ? 0.2
8.5 ? 0.4
7.7 ? 0.4
11.6 ? 0.8
23.6 ? 3.1
46.4
29.3
55.9
30.7
30.9
26.8
17.5
5.4
17.9
6.3
16.5
16.1
9.2 ? 0.2
7.4 ? 0.3
7.6 ? 0.3
5.8 ? 0.2
5.8 ? 0.1
36.2
26.8
33.6
28.4
26.8
13.5
6.1
11.4
4.3
2.4
overlaps the mean of the best result). On the right we also show the average rank across datasets,
relative to unsupervised BOW (bold indicates the best method). We highlight the unsupervised
WMD in blue (WMD) and our new result in red (S-WMD). Despite the very large number of competitive baselines, S-WMD achieves the lowest kNN test error on 5/8 datasets, with the exception
of BBCSPORT, CLASSIC and AMAZON. On these datasets it achieves the 4th lowest on BBCSPORT
and CLASSIC, and tied at 2nd on 20 NEWS. On average across all datasets it outperforms all other
26 methods. Another observation is that S-WMD right after initialization (S-WMD init.) performs
quite well. However, as training S-WMD is efficient (shown in Table 3), it is often well worth the
training time.
For unsupervised baselines, on datasets BBCSPORT
and OHSUMED, where the previous state-of-the-art
MazdaTIFFBoone
fit motherboardhappening
WMD was beaten by LSI, S-WMD reduces the eraskedautomotivehomosexuals
playoff motorcycles
ror of LSI relatively by 51% and 22%, respectively.
gay
dolphins computer animation
western
atheism
moon
In general, supervision seems to help all methods
Keith
controller
tappedmotif graphics clippersecurity
on average. One reason why NCA with a TF-IDF
orbitIsrael pro
lists Rutgershell
document representation may be performing better
talkNHL driver
hockeybiblical
firedbikerguns
auto saint
than S-WMD could be because of the long docu- virtual
gunaltcircuit
autos
sell
cute
ment lengths in BBCSPORT and OHSUMED. Havlabelride
riderrocket
Islamic
offerflight
ing denser BOW vectors may improve the inverse riding
keyDOD
cardrivers
document frequency weights, which in turn may be IDEshipping
baseballcrypto
image
bikes
a good initialization for NCA to further fine-tune.
monitor
story
Armenian card
polygon
Israeli
forsalefirearmsspace
On datasets with smaller documents such as TWITwarning
Turkishcopyencryption RISC
bus mouse
TER , CLASSIC , and REUTERS, S-WMD outperforms
compatiblemotorcycle summarized
powerbookelectronics diamond
NCA with TF-IDF relatively by 10%, 42%, and
SCSI government chip
sun doctorNASA
15%, respectively. On CLASSIC WMD outperforms
DOS
S-WMD possibly because of a poor initialization
and that S-WMD uses the squared Euclidean dis- Figure 2: The Top-100 words upweighted by
tance between word vectors, which may be subop- S-WMD on 20 NEWS.
timal for this dataset. This however, does not occur
for any other dataset.
apple
bike
windows
sale
mac
Visualization. Figure 1 shows a 2D embedding of the test split of each dataset by WMD and
S-WMD using t-Stochastic Neighbor Embedding (t-SNE) [33]. The quality of a distance can be
visualized by how clustered points in the same class are. Using this metric, S-WMD noticeably
improves upon WMD on almost all the 8 datasets. Figure 2 visualizes the top 100 words with
7
1/1
largest weights learned by S-WMD on the 20 NEWS dataset. The size of each word is proportional
its learned weight. We can observe that these upweighted words are indeed most representative for
the true classes of this dataset. More detailed results and analysis can be found in the supplementary.
Training time. Table 3 shows the training
times for S-WMD. Note that the time to learn
the initial metric A is not included in time
shown in the second column. Relative to the
initialization, S-WMD is surprisingly fast. This
is due to the fast gradient approximation and
the batch gradient descent introduced in Section 3.1 and 3.2. We note that these times are
comparable or even faster than the time it takes
to train a linear metric on the baseline methods
after PCA.
5
Table 3: Distance computation times.
F ULL T RAINING T IMES
DATASET
BBCSPORT
TWITTER
RECIPE
OHSUMED
CLASSIC
REUTERS
AMAZON
20 NEWS
Related Work
METRICS
S - WCD / S - WMD INIT.
1M 25S
28M 59S
23M 21S
46M 18S
1H 18M
2H 7M
2H 15M
14M 42S
S - WMD
4M 56S
7M 53S
23M 58S
29M 12S
36M 22S
34M 56S
20M 10S
1H 55M
Metric learning is a vast field that includes both
supervised and unsupervised techniques (see
Yang & Jin [37] for a large survey). Alongside NCA [15], described in Section 2, there are a number of popular methods for generalized Euclidean metric learning. Large Margin Nearest Neighbors
(LMNN) [36] learns a metric that encourages inputs with similar labels to be close in a local region,
while encouraging inputs with different labels to be farther by a large margin. Information-Theoretic
Metric Learning (ITML) [10] learns a metric by minimizing a KL-divergence subject to generalized
Euclidean distance constraints. Cuturi & Avis [7] was the first to consider learning the ground distance in the Earth Mover?s Distance (EMD). In a similar work, Wang & Guibas [34] learns a ground
distance that is not a metric, with good performance in certain vision tasks. Most similar to our
work Wang et al. [35] learn a metric within a generalized Euclidean EMD ground distance using
the framework of ITML for image classification. They do not, however, consider re-weighting the
histograms, which allows our method extra flexibility. Until recently, there has been relatively little
work towards learning supervised word embeddings, as state-of-the-art results rely on making use
of large unlabeled text corpora. Tang et al. [32] propose a neural language model that uses label
information from emoticons to learn sentiment-specific word embeddings.
6
Conclusion
We proposed a powerful method to learn a supervised word mover?s distance, and demonstrated
that it may well be the best performing distance metric for documents to date. Similar to WMD,
our S-WMD benefits from the large unsupervised corpus, which was used to learn the word2vec
embedding [22, 23]. The word embedding gives rise to a very good document distance, which
is particularly forgiving when two documents use syntactically different but conceptually similar
words. Two words may be similar in one sense but dissimilar in another, depending on the articles in
which they are contained. It is these differences that S-WMD manages to capture through supervised
training. By learning a linear metric and histogram re-weighting through the optimal transport of
the word mover?s distance, we are able to produce state-of-the-art classification results efficiently.
Acknowledgments
The authors are supported in part by the, III-1618134, III-1526012, IIS-1149882 grants from the
National Science Foundation and the Bill and Melinda Gates Foundation. We also thank Dor Kedem
for many insightful discussions.
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9
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5,680 | 614 | Explanation-Based Neural Network Learning
for Robot Control
Tom M. Mitchell
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
E-mail: mitchell@cs.cmu.edu
Sebastian B. Thrun
University of Bonn
Institut fUr Infonnatik III
ROmerstr. 164, D-5300 Bonn, Germany
thrnn@uran.informatik.uni-bonn.de
Abstract
How can artificial neural nets generalize better from fewer examples? In order
to generalize successfully, neural network learning methods typically require
large training data sets. We introduce a neural network learning method that
generalizes rationally from many fewer data points, relying instead on prior
knowledge encoded in previously learned neural networks. For example, in robot
control learning tasks reported here, previously learned networks that model the
effects of robot actions are used to guide subsequent learning of robot control
functions. For each observed training example of the target function (e.g. the
robot control policy), the learner explains the observed example in terms of its
prior knowledge, then analyzes this explanation to infer additional information
about the shape, or slope, of the target function. This shape knowledge is used
to bias generalization when learning the target function. Results are presented
applying this approach to a simulated robot task based on reinforcement learning.
1 Introduction
Neural network learning methods generalize from observed training data to new cases based
on an inductive bias that is similar to smoothly interpolating between observed training
points. Theoretical results [Valiant, 1984], [Baum and Haussler, 1989] on learnability, as
well as practical experience, show that such purely inductive methods require significantly
larger training data sets to learn functions of increasing complexity. This paper introduces
explanation-based neural network learning (EBNN), a method that generalizes successfully
from fewer training examples, relying instead on prior knowledge encoded in previously
learned neural networks.
EBNN is a neural network analogue to symbolic explanation-based learning methods (EBL)
[DeJong and Mooney, 1986], [Mitchell et al., 19861 Symbolic EBL methods generalize
based upon pre-specified domain knowledge represented by collections of symbolic rules.
287
288
Mitchell and Thrun
For example. in the task of learning general rules for robot control EBL can use prior
know ledge about the effects of robot actions to analytically generalize from specific training
examples of successful control actions. This is achieved by a. observing a sequence of states
and actions leading to some goal. b. explaining (i.e.? post-facto predicting) the outcome
of this sequence using the domain theory. then c. analyzing this explanation in order
to determine which features of the initial state are relevant to achieving the goal of the
sequence. and which are nol In previous approaches to EBL. the initial domain knowledge
has been represented symbolically. typically by propositional rules or hom clauses. and has
typically been assumed to be complete and correct
2 EBNN: Integrating inductive and analytical learning
EBNN extends explanation-based learning to cover situations in which prior knowledge
(also called the domain theory) is approximate and is itself1earned from scratch. In EBNN,
this domain theory is represented by real-valued neural networks. By using neural network
representations. it becomes possible to learn the domain theory using training algorithms
such as the Backpropagation algorithm [Rumelhart et al., 19861 In the robot domains
addressed in this paper. such domain theory networks correspond to action models. i.e.,
networks that model the effect of actions on the state of the world M:s x a - + Sf (here
a denotes an action, s a state. and Sf the successor state). This domain theory is used by
EBNN to bias the learning of the 'robot control function. Because the action models may be
only approximately correct. we require that EBNN be robust with respect to severe errors
in the domain theory.
The remainder of this section describes the EBNN learning algorithm. Assume that the
robot agent's action space is discrete. and that its domain knowledge is represented by a
collection of pre-trained action models Mi:S - + Sf. one for each discrete action i. The
learning task of the robot is to learn a policy for action selection that maximizes the reward,
denoted by R. which defines the task. More specifically. the agent has to learn an evaluation
function Q(s, a). which measures the cumulativefuture expected reward when action a is
executed at state s. Once learned. the function Q(s, a) allows the agent to select actions
that maximize the reward R (greedy policy). Hence learning control reduces to learning
the evaluation function Q. 1
How can the agent use its previously learned action models to focus its learning of Q?
To illustrate. consider the episode shown in Figure 1. The EBNN learning algorithm for
learning the target function Q consists of two components, an inductive learning component
and an analytical learning component.
2.1 The inductive component of EBNN
The observed episode is used by the agent to construct training examples, denoted by
for the evaluation function Q:
Q(sl,ad:= R
Q(s2,a2):= R
Q,
Q(s3,a3):= R
Q could for example be realized by a monolithic neural network, or by a collection of networks trained with the Backpropagation training procedure. As observed training episodes
are accumulated, Q will become increasingly accurate. Such pure inductive learning typIThis approach to learning a policy is adopted from recent research on reinforcenuml learning
[Barto et al., 1991].
Explanation-Based Neural Network Learning for Robot Control
_-----~~ reward: R
(goal state)
Figure 1: Episode: Starting with the initial state SI. the action sequence aI, az, a3 was observed to
produce the final reward R. The domain knowledge represented by neural network action models is
used to post-facto predict and analyze each step of the observed episode.
ically requires large amounts of training data (which will be costly in the case of robot
learning).
2.2
The analytical component or EBNN
In EBNN, the agent exploits its domain lrnowledge to extract additional shape lrnowledge
about the target function Q. to speed convexgence and reduce the number of training
examples required. This shape lrnowledge. represented by the estimated slope of the target
function Q. is then used to guide the generalization process. More specifically. EBNN
combines the above inductive learning component with an analytical learning component
that performs the following three steps for each observed training episode:
1. Explain: Post-facto predict the obsexved episode (states and final reward), using the
action models Mi (c.f. Fig. 1). Note that thexe may be a deviation between predicted
and observed states. since the domain lrnowledge is only approximately correct.
2. Analyze: Analyze the explanation to estimate the slope of the target function for
1..3). i.e.? extract the derivative of the
each observed state-action pair (81:, a1:) (k
final reward R with respect to the features of the states 81:. according to the action
models Mi. For instance. consider the explanation of the episode shown in Fig. 1.
The domain theory networks Mi represent differentiable functions. Therefore it is
possible to extract the derivative of the final reward R with respect to the preceding
state 83. denoted by "V '3R. Using the chain rule of differentiation. the derivatives of
the final reward R with respect to all states 81: can be extracted. These derivatives
"V,,, R describe the dependence of the final reward upon features of the previous states.
They provide the target slopes. denoted by "V,,, Q. for the target function Q:
=
-
(
"V '3 Q 83, a3)
"V '3 R
"V'2 R
8M43 (S3)
0
83
oM4,(83) OM42 (82)
OS3
082
OM43 (83) 8M42 (82) 8M41 (81)
883
082
881
3. Learn: Update the learned target function to better fit both the target values and target
slopes. Fig. 2 illustrates training information extracted by both the inductive (values)
and the analytical (slopes) components ofEBNN. Assume that the "true" Q-function
289
290
Mitchell and Thrun
Figure2: Fitting slopes: Let/bea target function for which tbreeexampies (Xl, I(xt)}. (X2, I(X2)).
and (X3, 1(X3)) are known. Based on these points the learner might generate the hypothesis g. If the
slopes are also known. the learner can do much better: h.
is shown in Fig. 2a, and that three training instances at Xl, X2 and X3 are given. When
only values are used for learning, i.e., as in standard inductive learning, the learner
might conclude the hypothesis g depicted in Fig. 2b. If the slopes are known as well,
the learner can better estimate the target function (Fig. 2c). From this example it is
clear that the analysis in EBNN may reduce the need for training data, provided that
the estimated slopes extracted from the explanations are sufficiently accurate.
In EBNN, the function Q is learned by a real-valued function approximator that fits both
the target values and target slopes. If this approximator is a neural network, an extended
version of the Backpropagation algorithm can be employed to fit these slope constraints
as well, as originally shown by [Simard et al., 19921 Their algorithm "Tangent Prop"
extends the Backpropagation error function by a second term measuring the mean
square error of the slopes. Gradient descent in slope space is then combined with
Backpropagation to minimize both error functions. In the experiments reported here,
however, we used an instance-based function approximation technique described in
Sect. 3.
2.3 Accommodating imperfect domain theories
Notice that the slopes extracted from explanations will be only approximately correct, since
they are derived from the approximate action models Mi. If this domain knowledge is
weak, the slopes can be arbitrarily poor, which may mislead generalization.
EBNN reduces this undesired effect by estimating the accuracy of the extracted slopes
and weighting the analytical component of learning by these estimated slope accuracies.
Generally speaking, the accuracy of slopes is estimated by the prediction accuracy of the
explanation (this heuristic has been named LOB *). More specifically, each time the domain
theory is used to post-facto predict a state sk+1, its prediction st~icted may deviate from the
observed state sr+ied ? Hence the I-step prediction accuracy at state Sk, denoted by Cl (i),
is defined as 1 minus the normalized prediction error:
( .)
Cl Z
:=
1 _ II st~cted - skb+red II
max..prediction...error
For a given episode we define the n-step accuracy cn(i) as the product of the I-step
accuracies in the next n steps. The n-step accuracy, which measures the accuracy of the
deri ved slopes n steps away from the end of the episode, posseses three desireable properties:
a. It is I if the learned domain theory is perfectly correct, b. it decreases monotonically as
the length of the chain of inferences increases, and c. it is bounded below by O. The n-step
accuracy is used to determine the ratio with which the analytical and inductive components
Explanation-Based Neural Network Learning for Robot Control
are weighted when learning the target concept. If an observation is n steps away from the
end of the episode. the analytically derived training information (slopes) is weighted by
the n-step accuracy times the weight of the inductive component (values). Although the
experimental results reported in section 3 are promising. the generality of this approach is
an open question. due to the heuristic nature of the assumption LOB *.
2.4
EBNN and Reinforcement Learning
To make EBNN applicable to robot learning, we extend it here to a more sophisticated
scheme for learning the evaluation function Q. namely Watkins' Q-Learning [Watkins,
1989] combined with Sutton's temporal difference methods [Sutton, 19881 The reason
for doing so is the problem 0/ suboptimal action choices in robot learning: Robots must
explore their environment. i.e., they must select non-optimal actions. Such non-optimal
actions can have a negative impact on the final reward of an episode which results in both
underestimating target values and misleading slope estimates.
Watkins' Q-Learning [Watkins, 1989] permits non-optimal actions during the course of
learning Q. In his algorithm targets for Q are constructed recursively, based on the
maximum possible Q-value at the next state: 2
...
Q(Sk, ak)
=
{ R
,
if k is the final step and R final reward
m~ Q(Sk+l, a) otherwise
a acuon
Here , (O~,~I) is a discount/actor that discounts reward over time, which is commonly
used for minimizing the number of actions. Sutton's TD(A) [Sutton, 1988] can be used to
combine both Watkins' Q-Learning and the non-recursive Q-estimation scheme underlying
the previous section. Here the parameter A(0 ~ A~ 1) determines the ratio between recursive
and non-recursive components:
...
Q(sk,ak)
=
{ R
(I-A),max a Q(sk+l,a)
+
if k final step )
A,Q(sk+l,ak+d otherwise (1
Eq. (1) describes the extended inductive component of the EBNN learning algoriLhm. The
extension of the analytical component in EBNN is straightforward. Slopes are extracted
via the derivative of Eq. (1), which is computed via the derivative of both the models !IIi
and the derivative of Q.
if k last step
otherwise
3 Experimental results
EBNN has been evaluated in a simulated robot navigation domain. The world and the
action space are depicted in Fig. 3a&b. The learning task is to find a Q function, for which
the greedy policy navigates the agent to its goal location (circle) from arbitrary starting
locations, while avoiding collisions with the walls or the obstacle (square). States are
210 order to simplify the notation. we assume that reward is only received at the end of the episode,
and is also modeled by the action models. The extension to more general cases is straightforward.
291
292
Mitchell and Thrun
(a)
(c)
(b)
error
0'*'
0.2
0.15
0.1
robot
0.05
0
number of
training examples
Figure 3: a. The simulated robot world. b. Actions. c. The squared generalization error of the
domain theory networks decreases monotonically as the amount of training data increases. These
nine alternative domain theories were used in the experiments.
described by the local view of the agent. in terms of distances and angles to the center of
the goal and to the center of the obstacle. Note that the world is deterministic in these
experiments, and that there is no sensor noise.
We applied Watkins' Q-Learning and TD(~) as described in the previous section with
A=0.7 and a discount factor ;=0.8. Each of the five actions was modeled by a separate
neural network (12 hidden units) and each had a separate Q evaluation function. The
latter functions were represented by a instance-based local approximation technique. In a
nutshell, this technique memorizes all training instances and their slopes explicitly, and fits
a local quadratic model over the [=3 nearest neighbors to the query point, fitting both target
values and target slopes. We found empirically that this technique outperformed Tangent
Prop in the domain at hand.3 We also applied an experience replay technique proposed by
Lin [Lin, 19911 in order to optimally exploit the information given by the observed training
episodes.
Fig. 4 shows average performance curves for EBNN using nine different domain theories
(action models) trained to different accuracies, with (Fig. 4a) and without (Fig. 4b) taking
the n-step accuracy of the slopes into account. Fig. 4a shows the main result. It shows
clearly that (1) EBNN outperfonns purely inductive learning, (2) more accurate domain
theories yield better performance than less accurate theories, and (3) EBNN learning degrades gracefully as the accuracy of the domain theory decreases, eventually matching the
performance of purely inductive learning. In the limit, as the size of the training data set
grows, we expect all methods to converge to the same asymptotic performance.
4 Conclusion
Explanation-based neural network learning. compared to purely inductive learning, generalizes more accurately from less training data. It replaces the need for large training
data sets by relying instead on a previously learned domain theory. represented by neural
networks. In this paper, EBNN has been described and evaluated in terms of robot learning
tasks. Because the learned action models Mi are independent of the particular control task
(reward function), this knowledge acquired during one task transfers directly to other tasks.
3Note that in a second experiment not reported here, we applied EBNN using neural network
representation for Q and Tangent Prop successfully in a real robot domain.
Explanation-Based Neural Network Learning for Robot Control
(a)
Prob(success)
1
0.8
0.6
0 .4
0.2
100
(b)
n\lI1lber of
epiSOdes
Prob(success)
1
0.8
0. 6
0.4
0.2
" ...... ..
~.
"
0
20
40
60
80
~
,"",_10
100
nU!Jlberof
episodes
Figure 4: How does domain knowledge improve generalization? a. Averaged results for EBNN
domain theories of differing accuracies, pre-trained with from 5 to 8 192 training examples for each
action model network. In contrast, the bold grey line reflects the learning curve for pure inductive
learning, i.e., Q-Leaming and TD(A). b. Same experiments, but without weighting the analytical
component of EBNN by its accuracy, illustrating the importance of the WB* heuristic. All curves
are averaged over 3 runs and are also locally window-averaged. The perfonnance (vertical axis) is
measured on an independent test set of starting positions.
EBNN differs from other approaches to knowledge-based neural network learning, such
as Shavlik/fowell's KBANNs [Shavlik and Towell, 1989]. in that the domain knowledge
and the target function are strictly separated, and that both are learned from scratch. A
major difference from other model-based approaches to robot learning, such as Sutton's
DYNA architecture [Sutton, 1990] or Jordan!Rumelhart's distal teacher method [Jordan
and Rumelhart, 1990], is the ability of EBNN to operate across the spectrum of strong to
weak domain theories (using LOB*). EBNN has been found to degrade gracefully as the
accuracy of the domain theory decreases.
We have demonstrated the ability of EBNN to transfer knowledge among robot learning
tasks. However, there are several open questions which will drive future research, the
most significant of which are: a. Can EBNN be extended to real-valued, parameterized
293
294
Mitchell and Thrun
action spaces? So far we assume discrete actions. b. Can EBNN be extended to handle
first-order predicate logic. which is common in symbolic approaches to EBL? c. How will
EBNN perform in highly stochastic domains? d. Can knowledge other than slopes (such
as higher order derivatives) be extracted via explanations? e. Is it feasible to automatically
partition/modularize the domain theory as well as the target function, as this is the case with
symbolic EBL methods? More research on these issues is warranted.
Acknowledgments
We thank Ryusuke Masuoka, Long-Ji Lin. the CMU Robot Learning GrouP. Jude Shavlik,
and Mike Jordan for invaluable discussions and suggestions. This research was sponsored in
part by the Avionics Lab, Wright Research and Development Center. Aeronautical Systems
Division (APSC). U. S. Air Force. Wright-Patterson AFB. OH 45433-6543 under Contract
F33615-90-C-1465.Arpa Order No. 7597 and by a grant from Siemens Corporation.
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| 614 |@word illustrating:1 version:1 open:2 grey:1 minus:1 recursively:1 initial:3 si:1 must:2 john:1 ronald:1 subsequent:1 partition:1 shape:4 sponsored:1 update:1 greedy:2 fewer:3 selected:1 underestimating:1 ebnn:35 location:2 five:1 constructed:1 become:1 consists:1 combine:2 fitting:2 introduce:1 acquired:1 expected:1 planning:1 relying:3 td:3 automatically:1 window:1 increasing:1 becomes:1 provided:1 estimating:1 bounded:1 underlying:1 maximizes:1 notation:1 what:1 dejong:3 differing:1 differentiation:1 corporation:1 temporal:2 nutshell:1 facto:4 control:13 unit:1 grant:1 monolithic:1 local:3 limit:1 sutton:9 ak:3 analyzing:1 reacting:1 approximately:3 might:2 discriminability:1 mateo:1 specifying:1 nol:1 averaged:3 practical:1 acknowledgment:1 lecun:1 recursive:3 differs:1 backpropagation:5 x3:3 procedure:1 significantly:1 matching:1 pre:3 integrating:1 cted:1 symbolic:5 selection:1 applying:1 deterministic:1 demonstrated:1 center:3 baum:3 straightforward:2 williams:1 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college:1 internal:1 latter:1 avoiding:1 |
5,681 | 6,140 | Graph Clustering: Block-models and model free
results
Marina Meil?a
Department of Statistics
University of Washington
Seattle, WA 98195-4322, USA
mmp@stat.washington.edu
Yali Wan
Department of Statistics
University of Washington
Seattle, WA 98195-4322, USA
yaliwan@washington.edu
Abstract
Clustering graphs under the Stochastic Block Model (SBM) and extensions are
well studied. Guarantees of correctness exist under the assumption that the data
is sampled from a model. In this paper, we propose a framework, in which we
obtain ?correctness? guarantees without assuming the data comes from a model.
The guarantees we obtain depend instead on the statistics of the data that can be
checked. We also show that this framework ties in with the existing model-based
framework, and that we can exploit results in model-based recovery, as well as
strengthen the results existing in that area of research.
1
Introduction: a framework for clustering with guarantees without model
assumptions
In the last few years, model-based clustering in networks has witnessed spectacular progress. At
the central of intact are the so-called block-models, the Stochastic Block Model (SBM), DegreeCorrected SBM (DC-SBM) and Preference Frame Model (PFM). The understanding of these models
has been advanced, especially in understanding the conditions when recovery of the true clustering
is possible with small or no error. The algorithms for recovery with guarantees have also been
improved. However, the impact of the above results is limited by the assumption that the observed
data comes from the model.
This paper proposes a framework to provide theoretical guarantees for the results of model based
clustering algorithms, without making any assumption about the data generating process. To describe the idea, we need some notation. Assume that a graph G on n nodes is observed. A modelbased algorithm clusters G, and outputs clustering C and parameters M(G, C).
The framework is as follows: if M(G, C) ?ts the data G well, then we shall prove that any other
clustering C ? of G that also ?ts G well will be a small perturbation of C. If this holds, then C with
model parameters M(G, C) can be said to capture the data structure in a meaningful way.
We exemplify our approach by obtaining model-free guarantees for the SBM and PFM models.
Moreover, we show that model-free and model-based results are intimately connected.
2
Background: graphs, clusterings and block models
Graphs, degrees, Laplacian, and clustering Let G be a graph on n nodes, described by its ad? De?ne d?i = ?n A?ij the degree of node i, and D
? = diag{d?i } the diagonal
jacency matrix A.
j=1
?=D
? ?1/2 A?D
? ?1/2 . In
matrix of the node degrees. The (normalized) Laplacian of G is de?ned as1 L
1
? [10].
Rigorously speaking, the normalized graph Laplacian is I ? L
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
extension, we de?ne the degree matrix D and the Laplacian L associated to any matrix A ? Rn?n ,
with Aij = Aji ? 0, in a similar way.
Let C be a partitioning (clustering) of the nodes of G into K clusters. We use the shorthand notation
i ? k for ?node i belongs to cluster k?. We will represent C by its n ? K indicator matrix Z, de?ned
by
Zik = 1 if i ? k, 0 otherwise, for i = 1, . . . n, k = 1, . . . K.
(1)
T
T ?
Note that Z Z = diag{nk } with nk counting the number of nodes in cluster k, and Z AZ =
[nkl ]K
k,l=1 with nkl counting the edges in G between clusters k and l. Moreover, for two indicator
matrices Z, Z ? for clusterings C, C ? , (Z T Z ? )kk? counts the number of points in the intersection of
?
? ? )kk? computes ?
cluster k of C with cluster k ? of C ? , and (Z T DZ
i?k?k? di the volume of the same
intersection.
?Block models? for random graphs (SBM, DC-SBM, PFM) This family of models contains
Stochastic Block Models (SBM) [1, 18], Degree-Corrected SBM (DC-SBM) [17] and Preference Frame Models (PFM) [20]. Under each of these model families, a graph G with adjacency matrix A? over n nodes is generated by sampling its edges independently following the law
A?ij ? Bernoulli(Aij ), for all i > j. The symmetric matrix A = [Aij ] describing the graph is the
edge probability matrix. The three model families differ in the constraints they put on an acceptable
A. Let C ? be a clustering. The entries of A are de?ned w.r.t C ? as follows (and we say that A is
compatible with C ? ).
SBM : Aij = Bkl whenever i ? k, j ? l, with B = [Bkl ] ? RK?K symmetric and nonnegative.
DC-SBM : Aij = wi wj Bkl whenever i ? k, j ? l, with B as above and w1 , . . . wn non-negative
weights associated with the graph nodes.
PFM : A satis?es D = diag(A1), D?1 AZ = ZR where 1 denotes the vector of all ones, Z
is the indicator matrix of C ? , and R is a stochastic matrix (R1 = 1, Rkl ? 0), the details
are in [20]
While perhaps not immediately obvious, the SBM is a subclass of the DC-SBM, and the latter a
subclass of the PFM. Another common feature of block-models, that will be signi?cant throughout
this work is that for all three, Spectral Clustering algorithms [15] have been proved to work well
estimating C ? .
3
Main theorem: blueprint and results for PFM, SBM
Let M be a model class, such as SBM, DC-SBM, PFM, and denote M(G, C) ? M to be a model
that is compatible with C and is ?tted in some way to graph G (we do not assume in general that this
?t is optimal).
Theorem 1 (Generic Theorem) We say that clustering C ?ts G well w.r.t M iff M(G, C) is ?close
to? G. If C ?ts G well w.r.t M, then (subject to other technical conditions) any other clustering C ?
which also ?ts G well is close to C, i.e. dist(C, C ? ) is small.
In what follows, we will instantiate this Generic Theorem, and the concepts therein; in
particular the following will be formally de?ned. (1) Model construction, i.e an algorithm
to ?t a model in M to (G, C).
This is necessary since we want our results to be
computable in practice.
(2) A goodness of ?t measure between M(C, G) and the data G.
(3) A distance between clusterings. We adopt the widely used Misclassi?cation Error (or Hamming)
distance de?ned below.
The Misclassi?cation Error (ME) distance between two clusterings C, C ? over the same set of n
points is
?
1
dist(C, C ? ) = 1 ? max
1,
(2)
n ??SK
i?k??(k)
where ? ranges over all permutations of K elements SK , and ?(k) indexes a cluster in C ? . If the
points are weighted by their degrees, a natural measure on the node set, the Weighted ME (wME)
2
distance is
?
1
distd?(C, C ? ) = 1 ? ?n
max
d?i .
? ??SK
i=1 di
i?k??(k)
(3)
?
In the above, i?k?k? d?i represents the total weight of the set of points assigned to cluster k by C
and to cluster k ? ( or ?(k)) by C ? . Note that in the indicator matrix representation of clusterings, this
? ? ? RK?K . While dist is more popular, we believe dist ?
is the (k, k ? ) element of the matrix Z T DZ
d
is more natural, especially when node degrees are dissimilar, as d? can be seen as a natural measure
on the set of nodes, and distd? is equivalent to the earth-mover?s distance.
3.1
Main result for PFM
Constructing a model Given a graph G and a clustering C of its nodes, we wish to construct a
? ? L|| is small.
PFM compatible with C, so that its Laplacian L satis?es that ||L
? be
Let the spectral decomposition of L
??
?
?
?
Y? T
?
0
T
? low Y?low
? Y? T + Y?low ?
?
?
?
= Y? ?
L = [Y Ylow ]
(4)
T
? low
Y?low
0 ?
?1, ? ? ? , ?
? K ), ?
? K+1 , ? ? ? , ?
? n ). To
? = diag(?
? low = diag(?
where Y? ? Rn?K , Y?low ? Rn?(n?K) , ?
?
?
?
ensure that the matrices Y , Ylow are uniquely de?ned we assume throughout the paper that L?s K-th
eigengap, i.e, |?K | ? |?K+1 |, is non-zero.
? 2 | ? . . . ? |?
? K | > |?
? K+1 | ? . . . |?
? n |.
? 1 = 1 ? |?
? satisfy ?
Assumption 1 The eigenvalues of L
Denote the subspace spanned by the columns of M , for any M matrix, by R(M ), and || || the
Euclidean or spectral norm.
PFM Estimation Algorithm
? D,
? L,
? Y? , ?,
? clustering C with indicator matrix Z.
Input Graph G with A,
Output (A, L) = P F M (G, C)
1. Construct an orthogonal matrix derived from Z.
? the column normalization of Z.
? 1/2 ZC ?1/2 , with C = Z TDZ
YZ = D
?
Note Ckk = i?k d?i is the volume of cluster k.
(5)
2. Project YZ on Y? and perform Singular Value Decomposition.
F = YZT Y? = U ?V T
(6)
3. Change basis in R(YZ ) to align with Y? .
Y = YZ U V T .
Complete Y to an orthonormal basis [Y B] of Rn .
(7)
4. Construct Laplacian L and edge probability matrix A.
T
? T + (BB T )L(BB
?
L = Y ?Y
),
? 1/2 LD
? 1/2 .
A = D
(8)
? D,
? L,
? Y? , ?
? and Z be de?ned as above, and (A, L) = P F M (G, C). Then,
Proposition 2 Let G, A,
? and L, or A de?ne a PFM with degrees d?1:n .
1. D
? 1:K .
2. The columns of Y are eigenvectors of L with eigenvalues ?
? 1 = 1.
? 1/2 1 is an eigenvector of both L and L
? with eigenvalue ?
3. D
The proof is relegated to the Supplement, as are all the omitted proofs.
P F M (G, C) is an estimator for the PFM parameters given the clustering. It is evidently not the
Maximum Likelihood estimator, but we can show that it is consistent in the following sense.
3
Proposition 3 (Informal) Assume that G is sampled from a PFM with parameters D? , L? and compatible with C ? , and let L = P F M (G, C ? ). Then, under standard recovery conditions for PFM (e.g
[20]) ||L? ? L|| = o(1) w.r.t. n.
? ? L|| ? ?.
Assumption 2 (Goodness of ?t for PFM) ||L
P F M (G, C) instantiates M(G, C), and Assumption 2 instantiates the goodness of ?t measure. It
remains to prove an instance of Generic Theorem 1 for these choices.
? 1:n as de?ned, and L
? L,
? ?
? satTheorem 4 (Main Result (PFM)) Let G be a graph with d?1:n , D,
isfy Assumption 1. Let C, C ? be two clusterings with K clusters, and L, L? be their corresponding
2
and
Laplacians, de?ned as in (8), and satisfy Assumption 2 respectively. Set ? = (|?? (K?1)?
?
|?|?
|)2
K
K+1
?0 = mink Ckk / maxk Ckk with C de?ned as in (5), where k indexes the clusters of C. Then,
whenever ? ? ?0 ,
maxk Ckk
?,
distd?(C, C ? ) ? ?
k Ckk
(9)
with distd? being the weighted ME distance (3).
In the remainder of this section we outline the proof steps, while the partial results of Proposition 5,
6, 7 are proved in the Supplement. First, we apply the perturbation bound called the Sinus Theorem
of Davis and Kahan, in the form presented in Chapter V of [19].
? 1:n , Y be de?ned as usual. If Assumptions 1 and 2 hold, then
Proposition 5 Let Y? , ?
|| diag(sin ?1:K (Y? , Y ))|| ?
?
= ??
?
? K+1 |
| ?K | ? | ?
(10)
where ?1:K are the canonical (or principal) angles between R(Y? ) and R(Y ) (see e.g [8]).
The next step concerns the closeness of Y, Y? in Frobenius norm. Since Proposition 5 bounds the
sinuses of the canonical angles, we exploit the fact that the cosines of the same angles are the singular
values of F = Y T Y? of (6).
? = Y? Y? T and F, ?? as above. Assumptions 1 and 2 imply that
Proposition 6 Let M = Y Y T , M
? T ? K ? (K ? 1)??2 .
1. ||F ||2F = trace M M
? ||2 ? 2(K ? 1)??2 .
2. ||M ? M
F
Now we show that all clusterings which satisfy Proposition 6 must be close to each other in the
weighted ME distance. For this, we ?rst need an intermediate result. Assume we have two clusterings C, C ? , with K clusters, for which we construct YZ , Y, L, M , respectively YZ? , Y ? , L? , M ? as
above. Then, the subspaces spanned by Y and Y ? will be close.
? satisfy Assumption 1 and let C, C ? represent two clusterings for which L, L?
Proposition 7 Let L
satisfy Assumption 2. Then, ||YZT YZ? ||2F ? K ? 4(K ? 1)??2 = K ? ?
The main result now follows from Proposition 7 and Theorem 9 of [13], as shown in the Supplement.
This proof approach is different from the existing perturbation bounds for clustering, which all use
counting arguments. The result of [13] is a local equivalence, which bounds the error we need in
terms of ? de?ned above (?local? meaning the result only holds for small ?).
4
3.2
Main Theorem for SBM
In this section, we offer an instantiation of Generic Theorem 1 for the case of the SBM. As before,
we start with a model estimator, which in this case is the Maximum Likelihood estimator.
SBM Estimation Algorithm
? clustering C with indicator matrix Z.
Input Graph with A,
Output A = SBM (G, C)
1. Construct an orthogonal matrix derived from Z: YZ = ZC ?1/2 with C = Z T Z.
?1
?
2. Estimate the edge probabilities: B = C ?1 Z T AZC
.
3. Construct A from B by A = ZBZ T .
? = C 1/2 BC 1/2 and denote the eigenvalues of B,
? ordered by decreasing magProposition 8 Let B
? be B
? = U ?U T , with U an orthogonal matrix
nitude, by ?1:K . Let the spectral decomposition of B
and ? = diag(?1:K ). Then
1. A is a SBM.
2. ?1:K are the K principal eigenvalues of A. The remaining eigenvalues of A are zero.
3. A = Y ?Y T where Y = YZ U .
Assumption 3 (Eigengap) B is non-singular (or, equivalently, |?K | > 0.
Assumption 4 (Goodness of ?t for SBM) ||A? ? A|| ? ?.
With the model (SBM), estimator, and goodness of ?t de?ned, we are ready for the main result.
? A the K-th
? and ?
Theorem 9 (Main Result (SBM)) Let G be a graph with incidence matrix A,
K
?
?
singular value of A. Let C, C be two clusterings with K clusters, satisfying Assumptions 3 and 4.
2
Set ? = |4K?
? A |2 and ?0 = mink nk / maxk nk , where k indexes the clusters of C. Then, whenever
?
K
? ? ?0 , dist(C, C ? ) ? ? maxk nk /n, where dist represents the ME distance (2).
? ?
? A is not bounded above, and neither is ?. Since the SBM is less
Note that the eigengap of A,
K
?exible than the PFM, we expect that for the same data G, Theorem 9 will be more restrictive than
Theorem 4.
4
4.1
The results in perspective
Cluster validation
Theorems like 4, 9 can provide model free guarantees for clustering. We exemplify this procedure in
the experimental Section 6, using standard spectral clustering as described in e.g [18, 17, 15]. What
is essential is that all the quantities such as ? and ? are computable from the data.
Moreover, if Y is available, then the bound in Theorem 4 can be improved.
? , M >F +(K ? 1)(?? )2 +
Proposition
10 Theorem 4 holds when ? is replaced by ?Y = K? < M
?
? ?
?
?
?
?
2 2(K ? 1)? ||M ? M ||F , with ? = ?/(|?K | ? |?K+1 |) and M, M de?ned in Proposition 6.
4.2
Using existing model-based recovery theorems to prove model-free guarantees
We exemplify this by using (the proof of) Theorem 3 of [20] to prove the following.
Theorem 11 (Alternative result based on [20] for PFM) Under the same conditions as in TheoK?2
.
rem 4, distd?(C, C ? ) ? ?WM , with ?WM = 128 (|?? |?|
?
?
|)2
K
5
K+1
It follows, too, that with the techniques in this paper, the error bound in [20] can be improved by a
factor of 128.
Similarly, if we use the results of [18] we obtain alternative model-free guarantee for the SBM.
? 2 ?L2 ||F ? ?, where L,
? L are the LaplaAssumption 5 (Alternative goodness of ?t for SBM) ||L
?
cians of A and A = SBM (G, C) respectively.
Theorem 12 (Alternative result based on [18] for SBM) Under the same conditions as in Theo2
k nk
.
rem 9, except for replacing Assumption 4 with 5, dist(C, C ? ) ? ?RCY with ?RCY = |??? |4 16 max
n
K
A problem with this result is that Assumption 5 is much stronger than 4 (being in Frobenius norm).
The more recent results of [17] (with unspeci?ed constants) in conjunction with our original As?2 )
sumptions 3, 4, and the assumption that all clusters have equal sizes, give a bound of O(K?2 /?
K
for the SBM; hence our model-free Theorem 9 matches this more restrictive model-based theorem.
4.3
Sanity checks and Extensions
It can be easily veri?ed that if indeed G is sampled from a SBM, or PFM, then for large enough n,
and large enough model eigengap, Assumptions 1 and 2 (or 3 and 4) will hold.
Some immediate extensions and variations of Theorems 4, 9 are possible. For example, one could
replace the spectral norm by the Frobenius norm in Assumptions 2 and 4, which would simplify
some of the proofs. However, using the Frobenius norm would be a much stronger assumption [18]
Theorem 4 holds not just for simple graphs, but in the more general case when A? is a weighted graph
(i.e. a similarity matrix). The theorems can be extended to cover the case when C ? is a clustering
? ? ||L ? L||(1
?
that is ?-worse than C, i.e when ||L? ? L||
? ?).
4.4
Clusterability and resilience
??
Our Theorems also imply the stability of a clustering to perturbations of the graph G. Indeed, let L
?
?
?
?
?
?
?
be the Laplacian of G , a perturbation of G. If ||L ? L|| ? ?, then ||L ? L|| ? 2?, and (1) G is
well ?tted by a PFM whenever G is, and (2) C is ? stable w.r.t G ? , hence C is what some authors [9]
call resilient.
A graph G is clusterable when G can be ?tted well by some clustering C ? . Much work [4, 7] has
been devoted to showing that clusterability implies that ?nding a C close to C ? is computationally
ef?cient. Such results can be obtained in our framework, by exploiting existing recovery theorems
such as [18, 17, 20], which give recovery guarantees for Spectral Clustering, under the assumption of
sampling from the model. For this, we can simply replace the model assumption with the assumption
that there is a C ? for which L (or A) satis?es Assumptions 1 and 2 (or 3 and 4).
5
Related work
To our knowledge, there is no work of the type of Theorem 1 in the literature on SBM, DC-SBM,
PFM. The closest work is by [6] which guarantees approximate recovery assuming G is close to a
DC-SBM.
Spectral clustering is also used for loss-based clustering in (weighted) graphs and some stability
results exist in this context. Even though they measure clustering quality by different criteria, so that
the ? values are not comparable, we review them here. The recent paper of [16], Theorem 1.2 states
? K+1 )/(cK 3 ) then the clustering error2
that if the K-way Cheeger constant of G is ?(k) ? (1 ? ?
opt
distd?(C, C ) ? C/c = ?P SZ . In the current proof, the constant C = 2 ? 105 ; moreover, ?(K)
cannot be computed tractably. In [14], the bound ?M SX depends on ?M SX , the Normalized Cut
scaled by the eigengap. Since both bounds refer to the result of spectral clustering, we can compare
the relationship between ?M SX and ?M SX ; for [14], this is ?M SX = 2?M SX [1 ? ?M SX /(K ? 1)],
2
The results is stronger, bounding the perturbation of each cluster individually by ?P SZ , but it also includes
a factor larger than 1, bounding the error of K-means algorithm.
6
which is about K ? 1 times larger than ? when ? = ?M SX . In [5], dist(C, C ? ) is de?ned in terms of
||YZT ? YZ? ||2F , and the loss is (closely related) to ||A? ? SBM (G, C)||2F . The bound does not take
into account the eigengap, that is, the stability of the subspace Y? itself.
Bootstrap for validating a clustering C was studied in [11] (see also references therein for earlier
work). In [3] the idea is to introduce a statistics, and large deviation bounds for it, conditioned on
sampling from a SBM (with covariates) and on a given C.
6
Experimental evaluation
Experiment Setup Given G, we obtain a clustering C0 by spectral clustering [15]. Then we
calculate clustering C by perturbing C0 with gradually increasing noise. For each C, we construct
PFM (C, G)and SBM(C, G) model, and further compute ?, ? and ?0 . If ? ? ?0 , C is guaranteed to be
stable by the theorems. In the remainder of this section, we describe the data generating process for
the simulated datasets and the results we obtained.
PFM Datasets We generate from PFM model with K = 5, n = 10000, ?1:K =
(1, 0.875, 0.75, 0.625, 0.5). eigengap = 0.48, n1:K = (2000, 2000, 2000, 2000, 2000). The
stochastic matrix R and its stationary distribution ? are shown below. We sample an adjacency
matrix A? from A (shown below).
A
?
25
.12
.17
.18
.28
.79
.02
.06
.03
.10
.71
.23
.00
.02
? .03
?
.16
.69
.00
.06
R = ? .09
? .04
.00
.00
.80
.16
.10
.01
.03
.11
.76
?=
?
?
A?
?
?
?
?
?
Perturbed PFM Datasets A is obtained from the previous model by perturbing its principal
subspace (details in Supplement). Then we sample A? from A.
Lancichinetti-Fortunato-Radicchi (LFR) simulated matrix [12] The LFR benchmark graphs
are widely used for community detection algorithms, due to heterogeneity in the distribution
of node degree and community size. A LFR matrix is simulated with n = 10000, K = 4,
nk = (2467, 2416, 2427, 2690) and ? = 0.2, where ? is the mixing parameter indicating the
fraction of edges shared between a node and the other nodes from outside its community.
? of hyperlinks between weblogs on US politics,
Political Blogs Dataset A directed network A
compiled from online directories by Adamic and Glance [2], where each blog is assigned a political
leaning, liberal or conservative, based on its blog content. The network A contains 1490 blogs.
? T A)
? 3 , which is a smoothed
After erasing the disconnected nodes, n = 983. We study A? = (A
T ?
?
undirected graph. For A A we ?nd no guarantees.
The ?rst two data sets are expected to ?t the PFM well, but not the SBM, while the LFR data is
expected to be a good ?t for a SBM. Since all bounds can be computed on weighted graphs as well,
we have run the experiments also on the edge probability matrices A used to generate the PFM and
perturbed PFM graphs.
The results of these experiments are summarized in Figure 1. For all of the experiments, the clustering C is ensured to be stable by Theorem 4 as the unweighted error grows to a breaking point, then
the assumptions of the theorem fail. In particular, the C0 is always stable in the PFM framework.
7
Comparing ? from Theorem 9 to that from Theorem 4, we ?nd that Theorem 9 (guarantees for SBM)
is much harder to satisfy. All ? values from Theorem 9 are above 1, and not shown.3 In particular,
for the SBM model class, the C cannot be proved stable even for the LFR data.
Note that part of the reason why with the PFM model very little difference from the clustering C0 can
be tolerated for a clustering to be stable is that the large eigengap makes P F M (G, C) differ from
P F M (G, C0 ) even for very small perturbations. By comparing the bounds for A? with the bounds
for the ?weighted graphs? A, we can evaluate that the sampling noise on ? is approximately equal
to that of the clustering perturbation. Of course, the sampling noise varies with n, decreasing for
larger graphs. Moreover, from Political Blogs data, we see that ?smoothing? a graph, by e.g. taking
powers of its adjacency matrix, has a stability inducing effect.
Figure 1:
? denotes a simple graph, while A denotes a
Quantities ?, ?, ?0 from Theorem 4 plotted vs dist(C, C0 ) for various datasets: A
weighted graph (i.e. a non-negative matrix). For the Political Blogs: Truth means C0 is true clustering of [2], spectral means C0 is obtained
from spectral clustering. For SBM, ? is always greater than ?0 .
7
Discussion
This paper makes several contributions. At a high level, it poses the problem of model free validation
in the area of community detection in networks. The stability paradigm is not entirely new, but
using it explicitly with model-based clustering (instead of cost-based) is. So is ?turning around? the
model-based recovery theorems to be used in a model-free framework.
All quantities in our theorems are computable from the data and the clustering C, i.e do not contain undetermined constants, and do not depend on parameters that are not available. As with
distribution-free results in general, making fewer assumptions allows for less con?dence in the conclusions, and the results are not always informative. Sometimes this should be so, e.g when the
data does not ?t the model well. But it is also possible that the ?t is good, but not good enough
to satisfy the conditions of the theorems as they are currently formulated. This happens with the
SBM bounds, and we believe tighter bounds are possible for this model. It would be particularly
interesting to study the non-spectral, sharp thresholds of [1] from the point of view of model-free
recovery. A complementary problem is to obtain negative guarantees (i.e that C is not unique up to
perturbations).
At the technical level, we obtain several different and model-speci?c stability results, that bound the
perturbation of a clustering by the perturbation of a model. They can be used both in model-free
and in existing or future model-based recovery guarantees, as we have shown in Section 3 and in the
experiments. The proof techniques that lead to these results are actually simpler, more direct, and
more elementary than the ones found in previous papers.
3
We also computed ?RCY but the bounds were not informative.
8
References
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[3] Edoardo M. Airoldi, David S. Choi, and Patrick J. Wolfe. Con?dence sets for network structure. Technical Report arXiv:1105.6245, 2011.
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9
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5,682 | 6,141 | An Architecture for Deep, Hierarchical Generative
Models
Philip Bachman
phil.bachman@maluuba.com
Maluuba Research
Abstract
We present an architecture which lets us train deep, directed generative models
with many layers of latent variables. We include deterministic paths between all
latent variables and the generated output, and provide a richer set of connections
between computations for inference and generation, which enables more effective
communication of information throughout the model during training. To improve
performance on natural images, we incorporate a lightweight autoregressive model
in the reconstruction distribution. These techniques permit end-to-end training of
models with 10+ layers of latent variables. Experiments show that our approach
achieves state-of-the-art performance on standard image modelling benchmarks,
can expose latent class structure in the absence of label information, and can
provide convincing imputations of occluded regions in natural images.
1
Introduction
Training deep, directed generative models with many layers of latent variables poses a challenging
problem. Each layer of latent variables introduces variance into gradient estimation which, given
current training methods, tends to impede the flow of subtle information about sophisticated structure
in the target distribution. Yet, for a generative model to learn effectively, this information needs to
propagate from the terminal end of a stochastic computation graph, back to latent variables whose
effect on the generated data may be obscured by many intervening sampling steps.
One approach to solving this problem is to use recurrent, sequential stochastic generative processes
with strong interactions between their inference and generation mechanisms, as introduced in the
DRAW model of Gregor et al. [5] and explored further in [1, 19, 22]. Another effective technique is
to use lateral connections for merging bottom-up and top-down information in encoder/decoder type
models. This approach is exemplified by the Ladder Network of Rasmus et al. [17], and has been
developed further for, e.g. generative modelling and image processing in [8, 23].
Models like DRAW owe much of their success to two key properties: they decompose the process
of generating data into many small steps of iterative refinement, and their structure includes direct
deterministic paths between all latent variables and the final output. In parallel, models with lateral
connections permit different components of a model to operate at well-separated levels of abstraction,
thus generating a hierarchy of representations. This property is not explicitly shared by DRAW-like
models, which typically reuse the same set of latent variables throughout the generative process. This
makes it difficult for any of the latent variables, or steps in the generative process, to individually
capture abstract properties of the data. We distinguish between the depth used by DRAW and the
depth made possible by lateral connections by describing them respectively as sequential depth and
hierarchical depth. These two types of depth are complimentary, rather than competing.
Our contributions focus on increasing hierarchical depth without forfeiting trainability. We combine
the benefits of DRAW-like models and Ladder Networks by developing a class of models which we
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
call Matryoshka Networks (abbr. MatNets), due to their deeply nested structure. In Section 2, we
present the general architecture of a MatNet. In the MatNet architecture we:
? Combine the ability of, e.g. LapGANs [3] and Diffusion Nets [21] to learn hierarchicallydeep generative models with the power of jointly-trained inference/generation1 .
? Use lateral connections, shortcut connections, and residual connections [7] to provide direct
paths through the inference network to the latent variables, and from the latent variables to
the generated output ? this makes hierarchically-deep models easily trainable in practice.
Section 2 also presents several extensions to the core architecture including: mixture-based prior
distributions, a method for regularizing inference to prevent overfitting in practical settings, and a
method for modelling the reconstruction distribution p(x|z) with a lightweight, local autoregressive
model. In Section 3, we present experiments showing that MatNets offer state-of-the-art performance
on standard benchmarks for modelling simple images and compelling qualitative performance
on challenging imputation problems for natural images. Finally, in Section 4 we provide further
discussion of related work and promising directions for future work.
2
The Matryoshka Network Architecture
Matryoshka Networks combine three components: a top-down network (abbr. TD network), a bottomup network (abbr. BU network), and a set of merge modules which merge information from the
BU and TD networks. In the context of stochastic variational inference [10], all three components
contribute to the approximate posterior distributions used during inference/training, but only the TD
network participates in generation. We first describe the MatNet model formally, and then provide a
procedural description of its three components. The full architecture is summarized in Fig. 1.
Latent Variables
Top-down
Network
latent mean
latent logvar
merge state
Merge
Modules
merge module
Bottom-up
Network
TD state
BU state
merge state
(a)
(b)
Figure 1: (a) The overall structure of a Matryoshka Network, and how information flows through
the network during training. First, we perform a feedforward pass through the bottom-up network to
generate a sequence of BU states. Next, we sample the initial latent variables conditioned on the final
BU state. We then begin a stochastic feedforward pass through the top-down network. Whenever
this feedforward pass requires sampling some latent variables, we get the sampling distribution by
passing the corresponding TD and BU states through a merge module. This module draws conditional
samples of the latent variables via reparametrization [10]. These latent samples are then combined
with the current TD state, and the feedforward pass continues. Intuitively, this approach allows the
TD network to invert the bottom-up network by tracking back along its intermediate states, and
eventually recover its original input. (b) Detailed view of a merge module from the network in (a).
This module stacks the relevant BU, TD, and merge states on top of each other, and then passes them
through a convolutional residual module, as described in Eqn. 10. The output has three parts ? the
first provides means for the latent variables, the second provides their log-variances, and the third
conveys updated state information to subsequent merge modules.
1
A significant downside of LapGANs and Diffusion Nets is that they define their inference mechanisms a
priori. This is computationally convenient, but prevents the model from learning abstract representations.
2
2.1
Formal Description
The distribution p(x) generated by a MatNet is encoded in its top-down network. To model p(x), the
TD network decomposes the joint distribution p(x, z) over an observation x and a sequence of latent
variables z ? {z0 , ..., zd } into a sequence of simpler conditional distributions:
X
p(x) =
p(x|zd , ..., z0 )p(zd |zd?1 , ..., z0 )...p(zi |zi?1 , ..., z0 )...p(z0 ),
(1)
(zd ,...,z0 )
which we marginalize with respect to the latent variables to get p(x). The TD network is designed so
that each conditional p(zi |zi?1 , ..., z0 ) can be truncated to p(x|hti ) using an internal TD state hti . See
Eqns. 7/8 in Sec. 2.2 for procedural details.
The distribution q(z|x) used for inference in an unconditional MatNet involves the BU network, TD
network, and merge modules. This distribution can be written:
q(zd , ..., z0 |x) = q(z0 |x)q(z1 |z0 , x)...q(zi |zi?1 , ..., z0 , x)...q(zd |zd?1 , ..., z0 , x),
(2)
q(zi |hm
i+1 )
where each conditional q(zi |zi?1 , ..., z0 , x) can be truncated to
using an internal merge
state hm
i+1 produced by the ith merge module. See Eqns. 10/11 in Sec. 2.2 for procedural details.
MatNets can also be applied to conditional generation problems like inpainting or pixel-wise segmentation. For, e.g. inpainting with known pixels xk and missing pixels xu , the predictive distribution of
a conditional MatNet is given by:
X
p(xu |xk ) =
p(xu |zd , ..., z0 , xk )p(zd |zd?1 , ..., z0 , xk )...p(z1 |z0 , xk )p(z0 |xk ). (3)
(zd ,...,z0 )
m:g
Each conditional p(zi |zi?1 , ..., z0 , xk ) can be truncated to p(zi |hm:g
i+1 ), where hi+1 indicates state in
a merge module belonging to the generator network. Crucially, conditional MatNets include BU
networks and merge modules that participate in generation, in addition to the BU networks and merge
modules used by both conditional and unconditional MatNets during inference/training.
The distribution used for inference in a conditional MatNet is given by:
q(zd , ..., z0 |xk , xu ) = q(zd |zd?1 , ..., z0 , xk , xu )...q(z1 |z0 , xk , xu )q(z0 |xk , xu ),
k
u
q(zi |hm:i
i+1 ),
(4)
hm:i
i+1
where each conditional q(zi |zi?1 , ..., z0 , x , x ) can be truncated to
where
indicates
state in a merge module belonging to the inference network. Note that, in a conditional MatNet the
distributions p(?|?) are not allowed to condition on xu , while the distributions q(?|?) can.
MatNets are well-suited to training with Stochastic Gradient Variational Bayes [10]. In SGVB, one
maximizes a lower-bound on the data log-likelihood based on the variational free-energy:
log p(x) ?
E
[log p(x|z)] ? KL(q(z|x) || p(z)),
z?q(z|x)
(5)
for which p and q must satisfy a few simple assumptions and KL(q(z|x) || p(z)) indicates the KL
divergence between the inference distribution q(z|x) and the model prior p(z). This bound is tight
when the inference distribution matches the true posterior p(z|x) in the model joint distribution
p(x, z) = p(x|z)p(z) ? in our case given by Eqns. 1/3.
For brevity, we only explicitly write the free-energy bound for a conditional MatNet, which is:
log p(xu |xk ) ?
E
log p(xu |zd , ..., z0 , xk ) ?
(6)
q(zd ,...,z0 |xk ,xu )
KL(q(zd , ..., z0 |xk , xu )||p(zd , ..., z0 |xk )).
With SGVB we can optimize the bound in Eqn. 6 using the ?reparametrization trick? to allow easy
backpropagation through the expectation over z ? q(z|xk , xu ). See [10, 18] for more details about
this technique. The bound for unconditional MatNets is nearly identical ? it just removes xk .
2.2
Procedural Description
Structurally, top-down networks in MatNets comprise sequences of modules in which each module
t
fit receives two inputs: a deterministic top-down state hti from the preceding module fi?1
, and some
3
latent variables zi . Module fit produces an updated state hti+1 = fit (hti , zi ; ?t ), where ?t indicates
the TD network?s parameters. By defining the TD modules appropriately, we can reproduce the
architectures for LapGANs, Diffusion Nets, and Probabilistic Ladder Networks [23]. Motivated by
the success of LapGANs and ResNets [7], we use TD modules in which the latent variables are
concatenated with the top-down state, then transformed, after which the transformed values are added
back to the top-down state prior to further processing. If the adding occurs immediately before,
e.g. a ReLU, then the latent variables can effectively gate the top-down state by knocking particular
elements below zero. This allows each stochastic module in the top-down network to apply small
refinements to the output of preceding modules. MatNets thus perform iterative stochastic refinement
through hierarchical depth, rather than through sequential depth as in DRAW2 .
More precisely, the top-down modules in our convolutional MatNets compute:
hti+1 = lrelu(hti + conv(lrelu(conv([hti ; zi ], vit )), wit )),
(7)
where [x; x0 ] indicates tensor concatenation along the ?feature? axis, lrelu(?) indicates the leaky
ReLU function, conv(h, w) indicates shape-preserving convolution of the input h with the kernel w,
and wit /vit indicate the trainable parameters for module i in the TD network. We elide bias terms
for brevity. When working with fully-connected models we use stochastic GRU-style state updates
rather than the stochastic residual updates in Eq. 7. Exhaustive descriptions of the modules can be
found in our code at: https://github.com/Philip-Bachman/MatNets-NIPS.
These TD modules represent each conditional p(zi |zi?1 , ..., z0 ) in Eq. 1 using p(zi |hti ). TD module
fit places a distribution over zi using parameters [?
?i ; log ?
?i2 ] computed as follows:
[?
?i ; log ?
?i2 ] = conv(lrelu(conv(hti , vit )), wit ),
(8)
where we use ?? to distinguish between Gaussian parameters from the generator network and those
from the inference network (see Eqn. 11). The distributions p(?) all depend on the parameters ?t .
Bottom-up networks in MatNets comprise sequences of modules in which each module receives input
only from the preceding BU module. Our BU networks are all deterministic and feedforward, but
sensibly augmenting them with auxiliary latent variables [16, 15] and/or recurrence is a promising
topic for future work. Each non-terminal module fib in the BU network computes an updated state:
hbi = fib (hbi+1 ; ?b ). The final module, f0b , provides means and log-variances for sampling z0 via
reparametrization [10]. To align BU modules with their counterparts in the TD network, we number
them in reverse order of evaluation. We structured the modules in our BU networks to take advantage
of residual connections. Specifically, each BU module fib computes:
hbi = lrelu(hbi+1 + conv(lrelu(conv(hbi+1 , vib )), wib )),
(9)
with operations defined as for Eq. 7. These updates can be replaced by GRUs, LSTMs, etc.
The updates described in Eqns. 7 and 9 both assume that module inputs and outputs are the same
shape. We thus construct MatNets using groups of ?meta modules?, within which module input/output
shapes are constant. To keep our network design (relatively) simple, we use one meta module for
each spatial scale in our networks (e.g. scales of 14x14, 7x7, and fully-connected for MNIST). We
connect meta modules using layers which may upsample, downsample, and change feature dimension
via strided convolution. We use standard convolution layers, possibly with up or downsampling, to
feed data into and out of the bottom-up and top-down networks.
During inference, merge modules compare the current top-down state with the state of the corresponding bottom-up module, conditioned on the current merge state, and choose a perturbation of the
top-down information to push it towards recovering the bottom-up network?s input (i.e. minimize
m b
t
m m
reconstruction error). The ith merge module outputs [?i ; log ?i2 ; hm
i+1 ] = fi (hi , hi , hi ; ? ), where
2
?i and log ?i are the mean and log-variance for sampling zi via reparametrization, and hm
i+1 gives
the updated merge state. As in the TD and BU networks, we use a residual update:
m
m
b
t
i
i
hm
i+1 = lrelu(hi + conv(lrelu(conv([hi ; hi ; hi ], ui )), vi ))
[?i ; log ?i2 ]
i
= conv(hm
i+1 , wi ),
(10)
(11)
2
Current DRAW-like models can be extended to incorporate hierarchical depth, and our models can be
extended to incorporate sequential depth.
4
in which the convolution kernels uii , vii , and wii constitute the trainable parameters of this module.
Each merge module thus computes an updated merge state and then reparametrizes a diagonal
Gaussian using a linear function of the updated merge state.
In our experiments all modules in all networks had their own trainable parameters. We experimented
with parameter sharing and GRU-style state in our convolutional models. The stochastic convolutional
GRU is particularly interesting when applied depth-wise (rather than time-wise as in [19]), as it
implements a stochastic Neural GPU [9] trainable by variational inference and capable of multi-modal
dynamics. We saw no performance gains with these changes, but they merit further investigation.
In unconditional MatNets, the top-most latent variables z0 follow a zero-mean, unit-variance Gaussian
prior, except in our experiments with mixture-based priors. In conditional MatNets, z0 follows a
distribution conditioned on the known values xk . Conditional MatNets use parallel sets of BU
and merge modules for the conditional generator and the inference network. BU modules in the
conditional generator observe a partial input xk , while BU modules in the inference network observe
both xk and the unknown values xu (which the model is trained to predict). The generative BU and
merge modules in a conditional MatNet interact with the TD modules analogously to the BU and
merge modules used for inference. Our models used independent Bernoullis, diagonal Gaussians, or
?integrated? Logistics (see [11]) for the final output distribution p(x|zd , ..., z0 )/p(xu |zd , ..., z0 , xk ).
2.3
Model Extensions
We also develop several extensions for the MatNet architecture. The first is to replace the zero-mean,
unit-variance Gaussian prior over z0 with a Gaussian Mixture Model, which we train simultaneously
with the rest of the model. When using a mixture prior, we use an analytical approximation to the
required KL divergence. For Gaussian distribution q, and Gaussian mixture p with components
{p1 , ..., pk } with uniform mixture weights, we use the KL approximation:
KL(q || p) ? log Pk
i=1
1
e? KL(q || pi )
.
(12)
Our tests with mixture-based priors are only concerned with qualitative behaviour, so we do not
worry about the approximation error in Eqn. 12.
The second extension is a technique for regularizing the inference model to prevent overfitting beyond
that which is present in the generator. This regularization is applied by optimizing:
maximize E
E [log p(x|z)] ? KL(q(z|x) || p(z)) .
(13)
q
x?p(x) z?q(z|x)
This maximizes the free-energy bound for samples drawn from our model, but without changing
their true log-likelihood. By maximizing Eqn. 13, we implicitly reduce KL(q(z|x) || p(z|x)), which
is the gap between the free-energy bound and the true log-likelihood. A similar regularizer can be
constructed for minimizing KL(p(z|x) || q(z|x)). We use (13) to reduce overfitting, and slightly
boost test performance, in our experiments with MNIST and Omniglot.
The third extension off-loads responsibility for modelling sharp local dynamics in images, e.g. precise
edge placements and small variations in textures, from the latent variables onto a local, deterministic
autoregressive model. We use a simplified version of the masked convolutions in the PixelCNN of
[25], modified to condition on the output of the final TD module in a MatNet. This modification is
easy ? we just concatenate the final TD module?s output and the true image, and feed this into a
PixelCNN with, e.g. five layers. A trick we use to improve gradient flow back to the MatNet is to
feed the MatNet?s output directly into each internal layer of the PixelCNN. In the masked convolution
layers, connections to the MatNet output are unrestricted, since they are already separated from the
ground truth by an appropriately-monitored noisy channel. Larger, more powerful mechanisms for
combining local autoregressions and conditioning information are explored in [26].
3
Experiments
We measured quantitative performance of MatNets on three datasets: MNIST, Omniglot [13], and
CIFAR 10 [12]. We used the 28x28 version of Omniglot described in [2], which can be found
at: https://github.com/yburda/iwae. All quantitative experiments measured performance in
5
Figure 2: MatNet performance on quantitative benchmarks. All tables except the lower-right table
describe standard unconditional generative NLL results. The lower-right table presents results from
the structured prediction task in [22], in which 1-3 quadrants of an MNIST digit are visible, and NLL
is measured on predictions for the unobserved quadrants.
Figure 3: Class-like structure learned by a MatNet trained on 28x28 Omniglot, without label
information. The model used a GMM prior over z0 with 50 mixture components. Each group of three
columns corresponds to a mixture component. The top row shows validation set examples whose
posterior over the mixture components placed them into each component. Subsequent rows show
samples drawn by freely resampling latent variables from the model prior, conditioned on the top
k layers of latent variables, i.e. {z0 , ..., zk?1 } being drawn from the approximate posterior for the
example at the top of the column. From the second row down, we show k = {1, 2, 4, 6, 8, 10}.
terms of negative log-likelihood, with the CIFAR 10 scores rescaled to bits-per-pixel and corrected for
discrete/continuous observations as described in [24]. We used the IWAE bound from [2] to evaluate
our models, with 2500 samples in the bound. We performed additional experiments measuring the
qualitative performance of MatNets using Omniglot, CelebA faces [14], LSUN 2015 towers, and
LSUN 2015 churches. The latter three datasets are 64x64 color images with significant detail and
non-trivial structure. Complete hyperparameters for model architecture and optimization can be
found in the code at https://github.com/Philip-Bachman/MatNets-NIPS.
We performed three quantitative tests using MNIST. The first tests measured generative performance
on dynamically-binarized images using a fully-connected model (for comparison with [2, 23]) and on
the fixed binarization from [20] using a convolutional model (for comparison with [25, 19]). MatNets
improved on existing results in both settings. See the tables in Fig. 2. Our third tests with MNIST
measured performance of conditional MatNets for structured prediction. For this, we recreated the
tests described in [22]. MatNet performance on these tests was also strong, though the prior results
were from a fully-connected model, which skews the comparison.
We also measured quantitative performance using the 32x32 color images of CIFAR 10. We trained
two models on this data ? one with a Gaussian reconstruction distribution and dequantization as
described in [24], and the other which added a local autoregression and used the ?integrated Logistic?
likelihood described in [11]. The Gaussian model fell just short of the best previously reported result
for a variational method (from [6]), and well short of the Pixel RNN presented in [25]. Performance
on this task seems very dependent on a model?s ability to predict pixel intensities precisely along
edges. The ability to efficiently capture global structure has a relatively weak benefit. Mistaking a cat
for a dog costs little when amortized over thousands of pixels, while misplacing a single edge can
spike the reconstruction cost dramatically. We demonstrate the strength of this effect in Fig. 4, where
we plot how the bits paid to encode observations are distributed among the modules in the network
over the course of training for MNIST, Omniglot, and CIFAR 10. The plots show a stark difference
between these distributions when modelling simple line drawings vs. when modelling more natural
6
(a)
(b)
(c)
t
Figure 4: This figure shows per module divergences KL(q(zi |hm
i+1 ) || p(zi |hi )) over the course of
training for models trained on MNIST, Omniglot, and CIFAR 10. The stacked area plots are grouped
by ?meta module? in the TD network. The MNIST and Omniglot models both had a single FC
module and meta modules at spatial dimension 7x7 and 14x14. The meta modules at 7x7 and 14x14
both comprised 5 TD modules. The CIFAR10 model (without autoregression) had one FC module,
and meta modules at spatial dimension 8x8, 16x16, and 32x32. These meta modules comprised 2,
4, and 4 modules respectively. Light lines separate modules, and dark lines separate meta modules.
The encoding cost on CIFAR 10 is clearly dominated by the low-level details encoded by the latent
variables in the full-resolution TD modules closest to the output.
images. For CIFAR 10, almost all of the encoding cost was spent in the 32x32 layers of the network
closest to the generated output. This was our motivation for adding a lightweight autoregression
to p(x|z), which significantly reduced the gap between our model and the PixelRNN. Fig. 5 shows
some samples from our model, which exhibit occasional glimpses of global and local structure.
Our final quantitative test used the Omniglot handwritten character dataset, rescaled to 28x28 as in [2].
These tests used the same convolutional architecture as on MNIST. Our model outperformed previous
results, as shown in Fig. 2. Using Omniglot we also experimented with placing a mixture-based prior
distribution over the top-most latent variables z0 . The purpose of these tests was to determine whether
the model could uncover latent class structure in the data without seeing any label information. We
visualize results of these tests in Fig. 3. Additional description is provided in the figure caption. We
placed a slight penalty on the entropy of the posterior distributions for each input to the model, to
encourage a stronger separation of the mixture components. The inputs assigned to each mixture
component (based on their posteriors) exhibit clear stylistic coherence.
In addition to qualitative tests exploring our model?s ability to uncover latent factors of variation
in Omniglot data, we tested the performance of our models at imputing missing regions of higher
resolution images. These tests used images of celebrity faces, churches, and towers. These images
include far more detail and variation than those in MNIST/Omniglot/CIFAR 10. We used two-stage
models for these tests, in which each stage was a conditional MatNet. The first stage formed an
initial guess for the missing image content, and the second stage then refined that guess. Both stages
used the same architectures for their inference and generator networks. We sampled imputation
problems by placing three 20x20 occluders uniformly at random in the image. Each stage had single
TD modules at scales 32x32, 16x16, 8x8, and fully-connected. We trained models for roughly 200k
updates, and show imputation performance on images from a test set that was held out during training.
Results are shown in Fig. 5.
4
Related Work and Discussion
Previous successful attempts to train hierarchically-deep models largely fall into a class of methods
based on deconstructing, and then reconstructing data. Such approaches are akin to solving mazes
by starting at the end and working backwards, or to learning how an object works by repeatedly
disassembling and reassembling it. Examples include LapGANs [3], which deconstruct an image by
repeatedly downsampling it, and Diffusion Nets [21], which deconstruct arbitrary data by subjecting
it to a long sequence of small random perturbations. The power of these approaches stems from the
way in which gradually deconstructing the data leaves behind a trail of crumbs which can be followed
back to a well-formed observation. In the generative models of [3, 21], the deconstruction processes
were defined a priori, which avoided the need for trained inference. This makes training significantly
7
(a) CIFAR 10 samples
(b) CelebA Faces
(c) LSUN Churches
(d) LSUN Towers
Figure 5: Imputation results on challenging, real-world images. These images show predictions
for missing data generated by a two stage conditional MatNet, trained as described in Section 3.
Each occluded region was 20x20 pixels. Locations for the occlusions were selected uniformly at
random within the images. One interesting behaviour which emerged in these tests was that our
model successfully learned to properly reconstruct the watermark for ?shutterstock?, which was a
source of many of the LSUN images ? see the second input/output pair in the third row of (b).
easier, but subverts one of the main motivations for working with latent variables and sample-based
approximate inference, i.e. the ability to capture salient factors of variation in the inferred relations
between latent variables and observed data. This deficiency is beginning to be addressed by, e.g. the
Probabilistic Ladder Networks of [23], which are a special case of our architecture in which the
deterministic paths from latent variables to observations are removed and the conditioning mechanism
in inference is more restricted.
Reasoning about data through the posteriors induced by an appropriate generative model motivates
some intriguing work at the intersection of machine learning and cognitive science. This work shows
that, in the context of an appropriate generative model, powerful inference mechanisms are capable
of exposing the underlying factors of variation in fairly sophisticated data. See, e.g. Lake et al. [13].
Techniques for training coupled generation and inference have now reached a level that makes it
possible to investigate these ideas while learning models end-to-end [4].
In future work we plan to apply our models to more ?interesting? generative modelling problems,
including more challenging image data and problems in language/sequence modelling. The strong
performance of our models on benchmark problems suggests their potential for solving difficult
structured prediction problems. Combining the hierarchical depth of MatNets with the sequential
depth of DRAW is also worthwhile.
8
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[26] A. van den Oord, N. Kalchbrenner, O. Vinyals, L. Espeholt, A. Graves, and K. Kavucuoglu. Conditional
image generation with pixelcnn decoders. arXiv:1606.05328 [cs.CV], 2016.
9
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5,683 | 6,142 | Data Poisoning Attacks on Factorization-Based
Collaborative Filtering
Bo Li ?
Vanderbilt University
bo.li.2@vanderbilt.edu
Aarti Singh
Carnegie Mellon University
aarti@cs.cmu.edu
Yining Wang ?
Carnegie Mellon University
ynwang.yining@gmail.com
Yevgeniy Vorobeychik
Vanderbilt University
yevgeniy.vorobeychik@vanderbilt.edu
Abstract
Recommendation and collaborative filtering systems are important in modern information and e-commerce applications. As these systems are becoming increasingly
popular in the industry, their outputs could affect business decision making, introducing incentives for an adversarial party to compromise the availability or
integrity of such systems. We introduce a data poisoning attack on collaborative
filtering systems. We demonstrate how a powerful attacker with full knowledge
of the learner can generate malicious data so as to maximize his/her malicious
objectives, while at the same time mimicking normal user behavior to avoid being
detected. While the complete knowledge assumption seems extreme, it enables a
robust assessment of the vulnerability of collaborative filtering schemes to highly
motivated attacks. We present efficient solutions for two popular factorizationbased collaborative filtering algorithms: the alternative minimization formulation
and the nuclear norm minimization method. Finally, we test the effectiveness of our
proposed algorithms on real-world data and discuss potential defensive strategies.
1
Introduction
Recommendation systems have emerged as a crucial feature of many electronic commerce systems. In
machine learning such problems are usually referred to as collaborative filtering or matrix completion,
where the known users? preferences are abstracted into an incomplete user-by-item matrix, and
the goal is to complete the matrix and subsequently make new item recommendations for each
user. Existing approaches in the literature include nearest-neighbor methods, where a user?s (item?s)
preference is determined by other users (items) with similar profiles [1], and factorization-based
methods where the incomplete preference matrix is assumed to be approximately low-rank [2, 3].
As recommendation systems play an ever increasing role in current information and e-commerce
systems, they are susceptible to a risk of being maliciously attacked. One particular form of attacks
is called data poisoning, in which a malicious party creates dummy (malicious) users in a recommendation system with carefully chosen item preferences (i.e., data) such that the effectiveness or
credibility of the system is maximally degraded. For example, an attacker might attempt to make
recommendations that are as different as possible from those that would otherwise be made by
the recommendation system. In another, more subtle, example, the attacker is associated with the
producer of a specific movie or product, who may wish to increase or decrease the popularity of a
certain item. In both cases, the credibility of a recommendation system is harmed by the malicious
activities, which could lead to significant economic loss. Due to the open nature of recommendation
?
Both authors contribute equally
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
systems and their reliance on user-specified judgments for building profiles, various forms of attacks
are possible and have been discussed, such as the random attack and random product push/nuke
attack [4, 5]. However, these attacks are not formally analyzed and cannot be optimized according
to specific collaborative filtering algorithms. As it is not difficult for attackers to determine the
defender?s filtering algorithm or even its parameters settings (e.g., through insider attacks), this can
lead one to significantly under-estimate the attacker?s ability and result in substantial loss.
We present a systematic approach to computing near-optimal data poisoning attacks for factorizationbased collaborative filtering/recommendation models. We assume a highly motivated attacker with
knowledge of both the learning algorithms and parameters of the learner following the Kerckhoffs?
principle to ensure reliable vulnerability analysis in the worst case. We focus on two most popular
algorithms: alternating minimization [6] and nuclear norm minimization [3]. Our main contributions
are as follows:
? Comprehensive characterization of attacker utilities: We characterize several attacker
utilities, which include availability attacks, where prediction error is increased, and integrity
attacks, where item-specific objectives are considered. Optimal attack strategies for all
utilities can be computed under a unified optimization framework.
? Novel gradient computations: Building upon existing gradient-based data poisoning frameworks [7, 8, 9], we develop novel methods for gradient computation based on first-order
KKT conditions for two widely used algorithms: alternating minimization [6] and nuclear
norm minimization [2]. The resulting derivations are highly non-trivial; in addition, to
our knowledge this work is the first to give systematic data poisoning attacks for problems
involving non-smooth nuclear norm type objectives.
? Mimicking normal user behaviors: For data poisoning attacks, most prior work focuses
on maximizing attacker?s utility. A less investigated problem is how to synthesize malicious
data points that are hard for a defender to detect. In this paper we provide a novel technique
based on stochastic gradient Langevin dynamics optimization [10] to produce malicious
users that mimic normal user behaviors in order to avoid detection, while achieving attack
objectives.
Related Work: There has been extensive prior research concerning the security of machine learning
algorithms [11, 12, 13, 14, 15]. Biggio et al. pioneered the research of optimizing malicious datadriven attacks for kernel-based learning algorithms such as SVM [16]. The key optimization technique
is to approximately compute implicit gradients of the solution of an optimization problem based on
first-order KKT conditions. Similar techniques were later generalized to optimize data poisoning
attacks for several other important learning algorithms, such as Lasso regression [7], topic modeling
[8], and autoregressive models [17]. The reader may refer to [9] for a general algorithmic framework
of the abovementioned methods.
In terms of collaborative filtering/matrix completion, there is another line of established research that
focuses on robust matrix completion, in which a small portion of elements or rows in the underlying
low-rank matrix is assumed to be arbitrarily perturbed [18, 19, 20, 21]. Specifically, the stability
of alternating minimization solutions was analyzed with respect to malicious data manipulations in
[22]. However, [22] assumes a globally optimal solution of alternating minimization can be obtained,
which is rarely true in practice.
2
Preliminaries
We first set up the collaborative filtering/matrix completion problem and give an overview of existing
low-rank factorization based approaches. Let M ? Rm?n be a data matrix consisting of m rows and
n columns. Mij for i ? [m] and j ? [n] would then correspond to the rating the ith user gives for the
jth item. We use ? = {(i, j) : Mij is observed} to denote all observable entries in M and assume
that |?| mn. We also use ?i ? [n] and ?0j ? [m] for columns (rows) that are observable at the
ith row (jth column). The goal of collaborative filtering (also referred to as matrix completion in the
statistical learning literature [2]) is then to recover the complete matrix M from few observations
M? .
The matrix completion problem is in general ill-posed as it is impossible to complete an arbitrary
matrix with partial observations. As a result, additional assumptions are imposed on the underlying
data matrix M. One standard assumption is that M is very close to an m ? n rank-k matrix with
2
k min(m, n). Under such assumptions, the complete matrix M can be recovered by solving the
following optimization problem:
min
X?Rm?n
kR? (M ? X)k2F ,
s.t. rank(X) ? k,
(1)
P
where kAk2F = i,j A2ij denotes the squared Frobenious norm of matrix A and [R? (A)]ij equals
Aij if (i, j) ? ? and 0 otherwise. Unfortunately, the feasible set in Eq. (1) is non-convex, making
the optimimzation problem difficult to solve. There has been an extensive prior literature on approximately solving Eq. (1) and/or its surrogates that lead to two standard approaches: alternating
minimization and nuclear norm minimization. For the first approach, one considers the following
problem:
min
kR? (M ? UV> )k2F +2?U kUk2F + 2?V kVk2F .
(2)
U?Rm?k ,V?Rn?k
Eq. (2) is equivalent to Eq. (1) when ?U = ?V = 0. In practice people usually set both regularization
parameters ?U and ?V to be small positive constants in order to avoid large entries in the completed
matrix and also improve convergence. Since Eq. (2) is bi-convex in U and V, an alternating
minimization procedure can be applied. Alternatively, one solves a nuclear-norm minimization
problem
min kR? (M ? X)k2F + 2?kXk? ,
(3)
X?Rm?n
Prank(X)
where ? > 0 is a regularization parameter and kXk? = i=1
|?i (X)| is the nuclear norm of X,
which acts as a convex surrogate of the rank function. Eq. (3) is a convex optimization function and
can be solved using an iterative singular value thresholding algorithm [3]. It can be shown that both
methods in Eq. (2) and (3) provably approximate the true underlying data matrix M under certain
conditions [6, 2].
3
The Attack Model
In this section we describe the data poisoning attack model considered in this paper. For a data
matrix consisting of m users and n items, the attacker is capable of adding ?m malicious users to the
training data matrix, and each malicious user is allowed to report his/her preference on at most B
items with each preference bounded in the range [??, ?].
Before proceeding to describe the attacker?s goals, we first introduce some notation to facilitate
f ? Rm0 ?n to denote
presentation. We use M ? Rm?n to denote the original data matrix and M
f and
e be the set of non-zero entries in M
the data matrix of all m0 = ?m malicious users. Let ?
e i ? [n] be all items that the ith malicious user rated. According to our attack models, |?
e i | ? B for
?
f max = max |M
f ij | ? ?. Let ?? (M;
f M) be the optimal solution
every i ? {1, ? ? ? , m0 } and kMk
f
computed jointly on the original and poisoned data matrices (M; M) using regularization parameters
?. For example, Eq. (2) becomes
f M) = arg min kR? (M?UV> )k2F +kR ? (M?
f UV
e > )k2F +2?U (kUk2F +kUk
e 2F )+2?V kVk2F
?? (M;
?
e
U,U,V
(4)
e for normal and malicious users as
where the resulting ? consists of low-rank latent factors U, U
well as V for items. Simiarly, for the nuclear norm minimization formulation in Eq. (3), we have
f M)
?? (M;
=
f ? X)k
e 2F + 2?k(X; X)k
e ?,
arg min kR? (M ? X)k2F + kR?? (M
(5)
e
X,X
c
e . Let M(?)
where ? = (X, X)
be the matrix estimated from learnt model ?. For example, for
c
c
Eq. (4) we have M(?)
= UV> and for Eq. (5) we have M(?)
= X. The goal of the attacker is to
?
f
find optimal malicious users M such that
m0 ?n
f ? ? argmax f R(M(?
c ? (M;
f M)), M).
M
M?M
(6)
f ? R
f max ? ?} is the set of all feasible poisoning attacks
? i | ? B, kMk
Here M = {M
: |?
c M) denotes the attacker?s utility for diverting the collabodiscussed earlier in this section and R(M,
c on an original data set M, with the help of few malicious users
rative filtering algorithm to predict M
f
M. Below we list several typical attacker utilities:
3
Availability attack the attacker wants to maximize the error of the collaborative filtering system,
and eventually render the system useless. Suppose M is the prediction of the collaborative filtering
system without data poisoning attacks.2 The utility function is then defined as the total amount of
c (predictions after poisoning attacks) on unseen entries
perturbation of predictions between M and M
C
? :
c M) = kR?C (M
c ? M)k2F .
Rav (M,
(7)
Integrity attack in this model the attacker wants to boost (or reduce) the popularity of a (subset)
of items. Suppose J0 ? [n] is the subset of items the attacker is interested in and w : J0 ? R is a
pre-specified weight vector by the attacker. The utility function is
c M) =
RJin0 ,w (M,
m X
X
i=1 j?J0
c ij .
w(j)M
(8)
Hybrid attack a hybrid loss function can also be defined:
c M) = ?1 RJav ,w (M,
c M) + ?2 Rin (M,
c M),
RJhybrid
(M,
0
0 ,w,?
(9)
where ? = (?1 , ?2 ) are coefficients that trade off the availability and integrity attack objectives. In
addition, ?1 could be negative, which models the case when the attacker wants to leave a ?light trace":
the attacker wants to make his item more popular while making the other recommendations of the
system less perturbed to avoid detection.
4
Computing Optimal Attack Strategies
We describe practical algorithms to solve the optimization problem in Eq. (6) for optimal attack
f ? that maximizes the attacker?s utility. We first consider the alternating minimization
strategy M
formulation in Eq. (4) and derive a projected gradient ascent method that solves for the corresponding
optimal attack strategy. Similar derivations are then extended to the nuclear norm minimization
formulation in Eq. (5). Finally, we discuss how to design malicious users that mimic normal user
behavior in order to avoid detection.
4.1
Attacking Alternating Minimization
We use the projected gradient ascent (PGA) method for solving the optimization problem in Eq. (6)
f (t) as
with respect to the alternating minimization formulation in Eq. (4): in iteration t we update M
follows:
f (t+1) = Proj M
f (t) + st ? ? f R(M,
c M) ,
M
M
M
(10)
where ProjM (?) is the projection operator onto the feasible region M and st is the step size in
f M) learnt on the joint
c depends on the model ?? (M;
iteration t. Note that the estimated matrix M
f
data matrix, which further depends on the malicious users M. Since the constraint set M is highly
non-convex, we generate B items uniformly at random for each malicious user to rate. The ProjM (?)
operator then reduces to projecting each malicious users? rating vector onto an `? ball of diameter ?,
f at the level of ??.
which can be easily evaluated by truncating all entries in M
c
We next show how to (approximately) compute ?M
f R(M, M). This is challenging because one of
the arguments in the loss function involves an implicit optimization problem. We first apply chain
rule to arrive at
c
f
c
?M
f R(M, M) = ?M
f ?? (M; M)?? R(M, M).
(11)
The second gradient (with respect to ?) is easy to evaluate, as all loss functions mentioned in the
c M) is deferred to
previous section are smooth and differentiable. Detailed derivation of ?? R(M,
Appendix A. On the other hand, the first gradient term term is much harder to evaluate because ?? (?)
is an optimization procedure. Inspired by [7, 8, 9], we exploit the KKT conditions of the optimization
f
problem ?? (?) to approximately compute ?M
f ?? (M; M). More specifically, the optimal solution
e
? = (U, U, V) of Eq. (4) satisfies
? U ui =
X
j??i
(Mij ? u>
i v j )v j ;
2
Note that when the collaborative filtering algorithm and its parameters are set, M is a function of observed
entries R? (M).
4
f via PGA
Algorithm 1 Optimizing M
1: Input: Original partially observed m ? n data matrix M, algorithm regularization parameter ?, attack
budget parameters ?, B and ?, attacker?s utility function R, step size {st }?
t=1 .
f (0) ? M with both ratings and rated items uniformly sampled at random; t = 0.
2: Initialization: random M
f (t) does not converge do
3: while M
f (t) ; M).
4:
Compute the optimal solution ?? (M
c
5:
Compute gradient ?M
f R(M, M) using Eq. (10).
f (t+1) = Proj (M
f (t) + st ? f R).
6:
Update: M
M
M
7:
t ? t + 1.
8: end while
f (t) .
9: Output: m0 ? n malicious matrix M
?i =
?U u
?V v j =
X
ei
j??
X
i??0j
f ij ? u
?>
(M
i v j )v j ;
(Mij ? u>
i v j )ui +
X
e0
i??
j
f ij ? u
?>
(M
ui ,
i v j )?
e and v j is the jth row (also of dimension k)
? i are the ith rows (of dimension k) in U or U
where ui , u
? i , v j } can be expressed as functions of the original and malicious data
in V. Subsequently, {ui , u
f Using the fact that (a> x)a = (aa> )x and M does not change with M,
f we
matrices M and M.
obtain
?1
f
f
?ui (M)
= 0;
f ij
?M
(i)
(j)
?1
f
?v j (M)
(j)
= ? V Ik + ?V
ui .
f ij
?M
Here ?U and ?V are defined as
(i)
?U =
? i (M)
?u
(i)
= ?U Ik + ?U
f ij
?M
X
(j)
vj v>
j ,
?V =
X
vj ;
ui u>
i .
(12)
e0
i??0j ??
j
ei
j??i ??
A framework of the proposed optimization algorithm is described in Algorithm 1.
4.2
Attacking Nuclear Norm Minimization
We extend the projected gradient ascent algorithm in Sec. 4.1 to compute optimal attack strategies
for the nuclear norm minimization formulation in Eq. (5). Since the objective in Eq. (5) is convex,
e can be obtained by conventional convex optimization
the global optimal solution ? = (X, X)
procedures such as proximal gradient descent (a.k.a. singular value thresholding [3] for nuclear norm
e is low rank due to the nuclear norm
minimization). In addition, the resulting estimation (X; X)
e
e V, ?) as an alternative
penalty [2]. Suppose (X; X) has rank ? ? min(m, n). We use ?0 = (U, U,
characterization of the learnt model with a reduced number of parameters. Here X = U?V> and
>
e = U?V
e
e that is, U ? Rm?? , U
e ? Rm0 ?? ,
X
are singular value decompositions of X and X;
V ? Rn?? have orthornormal columns and ? = diag(?1 , ? ? ? , ?? ) is a non-negative diagonal
matrix.
c
To compute the gradient ?M
f R(M, M), we again apply the chain rule to decompose the gradient
into two parts:
0 f
c
c
?M
f R(M, M) = ?M
f ?? (M; M)??0 R(M, M).
(13)
c M) is relatively easier to evaluate. Its
Similar to Eq. (11), the second gradient term ??0 R(M,
derivation details are deferred to the Appendix. In the remainder of this section we shall focus on the
e V, ?) with
computation of the first gradient term, which involves partial derivatives of ?0 = (U, U,
f
respect to malicious users M.
We begin with the KKT condition at the optimal solution ?0 of Eq. (5). Unlike the alternating
minimization formulation, the nuclear norm function k ? k? is not everywhere differentiable. As a
5
f via SGLD
Algorithm 2 Optimizing M
1: Input: Original partially observed m ? n data matrix M, algorithm regularization parameter ?, attack
budget parameters ?, B and ?, attacker?s utility function R, step size {st }?
t=1 , tuning parameter ?, number
of SGLD iterations T .
P
P
m
m
2
2
1
1
2: Prior setup: compute ?j = m
i=1 Mij and ?j = m
i=1 (Mij ? ?j ) for every j ? [n].
(0)
2
0
f ? N (?j , ?j ) for i ? [m ] and j ? [n].
3: Initialization: sample M
ij
4: for t = 0 to T do
f (t) ; M).
5:
Compute the optimal solution ?? (M
c
6:
Compute gradient ?M
f R(M, M) using Eq. (10).
(t+1)
f
7:
Update M
according to Eq. (17).
8: end for
f ? ? arg min f kM
f?M
f (t) k2F . Details in the main text.
9: Projection: find M
M?M
0
?
f
10: Output: m ? n malicious matrix M .
result, the KKT condition relates the subdifferential of the nuclear norm function ?k ? k? as
f ? [X; X]
e ? ??k[X; X]k
e ?.
R?,?? [M; M]
(14)
e The subdifferential of the nuclear
e is the concatenated (m + m0 ) ? n matrix of X and X.
Here [X; X]
norm function ?k ? k? is also known [2]:
n
o
?kXk? = UV> + W : U> W = WV = 0, kWk2 ? 1 ,
where X = U?V> is the singular value decomposition of X. Suppose {ui }, {?
ui } and {v j } are
e
rows of U, U, V and W = {wij }. We can then re-formulate the KKT condition Eq. (14) as follows:
?(i, j) ? ?,
e
?(i, j) ? ?,
Mij
f ij
M
=
u>
i (? + ?I? )v j + ?wij ;
=
?>
u
?ij .
i (? + ?I? )v j + ?w
? , v, ?); the full derivation is deferred to the extended version 3 .
Now we derive ?M
f ? = ?M
f (u, u
4.3
Mimicing Normal User Behaviors
Normal users generally do not rate items uniformly at random. For example, some movies are
significantly more popular than others. As a result, malicious users that pick rated movies uniformly
at random can be easily identified by running a t-test against a known database consisting of only
normal users, as shown in Sec. 5. To alleviate this issue, in this section we propose an alternative
f mimics normal
approach to compute data poisoning attacks such that the resulting malicious users M
c M) for
users M to avoid potential detection, while still achieving reasonably large utility R(M,
the attacker. We use a Bayesian formulation to take both data poisoning and detection avoidance
f captures normal user behaviors and is
objectives into consideration. The prior distribution p0 (M)
defined as a multivariate normal distribution
f =
p0 (M)
0
m Y
n
Y
i=1 j=1
f ij ; ?j , ?j2 ),
N (M
where ?j and ?j2 are mean and variance parameters for the rating of the jth item provided by normal
Pm
1
users. In practice both parameters can be estimated using normal user matrix M as ?j = m
i=1 Mij
P
m
1
2
f
and ? 2 = m
(M
?
?
)
.
On
the
other
hand,
the
likelihood
p(M|
M)
is
defined
as
ij
j
i=1
f =
p(M|M)
1
c M) ,
exp ? ? R(M,
Z
(15)
c M) = R(M(?
c ? (M;
f M)), M) is one of the attacker utility functions defined in Sec. 3,
where R(M,
Z is a normalization constant and ? > 0 is a tuning parameter that trades off attack performance and
3
http://arxiv.org/abs/1608.08182
6
(a)
(b)
(c)
(d)
Figure 1: RMSE/Average ratings for alternating minimization with different percentage of malicious
profiles; (a) ?1 = 1, ?2 = 0, (b) ?1 = 1, ?2 = ?1, (c)?1 = 0, ?2 = 1, (d)?1 = ?1, ?2 = 1.
f toward its prior, which makes the resulting
detection avoidance. A small ? shifts the posterior of M
attack strategy less effective but harder to detect, and vice versa.
f can be
Given both prior and likelihood functions, an effective detection-avoiding attack strategy M
obtained by sampling from its posterior distribution:
f
p(M|M)
=
f
f
p0 (M)p(M|
M)/p(M)
?
?
?
m0 X
n
X
f ij ? ?j )2
(
M
c M)? .
exp ??
+ ?R(M,
2?j2
i=1 j=1
(16)
Posterior sampling of Eq. (16) is clearly intractable due to the implicit and complicated dependency
c on the malicious data M,
f that is, M
c = M(?
c ? (M;
f M))). To circumvent
of the estimated matrix M
this problem, we apply Stochastic Gradient Langevin Dynamics (SGLD, [10]) to approximately
f from its posterior distribution in Eq. (16). More specfically, the SGLD algorithm iteratively
sample M
f (t) }t?0 and in iteration t the new sample M
f (t+1) is
computes a sequence of posterior samples {M
computed as
f (t+1) = M
f (t) + st ? f log p(M|M)
f
M
+ ?t ,
M
2
(17)
where {st }t?0 are step sizes and ?t ? N (0, st I) are independent Gaussian noises injected at each
f
SGLD iteration. The gradient ?M
f log p(M|M) can be computed as
?1
f
f
c
?M
+ ??M
f log p(M|M) = ?(M ? ?)?
f R(M, M),
where ? = diag(?12 , ? ? ? , ?n2 ) and ? is an m0 ? n matrix with ?ij = ?j for i ? [m0 ] and j ? [n]. The
c
other gradient ?M
f R(M, M) can be computed using the procedure in Sections 4.1 and 4.2. Finally,
f (t) is projected back onto the feasible set M by selecting B items
the sampled malicious matrix M
per user with the largest absolute rating and truncating ratings to the level of {??}. A high-level
description of the proposed method is given in Algorithm 2.
5
Experimental Results
To evaluate the effectiveness of our proposed poisoning attack strategy, we use the publicly available
MovieLens dataset which contains 20 millions ratings and 465,000 tag applications applied to 27,000
movies by 138,000 users [23]. We shift the rating range to [?2, 2] for computation convenience. To
avoid the ?cold-start? problem, we consider users who have rated at least 20 movies. Two metrics are
employed to measure the relative performance of the systems before and after data poisoning attacks:
root mean square error (RMSE) for the predicted unseen entries4 and average rating for specific items.
We then analyze the tradeoff between attack performance and detection avoidance, which is controled
by the ? parameter in Eq. (15). This serves as a guide for how ? should be set in later experiments.
We use a paired t-test to compare the distributions of rated items between normal and malicious users.
We present the trend of p-value against different values of ? in the extended version of the paper. To
strive for a good tradeoff, we set ? = 0.6 at which the p-value stablizes around 0.7 and the poisoning
attack performance is not significantly sacrificed.
We employ attack models specified in Eq. (9), where the utility parameters ?1 and ?2 balance two
different malicious goals (availability and integrity) an attacker wishes to achieve. For the integrity
qP
C
c 2
defined as RMSE =
(i,j)??C (Mij ? Mij ) /|? |, where M is the prediction of model trained on
clean data R? (M) only (i.e., without data poisoning attacks).
4
7
(a)
(b)
(c)
(d)
Figure 2: RMSE/Average ratings for nuclear norm minimization with different percentage of malicious profiles; (a) ?1 = 1, ?2 = 0, (b) ?1 = 1, ?2 = ?1, (c)?1 = 0, ?2 = 1, (d)?1 = ?1, ?2 = 1.
utility RJin0 ,w , the J0 set contains only one item j0 selected randomly from all items whose average
predicted ratings are around 0.8. The weight wj0 is set as wj0 = 2. Figure 1 (a) (b) plots the RMSE
after data poisoning attacks. When ?1 = 1, ?2 = 0, the attacker is interested in increasing the RMSE
of the collaborative filtering system and hence reducing the system?s availability. On the other hand,
when ?1 = 1, ?2 = ?1 the attacker wishes to increase RMSE while at the same time keeping the
rating of specific items (j0 ) as low as possible for certain malicious purposes. Figure 1 (b) shows
that when the attackers consider to both objectives (?1 = 1, ?2 = ?1), the RMSE after poisoning is
slightly lower than that if only availability is targeted (?1 = 1, ?2 = 0). In addition, the projected
gradient ascent (PGA) strategy generates the largest RMSE score compared with the other methods.
However, PGA requires malicious users to rate each item uniformly at random, which might expose
the malicious profiles to an informed defender. More specifically, the paired t-test on those malicious
profiles produced by PGA rejects the null hypothesis that the items rated by the attacker strategies
are the same as those obtained from normal users (p < 0.05). In contrast, the SGLD method leads
to slightly worse attacker utility but generates malicious users that are hard to distinguish from the
normal users (for example, the paired t-test leads to inconclusive p-values (larger than 0.7) with
? = 0.6. Finally, both PGA and SGLD result in higher attacker utility compared to uniform attacks,
where both ratings and rated items are sampled uniformly at random for malicious profiles.
Apart from the RMSE scores, we also plot ratings of specific items against percentage of malicious
profiles in Figure 1 (c) (d). We consider two additional attack utility settings: ?1 = 0, ?2 = 1, in
which the attacker wishes to push the ratings of some particular items (specified in w and J0 of Rin )
as high as possible; and ?1 = ?1, ?2 = 1, where the attacker also wants to leave a ?light trace" by
reducing the impact on the entire system resulted from malicious activities. It is clear that targeted
attackes (both PGA and SGLD) are indeed more effective at manipulating ratings of specific items
for integrity attacks.
We also plot RMSE/Average ratings against malicious user percentage in Figure 2 for the nuclear norm
minimization under similar settings based on a subset of 1000 users and 1700 movies (items), since
it is more computationally expensive than alternating minimization. In general, we observe similar
behavior of both RMSE/Average ratings under different attacking models ?1 , ?2 with alternating
minimization.
6
Discussion and Concluding Remarks
Our ultimate goal for the poisoning attack analysis is to develop possible defensive strategies based
on the careful analysis of adversarial behaviors. Since the poisoning data is optimized based on the
attacker?s malicious objectives, the correlations among features within a feature vector may change
to appear different from normal instances. Therefore, tracking and detecting deviations in the feature
correlations and other accuracy metrics can be one potential defense. Additionally, defender can also
apply the combinational models or sampling strategies, such as bagging, to reduce the influence of
poisoning attacks.
Acknowledgments
This research was partially supported by the NSF (CNS-1238959, IIS-1526860), ONR (N00014-151-2621), ARO (W911NF-16-1-0069), AFRL (FA8750-14-2-0180), Sandia National Laboratories,
and Symantec Labs Graduate Research Fellowship.
8
References
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9
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5,684 | 6,143 | DISCO Nets: DISsimilarity COefficient Networks
Diane Bouchacourt
University of Oxford
diane@robots.ox.ac.uk
M. Pawan Kumar
University of Oxford
pawan@robots.ox.ac.uk
Sebastian Nowozin
Microsoft Research Cambridge
sebastian.nowozin@microsoft.com
Abstract
We present a new type of probabilistic model which we call DISsimilarity COefficient Networks (DISCO Nets). DISCO Nets allow us to efficiently sample from a
posterior distribution parametrised by a neural network. During training, DISCO
Nets are learned by minimising the dissimilarity coefficient between the true distribution and the estimated distribution. This allows us to tailor the training to the loss
related to the task at hand. We empirically show that (i) by modeling uncertainty on
the output value, DISCO Nets outperform equivalent non-probabilistic predictive
networks and (ii) DISCO Nets accurately model the uncertainty of the output,
outperforming existing probabilistic models based on deep neural networks.
1
Introduction
We are interested in the class of problems that require the prediction of a structured output y ? Y
given an input x ? X . Complex applications often have large uncertainty on the correct value of y.
For example, consider the task of hand pose estimation from depth images, where one wants to
accurately estimate the pose y of a hand given a depth image x. The depth image often has some
occlusions and missing depth values and this results in some uncertainty on the pose of the hand. It is,
therefore, natural to use probabilistic models that are capable of representing this uncertainty. Often,
the capacity of the model is restricted and cannot represent the true distribution perfectly. In this case,
the choice of the learning objective influences final performance. Similar to Lacoste-Julien et al. [12],
we argue that the learning objective should be tailored to the evaluation loss in order to obtain the best
performance with respect to this loss. In details, we denote by ?training the loss function employed
during model training, and by ?task the loss employed to evaluate the model?s performance.
We present a simple example to illustrate the point made above. We consider a data distribution that is a mixture of two bidimensional Gaussians. We now consider two models to capture
the data probability distribution. Each model is able to represent a bidimensional Gaussian
distribution with diagonal covariance parametrised by (?1 , ?2 , ?1 , ?2 ). In this case, neither of
the models will be able to recover the true data distribution since they do not have the ability to
represent a mixture of Gaussians. In other words, we cannot avoid model error, similarly to the
real data scenario. Each model uses its own training loss ?training . Model A employs a loss that
emphasises on the first dimension of the data, specified for x = (x1 , x2 ), x0 = (x01 , x02 ) ? R2
1
by ?A (x ? x0 ) = (10 ? (x1 ? x01 )2 + 0.1 ? (x2 ? x02 )2 ) /2 . Model B does the opposite and employs
1
the loss function ?B (x ? x0 ) = (0.1 ? (x1 ? x01 )2 + 10 ? (x2 ? x02 )2 ) /2 . Each model performs a
grid search over the best parameters values for (?1 , ?2 , ?1 , ?2 ). Figure 1 shows the contours of the
Mixture of Gaussians distribution of the data (in black), and the contour of the Gaussian fitted by
each model (in red and green). Detailed setting of this example is available in the supplementary
material.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Table 1: ?task ? SEM (standard error of
the mean) with respect to ?training employed.
Evaluation is done the test set.
?task
?training
?A
?B
?A
?B
11.6 ? 0.287 13.7 ? 0.331
12.1 ? 0.305 11.0 ? 0.257
Figure 1: Contour lines of the Gaussian distribution fitted by each
model on the Mixture of Gaussians data distribution. Best viewed
in color.
As expected, the fitted Gaussian distributions differ according to ?training employed. Table 1 shows
that the loss on the test set, evaluated with ?task , is minimised if ?training = ?task . This simple
example illustrates the advantage to being able to tailor the model?s training objective function to
have ?training = ?task . This is in contrast to the commonly employed learning objectives we present
in Section 2, that are agnostic of the evaluation loss.
In order to alleviate the aforementioned deficiency of the state-of-the-art, we introduce DISCO Nets,
a new class of probabilistic model. DISCO Nets represent P , the true posterior distribution of the
data, with a distribution Q parametrised by a neural network. We design a learning objective based
on a dissimilarity coefficient between P and Q. The dissimilarity coefficient we employ was first
introduced by Rao [23] and is defined for any non-negative symmetric loss function. Thus, any such
loss can be incorporated in our setting, allowing the user to tailor DISCO Nets to his or her needs.
Finally, contrarily to existing probabilistic models presented in Section 2, DISCO Nets do not require
any specific architecture or training procedure, making them an efficient and easy-to-use class of
model.
2
Related Work
Deep neural networks, and in particular, Convolutional Neural Networks (CNNs) are comprised of
several convolutional layers, followed by one or more fully connected (dense) layers, interleaved by
non-linear function(s) and (optionally) pooling. Recent probabilistic models use CNNs to represent
non-linear functions of the data. We observe that such models separate into two types. The first
type of model does not explicitly compute the probability distribution of interest. Rather, these
models allow the user to sample from this distribution by feeding the CNN with some noise z.
Among such models, Generative Adversarial Networks (GAN) presented in Goodfellow et al. [7] are
very popular and have been used in several computer vision applications, for example in Denton
et al. [1], Radford et al. [22], Springenberg [25] and Yan et al. [28]. A GAN model consists of
two networks, simultaneously trained in an adversarial manner. A generative model, referred as
the Generator G, is trained to replicate the data from noise, while an adversarial discriminative
model, referred as the Discriminator D, is trained to identify whether a sample comes from the
true data or from G. The GAN training objective is based on a minimax game between the two
networks and approximately optimizes a Jensen-Shannon divergence. However, as mentioned
in Goodfellow et al. [7] and Radford et al. [22], GAN models require very careful design of the
networks? architecture. Their training procedure is tedious and tends to oscillate. GAN models have
been generalized to conditional GAN (cGAN) in Mirza and Osindero [16], where some additional
input information can be fed to the Generator and the Discriminator. For example in Mirza and
Osindero [16] a cGAN model generates tags corresponding to an image. Gauthier [4] applies cGAN
to face generation. Reed et al. [24] propose to generate images of flowers with a cGAN model, where
the conditional information is a word description of the flower to generate1 . While the application of
cGAN is very promising, little quantitative evaluation has been done. Furthermore, cGAN models
suffer from the same difficulties we mentioned for GAN. Another line of work has developed towards
the use of statistical hypothesis testing to learn probabilistic models. In Dziugaite et al. [2] and Li
et al. [14], the authors propose to train generative deep networks with an objective function based on
the Maximum Mean Discrepancy (MMD) criterion. The MMD method (see Gretton et al. [8, 9]) is
a statistical hypothesis test assessing if two probabilistic distributions are similar. As mentioned
in Dziugaite et al. [2], the MMD test can been seen as playing the role of an adversary.
1
At the time writing, we do not have access to the full paper of Reed et al. [24] and therefore cannot take
advantage of this work in our experimental comparison.
2
The second type of model approximates intractable posterior distributions with use of variational inference. The Variational Auto-Encoders (VAE) presented in Kingma and Welling [10] is
composed of a probabilistic encoder and a probabilistic decoder. The probabilistic encoder is
fed with the input x ? X and produces a posterior distribution P (z|x) over the possible values
of noise z that could have generated x. The probabilistic decoder learns to map the noise z back
to the data space X . The training of VAE uses an objective function based on a Kullback-Leibler
Divergence. VAE and GAN models have been combined in Makhzani et al. [15], where the authors
propose to regularise autoencoders with an adversarial network. The adversarial network ensures that
the posterior distribution P (z|x) matches an arbitrary prior P (z).
In hand pose estimation, imagine the user wants to obtain accurate positions of the thumb
and the index finger but does not need accurate locations of the other fingers. The task loss ?task
might be based on a weighted L2-norm between the predicted and the ground-truth poses, with high
weights on the thumb and the index. Existing probabilistic models cannot be tailored to task-specific
losses and we propose the DISsimilarity COefficient Networks (DISCO Nets) to alleviate this
deficiency.
3
DISCO Nets
We begin the description of our model by specifying how it can be used to generate samples from the
posterior distribution, and how the samples can in turn be employed to provide a pointwise estimate.
In the subsequent subsection, we describe how to estimate the parameters of the model.
3.1 Prediction
Sampling. A DISCO Net consists of several convolutional and dense layers (interleaved by nonlinear function(s) and possibly pooling) and takes as input a pair (x, z) ? X ? Z, where x is input
data and z is some random noise. Given one pair (x, z), the DISCO Net produces a value for the
output y. In the example of hand pose estimation, the input depth image x is fed to the convolutional
layers. The output of the last convolutional layer is flattened and concatenated with a noise sample z.
The resulting vector is fed to several dense layers, and the last dense layer outputs a pose y. From
a single depth image x, by using different noise samples, the DISCO Net produces different pose
candidates for the depth image. This process is illustrated in Figure 2. Importantly, DISCO Nets are
flexible in the choice of the architecture. For example, the noise could be concatenated at any stage
of the network, including at the start.
Figure 2: For a single depth image x, using 3 different noise samples (z1 , z2 , z3 ), DISCO Nets output 3 different
candidate poses (y1 , y2 , y3 ) (shown superimposed on the depth image). The depth image is from the NYU Hand
Pose Dataset of Tompson et al. [27], preprocessed as in Oberweger et al. [17]. Best viewed in color.
We denote Q the distribution that is parametrised by the DISCO Net?s neural network. For a given
input x, DISCO Nets provide the user with samples y drawn from Q(y|x) without requiring the
expensive computation of the (often intractable) partition function. In the remainder of the paper we
consider x ? Rdx , y ? Rdy and z ? Rdz .
Pointwise Prediction. In order to obtain a single prediction y for a given input x, DISCO Nets
use the principle of Maximum Expected Utility (MEU), similarly to Premachandran et al. [21].
The prediction y?task maximises the expected utility, or rather minimises the expected task-specific
loss ?task , estimated using the sampled candidates. Formally, the prediction is made as follows:
3
y?task = argmax EU(yk ) = argmin
k?[1,K]
K
X
?task (yk , yk0 )
(1)
k?[1,K] k0 =1
where (y1 , ..., yK ) are the candidate outputs sampled for the single input x. Details on the MEU
method are in the supplementary material.
3.2
Learning DISCO Nets
Objective Function. We want DISCO Nets to accurately model the true probability P (y|x)
via Q(y|x). In other words, Q(y|x) should be as similar as possible to P (y|x). This similarity is evaluated with respect to the loss specific to the task at hand. Given any non-negative symmetric
loss function between two outputs ?(y, y 0 ) with (y, y 0 ) ? Y ? Y, we employ a diversity coefficient
that is the expected loss between two samples drawn randomly from the two distributions. Formally,
the diversity coefficient is defined as:
DIV? (P, Q, D) = Ex?D(x) [Ey?P (y|x) [Ey0 ?Q(y0 |x) [?(y, y 0 )]]]
(2)
Intuitively, we should minimise DIV? (P, Q, D) so that Q(y|x) is as similar as possible to P (y|x).
However there is uncertainty on the output y to predict for a given x. In other words, P (y|x) is
diverse and Q(y|x) should be diverse as well. Thus we encourage Q(y|x) to provide sample outputs,
for a given x, that are diverse by minimising the following dissimilarity coefficient:
DISC? (P, Q, D) = DIV? (P, Q, D) ? ?DIV? (Q, Q, D) ? (1 ? ?)DIV? (P, P, D)
(3)
with ? ? [0, 1]. The dissimilarity DISC? (P, Q, D) is the difference between the diversity between P
and Q, and an affine combination of the diversity of each distribution, given x ? D(x). These
coefficients were introduced by Rao [23] with ? = 1/2 and used for latent variable models by Kumar
et al. [11]. We do not need to consider the term DIV? (P, P, D) as it is a constant in our problem,
and thus the DISCO Nets objective function is defined as follows:
F = DIV? (P, Q, D) ? ?DIV? (Q, Q, D)
(4)
When minimising F , the term ?DIV? (Q, Q, D) encourages Q(y|x) to be diverse. The value of ?
balances between the two goals of Q(y|x) that are providing accurate outputs while being diverse.
Optimisation. Let us consider a training dataset composed of N examples input-output pairs D =
{(xn , yn ), n = 1..N }. In order to train DISCO Nets, we need to compute the objective function of equation (4). We do not have knowledge of the true probability distributions P (y, x)
and P (x). To overcome this deficiency, we construct estimators of each diversity term DIV? (P, Q)
and DIV? (Q, Q). First, we take an empirical distribution of the data, that is, taking ground-truth
pairs (xn , yn ). We then estimate each distribution Q(y|xn ) by sampling K outputs from our model
for each xn . This gives us an unbiased estimate of each diversity term, defined as:
N
K
X
1 X
d ? (P, Q, D) = 1
DIV
?(yn , G(zk , xn ; ?))
N n=1 K
k=1
N
K
X
X
1
d ? (Q, Q, D) = 1
DIV
N n=1 K(K ? 1)
K
X
(5)
?(G(zk , xn ; ?), G(zk0 , xn ; ?))
k=1 k0 =1,k0 6=k
We have an unbiased estimate of the DISCO Nets? objective function of equation (4):
d ? (P, Q, D) ? ? DIV
d ? (Q, Q, D)
Fb(?, ?) = DIV
(6)
where yk = G(zk , xn ; ?) is a candidate output sampled from DISCO Nets for (xn ,zk ), and ? are the
parameters of DISCO Nets. It is important to note that the second term of equation (6) is summing
over k and k 0 6= k to have an unbiased estimate, therefore we compute the loss between pairs of
different samples G(zk , xn ; ?) and G(zk0 , xn ; ?). The parameters ? are learned by Gradient Descent.
Algorithm 1 shows the training of DISCO Nets. In steps 4 and 5 of Algorithm 1, we draw K random
noise vectors (zn,1 , ...zn,k ) per input example xn , and generate K candidate outputs G(zn,k , xn ; ?)
per input. This allow us to compute an unbiased estimate of the gradient in step 7. For clarity, in the
remainder of the paper we do not explicitely write the parameters ? and write G(zk , xn ).
4
Algorithm 1: DISCO Nets Training algorithm.
for t=1...T epochs do
Sample minibatch of b training example pairs {(x1 , y1 )...(xb , yb )}.
for n=1...b do
Sample K random noise vectors (zn,1 , ...zn,k ) for training example xn
Generate K candidate outputs G(zn,k , xn ; ?), k = 1..K for training example xn
end
Update parameters ? t ? ? t?1 by descending the gradient of equation (6) : ?? Fb(?, ?).
end
1
2
3
4
5
6
7
8
3.3 Strictly Proper Scoring Rules.
Scoring Rule for Learning. A scoring rule S(Q, P ), as defined in Gneiting and Raftery [5],
evaluates the quality of a predictive distribution Q with respect to a true distribution P . When using
a scoring rule one should ensure that it is proper, which means it is maximised when P = Q. A
scoring rule is said to be strictly proper if P = Q is the unique maximiser of S. Hence maximising a
proper scoring rule ensures that the model aims at predicting relevant forecast. Gneiting and Raftery
[5] define score divergences corresponding to a proper scoring rule S:
d(Q, P ) = S(P, P ) ? S(Q, P )
(7)
If S is proper, d is a valid non-negative divergence function, with value 0 if (and only if, in the case
of strictly proper) Q = P . For example the MMD criterion (see Gretton et al. [8, 9]) mentioned
in Section 2 is an example of this type of divergence. In our case, any loss ? expressed with an
universal kernel will define the DISCO Nets? objective function as such divergence (see Zawadzki
and Lahaie [29]). For example, by Theorem 5 of Gneiting and Raftery [5], if we take as loss
Pdy
?
function ?? (y, y 0 ) = ||y ? y 0 ||?2 = i=1
|(y i ? y 0i |2 ) /2 with ? ? [0, 2] excluding 0 and 2, our
training objective is (the negative of) a strictly proper scoring rule, that is:
i
P P
1 PN h 1 P
1
1
?
?
0
Fb(?, ?) =
n=1
k ||yn ? G(zk , xn )||2 ?
k
k0 6=k ||G(zk , xn ) ? G(zk , xn )||2
N
K
2 K(K ? 1)
(8)
This score has been referred in the litterature as the Energy Score in Gneiting and Raftery
[5], Gneiting et al. [6], Pinson and Tastu [19].
By employing a (strictly) proper scoring rule we ensure that our objective function is (only)
minimised at the true distribution P , and expect DISCO Nets to generalise better on unseen data.
We show below that strictly proper scoring rules are also relevant to assess the quality of the
distribution Q captured by the model.
Discriminative power of proper scoring rules. As observed in Fukumizu et al. [3], kernel density
estimation (KDE) fails in high dimensional output spaces. Our goal is to compare the quality of the
distribution captured between two models, Q1 and Q2 . In our setting Q1 better models P than Q2
according to the scoring rule S and its associated divergence d if d(Q1 , P ) < d(Q2 , P ). As noted
in Pinson and Tastu [19], S being proper does not ensure d(Q1 , y) < d(Q2 , y) for all observations y
drawn from P . However if the scoring rule is strictly proper scoring rule, this property should be
ensured in the neighbourhood of the true distribution.
4
Experiments : Hand Pose Estimation
Given a depth image x, which often contains occlusions and missing values, we wish to predict the
hand pose y. We use the NYU Hand Pose dataset of Tompson et al. [27] to estimate the efficiency of
DISCO Nets for this task.
4.1 Experimental Setup
NYU Hand Pose Dataset. The NYU Hand Pose dataset of Tompson et al. [27] contains 8252
testing and 72,757 training frames of captured RGBD data with ground-truth hand pose information.
The training set is composed of images of one person whereas the testing set gathers samples from
two persons. For each frame, the RGBD data from 3 Kinects is provided: a frontal view and 2 side
views. In our experiments we use only the depth data from the frontal view. While the ground truth
5
contains J = 36 annotated joints, we follow the evaluation protocol of Oberweger et al. [17, 18] and
use the same subset of J = 14 joints. We also perform the same data preprocessing as in Oberweger
et al. [17, 18], and extract a fixed-size metric cube around the hand from the depth image. We resize
the depth values within the cube to a 128 ? 128 patch and normalized them in [?1, 1]. Pixels deeper
than the back of the cube and missing depth values are both set to a depth of 1.
Methods. We employ loss functions between two outputs of the form of the Energy score (8), that
is, ?training = ?? (y, y 0 ) = ||y ? y 0 ||?2 . Our first goal is to assess the advantages of DISCO Nets
with respect to non-probabilistic deep networks. One model, referred as DISCO?,? , is a DISCO Nets
probabilistic model, with ? 6= 0 in the dissimilarity coefficient of equation (6). When taking ? = 0,
noise is injected and the model capacity is the same as DISCO?,?6=0 . The model BASE? , is a
non-probabilistic model, by taking ? = 0 in the objective function of equation (6) and no noise is
concatenated. This corresponds to a classic deep network which for a given input x generates a single
output y = G(x). Note that we write G(x) and not G(z, x) since no noise is concatenated.
Evaluation Metrics. We report classic non-probabilistic metrics for hand pose estimation employed
in Oberweger et al. [17, 18] and Taylor et al. [26], that are, the Mean Joint Euclidean Error (MeJEE),
the Max Joint Euclidean Error (MaJEE) and the Fraction of Frames within distance (FF). We refer
the reader to the supplementary material for detailed expression of these metrics. These metrics use
the Euclidean distance between the prediction and the ground-truth and require a single pointwise
prediction. This pointwise prediction is chosen with the MEU method among K candidates. We
added the probabilistic metric ProbLoss. ProbLoss is defined as in Equation 8 with the Euclidean
norm and is the divergence associated with a strictly proper scoring rule. Thus, ProbLoss ranks the
ability of the models to represent the true distribution. ProbLoss is computed using K candidate
poses for a given depth image. For the non-probabilistic model BASE? , only a single pointwise
predicted output y is available. We construct the K candidates by adding some Gaussian random
noise of mean 0 and diagonal covariance ? = ?1, with ? ? {1mm, 5mm, 10mm} and refer to the
model as BASE?,? . 2
Loss functions. As we employ standard evaluation metrics based on the Euclidean norm, we train
with the Euclidean norm (that is, ?training (y, y 0 ) = ||y ? y 0 ||?2 with ? = 1). When ? = 12 our
objective function coincides with ProbLoss.
Architecture. The novelty of DISCO Nets resides in their objective function. They do not require
the use of a specific network architecture. This allows us to design a simple network architecture
inspired by Oberweger et al. [18]. The architecture is shown in Figure 2. The input depth image x
is fed to 2 convolutional layers, each having 8 filters, with kernels of size 5 ? 5, with stride 1,
followed by Rectified Linear Units (ReLUs) and Max Pooling layers of kernel size 3 ? 3. A third
and last convolutional layer has 8 filters, with kernels of size 5 ? 5, with stride 1, followed by a
Rectified Linear Unit. The ouput of the convolution is concatenated to the random noise vector z
of size dz = 200, drawn from a uniform distribution in [?1, 1]. The result of the concatenation
is fed to 2 dense layers of output size 1024, with ReLUs, and a third dense layer that outputs the
candidate pose y ? R3?J . For the non-probabilistic BASE?,? model no noise is concatenated as
only a pointwise estimate is produced.
Training. We use 10,000 examples from the 72,757 training frames to construct a validation
dataset and train only on 62,757 examples. Back-propagation is used with Stochastic Gradient
Descent with a batchsize of 256. The learning rate is fixed to ? = 0.01 and we use a momentum
of m = 0.9 (see Polyak [20]). We also add L2-regularisation controlled by the parameter C. We
use C = [0.0001, 0.001, 0.01] which is a relevant range as the comparative model BASE? is best
performing for C = 0.001. Note that DISCO Nets report consistent performances across the different
values C, contrarily to BASE? . We use 3 different random seeds to initialize each model network
parameters. We report the performance of each model with its best cross-validated seed and C. We
train all models for 400 epochs as it results in a change of less than 3% in the value of the loss on the
validation dataset for BASE? . We refer the reader to the supplementary material for details on the
setting.
2
We also evaluate the non-probabilistic model BASE? using its pointwise prediction rather than the MEU
method. Results are consistent and detailed in the supplementary material.
6
Table 2: Metrics values on the test set ? SEM. Best Table 3: Metrics values on the test set ? SEM for
performances in bold.
cGAN.
Model
ProbLoss (mm) MeJEE (mm) MaJEE (mm) FF (80mm)
BASE?=1,?=1
103.8?0.627 25.2?0.152 52.7?0.290 86.040
BASE?=1,?=5
99.3?0.620 25.5?0.151 52.9?0.289 85.773
BASE?=1,?=10
96.3?0.612 25.7?0.149 53.2?0.288 85.664
DISCO?=1,?=0
92.9?0.533 21.6?0.128 46.0?0.251 92.971
DISCO?=1,?=0.25 89.9?0.510 21.2?0.122 46.4?0.252 93.262
DISCO?=1,?=0.5 83.8 ?0.503 20.9?0.124 45.1?0.246 94.438
4.2
Model
ProbLoss (mm) MeJEE (mm) MaJEE (mm) FF (80mm)
cGAN
442.7?0.513 109.8?0.128 201.4?0.320 0.000
cGANinit, fixed 128.9?0.480 31.8?0.117 64.3?0.230 78.454
Results.
Quantitative Evaluation. Table 2 reports performances on the test dataset, with parameters crossvalidated on the validation set. All versions of the DISCO Net model outperform the BASE? model.
Among the different values of ?, we see that ? = 0.5 better captures the true distribution (lower
ProbLoss) while retaining accurate performance on the standard pointwise metrics. Interestingly,
using an all-zero noise at test-time gives similar performances on pointwise metrics. We link this to
the observation that both the MEAN and the MEU method perform equivalently on these metrics
(see supplementary material).
Qualitative Evaluation. In Figure 3 we show candidate poses generated by DISCO?=1,?=0.5 for
3 testing examples. The left image shows the input depth image, and the right image shows the
ground-truth pose (in grey) with 100 candidate outputs (superimposed in transparent red). The model
predict the joint locations and we interpolate the joints with edges. If an edge is thinner and more
opaque, it means the different predictions overlap and that the uncertainty on the location of the
edge?s joints is low. We can see that DISCO?=1,?=0.5 captures relevant information on the structure
of the hand.
(a) When there are no occlusions,
DISCO Nets model low uncertainty on all joints.
(b) When the hand is half-fisted,
(c) Here the fingertips of all finDISCO Nets model the uncergers but the forefinger are octainty on the location of the fincluded and DISCO Nets model
gertips.
high uncertainty on them.
Figure 3: Visualisation of DISCO?=1,?=0.5 predictions for 3 examples from the testing dataset. The left image
shows the input depth image, and the right image shows the ground-truth pose in grey with 100 candidate outputs
superimposed in transparent red. Best viewed in color.
P PR PL TR TM TT IM IT MM MT RM RT PM PT
P PR PL TR TM TT IM IT MM MT RM RT PM PT
Figure 4 shows the matrices of Pearson product-moment correlation coefficients between joints. We
note that DISCO Net with ? = 0.5 better captures the correlation between the joints of a finger and
between the fingers.
P PR PL TR TM TT IM IT MM MT RM RT PM PT
P PR PL TR TM TT IM IT MM MT RM RT PM PT
?=0
? = 0.5
Figure 4: Pearson coefficients matrices of the joints: Palm (no value as the empirical variance is null), Palm
Right, Palm Left, Thumb Root, Thumb Mid, Index Mid, Index Tip, Middle Mid, Middle Tip, Ring Mid, Ring Tip,
Pinky Mid, Pinky Tip.
7
4.3 Comparison with existing probabilistic models.
To the best of our knowledge the conditional Generative Adversarial Networks (cGAN) from Mirza
and Osindero [16] has not been applied to pose estimation. In order to compare cGAN to DISCO Nets,
several issues must be overcome. First, we must design a network architecture for the Discriminator.
This is a first disadvantage of cGAN compared to DISCO Nets which require no adversary. Second,
as mentioned in Goodfellow et al. [7] and Radford et al. [22], GAN (and thus cGAN) require very
careful design of the networks? architecture and training procedure. In order to do a fair comparison,
we followed the work in Mirza and Osindero [16] and practical advice for GAN presented in Larsen
and S?nderby [13]. We try (i) cGAN, initialising all layers of D and G randomly, and (ii) cGANinit, fixed
initialising the convolutional layers of G and D with the trained best-performing DISCO?=1,?=0.5
of Section 4.2, and keeping these layers fixed. That is, the convolutional parts of G and D are fixed
feature extractors for the depth image. This is a setting similar to the one employed for tag-annotation
of images in Mirza and Osindero [16]. Details on the setting can be found in the supplementary
material. Table 3 shows that the cGAN model obtains relevant results only when the convolutional
layers of G and D are initialised with our trained model and kept fixed, that is cGANinit, fixed . These
results are still worse than DISCO Nets performances. While there may be a better architecture for
cGAN, our experiments demonstrate the difficulty of training cGAN over DISCO Nets.
4.4 Reference state-of-the-art values.
We train the best-performing DISCO?=1,?=0.5 of Section 4.2 on the entire dataset, and compare
performances with state-of-the-art methods in Table 4 and Figure 5. These state-of-the-art methods
are specifically designed for hand pose estimation. In Oberweger et al. [17] a constrained prior hand
model, referred as NYU-Prior, is refined on each hand joint position to increase accuracy, referred
as NYU-Prior-Refined. In Oberweger et al. [18], the input depth image is fed to a first network
NYU-Init, that outputs a pose used to synthesize an image with a second network. The synthesized
image is used with the input depth image to derive a pose update. We refer to the whole model as
NYU-Feedback. On the contrary, DISCO Nets uses a single network whose architecture is similar
to NYU-Prior (without constraining on a pose prior). By accurately modeling the distribution of
the pose given the depth image, DISCO Nets obtain comparable performances to NYU-Prior and
NYU-Prior-Refined. Without any extra effort, DISCO Nets could be embedded in the presented
refinement and feedback methods, possibly boosting state-of-the-art performances.
Table 4: DISCO Nets compared to stateof-the-art performances ? SEM.
Model
NYU-Prior
NYU-Prior-Refined
NYU-Init
NYU-Feedback
DISCO?=1,?=0.5
5
MeJEE (mm)
20.7?0.150
19.7?0.157
27.4?0.152
16.0?0.096
20.7?0.121
MaJEE (mm)
44.8?0.289
44.7?0.327
55.4?0.265
36.1?0.208
45.1?0.246
FF (80mm)
91.190
88.148
86.537
97.334
93.250
Discussion.
Figure 5: Fractions of frames within distance d in mm (by 5 mm). Best
viewed in color.
We presented DISCO Nets, a new family of probabilistic model based on deep networks. DISCO Nets
employ a prediction and training procedure based on the minimisation of a dissimilarity coefficient.
Theoretically, this ensures that DISCO Nets accurately capture uncertainty on the correct output
to predict given an input. Experimental results on the task of hand pose estimation consistently
support our theoretical hypothesis as DISCO Nets outperform non-probabilistic equivalent models,
and existing probabilistic models. Furthermore, DISCO Nets can be tailored to the task to perform.
This allows a possible user to train them to tackle different problems of interest. As their novelty
resides mainly in their objective function, DISCO Nets do not require any specific architecture and
can be easily applied to new problems. We contemplate several directions for future work. First, we
will apply DISCO Nets to other prediction problems where there is uncertainty on the output. Second,
we would like to extend DISCO Nets to latent variables models, allowing us to apply DISCO Nets to
diverse dataset where ground-truth annotations are missing or incomplete.
6
Acknowlegements.
This work is funded by the Microsoft Research PhD Scholarship Programme. We would like to thank
Pankaj Pansari, Leonard Berrada and Ondra Miksik for their useful discussions and insights.
8
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9
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5,685 | 6,144 | Higher-Order Factorization Machines
Mathieu Blondel, Akinori Fujino, Naonori Ueda
NTT Communication Science Laboratories
Japan
Masakazu Ishihata
Hokkaido University
Japan
Abstract
Factorization machines (FMs) are a supervised learning approach that can use
second-order feature combinations even when the data is very high-dimensional.
Unfortunately, despite increasing interest in FMs, there exists to date no efficient
training algorithm for higher-order FMs (HOFMs). In this paper, we present
the first generic yet efficient algorithms for training arbitrary-order HOFMs. We
also present new variants of HOFMs with shared parameters, which greatly reduce model size and prediction times while maintaining similar accuracy. We
demonstrate the proposed approaches on four different link prediction tasks.
1
Introduction
Factorization machines (FMs) [13, 14] are a supervised learning approach that can use second-order
feature combinations efficiently even when the data is very high-dimensional. The key idea of FMs is
to model the weights of feature combinations using a low-rank matrix. This has two main benefits.
First, FMs can achieve empirical accuracy on a par with polynomial regression or kernel methods
but with smaller and faster to evaluate models [4]. Second, FMs can infer the weights of feature
combinations that were not observed in the training set. This second property is crucial for instance
in recommender systems, a domain where FMs have become increasingly popular [14, 16]. Without
the low-rank property, FMs would fail to generalize to unseen user-item interactions.
Unfortunately, although higher-order FMs (HOFMs) were briefly mentioned in the original work of
[13, 14], there exists to date no efficient algorithm for training arbitrary-order HOFMs. In fact, even
just computing predictions given the model parameters naively takes polynomial time in the number
of features. For this reason, HOFMs have, to our knowledge, never been applied to any problem. In
addition, HOFMs, as originally defined in [13, 14], model each degree in the polynomial expansion
with a different matrix and therefore require the estimation of a large number of parameters.
In this paper, we propose the first efficient algorithms for training arbitrary-order HOFMs. To do
so, we rely on a link between FMs and the so-called ANOVA kernel [4]. We propose linear-time
dynamic programming algorithms for evaluating the ANOVA kernel and computing its gradient.
Based on these, we propose stochastic gradient and coordinate descent algorithms for arbitrary-order
HOFMs. To reduce the number of parameters, as well as prediction times, we also introduce two new
kernels derived from the ANOVA kernel, allowing us to define new variants of HOFMs with shared
parameters. We demonstrate the proposed approaches on four different link prediction tasks.
2
Factorization machines (FMs)
Second-order FMs. Factorization machines (FMs) [13, 14] are an increasingly popular method
for efficiently using second-order feature combinations in classification or regression tasks even
when the data is very high-dimensional. Let w ? Rd and P ? Rd?k , where k ? N is a rank
? j and its columns by ps , for j ? [d] and s ? [k],
hyper-parameter. We denote the rows of P by p
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
where [d] := {1, . . . , d}. Then, FMs predict an output y ? R from a vector x = [x1 , . . . , xd ]T by
X
? j 0 ixj xj 0 .
y?FM (x) := hw, xi +
h?
pj , p
(1)
j 0 >j
An important characteristic of (1) is that it considers only combinations of distinct features (i.e., the
squared features x21 , . . . , x2d are ignored). The main advantage of FMs compared to naive polynomial
regression is that the number of parameters to estimate is O(dk) instead of O(d2 ). In addition, we
can compute predictions in O(2dk) time1 using
y?FM (x) = wT x +
k
1 T 2 X
kP xk ?
kps ? xk2 ,
2
s=1
where ? indicates element-wise product [3]. Given a training set X = [x1 , . . . , xn ] ? Rd?n and
y = [y1 , . . . , yn ]T ? Rn , w and P can be learned by minimizing the following non-convex objective
n
1X
?1
?2
` (yi , y?FM (xi )) + kwk2 + kP k2 ,
n i=1
2
2
(2)
where ` is a convex loss function and ?1 > 0, ?2 > 0 are hyper-parameters. The popular libfm
library [14] implements efficient stochastic gradient and coordinate descent algorithms for obtaining
a stationary point of (2). Both algorithms have a runtime complexity of O(2dkn) per epoch.
Higher-order FMs (HOFMs). Although no training algorithm was provided, FMs were extended
to higher-order feature combinations in the original work of [13, 14]. Let P (t) ? Rd?kt , where
t ? {2, . . . , m} is the order or degree of feature combinations considered, and kt ? N is a rank
(t)
? j be the j th row of P (t) . Then m-order HOFMs can be defined as
hyper-parameter. Let p
X (2) (2)
X
(m)
(m)
? j 0 ixj xj 0 + ? ? ? +
? jm ixj1 xj2 . . . xjm (3)
y?HOFM (x) := hw, xi +
h?
pj , p
h?
pj1 , . . . , p
j 0 >j
(t)
jm >???>j1
(t)
(t)
(t)
? jt i := sum(?
? jt ) (sum of element-wise products). The
where we defined h?
pj1 , . . . , p
pj1 ? ? ? ? ? p
objective function of HOFMs can be expressed in a similar way as for (2):
n
m
X
1X
?1
?t
` (yi , y?HOFM (xi )) + kwk2 +
kP (t) k2 ,
n i=1
2
2
t=2
(4)
where ?1 , . . . , ?m > 0 are hyper-parameters. To avoid the combinatorial explosion of hyperparameter combinations to search, in our experiments we will simply set ?1 = ? ? ? = ?m and
k2 = ? ? ? = km . While (3) looks quite daunting, [4] recently showed that FMs can be expressed from
a simpler kernel perspective. Let us define the ANOVA2 kernel [19] of degree 2 ? m ? d by
Am (p, x) :=
X
m
Y
pjt xjt .
(5)
jm >???>j1 t=1
For later convenience, we also define A0 (p, x) := 1 and A1 (p, x) := hp, xi. Then it is shown that
k2
km
X
X
y?HOFM (x) = hw, xi +
A2 p(2)
Am p(m)
(6)
s ,x + ??? +
s ,x ,
s=1
(t)
ps
th
s=1
(t)
where
is the s column of P . This perspective shows that we can view FMs and HOFMs
as a type of kernel machine whose ?support vectors? are learned directly from data. Intuitively, the
ANOVA kernel can be thought as a kind of polynomial kernel that uses feature combinations without
replacement (i.e., of distinct features). A key property of the ANOVA kernel is multi-linearity [4]:
Am (p, x) = Am (p?j , x?j ) + pj xj Am?1 (p?j , x?j ),
(7)
where p?j denotes the (d ? 1)-dimensional vector with pj removed and similarly for x?j . That
is, everything else kept fixed, Am (p, x) is an affine function of pj ?j ? [d]. Although no training
1
2
We include the constant factor for fair later comparison with arbitrary-order HOFMs.
The name comes from the ANOVA decomposition of functions. [20, 19]
2
algorithm was provided, [4] showed based on (7) that, although non-convex, the objective function of
arbitrary-order HOFMs is convex in w and in each row of P (2) , . . . , P (m) , separately.
Interpretability of HOFMs. An advantage of FMs and HOFMs is their interpretability. To see why
this is the case, notice that we can rewrite (3) as
X
X
(2)
(m)
y?HOFM (x) = hw, xi +
W j,j 0 xj xj 0 + ? ? ? +
W j1 ,...,jm xj1 xj2 . . . xjm ,
j 0 >j
where we defined W
(t)
:=
Pkt
jm >???>j1
(t)
s=1 ps
|
? ??? ?
{z
p(t)
s .
}
t times
t
Intuitively, W (t) ? Rd is a low-rank t-way
tensor which contains the weights of feature combinations of degree t. For instance, when t = 3,
(3)
W i,j,k is the weight of xi xj xk . Similarly to the ANOVA decomposition of functions, HOFMs
consider only combinations of distinct features (i.e., xj1 xj2 . . . xjm for jm > ? ? ? > j2 > j1 ).
This paper. Unfortunately, there exists to date no efficient algorithm for training arbitrary-order
HOFMs. Indeed, computing (5) naively takes O(dm ), i.e., polynomial time. In the following, we
present linear-time algorithms. Moreover, HOFMs, as originally defined in [13, 14] require the
estimation of m ? 1 matrices P (2) , . . . , P (m) . Thus, HOFMs can produce large models when m is
large. To address this issue, we propose new variants of HOFMs with shared parameters.
3
Linear-time stochastic gradient algorithms for HOFMs
The kernel view presented in Section 2 allows us to focus on the ANOVA kernel as the main
?computational unit? for training HOFMs. In this section, we develop dynamic programming (DP)
algorithms for evaluating the ANOVA kernel and computing its gradient in only O(dm) time.
Evaluation. The main observation (see also [18, Section 9.2]) is that we can use (7) to recursively
remove features until computing the kernel becomes trivial. Let us denote a subvector of p by
p1:j ? Rj and similarly for x. Let us introduce the shorthand aj,t := At (p1:j , x1:j ). Then, from (7),
aj,t = aj?1,t + pj xj aj?1,t?1 ?d ? j ? t ? 1.
(8)
For convenience, we also define aj,0 = 1 ?j ? 0 since A0 (p, x) = 1 and aj,t = 0 ?j < t since
there does not exist any t-combination of features in a j < t dimensional vector.
The quantity we want to compute is Am (p, x) = ad,m .
Instead of naively using recursion (8), which would lead
to many redundant computations, we use a bottom-up approach and organize computations in a DP table. We start
from the top-left corner to initialize the recursion and go
through the table to arrive at the solution in the bottomright corner. The procedure, summarized in Algorithm
1, takes O(dm) time and memory.
Table 1: Example of DP table
t=0
t=1
t=2
.
.
.
t=m
j=0
1
0
0
.
.
.
0
j=1
1
a1,1
0
.
.
.
0
j=2
1
a2,1
a2,2
.
.
.
0
...
1
...
...
..
.
...
j=d
1
ad,1
ad,2
.
.
.
ad,m
Gradients. For computing the gradient of Am (p, x) w.r.t. p, we use reverse-mode differentiation
[2] (a.k.a. backpropagation in a neural network context), since it allows us to compute the entire
gradient in a single pass. We supplement each variable aj,t in the DP table by a so-called adjoint
?a
a
?j,t := ?ad,m
, which represents the sensitivity of ad,m = Am (p, x) w.r.t. aj,t . From recursion (8),
j,t
except for edge cases, aj,t influences aj+1,t+1 and aj+1,t . Using the chain rule, we then obtain
?ad,m ?aj+1,t
?ad,m ?aj+1,t+1
+
=a
?j+1,t +pj+1 xj+1 a
?j+1,t+1 ?d?1 ? j ? t ? 1.
a
?j,t =
?aj+1,t ?aj,t
?aj+1,t+1 ?aj,t
(9)
?ad,m
Similarly, we introduce the adjoint p?j := ?pj ?j ? [d]. Since pj influences aj,t ?t ? [m], we have
p?j =
m
X
?ad,m ?aj,t
t=1
?aj,t ?pj
=
m
X
a
?j,t aj?1,t?1 xj .
t=1
?a
We can run recursion (9) in reverse order of the DP table starting from a
?d,m = ?ad,m
= 1. Using
d,m
m
T
this approach, we can compute the entire gradient ?A (p, x) = [?
p1 , . . . , p?d ] w.r.t. p in O(dm)
time and memory. The procedure is summarized in Algorithm 2.
3
Algorithm 1 Evaluating Am (p, x) in O(dm)
Input: p ? Rd , x ? Rd
aj,t ? 0 ?t ? [m], j ? [d] ? {0}
aj,0 ? 1 ?j ? [d] ? {0}
Algorithm 2 Computing ?Am (p, x) in O(dm)
Input: p ? Rd , x ? Rd , {aj,t }d,m
j,t=0
a
?j,t ? 0 ?t ? [m + 1], j ? [d]
a
?d,m ? 1
for t := m, . . . , 1 do
for j := d ? 1, . . . , t do
a
?j,t ? a
?j+1,t + a
?j+1,t+1 pj+1 xj+1
end for
end for
P
?j,t aj?1,t?1 xj ?j ? [d]
p?j := m
t=1 a
Output: ?Am (p, x) = [?
p1 , . . . , p?d ]T
for t := 1, . . . , m do
for j := t, . . . , d do
aj,t ? aj?1,t + pj xj aj?1,t?1
end for
end for
Output: Am (p, x) = ad,m
Stochastic gradient (SG) algorithms. Based on Algorithm 1 and 2, we can easily learn arbitraryorder HOFMs using any gradient-based optimization algorithm. Here we focus our discussion on SG
algorithms. If we alternatingly minimize (4) w.r.t P (2) , . . . , P (m) , then the sub-problem associated
with degree m is of the form
!
k
n
X
1X
?
m
F (P ) :=
A (ps , xi ) + oi + kP k2 ,
` yi ,
(10)
n i=1
2
s=1
where o1 , . . . , on ? R are fixed offsets which account for the contribution of degrees other than m
to the predictions. The sub-problem is convex in each row of P [4]. A SG update for (10) w.r.t. ps
for some instance xi can be computed by ps ? ps ? ?`0 (yi , y?i )?Am (ps , xi ) ? ??ps , where ? is
Pk
a learning rate and where we defined y?i := s=1 Am (ps , xi ) + oi . Because evaluating Am (p, x)
and computing its gradient both take O(dm), the cost per epoch, i.e., of visiting all instances, is
O(mdkn). When m = 2, this is the same cost as the SG algorithm implemented in libfm.
Sparse data. We conclude this section with a few useful remarks on sparse data. Let us denote
the support of a vector x = [x1 , . . . , xd ]T by supp(x) := {j ? [d] : xj 6= 0} and let us define
xS := [xj : j ? S]T . It is easy to see from (7) that the gradient and x have the same support, i.e.,
supp(?Am (p, x)) = supp(x). Another useful remark is that Am (p, x) = Am (psupp(x) , xsupp(x) ),
provided that m ? nz (x), where nz (x) is the number of non-zero elements in x. Hence, when the
data is sparse, we only need to iterate over non-zero features in Algorithm 1 and 2. Consequently,
their time and memory cost is only O(nz (x)m) and thus the cost per epoch of SG algorithms is
O(mknz (X)).
4
Coordinate descent algorithm for arbitrary-order HOFMs
We now describe a coordinate descent (CD) solver for arbitrary-order HOFMs. CD is a good choice
for learning HOFMs because their objective function is coordinate-wise convex, thanks to the multilinearity of the ANOVA kernel [4]. Our algorithm can be seen as a generalization to higher orders of
the CD algorithms proposed in [14, 4].
An alternative recursion. Efficient CD implementations typically require maintaining statistics for
each training instance, such as the predictions at the current iteration. When a coordinate is updated,
the statistics then need to be synchronized. Unfortunately, the recursion we used in the previous
section is not suitable for a CD algorithm because it would require to store and synchronize the DP
table for each training instance upon coordinate-wise updates. We therefore turn to an alternative
recursion:
m
1 X
Am (p, x) =
(?1)t+1 Am?t (p, x)Dt (p, x),
(11)
m t=1
Pd
where we defined Dt (p, x) := j=1 (pj xj )t . Note that the recursion was already known in the
context of traditional kernel methods (c.f., [19, Section 11.8]) but its application to HOFMs is novel.
Since we know that A0 (p, x) = 1 and A1 (p, x) = hp, xi, we can use (11) to compute A2 (p, x),
then A3 (p, x), and so on. The overall evaluation cost for arbitrary m ? N is O(md + m2 ).
4
Coordinate-wise derivatives. We can apply reverse-mode differentiation to recursion (11) in order
to compute the entire gradient (c.f., Appendix C). However, in CD, since we only need the derivative
of one variable at a time, we can simply use forward-mode differentiation:
m
?Am (p, x)
1 X
?Am?t (p, x) t
?Dt (p, x)
=
(?1)t+1
D (p, x) + Am?t (p, x)
,
(12)
?pj
m t=1
?pj
?pj
t
t
t
where ?D?p(p,x)
= tpt?1
j xj . The advantage of (12) is that we only need to cache D (p, x) for t ? [m].
j
Hence the memory complexity per sample is only O(m) instead of O(dm) for (8).
Use in a CD algorithm. Similarly to [4], we assume that the loss function ` is ?-smooth and update
?1 ?F (P )
the elements pj,s of P in cyclic order by pj,s ? pj,s ? ?j,s
?pj,s , where we defined
n
?j,s :=
?X
n i=1
?Am (ps , xi )
?pj,s
2
n
+?
and
?F (P )
1X 0
?Am (ps , xi )
=
` (yi , y?i )
+ ?pj,s .
?pj,s
n i=1
?pj,s
The update guarantees that the objective value is monotonically non-increasing and is the exact
coordinate-wise minimizer when ` is the squared loss. Overall, the total cost per epoch, i.e., updating
all coordinates once, is O(? (m)knz (X)), where ? (m) is the time it takes to compute (12). Assuming
Dt (ps , xi ) have been previously cached, for t ? [m], computing (12) takes ? (m) = m(m + 1)/2 ? 1
operations. For fixed m, if we unroll the two loops needed to compute (12), modern compilers can
often further reduce the number of operations needed. Nevertheless, this quadratic dependency on m
means that our CD algorithm is best for small m, typically m ? 4.
5
HOFMs with shared parameters
HOFMs, as originally defined in [13, 14], model each degree with separate matrices P (2) , . . . , P (m) .
Assuming that we use the same rank k for all matrices, the total model size of m-order HOFMs is
therefore O(kdm). Moreover, even when using our O(dm) DP algorithm, the cost of computing
predictions is O(k(2d + ? ? ? + md)) = O(kdm2 ). Hence, HOFMs tend to produce large, expensiveto-evaluate models. To reduce model size and prediction times, we introduce two new kernels which
allow us to share parameters between each degree: the inhomogeneous ANOVA kernel and the
all-subsets kernel. Because both kernels are derived from the ANOVA kernel, they share the same
appealing properties: multi-linearity, sparse gradients and sparse-data friendliness.
5.1
Inhomogeneous ANOVA kernel
It is well-known that a sum of kernels is equivalent to concatenating their associated feature maps
[18, Section 3.4]. Let ? = [?1 , . . . , ?m ]T . To combine different degrees, a natural kernel is therefore
A1?m (p, x; ?) :=
m
X
t=1
?t At (p, x).
(13)
The kernel uses all feature combinations of degrees 1 up to m. We call it inhomogeneous ANOVA
kernel, since it is an inhomogeneous polynomial of x. In contrast, Am (p, x) is homogeneous. The
main difference between (13) and (6) is that all ANOVA kernels in the sum share the same parameters.
However, to increase modeling power, we allow each kernel to have different weights ?1 , . . . , ?m .
Evaluation. Due to the recursive nature of Algorithm 1, when computing Am (p, x), we also get
A1 (p, x), . . . , Am?1 (p, x) for free. Indeed, lower-degree kernels are available in the last column of
the DP table, i.e., At (p, x) = ad,t ?t ? [m]. Hence, the cost of evaluating (13) is O(dm) time. The
Pk
total cost for computing y? = s=1 A1?m (ps , x; ?) is O(kdm) instead of O(kdm2 ) for y?HOFM (x).
Learning. While it is certainly possible to learn P and ? by directly minimizing some objective
function, here we propose an easier solution, which works well in practice. Our key observation is
that we can easily turn Am into A1?m by adding dummy values to feature vectors. Let us denote the
concatenation of p with a scalar ? by [?, p] and similarly for x. From (7), we easily obtain
Am ([?1 , p], [1, x]) = Am (p, x) + ?1 Am?1 (p, x).
5
Table 2: Datasets used in our experiments. For a detailed description, c.f. Appendix A.
Dataset
NIPS [17]
Enzyme [21]
GD [10]
Movielens 100K [6]
n+
4,140
2,994
3,954
21,201
Columns of A
Authors
Enzymes
Diseases
Users
nA
2,037
668
3,209
943
dA
13,649
325
3,209
49
Columns of B
nB
dB
Genes
Movies
12,331
1,682
25,275
29
Similarly, if we apply (7) twice, we obtain:
Am ([?1 , ?2 , p], [1, 1, x]) = Am (p, x) + (?1 + ?2 )Am?1 (p, x) + ?1 ?2 Am?2 (p, x).
Applying the above to m = 2 and m = 3, we obtain
A2 ([?1 , p], [1, x]) = A1?2 (p, x; [?1 , 1]) and
A3 ([?1 , ?2 , p], [1, 1, x]) = A1?3 (p, x; [?1 ?2 , ?1 +?2 , 1]).
More generally, by adding m ? 1 dummy features to p and x, we can convert Am to A1?m . Because
p is learned, this means that we can automatically learn ?1 , . . . , ?m?1 . These weights can then be
converted to ?1 , . . . , ?m by ?unrolling? recursion (7). Although simple, we show in our experiments
that this approach works favorably compared to directly learning P and ?. The main advantage of
this approach is that we can use the same software unmodified (we simply need to minimize (10)
with the augmented data). Moreover, the cost of computing the entire gradient by Algorithm 2 using
the augmented data is just O(dm + m2 ) compared to O(dm2 ) for HOFMs with separate parameters.
5.2
All-subsets kernel
We now consider a closely related kernel called all-subsets kernel [18, Definition 9.5]:
S(p, x) :=
d
Y
(1 + pj xj ).
j=1
The main difference with the traditional use of this kernel is that we learn p. Interestingly, it can
be shown that S(p, x) = 1 + A1?d (p, x; 1) = 1 + A1?nz (x) (p, x; 1), where nz (x) is the number
of non-zero features in x. Hence, the kernel uses all combinations of distinct features up to order
nz (x) with uniform weights. Even if d is very large, the kernel can be a good choice if each training
instance contains only a few non-zero elements. To learn the parameters, we simply substitute Am
with S in (10). In SG or CD algorithms, all it entails is to substitute ?Am (p, x) with ?S(p, x).
For computing ?S(p, x), it is easy to verify that S(p, x) = S(p?j , x?j )(1 + pj xj ) ?j ? [d] and
therefore we have
T
T
x1 S(p, x)
xd S(p, x)
?S(p, x) = x1 S(p?1 , x?1 ), . . . , xd S(p?d , x?d ) =
,...,
.
1 + p 1 x1
1 + p d xd
Therefore, the main advantage of the all-subsets kernel is that we can evaluate it and compute its
Pk
gradient in just O(d) time. The total cost for computing y? = s=1 S(ps , x) is only O(kd).
6
6.1
Experimental results
Application to link prediction
Problem setting. We now demonstrate a novel application of HOFMs to predict the presence or
absence of links between nodes in a graph. Formally, we assume two sets of possibly disjoint nodes
of size nA and nB , respectively. We assume features for the two sets of nodes, represented by
matrices A ? RdA ?nA and B ? RdB ?nB . For instance, A can represent user features and B movie
features. We denote the columns of A and B by ai and bj , respectively. We are given a matrix
Y ? {0, 1}nA ?nB , whose elements indicate presence (positive sample) or absence (negative sample)
of link between two nodes ai and bj . We denote the number of positive samples by n+ . Using this
data, our goal is to predict new associations. Datasets used in our experiments are summarized in
Table 2. Note that for the NIPS and Enzyme datasets, A = B.
Conversion to a supervised problem. We need to convert the above information to a format FMs
and HOFMs can handle. To predict an element yi,j of Y , we simply form xi,j to be the concatenation
6
Table 3: Comparison of area under the ROC curve (AUC) as measured on the test sets.
HOFM (m = 2)
HOFM (m = 3)
HOFM (m = 4)
HOFM (m = 5)
HOFM-shared-augmented (m = 2)
HOFM-shared-augmented (m = 3)
HOFM-shared-augmented (m = 4)
HOFM-shared-augmented (m = 5)
HOFM-shared-simplex (m = 2)
HOFM-shared-simplex (m = 3)
HOFM-shared-simplex (m = 4)
HOFM-shared-simplex (m = 5)
All-subsets
Polynomial network (m = 2)
Polynomial network (m = 3)
Polynomial network (m = 4)
Polynomial network (m = 5)
Low-rank bilinear regression
NIPS
0.856
0.875
0.874
0.874
0.858
0.874
0.836
0.824
0.716
0.777
0.758
0.722
0.730
0.725
0.789
0.782
0.543
0.855
Enzyme
0.880
0.888
0.887
0.887
0.876
0.887
0.824
0.795
0.865
0.870
0.870
0.869
0.840
0.879
0.853
0.873
0.524
0.694
GD
0.717
0.717
0.717
0.717
0.704
0.704
0.663
0.600
0.721
0.721
0.721
0.721
0.721
0.721
0.719
0.717
0.648
0.611
Movielens 100K
0.778
0.786
0.786
0.786
0.778
0.787
0.779
0.621
0.701
0.709
0.709
0.709
0.714
0.761
0.696
0.708
0.501
0.718
of ai and bj and feed this to a HOFM in order to compute a prediction y?i,j . Because HOFMs use
feature combinations in xi,j , they can learn the weights of feature combinations between ai and bj .
At training time, we need both positive and negative samples. Let us denote the set of positive and
negative samples by ?. Then our training set is composed of (xi,j , yi,j ) pairs, where (i, j) ? ?.
Models compared.
? HOFM: y?i,j = y?HOFM (xi,j ) as defined in (3) and as originally proposed in [13, 14]. We minimize
(4) by alternating minimization of (10) for each degree.
Pk
? HOFM-shared: y?i,j = s=1 A1?m (ps , xi,j ; ?). We learn P and ? using the simple augmented
data approach described in Section 5.1 (HOFM-shared-augmented). Inspired
Pby SimpleMKL [12],
1
we also report results when learning P and ? directly by minimizing |?|
?i,j ) +
(i,j)?? `(yi,j , y
?
2
2 kP k
subject to ? ? 0 and h?, 1i = 1 (HOFM-shared-simplex).
Pk
? All-subsets: y?i,j = s=1 S(ps , xi,j ). As explained in Section 5.2, this model is equivalent to the
HOFM-shared model with m = nz (xi,j ) and ? = 1.
Pk
? Polynomial network: y?i,j = s=1 (?s + hps , xi,j i)m . This model can be thought as factorization
machine variant that uses a polynomial kernel instead of the ANOVA kernel (c.f., [8, 4, 22]).
? Low-rank bilinear regression: y?i,j = ai U V T bj , where U ? RdA ?k and V ? RdB ?k . Such
model
for link prediction in [9] and [10]. We learn U and V by minimizing
P was shown to work well
?
1
`(y
,
y
?
)
+
(kU
k2 + kV k2 ).
i,j
i,j
(i,j)??
|?|
2
Experimental setup and evaluation. In this experiment, for all models above, we use CD rather than
SG to avoid the tuning of a learning rate hyper-parameter. We set ` to be the squared loss. Although
we omitted it from our notation for clarity, we also fit a bias term for all models. We evaluated
the compared models using the area under the ROC curve (AUC), which is the probability that the
model correctly ranks a positive sample higher than a negative sample. We split the n+ positive
samples into 50% for training and 50% for testing. We sample the same number of negative samples
as positive samples for training and use the rest for testing. We chose ? from 10?6 , 10?5 , . . . , 106
by cross-validation and following [9] we empirically set k = 30. Throughout our experiments, we
initialized the elements of P randomly by N (0, 0.01).
Results are indicated in Table 3. Overall the two best models were HOFM and HOFM-sharedaugmented, which achieved the best scores on 3 out of 4 datasets. The two models outperformed
low-rank bilinear regression on 3 out 4 datasets, showing the benefit of using higher-order feature
combinations. HOFM-shared-augmented achieved similar accuracy to HOFM, despite using a smaller
model. Surprisingly, HOFM-shared-simplex did not improve over HOFM-shared-augmented except
7
(a) Convergence when m = 2
(b) Convergence when m = 3
(c) Convergence when m = 4
(d) Scalability w.r.t. degree m
Figure 1: Solver comparison for minimizing (10) when varying the degree m on the NIPS dataset
with ? = 0.1 and k = 30. Results on other datasets are in Appendix B.
on the GD dataset. We conclude that our augmented data approach is convenient yet works well in
practice. All-subsets and polynomial networks performed worse than HOFM and HOFM-sharedaugmented, except on the GD dataset where they were the best. Finally, we observe that HOFM were
quite robust to increasing m, which is likely a benefit of modeling each degree with a separate matrix.
6.2
Solver comparison
We compared AdaGrad [5], L-BFGS and coordinate descent (CD) for minimizing (10) when varying
the degree m on the NIPS dataset with ? = 0.1 and k = 30. We constructed the data in the same
way as explained in the previous section and added m ? 1 dummy features, resulting in n = 8, 280
sparse samples of dimension d = 27, 298 + m ? 1. For AdaGrad and L-BFGS, we computed the
(stochastic) gradients using Algorithm 2. All solvers used the same initialization.
Results are indicated in Figure 1. We see that our CD algorithm performs very well when m ? 3
but starts to deteriorate when m ? 4, in which case L-BFGS becomes advantageous. As shown in
Figure 1 d), the cost per epoch of AdaGrad and L-BFGS scales linearly with m, a benefit of our DP
algorithm for computing the gradient. However, to our surprise, we found that AdaGrad is quite
sensitive to the learning rate ?. AdaGrad diverged for ? ? {1, 0.1, 0.01} and the largest value to work
well was ? = 0.001. This explains why AdaGrad did not outperform CD despite the lower cost per
epoch. In the future, it would be useful to create a CD algorithm with a better dependency on m.
7
Conclusion and future directions
In this paper, we presented the first training algorithms for HOFMs and introduced new HOFM
variants with shared parameters. A popular way to deal with a large number of negative samples is
to use an objective function that directly maximize AUC [9, 15]. This is especially easy to do with
SG algorithms because we can sample pairs of positive and negative samples from the dataset upon
each SG update. We therefore expect the algorithms developed in Section 3 to be especially useful
in this setting. Recently, [7] proposed a distributed SG algorithm for training second-order FMs. It
should be straightforward to extend this algorithm to HOFMs based on our contributions in Section 3.
Finally, it should be possible to integrate Algorithm 1 and 2 into a deep learning framework such as
TensorFlow [1], in order to easily compose ANOVA kernels with other layers (e.g., convolutional).
8
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5,686 | 6,145 | A Multi-Batch L-BFGS Method for Machine
Learning
Albert S. Berahas
Northwestern University
Evanston, IL
albertberahas@u.northwestern.edu
Jorge Nocedal
Northwestern University
Evanston, IL
j-nocedal@northwestern.edu
Martin Tak??c
Lehigh University
Bethlehem, PA
takac.mt@gmail.com
Abstract
The question of how to parallelize the stochastic gradient descent (SGD) method
has received much attention in the literature. In this paper, we focus instead on batch
methods that use a sizeable fraction of the training set at each iteration to facilitate
parallelism, and that employ second-order information. In order to improve the
learning process, we follow a multi-batch approach in which the batch changes
at each iteration. This can cause difficulties because L-BFGS employs gradient
differences to update the Hessian approximations, and when these gradients are
computed using different data points the process can be unstable. This paper shows
how to perform stable quasi-Newton updating in the multi-batch setting, illustrates
the behavior of the algorithm in a distributed computing platform, and studies its
convergence properties for both the convex and nonconvex cases.
1
Introduction
It is common in machine learning to encounter optimization problems involving millions of parameters
and very large datasets. To deal with the computational demands imposed by such applications, high
performance implementations of stochastic gradient and batch quasi-Newton methods have been
developed [1, 11, 9]. In this paper we study a batch approach based on the L-BFGS method [20] that
strives to reach the right balance between efficient learning and productive parallelism.
In supervised learning, one seeks to minimize empirical risk,
n
F (w) :=
n
X
1X
def 1
f (w; xi , y i ) =
fi (w),
n i=1
n i=1
where (xi , y i )ni=1 denote the training examples and f (?; x, y) : Rd ? R is the composition of a
prediction function (parametrized by w) and a loss function. The training problem consists of finding
an optimal choice of the parameters w ? Rd with respect to F , i.e.,
n
min F (w) =
w?Rd
1X
fi (w).
n i=1
(1.1)
At present, the preferred optimization method is the stochastic gradient descent (SGD) method [23, 5],
and its variants [14, 24, 12], which are implemented either in an asynchronous manner (e.g. when
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
using a parameter server in a distributed setting) or following a synchronous mini-batch approach that
exploits parallelism in the gradient evaluation [2, 22, 13]. A drawback of the asynchronous approach
is that it cannot use large batches, as this would cause updates to become too dense and compromise
the stability and scalability of the method [16, 22]. As a result, the algorithm spends more time in
communication as compared to computation. On the other hand, using a synchronous mini-batch
approach one can achieve a near-linear decrease in the number of SGD iterations as the mini-batch
size is increased, up to a certain point after which the increase in computation is not offset by the
faster convergence [26].
An alternative to SGD is a batch method, such as L-BFGS, which is able to reach high training
accuracy and allows one to perform more computation per node, so as to achieve a better balance
with communication costs [27]. Batch methods are, however, not as efficient learning algorithms as
SGD in a sequential setting [6]. To benefit from the strength of both methods some high performance
systems employ SGD at the start and later switch to a batch method [1].
Multi-Batch Method. In this paper, we follow a different approach consisting of a single method
that selects a sizeable subset (batch) of the training data to compute a step, and changes this batch at
each iteration to improve the learning abilities of the method. We call this a multi-batch approach
to differentiate it from the mini-batch approach used in conjunction with SGD, which employs a
very small subset of the training data. When using large batches it is natural to employ a quasiNewton method, as incorporating second-order information imposes little computational overhead
and improves the stability and speed of the method. We focus here on the L-BFGS method, which
employs gradient information to update an estimate of the Hessian and computes a step in O(d) flops,
where d is the number of variables. The multi-batch approach can, however, cause difficulties to
L-BFGS because this method employs gradient differences to update Hessian approximations. When
the gradients used in these differences are based on different data points, the updating procedure can
be unstable. Similar difficulties arise in a parallel implementation of the standard L-BFGS method, if
some of the computational nodes devoted to the evaluation of the function and gradient are unable to
return results on time ? as this again amounts to using different data points to evaluate the function
and gradient at the beginning and the end of the iteration. The goal of this paper is to show that stable
quasi-Newton updating can be achieved in both settings without incurring extra computational cost, or
special synchronization. The key is to perform quasi-Newton updating based on the overlap between
consecutive batches. The only restriction is that this overlap should not be too small, something that
can be achieved in most situations.
Contributions. We describe a novel implementation of the batch L-BFGS method that is robust in
the absence of sample consistency; i.e., when different samples are used to evaluate the objective
function and its gradient at consecutive iterations. The numerical experiments show that the method
proposed in this paper ? which we call the multi-batch L-BFGS method ? achieves a good balance
between computation and communication costs. We also analyze the convergence properties of the
new method (using a fixed step length strategy) on both convex and nonconvex problems.
2
The Multi-Batch Quasi-Newton Method
In a pure batch approach, one applies a gradient based method, such as L-BFGS [20], to the
deterministic optimization problem (1.1). When the number n of training examples is large, it is
natural to parallelize the evaluation of F and ?F by assigning the computation of the component
functions fi to different processors. If this is done on a distributed platform, it is possible for some
of the computational nodes to be slower than the rest. In this case, the contribution of the slow (or
unresponsive) computational nodes could be ignored given the stochastic nature of the objective
function. This leads, however, to an inconsistency in the objective function and gradient at the
beginning and at the end of the iteration, which can be detrimental to quasi-Newton methods. Thus,
we seek to find a fault-tolerant variant of the batch L-BFGS method that is capable of dealing with
slow or unresponsive computational nodes.
A similar challenge arises in a multi-batch implementation of the L-BFGS method in which the entire
training set T = {(xi , y i )ni=1 } is not employed at every iteration, but rather, a subset of the data is
used to compute the gradient. Specifically, we consider a method in which the dataset is randomly
divided into a number of batches ? say 10, 50, or 100 ? and the minimization is performed with
respect to a different batch at every iteration. At the k-th iteration, the algorithm chooses a batch
2
Sk ? {1, . . . , n}, computes
1 X
fi (wk ) ,
F Sk (wk ) =
|Sk |
i?Sk
?F Sk (wk ) = gkSk =
1 X
?fi (wk ) ,
|Sk |
(2.2)
i?Sk
and takes a step along the direction ?Hk gkSk , where Hk is an approximation to ?2 F (wk )?1 . Allowing the sample Sk to change freely at every iteration gives this approach flexibility of implementation
and is beneficial to the learning process, as we show in Section 4. (We refer to Sk as the sample of
training points, even though Sk only indexes those points.)
The case of unresponsive computational nodes and the multi-batch method are similar. The main
difference is that node failures create unpredictable changes to the samples Sk , whereas a multi-batch
method has control over sample generation. In either case, the algorithm employs a stochastic approximation to the gradient and can no longer be considered deterministic. We must, however, distinguish
our setting from that of the classical SGD method, which employs small mini-batches and noisy
gradient approximations. Our algorithm operates with much larger batches so that distributing the
function evaluation is beneficial and the compute time of gkSk is not overwhelmed by communication
costs. This gives rise to gradients with relatively small variance and justifies the use of a second-order
method such as L-BFGS.
Robust Quasi-Newton Updating. The difficulties created by the use of a different sample Sk at each
iteration can be circumvented if consecutive samples Sk and Sk+1 overlap, so that Ok = Sk ?Sk+1 6=
?. One can then perform stable quasi-Newton updating by computing gradient differences based on
this overlap, i.e., by defining
Ok
yk+1 = gk+1
? gkOk ,
sk+1 = wk+1 ? wk ,
(2.3)
in the notation given in (2.2). The correction pair (yk , sk ) can then be used in the BFGS update.
When the overlap set Ok is not too small, yk is a useful approximation of the curvature of the objective
function F along the most recent displacement, and will lead to a productive quasi-Newton step. This
observation is based on an important property of Newton-like methods, namely that there is much
more freedom in choosing a Hessian approximation than in computing the gradient [7, 3]. Thus, a
smaller sample Ok can be employed for updating the inverse Hessian approximation Hk than for
computing the batch gradient gkSk in the search direction ?Hk gkSk . In summary, by ensuring that
unresponsive nodes do not constitute the vast majority of all working nodes in a fault-tolerant parallel
implementation, or by exerting a small degree of control over the creation of the samples Sk in the
multi-batch method, one can design a robust method that naturally builds upon the fundamental
properties of BFGS updating.
We should mention in passing that a commonly used strategy for ensuring stability of quasi-Newton
updating in machine learning is to enforce gradient consistency [25], i.e., to use the same sample
Sk to compute gradient evaluations at the beginning and the end of the iteration. Another popular
remedy is to use the same batch Sk for multiple iterations [19], alleviating the gradient inconsistency
problem at the price of slower convergence. In this paper, we assume that achieving such sample
consistency is not possible (in the fault-tolerant case) or desirable (in a multi-batch framework), and
wish to design a new variant of L-BFGS that imposes minimal restrictions in the sample changes.
2.1
Specification of the Method
At the k-th iteration, the multi-batch BFGS algorithm chooses a set Sk ? {1, . . . , n} and computes a
new iterate
wk+1 = wk ? ?k Hk gkSk ,
(2.4)
where ?k is the step length, gkSk is the batch gradient (2.2) and Hk is the inverse BFGS Hessian
matrix approximation that is updated at every iteration by means of the formula
Hk+1 = VkT Hk Vk + ?k sk sTk ,
?k =
1
Ts ,
yk
k
Vk = I ? ?k yk sTk .
To compute the correction vectors (sk , yk ), we determine the overlap set Ok = Sk ? Sk+1 consisting
of the samples that are common at the k-th and k + 1-st iterations. We define
1 X
1 X
F Ok (wk ) =
fi (wk ) ,
?F Ok (wk ) = gkOk =
?fi (wk ) ,
|Ok |
|Ok |
i?Ok
i?Ok
3
and compute the correction vectors as in (2.3). In this paper we assume that ?k is constant.
In the limited memory version, the matrix Hk is defined at each iteration as the result of applying
m BFGS updates to a multiple of the identity matrix, using a set of m correction pairs {si , yi }
kept in storage. The memory parameter m is typically in the range 2 to 20. When computing the
matrix-vector product in (2.4) it is not necessary to form that matrix Hk since one can obtain this
product via the two-loop recursion [20], using the m most recent correction pairs {si , yi }. After the
step has been computed, the oldest pair (sj , yj ) is discarded and the new curvature pair is stored.
A pseudo-code of the proposed method is given below, and depends on several parameters. The
parameter r denotes the fraction of samples in the dataset used to define the gradient, i.e., r = |S|
n .
The parameter o denotes the length of overlap between consecutive samples, and is defined as a
fraction of the number of samples in a given batch S, i.e., o = |O|
|S| .
Algorithm 1 Multi-Batch L-BFGS
Input: w0 (initial iterate), T = {(xi , y i ), for i = 1, . . . , n} (training set), m (memory parameter), r
(batch, fraction of n), o (overlap, fraction of batch), k ? 0 (iteration counter).
1: Create initial batch S0
. As shown in Firgure 1
2: for k = 0, 1, 2, ... do
3:
Calculate the search direction pk = ?Hk gkSk
. Using L-BFGS formula
4:
Choose the step length ?k > 0
5:
Compute wk+1 = wk + ?k pk
6:
Create the next batch Sk+1
Ok
7:
Compute the curvature pairs sk+1 = wk+1 ? wk and yk+1 = gk+1
? gkOk
8:
Replace the oldest pair (si , yi ) by sk+1 , yk+1
9: end for
2.2
Sample Generation
We now discuss how the sample Sk+1 is created at each iteration (Line 8 in Algorithm 1).
Distributed Computing with Faults. Consider a distributed implementation in which slave nodes
read the current iterate wk from the master node, compute a local gradient on a subset of the
dataset, and send it back to the master node for aggregation in the calculation (2.2). Given a time
(computational) budget, it is possible for some nodes to fail to return a result. The schematic in
Figure 1a illustrates the gradient calculation across two iterations, k and k+1, in the presence of faults.
Here Bi , i = 1, ..., B denote the batches of data that each slave node i receives (where T = ?i Bi ),
? (w) is the gradient calculation using all nodes that responded within the preallocated time.
and ?f
n
MASTER
NODE
wk
d
wk+1
S0
B1
B2
? B1 (wk )
rf
B3
???
SLAVE
NODES
BB
? B3 (wk ) rf
? BB (wk )
rf
wk
B1
B2
B3
? B1 (wk+1 )
rf
MASTER
NODE
? (wk )
rf
???
SHUFFLED DATA
O0
SHUFFLED DATA
S3 O 2
O3
S6 O 5
O6
BB
S1 O 0
O1
S4 O 3
O4
? BB (wk+1 )
rf
S2 O 1
wk+1
O2
S5 O 4
O5
? (wk+1 )
rf
(a)
(b)
Figure 1: Sample and Overlap formation.
Let Jk ? {1, 2, ..., B} and Jk+1 ? {1, 2, ..., B} be the set of indices of all nodes that returned a
gradient at the k-th and k + 1-st iterations, respectively. Using this notation Sk = ?j?Jk Bj and
Sk+1 = ?j?Jk+1 Bj , and we define Ok = ?j?Jk ?Jk+1 Bj . The simplest implementation in this
setting preallocates the data on each compute node, requiring minimal data communication, i.e., only
4
one data transfer. In this case the samples Sk will be independent if node failures occur randomly.
On the other hand, if the same set of nodes fail, then sample creation will be biased, which is harmful
both in theory and practice. One way to ensure independent sampling is to shuffle and redistribute the
data to all nodes after a certain number of iterations.
Multi-batch Sampling. We propose two strategies for the multi-batch setting.
Figure 1b illustrates the sample creation process in the first strategy. The dataset is shuffled and
batches are generated by collecting subsets of the training set, in order. Every set (except S0 ) is
of the form Sk = {Ok?1 , Nk , Ok }, where Ok?1 and Ok are the overlapping samples with batches
Sk?1 and Sk+1 respectively, and Nk are the samples that are unique to batch Sk . After each pass
through the dataset, the samples are reshuffled, and the procedure described above is repeated. In our
implementation samples are drawn without replacement, guaranteeing that after every pass (epoch)
all samples are used. This strategy has the advantage that it requires no extra computation in the
Ok
evaluation of gkOk and gk+1
, but the samples {Sk } are not independent.
The second sampling strategy is simpler and requires less control. At every iteration k, a batch Sk is
created by randomly selecting |Sk | elements from {1, . . . n}. The overlapping set Ok is then formed
by randomly selecting |Ok | elements from Sk (subsampling). This strategy is slightly more expensive
Ok
since gk+1
requires extra computation, but if the overlap is small this cost is not significant.
3
Convergence Analysis
In this section, we analyze the convergence properties of the multi-batch L-BFGS method (Algorithm
1) when applied to the minimization of strongly convex and nonconvex objective functions, using a
fixed step length strategy. We assume that the goal is to minimize the empirical risk F given in (1.1),
but note that a similar analysis could be used to study the minimization of the expected risk.
3.1
Strongly Convex case
Due to the stochastic nature of the multi-batch approach, every iteration of Algorithm 1 employs a
gradient that contains errors that do not converge to zero. Therefore, by using a fixed step length
strategy one cannot establish convergence to the optimal solution w? , but only convergence to a
neighborhood of w? [18]. Nevertheless, this result is of interest as it reflects the common practice of
using a fixed step length and decreasing it only if the desired testing error has not been achieved. It
also illustrates the tradeoffs that arise between the size of the batch and the step length.
In our analysis, we make the following assumptions about the objective function and the algorithm.
Assumptions A.
1. F is twice continuously differentiable.
? and ?
? ?2 F O (w) ?I
? such that ?I
? for all w ? Rd and all
2. There exist positive constants ?
sets O ? {1, 2, . . . , n}.
2
3. There is a constant ? such that ES k?F S (w)k ? ? 2 for all w ? Rd and all sets S ?
{1, 2, . . . , n}.
4. The samples S are drawn independently and ?F S (w) is an unbiased estimator of the true
gradient ?F (w) for all w ? Rd , i.e., ES [?F S (w)] = ?F (w).
Note that Assumption A.2 implies that the entire Hessian ?2 F (w) also satisfies
?I ?2 F (w) ?I,
?w ? Rd ,
for some constants ?, ? > 0. Assuming that every sub-sampled function F O (w) is strongly convex
is not unreasonable as a regularization term is commonly added in practice when that is not the case.
We begin by showing that the inverse Hessian approximations Hk generated by the multi-batch
L-BFGS method have eigenvalues that are uniformly bounded above and away from zero. The proof
technique used is an adaptation of that in [8].
Lemma 3.1. If Assumptions A.1-A.2 above hold, there exist constants 0 < ?1 ? ?2 such that the
Hessian approximations {Hk } generated by Algorithm 1 satisfy
?1 I Hk ?2 I,
for k = 0, 1, 2, . . .
5
Utilizing Lemma 3.1, we show that the multi-batch L-BFGS method with a constant step length
converges to a neighborhood of the optimal solution.
Theorem 3.2. Suppose that Assumptions A.1-A.4 hold and let F ? = F (w? ), where w? is the
minimizer of F . Let {wk } be the iterates generated by Algorithm 1 with ?k = ? ? (0, 2?11 ? ), starting
from w0 . Then for all k ? 0,
E[F (wk ) ? F ? ] ? (1 ? 2??1 ?)k [F (w0 ) ? F ? ] + [1 ? (1 ? ??1 ?)k ]
k??
?????
??22 ? 2 ?
4?1 ?
??22 ? 2 ?
.
4?1 ?
The bound provided by this theorem has two components: (i) a term decaying linearly to zero, and
(ii) a term identifying the neighborhood of convergence. Note that a larger step length yields a
more favorable constant in the linearly decaying term, at the cost of an increase in the size of the
neighborhood of convergence. We will consider again these tradeoffs in Section 4, where we also
note that larger batches increase the opportunities for parallelism and improve the limiting accuracy
in the solution, but slow down the learning abilities of the algorithm.
One can establish convergence of the multi-batch L-BFGS method to the optimal solution w? by
employing a sequence of step lengths {?k } that converge to zero according to the schedule proposed
by Robbins and Monro [23]. However, that provides only a sublinear rate of convergence, which is of
little interest in our context where large batches are employed and some type of linear convergence is
expected. In this light, Theorem 3.2 is more relevant to practice.
3.2
Nonconvex case
The BFGS method is known to fail on noconvex problems [17, 10]. Even for L-BFGS, which
makes only a finite number of updates at each iteration, one cannot guarantee that the Hessian
approximations have eigenvalues that are uniformly bounded above and away from zero. To establish
convergence of the BFGS method in the nonconvex case cautious updating procedures have been
proposed [15]. Here we employ a cautious strategy that is well suited to our particular algorithm; we
skip the update, i.e., set Hk+1 = Hk , if the curvature condition
ykT sk ? ksk k2
(3.5)
is not satisfied, where > 0 is a predetermined constant. Using said mechanism we show that the
eigenvalues of the Hessian matrix approximations generated by the multi-batch L-BFGS method are
bounded above and away from zero (Lemma 3.3). The analysis presented in this section is based on
the following assumptions.
Assumptions B.
1. F is twice continuously differentiable.
2. The gradients of F are ?-Lipschitz continuous, and the gradients of F O are ?O -Lipschitz
continuous for all w ? Rd and all sets O ? {1, 2, . . . , n}.
3. The function F (w) is bounded below by a scalar Fb .
2
4. There exist constants ? ? 0 and ? > 0 such that ES k?F S (w)k ? ? 2 + ?k?F (w)k2 for all
w ? Rd and all sets S ? {1, 2, . . . , n}.
5. The samples S are drawn independently and ?F S (w) is an unbiased estimator of the true
gradient ?F (w) for all w ? Rd , i.e., E[?F S (w)] = ?F (w).
Lemma 3.3. Suppose that Assumptions B.1-B.2 hold and let > 0 be given. Let {Hk } be the
Hessian approximations generated by Algorithm 1, with the modification that Hk+1 = Hk whenever
(3.5) is not satisfied. Then, there exist constants 0 < ?1 ? ?2 such that
?1 I Hk ?2 I,
for k = 0, 1, 2, . . .
We can now follow the analysis in [4, Chapter 4] to establish the following result about the behavior
of the gradient norm for the multi-batch L-BFGS method with a cautious update strategy.
Theorem 3.4. Suppose that Assumptions B.1-B.5 above hold, and let > 0 be given. Let {wk } be
1
the iterates generated by Algorithm 1, with ?k = ? ? (0, ?2???
), starting from w0 , and with the
2
6
modification that Hk+1 = Hk whenever (3.5) is not satisfied. Then,
i ??2 ? 2 ? 2[F (w ) ? Fb]
h 1 L?1
X
0
2
k?F (wk )k2 ?
+
E
L
?1
??1 L
k=0
L??
?????
??22 ? 2 ?
.
?1
This result bounds the average norm of the gradient of F after the first L ? 1 iterations, and shows
that the iterates spend increasingly more time in regions where the objective function has a small
gradient.
4
Numerical Results
In this Section, we present numerical results that evaluate the proposed robust multi-batch L-BFGS
scheme (Algorithm 1) on logistic regression problems. Figure 2 shows the performance on the
webspam dataset1 , where we compare it against three methods: (i) multi-batch L-BFGS without
enforcing sample consistency (L-BFGS), where gradient differences are computed using different
Sk+1
samples, i.e., yk = gk+1
? gkSk ; (ii) multi-batch gradient descent (Gradient Descent), which is
obtained by setting Hk = I in Algorithm 1; and, (iii) serial SGD, where at every iteration one
sample is used to compute the gradient. We run each method with 10 different random seeds, and,
where applicable, report results for different batch (r) and overlap (o) sizes. The proposed method
is more stable than the standard L-BFGS method; this is especially noticeable when r is small. On
the other hand, serial SGD achieves similar accuracy as the robust L-BFGS method and at a similar
1
rate (e.g., r = 1%), at the cost of n communications per epochs versus r(1?o)
communications per
epoch. Figure 2 also indicates that the robust L-BFGS method is not too sensitive to the size of
overlap. Similar behavior was observed on other datasets, in regimes where r ? o was not too small.
We mention in passing that the L-BFGS step was computed using the a vector-free implementation
proposed in [9].
webspam ? = 1 r= 1% K = 16 o=20%
2
10
0
10
?2
?2
?4
0.5
1
1.5
Epochs
2
2.5
3
10
webspam ? = 1 r= 1% K = 16 o=5%
1
?3
10
0
?4
0.5
1
1.5
Epochs
2
2.5
3
10
webspam ? = 1 r= 1% K = 16 o=10%
2
10
Robust L?BFGS
L?BFGS
Gradient Descent
SGD
10
0
0
0
0.5
1
1.5
Epochs
2
2.5
3
webspam ? = 1 r= 1% K = 16 o=30%
0
10
Robust L?BFGS
L?BFGS
Gradient Descent
SGD
Robust L?BFGS
L?BFGS
Gradient Descent
SGD
?1
10
10
?1
k?F (w)k
10
?2
10
k?F (w)k
0
?2
10
?3
10
k?F (w)k
10
10
?4
10
Robust L?BFGS
L?BFGS
Gradient Descent
SGD
?1
10
10
webspam ? = 1 r= 10% K = 16 o=20%
0
10
Robust L?BFGS
L?BFGS
Gradient Descent
SGD
?1
k?F (w)k
k?F (w)k
10
webspam ? = 1 r= 5% K = 16 o=20%
0
10
Robust L?BFGS
L?BFGS
Gradient Descent
SGD
k?F (w)k
4
10
?2
10
?2
10
?3
10
?3
10
?4
10
0
?4
0.5
1
1.5
Epochs
2
2.5
3
10
0
?4
0.5
1
1.5
Epochs
2
2.5
3
10
0
0.5
1
1.5
Epochs
2
2.5
3
Figure 2: webspam dataset. Comparison of Robust L-BFGS, L-BFGS (multi-batch L-BFGS without
enforcing sample consistency), Gradient Descent (multi-batch Gradient method) and SGD for various
batch (r) and overlap (o) sizes. Solid lines show average performance, and dashed lines show worst
and best performance, over 10 runs (per algorithm). K = 16 MPI processes.
We also explore the performance of the robust multi-batch L-BFGS method in the presence of node
failures (faults), and compare it to the multi-batch variant that does not enforce sample consistency
(L-BFGS). Figure 3 illustrates the performance of the methods on the webspam dataset, for various
1
LIBSVM: https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/binary.html.
7
probabilities of node failures p ? {0.1, 0.3, 0.5}, and suggests that the robust L-BFGS variant is
more stable.
webspam ? = 0.1 p= 0.1 K = 16
0
10
webspam ? = 0.1 p= 0.3 K = 16
0
10
Robust L?BFGS
L?BFGS
webspam ? = 0.1 p= 0.5 K = 16
0
10
Robust L?BFGS
L?BFGS
?1
Robust L?BFGS
L?BFGS
?1
10
10
?2
?2
?3
10
?4
?2
10
k?F (w)k
10
k?F (w)k
k?F (w)k
10
?3
10
10
?4
?4
10
?6
10
0
10
?5
50
100
150
200
Iterations/Epochs
250
300
10
0
?5
50
100
150
200
Iterations/Epochs
250
300
10
0
50
100
150
200
Iterations/Epochs
250
300
Figure 3: webspam dataset. Comparison of Robust L-BFGS and L-BFGS (multi-batch L-BFGS
without enforcing sample consistency), for various node failure probabilities p. Solid lines show
average performance, and dashed lines show worst and best performance, over 10 runs (per algorithm).
K = 16 MPI processes.
Lastly, we study the strong and weak scaling properties of the robust L-BFGS method on artificial
data (Figure 4). We measure the time needed to compute a gradient (Gradient) and the associated
communication (Gradient+C), as well as, the time needed to compute the L-BFGS direction (LBFGS) and the associated communication (L-BFGS+C), for various batch sizes (r). The figure
on the left shows strong scaling of multi-batch LBFGS on a d = 104 dimensional problem with
n = 107 samples. The size of input data is 24GB, and we vary the number of MPI processes,
K ? {1, 2, . . . , 128}. The time it takes to compute the gradient decreases with K, however, for small
values of r, the communication time exceeds the compute time. The figure on the right shows weak
scaling on a problem of similar size, but with varying sparsity. Each sample has 10 ? K non-zero
elements, thus for any K the size of local problem is roughly 1.5GB (for K = 128 size of data
192GB). We observe almost constant time for the gradient computation while the cost of computing
the L-BFGS direction decreases with K; however, if communication is considered, the overall time
needed to compute the L-BFGS direction increases slightly.
Strong Scaling
Elapsed Time [s]
?2
10
?1
Gradient
Gradient+C
L?BFGS
L?BFGS+C
r = 10.00%
r = 5.00%
r = 2.50%
r = 1.25%
r = 0.63%
r = 0.32%
r = 0.16%
r = 0.08%
r = 0.04%
?4
10
10
?2
10
Elapsed Time [s]
0
10
?3
10
?4
10
?5
10
?6
10
?6
1
10
Number of MPI processes ? K
10
2
10
Weak Scaling ? Fix problem dimensions
r = 10.00%
r = 5.00%
r = 2.50%
r = 1.25%
r = 0.63%
r = 0.32%
r = 0.16%
r = 0.08%
r = 0.04%
Gradient
Gradient+C
L?BFGS
L?BFGS+C
1
10
Number of MPI processes ? K
2
10
Figure 4: Strong and weak scaling of multi-batch L-BFGS method.
5
Conclusion
This paper describes a novel variant of the L-BFGS method that is robust and efficient in two settings.
The first occurs in the presence of node failures in a distributed computing implementation; the second
arises when one wishes to employ a different batch at each iteration in order to accelerate learning.
The proposed method avoids the pitfalls of using inconsistent gradient differences by performing
quasi-Newton updating based on the overlap between consecutive samples. Numerical results show
that the method is efficient in practice, and a convergence analysis illustrates its theoretical properties.
Acknowledgements
The first two authors were supported by the Office of Naval Research award N000141410313, the
Department of Energy grant DE-FG02-87ER25047 and the National Science Foundation grant
DMS-1620022. Martin Tak??c was supported by National Science Foundation grant CCF-1618717.
8
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9
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5,687 | 6,146 | SoundNet: Learning Sound
Representations from Unlabeled Video
Yusuf Aytar?
MIT
yusuf@csail.mit.edu
Carl Vondrick?
MIT
vondrick@mit.edu
Antonio Torralba
MIT
torralba@mit.edu
Abstract
We learn rich natural sound representations by capitalizing on large amounts of
unlabeled sound data collected in the wild. We leverage the natural synchronization
between vision and sound to learn an acoustic representation using two-million
unlabeled videos. Unlabeled video has the advantage that it can be economically
acquired at massive scales, yet contains useful signals about natural sound. We
propose a student-teacher training procedure which transfers discriminative visual
knowledge from well established visual recognition models into the sound modality
using unlabeled video as a bridge. Our sound representation yields significant
performance improvements over the state-of-the-art results on standard benchmarks
for acoustic scene/object classification. Visualizations suggest some high-level
semantics automatically emerge in the sound network, even though it is trained
without ground truth labels.
1
Introduction
The fields of object recognition, speech recognition, machine translation have been revolutionized by
the emergence of massive labeled datasets [31, 42, 10] and learned deep representations [17, 33, 10,
35]. However, there has not yet been the same corresponding progress in natural sound understanding
tasks. We attribute this partly to the lack of large labeled datasets of sound, which are often both
expensive and ambiguous to collect. We believe that large-scale sound data can also significantly
advance natural sound understanding. In this paper, we leverage over one year of sounds collected
in-the-wild to learn semantically rich sound representations.
We propose to scale up by capitalizing on the natural synchronization between vision and sound
to learn an acoustic representation from unlabeled video. Unlabeled video has the advantage that
it can be economically acquired at massive scales, yet contains useful signals about sound. Recent
progress in computer vision has enabled machines to recognize scenes and objects in images and
videos with good accuracy. We show how to transfer this discriminative visual knowledge into sound
using unlabeled video as a bridge.
We present a deep convolutional network that learns directly on raw audio waveforms, which is
trained by transferring knowledge from vision into sound. Although the network is trained with
visual supervision, the network has no dependence on vision during inference. In our experiments,
we show that the representation learned by our network obtains state-of-the-art accuracy on three
standard acoustic scene classification datasets. Since we can leverage large amounts of unlabeled
sound data, it is feasible to train deeper networks without significant overfitting, and our experiments
suggest deeper models perform better. Visualizations of the representation suggest that the network is
also learning high-level detectors, such as recognizing bird chirps or crowds cheering, even though it
is trained directly from audio without ground truth labels.
?
contributed equally
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Visual
?Recognition
?Networks
Unlabeled
?
Video
Object
?Distribution
RGB
?Frames
KL
ImageNet CNN
Scene
?Distribution
KL
Places
?CNN
Raw
?
Waveform
conv4
pool2
conv1 pool1
Input
conv2
conv3
pool5
conv5
conv6
conv7
conv8
SoundNet Architecture
Deep
?1D
?Convolutional
?Network
Figure 1: SoundNet: We propose a deep convolutional architecture for natural sound recognition.
We train the network by transferring discriminative knowledge from visual recognition networks into
sound networks. Our approach capitalizes on the synchronization of vision and sound in video.
The primary contribution of this paper is the development of a large-scale and semantically rich
representation for natural sound. We believe large-scale models of natural sounds can have a large
impact in many real-world applications, such as robotics and cross-modal understanding. The
remainder of this paper describes our method and experiments in detail. We first review related work.
In section 2, we describe our unlabeled video dataset and in section 3 we present our network and
training procedure. Finally in section 4 we conclude with experiments on standard benchmarks and
show several visualizations of the learned representation. Code, data, and models will be released.
1.1 Related Work
Sound Recognition: Although large-scale audio understanding has been extensively studied in the
context of music [5, 37] and speech recognition [10], we focus on understanding natural, in-the-wild
sounds. Acoustic scene classification, classifying sound excerpts into existing acoustic scene/object
categories, is predominantly based on applying a variety of general classifiers (SVMs, GMMs, etc.)
to the manually crafted sound features (MFCC, spectrograms, etc.) [4, 29, 21, 30, 34, 32]. Even
though there are unsupervised [20] and supervised [27, 23, 6, 12] deep learning methods applied to
sound classification, the models are limited by the amount of available labeled natural sound data.
We distinguish ourselves from the existing literature by training a deep fully convolutional network
on a large scale dataset (2M videos). This allows us to train much deeper networks. Another key
advantage of our approach is that we supervise our sound recognition network through semantically
rich visual discriminative models [33, 17] which proved their robustness on a variety of large scale
object/scene categorization challenges[31, 42]. [26] also investigates the relation between vision and
sound modalities, but focuses on producing sound from image sequences. Concurrent work [11] also
explores video as a form of weak labeling for audio event classification.
Transfer Learning: Transfer learning is widely studied within computer vision such as transferring
knowledge for object detection [1, 2] and segmentation [18], however transferring from vision to
other modalities are only possible recently with the emergence of high performance visual models
[33, 17]. Our method builds upon teacher-student models [3, 9] and dark knowledge transfer [13].
In [3, 13] the basic idea is to compress (i.e. transfer) discriminative knowledge from a well-trained
complex model to a simpler model without loosing considerable accuracy. In [3] and [13] both the
teacher and the student are in the same modality, whereas in our approach the teacher operates on
vision to train the student model in sound. [9] also transfer visual supervision into depth models.
Cross-Modal Learning and Unlabeled Video: Our approach is broadly inspired by efforts to
model cross-modal relations [24, 14, 7, 26] and works that leverage large amounts of unlabeled video
[25, 41, 8, 40, 39]. In this work, we leverage the natural synchronization between vision and sound
to learn a deep representation of natural sounds without ground truth sound labels.
2
Beach
Classroom
Construction
River
Club
Forrest
Hockey
Playroom
Engine
Vegetation
Figure 2: Unlabeled Video Dataset: Sample frames from our 2+ million video dataset. For visualization purposes, each frame is automatically categorized by object and scene vision networks.
2
Large Unlabeled Video Dataset
We seek to learn a representation for sound by leveraging massive amounts of unlabeled videos.
While there are a variety of sources available on the web (e.g., YouTube, Flickr), we chose to use
videos from Flickr because they are natural, not professionally edited, short clips that capture various
sounds in everyday, in-the-wild situations. We downloaded over two million videos from Flickr by
querying for popular tags [36] and dictionary words, which resulted in over one year of continuous
natural sound and video, which we use for training. The length of each video varies from a few
seconds to several minutes. We show a small sample of frames from the video dataset in Figure 2.
We wish to process sound waves in the raw. Hence, the only post-processing we did on the videos
was to convert sound to MP3s, reduce the sampling rate to 22 kHz, and convert to single channel
audio. Although this slightly degrades the quality of the sound, it allows us to more efficiently operate
on large datasets. We also scaled the waveform to be in the range [?256, 256]. We did not need to
subtract the mean because it was naturally near zero already.
3
3.1
Learning Sound Representations
Deep Convolutional Sound Network
Convolutional Network: We present a deep convolutional architecture for learning sound representations. We propose to use a series of one-dimensional convolutions followed by nonlinearities (i.e.
ReLU layer) in order to process sound. Convolutional networks are well-suited for audio signals for
a couple of reasons. Firstly, like images [19], we desire our network to be invariant to translations, a
property that reduces the number of parameters we need to learn and increases efficiency. Secondly,
convolutional networks allow us to stack layers, which enables us to detect higher-level concepts
through a series of lower-level detectors.
Variable Length Input/Output: Since sound can vary in temporal length, we desire our network to
handle variable-length inputs. To do this, we use a fully convolutional network. As convolutional
layers are invariant to location, we can convolve each layer depending on the length of the input.
Consequently, in our architecture, we only use convolutional and pooling layers. Since the representation adapts to the input length, we must design the output layers to work with variable length inputs
as well. While we could have used a global pooling strategy [37] to down-sample variable length
inputs to a fixed dimensional vector, such a strategy may unnecessarily discard information useful
for high-level representations. Since we ultimately aim to train this network with video, which is
also variable length, we instead use a convolutional output layer to produce an output over multiple
timesteps in video. This strategy is similar to a spatial loss in images [22], but instead temporally.
Network Depth: Since we will use a large amount of video to train, it is feasible to use deep architectures without significant over-fitting. We experiment with both five-layer and eight-layer networks.
3
Layer
conv1
pool1 conv2 pool2 conv3 conv4 conv5 pool5 conv6 conv7 conv8
Dim.
220,050 27,506 13,782 1,722 862
432
217
54
28
15
4
# of Filters
16
16
32
32
64
128
256
256
512
1024 1401
Filter Size
64
8
32
8
16
8
4
4
4
4
8
Stride
2
1
2
1
2
2
2
1
2
2
2
Padding
32
0
16
0
8
4
2
0
2
2
0
Table 1: SoundNet (8 Layer): The configuration of the layers for the 8-layer SoundNet.
conv1
pool1 conv2 pool2 conv3 pool3 conv4 conv5
220,050 27,506 13,782 1,722 862
432
217
54
32
32
64
64
128
128
256
1401
8
32
8
16
8
8
16
64
2
8
2
8
2
8
2
12
0
16
0
8
0
4
4
32
Table 2: SoundNet (5 Layer): The configuration for the 5-layer SoundNet.
We visualize the eight-layer network architecture in Figure 1, which conists of 8 convolutional layers
and 3 max-pooling layers. We show the layer configuration in Table 1 and Table 2.
3.2
Visual Transfer into Sound
The main idea in this paper is to leverage the natural synchronization between vision and sound in
unlabeled video in order to learn a representation for sound. We model the learning problem from a
student-teacher perspective. In our case, state-of-the-art networks for vision will teach our network
for sound to recognize scenes and objects.
Let xi ? RD be a waveform and yi ? R3?T ?W ?H be its corresponding video for 1 ? i ? N , where
W, H, T are width, height and number of sampled frames in the video, respectively. During learning,
we aim to use the posterior probabilities from a teacher vision network gk (yi ) in order to train our
student network fk (xi ) to recognize concepts given sound. As we wish to transfer knowledge from
both object and scene networks, k enumerates the concepts we are transferring. During learning, we
PK PN
P
Pj
optimize min?
k=1
i=1 DKL (gk (yi )||fk (xi ; ?)) where DKL (P ||Q) =
j Pj log Qj is the KLdivergence. While there are a variety of distance metrics we could have use, we chose KL-divergence
because the outputs from the vision network gk can be interpreted as a distribution of categories. As
KL-divergence is differentiable, we optimize it using back-propagation [19] and stochastic gradient
descent. We transfer from both scene and object visual networks (K = 2).
3.3
Sound Classification
Although we train SoundNet to classify visual categories, the categories we wish to recognize may not
appear in visual models (e.g., sneezing). Consequently, we use a different strategy to attach semantic
meaning to sounds. We ignore the output layer of our network and use the internal representation as
features for training classifiers, using a small amount of labeled sound data for the concepts of interest.
We pick a layer in the network to use as features and train a linear SVM. For multi-class classification,
we use a one-vs-all strategy. We perform cross-validation to pick the margin regularization hyperparameter. For robustness, we follow a standard data augmentation procedure where each training
sample is split into overlapping fixed length sound excerpts, which we compute features on and use
for training. During inference, we average predictions across all windows.
3.4
Implementation
Our approach is implemented in Torch7. We use the Adam [16] optimizer and a fixed learning rate of
0.001 and momentum term of 0.9 throughout our experiments. We experimented with several batch
sizes, and found 64 to produce good results. We initialized all the weights to zero mean Gaussian
noise with a standard deviation of 0.01. After every convolution, we use batch normalization [15]
and rectified linear activation units [17]. We train the network for 100, 000 iterations. Optimization
typically took 1 day on a GPU.
4
Experiments
Experimental Setup: We split the unlabeled video dataset into a training set and a held-out validation
set. We use 2, 000, 000 videos for training, and the remaining 140, 000 videos for validation. After
training the network, we use the hidden representation as a feature extractor for learning on smaller,
4
Method
Accuracy
RG [29]
69%
LTT [21]
72%
RNH [30]
77%
Ensemble [34]
78%
SoundNet
88%
Table 3: Acoustic Scene Classification
on DCASE: We evaluate classification
accuracy on the DCASE dataset. By
leveraging large amounts of unlabeled
video, SoundNet generally outperforms
hand-crafted features by 10%.
Accuracy on
Method
ESC-50 ESC-10
SVM-MFCC [28]
39.6%
67.5%
Convolutional Autoencoder 39.9%
74.3%
Random Forest [28]
44.3%
72.7%
Piczak ConvNet [27]
64.5%
81.0%
SoundNet
74.2%
92.2%
Human Performance [28]
81.3%
95.7%
Table 4: Acoustic Scene Classification on ESC-50
and ESC-10: We evaluate classification accuracy on
the ESC datasets. Results suggest that deep convolutional sound networks trained with visual supervision
on unlabeled data outperforms baselines.
labeled sound only datasets. We extract features for a given layer, and train an SVM on the task of
interest. For training the SVM, we use the standard training/test splits of the datasets. We report
classification accuracy.
Baselines:: In addition to published baselines on standard datasets, we explored an additional baseline
trained on our unlabeled videos. We experimented using a convolutional autoencoder for sound,
trained over our video dataset. We use an autoencoder with 4 encoder layers and 4 decoder layers. For
the encoder layers, we used the same first four convolutional layers as SoundNet. For the decoders,
we used a fractionally strided convolutional layers (in order to upsample instead of downsample).
Note that we experimented with deeper autoencoders, but they performed worse. We used mean
squared error for the reconstruction loss, and trained the autoencoders for several days.
4.1
Acoustic Scene Classification
We evaluate the SoundNet representation for acoustic scene classification. The aim in this task is to
categorize sound clips into one of the many acoustic scene categories. We use three standard, publicly
available datasets: DCASE Challenge[34], ESC-50 [28], and ESC-10 [28].
DCASE[34]: One of the tasks in the Detection and Classification of Acoustic Scenes and Events
Challenge (DCASE)[34] is to recognize scenes from natural sounds. In the challenge, there are 10
acoustic scene categories, 10 training examples per category, and 100 held-out testing examples.
Each example is a 30 seconds audio recording. The task is to categorize natural sounds into existing
10 acoustic scene categories. Multi-class classification accuracy is used as the performance metric.
ESC-50 and ESC-10 [28]: The ESC-50 dataset
is a collection of 2000 short (5 seconds) environmental sound recordings of equally balanced 50 categories selected from 5 major
groups (animals, natural soundscapes, human
non-speech sounds, interior/domestic sounds,
and exterior/urban noises). Each category has
40 samples. The data is prearranged into 5 folds
and the accuracy results are reported as the mean
of 5 leave-one-fold-out evaluations. The performance of untrained human participants on
this dataset is 81.3% [28]. ESC-10 is a subset
of ESC-50 which consists of 10 classes (dog
bark, rain, sea waves, baby cry, clock tic, person sneeze, helicopter, chainsaw, rooster, and
fire cracking). The human performance on this
dataset is 95.7%.
We have two major evaluations on this section:
(a) comparison with the existing state of the art
results, (b) diagnostic performance evaluation
Figure 3: SoundNet confusions on ESC-50
of inner layers of SoundNet as generic features
for this task. In DCASE we used 5 second excerpts, and in ESC datasets we used 1 second windows.
In both evaluations a multi-class SVM (multiple one-vs all classifiers) is trained over extracted
5
Accuracy on
Comparison of SoundNet Model
ESC-50 ESC-10
Table 5: Ablation Analysis:
8 Layer, `2 Loss
47.8% 81.5%
We breakdown accuracy of
Loss
8 Layer, KL Loss
72.9% 92.2%
various configurations using
pool5 from SoundNet trained
8 Layer, ImageNet Only
69.5% 89.8%
with VGG. Results suggest that
Teacher Net
8 Layer, Places Only
71.1% 89.5%
deeper convolutional sound
8 Layer, Both
72.9% 92.2%
networks trained with visual
5 Layer, Scratch Init
65.0% 82.3%
supervision on unlabeled data
Depth and
8 Layer, Scratch Init
51.1% 75.5%
helps recognition.
Visual Transfer 5 Layer, Unlabeled Video 66.1% 86.8%
8 Layer, Unlabeled Video 72.9% 92.2%
Dataset
Model
conv4 conv5 pool5 conv6 conv7 conv8
8 Layer, AlexNet
84%
85%
84%
83%
78%
68%
DCASE [34]
8 Layer, VGG
77%
88%
88%
87%
84%
74%
8 Layer, AlexNet 66.0% 71.2% 74.2%
74%
63.8% 45.7%
ESC50 [28]
8 Layer, VGG
66.0% 69.3% 72.9% 73.3% 59.8% 43.7%
Table 6: Which layer and teacher network gives better features? The performance comparison
of extracting features at different SoundNet layers on acoustic scene/object classification tasks.
SoundNet features. Same data augmentation procedure is also applied during testing and the mean
score of all sound excerpts is used as the final score of a test recording for any particular category.
Comparison to State-of-the-Art: Table 3 and 4 compare recognition performance of SoundNet
features versus previous state-of-the-art features on three datasets. In all cases SoundNet features
outperformed the existing results by around 10%. Interestingly, SoundNet features approach human
performance on ESC-10 dataset, however we stress that this dataset may be easy. We report the
confusion matrix across all folds on ESC-50 in Figure 3. The results suggest our approach obtains
very good performance on categories such as toilet flush (97% accuracy) or door knocks (95%
accuracy). Common confusions are laughing confused as hens, foot steps confused as door knocks,
and insects confused as washing machines.
4.2
Ablation Analysis
To better understand our approach, we perform an ablation analysis in Table 5 and Table 6.
Comparison of Loss and Teacher Net (Table 5): We tried training with different subsets of target
categories. In general, performance generally improves with increasing visual supervision. As
expected, our results suggest that using both ImageNet and Places networks as supervision performs
better than a single one. This indicates that progress in sound understanding may be furthered by
building stronger vision models. We also experimented with using `2 loss on the target outputs
instead of KL loss, which performed significantly worse.
Comparison of Network Depth (Table 5): We quantified the impact of network depth. We use five
layer version of SoundNet (instead of the full eight) as a feature extractor instead. The five-layer
SoundNet architecture performed 8% worse than the eight-layer architecture, suggesting depth is
helpful for sound understanding. Interestingly, the five-layer network still generally outperforms
previous state-of-the-art baselines, but the margin is less. We hypothesize even deeper networks may
perform better, which can be trained without significant over-fitting by leveraging large amounts of
unlabeled video.
Comparison of Supervision (Table 5): We also experimented with training the network without
video by using only the labeled target training set, which is relatively small (thousands of examples).
We simply change the network to output the class probabilities, and train it from random initialization
with a cross entropy loss. Hence, the only change is that this baseline does not use any unlabeled
video, allowing us to quantify the contribution of unlabeled video. The five layer SoundNet achieves
slightly better results than [27] which is also a convolutional network trained with same data but with
a different architecture, suggesting our five layer architecture is similar. Increasing the depth from
five layers to eight layers decreases the performance from 65% to 51%, probably because it overfits
to the small training set. However, when trained with visual transfer from unlabeled video, the eight
layer SoundNet achieves a significant gain of around 20% compared to the five layer version. This
6
(a) t-SNE embedding of visual features
(b) t-SNE embedding of sound features
Figure 4: t-SNE embeddings using visual features and sound features (SoundNet conv7). The visual
features are concatenated fc7 features from the VGG networks for ImageNet and Places2. Note that
t-SNE embeddings do not use the class labels. Labels are only used during final visualization.
Feature
8 Layer, conv7
8 Layer, conv8
sound
32.4%
32.3%
vision
49.4%
49.4%
vision+sound
51.4%
50.5%
Table 7: Multi-Modal Recognition: We
report classification accuracy on ? 4K labeled test videos over 44 categories.
suggests that unlabeled video is a powerful signal for sound understanding, and it can be acquired at
large enough scales to support training high-capacity deep networks.
Comparison of Layer and Teacher Network (Table 6): We analyze the discriminative performance
of each SoundNet layer. Generally, features from the pool5 layer gives the best performance. We
also compared different teacher networks for visual supervision (either VGGNet or AlexNet). The
results are inconclusive on which teacher network to use: VGG is a better teacher network for DCASE
while AlexNet is a better teacher network for ESC50.
4.3
Multi-Modal Recognition
In order to compare sound features with visual features on scene/object categorization, we annotated
additional 9,478 videos (vision+sound) which are not seen by the trained networks before. This new
dataset consists of 44 categories from 6 major groups of concepts (i.e. urban, nature, work/home,
music/entertainment, sports, and vehicles). It is annotated by Amazon Mechanical Turk workers. The
frequency of categories depend on natural occurrences on the web, hence unbalanced.
Vision vs. Sound Embeddings: In order to show the semantic relevance of the features, we performed a two dimensional t-SNE [38] embedding and visualized our dataset in figure 4. The visual
features are concatenated fc7 features of the two VGG networks trained using ImageNet and Places2
datasets. We computed the visual features from uniformly selected 4 frames for each video and
computed the mean feature as the final visual representation. The sound features are the conv7
features extracted using SoundNet trained with VGG supervision. This visualizations suggests that
sound features alone also contain considerable amount of semantic information.
Object and Scene Classification: We also performed a quantitative comparison between sound
features and visual features. We used 60% of our dataset for training and the rest for the testing.
The chance level of the task is 2.2% and choosing always the most common category (i.e. music
performance) yields 14% accuracy. Similar to acoustic scene classification methods, we trained a
multi-class SVM over both sound and visual features individually and then jointly. The results are
displayed in Table 7. Visual features alone obtained an accuracy of 49.4%. The SoundNet features
obtained 32.4% accuracy. This suggests that even though sound is not as informative as vision, it still
contains considerable amount of discriminative information. Furthermore, sound and vision together
resulted in a modest improvement of 2% over vision only models.
4.4
Visualizations
In order to have a better insight on what network learned, we visualize its representation. Figure 5
displays the first 16 convolutional filters applied to the raw input audio. The learned filters are diverse,
including low and high frequencies, wavelet-like patterns, increasing and decreasing amplitude filters.
We also visualize some of the hidden units in the last hidden layer (conv7) of our sound representation
7
Figure 5: Learned filters in conv1: We visualize the filters for raw audio in the first layer of the
deep convolutional network.
Baby Talk
Bubbles
Cheering
Bird Chirps
Figure 6: What emerges in sound hidden units? We visualize some of the hidden units in the last
hidden layer of our sound representation by finding inputs that maximally activate a hidden unit.
Above, we illustrate what these units capture by showing the corresponding video frames. No vision
is used in this experiment; we only show frames for visualization purposes only.
by finding inputs that maximally activate a hidden unit. These visualization are displayed on Figure
6. Note that visual frames are not used during computation of activations; they are only included in
the figure for visualization purposes.
5
Conclusion
We propose to train deep sound networks (SoundNet) by transferring knowledge from established
vision networks and large amounts of unlabeled video. The synchronous nature of videos (sound +
vision) allow us to perform such a transfer which resulted in semantically rich audio representations
for natural sounds. Our results show that transfer with unlabeled video is a powerful paradigm for
learning sound representations. All of our experiments suggest that one may obtain better performance
simply by downloading more videos, creating deeper networks, and leveraging richer vision models.
Acknowledgements: We thank MIT TIG, especially Garrett Wollman, for helping store 26 TB of
video. We are grateful for the GPUs donated by NVidia. This work was supported by NSF grant
#1524817 to AT and the Google PhD fellowship to CV.
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5,688 | 6,147 | Towards Unifying Hamiltonian Monte Carlo
and Slice Sampling
Yizhe Zhang, Xiangyu Wang, Changyou Chen, Ricardo Henao, Kai Fan, Lawrence Carin
Duke University
Durham, NC, 27708
{yz196,xw56,changyou.chen, ricardo.henao, kf96 , lcarin} @duke.edu
Abstract
We unify slice sampling and Hamiltonian Monte Carlo (HMC) sampling, demonstrating their connection via the Hamiltonian-Jacobi equation from Hamiltonian
mechanics. This insight enables extension of HMC and slice sampling to a broader
family of samplers, called Monomial Gamma Samplers (MGS). We provide a
theoretical analysis of the mixing performance of such samplers, proving that in
the limit of a single parameter, the MGS draws decorrelated samples from the
desired target distribution. We further show that as this parameter tends toward this
limit, performance gains are achieved at a cost of increasing numerical difficulty
and some practical convergence issues. Our theoretical results are validated with
synthetic data and real-world applications.
1
Introduction
Markov Chain Monte Carlo (MCMC) sampling [1] stands as a fundamental approach for probabilistic
inference in many computational statistical problems. In MCMC one typically seeks to design
methods to efficiently draw samples from an unnormalized density function. Two popular auxiliaryvariable sampling schemes for this task are Hamiltonian Monte Carlo (HMC) [2, 3] and the slice
sampler [4]. HMC exploits gradient information to propose samples along a trajectory that follows
Hamiltonian dynamics [3], introducing momentum as an auxiliary variable. Extending the random
proposal associated with Metropolis-Hastings sampling [4], HMC is often able to propose large
moves with acceptance rates close to one [2]. Recent attempts toward improving HMC have leveraged
geometric manifold information [5] and have used better numerical integrators [6]. Limitations of
HMC include being sensitive to parameter tuning and being restricted to continuous distributions.
These issues can be partially solved by using adaptive approaches [7, 8], and by transforming sampling
from discrete distributions into sampling from continuous ones [9, 10].
Seemingly distinct from HMC, the slice sampler [4] alternates between drawing conditional samples
based on a target distribution and a uniformly distributed slice variable (the auxiliary variable). One
problem with the slice sampler is the difficulty of solving for the slice interval, i.e., the domain of
the uniform distribution, especially in high dimensions. As a consequence, adaptive methods are
often applied [4]. Alternatively, one recent attempt to perform efficient slice sampling on latent
Gaussian models samples from a high-dimensional elliptical curve parameterized by a single scalar
[11]. It has been shown that in some cases slice sampling is more efficient than Gibbs sampling and
Metropolis-Hastings, due to the adaptability of the sampler to the scale of the region currently being
sampled [4].
Despite the success of slice sampling and HMC, little research has been performed to investigate
their connections. In this paper we use the Hamilton-Jacobi equation from classical mechanics to
show that slice sampling is equivalent to HMC with a (simply) generalized kinetic function. Further,
we also show that different settings of the HMC kinetic function correspond to generalized slice
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
sampling, with a non-uniform conditional slicing distribution. Based on this relationship, we develop
theory to analyze the newly proposed broad family of auxiliary-variable-based samplers. We prove
that under this special family of distributions for the momentum in HMC, as the distribution becomes
more heavy-tailed, the one-step autocorrelation of samples from the target distribution converges
asymptotically to zero, leading to potentially decorrelated samples. While of limited practical impact,
this theoretical result provides insights into the properties of the proposed family of samplers. We
also elaborate on the practical tradeoff between the increased computational complexity associated
with improved theoretical sampling efficiency. In the experiments, we validate our theory on both
synthetic data and with real-world problems, including Bayesian Logistic Regression (BLR) and
Independent Component Analysis (ICA), for which we compare the mixing performance of our
approach with that of standard HMC and slice sampling.
2
Solving Hamiltonian dynamics via the Hamilton-Jacobi equation
A Hamiltonian system consists of a kinetic function K(p) with momentum variable p 2 R, and a
potential energy function U (x) with coordinate x 2 R. We elaborate on multivariate cases in the
Appendix. The dynamics of a Hamiltonian system are completely determined by a set of first-order
Partial Differential Equations (PDEs) known as Hamilton?s equations [12]:
@p
=
@?
@H(x, p, ? )
,
@x
@x
@H(x, p, ? )
=
,
@?
@p
(1)
where H(x, p, ? ) = K(p(? )) + U (x(? )) is the Hamiltonian, and ? is the system time. Solving
(1) gives the dynamics of x(? ) and p(? ) as a function of system time ? . In a Hamiltonian system
governed by (1), H(?) is a constant for every ? [12]. A specified H(?), together with the initial point
{x(0), p(0)}, defines a Hamiltonian trajectory {{x(? ), p(? )} : 8? }, in {x, p} space.
It is well known that in many practical cases, a direct solution to (1) may be difficult [13]. Alternatively,
one might seek to transform the original HMC system {H(?), x, p, ? } to a dual space {H 0 (?), x0 , p0 , ? }
in hope that the transformed PDEs in the dual space becomes simpler than the original PDEs in
(1). One promising approach consists of using the Legendre transformation [12]. This family of
transformations defines a unique mapping between primed and original variables, where the system
time, ? , is identical. In the transformed space, the resulting dynamics are often simpler than the
original Hamiltonian system.
An important property of the Legendre transformation is that the form of (1) is preserved in the new
space [14], i.e., @p0 /@? = @H 0 (x0 , p0 , ? )/@x0 , @x0 /@? = @H 0 (x0 , p0 , ? )/@p0 . To guarantee a valid
Legendre transformation between the original Hamiltonian system {H(?), x, p, ? } and the transformed
Hamiltonian system {H 0 (?), x0 , p0 , ? }, both systems should satisfy the Hamilton?s principle [13],
which equivalently express Hamilton?s equations (1). The form of this Legendre transformation is not
unique. One possibility is to use a generating function approach [13], which requires the transformed
variables to satisfy p ? @x/@? H(x, p, ? ) = p0 ? @x0 /@? H(x0 , p0 , ? )0 + dG(x, x0 , p0 , ? )/d? ,
where dG(x, x0 , p0 , ? )/d? follows from the chain rule and G(?) is a Type-2 generating function
defined as G(?) , x0 ? p0 + S(x, p0 , ? ) [14], with S(x, p0 , ? ) being the Hamilton?s principal function
[15], defined below. The following holds due to the independency of x, x0 and p0 in the previous
transformation (after replacing G(?) by its definition):
p=
@S(x, p0 , ? )
,
@x
x0 =
@S(x, p0 , ? )
,
@p0
H 0 (x0 , p0 , ? ) = H(x, p, ? ) +
@S(x, p0 , ? )
. (2)
@?
We then obtain the desired Legendre transformation by setting H 0 (x0 , p0 , ? ) = 0. The resulting
(2) is known as the Hamilton-Jacobi equation (HJE). We refer the reader to [13, 12] for extensive
discussions on the Legendre transformation and HJE.
Recall from above that the Legendre transformation preserves the form of (1). Since H 0 (x0 , p0 , ? ) = 0,
{x0 , p0 } are time-invariant (constant for every ? ). Importantly, the time-invariant point {x0 , p0 } corresponds to a Hamiltonian trajectory in the original space, and it defines the initial point {x(0), p(0)}
in the original space {x, p}; hence, given {x0 , p0 }, one may update the point along the trajectory
by specifying the time ? . A new point {x(? ), p(? )} in the original space along the Hamiltonian
trajectory, with system time ? , can be determined from the transformed point {x0 , p0 } via solving (2).
One typically specifies the kinetic function as K(p) = p2 [2], and Hamilton?s principal function as
S(x, p0 , ? ) = W (x) p0 ? , where W (x) is a function to be determined (defined below). From (2),
2
and the definition of S(?), we can write
H(x, p, ? ) +
@S
= H(x, p, ? )
@?
p0 = U (x) +
?
@S
@x
2
p0 = U (x) +
?
dW (x)
dx
2
p0 = 0 , (3)
where the second equality is obtained by replacing H(x, p, ? ) = U (x(? )) + K(p(? )) and the third
equality by replacing p from (2) into K(p(? )). From (3), p0 = H(x, p, ? ) represents the total
Hamiltonian in the original space {x, p}, and uniquely defines a Hamiltonian trajectory in {x, p}.
Define X , {x : H(?) U (x)
0} as the slice interval, which for constant p0 = H(x, p, ? )
corresponds to a set of valid coordinates in the original space {x, p}. Solving (3) for W (x) gives
?
Z x(? )
1
H(?) U (z), z 2 X
2
W (x) =
f (z) dz + C ,
f (z) =
,
(4)
0,
z 62 X
xmin
where xmin = min{x : x 2 X} and C is a constant. In addition, from (2) we have
Z
1
@S(x, p0 , ? )
@W (x)
1 x(? )
0
x =
=
?=
f (z) 2 dz ? ,
@p0
@H
2 xmin
(5)
where the second equality is obtained by substituting S(?) by its definition and the third equality is
obtained by applying Fubini?s theorem on (4). Hence, for constant {x0 , p0 = H(x, p, ? )}, equation
(5) uniquely defines x(? ) in the original space, for a specified system time ? .
3
3.1
Formulating HMC as a Slice Sampler
p
xt+1 (0), pt+1 (0)
Revisiting HMC and Slice Sampling
xt (?t ), pt (?t )
Suppose we are interested in sampling a random variable x from
an unnormalized density function f (x) / exp[ U (x)], where
xt (0), pt (0)
x
U (x) is the potential energy function. Hamiltonian Monte Carlo
(HMC) augments the target density with an auxiliary momentum Figure 1: Representation of HMC
random variable p, that is independent of x. The distribution of p sampling. Points {xt (0), pt (0)}
is specified as / exp[ K(p)], where K(p) is the kinetic energy and {xt+1 (0), pt+1 (0)} represent
function. Define H(x, p) = U (x) + K(p) as the Hamiltonian. HMC samples at iterations t and
We have omitted the dependency of H(?), x and p on the system t + 1, respectively. The trajectotime ? for simplicity. HMC iteratively performs dynamic evolv- ries for t and t + 1 correspond to
ing and momentum resampling steps, by sampling xt from the distinct Hamiltonian levels Ht (?)
target distribution and pt from the momentum distribution (Gaus- and Ht+1 (?), denoted as black and
sian as K(p) = p2 ), respectively, for t = 1, 2, . . . iterations. red lines, respectively.
Figure 1 illustrates two iterations of this procedure. Starting
from point {xt (0), pt (0)} at the t-th (discrete) iteration, HMC leverages the Hamiltonian dynamics,
governed by Hamilton?s equations in (1) to propose the next sample {xt (?t ), pt (?t )}, at system time
?t . The position in HMC at iteration t + 1 is updated as xt+1 (0) = xt (?t ) (dynamic evolving). A new
momentum pt+1 (0) is resampled independently from a Gaussian distribution (assuming K(p) = p2 ),
establishing the next initial point {xt+1 (0), pt+1 (0)} for iteration t + 1 (momentum resampling).
The latter point corresponds to the initial point of a new trajectory because the Hamiltonian H(?) is
commensurately updated. This means that trajectories correspond to distinct values of H(?).
Typically, numerical integrators such as the leap-frog method [2] are employed to numerically
approximate the Hamiltonian dynamics. In practice, a random number (uniformly drawn from a
fixed range) of discrete numerical integration steps (leap-frog steps) are often used (corresponding to
random time ?t along the trajectory), which has been shown to have better convergence properties
than a single leap-frog step [16]. The discretization error introduced by the numerical integration is
corrected by a Metropolis Hastings (MH) step.
Slice sampling is conceptually simpler than HMC. It augments the target unnormalized density f (x)
with a randomR variable y, with joint distribution expressed as p(x, y) = Z1 1 , s.t. 0 < y < f (x),
where Z1 = f (x)dx is the normalization constant, and the marginal distribution of x exactly
recovers the target normalized distribution f (x)/Z1 . To sample from the target density, slice sampling
iteratively performs a conditional sampling step from p(x|y) and sampling a slice from p(y|x). At
iteration t, starting from xt , a slice yt is uniformly drawn from (0, f (xt )). Then, the next sample
xt+1 , at iteration t + 1, is uniformly drawn from the slice interval {x : f (x) > yt }.
3
HMC and slice sampling both augment the target distribution with auxiliary variables and can
propose long-range moves with high acceptance probability.
3.2
Formulating HMC as a Slice Sampler
Consider the dynamic evolving step in HMC, i.e., {xt (0), pt (0)} 7! {xt (? ), pt (? )} in Figure 1.
From Section 2, the Hamiltonian dynamics in {x, p} space with initial point {x(0), p(0)} can be
performed by mapping to {x0 , p0 } space and updating {x(? ), p(? )} via selecting a ? and solving
(5). AsRwe show in the Appendix, from (5) and in univariate cases? the Hamiltonian dynamics has
1
period X [H(?) U (z)] 2 dz and is symmetric along p = 0 (due to the symmetric form of the kinetic
function). Also from (5), the system time, ? , is? specified uniformly sampled from?a half-period of
R
1
the Hamiltonian dynamics. i.e., ? ? Uniform
x0 , x0 + 12 X [H(?) U (z)] 2 . Intuitively, x0
is
R the ?anchor? of 1the initial point {x(0), p(0)}, w.r.t. the start of the first half period, i.e, when
[H(?) U (z)] 2 = 0. Further, we only need consider half a period because for a symmetric
X
kinetic function, K(p) = p2 , the Hamiltonian dynamics for the two half-periods are mirrored [14].
For the same reason, Figure 1 only shows half of the {x, p} space, when p 0.
Given the sampled ? and the constant {x0 , p0 }, equation (5) can be solved for x? , x(? ), i.e., the
value of x at time ? . Interestingly, the integral in (5) can be interpreted as (up to normalization
constant) a cumulative density function (CDF) of x(? ). From the inverse CDF transform sampling
method, uniformly sampling ? from half of a period and solving for x? from (5), are equivalent to
directly sampling x? from the following density
p(x? |H(?)) / [H(?)
U (x? )]
1
2
,
s.t., H(?)
U (x? )
0.
(6)
We note that this transformation does not make the analytic solution of x(? ) generally tractable.
However, it provides the basic setup to reveal the connection between the slice sampler and HMC.
In the momentum resampling step of HMC, i.e., {xt (? ), pt (? )} 7! {xt+1 (0), pt+1 (0)} in Figure 1,
and using the previously described kinetic function, K(p) = p2 , resampling corresponds to drawing
p from a Gaussian distribution [2].
The algorithm to analytically sample from the HMC (analytic HMC) proceeds as follows: at iteration
t, momentum pt is drawn from a Gaussian distribution. The previously sampled value of xt 1 and
the newly sampled pt yield a Hamiltonian Ht (?). Then, the next sample xt is drawn from (6). This
procedure relates HMC to the slice sampler. To clearly see the connection, we denote yt = e Ht (?) .
Instead of directly sampling {p, x} as just described, we sample {y, x} instead. By substituting Ht (?)
with yt in (6), the conditional updates for this new sampling procedure can be rewritten as below,
yielding the HMC slice sampler (HMC-SS), with conditional distributions defined as
1
Sampling a slice: p(yt |xt ) =
[log f (xt ) log yt ]1 a , s.t. 0 < yt < f (xt ) , (7)
(a)f (xt )
1
Conditional sampling: p(xt+1 |yt ) =
[log f (xt+1 ) log yt ]1 a , s.t. f (xt ) > yt , (8)
Z2 (yt )
where Ra = 1/2 (other values ofR a considered below), f (x) = e U (x) is an unnormalized density, and
1
Z1 , f (x)dx and Z2 (y) , f (x)>y [log f (x) log y] 2 dx are the normalization constants.
Comparing these two procedures, analytic HMC and HMC-SS, we see that the resampling momentum
in analytic HMC corresponds to sampling a slice in HMC-SS. Further, the dynamic evolving in
HMC corresponds to the conditional sampling in MG-SS. We have thus shown that HMC can be
equivalently formulated as a slice sampler procedure via (7) and (8).
3.3
Reformulating Standard Slice Sampler from HMC-SS
In standard slice sampling (described in Section 3.1), both conditional sampling and sampling a
slice are drawn from uniform distributions. However those for HMC-SS in (7) and (8) represent
non-uniform distributions. Interestingly, if we change a in (7) and (8) from a = 1/2 to a = 1, we
obtain the desired uniform distributions for standard slice sampling. This key observation leads us to
consider a generalized form of the kinetic function for HMC, described below.
?
For multidimensional cases, the Hamiltonian dynamics are semi-periodic, yet a similar conclusion still
holds. Details are discussed in the Appendix.
4
Consider the generalized family of kinetic functions K(p) = |p|1/a with a > 0. One may rederive
equations (3)-(8) using this generalized kinetic energy. As shown in the Appendix, these equations
remained unchanged, with the update that each isolated 2 in these equations is replaced by 1/a, and
1/2 is replaced by a 1.
Sampling p (for the momentum resampling step) with the generalized kinetics, corresponds to drawing
p from ?(p; m, a) = 12 m a / (a + 1) exp[ |p|1/a /m], with m = 1. All the formulation in the paper
still holds for arbitrary m, see Appendix for details. We denote this distribution the monomial Gamma
(MG) distribution, MG(a, m), where m is the mass parameter, and a is the monomial parameter.
Note that this is equivalent to the exponential power distribution with zero-mean, described in [17].
We summarize some properties of the MG distribution in the Appendix.
To generate random samples from the MG distribution, one can draw G ? Gamma(a, m) and a
uniform sign variable S ? { 1, 1}, then S ? Ga follows the MG(a, m) distribution. We call the HMC
sampler based on the generalized kinetic function, K(p; a, m): Monomial Gamma Hamiltonian
Monte Carlo (MG-HMC). The algorithm to analytically sample from the MG-HMC is shown
in Algorithm 1. The only difference between this procedure and the previously described is the
momentum resampling step, in that for analytic HMC, p is drawn Gaussian instead of MG(a, m).
However, note that the Gaussian distribution is a special case of MG(a, m) when a = 1/2.
Algorithm 1: MG-HMC with HJE
for t = 1 to T do
Resample momentum: pt ? MG(m, a).
Compute Hamiltonian: Ht = U (xt 1 ) + K(pt ).
Find X , {x : x 2 R; U (x) ? Ht (?)}.
Dynamic evolving: xt |Ht (?) / [Ht (?) U (xt )]a
Algorithm 2: MG-SS
1
; x 2 X.
for t = 1 to T do
Sampling a slice:
Sample yt from (7).
Conditional sampling:
Sample xt from (8).
Interestingly, when a = 1, the Monomial Gamma Slice sampler (MG-SS) in Algorithm 2 recovers
exactly the same update formulas as in standard slice sampling, described in Section 3.1, where the
conditional distributions in (7) and (8) are both uniform. When a 6= 1, we have to iteratively alternate
between sampling from non-uniform distributions (7) and (8), for both auxiliary (slicing) variable y
and target variable x.
Using the same argument from the convergence analysis of standard slice sampling [4], the iterative
sampling procedure in (7) and (8), converges to an invariant joint distribution (detailed in the
Appendix). Further, the marginal distribution of x recovers the target distribution as f (x)/Z1 , while
the marginal distribution of y is given by p(y) = Z2 (y)/[ (a)Z1 ].
The MG-SS can be divided into three broad regimes: 0 < a < 1, a = 1 and a > 1 (illustrated in the
Appendix). When 0 < a < 1, the conditional distribution p(yt |xt ) is skewed towards the current
unnormalized density value f (xt ). The conditional draw of p(xt+1 |yt ) encourages taking samples
with smaller density value (inefficient moves), within the domain of the slice interval X. On the other
hand, when a > 1, draws of yt tend to take smaller values, while draws of xt+1 encourage sampling
from those with large density function values (efficient moves). The case a = 1 corresponds to the
conventional slice sampler. Intuitively, setting a to be small makes the auxiliary variable, yt , stay
close to f (xt ), thus f (xt+1 ) is close to f (xt ). As a result, a larger a seems more desirable. This
intuition is justified in the following sections.
4
Theoretical analysis
We analyze theoretical properties of the MG sampler. All the proofs as well as the ergodicity
properties of analytic MG-SS are given in the Appendix.
One-step autocorrelation of analytic MG-SS We present results on the univariate distribution
case: p(x) / e U (x) . We first investigate the impact of the monomial parameter a on the one-step
autocorrelation function (ACF), ?x (1) , ?(xt , xt+1 ) = [Ext xt+1 (Ex)2 ]/Var(x), as a ! 1.
Theorem 1 characterizes the limiting behavior of ?(xt , xt+1 ).
Theorem 1 For a univariate target distribution, i.e. exp[ U (x)] has finite integral over R, under certain regularity conditions, the one-step autocorrelation of the MG-SS parameterized by a,
asymptotically approaches zero as a ! 1, i.e., lima!0 ?x (1) = 0.
5
In the Appendix we also show that lima!1 ?(yt , yt+1 ) = 0. In addition, we show that ?(yt , yt+h )
is a non-negative decreasing function of the time lag in discrete steps h.
Effective sample size The variance of a Monte Carlo
P1estimator is determined by its Effective
Sample Size (ESS) [18], defined as ESS = N/(1 + 2 ? h=1 ?x (h)), where N is the total number of
samples, ?x (h) is the h-step autocorrelation function, which can be calculated in a recursive manner.
We prove in the Appendix that ?x (h) is non-negative. Further, assuming the MG sampler is uniformly
ergodic and ?x (h) is monotonically decreasing, it can be shown that lima!1 ESS = N . When ESS
approaches full sample size, N , the resulting sampler delivers excellent mixing efficiency [5]. Details
and further discussion are provided in the Appendix.
Case study To examine a specific 1D example, we consider sampling from the exponential
distribution, Exp(?), with energy function given by U (x) = x/?, where x
0. This case has
analytic ?x (h) and ESS. After some algebra (details in the Appendix),
?x (1) =
1
1
Na
x0 ?
, ?x (h) =
, ESS =
,x
?h (x0 ) , E?h (xh |x0 ) xh = ? +
.
a+1
(a + 1)h
a+2
(a + 1)h
These results are in agreement with Theorem 1 and related arguments of ESS and monotonicity of
autocorrelation w.r.t. a. Here x
?h (x0 ) denotes the expectation of the h-lag sample, starting from any
1
x0 . The relative difference x?hx(x0 0 )? ? decays exponentially in h, with a factor of a+1
. In fact, the ?x (1)
for the exponential family class of models introduced in [19], with potential energy U (x) = x! /?,
where x 0, !, ? > 0, can be analytically calculated. The result, provided in the Appendix, indicates
that for this family, ?x (1) decays at a rate of O(a 1 ).
MG-HMC mixing performance In theory, the analytic MG-HMC (the dynamics in (5) can be
solved exactly) is expected to have the same theoretical properties of the analytic MG-SS for unimodal
cases, since they are derived from the same setup. However, the mixing performance of the two
methods could differ significantly when sampling from a multimodal distribution, due to the fact
that the Hamiltonian dynamics may get ?trapped? into a single closed trajectory (one of the modes)
with low energy, whereas the analytic MG-SS does not suffer from this problem as is able to sample
from disjoint slice intervals (one per mode). This is a well-known property of slice sampling [4] that
arises from (7) and (8). However, if a is large enough, as we show in the Appendix, the probability of
getting into a low-energy level associated with more than one Hamiltonian trajectory, which restrict
movement between modes, is arbitrarily small. As a result, the analytic MG-HMC with large value
of a is able to approach the stationary mixing performance of MG-SS.
5
MG sampling in practice
MG-HMC with numerical integrator In practice, MG-SS (performing Algorithm 2) requires: 1)
analytically solving for the slice interval X, which is typically infeasible for multivariate cases [4]; or
2) analytically computing the integral Z2 (y) over X, implied by the non-uniform conditionals from
MG-SS. These are usually computationally infeasible, though adaptive estimation of X could be done
using schemes like ?doubling? and ?shrinking? strategies from the slice sampling literature [4].
It is more convenient to perform approximate MG-HMC using a numerical integrator like in traditional
HMC, i.e., in each iteration, the momentum p is first initialized by sampling from MG(m, a), then
second order St?rmer-Verlet integration [2] is performed for the Hamiltonian dynamics updates:
pt+1/2 = pt
?
2 rU (xt ) ,
xt+1 = xt + ?rK(pt+1/2 ) , pt+1 = pt+1/2
1
1/a 1
.
ma |p|
?
2 rU (xt+1 ) ,
(9)
where rK(p) = sign(p) ?
When a = 1, [rK(p)]d = 1/m for any dimension d,
independent of x and p. To avoid moving on a grid when a = 1, we employ a random step-size ?
from a uniform distribution within non-negative range (r1 , r2 ), as suggested in [2].
No free lunch With a numerical integrator for MG-HMC, however, the argument about choosing
large a (of great theoretical advantage as discussed in the previous section) may face practical issues.
First, a large value of a will lead to a less accurate numerical integrator. This is because as a gets
larger, the trajectory of the total Hamiltonian becomes ?stiffer?, i.e., that the maximum curvature
becomes larger. When a > 1/2, the Hamiltonian trajectory in the phase space, (x, p), has at least
2D (D denotes the total dimension) non-differentiable points (?turnovers?), at each intersection
point with the hyperplane p(d) = 0, d 2 {1 ? ? ? D}. As a result, directly applying St?rmer-Verlet
integration would lead to high integration error as D becomes large.
6
Second, if the sampler is initialized in the tail region of a light-tailed target distribution, MG-HMC
with a > 1 may converge arbitrarily slow to the true target distribution, i.e., the burn-in period could
take arbitrarily long time. For example, with a > 1, rU (x0 ) can be very large when x0 is in the
light-tailed region, leading the update x0 + rK(p0 + rU (x0 )) to be arbitrary close to x0 , i.e., the
sampler does not move.
To ameliorate these issues, we provide mitigating strategies. For the first (numerical) issue, we
propose two possibilities: 1) As an analog to the ?reflection? action of [2], in (9), whenever the
d-th dimension(s) of the momentum changes sign, we ?recoil? the point of these dimension(s) to the
(d)
(d)
(d)
(d)
previous iteration, and negate the momentum of these dimension(s), i.e., xt+1 = xt , pt+1 = pt .
2) Substituting the kinetic function K(p) with a ?softened? kinetic function, and use importance
sampling to sample the momentum. The details and comparison between the ?reflection? action and
?softened? kinetics are discussed in the Appendix.
For the second (convergence) issue, we suggest using a step-size decay scheme, e.g., ? =
max(?1 ?t , ?0 ). In our experiments we use (?1 , ?) = (106 , 0.9), where ?0 is problem-specific. This approach empirically alleviates the slow convergence problem, however we note that a more principled
way would be adaptively selecting a during sampling, which is left for further investigation.
As a compromise between theoretical gains and practical issues, we suggest setting a = 1 (HMC
implementation of a slice sampler) when the dimension is relatively large. This is because in our
experiments, when a > 1, numerical errors and convergence issues tend to overwhelm the theoretical
mixing performance gains described in Section 4.
Theoretical
MG-SS
MG-HMC
(c)
#104
0.4
0.2
1
Theoretical
MG-SS
MG-HMC
0.5
0
0
1
2
3
Theoretical
MG-SS
MG-HMC
4
Monomial parameter a
0
0
1
2
3
4
(e)
#104
0.6
0.4
1.5
Theoretical
MG-SS
MG-HMC
1
0.5
0
0
1
2
3
1
4
Monomial parameter a
Theoretical
0.8
2
0.2
Monomial parameter a
2.5
ESS
1.5
0.6
(d)
1
0.8
ESS
;(1)
2
;(1)
(b)
1
0.8
;(1)
(a)
0
0
1
2
3
MG-HMC
0.6
0.4
0.2
4
Monomial parameter a
0
0
1
2
3
4
Monomial parameter a
Figure 2: Theoretical and empirical ?x (1) and ESS of exponential distribution (a,b), N+ (c,d) and Gamma (e).
6
6.1
Experiments
Simulation studies
1D unimodal problems We first evaluate the performance of the MG sampler with several univariate
distributions: 1) Exponential distribution, U (x) = ?x, x
0. 2) Truncated Gaussian, U (x) =
?x2 , x 0. 3) Gamma distribution, U (x) = (r 1) log x + ?x. Note that the performance of
the sampler does not depend on the scale parameter ? > 0. We compare the empirical ?x (1) and
ESS of the analytic MG-SS and MG-HMC with their theoretical values. In the Gamma distribution
case, analytic derivations of the autocorrelations and ESS are difficult, thus we resort to a numerical
approach to compute ?x (1) and ESS. Details are provided in the Appendix. Each method is run for
30,000 iterations with 10,000 burn-in samples. The number of leap-frog steps is set to be uniformly
drawn from (100 l, 100 + l) with l = 20, as suggested by [16]. We also compared MG-HMC
(a = 1) with standard slice sampling using doubling and shrinking scheme [4] As expected, the
resulting ESS (not shown) for these two methods is almost identical. The experiment settings and
results are provided in the Appendix. The acceptance rates decrease from around 0.98 to around 0.77
for each case, when a grows from 0.5 to 4, as shown in Figure 2(a)-(d),
The results for analytic MG-SS match well with the theoretical results, however MG-HMC seems to
suffer from practical difficulties when a is large, evidenced by results gradually deviating from the
theoretical values. This issue is more evident in the Gamma case (see Figure 2(e)), where ?x (1) first
decreases then increases. Meanwhile, the acceptance rates decreases from 0.9 to 0.5.
1D and 2D bimodal problems We further conduct simulation studies to evaluate the efficiency of
MG-HMC when sampling 1D and 2D multimodal distributions. For the univariate case, the potential
energy is given by U (x) = x4 2x2 ; whereas U (x) = 0.2 ? (x1 + x2 )2 + 0.01 ? (x1 + x2 )4
0.4 ? (x1 x2 )2 in the bivariate case. We show in the Appendix that if the energy functions are
symmetric along x = C, where C is a constant, in theory, the analytic MG-SS will have ESS equal
to the total sample size. However, as shown in Section 4, the analytic MG-HMC is expected to have
an ESS less than its corresponding analytic MG-SS, and the gap between the analytic MG-HMC
7
and analytic MG-SS counterpart should decrease with a. As a result, despite numerical difficulties,
we expect the MG-HMC based on numerical integration to have better mixing performance with
large a. To verify our theory, we run MG-HMC for a = {0.5, 1, 2} for 30,000 iterations with 10,000
burn-in samples. The parameter settings and the acceptance rates are detailed in the Appendix.
Empirically, we find that the efficiency of HMC is significantly improved with a large a as shown in
Table 1, which coincides with the theory in Section 4. From Figure 3, we observe that the MG-HMC
sampler with monomial parameter a = {1, 2} performs better at jumping between modes of the
target distribution, when compared to standard HMC, which confirms the theory in Section 4. We
also compared MG-HMC (a = 1) with standard SS [4]. As expected, in the 1D case, the standard SS
yields ESS close to full sample size, while in 2D case, the resulting ESS is lower than MG-HMC
(a = 1) (details are provided in the Appendix).
Figure 3: 10 MC samples Table 1: ESS of MG-HMC
by MG-HMC from a 2D for 1D and 2D bimodal disBayesian logistic regression We evalu- distribution and different a. tributions.
ate our methods on 6 real-world datasets
1D
ESS
?x (1)
from the UCI repository [20]: German
3
a = 0.5
5175
0.60
credit (G), Australian credit (A), Pima In2
a=1
10157
0.43
dian (P), Heart (H), Ripley (R) and Car1
a=2
24298
0.11
0
avan (C) [21]. Feature dimensions range
2D
ESS
?
x (1)
-1
from 7 to 87, and total data instances are
MG-HMC (a=0.5)
a
=
0.5
4691
0.67
-2
MG-HMC (a=1)
between 250 to 5822. All datasets are
a=1
16349
0.60
MG-HMC (a=2)
-3
Density contour
normalized to have zero mean and unit
a=2
18007
0.53
-3.5
-2
-0.5 x
1
2.5 3.5
variance. Gaussian priors N (0, 100I)
are imposed on the regression coefficients. We draw 5000 iterations with 1000 burn-in samples for
each experiment. The leap-frog steps are set to be uniformly drawn from (100 l, 100 + l) with
l = 20. Other experimental settings (m and ?) are provided in the Appendix.
Real data
x
2
6.2
1
Results in terms of minimum ESS are summarized in Table 2. Prediction accuracies estimated
via cross-validation are almost identical all across (reported in the Appendix). It can be seen that
MG-HMC with a = 1 outperforms (in terms of ESS) the other two settings with a = 0.5 and a = 2,
indicating increased numerical difficulties counter the theoretical gains when a becomes large. This
can be also seen by noting that the acceptance rates drop from around 0.9 to around 0.7 as a increases
from 0.5 to 2. The dimensionality also seems to have an impact on the optimal setting of a, since in
the high-dimensional dataset Cavaran, the improvement of MG-HMC with a = 1 is less significant
compared with other datasets, and a = 2 seems to suffer more of numerical difficulties. Comparisons
between MG-HMC (a = 1) and standard slice sampling are provided in the Appendix. In general,
standard slice sampling with adaptive search underperforms relative to MG-HMC (a = 1).
Table 2: Minimum ESS for each method (dimensionality indicated in parenthesis). Left: BLR; Right: ICA
Dataset (dim) A (15) G (25) H (14) P (8) R (7)
C (87)
ICA (25)
a = 0.5
3124
3447
3524
3434 3317 33 (median 3987)
2677
a=1
4308
4353
4591
4664 4226 36 (median 4531)
3029
a=2
1490
3646
4315
4424 1490
7 (median 740)
1534
ICA We finally evaluate our methods on the MEG [22] dataset for Independent Component Analysis
(ICA), with 17,730 time points and 25 feature dimension. All experiments are based on 5000 MCMC
samples. The acceptance rates for a = (0.5, 1, 2) are (0.98, 0.97, 0.77). Running time is almost
identical for different a. Settings (including m and ?) are provided in the Appendix. As shown in
Table 2, when a = 1, MG-HMC has better mixing performance compared with other settings.
7
Conclusion
We demonstrated the connection between HMC and slice sampling, introducing a new method for
implementing a slice sampler via an augmented form of HMC. With few modifications to standard
HMC, our MG-HMC can be seen as a drop-in replacement for any scenario where HMC and its
variants apply, for example, Hamiltonian Variational Inference (HVI) [23]. We showed the theoretical
advantages of our method over standard HMC, as well as numerical difficulties associated with it.
Several future extensions can be explored to mitigate numerical issues, e.g., performing MG-HMC
on the Riemann manifold [5] so that step-sizes can be adaptively chosen, and using a high-order
symplectic numerical method [24, 25] to reduce the discretization error introduced by the integrator.
8
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5,689 | 6,148 | Interpretable Distribution Features
with Maximum Testing Power
Wittawat Jitkrittum, Zolt?n Szab?, Kacper Chwialkowski, Arthur Gretton
wittawatj@gmail.com
zoltan.szabo.m@gmail.com
kacper.chwialkowski@gmail.com
arthur.gretton@gmail.com
Gatsby Unit, University College London
Abstract
Two semimetrics on probability distributions are proposed, given as the sum of
differences of expectations of analytic functions evaluated at spatial or frequency
locations (i.e, features). The features are chosen so as to maximize the distinguishability of the distributions, by optimizing a lower bound on test power for a
statistical test using these features. The result is a parsimonious and interpretable
indication of how and where two distributions differ locally. We show that the
empirical estimate of the test power criterion converges with increasing sample size,
ensuring the quality of the returned features. In real-world benchmarks on highdimensional text and image data, linear-time tests using the proposed semimetrics
achieve comparable performance to the state-of-the-art quadratic-time maximum
mean discrepancy test, while returning human-interpretable features that explain
the test results.
1
Introduction
We address the problem of discovering features of distinct probability distributions, with which they
can most easily be distinguished. The distributions may be in high dimensions, can differ in non-trivial
ways (i.e., not simply in their means), and are observed only through i.i.d. samples. One application
for such divergence measures is to model criticism, where samples from a trained model are compared
with a validation sample: in the univariate case, through the KL divergence (Cinzia Carota and Polson,
1996), or in the multivariate case, by use of the maximum mean discrepancy (MMD) (Lloyd and
Ghahramani, 2015). An alternative, interpretable analysis of a multivariate difference in distributions
may be obtained by projecting onto a discriminative direction, such that the Wasserstein distance on
this projection is maximized (Mueller and Jaakkola, 2015). Note that both recent works require low
dimensionality, either explicitly (in the case of Lloyd and Gharamani, the function becomes difficult
to plot in more than two dimensions), or implicitly in the case of Mueller and Jaakkola, in that a
large difference in distributions must occur in projection along a particular one-dimensional axis.
Distances between distributions in high dimensions may be more subtle, however, and it is of interest
to find interpretable, distinguishing features of these distributions.
In the present paper, we take a hypothesis testing approach to discovering features which best
distinguish two multivariate probability measures P and Q, as observed by samples X := {xi }ni=1
drawn independently and identically (i.i.d.) from P , and Y := {yi }ni=1 ? Rd from Q. Nonparametric two-sample tests based on RKHS distances (Eric et al., 2008; Fromont et al., 2012;
Gretton et al., 2012a) or energy distances (Sz?kely and Rizzo, 2004; Baringhaus and Franz, 2004)
have as their test statistic an integral probability metric, the Maximum Mean Discrepancy (Gretton
et al., 2012a; Sejdinovic et al., 2013). For this metric, a smooth witness function is computed, such
that the amplitude is largest where the probability mass differs most (e.g. Gretton et al., 2012a,
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Figure 1). Lloyd and Ghahramani (2015) used this witness function to compare the model output of
the Automated Statistician (Lloyd et al., 2014) with a reference sample, yielding a visual indication
of where the model fails. In high dimensions, however, the witness function cannot be plotted, and
is less helpful. Furthermore, the witness function does not give an easily interpretable result for
distributions with local differences in their characteristic functions. A more subtle shortcoming is
that it does not provide a direct indication of the distribution features which, when compared, would
maximize test power - rather, it is the witness function norm, and (broadly speaking) its variance
under the null, that determine test power.
Our approach builds on the analytic representations of probability distributions of Chwialkowski
et al. (2015), where differences in expectations of analytic functions at particular spatial or frequency
locations are used to construct a two-sample test statistic, which can be computed in linear time.
Despite the differences in these analytic functions being evaluated at random locations, the analytic
tests have greater power than linear time tests based on subsampled estimates of the MMD (Gretton
et al., 2012b; Zaremba et al., 2013). Our first theoretical contribution, in Sec. 3, is to derive a lower
bound on the test power, which can be maximized over the choice of test locations. We propose two
novel tests, both of which significantly outperform the random feature choice of Chwialkowski et al..
The (ME) test evaluates the difference of mean embeddings at locations chosen to maximize the test
power lower bound (i.e., spatial features); unlike the maxima of the MMD witness function, these
features are directly chosen to maximize the distinguishability of the distributions, and take variance
into account. The Smooth Characteristic Function (SCF) test uses as its statistic the difference of
the two smoothed empirical characteristic functions, evaluated at points in the frequency domain so
as to maximize the same criterion (i.e., frequency features). Optimization of the mean embedding
kernels/frequency smoothing functions themselves is achieved on a held-out data set with the same
consistent objective.
As our second theoretical contribution in Sec. 3, we prove that the empirical estimate of the test
power criterion asymptotically converges to its population quantity uniformly over the class of
Gaussian kernels. Two important consequences follow: first, in testing, we obtain a more powerful
test with fewer features. Second, we obtain a parsimonious and interpretable set of features that best
distinguish the probability distributions. In Sec. 4, we provide experiments demonstrating that the
proposed linear-time tests greatly outperform all previous linear time tests, and achieve performance
that compares to or exceeds the more expensive quadratic-time MMD test (Gretton et al., 2012a).
Moreover, the new tests discover features of text data (NIPS proceedings) and image data (distinct
facial expressions) which have a clear human interpretation, thus validating our feature elicitation
procedure in these challenging high-dimensional testing scenarios.
2
ME and SCF tests
In this section, we review the ME and SCF tests (Chwialkowski et al., 2015) for two-sample testing.
In Sec. 3, we will extend these approaches to learn features that optimize the power of these tests.
Given two samples X := {xi }ni=1 , Y := {yi }ni=1 ? Rd independently and identically distributed
(i.i.d.) according to P and Q, respectively, the goal of a two-sample test is to decide whether P is
different from Q on the basis of the samples. The task is formulated as a statistical hypothesis test
proposing a null hypothesis H0 : P = Q (samples are drawn from the same distribution) against
an alternative hypothesis H1 : P 6= Q (the sample generating distributions are different). A test
? n from X and Y, and rejects H0 if ?
? n exceeds a predetermined test threshold
calculates a test statistic ?
?n
(critical value). The threshold is given by the (1 ? ?)-quantile of the (asymptotic) distribution of ?
under H0 i.e., the null distribution, and ? is the significance level of the test.
? n , a form of Hotelling?s T-squared statistic, defined
ME test The ME test uses as its test statistic ?
Pn
Pn
> ?1
1
1
>
?
as ?n := nzn Sn zn , where zn := n i=1 zi , Sn := n?1
i=1 (zi ? zn )(zi ? zn ) , and zi :=
J
J
(k(xi , vj ) ? k(yi , vj ))j=1 ? R . The statistic depends on a positive definite kernel k : X ? X ? R
(with X ? Rd ), and a set of J test locations V = {vj }Jj=1 ? Rd . Under H0 , asymptotically
? n follows ?2 (J), a chi-squared distribution with J degrees of freedom. The ME test rejects H0
?
? n > T? , where the test threshold T? is given by the (1 ? ?)-quantile of the asymptotic null
if ?
? n under H1 was not derived, Chwialkowski et al.
distribution ?2 (J). Although the distribution of ?
(2015) showed that if k is analytic, integrable and characteristic (in the sense of Sriperumbudur et al.
? n can be arbitrarily large as n ? ?, allowing the test to correctly reject H0 .
(2011)), under H1 , ?
2
One can intuitively think of the ME test statistic as a squared normalized (by the inverse covariance
2
, V ) distance of the mean embeddings (Smola et al., 2007) of the empirical measures
S?1
n ) L (X
Pn J
Pn
PJ
1
Pn := n i=1 ?xi , and Qn := n1 i=1 ?yi where VJ := J1 i=1 ?vi , and ?x is the Dirac measure
concentrated at x. The unnormalized counterpart (i.e., without S?1
n ) was shown by Chwialkowski
et al. (2015) to be a metric on the space of probability measures for any V. Both variants behave
similarly for two-sample testing, with the normalized version being a semimetric having a more
computationally tractable null distribution, i.e., ?2 (J).
SCF test The SCF uses the test statistic which has the same form as the ME test statistic with
>
>
>
J
?
?
?
a modified zi := [?l(xi ) sin(x>
i vj ) ? l(yi ) sin(yi vj ), l(xi ) cos(xi vj ) ? l(yi ) cos(yi vj )]j=1 ?
R
R2J , where ?l(x) = Rd exp(?iu> x)l(u) du is the Fourier transform of l(x), and l : Rd ? R
is an analytic translation-invariant kernel i.e., l(x ? y) defines a positive definite kernel for x
and y. In contrast to the ME test defining the statistic in terms of spatial locations, the locations
V = {vj }Jj=1 ? Rd in the SCF test are in the frequency domain. As a brief description, let
>
?P (w) := Ex?P exp(iw
R x) be the characteristic function of P . Define a smooth characteristic
function as ?P (v) = Rd ?P (w)l(v ? w) dw (Chwialkowski et al., 2015, Definition 2). Then,
similar to the ME test, the statistic defined by the SCF test can be seen as a normalized (by S?1
n )
version of L2 (X , VJ ) distance of empirical ?P (v) and ?Q (v). The SCF test statistic has asymptotic
distribution ?2 (2J) under H0 . We will use J 0 to refer to the degrees of freedom of the chi-squared
distribution i.e., J 0 = J for the ME test, and J 0 = 2J for the SCF test.
? n :=
In this work, we modify the statistic with a regularization parameter ?n > 0, giving ?
?1
zn , for stability of the matrix inverse. Using multivariate Slutsky?s theorem,
nz>
n (Sn + ?n I)
?
under H0 , ?n still asymptotically follows ?2 (J 0 ) provided that ?n ? 0 as n ? ?.
3
Lower bound on test power, consistency of empirical power statistic
This section contains our main results. We propose to optimize the test locations V and kernel
parameters (jointly referred to as ?) by maximizing a lower bound on the test power in Proposition 1.
This criterion offers a simple objective function for fast parameter tuning. The bound may be of
independent interest in other Hotelling?s T-squared statistics, since apart from the Gaussian case
(e.g. Bilodeau and Brenner, 2008, Ch. 8), the characterization of such statistics under the alternative
distribution is challenging. The optimization procedure is given in Sec. 4. We use Exy as a shorthand
for Ex?P Ey?Q and let k ? kF be the Frobenius norm.
Proposition 1 (Lower bound on ME test power). Let K be a uniformly bounded (i.e., ?B < ?
such that supk?K sup(x,y)?X 2 |k(x, y)| ? B) family of k : X ? X ? R measurable kernels. Let V be a collection in which each element is a set of J test locations. Assume that
? n ? T? of the ME test satisfies
c? := supV?V,k?K k??1 kF < ?. Then, the test power P ?
? n ? T? ? L(?n ) where
P ?
2
?
[?n (?n ?T? )(n?1)??2 n]2
2
2
?3 n(2n?1)2
? 2e?[(?n ?T? )/3?c3 n?n ] ?n /?4 ,
L(?n ) := 1 ? 2e??1 (?n ?T? ) /n ? 2e
and c3 , ?1 , . . . ?4 are positive constants depending on only B, J and c?. The parameter ?n :=
? n := nz> (Sn + ?n I)?1 zn where ? = Exy [z1 ] and
n?> ??1 ? is the population counterpart of ?
n
>
? = Exy [(z1 ? ?)(z1 ? ?) ]. For large n, L(?n ) is increasing in ?n .
? n ? ?n | which involves bounding kzn ? ?k2
Proof (sketch). The idea is to construct a bound for |?
and kSn ? ?kF separately using Hoeffding?s inequality.The result follows after a reparameterization
? n ? ?n | ? t) to have P ?
? n ? T? . See Sec. F for details.
of the bound on P(|?
Proposition 1 suggests that for large n it is sufficient to maximize ?n to maximize a lower bound
on the ME test power. The same conclusion holds for the SCF test (result omitted due to space
constraints). Assume that k is characteristic (Sriperumbudur et al., 2011). It can be shown that
?n = 0 if and only if P = Q i.e., ?n is a semimetric for P and Q. In this sense, one can see ?n as
encoding the ease of rejecting H0 . The higher ?n , the easier for the test to correctly reject H0 when
H1 holds. This observation justifies the use of ?n as a maximization objective for parameter tuning.
3
? n for both ME and SCF tests depends on a set of test locations V and
Contributions The statistic ?
a kernel parameter ?. We propose to set ? := {V, ?} = arg max? ?n = arg max? ?> ??1 ?. The
optimization of ? brings two benefits: first, it significantly increases the probability of rejecting
H0 when H1 holds; second, the learned test locations act as discriminative features allowing an
interpretation of how the two distributions differ. We note that optimizing parameters by maximizing
a test power proxy (Gretton et al., 2012b) is valid under both H0 and H1 as long as the data used
for parameter tuning and for testing are disjoint. If H0 holds, then ? = arg max 0 is arbitrary. Since
the test statistic asymptotically follows ?2 (J 0 ) for any ?, the optimization does not change the null
distribution. Also, the rejection threshold T? depends on only J 0 and is independent of ?.
To avoid creating a dependency between ? and the data used for testing (which would affect the null
distribution), we split the data into two disjoint sets. Let D := (X, Y) and Dtr , Dte ? D such that
? tr in place of
Dtr ? Dte = ? and Dtr ? Dte = D. In practice, since ? and ? are unknown, we use ?
n/2
? tr is the test statistic computed on the training set Dtr . For simplicity, we assume that
?n , where ?
n/2
each of Dtr and Dte has half of the samples in D. We perform an optimization of ? with gradient
? tr (?). The actual two-sample test is performed using the test statistic ?
? te (?)
ascent algorithm on ?
n/2
n/2
te
computed on D . The full procedure from tuning the parameters to the actual two-sample test is
summarized in Sec. A.
? tr in place of ?n for parameter optimization, we give a finiteSince we use an empirical estimate ?
n/2
?1
sample bound in Theorem 2 guaranteeing the convergence of z>
zn to ?> ??1 ? as
n (Sn + ?n I)
n increases, uniformly over all kernels k ? K (a family of uniformly bounded kernels) and all test
locations in an appropriate class. Kernel classes
the widely
satisfying conditions of Theorem 2 include
used isotropic Gaussian kernel class Kg = k? : (x, y) 7? exp ?(2? 2 )?1 kx ? yk2 | ? > 0 , and
the more general full Gaussian kernel class Kfull = {k : (x, y) 7? exp ?(x ? y)> A(x ? y) |
A is positive definite} (see Lemma 5 and Lemma 6).
? n in the ME test). Let X ? Rd be a measurable set, and V be a
Theorem 2 (Consistency of ?
collection in which each element is a set of J test locations. All suprema over V and k are to be
understood as supV?V and supk?K respectively. For a class of kernels K on X ? Rd , define
F1 := {x 7? k(x, v) | k ? K, v ? X },
F2 := {x 7? k(x, v)k(x, v0 ) | k ? K, v, v0 ? X }, (1)
F3 := {(x, y) 7? k(x, v)k(y, v0 ) | k ? K, v, v0 ? X }.
(2)
Assume that (1) K is a uniformly bounded (by B) family of k : X ? X ? R measurable kernels,
(2) c? := supV,k k??1 kF < ?, and (3) Fi = {f?i | ?i ? ?i } is VC-subgraph with VC-index
?
?
V C(Fi ), and ? 7? f?i (m) is continuous (?m, i = 1, 2, 3). Let c1 := 4B 2 J J c?, c2 := 4B J c?,
and c3 := 4B 2 J c?2 . Let Ci -s (i = 1, 2, 3) be the universal constants associated to Fi -s according to
Theorem 2.6.7 in van der Vaart and Wellner (2000). Then for any ? ? (0, 1) with probability at least
1 ? ?,
?1
sup z>
zn ? ?> ??1 ?
n (Sn + ?n I)
V,k
?
2n ? 1
2
2
8 c1 B 2 J
c1 BJ
+ c2 J +
c1 J(TF2 + TF3 ) +
+ c3 ?n , where
?n
n?1
?n
?n n ? 1
!
r
p
?
q
2?[V C(Fj ) ? 1]
2 log(5/?)
16 2B ?j
?j
V C(Fj )
?
=
2 log Cj ? V C(Fj )(16e)
+
+B
,
2
n
n
? 2TF1
TFj
for j = 1, 2, 3 and ?1 = 1, ?2 = ?3 = 2.
Proof (sketch). The idea is to lower bound the difference with an expression involving supV,k kzn ?
?k2 and supV,k kSn ? ?kF . These two quantities can be seen as suprema of empirical processes,
and can be bounded by Rademacher complexities of their respective function classes (i.e., F1 , F2 ,
and F3 ). Finally, the Rademacher complexities can be upper bounded using Dudley entropy bound
and VC subgraph properties of the function classes. Proof details are given in Sec. D.
Theorem
2 implies that if we set ?n
=
O(n?1/4 ),
then we
>
?1
> ?1
?1/4
supV,k zn (Sn + ?n I) zn ? ? ? ? = Op (n
) as the rate of convergence.
4
have
Both
Proposition 1 and Theorem 2 require c? := supV?V,k?K k??1 kF < ? as a precondition.
To guarantee that c? < ?, a concrete construction of K is the isotropic Gaussian kernel class Kg , where ? is constrained to be in a compact set. Also, consider V := {V |
any two locations are at least distance apart, and all test locations have their norms bounded by ?}
for some , ? > 0. Then, for any non-degenerate P, Q, we have c? < ? since (?, V) 7? ?n is
continuous, and thus attains its supremum over compact sets K and V.
4
Experiments
In this section, we demonstrate the effectiveness Table 1: Four toy problems. H0 holds only in SG.
of the proposed methods on both toy and real
Data
P
Q
problems. We consider the isotropic Gaussian
kernel class Kg in all kernel-based tests. We
SG
N (0d , Id ) N (0d , Id )
study seven two-sample test algorithms. For the
GMD N (0d , Id ) N ((1, 0, . . . , 0)> , Id )
SCF test, we set ?l(x) = k(x, 0). Denote by ME- GVD N (0d , Id ) N (0d , diag(2, 1, . . . , 1))
full and SCF-full the ME and SCF tests whose
Blobs Gaussian mixtures in R2 as studied in
test locations and the Gaussian width ? are fully
Chwialkowski et al. (2015); Gretton
optimized using gradient ascent on a separate
et al. (2012b).
Blobs data. Sample from P.
Blobs data. Sample from Q.
training sample (Dtr ) of the same size as the
test set (Dte ). ME-grid and SCF-grid are as in
Chwialkowski et al. (2015) where only the Gaussian width is optimized by a grid search,1 and
the test locations are randomly drawn from a
multivariate normal distribution. MMD-quad
(quadratic-time) and MMD-lin (linear-time) refer to the nonparametric tests based on maximum mean discrepancy of Gretton et al. (2012a), where
to ensure a fair comparison, the Gaussian kernel width is also chosen so as to maximize a criterion
for the test power on training data, following the same principle as (Gretton et al., 2012b). For MMDquad, since its null distribution is given by an infinite sum of weighted chi-squared variables (no
closed-form quantiles), in each trial we randomly permute the two samples 400 times to approximate
the null distribution. Finally, T 2 is the standard two-sample Hotelling?s T-squared test, which serves
as a baseline with Gaussian assumptions on P and Q.
10
10
5
5
0
0
?5
?5
?10
?10
?5
0
5
10
?10
?10
?5
0
5
10
In all the following experiments, each problem is repeated for 500 trials. For toy problems, new
samples are generated from the specified P, Q distributions in each trial. For real problems, samples
are partitioned randomly into training and test sets in each trial. In all of the simulations, we report an
? te
? te ? T? ) which is the proportion of the number of times the statistic ?
empirical estimate of P(?
n/2
n/2
is above T? . This quantity is an estimate of type-I error under H0 , and corresponds to test power
when H1 is true. We set ? = 0.01 in all the experiments. All the code and preprocessed data are
available at https://github.com/wittawatj/interpretable-test.
? tr (?) is a function of ? consisting of one real-valued
Optimization The parameter tuning objective ?
n/2
? and J test locations each of d dimensions. The parameters ? can thus be regarded as a Jd + 1
? tr (?) with respect to ?, and use gradient ascent to
Euclidean vector. We take the derivative of ?
n/2
maximize it. J is pre-specified and fixed. For the ME test, we initialize the test locations with
realizations from two multivariate normal distributions fitted to samples from P and Q; this ensures
that the initial locations are well supported by the data. For the SCF test, initialization using the
standard normal distribution is found to be sufficient. The parameter ?n is not optimized; we set
the regularization parameter ?n to be as small as possible while being large enough to ensure that
(Sn + ?n I)?1 can be stably computed. We emphasize that both the optimization and testing are
linear in n. The testing cost O(J 3 + J 2 n + dJn) and the optimization costs O(J 3 + dJ 2 n) per
gradient ascent iteration. Runtimes of all methods are reported in Sec. C in the appendix.
1. Informative features: simple demonstration We begin with a demonstration that the proxy
? tr (?) for the test power is informative for revealing the difference of the two samples in the ME
?
n/2
1
Chwialkowski et al. (2015) chooses the Gaussian width that minimizes the median of the p-values, a
heuristic that does not directly address test power. Here, we perform a grid search to choose the best Gaussian
? tr as done in ME-full and SCF-full.
width by maximizing ?
n/2
5
1.0
0.8
0.8
0.005
0.000
1000
2000 3000 4000
Test sample size
(a) SG. d = 50.
5000
0.6
0.4
0.2
0.0
1000
2000 3000 4000
Test sample size
(b) GMD. d = 100.
5000
1.0
ME-full
ME-grid
SCF-full
SCF-grid
MMD-quad
MMD-lin
0.8
0.6
Test power
0.010
Test power
1.0
0.015
Test power
Type-I error
0.020
0.4
0.2
0.0
1000
2000 3000 4000
Test sample size
0.6
0.4
0.2
5000 0.0
1000
(c) GVD. d = 50.
2000 3000 4000
Test sample size
5000
T2
(d) Blobs.
te
Figure 2: Plots of type-I error/test power against the test sample size n in the four toy problems.
test. We consider the Gaussian Mean Difference (GMD) problem (see Table 1), where both P and Q
are two-dimensional normal distributions with the difference in means. We use J = 2 test locations
v1 and v2 , where v1 is fixed to the location indicated by the black triangle in Fig. 1. The contour plot
? tr (v1 , v2 ).
shows v2 7? ?
n/2
? tr is maximized when v2 is placed in either of the two regions that
Fig. 1 (top) suggests that ?
n/2
captures the difference of the two samples i.e., the region in which the probability masses of P and
Q have less overlap. Fig. 1 (bottom), we consider placing v1 in one of the two key regions. In this
? tr , implying
case, the contour plot shows that v2 should be placed in the other region to maximize ?
n/2
that placing multiple test locations in the same neighborhood will not increase the discriminability.
The two modes on the left and right suggest two ways to place the test location in a region that
? tr is an indication of many informative ways to
reveals the difference. The non-convexity of the ?
n/2
detect differences of P and Q, rather than a drawback. A convex objective would not capture this
multimodality.
2. Test power vs. sample size n We now demonstrate the rate of increase of test power with sample size. When the null hypothesis holds, the
type-I error stays at the specified level ?. We consider the following four
toy problems: Same Gaussian (SG), Gaussian mean difference (GMD),
Gaussian variance difference (GVD), and Blobs. The specifications of
P and Q are summarized in Table. 1. In the Blobs problem, P and Q are
defined as a mixture of Gaussian distributions arranged on a 4 ? 4 grid in
R2 . This problem is challenging as the difference of P and Q is encoded
at a much smaller length scale compared to the global structure (Gretton
et al., 2012b). Specifically, the eigenvalue ratio for the covariance of each
Gaussian distribution is 2.0 in P , and 1.0 in Q. We set J = 5 in this
experiment.
The results are shown in Fig. 2 where type-I error (for SG problem), and
test power (for GMD, GVD and Blobs problems) are plotted against test
sample size. A number of observations are worth noting. In the SG
problem, we see that the type-I error roughly stays at the specified level:
the rate of rejection of H0 when it is true is roughly at the specified level
? = 0.01.
v 2 ? ?^tr
n=2 (v 1 ; v 2 )
160
140
120
100
80
60
40
20
v 2 ? ?^tr
n=2 (v 1 ; v 2 )
0
192
184
176
168
160
152
144
136
128
Figure 1: A contour plot
? tr as a function of
of ?
n/2
v2 when J = 2 and
v1 is fixed (black trian? tr
gle). The objective ?
n/2
is high in the regions that
GMD with 100 dimensions turns out to be an easy problem for all the
reveal the difference of
tests except MMD-lin. In the GVD and Blobs cases, ME-full and SCFthe two samples.
full achieve substantially higher test power than ME-grid and SCF-grid,
respectively, suggesting a clear advantage from optimizing the test locations. Remarkably, ME-full
consistently outperforms the quadratic-time MMD across all test sample sizes in the GVD case. When
the difference of P and Q is subtle as in the Blobs problem, ME-grid, which uses randomly drawn
test locations, can perform poorly (see Fig. 2d) since it is unlikely that randomly drawn locations will
be placed in the key regions that reveal the difference. In this case, optimization of the test locations
can considerably boost the test power (see ME-full in Fig. 2d). Note also that SCF variants perform
significantly better than ME variants on the Blobs problem, as the difference in P and Q is localized
in the frequency domain; ME-full and ME-grid would require many more test locations in the spatial
domain to match the test powers of the SCF variants. For the same reason, SCF-full does much better
than the quadratic-time MMD across most sample sizes, as the latter represents a weighted distance
between characteristic functions integrated across the entire frequency domain (Sriperumbudur et al.,
2010, Corollary 4).
6
Test power
Type-I error
0.020
0.015
0.010
0.005
0.0005
300 600 900 1200 1500
Dimension
(a) SG
1.0
0.8
0.6
0.4
0.2
0.05
1.0
ME-full
ME-grid
SCF-full
SCF-grid
MMD-lin
0.8
Test power
0.025
300 600 900 1200 1500
Dimension
0.6
0.4
0.2
0.0 5
100
(b) GMD
200 300
Dimension
400
500
T2
(c) GVD
Figure 3: Plots of type-I error/test power against the dimensions d in the four toy problems in Table 1.
Table 2: Type-I errors and powers of various tests in the problem of distinguishing NIPS papers from
two categories. ? = 0.01. J = 1. nte denotes the test sample size of each of the two samples.
Problem
nte ME-full ME-grid SCF-full SCF-grid MMD-quad MMD-lin
Bayes-Bayes
Bayes-Deep
Bayes-Learn
Bayes-Neuro
Learn-Deep
Learn-Neuro
215
216
138
394
149
146
.012
.954
.990
1.00
.956
.960
.018
.034
.774
.300
.052
.572
.012
.688
.836
.828
.656
.590
.004
.180
.534
.500
.138
.360
.022
.906
1.00
.952
.876
1.00
.008
.262
.238
.972
.500
.538
3. Test power vs. dimension d We next investigate how the dimension (d) of the problem can
affect type-I errors and test powers of ME and SCF tests. We consider the same artificial problems:
SG, GMD and GVD. This time, we fix the test sample size to 10000, set J = 5, and vary the
dimension. The results are shown in Fig. 3. Due to the large dimensions and sample size, it is
computationally infeasible to run MMD-quad.
We observe that all the tests except the T-test can maintain type-I error at roughly the specified
significance level ? = 0.01 as dimension increases. The type-I performance of the T-test is incorrect
at large d because of the difficulty in accurately estimating the covariance matrix in high dimensions.
It is interesting to note the high performance of ME-full in the GMD problem in Fig. 3b. ME-full
achieves the maximum test power of 1.0 throughout and matches the power T-test, in spite of being
nonparametric and making no assumption on P and Q (the T-test is further advantaged by its excessive
Type-I error). However, this is true only with optimization of the test locations. This is reflected in
the test power of ME-grid in Fig. 3b which drops monotonically as dimension increases, highlighting
the importance of test location optimization. The performance of MMD-lin degrades quickly with
increasing dimension, as expected from Ramdas et al. (2015).
4. Distinguishing articles from two categories We now turn to performance on real data. We first
consider the problem of distinguishing two categories of publications at the conference on Neural
Information Processing Systems (NIPS). Out of 5903 papers published in NIPS from 1988 to 2015,
we manually select disjoint subsets related to Bayesian inference (Bayes), neuroscience (Neuro),
deep learning (Deep), and statistical learning theory (Learn) (see Sec. B). Each paper is represented
as a bag of words using TF-IDF (Manning et al., 2008) as features. We perform stemming, remove
all stop words, and retain only nouns. A further filtering of document-frequency (DF) of words that
satisfies 5 ? DF ? 2000 yields approximately 5000 words from which 2000 words (i.e., d = 2000
dimensions) are randomly selected. See Sec. B for more details on the preprocessing. For ME
and SCF tests, we use only one test location i.e., set J = 1. We perform 1000 permutations to
approximate the null distribution of MMD-quad in this and the following experiments.
Type-I errors and test powers are summarized in Table. 2. The first column indicates the categories of
the papers in the two samples. In Bayes-Bayes problem, papers on Bayesian inference are randomly
partitioned into two samples in each trial. This task represents a case in which H0 holds. Among all
the linear-time tests, we observe that ME-full has the highest test power in all the tasks, attaining a
maximum test power of 1.0 in the Bayes-Neuro problem. This high performance assures that although
different test locations V may be selected in different trials, these locations are each informative. It
is interesting to observe that ME-full has performance close to or better than MMD-quad, which
requires O(n2 ) runtime complexity. Besides clear advantages of interpretability and linear runtime
of the proposed tests, these results suggest that evaluating the differences in expectations of analytic
functions at particular locations can yield an equally powerful test at a much lower cost, as opposed to
7
Table 3: Type-I errors and powers in the problem of distinguishing positive (+) and negative (-) facial
expressions. ? = 0.01. J = 1.
Problem nte ME-full ME-grid SCF-full SCF-grid MMD-quad MMD-lin
? vs. ?
+ vs. ?
201
201
.010
.998
.012
.656
.014
1.00
.002
.750
.018
1.00
.008
.578
computing the RKHS norm of the witness function as done in MMD. Unlike Blobs, however, Fourier
features are less powerful in this setting.
We further investigate the interpretability of the ME test by the following procedure. For the learned
? t = (?
test location vt ? Rd (d = 2000) in trial t, we construct v
v1t , . . . , v?dt ) such that v?jt = |vjt |.
t
t
Let ?j ? {0, 1} be an indicator variable taking value 1 if v?j is among the top five largest for all
P
j ? {1, . . . , d}, and 0 otherwise. Define ?j := t ?jt as a proxy indicating the significance of word
j i.e., ?j is high if word j is frequently among the top five largest as measured by v?jt . The top seven
words as sorted in descending order by ?j in the Bayes-Neuro problem are spike, markov, cortex,
dropout, recurr, iii, gibb, showing that the learned test locations are highly interpretable. Indeed,
?markov? and ?gibb? (i.e., stemmed from Gibbs) are discriminative terms in Bayesian inference
category, and ?spike? and ?cortex? are key terms in neuroscience. We give full lists of discriminative
terms learned in all the problems in Sec. B.1. To show that not all the randomly selected 2000 terms
are informative, if the definition of ?jt is modified to consider the least important words (i.e., ?j
is high if word j is frequently among the top five smallest as measured by v?jt ), we instead obtain
circumfer, bra, dominiqu, rhino, mitra, kid, impostor, which are not discriminative.
5. Distinguishing positive and negative emotions In the final experiment, we study how well ME and SCF tests can distinguish two samples of photos of people showing positive and
negative facial expressions. Our emphasis is on the discriminative features of the faces identified by ME test showing how
(a) HA (b) NE (c) SU
the two groups differ. For this purpose, we use Karolinska Directed Emotional Faces (KDEF) dataset (Lundqvist et al., 1998)
containing 5040 aligned face images of 70 amateur actors, 35 females and 35 males. We use only photos showing front views of
the faces. In the dataset, each actor displays seven expressions:
happy (HA), neutral (NE), surprised (SU), sad (SA), afraid (AF), (d) AF (e) AN (f) DI (g) v
1
angry (AN), and disgusted (DI). We assign HA, NE, and SU
faces into the positive emotion group (i.e., samples from P ), and Figure 4: (a)-(f): Six facial expresAF, AN and DI faces into the negative emotion group (samples sions of actor AM05 in the KDEF
from Q). We denote this problem as ?+ vs. ??. Examples of data. (g): Average across trials of
six facial expressions from one actor are shown in Fig. 4. Photos the learned test locations v .
1
of the SA group are unused to keep the sizes of the two samples
the same. Each image of size 562 ? 762 pixels is cropped to exclude the background, resized to
48 ? 34 = 1632 pixels (d), and converted to grayscale.
We run the tests 500 times with the same setting used previously i.e., Gaussian kernels, and J = 1.
The type-I errors and test powers are shown in Table 3. In the table, ?? vs. ?? is a problem in which
all faces expressing the six emotions are randomly split into two samples of equal sizes i.e., H0 is
true. Both ME-full and SCF-full achieve high test powers while maintaining the correct type-I errors.
As a way to interpret how positive and negative emotions differ, we take an average across trials of
the learned test locations of ME-full in the ?+ vs. ?? problem. This average is shown in Fig. 4g.
We see that the test locations faithfully capture the difference of positive and negative emotions by
giving more weights to the regions of nose, upper lip, and nasolabial folds (smile lines), confirming
the interpretability of the test in a high-dimensional setting.
Acknowledgement
We thank the Gatsby Charitable Foundation for the financial support.
8
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5,690 | 6,149 | Threshold Bandit, With and Without Censored
Feedback
Jacob Abernethy
Department of Computer Science
University of Michigan
Ann Arbor, MI 48109
jabernet@umich.edu
Kareem Amin
Department of Computer Science
University of Michigan
Ann Arbor, MI 48109
amkareem@umich.edu
Ruihao Zhu
AeroAstro&CSAIL
MIT
Cambridge, MA 02139
rzhu@mit.edu
Abstract
We consider the Threshold Bandit setting, a variant of the classical multi-armed
bandit problem in which the reward on each round depends on a piece of side
information known as a threshold value. The learner selects one of K actions
(arms), this action generates a random sample from a fixed distribution, and the
action then receives a unit payoff in the event that this sample exceeds the threshold
value. We consider two versions of this problem, the uncensored and censored case,
that determine whether the sample is always observed or only when the threshold is
not met. Using new tools to understand the popular UCB algorithm, we show that
the uncensored case is essentially no more difficult than the classical multi-armed
bandit setting. Finally we show that the censored case exhibits more challenges, but
we give guarantees in the event that the sequence of threshold values is generated
optimistically.
1
Introduction
The classical Multi-armed Bandit (MAB) problem provides a framework to reason about sequential
decision settings, but specifically where the learner?s chosen decision is intimately tied to information
content received as feedback. MAB problems have generated much interest in the Machine Learning
research literature in recent years, particularly as a result of the changing nature in which learning
and estimation algorithms are employed in practice. More and more we encounter scenarios in which
the procedure used to make and exploit algorithmic predictions is exactly the same procedure used to
capture new data to improve prediction performance. In other words it is increasingly harder to view
training and testing as distinct entities.
MAB problems generally involve repeatedly making a choice between one of a finite (or even infinite)
set of actions, and these actions have historically been referred to as arms of the bandit. If we ?pull?
arm i at round t, then we receive a reward Rti 2 [0, 1] which is frequently assumed to be a stochastic
quantity that is drawn according to distribution Di . Typically we assume that Di are heterogeneous
across the arms i, whereas we assume the samples {Rti }t=1,...,T are independently and identically
distributed according to the fixed Di across all times t.1 Of course, were the learner to have full
knowledge of the distributions Di from the outset, she would presumably choose to pull the arm
whose expected reward ?i is highest. With that in mind, we tend to consider the (expected) regret of
the learner, defined to be the (expected) reward of the best arm minus the (expected) reward of the
actual arms selected by the learner.
Early work on MAB problems (Robbins, 1952; Lai and Robbins, 1985; Gittins et al., 2011) tended to
be more focused on asymptotic guarantees, whereas more recent work (Auer et al., 2002; Auer, 2003)
1 Note
that in much of our notation we use superscript t to denote the time period rather than as an exponent.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
has been directed towards a more ?finite time? approach in which we can bound regret for fixed time
horizons T . One of the best-known and well-studied techniques is known as the Upper Confidence
Bound (UCB) algorithm (Auer et al., 2002; Auer and Ortner, 2010). The magic of UCB relies on a
very intuitive policy framework, that a learner should select decisions by maximizing over rewards
estimated from previous data but only after biasing each estimate according to its uncertainty. Simply
put, one should choose the arm that maximizes the ?mean plus confidence interval,? hence the name
Upper Confidence Bound.
In the present paper we focus on the Threshold Bandit setting, described as follows. On each round t,
a piece of side information is given to the learner in the form of a real number ct , the learner must
then choose arm i out of K arms, and this arm produces a value Xit drawn from a survival distribution
with survival function Fi (x) = Pr(Xit x). The reward to the learner is not Xit itself but is instead
the binary value Rti = I[Xit ct ]; that is, we receive a unit reward when the sample Xit exceeds the
threshold value ct , and otherwise we receive no reward. For a fixed value of ct , each arm i has
expected payoff E[Rti ] = Fi (ct ). Notice, crucially, that the arm with the greatest expected payoff can
vary significantly across different threshold values.
This abstract model has a number of very natural applications:
1. Packet Delivery with Deadlines: FedEx receives a stream of packages that need to be
shipped from source to destination, and each package is supplied with a delivery deadline.
The goal of the FedEx routing system is to select a transportation route (via air or road or
ship, etc.) that has the highest probability of on-time arrival. Of course some transportation
schemes are often faster (e.g. air travel) but have higher volatility (e.g. due to poor weather).
2. Supplier Selection: Customers approach a manufacturing firm to produce a product with
specific quality demands. The firm must approach one of several suppliers to contract out
the work, but the firm is uncertain as to the capabilities and variabilities of the products each
supplier produces.
3. Dark Pool Brokerage: A financial brokerage firm is asked to buy or sell various sized
bundles of shares, and the brokerage aims to offload the transactions onto one of many
dark pools, i.e. financial exchanges that match buyers and sellers in a confidential manner
(Ganchev et al., 2010; Amin et al., 2012; Agarwal et al., 2010). A standard dark pool
mechanism will simply execute the transaction if there is suitable liquidity, or will reject the
transaction when no match is made. Of course the brokerage gets paid on commission, and
simply wants to choose the pool that has the highest probability of completion.
What distinguishes the Threshold Bandit problem from the standard stochastic multi-armed bandit
setting are two main features:
1. The regret of the learner will be measured in comparison to the best policy rather than to
simply the best arm. Note that the optimal offline policy may incorporate the threshold
value ct before selecting an arm I t .
2. Whereas the standard stochastic bandit setting assumes that we observe the reward RtI t of
the chosen arm I t , in the Threshold Bandit setting we consider two types of feedback.
(a) Uncensored Feedback: After playing arm I t , the learner observes the sample XIt t
regardless of the threshold value ct . This is a natural model for the FedEx routing
problem above, wherein one learns the travel time of a package regardless of the
deadline having been met.
(b) Censored Feedback: After playing I t , the learner observes a null value when XIt t ct ,
and otherwise observes XIt t . This is a natural model for the Supplier Selection problem
above, as we would only learn the product?s quality value when the customer rejects
what is received from the supplier.
In the present paper we present roughly three primary results. First, we provide a new perspective
on the classical UCB algorithm, giving an alternative proof that relies on an interesting potential
function argument; we believe this technique may be of independent interest. Second, we analyze
the Threshold Bandit setting when given uncensored feedback, and we give a novel algorithm called
DKWUCB based on the Dvoretzky-Kiefer-Wolfowitz inequality (Dvoretzky et al., 1956). We show,
somewhat surprisingly, that with uncensored feedback the regret bound is no worse than the standard
2
stochastic MAB setting, suggesting that despite the much richer policy class one has nearly the same
learning challenge. Finally, we consider learning in the censored feedback setting, and propose an
algorithm KMUCB, akin to the Kaplan-Meier estimator (Kaplan and Meier, 1958). Learning with
censored feedback is indeed more difficult, as the performance can depend significantly on the order
of the threshold values. In the worst case, when threshold values are chosen in an adversarial order,
the cost of learning scales with the number of unique threshold values, whereas one can perform
significantly better under certain constraints on optimistic assumptions on the order or even a random
order.
2
A New Perspective on UCB
Before focusing on the Threshold Bandit problem, let us turn our attention to the classical stochastic
MAB setting and give another look at the UCB algorithm. We will now provide a modified proof
of the regret bound of UCB that relies on a potential function. Potential arguments have proved
quite popular in studying adversarial bandit problems (Auer et al., 2003; Audibert and Bubeck, 2009;
Abernethy et al., 2012; Neu and Bart?k, 2013; Abernethy et al., 2015), but have received less use
in the stochastic setting. This potential trick is the basis for forthcoming results on the Threshold
Bandit.
Let Di be a distribution on the reward Rti , with support on [0, 1]. We imagine the rewards
i.i.d.
R1i , . . . , RTi ? Di , whose mean E[Rti ] = ?i . A bandit algorithm is simply a procedure that chooses
a random arm/action I t on round t as a function of the set of past observed (action, reward) pairs,
(I 1 , R1I 1 ), . . . , (I t 1 , RtI t 11 ). Finally, let Nit := ?tt=11 I[I t = i] and define the empirical mean estimator
at time t to be ?? ti :=
1
I[I t =i]RtI t
?tt=1
.
Nit
We assume we are given a particular deviation bound which provides the following guarantee,
Pr |?i
?? ti | > e
Nit
N ? f (N, e),
where f (?) is some function, continuous in e and monotonically decreasing in both parameters, that
controls the probability of a large deviation. While UCB relies specifically on the Hoeffding-Azuma
inequality (Cesa-Bianchi and Lugosi, 2006), for now we leave the deviation bound in generic form.
This will be useful in following sections.
Given f (?, ?), what is of interest to our present work is a pair of functions that allow us to convert
between values of e and N in order to guarantee that f (N, e) ? d for a given d > 0. To this end define
](e, d) :=
V(N, d) :=
min{N : f (N, e/2) ? d},
?
inf{e : f (N, e) ? d} if N > 0;
1
otherwise,
We will often omit the d in the argument to ](?), V(?). Note the key property that V(N, d) ? e/2 for
any N ](e, d).
We can now define our variant of the UCB algorithm for a fixed choice of d > 0.
UCB Algorithm:
on round t play I t = arg max ?? ti + V(Nit , d)
i
(1)
We will make the simplifying assumption that the largest ?i is unique and, without loss of generality, let
us assume that the coordinates are permuted in order that ?1 is the largest mean reward. Furthermore,
define Di := ?1 ?i for i = 2, . . . , K.
A central piece of the analysis relies on the following potential function, which depends on the current
number of plays of each arm i = 2, . . . , K.
F(N2t , . . . , NKt )
t
K Ni 1
:= 2 ?
?
V(N, d)
i=2 N=0
Lemma 1. The expected regret of UCB is bounded as
T +1
E[RegretT (UCB)] ? E[F(N2
3
, . . . , NKT +1 )] + O(T d)
(2)
Proof. The (random) additional regret suffered on round t of UCB is exactly ?1 ?I t . By virtue of
our given deviation bound, we know that
?1 ? ?? t1 + V(N1t , d)
and
?? tI t ? ?I t + V(NIt t , d),
each w.p. > 1 d.
(3)
t
Also, let x be the indicator variable that one of the above two inequalities fails to hold. Of course we
chose V(?) in order that E[xt ] ? 2d via a simple union bound.
Note that, by virtue of using the UCB selection rule for I t , it is clear that we have
?? t1 + V(N1t , d) ? ?? tI t + V(NIt t , d)
If we combine Equations 3 and 4, and consider the event that xt = 0, then we obtain
?1 ? ?? t1 + V(N1t , d) ? ?? tI t + V(NIt t , d) ? ?I t + 2V(NIt t , d).
Even in the event that xt = 1 we have that ?1 ?I t ? 1. Hence, it follows immediately that ?1
2V(NIt t , d) + xt .
(4)
?I t ?
Finally, we observe that the potential function was chosen so that F(N2t+1 , . . . , NKt+1 )
F(N2t , . . . , NKt ) = 2V(NIt t , d). Recalling that F(0, . . . , 0) = 0, a simple telescoping argument gives that
"
#
T
T +1
T +1
t
T +1
T +1
E[RegretT (UCB)] ? E F(N2 , . . . , NK ) + ? x = E[F(N2 , . . . , NK )] + 2T d.
t=1
The final piece we need to establish is that the number of pulls Nit of arm i, for i = 2, . . . , K, is unlikely
to exceed ](Di , d). This result uses some more standard techniques from the original UCB analysis
(Auer et al., 2002), and we defer it to the appendix.
Lemma 2. For any T > 0 we have E[F(N2T +1 , . . . , NKT +1 )] ? F(](D2 , d), . . . , ](DK , d)) + O(T 2 d).
We are now able to combine the above results for the final bound.
Theorem 1. If we set d = T 2 /2, the expected regret of UCB is bounded as
K
log(T )
+ O(1).
Di
i=2
E[RegretT (UCB)] ? 8 ?
Proof. Note that a very standard deviation bound that holds for all distributions supported on
[0, 1] is the Hoeffding-Azuma inequality (Cesa-Bianchi and Lugosi, 2006), wherelthe bound
m is
2 log 2/d
2
given by f (N, e) = 2 exp( 2Ne ). Utilizing Hoeffding-Azuma we have ](e, d) =
and
e2
q
p
V(N, d) = log(2/d)
for N > 0. If we utilize the fact that ?Yy=1 p1y ? 2 Y , then we see that
2N
r
K ](Di ,d)
K
K
log(2/d)](Di , d)
log(2/d)
F(](D2 , d), . . . , ](DK , d)) = 2 ? ? V(N, d) = 2 ? 2
=4?
.
2
Di
i=2 N=0
i=2
i=2
Combining the Lemma 1 and Lemma 2, setting d = T
3
2 /2,
we conclude the theorem.
The Threshold Bandits Model
In the preceding, we described a potential-based proof for the UCB algorithm in the classic stochastic
bandit problem. We now return to the Threshold Bandit setting, our problem of interest.
A K-armed Threshold Bandit problem is defined by random variables Xit and a sequence of threshold
values ct for 1 ? i ? K and 1 ? t ? T, where i is the index for arms. Successive pulling of arm
i generates the values Xi1 , Xi2 , . . . , XiT , which are drawn i.i.d. from an unknown distribution. The
threshold values c1 , c2 , . . . , cT are drawn from M = {1, 2, . . . , m} (according to rules specified later).
The threshold value ct is observed at the beginning of round t, and the learner follows a policy P to
choose the arm to play based on its past selections and previously observed feedbacks. Suppose the
arm pulled at round t is I t , the observed reward is then RtI t = I[XIt t ct ]; that is, we receive a unit
reward when the sample XIt t exceeds the threshold value ct , and otherwise we receive no reward. We
distinguish two different types of feedback.
4
1. Uncensored Feedback: After playing arm I t , the learner observes the sample XIt t regardless
of the threshold value ct .
?
0/
if XIt t ct ,
2. Censored Feedback: After playing I t , the learner observes2
.
t
XI t otherwise
In this case, we refer to the threshold value as a censor value.
Let Fi (x) denote the survival function of the distribution on arm i. That is, Fi (x) = Pr(Xit x). We
measure regret against the optimal policy with full knowledge of F1 , . . . , Fn i.e.,
" ?
" ?
?#
?#
T
T
t
t
t
t
t
t
RegretT (P) = E ? max Ri RI t
= E ? max I Xi c
I XI t c
.
t=1
i2[n]
t=1
i2[n]
Notice that for a fixed value of ct , each arm i has expected payoff E[Rti ] = Fi (ct ), the regret can also
be written as
? T
?
RegretT (P) = E ?t=1
maxi2[n] Fi (ct ) FI t (ct ) .
Our goal is to design a policy that minimizes the regret.
4
DKWUCB: Dvoretzky-Kiefer-Wolfowitz Inequality based Upper
Confidence Bound algorithm
In this section, we study the uncensored feedback setting in which the value XIt t is always observed
regardless of ct . We assume that the largest Fi ( j) is unique for all j 2 M, and define i? ( j) =
arg maxi Fi ( j), Di ( j) = Fi? ( j) ( j) Fi ( j) for all i = 1, 2, . . . , K and j 2 M.
Under this setting, the algorithm will use the empirical distribution as an estimate for the true
distribution. Specifically, we want to estimate the true survival function Fi via:
?t 1 I[X tt j, I t = i]
F?it ( j) = t=1 I t
8j 2 M
Ni
(5)
The key tool in our analysis is a deviation bound on the empirical CDF of a distribution, and we
note that this bound holds uniformly over the support of the distribution. The Dvoretzky-KieferWolfowitz (DKW) inequality (Dvoretzky et al., 1956) allows us to bound the error on F?it ( j) :
Lemma 3. At a time t, let F?it be the empirical distribution function of Fi as given in equation 5. The
probability that the maximum of the difference between F?it and Fi over all j 2 M is at least e is less
than 2 exp 2e2 Nit , i.e.,
Pr sup j2M |F?it ( j) Fi ( j)| e Nit N ? 2 exp 2e2 N .
The proof of the lemma can be found in Dvoretzky et al. (1956). The key insight is that the estimate F?i
converges to Fi point-wise at the same rate as the Hoeffding-Asumza inequality. That is, one does not
pay an additional M factor from applying a union bound. The fact that we have uniform convergence
of the CDF with the same rate as the Hoeffding-Azuma inequality allows us to immediately apply
the potential function argument from Section 2. In particular, we define f (N, e) = 2 exp 2e2 N , as
well as the pair of functions ](e, d) and V(N, d) exactly the same as the previous section, i.e.,
?
?
2 log 2/d
](e, d) :=
,
e2
( q
log(2/d)
if N > 0;
2N
V(N, d) :=
1 otherwise.
We are now ready to define our DKWUCB algorithm for a fixed choice of parameter d > 0 to solve
the problem.
DKWUCB Algorithm:
on round t play I t
arg max F?it (ct ) + V(Nit , d) .
(6)
i
2 Existing
literature often refers to this as right-censoring. With right-censored feedback, samples from
playing arms at high threshold values can inform decisions at low threshold values but not vice versa.
5
To analyze DKWUCB, we use a slight variant of the potential function defined in Section 2. Let
i? ( j) = arg maxi Fi ( j) denote the optimal arm for threshold value j, and N? it denote the number of
rounds arm i is pulled when it is not optimal, N? it = ?tt=11 I[I t = i, I t 6= i? (ct )]. Notice that N? it ? Nit .
Define the potential function as:
t
K N? i 1
F(N? 1t , . . . , N? Kt ) := 2 ?
?
V(N, d)
(7)
i=1 N=0
Theorem 2. Setting d = T
2 /2,
the expected regret of DKWUCB is bounded as
K
log T
+ O(1),
min
j2M Di ( j)
i=1
E[RegretT (DKWUCB)] ? 8 ?
We defer the proof of this theorem to the appendix.
We pause now to comment on some of the strengths of this type of analysis. At a high-level, the
typical analysis to the UCB algorithm for the standard multi-armed bandit problem (Auer
? et al.,
?
)
2002) is the following: (1) at some finite time T , the number of pulls of a bad arm i is O log(T
D2
i
with high probability, and (2) the
? regret
? suffered by any such pull is O(Di ). The contribution of arm
log(T )
i to total regret is therefore O
. In contrast, we analyzed the UCB algorithm in Section 2
Di
by observing that the expected regret suffered on round t is bounded by the difference between the
empirical mean estimator and the true mean for the payoff of arm I t . Of course by design this quantity
is almost certainly (w.p. at least 1 d) less than V(NItt ). The potential function F(?, . . . , ?) tracks the
accumulation of these values V(Nit ) for each arm i, and the final regret bound is a consequence of the
summation properties of V, ] for the particular estimator being used.
While these two approaches lead to the same bound in the standard multi-armed bandit problem,
the potential function approach bears fruit in the Threshold Bandit setting. Because the uniform
convergence rate promised by the DKW inequality matches that of the Hoeffding-Azume inequality,
Theorem 2 should not be surprising; the ith arm?s contribution to DKWUCB?s regret should be
idenitical to UCB, but with the suboptimality gap now equal to min j Di ( j).
However, following the program
analysis of UCB, one would naively argue that
? for the standard
?
arm i is incorrectly pulled O (min log(TD)( j))2 times. These pulls might come in the face of any
j2M
i
t
number
of threshold
?
? values c , suffering as much as max j2M Di ( j) regret,? yielding?a bound of
max j2M Di ( j) log(T )
max D ( j)
O (min
on the ith arm?s regret contribution, which is a factor O min jj Dii( j) worse than
2
j2M Di ( j))
the derived result. By tracking the convergence of the underlying estimator, we circumvent this
problem entirely.
5
KMUCB: Kaplan-Meier based Upper Confidence Bound Algorithm
We now turn to the censored feedback setting, in which the feedback of pulling arm I t is observed
only when XIt t is less than ct . For ease of presentation, we assume that the largest Fi ( j) is unique for
all j 2 M, and define i? ( j) = arg maxi Fi ( j), Di ( j) = Fi? ( j) ( j) Fi ( j) for all i = 1, 2, . . . , K and j 2 M.
One prevalent non-parametric estimator for censored data is the Kaplan-Meier maximum likelihood
estimator Kaplan and Meier (1958); Peterson (1983). Most of existing works have studied the uniform
error bound of Kaplan-Meier estimator in the case that the threshold values are drawn i.i.d. from a
known distribution Foldes and Rejto (1981) or asymptotic error bound for the non-i.i.d. case Huh
et al. (2009). The only known uniform error bound of Kaplan-Meier estimator is proposed in Ganchev
et al. (2010).
Noting that for a given threshold value, all the feedbacks from larger threshold values are useful, we
propose a new estimator with tighter uniform error bound based on the Kaplan-Meier estimator as
following:
Dt ( j)
F?it = ti
(8)
Ni ( j)
6
where Dti ( j) and Nit ( j) is defined as follows
At := min{XIt t , ct },
Dti ( j) :=
t 1
? I[At
j, I t = i],
t=1
Nit ( j) :=
t 1
? I[ct
t=1
j, I t = i].
We first present an error bound for the modified Kaplan-Meier estimate of Fi ( j) :
Lemma 4. At time t, let F?it be the modified Kaplan-Meier estimate of Fi as given in equation 8.
For any j 2 M, the probability that the difference between F?it ( j) and Fi ( j) is at least e is less than
? 2 t ?
e Ni ( j)
2 exp
, i.e.,
2
? 2 t ?
e Ni ( j)
t
?
Pr |Fi ( j) Fi ( j)| e ? 2 exp
.
2
We defer the proof of this lemma to the appendix.
Different to the stochastic uncensored MAB setting, we show that the cost of learning with censored
feedback depends significantly on the order of the threshold values. To illustrate this point, we first
show a comparison between the regret of adversarial setting and optimistic setting. In the adversarial
setting, the threshold values are chosen to arrive in a non-decreasing order 1, 1, . . . , 1, 2, . . . , 2, 3, . . . , m,
the problem becomes playing m independent copies of bandits, and the regret scales with m;
while in the optimistic setting, the threshold values are chosen to arrive in a non-increasing order m, m, . . . , m, m 1, . . . , m 1, . . . , 1, . . . , 1, which means the learner can make full use of the
samples, and can thus perform significantly better. Afterwards, we show that if the order of the
threshold values is close to uniformly random, the regret only scales with log m.
5.1
Adversarial vs. Optimistic Setting
For the simplicity of presentation, we assume that in both settings, the time horizon could be divided
in to m stages, each with length bT /mc.. In the adversarial setting, threshold value j comes during
stage j; while in the optimistic setting threshold value m j + 1 comes during stage j.
For the adversarial setting, due to the censored feedback structure, only the samples observed within
the same stage can help to inform decision making. From the perspective of the learner, this is
equivalent to facing m independent copies of stochastic MAB problems, and thus, the regret scales
with m. Making use of the lower bound of stochastic MAB problems Lai and Robbins (1985), we can
conclude the following theorem.
Theorem 3. If the threshold values arrive according to the adversarial order specified above, no
/m)
learning algorithm can achive a regret bound better than ?mj=1 ?Ki=1 KL(B(Filog(T
( j)||B(F ? ( j))) , where
i ( j)
KL(?||?) is the Kullback-Leibler divergence Lai and Robbins (1985) and B(?) is the probability
distribution function of Bernoulli distribution.
For the optimistic setting, although the feedbacks are right censored, we note that every sample
observed in the previous rounds are useful in later rounds. This is because the threshold values arrive
in non-increasing order. Therefore, we can reduce the optimistic setting to the Threshold Bandit
problem with uncensored feedback, and use the DKWUCB proposed in Section 4 to solve it. More
specifically, we can set
f (N, e) := 2 exp( e2 N/2),
?
?
8 log 2/d
](e, d) :=
,
e2
( q
2 log(2/d)
N
if N 1; ,
1 otherwise.
and on every round, the learner plays the same strategy as DKWUCB. We call this strategy OPTIM.
Following the same procedure in Section 4, we can provide a regret for OPTIM.
Theorem 4. Let d = T 2 /2 and assume T mK. The regret of the optimistic setting satisfies
V(N, d) :=
K
log T
+ O(1).
i=1 min j2M Di ( j)
E[RegretT (OPTIM )] ? 32 ?
7
5.2
Cyclic Permutation Setting
In this subsection, we show that if the order of threshold values is close to uniformly random, we can
perform significantly better than the adversarial setting. To be precise, we assume that the threshold
values are a cyclic permutation order of 1, 2, . . . , m. We define the set M = {ckm , ckm+1 , . . . , ck(m+1) 1 }
for any non-negative integer k ? T /m.
We are now ready to present KMUCB, which is a modified Kaplan-Meier-based UCB algorithm.
KMUCB divides the time horizon into epochs of length Km and, for each epoch, pulls each arm once
for each threshold value. KMUCB then performs an ?arm elimination? process, and once all but one
arm has been eliminated, it proceeds to pull the single remaining arm for the given threshold value.
KMUCB?s estimation procedure leverages information across threshold values, where observations
from higher thresholds are utilized to estimate mean payoffs for lower thresholds; information does
not flow in the other direction, however, as a result of the censoring assumption. Specifically, for a
given threshold index j, KMUCB tracks the arm elimination process as follows: for any threshold
values below j, KMUCB believes that we have determined the best arm, and plays that arm constantly.
For threshold values greater than or equal to j, KMUCB explores all arms uniformly. Note that by
uniform exploration over all arms for?threshold value j, all?sub-optimal arms can be detected with
T
probability at least 1 O T1 after O (m j+1)log
epochs. KMUCB then removes all the
min
D2 ( j)
i2[K] i
sub-optimal arms for threshold value j, and increments j by 1. Denoting the last time unit of epoch k
as tk = kKm, the detailed description of KMUCB is shown in Algorithm 1.
Algorithm 1 KMUCB
1: Input: A set of arms 1, 2, . . . , K.
2: Initialization: L j
[K] 8 j 2 M, k 1, j 1
3: for epoch k = 1, 2, . . . , T /Km do
4:
count[ j0 ] 0 8 j0 2 M
5:
for t from (tk 1 + 1) to tk do
6:
Observe ct = j0 and set count[ j0 ] count[ j0 ] + 1
7:
if j0 < j then
8:
I t index of the single arm remaining in L j0
9:
else
10:
I t count[ j0 ].
11:
end if
12:
end for
q
t
t
16 log(T k)
? tk
13:
if j ? m and maxi0 2[K] F?i0k ( j) F?i k ( j)
(m j+1)k 8i 2 L j \ {arg maxi0 2[K] Fi0 ( j)} then
14:
Lj
(
)
t
arg max F?i0k ( j)
0
i 2[K]
,
j
j+1
15:
end if
16: end for
Theorem 5. The expected regret of KMUCB is bounded as
K
? log m
i=1
128 max j2M Di ( j) log T
+ O(1).
mini2[K], j2M D2i ( j)
We defer the proof of this theorem to the appendix.
We note two directions of future research. First, we believe the above bound can likely be made
stronger by either improving upon the minimization in the denominator or the maximization in the
numerator. Second, we believe the ?cyclic permutation? assumption can be weakened to ?uniformly
randomly sequence of thresholds,? but we were unable to make progress in this direction. We
welcome further investigation along these lines.
8
References
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4164?4175.
Jacob D Abernethy, Chansoo Lee, and Ambuj Tewari. 2015. Fighting Bandits with a New Kind of
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Multiarmed Bandit Problem. SIAM Journal of Computuataion 32, 1 (2003), 48?77.
Peter Auer and Ronald Ortner. 2010. UCB revisited: Improved regret bounds for the stochastic
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Nicol? Cesa-Bianchi and G?bor Lugosi. 2006. Prediction, Learning, and Games. Cambridge
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A. Dvoretzky, J. Kiefer, and J. Wolfowitz. 1956. Asymptotic Minimax Character of the Sample
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John Wiley & Sons.
W. T. Huh, R. Levi, P. Rusmevichientong, and J. Orlin. 2009. Adaptive data-driven inventory control
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E. L. Kaplan and P. Meier. 1958. Nonparametric Estimation from Incomplete Observations. In JASA.
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Gergely Neu and G?bor Bart?k. 2013. An efficient algorithm for learning with semi-bandit feedback.
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58, 5 (1952), 527?535.
9
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5,691 | 615 | Neural Network On-Line Learning Control
of Spacecraft Smart Structures
Dr. Christopher Bowman
Ball Aerospace Systems Group
P.O. Box 1062
Boulder. CO 80306
Abstract
The overall goal is to reduce spacecraft weight. volume, and cost by online adaptive non-linear control of flexible structural components. The
objective of this effort is to develop an adaptive Neural Network (NN)
controller for the Ball C-Side 1m x 3m antenna with embedded actuators
and the RAMS sensor system. A traditional optimal controller for the
major modes is provided perturbations by the NN to compensate for
unknown residual modes. On-line training of recurrent and feed-forward
NN architectures have achieved adaptive vibration control with
unknown modal variations and noisy measurements. On-line training
feedback to each actuator NN output is computed via Newton's method
to reduce the difference between desired and achieved antenna positions.
1 ADAPTIVE CONTROL BACKGROUND
The two traditional approaches to adaptive control are 1) direct control (such as perfonned
in direct model reference adaptive controllers) and 2) indirect control (such as performed by
explicit self-tuning regulators). Direct control techniques (e.g. model-reference adaptive
cootrul) provide good stability however are susceptible to noise. Whereas indirect control
techn;'q~es (e.g. explicit self-tuning regulators) have low noise susceptibility and good
convergence rate. However they require more control effort and have worse stability and
are less roblistto mismodeling. NNs synergistically augment traditional adaptive control
techniques by providing improved mismodeling robustness both adaptively on-line for
time-varying dynamics as well as in a learned control mode at a slower rate.
The NN control approaches which correspond to direct and indirect adaptive control are
commonly known as inverse and forward modeling. respectively. More specifically, aNN
which maps the plant state and its desired perfonnance to the control command is called
an inverse model, a NN mapping both the current plant state and control to the next state
and its performance is called the forward model.
When given a desired performwce and the current state. the inverse model generates the
control. see Figure 1. The actual perfonnance is observed and is used to train/update the
inverse model. A significant problem occurs when the desired and achieved perfonnance
differ greatly since the model near the desired slate is not changed. This condition is
corrected by adding random noise to the control outputs so as to extend the state space
303
304
Bowman
being explored. However, this correction has the effect of slowing the learning and
reducing broadband stability.
easurements
Trainin
,..---=.,:-:-=.....n..:=;
__~-----'
,."
""
n;o:uu;u.;k
lnV_~;""1
Con
~-~
Nel!Pll'Controller
..."
1 Nonlinear r~ s....""' 1 L.!!.:.~-Structures
x
Filters
Y
ents
Previous controls and stale measurements
Figure 1: Direct Adaptive Control Using Inverse Modeling Neural Network Controller
Trainin
"
"
""
Feedback
ements
y
II~F~=fI-..1 Current and
""
-lnv-I-~-'M"'I""ode-I" Control
I':""
Net.Tl!JtControlier
I'
u
Previous controls
Provisions state
pr;.::ev.;.;l;.;;,O.::.;Us;..;c;.;o;;.;n.;;.tro~_ _ _.....;;meas=;.;;urements
1
N_
Structures
~i
x
-I ~M=eas=ur=e=m=en=ts=~
Fillers
y
and stale measurements
Figure 2: Dual (Indirect and Direct) Adaptive Control Using Forward Modeling Neural
Network State Predictor To Aid Inverse Model Convergence
For forward modeling the map from the current control and state to the resulting state and
performance is learned, see Figure 2. For cases where the performance is evaluated at a
future time (i.e. distal in time), a predictive critic [Barto and Sutton, 1989] NN model is
learned. In both cases the Jacobian of this performance can be computed to iteratively
generate the next control action. However, this differentiating of the critic NN for backpropagation training of the controller network is very slow and in some cases steers the
searching the wrong direction due to initial erroneous forward model estimates. As the NN
adapts itself the performance flattens which results in the slow halting of learning at an
Neural Network On-Line Learning Control of Spacecraft Smart Structures
unacceptable solution. Adding noise to the controller's output [Jordan and Jacobs, 1990]
breaks the redundancy but forces the critic to predict the effects of future noise. This
problem has been solved by using a separately trained intermediate plant model to predict
the next state from the prior state and control while having an independent predictor model
generate the performance evaluation from the plant model predicted state [Werbos, 1990]
and [Brody, 1991]. The result is a 50-100 fold learning speed improvement over
reinforcement training of the forward model controller NN.
However, this method still relies on a "good" forward model to incrementally train the
inverse model. These incremental changes can still lead to undesirable solutions. For
control systems which follow the stage 1,2 or 3 models given in [Narendra, 1991) the
control can be analytically computed from a forward-only model. For the most general,
non-linear (stage 4) systems, an alternative is the memory-based forward model [Moore,
1992]. Using only a forward NN model, a direct hill-climbing or Newton's method search
of candidate actions can be applied until a control decision is reached. The resulting state
and its performance are used for on-line training of the forward model. Judicial random
control actions are applied to improve behavior only where the forward model error is
predicted to be large (e.g. via cross-validation). Also using robust regression, experiences
can be deweighted according to their quality and their age. The high computational burden
of these cross-validation techniques can be reduced by parallel on-line processing
providing the "policy" parameters for fast on-line NN control.
For control problems which are distal in time and space, a hybrid of these two forwardmodeling approaches can be used. Namely, a NN plant model is added which is trained
off-line in real-time and updated as necessary at a slower rate than the on-line forward
model which predicts performance based upon the current plant model. This slower rate
trained forward-model NN supports learned control (e.g. via numerical inversion) whereas
the on-line forward model provides the faster response adaptive control. Other NN control
techniques such as using a Hopfield net to solve the optimal-control quadraticprogramming problem or the supervised training of ART II off-line with adaptive
vigilance for on-line pole placement have been proposed. However, their on-line
robustness appears limited due to their sensitivity to a priori parameter assumptions.
A forward model NN which augments a traditional controller for unmodeled modes and
unforeseen situations is presented in the following section. Performance results for both
feed-faward and current learning versions are compared in Section 3.
2
RESIDUAL FORWARD MODEL NEURAL NETWORK
(RFM-NN) CONTROLLER
A type of forward model NN which acts as a residual mode mter to support a reduced-order
model (ROM) traditional optimal state controller has been evaluated. see Figure 3. The
ROM determines the control based upon its model coordinate approximate representation
of the structure. Model coordinates are obtained by a transformation using known primary
vibration modes, [Young, 1990]. The transformation operator is a set of eigenvectors
(mode shapes) generated by finite element modeling. The ROM controller is traditionally
augmented by a residual-mode mter (RMF). Ball's RFM-NN Ball's RFM-NN replaces the
RMF in order to better capture the mismodeled. unmodeled and changing modes.
The objective of the RFM-NN is to provide ROM controller with ROM derivative state
perturbations, so that the ROM controls the structure as desired by the user. The RFMNN is trained on-line using scored supervised feedback to generate these desired ROM
state perturbations. The scored supervised training provides a score for each perturbation
output based upon the measured position of the structure. The measured deviations, Y*(t),
from the desired structure position are converted to errors in the estimated ROM state
using the ROM. transformation. Specifically, the training score, S(t), for each ROM
derivative state XN (t) is expressed in the following discrete equation:
.
S(t) = BN Y * (t) - xN(t)
where *N(t) = [AN + BNGN - KNCN]XN(t -1) + KN Y(t -1)
305
306
Bowman
"
,
. .;
;"
.}
,"
..:,::..-:.;"
.. ::
.
::: ': ..: ",
Figure 3: Residual Forward Model Neural Network Adaptive Controller Replaces
Traditional Residual Mode Filter
0*=
(1) ROM state ~ations which zero
Newton's method is then applied to find lbe
the score. First, the score is smoothed, Set) ~S(t -1) + (1- o)S(t) and the neural
network output is smoothed similarly. Second, Newton's method computes the
adjusbnents needed to zero the scores,
~(O*N(t?
=-S(t)(8i N(t) -
-1?
8iN(t
I [S(t) - Set -1)]
= -EXN(t) (if either difference = 0)
Third, the NN is trained, 8*T(t + 1) ~(8iN(t? + 8i N(t) with the appropriate
learning rate, a (e.g. approximation to inverse of largest eigenvalue of the Hessian
weight matrix).
=
3
RFM-NN ON-LINE LEARNING RESULTS
Both feed-forward and recurrent RFM-NNs have been incorporated into an interactive
simulation of Ball's Control-Structure Interaction Demonstration Experiment (C-SIDE)
see Figure 4. This 1m x 3m lightweight antenna facesheet has 8 embedded actuators plus
three auxiliary input actuators and uses 8 remote angular measurement sensors (RAMS)
plus 4 displacement and 3 velocity auxiliary sensors. In order to evaluate the on-line
performance of the RFM-NNs the ROM controller was given insufficient and partially
incorrect modes. The ROM without the RFM-NN grew unstable (i.e. greater than 10
millimeter C-SIDE displacements) in 13 seconds. The initial feed-forward RFM-NN used
8 sensor and 6 ROM state feedback estimate inputs as well as 5 hidden units and 3 ROM
velocity state perturbation outputs. This RFM-NN had random initial weights, logistic
Neural Network On-Line Learning Control of Spacecraft Smart Structures
activation functions. and back-propagation training using one sixth the learning rate for
the output layers (e.g..06 and .01). Newton RFM-NN training search used a step size of
one with smoothing factor of one tenth.
Figure 4: 1m x 3m C-SIDE Antenna Facesheet With Embedded Actuators.
This RFM-NN learned on-line to stabilized and reduce vibration to less than ?Imm
within 20 seconds, see Figure 5. A five Newton force applied a few seconds later is
compensated for within nine seconds, see Figure 6. This is accomplished with learning
off as well as when on. To test the necessity of the RFM-NN the ROM was given the
scored supervised training (Le. Newton's search estimates) directly instead of the RFMNN outputs. This caused immediate unstable behavior. To test the RFM-NN sensitivity
to measurement accuracy a unifonn error of ?5% was added. Starting from the same
random weight start the RFM-NN required 25 seconds to learn to stabilize the antenna,
see Figure 7. The best stability was achieved when the product of the Newton and BPN
steps was approximately .01. This feed-forward NN was compared to an Elman-type
recurrent NN (i.e. hidden layer feedback to itself with one-step BP training). The recurrent
RFM-NN on-line learning stability was much less sensitive to initial weights. The
recurrent RFM-NN stabilized C-SIDE with up to 10% - 20% measurement noise versus
5% - 10% limit for feed-forward RFM-NN.
4 SUMMARY AND RECOMMENDATIONS
Adaptive smart sbUctures promise to reduce spacecraft weight and dependence on extensive
ground monitoring. A recurrent forward model NN is used as a residual mode fllter to
augment a traditional reduced-order model (ROM) controller. It was more robust than the
feed-forward NN and the traditional-only controller in the presence of unmodeled modes
and noisy measurements. Further analyses and hardware implementations will be
perfonned to better quantify this robustness including the sensitivity to the ROM
controller mode fidelity, number of output modes. learning rates, measurement-to-state
errors, and time quantization effects.
To improve robustness to ROM mode changes a comparison to the dual forward/inverse
NN control approach is recommended. The forward model will adjust the search used to
train an inverse model which provides control augmentations to the ROM controller. This
will enable control searches to occur both off-line faster than real-time using the forward
model (Le. imagination) and on-line using direct search trials with varying noise levels.
The forward model will adapt using quality experiences (e.g. via cross validation) which
improves inverse models searches. The inverse model reliance on forward model will
reduce until forward model prediction errors increase. Future challenges. include solving
307
308
Bowman
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the temporal credit assignment problem, partitioning to restricted chip sizes, combining
with incomplete a priori knowledge, and balancing adaptivity of response with long-term
learning. The goal is to extend stabiJity-dominated, fixed-goal traditional control with
adaptive robotic-type neural control to enable better autonomous control where fullyjustified fixed models and complete system knowledge are not required. The resultant
robust autonomous control will capitalize on the speed of massively parallel analog
neural-like computations (e.g. with NN pulse stream chips).
Neural Network On-Line Learning Control of Spacecraft Smart Structures
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Displacement Measurements
309
310
Bowman
5 REFERENCES
Barto, A.G., Sutton, R.S .? and Watkins, CJ.C.H., Learning and Sequential Decision
Making. Univ. of Mass. at Amherst COINS Technical Report 89-95, September 1989
Bowman, C.L., Adaptive Neural Networks Applied to Signal Recognition, 3rd TriService Data fusion Symposium, May 1989
Brody, Carlos, Fast Learning With Predictive Forward Models. Neural Information
Processing Systems 4 (NIPS4), 1992
Jorden, M.I., and Jacobs, R.A., Learning to Control and Unstable System with Forward
Modeling, in D.S. Touretzky, ed., Advances in NIPS 2, Morgan Kaufmann 1990.
Moore, A.W., Fast. Robust Adaptive Control by Learning Only Forward Models.
NIPS 4, 1992
Mukhopadhyay S. and Narendra, D.S .? Disturbance Rejection in Nonlinear Systems
Using Neural Networks Yale University Report No. 9114 December 1991
Werbos, P., Architectures For Reinforcement Learning, in Miller, Sutton and Werbos.
ed., Neural Networks for Control, MIT Press 1990
Young, D.O., Distributed Finite-Element Modeling and Control Approach for Large
Flexible Structures, J. of Guidance, Control and Dynamics, Vol. 13 (4),703-713,1990
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5,692 | 6,150 | Learning from Rational Behavior:
Predicting Solutions to Unknown Linear Programs
Shahin Jabbari, Ryan Rogers, Aaron Roth, Zhiwei Steven Wu
University of Pennsylvania
{jabbari@cis, ryrogers@sas, aaroth@cis, wuzhiwei@cis}.upenn.edu
Abstract
We define and study the problem of predicting the solution to a linear program (LP)
given only partial information about its objective and constraints. This generalizes
the problem of learning to predict the purchasing behavior of a rational agent who
has an unknown objective function, that has been studied under the name ?Learning
from Revealed Preferences". We give mistake bound learning algorithms in two
settings: in the first, the objective of the LP is known to the learner but there is an
arbitrary, fixed set of constraints which are unknown. Each example is defined by
an additional known constraint and the goal of the learner is to predict the optimal
solution of the LP given the union of the known and unknown constraints. This
models the problem of predicting the behavior of a rational agent whose goals
are known, but whose resources are unknown. In the second setting, the objective
of the LP is unknown, and changing in a controlled way. The constraints of the
LP may also change every day, but are known. An example is given by a set of
constraints and partial information about the objective, and the task of the learner
is again to predict the optimal solution of the partially known LP.
1
Introduction
We initiate the systematic study of a general class of multi-dimensional prediction problems, where
the learner wishes to predict the solution to an unknown linear program (LP), given some partial
information about either the set of constraints or the objective. In the special case in which there is a
single known constraint that is changing and the objective that is unknown and fixed, this problem
has been studied under the name learning from revealed preferences [1, 2, 3, 16] and captures the
following scenario: a buyer, with an unknown linear utility function over d goods u : Rd ! R
defined as u(x) = c ? x faces a purchasing decision every day. On day t, she observes a set of prices
pt 2 Rd 0 and buys the bundle of goods that maximizes her unknown utility, subject to a budget b:
x(t) = argmax c ? x
x
such that pt ? x ? b
In this problem, the goal of the learner is to predict the bundle that the buyer will buy, given the
prices that she faces. Each example at day t is specified by the vector pt 2 Rd 0 (which fixes the
constraint), and the goal is to accurately predict the purchased bundle x(t) 2 [0, 1]d that is the result
of optimizing the unknown linear objective.
It is also natural to consider the class of problems in which the goal is to predict the outcome to a LP
broadly e.g. suppose the objective c ? x is known but there is an unknown set of constraints Ax ? b.
An instance is again specified by a changing known constraint (pt , bt ) and the goal is to predict:
x(t) = argmax c ? x
x
such that Ax ? b and pt ? x ? bt .
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
(1)
This models the problem of predicting the behavior of an agent whose goals are known, but whose
resource constraints are unknown.
Another natural generalization is the problem in which the objective is unknown, and may vary in a
specified way across examples, and in which there may also be multiple arbitrary known constraints
which vary across examples. Specifically, suppose that there are n distinct, unknown linear objective
functions v1 , . . . , vn . An instance on day t is specified by a subset of the unknown objective
functions, S t ? [n] := {1, . . . , n} and a convex feasible region P t , and the goal is to predict:
X
x(t) = argmax
vi ? x
such that x 2 P t .
(2)
x
i2S t
When the changing feasible regions P t correspond simply to varying prices as in the revealed
preferences problem, this models a setting in which at different times, purchasing decisions are made
by different members of an organization, with heterogeneous preferences ? but are still bound by
an organization-wide budget. The learner?s problem is, given the subset of decision makers and the
prices at day t, to predict which bundle they will purchase. This generalizes some of the preference
learning problems recently studied by Blum et al [6]. Of course, in this generality, we may also
consider a richer set of changing constraints which represent things beyond prices and budgets.
In all of the settings we study, the problem can be viewed as the task of predicting the behavior of a
rational decision maker, who always chooses the action that maximizes her objective function subject
to a set of constraints. Some part of her optimization problem is unknown, and the goal is to learn,
through observing her behavior, that unknown part of her optimization problem sufficiently so that
we may reliably predict her future actions.
1.1
Our Results
We study both variants of the problem (see below) in the strong mistake bound model of learning
[13]. In this model, the learner encounters an arbitrary adversarially chosen sequence of examples
online and must make a prediction for the optimal solution in each example before seeing future
examples. Whenever the learner?s prediction is incorrect, the learner encounters a mistake, and
the goal is to prove an upper bound on the number of mistakes the learner can make, in the worst
case over the sequence of examples. Mistake bound learnability is stronger than (and implies) PAC
learnability [15].
Known Objective and Unknown Constraints We first study this problem under the assumption
that there is a uniform upper bound on the number of bits of precision used to specify the constraint
defining each example. In this case, we show that there is a learning algorithm with both running time
and mistake bound linear in the number of edges of the polytope formed by the unknown constraint
matrix Ax ? b. We note that this is always polynomial in the dimension d when the number of
unknown constraints is at most d + O(1). (In the supplementary material, we show that by allowing
the learner to run in time exponential in d, we can give a mistake bound that is always linear in
the dimension and the number of rows of A, but we leave as an open question whether or not this
mistake bound can be achieved by an efficient algorithm.) We then show that our bounded precision
assumption is necessary ? i.e. we show that when the precision to which constraints are specified
need not be uniformly upper bounded, then no algorithm for this problem in dimension d 3 can
have a finite mistake bound.
This lower bound motivates us to study a PAC style variant of the problem, where the examples are
not chosen in an adversarial manner, but instead are drawn independently at random from an arbitrary
unknown distribution. In this setting, we show that even if the constraints can be specified to arbitrary
(even infinite) precision, there is a learner that requires sample complexity only linear in the number
of edges of the unknown constraint polytope. This learner can be implemented efficiently when the
constraints are specified with finite precision.
Known Constraints and Unknown Objective For the variant of the problem in which the objective is unknown and changing and the constraints are known but changing, we give an algorithm
that has a mistake bound and running time polynomial in the dimension d. Our algorithm uses the
Ellipsoid algorithm to learn the coefficients of the unknown objective by implementing a separation
oracle that generates separating hyperplanes given examples on which our algorithm made a mistake.
2
We leave the study of either of our problems under natural relaxations (e.g. under a less demanding
loss function) and whether it is possible to substantially improve our results in these relaxations as an
interesting open problem.
1.2
Related Work
Beigman and Vohra [3] were the first to study revealed preference problems (RPP) as a learning
problems and to relate them to multi-dimensional classification. They derived sample complexity
bounds for such problems by computing the fat shattering dimension of the class of target utility
functions, and showed that the set of Lipschitz-continuous valuation functions had finite fat-shattering
dimension. Zadimoghaddam and Roth [16] gave efficient algorithms with polynomial sample
complexity for PAC learning of the RPP over the class of linear (and piecewise linear) utility
functions. Balcan et al. [2] showed a connection between RPP and the structured prediction problem
of learning d-dimensional linear classes [7, 8, 12], and use an efficient variant of the compression
techniques given by Daniely and Shalev-Shwartz [9] to give efficient PAC algorithms with optimal
sample complexity for various classes of economically meaningful utility functions. Amin et al. [1]
study the RPP for linear valuation functions in the mistake bound model, and in the query model
in which the learner gets to set prices and wishes to maximize profit. Roth et al. [14] also study
the query model of learning and give results for strongly concave objective functions, leveraging an
algorithm of Belloni et al. [4] for bandit convex optimization with adversarial noise.
All of the works above focus on the setting of predicting the optimizer of a fixed unknown objective
function, together with a single known, changing constraint representing prices. This is the primary
point of departure for our work ? we give algorithms for the more general settings of predicting the
optimizer of a LP when there may be many unknown constraints, or when the unknown objective
function is changing. Finally, the literature on preference learning (see e.g. [10]) has similar goals,
but is technically quite distinct: the canonical problem in preference learning is to learn a ranking on
distinct elements. In contrast, the problem we consider here is to predict the outcome of a continuous
optimization problem as a function of varying constraints.
2
Model and Preliminaries
We first formally define the geometric notions used throughout this paper. A hyperplane and a
halfspace in Rd are the set of points satisfying the linear equation a1 x1 + . . . ad xd = b and the
linear inequality a1 x1 + . . . + ad xd ? b for a set of ai s respectively, assuming that not all ai ?s are
simultaneously zero. A set of hyperplanes are linearly independent if the normal vectors to the
hyperplanes are linearly independent. A polytope (denoted by P ? Rd ) is the bounded intersection
of finitely many halfspaces, written as P = {x | Ax ? b}. An edge-space e of a polytope P is a one
dimensional subspace that is the intersection of d 1 linearly independent hyperplanes of P, and an
edge is the intersection between an edge-space e and the polytope P.We denote the set of edges of
polytope P by EP . A vertex of P is a point where d linearly independent hyperplanes of P intersect.
Equivalently, P can be written as the convex hull of its vertices V denoted by Conv(V ). Finally, we
define a set of points to be collinear if there exists a line that contains all the points in the set.
We study an online prediction problem with the goal of predicting the optimal solution of a changing
LP whose parameters are only partially known. Formally, in each day t = 1, 2, . . . an adversary
chooses a LP specified by a polytope P (t) (a set of linear inequalities) and coefficients c(t) 2 Rd
of the linear objective function. The learner?s goal is to predict the solution x(t) where x(t) =
? (t) , the learner observes the optimal x(t) and
argmaxx2P (t) c(t) ? x. After making the prediction x
(t)
(t)
learns whether she has made a mistake (?
x 6= x ). The mistake bound is defined as follows.
Definition 1. Given a LP with feasible polytope P and objective function c, let (t) denote the
parameters of the LP that are revealed to the learner on day t. A learning algorithm A takes as
input the sequence { (t) }t , the known parameters of an adaptively chosen sequence {(P (t) , c(t) )}t
of LPs and outputs a sequence
predictions
{?
x(t) }t . We say that A has mistake bound M if
? (t) of (t)
?
1
? 6= x
max{(P (t) ,c(t) )}t ?t=1 1 x
? M, where x(t) = argmaxx2P (t) c(t) ? x on day t.
We consider two different instances of the problem described above. First, in Section 3, we study
the problem given in (1) in which c(t) = c is fixed and known to the learner but the polytope P (t) =
3
P \ N (t) consists of an unknown fixed polytope P and a new constraint N (t) = {x | p(t) ? x ? b(t) }
which is revealed to the learner on day t i.e. (t) = (N (t) , c). We refer to this as the Known Objective
problem. Then, in Section 4, we study
in which the polytope P (t) is changing and known
P the problem
(t)
i
but the objective function c = i2S (t) v is unknown and changing as in (2) where the set S (t) is
known i.e. (t) = (P (t) , S (t) ). We refer to this as the Known Constraints problem.
In order for our prediction problem to be well defined, we make Assumption 1 about the observed
solution x(t) in each day. Assumption 1 guarantees that each solution is on a vertex of P (t) .
Assumption 1. The optimal solution to the LP: maxx2P (t) c(t) ? x is unique for all t.
3
The Known Objective Problem
In this section, we focus on the Known Objective Problem where the coefficients of the objective
function c are fixed and known to the learner but the feasible region P (t) on day t is unknown and
changing. In particular, P (t) is the intersection of a fixed and unknown polytope P = {x | Ax ?
b, A ? Rm?d } and a known halfspace N (t) = {x | p(t) ? x ? b(t) } i.e. P (t) = P \ N (t) .
Throughout this section we make the following assumptions. First, we assume w.l.o.g. (up to scaling)
that the points in P have `1 -norm bounded by 1.
Assumption 2. The unknown polytope P lies inside the unit `1 -ball i.e. P ? {x | ||x||1 ? 1}.
We also assume that the coordinates of the vertices in P can be written with finite precision (this is
implied if the halfspaces defining P can be described with finite precision). 1
Assumption 3. The coordinates of each vertex of P can be written with N bits of precision.
We show in Section 3.3 that Assumption 3 is necessary ? without any upper bound on precision,
there is no algorithm with a finite mistake bound. Next, we make some non-degeneracy assumptions
on polytopes P and P (t) , respectively. We require these assumptions to hold on each day.
Assumption 4. Any subset of d 1 rows of A have rank d 1 where A is the constraint matrix in
P = {x | Ax ? b}.
Assumption 5. Each vertex of P (t) is the intersection of exactly d-hyperplanes of P (t) .
The rest of this section is organized as follows. We present LearnEdge for the Known Objective
Problem and analyze its mistake bound in Sections 3.1 and 3.2, respectively. Then in Section 3.3,
we prove the necessity of Assumption 3 to get a finite mistake bound. Finally in Section 3.4, we
present the LearnHull in a PAC style setting where the new constraint each day is drawn i.i.d. from
an unknown distribution, rather than selected adversarially.
3.1
LearnEdge Algorithm
In this section we introduce LearnEdge and show in Theorem 1 that the number of mistakes of
LearnEdge depends linearly on the number of edges EP and the precision parameter N and only
logarithmically on the dimension d. We defer all the missing proofs to the supplementary material.
Theorem 1. The number of mistakes and per day running time of LearnEdge in the Known Objective
Problem are O(|EP |N log(d)) and poly(m, d, |EP |) respectively when A ? Rm?d .
At a high level, LearnEdge maintains a set of prediction information I (t) about the prediction history
up to day t, and makes prediction in each day based on I (t) and a set of prediction rules (P.1 P.4).
After making a mistake, LearnEdge updates the information with a set of update rules (U.1 U.4).
Prediction Information It is natural to ask ?What information is useful for prediction?" Lemma 2
establishes the importance of the set of edges EP by showing that all the observed solutions will be
on an element of EP .
1
Lemma 6.2.4 from Grotschel et al. [11] states that if each constraint in P ? Rd has encoding length at most
N then each vertex of P has encoding length at most 4d2 N . Typically the finite precision assumption is made
on the constraints of the LP. However, since this assumption implies that the vertices can be described with finite
precision, for simplicity, we make our assumption directly on the vertices.
4
Lemma 2. On any day t, the observed solution x(t) lies on an edge in EP .
In the proof of Lemma 2 we also show that when x(t) does not bind the new constraint N (t) , then
x(t) is the solution for the underlying LP: argmaxx2P c ? x.
Corollary 1. If x(t) 2 {x | p(t) x < b(t) } then x(t) = x? ? argmaxx2P c ? x.
We then show how an edge-space e of P can be recovered after seeing 3 collinear observed solutions.
Lemma 3. Let x, y, z be 3 distinct collinear points on edges of P. Then they are all on the same
edge of P and the 1-dimensional subspace containing them is an edge-space of P.
Given the relation between observed solutions and edges, the information I (t) is stored as follows:
Me0
Me1
}
}
}
}
}
Ye0
Q e0
Fe
Q e1
Ye1
Figure 1: Regions on an edge-space e: feasible
region Fe (blue), questionable intervals Q0e and
Q1e (green) with their mid-points Me0 and Me1 and
infeasible regions Ye0 and Ye1 (dashed).
I.1 (Observed Solutions) LearnEdge keeps track of the set of observed solutions that were
? (? ) 6= x(? ) } and also the solution for
predicted incorrectly so far X (t) = {x(? ) : ? ? t x
the underlying unknown polytope x? ? argmaxx2P c ? x if it is observed.
I.2 (Edges) LearnEdge keeps track of the set of edge-spaces E (t) given by any 3 collinear
points in X (t) . For each e 2 E (t) , it also maintains the regions on e that are certainly
feasible or infeasible. The remaining parts of e called the questionable region is where
LearnEdge cannot classify as infeasible or feasible with certainty (see Figure 1). Formally,
1. (Feasible Interval) The feasible interval Fe is an interval along e that is identified to be on
the boundary of P. More formally, Fe = Conv(X (t) \ e).
2. (Infeasible Region) The infeasible region Ye = Ye0 [ Ye1 is the union of two disjoint
intervals Ye0 and Ye1 that are identified to be outside of P. By Assumption 2, we initialize
the infeasible region Ye to {x 2 e | kxk1 > 1} for all e.
3. (Questionable Region) The questionable region Qe = Q0e [ Q1e on e is the union of two
disjoint questionable intervals along e. Formally, Qe = e \ (Fe [ Ye ). The points in Qe
cannot be certified to be either inside or outside of P by LearnEdge.
4. (Midpoints in Qe ) For each questionable interval Qie , let Mei denote the midpoint of Qie .
We add the superscript (t) to show the dependence of these quantities on days. Furthermore, we
S
(t)
eliminate the subscript e when taking the union over all elements in E (t) , e.g. F (t) = e2E (t) Fe .
(t)
(t)
(t)
(t)
(t)
(t)
(t)
(t)
So the information I can be written as follows: I = X , E , F , Y , Q , M
.
e (t) = {x |
Prediction Rules We now focus on the prediction rules of LearnEdge. On day t, let N
(t)
(t)
(t)
(t)
e (t) , then
p ? x = b } be the hyperplane specified by the additional constraint N . If x 2
/N
(t)
?
?
?
x = x by Corollary 1. So whenever the algorithm observes x , it will store x and predict it in
the future days when x? 2 N (t) . This is case P.1. So in the remaining cases we know x? 2
/ N (t) .
e (t) and the edges EP ,
The analysis of Lemma 2 shows that x(t) must be in the intersection between N
(t)
so x = argmaxx2Ne (t) \EP c ? x. Hence, LearnEdge can restrict its prediction to the following
? (t) } \ N
e (t) where E
? (t) = {e 2 E (t) | e ? N
e (t) }. As
candidate set: Cand(t) = {(E (t) [ X (t) ) \ E
(t)
(t)
(t)
(t)
?
?
we show in Lemma 4, x will not be in E , so it is safe to remove E from Cand .
e (t) , then x(t) 62 e.
Lemma 4. Let e be an edge-space of P such that e ? N
However, Cand(t) can be empty or only contain points in the infeasible regions of the edge-spaces. If
so, then there is simply not enough information to predict a feasible point in P. Hence, LearnEdge
predicts an arbitrary point outside of Cand(t) . This is case P.2.
5
Otherwise Cand(t) contains points from the feasible and questionable regions of the edge-spaces.
LearnEdge predicts from a subset of Cand(t) called the extended feasible region Ext(t) instead of
directly predicting from Cand(t) . Ext(t) contains the whole feasible region and only parts of the
? (t) . We will show later that this guarantees
questionable region on all the edge-spaces in E (t) \ E
LearnEdge makes progress in learning the true feasible region on some edge-space upon making a
e (t) with the union of intervals between the
mistake. More formally, Ext(t) is the intersection of N
0 (t)
1 (t)
? (t) and all points in X (t) :
two mid-points (Me ) and (Me ) on every edge-space e 2 E (t) \ E
e (t) .
Ext(t) = X (t) [ [e2E (t) \E? (t) Conv (Me0 )(t) , (Me1 )(t)
\N
In P.3, if Ext(t) 6= ; then LearnEdge predicts the point with the highest objective value in Ext(t) .
e (t) only intersects within the questionable regions of the
Finally, if Ext(t) = ;, then we know N
learned edge-spaces. In this case, LearnEdge predicts the intersection point with the lowest objective
value, which corresponds to P.4. Although it might seem counter-intuitive to predict the point with the
lowest objective value, this guarantees that LearnEdge makes progress in learning the true feasible
region on some edge-space upon making a mistake. The prediction rules are summarized as follows:
? (t)
P.1 First, if x? is observed and x? 2 N (t) , then predict x
x? ;
S
(t)
P.2 Else if Cand = ; or Cand(t) ? e2E (t) Ye , then predict any point outside Cand(t) ;
? (t) = argmaxx2Ext(t) c ? x;
P.3 Else if Ext(t) 6= ;, then predict x
? (t) = argminx2Cand(t) c ? x.
P.4 Else, predict x
Update Rules Next we describe how LearnEdge updates its information. Upon making a mistake,
LearnEdge adds x(t) to the set of previously observed solutions X (t) i.e. X (t+1)
X (t) [ {x(t) }.
Then it performs one of the following four mutually exclusive update rules (U.1-U.4) in order.
e (t) , then LearnEdge records x(t) as the unconstrained optimal solution x? .
U.1 If x(t) 2
/N
U.2 Then if x(t) is not on any learned edge-space in E (t) , LearnEdge will try to learn a new
edge-space by checking the collinearity of x(t) and any couple of points in X (t) . So after
this update LearnEdge might recover a new edge-space of the polytope.
If the previous updates were not invoked, then x(t) was on some learned edge-space e. LearnEdge
? (t) and x(t) (we know c ? x
? (t) 6= c ? x(t) by Assumption 1):
then compares the objective values of x
? (t) > c ? x(t) , then x
? (t) must be infeasible and LearnEdge then updates the questionU.3 If c ? x
able and infeasible regions for e.
? (t) < c ? x(t) then x(t) was outside of the extended feasible region of e. LearnEdge
U.4 If c ? x
then updates the questionable region and feasible interval on e.
In both of U.3 and U.4, LearnEdge will shrink some questionable interval substantially till the
interval has length less than 2 N in which case Assumption 3 implies that the interval contains no
points. So LearnEdge can update the adjacent feasible region and infeasible interval accordingly.
3.2
Analysis of LearnEdge
Whenever LearnEdge makes a mistake, one of the update rules U.1 - U.4 is invoked. So the number
of mistakes of LearnEdge is bounded by the number of times each update rule is invoked. The
mistake bound of LearnEdge in Theorem 1 is hence the sum of mistakes bounds in Lemmas 5-7.
Lemma 5. Update U.1 is invoked at most 1 time.
Lemma 6. Update U.2 is invoked at most 3|EP | times.
2
Lemma 7. Updates U.3 and U.4 are invoked at most O(|EP |N log(d)) times.
2
The dependency on |EP | can be improved by replacing it with the set of edges of P on which an optimal
solution is observed. This applies to all the dependencies on |EP | in our bounds.
6
3.3
Necessity of the Precision Bound
We show the necessity of Assumption 3 by showing that the dependence on the precision parameter
N in our mistake bound is tight. We show that subject to Assumption 3, there exist a polytope and a
sequence of additional constraints such that any learning algorithm will make ?(N ) mistakes. This
implies that without any upper bound on precision, it is impossible to learn with finite mistakes.
Theorem 8. For any learning algorithm A in the Known Objective Problem and any d 3, there
exists a polytope P and a sequence of additional constraints {N (t) }t such that the number of mistakes
made by A is at least ?(N ). 3
3.4
Stochastic Setting
Given the lower bound in Theorem 8, we ask ?In what settings we can still learn without an upper
bound on the precision to which constraints are specified?? The lower bound implies we must
abandon the adversarial setting so we consider a PAC style variant. In this variant, the additional
constraint at each day t is drawn i.i.d. from some fixed but unknown distribution D over Rd ? R such
that each point (p, b) drawn from D corresponds to the halfspace N = {x | p ? x ? b}. We make no
assumption on the form of D and require our bounds to hold in the worst case over all choices of D.
We describe LearnHull an algorithm based on the following high level idea: LearnHull keeps track
of the convex hull C (t 1) of all the solutions observed up to day t. LearnHull then behaves as if this
convex hull is the entire feasible region. So at day t, given the constraint N (t) = {x | p(t) ? x ? b(t) },
? (t) where x
? (t) = argmaxx2C (t 1) \N (t) c ? x.
LearnHull predicts x
LearnHull?s hypothetical feasible region is therefore always a subset of the true feasible region ?
i.e. it can never make a mistake because its prediction was infeasible, but only because its prediction
was sub-optimal. Hence, whenever LearnHull makes a mistake, it must have observed a point that
expands the convex hull. Hence, whenever it fails to predict x(t) , LearnHull will enlarge its feasible
region by adding the point x(t) to the convex hull: C (t)
Conv(C (t 1) [ {x(t) }), otherwise it
(t)
(t 1)
will simply set C
C
for the next day. We show that the expected number of mistakes of
LearnHull over T days is linear in the number of edges of P and only logarithmic in T . 4
Theorem 9. For any T > 0 and any constraint distribution D, the expected number of mistakes of
LearnHull after T days is bounded by O (|EP | log(T )).
To prove Theorem 9, first in Lemma 10 we bound the probability that the solution observed at day t
falls outside of the convex hull of the previously observed solutions. This is the only event that can
cause LearnHull to make a mistake. In Lemma 10, we abstract away the fact that the point observed
at each day is the solution to some optimization problem.
Lemma 10. Let P be a polytope and D a distribution over points on EP . Let X = {x1 , . . . , xt 1 } be
t 1 i.i.d. draws from D and xt an additional independent draw from D. Then Pr[xt 62 Conv(X)] ?
2|EP |/t where the probability is taken over the draws of points x1 , . . . , xt from D.
Finally in Theorem 11 we convert the bound on the expected number of mistakes of LearnHull in
Theorem 9 to a high probability bound. 5
Theorem 11. There exists a deterministic procedure such that after T = O (|EP | log (1/ )) days,
the probability (over the randomness of the additional constraint) that the procedure makes a mistake
on day T + 1 is at most for any 2 (0, 1/2).
4
The Known Constraints Problem
We now consider the Known Constraints Problem in which the learner observes the changing
constraint polytope P (t) at each day, but does not know the changing objective function which we
3
We point out that the condition d 3 is necessary in the statement of Theorem 8 since there exists learning
algorithms for d = 1 and d = 2 with finite mistake bounds independent of N . See the supplementary material.
4
LearnHull can be implemented efficiently in time poly(T, N, d) if all of the coefficients in the unknown
constraints in P are represented in N bits. Note that given the observed solutions so far and a new point, a
separation oracle can be implemented in time poly(T, N, d) using a LP solver.
5
LearnEdge fails to give any non-trivial mistake bound in the adversarial setting.
7
P
assume to be written as c(t) = i2S (t) vi , where {vi }i2[n] are fixed but unknown. Given P (t) and
? (t) on each day. Inspired by Bhaskar et
the subset S (t) ? [n], the learner must make a prediction x
al. [5], we use the Ellipsoid algorithm to learn the coefficients {vi }i2[n] , and show that the mistake
bound of the resulting algorithm is bounded by the (polynomial) running time of the Ellipsoid. We
use V 2 Rd?n to denote the matrix whose columns are vi and make the following assumption on V .
Assumption 6. Each entry in V can be written with N bits of precision. Also w.l.o.g. ||V ||F ? 1.
Similar to Section 3 we assume the coordinates of P (t) ?s vertices can be written with finite precision.6
Assumption 7. The coordinates of each vertex of P (t) can be written with N bits of precision.
We first observe that the coefficients of the objective function represent a point that is guaranteed to
lie in a region F (described below) which may be written as the intersection of possibly infinitely
many halfspaces. Given a subset S ? [n] and a polytope P, let xS,P denote the optimal solution to
the instance defined by S and P. Informally, the halfspaces defining F ensure that for any problem
instance defined by arbitrary choices of S and P, the objective value of the optimal solution xS,P
must be at least as high as the objective value of any feasible point in P. Since the convergence rate
of the Ellipsoid algorithm depends on the precision to which constraints are specified, we do not in
fact consider a hyperplane for every feasible solution but only for those solutions that are vertices of
the feasible polytope P. This is not a relaxation, since LPs always have vertex-optimal solutions.
We denote the set of all vertices of polytope P by vert(P), and the set of polytopes P satisfying
Assumption 7 by . We then define F as follows:
(
)
X
1
n
n?d
i
S,P
F = W = (w , . . . , w ) 2 R
| 8S ? [n], 8P 2 ,
w ? x
x
0, 8x 2 vert(P)
i2S
The idea behind our LearnEllipsoid algorithm is that we will run a copy of the Ellipsoid algorithm
with variables w 2 Rd?n , as if we were solving the feasibility LP defined by the constraints defining
F. We will always predict according to the centroid of the ellipsoid maintained by the Ellipsoid
algorithm (i.e. its candidate solution). Whenever a mistake occurs, we are able to find one of the
constraints that define F such that our prediction violates the constraint ? exactly what is needed to
take a step in solving the feasibility LP. Since we know F is non-empty (at least the true objective
function V lies within it) we know that the LP we are solving is feasible. Given the polynomial
convergence time of the Ellipsoid algorithm, this gives a polynomial mistake bound for our algorithm.
The Ellipsoid algorithm will generate a sequence of ellipsoids with decreasing volume
such that each one contains feasible region F.
Given the ellipsoid E (t) at day t,
(t)
LearnEllipsoid uses the centroid of E as its hypothesis for the objective function W (t) =
(w1 )(t) , . . . , (wn )(t) . Given the subset S (t) and polytope P (t) , LearnEllipsoid predicts
P
? (t) 2 argmaxx2P (t) { i2S (t) (wi )(t) ? x}. When a mistake occurs, LearnEllipsoid finds the
x
P
? (t) ) > 0 that separates
hyperplane H(t) = W = (w1 , . . . , wn ) 2 Rn?d : i2S (t) wi ? (x(t) x
the centroid of the current ellipsoid (the current candidate objective) from F.
After the update, we use the Ellipsoid algorithm to compute the minimum-volume ellipsoid E (t+1)
that contains H(t) \ E (t) . On day t + 1, LearnEllipsoid sets W (t+1) to be the centroid of E (t+1) .
We left the procedure used to solve the LP in the prediction rule of LearnEllipsoid unspecified. To
? (t) which is a vertex of P (t) .
simplify our analysis, we use a specific LP solver to obtain a prediction x
Theorem 12 (Theorem 6.4.12 and Remark 6.5.2 [11]). There exists a LP solver that runs in time
polynomial in the length of its input and returns an exact solution that is a vertex of P (t) .
In Theorem 13, we show that the number of mistakes made by LearnEllipsoid is at most the
number of updates that the Ellipsoid algorithm makes before it finds a point in F and the number of
updates of the Ellipsoid algorithm can be bounded by well-known results from the literature on LP.
Theorem 13. The total number of mistakes and the running time of LearnEllipsoid in the Known
Constraints Problem is at most poly(n, d, N ).
6
We again point out that this is implied if the halfspaces defining the polytope are described with finite
precision [11].
8
References
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9
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5,693 | 6,151 | A Forward Model at Purkinje Cell Synapses
Facilitates Cerebellar Anticipatory Control
Ivan Herreros-Alonso
SPECS lab
Universitat Pompeu Fabra
Barcelona, Spain
ivan.herreros@upf.edu
Xerxes D. Arsiwalla
SPECS lab
Universitat Pompeu Fabra
Barcelona, Spain
Paul F.M.J. Verschure
SPECS, UPF
Catalan Institution of Research
and Advanced Studies (ICREA)
Barcelona, Spain
Abstract
How does our motor system solve the problem of anticipatory control in spite
of a wide spectrum of response dynamics from different musculo-skeletal systems, transport delays as well as response latencies throughout the central nervous
system? To a great extent, our highly-skilled motor responses are a result of a
reactive feedback system, originating in the brain-stem and spinal cord, combined
with a feed-forward anticipatory system, that is adaptively fine-tuned by sensory
experience and originates in the cerebellum. Based on that interaction we design
the counterfactual predictive control (CFPC) architecture, an anticipatory adaptive
motor control scheme in which a feed-forward module, based on the cerebellum,
steers an error feedback controller with counterfactual error signals. Those are
signals that trigger reactions as actual errors would, but that do not code for any current or forthcoming errors. In order to determine the optimal learning strategy, we
derive a novel learning rule for the feed-forward module that involves an eligibility
trace and operates at the synaptic level. In particular, our eligibility trace provides
a mechanism beyond co-incidence detection in that it convolves a history of prior
synaptic inputs with error signals. In the context of cerebellar physiology, this
solution implies that Purkinje cell synapses should generate eligibility traces using
a forward model of the system being controlled. From an engineering perspective,
CFPC provides a general-purpose anticipatory control architecture equipped with a
learning rule that exploits the full dynamics of the closed-loop system.
1
Introduction
Learning and anticipation are central features of cerebellar computation and function (Bastian, 2006):
the cerebellum learns from experience and is able to anticipate events, thereby complementing a
reactive feedback control by an anticipatory feed-forward one (Hofstoetter et al., 2002; Herreros
and Verschure, 2013). This interpretation is based on a series of anticipatory motor behaviors that
originate in the cerebellum. For instance, anticipation is a crucial component of acquired behavior in
eye-blink conditioning (Gormezano et al., 1983), a trial by trial learning protocol where an initially
neutral stimulus such as a tone or a light (the conditioning stimulus, CS) is followed, after a fixed
delay, by a noxious one, such as an air puff to the eye (the unconditioned stimulus, US). During early
trials, a protective unconditioned response (UR), a blink, occurs reflexively in a feedback manner
following the US. After training though, a well-timed anticipatory blink (the conditioned response,
CR) precedes the US. Thus, learning results in the (partial) transference from an initial feedback
action to an anticipatory (or predictive) feed-forward one. Similar responses occur during anticipatory
postural adjustments, which are postural changes that precede voluntary motor movements, such
as raising an arm while standing (Massion, 1992). The goal of these anticipatory adjustments is to
counteract the postural and equilibrium disturbances that voluntary movements introduce. These
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
behaviors can be seen as feedback reactions to events that after learning have been transferred to
feed-forward actions anticipating the predicted events.
Anticipatory feed-forward control can yield high performance gains over feedback control whenever
the feedback loop exhibits transmission (or transport) delays (Jordan, 1996). However, even if a
plant has negligible transmission delays, it may still have sizable inertial latencies. For example,
if we apply a force to a visco-elastic plant, its peak velocity will be achieved after a certain delay;
i.e. the velocity itself will lag the force. An efficient way to counteract this lag will be to apply
forces anticipating changes in the desired velocity. That is, anticipation can be beneficial even when
one can act instantaneously on the plant. Given that, here we address two questions: what is the
optimal strategy to learn anticipatory actions in a cerebellar-based architecture? and how could it be
implemented in the cerebellum?
To answer that we design the counterfactual predictive control (CFPC) scheme, a cerebellar-based
adaptive-anticipatory control architecture that learns to anticipate performance errors from experience.
The CFPC scheme is motivated from neuro-anatomy and physiology of eye-blink conditioning.
It includes a reactive controller, which is an output-error feedback controller that models brain
stem reflexes actuating on eyelid muscles, and a feed-forward adaptive component that models the
cerebellum and learns to associate its inputs with the error signals driving the reactive controller.
With CFPC we propose a generic scheme in which a feed-forward module enhances the performance
of a reactive error feedback controller steering it with signals that facilitate anticipation, namely,
with counterfactual errors. However, within CFPC, even if these counterfactual errors that enable
predictive control are learned based on past errors in behavior, they do not reflect any current or
forthcoming error in the ongoing behavior.
In addition to eye-blink conditioning and postural adjustments, the interaction between reactive
and cerebellar-dependent acquired anticipatory behavior has also been studied in paradigms such
as visually-guided smooth pursuit eye movements (Lisberger, 1987). All these paradigms can be
abstracted as tasks in which the same predictive stimuli and disturbance or reference signal are
repeatedly experienced. In accordance to that, we operate our control scheme in trial-by-trial (batch)
mode. With that, we derive a learning rule for anticipatory control that modifies the well-known
least-mean-squares/Widrow-Hoff rule with an eligibility trace. More specifically, our model predicts
that to facilitate learning, parallel fibers to Purkinje cell synapses implement a forward model that
generates an eligibility trace. Finally, to stress that CFPC is not specific to eye-blink conditioning, we
demonstrate its application with a smooth pursuit task.
2
2.1
Methods
Cerebellar Model
w1
x1
xj
wj
xN
wN
e
o
Figure 1: Anatomical scheme of a Cerebellar Purkinje cell. The xj denote parallel fiber inputs to
Purkinje synapses (in red) with weights wj . o denotes the output of the Purkinje cell. The error signal
e, through the climbing fibers (in green), modulates synaptic weights.
We follow the simplifying approach of modeling the cerebellum as a linear adaptive filter, while
focusing on computations at the level of the Purkinje cells, which are the main output cells of the
cerebellar cortex (Fujita, 1982; Dean et al., 2010). Over the mossy fibers, the cerebellum receives
a wide range of inputs. Those inputs reach Purkinke cells via parallel fibers (Fig. 1), that cross
2
dendritic trees of Purkinje cells in a ratio of up to 1.5 ? 106 parallel fiber synapses per cell (Eccles
et al., 1967). We denote the signal carried by a particular fiber as xj , j ? [1, G], with G equal to the
total number of inputs fibers. These inputs from the mossy/parallel fiber pathway carry contextual
information (interoceptive or exteroceptive) that allows the Purkinje cell to generate a functional
output. We refer to these inputs as cortical bases, indicating that they are localized at the cerebellar
cortex and that they provide a repertoire of states and inputs that the cerebellum combines to generate
its output o. As we will develop a discrete time analysis of the system, we use n to indicate time (or
time-step). The output of the cerebellum at any time point n results from a weighted sum of those
cortical bases. wj indicates the weight or synaptic efficacy associated with the fiber j. Thus, we
|
|
have x[n] = [x1 [n], . . . , xG [n]] and w[n] = [w1 [n], . . . , wG [n]] (where the transpose, | , indicates
that x[n] and w[n] are column vectors) containing the set of inputs and synaptic weights at time n,
respectively, which determine the output of the cerebellum according to
o[n] = x[n]| w[n]
(1)
The adaptive feed-forward control of the cerebellum stems from updating the weights according to a
rule of the form
?wj [n + 1] = f (xj [n], . . . , xj [1], e[n], ?)
(2)
where ? denotes global parameters of the learning rule; xj [n], . . . , xj [1], the history of its presynaptic inputs of synapse j; and e[n], an error signal that is the same for all synapses, corresponding
to the difference between the desired, r, and the actual output, y, of the controlled plant. Note that in
drawing an analogy with the eye-blink conditioning paradigm, we use the simplifying convention
of considering the noxious stimulus (the air-puff) as a reference, r, that indicates that the eyelids
should close; the closure of the eyelid as the output of the plant, y; and the sensory response to the
noxious stimulus as an error, e, that encodes the difference between the desired, r, and the actual
eyelid closures, y. Given this, we advance a new learning rule, f , that achieves optimal performance
in the context of eye-blink conditioning and other cerebellar learning paradigms.
2.2
Cerebellar Control Architecture
Cerebellum (cortex and nuclei)
and Inferior olive [FF]
Trigeminal
nucleus
[o]
[e]
[e]
[u]
+
-
[r]
US
(airpuff)
[y]
Facial
nucleus
[C]
Eyelids
(Blink)
[P]
x
ADAPTIVE-ANTICIPATORY
(FEED-FORWARD) LAYER
FF
o
[x]
r
Pons
+ e
-
+
C
u
P
y
REACTIVE
(FEEDBACK) LAYER
CS
(Context,
e.g.: sound, light)
FEEDBACK CLOSEDLOOP SYSTEM
Figure 2: Neuroanatomy of eye-blink conditioning and the CFPC architecture. Left: Mapping of
signals to anatomical structures in eye-blink conditioning (De Zeeuw and Yeo, 2005); regular arrows
indicate external inputs and outputs, arrows with inverted heads indicate neural pathways. Right:
CFPC architecture. Note that the feedback controller, C, and the feed-forward module, F F , belong
to the control architecture, while the plant, P , denotes an object controlled. Other abbreviations: r,
reference signal; y, plant?s output; e, output error; x, basis signals; o, feed-forward signal; and u,
motor command.
We embed the adaptive filter cerebellar module in a layered control architecture, namely the CFPC
architecture, based on the interaction between brain stem motor nuclei driving motor reflexes and
the cerebellum, such as the one established between the cerebellar microcircuit responsible for
conditioned responses and the brain stem reflex circuitry that produces unconditioned eye-blinks
(Hesslow and Yeo, 2002) (Fig. 2 left). Note that in our interpretation of this anatomy we assume
that cerebellar output, o, feeds the lower reflex controller (Fig. 2 right). Put in control theory terms,
within the CFPC scheme an adaptive feed-forward layer supplements a negative feedback controller
steering it with feed-forward signals.
3
Our architecture uses a single-input single-output negative-feedback controller. The controller
receives as input the output error e = r ? y. For the derivation of the learning algorithm, we assume
that both plant and controller are linear and time-invariant (LTI) systems. Importantly, the feedback
controller and the plant form a reactive closed-loop system, that mathematically can be seen as a
system that maps the reference, r, into the plant?s output, y. A feed-forward layer that contains the
above-mentioned cerebellar model provides the negative feedback controller with an additional input
signal, o. We refer to o as a counter-factual error signal, since although it mechanistically drives the
negative feedback controller analogously to an error signal it is not an actual error. The counterfactual
error is generated by the feed-forward module that receives an output error, e, as its teaching signal.
Notably, from the point of view of the reactive layer closed-loop system, o can also be interpreted as
a signal that offsets r. In other words, even if r remains the reference that sets the target of behavior,
r + o functions as the effective reference that drives the closed-loop system.
3
3.1
Results
Derivation of the gradient descent update rule for the cerebellar control architecture
We apply the CFPC architecture defined in the previous section to a task that consists in following
a finite reference signal r ? RN that is repeated trial-by-trial. To analyze this system, we use the
discrete time formalism and assume that all components are linear time-invariant (LTI). Given this,
both reactive controller and plant can be lumped together into a closed-loop dynamical system, that
can be described with the dynamics A, input B, measurement C and feed-through D matrices. In
general, these matrices describe how the state of a dynamical system autonomously evolves with
time, A; how inputs affect system states, B; how states are mapped into outputs, C; and how inputs
instantaneously affect the system?s output D (Astrom and Murray, 2012). As we consider a reference
of a finite length N , we can construct the N -by-N transfer matrix T as follows (Boyd, 2008)
?
?
D
0
0
... 0
CB
D
0
... 0 ?
?
? CAB
CB
D
... 0 ?
?
T =?
?
.. ?
..
..
..
..
?
.
.
.
.
. ?
CAN ?2 B
CAN ?3 B
CAN ?4 B
...
D
With this transfer matrix we can map any given reference r into an output yr using yr = T r, obtaining
what would have been the complete output trajectory of the plant on an entirely feedback-driven trial.
Note that the first column of T contains the impulse response curve of the closed-loop system, while
the rest of the columns are obtained shifting that impulse response down. Therefore, we can build
the transfer matrix T either in a model-based manner, deriving the state-space characterization of
the closed-loop system, or in measurement-based manner, measuring the impulse response curve.
Additionally, note that (I ? T )r yields the error of the feedback control in following the reference, a
signal which we denote with e0 .
Let o ? RN be the entire feed-forward signal for a given trial. Given commutativity, we can consider
that from the point of view of the closed-loop system o is added directly to the reference r, (Fig. 2
right). In that case, we can use y = T (r + o) to obtain the output of the closed-loop system
when it is driven by both the reference and the feed-forward signal. The feed-forward module only
outputs linear combinations of a set of bases. Let X ? RN ?G be a matrix with the content of the
G bases during all the N time steps of a trial. The feed-forward signal becomes o = Xw, where
w ? RG contains the mixing weights. Hence, the output of the plant given a particular w becomes
y = T (r + Xw).
We implement learning as the process of adjusting the weights w of the feed-forward module in a
trial-by-trial manner. At each trial the same reference signal, r, and bases, X, are repeated. Through
learning we want to converge to the optimal weight vector w? defined as
1
1
w? = arg min c(w) = arg min e| e = arg min (r ? T (r + Xw))| (r ? T (r + Xw))
2
2
w
w
w
(3)
where c indicates the objective function to minimize, namely the L2 norm or sum of squared errors.
? = T X and using e0 = (I ? T )r, the minimization problem can be cast as a
With the substitution X
4
canonical linear least-squares problem:
1
? | (e0 ? Xw)
?
w? = arg min (e0 ? Xw)
2
w
(4)
? ? e0 ,
One the one hand, this allows to directly find the least squares solution for w? , that is, w? = X
where ? denotes the Moore-Penrose pseudo-inverse. On the other hand, and more interestingly, with
?
w[k] being the weights at trial k and having e[k] = e0 ? Xw[k],
we can obtain the gradient of the
error function at trial k with relation to w as follows:
? | e[k] = ?X| T | e[k]
?w c = ?X
Thus, setting ? as a properly scaled learning rate (the only global parameter ? of the rule), we can
derive the following gradient descent strategy for the update of the weights between trials:
w[k + 1] = w[k] + ?X| T | e[k]
(5)
This solves for the learning rule f in eq. 2. Note that f is consistent with both the cerebellar anatomy
(Fig. 2left) and the control architecture (Fig. 2right) in that the feed-forward module/cerebellum only
requires two signals to update its weights/synaptic efficacies: the basis inputs, X, and error signal, e.
3.2
T | facilitates a synaptic eligibility trace
The standard least mean squares (LMS) rule (also known as Widrow-Hoff or decorrelation learning
rule) can be represented in its batch version as w[k + 1] = w[k] + ?X| e[k]. Hence, the only
difference between the batch LMS rule and the one we have derived is the insertion of the matrix
factor T | . Now we will show how this factor acts as a filter that computes an eligibility trace at each
weight/synapse. Note that the update of a single weight, according Eq. 5 becomes
wj [k + 1] = wj [k] + ?x|j T | e[k]
(6)
where xj contains the sequence of values of the cortical basis j during the entire trial. This can be
rewritten as
wj [k + 1] = wj [k] + ?h|j e[k]
(7)
with hj ? T xj . The above inner product can be expressed as a sum of scalar products
wj [k + 1] = wj [k] + ?
N
X
hj [n]e[k, n]
(8)
n=1
where n indexes the within trial time-step. Note that e[k] in Eq. 7 refers to the whole error signal
at trial k whereas e[k, n] in Eq. 8 refers to the error value in the n-th time-step of the trial k. It is
now clear that each hj [n] weighs how much an error arriving at time n should modify the weight
wj , which is precisely the role of an eligibility trace. Note that since T contains in its columns/rows
shifted repetitions of the impulse response curve of the closed-loop system, the eligibility trace codes
at any time n, the convolution of the sequence of previous inputs with the impulse-response curve of
the reactive layer closed-loop. Indeed, in each synapse, the eligibility trace is generated by a forward
model of the closed-loop system that is exclusively driven by the basis signal.
Consequently, our main result is that by deriving a gradient descent algorithm for the CFPC cerebellar
control architecture we have obtained an exact definition of the suitable eligibility trace. That
definition guarantees that the set of weights/synaptic efficacies are updated in a locally optimal
manner in the weights? space.
3.3
On-line gradient descent algorithm
The trial-by-trial formulation above allowed for a straightforward derivation of the (batch) gradient
descent algorithm. As it lumped together all computations occurring in a same trial, it accounted for
time within the trial implicitly rather than explicitly: one-dimensional time-signals were mapped onto
points in a high-dimensional space. However, after having established the gradient descent algorithm,
we can implement the same rule in an on-line manner, dropping the repetitiveness assumption inherent
to trial-by-trial learning and performing all computations locally in time. Each weight/synapse must
5
have a process associated to it that outputs the eligibility trace. That process passes the incoming
(unweighted) basis signal through a (forward) model of the closed-loop as follows:
sj [n + 1] = Asj [n] + Bxj [n]
hj [n] = Csj [n] + Dxj [n]
where matrices A, B, C and D refer to the closed-loop system (they are the same matrices that we
used to define the transfer matrix T ), and sj [n] is the state vector of the forward model of the synapse
j at time-step n. In practice, each ?synaptic? forward model computes what would have been the
effect of having driven the closed-loop system with each basis signal alone. Given the superposition
principle, the outcome of that computation can also be interpreted as saying that hj [n] indicates what
would have been the displacement over the current output of the plant, y[n], achieved feeding the
closed-loop system with the basis signal xj . The process of weight update is completed as follows:
wj [n + 1] = wj [n] + ?hj [n]e[n]
(9)
At each time step n, the error signal e[n] is multiplied by the current value of the eligibility trace
hj [n], scaled by the learning rate ?, and subtracted to the current weight wj [n]. Therefore whereas
the contribution of each basis to the output of the adaptive filter depends only on its current value and
weight, the change in weight depends on the current and past values passed through a forward model
of the closed-loop dynamics.
3.4
Simulation of a visually-guided smooth pursuit task
0.2
r
y[1]
y[50]
1
0.8
angular position (a.u.)
angular position (a.u.)
We demonstrate the CFPC approach in an example of a visual smooth pursuit task in which the
eyes have to track a target moving on a screen. Even though the simulation does not capture all the
complexity of a smooth pursuit task, it illustrates our anticipatory control strategy. We model the
plant (eye and ocular muscles) with a two-dimensional linear filter that maps motor commands into
angular positions. Our model is an extension of the model in (Porrill and Dean, 2007), even though
in that work the plant was considered in the context of the vestibulo-ocular reflex. In particular, we
use a chain of two leaky integrators: a slow integrator with a relaxation constant of 100 ms drives the
eyes back to the rest position; the second integrator, with a fast time constant of 3 ms ensures that
the change in position does not occur instantaneously. To this basic plant, we add a reactive control
layer modeled as a proportional-integral (PI) error-feedback controller, with proportional gain kp and
integral gain ki . The control loop includes a 50 ms delay in the error feedback, to account for both
the actuation and the sensing latency. We choose gains such that reactive tracking lags the target by
approximately 100 ms. This gives kp = 20 and ki = 100. To complete the anticipatory and adaptive
control architecture, the closed-loop system is supplemented by the feed-forward module.
0.6
0.4
0.2
0
0
0.5
1
1.5
time (s)
2
2.5
e[1]
e[50]
o[50]
0.1
0
?0.1
0
0.5
1
1.5
time (s)
2
2.5
Figure 3: Behavior of the system. Left: Reference (r) and output of the system before (y[1]) and
after learning (y[50]). Right: Error before e[1] and after learning e[50] and output acquired by
cerebellar/feed-forward component (o[50])
The architecture implementing the forward model-based gradient descent algorithm is applied to a
task structured in trials of 2.5 sec duration. Within each trial, a target remains still at the center of
the visual scene for a duration 0.5 sec, next it moves rightwards for 0.5 sec with constant velocity,
remains still for 0.5 sec and repeats the sequence of movements in reverse, returning to the center.
The cerebellar component receives 20 Gaussian basis signals (X) whose receptive fields are defined
in the temporal domain, relative to trial onset, with a width (standard-deviation) of 50 ms and spaced
by 100 ms. The whole system is simulated using a 1 ms time-step. To construct the matrix T we
computed closed-loop system impulse response.
6
At the first trial, before any learning, the output of the plant lags the reference signal by approximately
100 ms converging to the position only when the target remains still for about 300 ms (Fig. 3 left). As
a result of learning, the plant?s behavior shifts from a reactive to an anticipatory mode, being able to
track the reference without any delay. Indeed, the error that is sizable during the target displacement
before learning, almost completely disappears by the 50th trial (Fig. 3 right). That cancellation
results from learning the weights that generate a feed-forward predictive signal that leads the changes
in the reference signal (onsets and offsets of target movements) by approximately 100 ms (Fig. 3
right). Indeed, convergence of the algorithm is remarkably fast and by trial 7 it has almost converged
to the optimal solution (Fig. 4).
WH
WH+50ms
WH+70ms
FM?ET
1
rRMSE
0.8
0.6
0.4
0.2
0
0
10
20
#trial
30
40
50
Figure 4: Performance achieved with different learning rules. Representative learning curves of the
forward model-based eligibility trace gradient descent (FM-ET), the simple Widrow-Hoff (WH) and
the Widrow-Hoff algorithm with a delta-eligibility trace matched to error feedback delay (WH+50
ms) or with an eligibility trace exceeding that delay by 20 ms (WH+70 ms). Error is quantified as the
relative root mean-squared error (rRMSE), scaled proportionally to the error in the first trial. Error of
the optimal solution, obtained with w? = (T X)? e0 , is indicated with a dashed line.
To assess how much our forward-model-based eligibility trace contributes to performance, we test
three alternative algorithms. In both cases we employ the same control architecture, changing the
plasticity rule such that we either use no eligibility trace, thus implementing the basic Widrow-Hoff
learning rule, or use the Widrow-Hoff rule extended with a delta-function eligibility trace that matches
the latency of the error feedback (50 ms) or slightly exceeds it (70 ms). Performance with the basic
WH model worsens rapidly whereas performance with the WH learning rule using a ?pure delay?
eligibility trace matched to the transport delay improves but not as fast as with the forward-modelbased eligibility trace (Fig. 4). Indeed, in this case, the best strategy for implementing a delayed
delta eligibility trace is setting a delay exceeding the transport delay by around 20 ms, thus matching
the peak of the impulse response. In that case, the system performs almost as good as with the
forward-model eligibility trace (70 ms). This last result implies that, even though the literature
usually emphasizes the role of transport delays, eligibility traces also account for response lags due
to intrinsic dynamics of the plant.
To summarize our results, we have shown with a basic simulation of a visual smooth pursuit task
that generating the eligibility trace by means of a forward model ensures convergence to the optimal
solution and accelerates learning by guaranteeing that it follows a gradient descent.
4
Discussion
In this paper we have introduced a novel formulation of cerebellar anticipatory control, consistent
with experimental evidence, in which a forward model has emerged naturally at the level of Purkinje
cell synapses. From a machine learning perspective, we have also provided an optimality argument
for the derivation of an eligibility trace, a construct that was often thought of in more heuristic terms
as a mechanism to bridge time-delays (Barto et al., 1983; Shibata and Schaal, 2001; McKinstry et al.,
2006).
The first seminal works of cerebellar computational models emphasized its role as an associative
memory (Marr, 1969; Albus, 1971). Later, the cerebellum was investigates as a device processing
correlated time signals(Fujita, 1982; Kawato et al., 1987; Dean et al., 2010). In this latter framework,
7
the use of the computational concept of an eligibility trace emerged as a heuristic construct that
allowed to compensate for transmission delays in the circuit(Kettner et al., 1997; Shibata and Schaal,
2001; Porrill and Dean, 2007), which introduced lags in the cross-correlation between signals.
Concretely, that was referred to as the problem of delayed error feedback, due to which, by the time
an error signal reaches a cell, the synapses accountable for that error are no longer the ones currently
active, but those that were active at the time when the motor signals that caused the actual error were
generated. This view has however neglected the fact that beyond transport delays, response dynamics
of physical plants also influence how past pre-synaptic signals could have related to the current
output of the plant. Indeed, for a linear plant, the impulse-response function of the plant provides the
complete description of how inputs will drive the system, and as such, integrates transmission delays
as well as the dynamics of the plant. Recently,
Even though cerebellar microcircuits have been used as models for building control architectures,
e.g., the feedback-error learning model (Kawato et al., 1987), our CFPC is novel in that it links
the cerebellum to the input of the feedback controller, ensuring that the computational features of
the feedback controller are exploited at all times. Within the domain of adaptive control, there are
remarkable similarities at the functional level between CFPC and iterative learning control (ILC)
(Amann et al., 1996), which is an input design technique for learning optimal control signals in
repetitive tasks. The difference between our CFPC and ILC lies in the fact that ILC controllers
directly learn a control signal, whereas, the CFPC learns a conterfactual error signal that steers a
feedback controller. However the similarity between the two approaches can help for extending
CFPC to more complex control tasks.
With our CFPC framework, we have modeled the cerebellar system at a very high level of abstraction:
we have not included bio-physical constraints underlying neural computations, obviated known
anatomical connections such as the cerebellar nucleo-olivary inhibition (Bengtsson and Hesslow,
2006; Herreros and Verschure, 2013) and made simplifications such as collapsing cerebellar cortex and
nuclei into the same computational unit. On the one hand, such a choice of high-level abstraction may
indeed be beneficial for deriving general-purpose machine learning or adaptive control algorithms.
On the other hand, it is remarkable that in spite of this abstraction our framework makes fine-grained
predictions at the micro-level of biological processes. Namely, that in a cerebellar microcircuit (Apps
and Garwicz, 2005), the response dynamics of secondary messengers (Wang et al., 2000) regulating
plasticity of Purkinje cell synapses to parallel fibers must mimic the dynamics of the motor system
being controlled by that cerebellar microcircuit. Notably, the logical consequence of this prediction,
that different Purkinje cells should display different plasticity rules according to the system that they
control, has been validated recording single Purkinje cells in vivo (Suvrathan et al., 2016).
In conclusion, we find that a normative interpretation of plasticity rules in Purkinje cell synapses
emerges from our systems level CFPC computational architecture. That is, in order to generate
optimal eligibility traces, synapses must include a forward model of the controlled subsystem. This
conclusion, in the broader picture, suggests that synapses are not merely components of multiplicative
gains, but rather the loci of complex dynamic computations that are relevant from a functional
perspective, both, in terms of optimizing storage capacity (Benna and Fusi, 2016; Lahiri and Ganguli,
2013) and fine-tuning learning rules to behavioral requirements.
Acknowledgments
The research leading to these results has received funding from the European Commission?s Horizon
2020 socSMC project (socSMC-641321H2020-FETPROACT-2014) and by the European Research
Council?s CDAC project (ERC-2013-ADG 341196).
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9
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5,694 | 6,152 | Learning Tree Structured Potential Games
Vikas K. Garg
CSAIL, MIT
vgarg@csail.mit.edu
Tommi Jaakkola
CSAIL, MIT
tommi@csail.mit.edu
Abstract
Many real phenomena, including behaviors, involve strategic interactions that can
be learned from data. We focus on learning tree structured potential games where
equilibria are represented by local maxima of an underlying potential function. We
cast the learning problem within a max margin setting and show that the problem
is NP-hard even when the strategic interactions form a tree. We develop a variant
of dual decomposition to estimate the underlying game and demonstrate with
synthetic and real decision/voting data that the game theoretic perspective (carving
out local maxima) enables meaningful recovery.
1
Introduction
Structured prediction methods [1; 2; 3; 4; 5] are widely adopted techniques for learning mappings
between context descriptions x ? X and configurations y ? Y. The variables specifying each configuration y (e.g., arcs in natural language parsing) are typically mutually dependent and it is therefore
beneficial to predict them jointly rather than individually. The predicted y often arises as the highest
scoring configuration with respect to a parameterized scoring function that decomposes into terms
that couple two or more variables together to model their interactions. Structured prediction methods
have been broadly useful across areas, from computational biology (e.g., molecular arrangements,
alignments), natural language processing (e.g., parsing, tagging), computer vision (e.g., segmentation,
matching), and many others. However, the setting is less suitable for modeling strategic interactions
that are better characterized in terms of local consistency constraints.
We consider the problem of predicting configurations y that represent game theoretic equilibria. Such
configurations are unlikely to coincide with the maximum of a global scoring function as in structured
prediction. Indeed, there may be many possible equilibria in a specific context, and the particular
choice may vary considerably. Each possible configuration is nevertheless characterized by local
constraints that represent myopic optimizations of individual players. For example, senators can
be thought to vote relative to give and take deals with other closely associated senators. Several
assumptions are necessary to make the game theoretic setting feasible.
We abstract the setting as a potential game [6; 7; 8] among the players, and define a stochastic process
to model the dynamics of the game. A game is said to be a potential game if the incentive of all
players to change their strategy can be expressed using a single global potential function. Every
potential game is guaranteed to have at least one (possibly multiple) pure strategy Nash equilibria
[9], and we will exploit this property in modeling and analyzing several real world scenarios. Note
that each pure Nash equilibrium corresponds to a local optimum of the underlying potential function
rather than the global optimum as in structured prediction.
We further restrict the setting by permitting the payoff of each player to depend only on their own
action and the actions of their neighbors (a subset of the other players). Thus, we may view our
setting as a graphical game [10; 11]. In this work, we investigate potential games where the graphical
structure of the interactions form a tree. The goal is to recover the tree structured potential function
that supports observed configurations of actions as locally optimal solutions. We prove that it is
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
NP-hard to recover such games under a max-margin setting. We then propose a variant of dual
decomposition (cf. [12; 13]) to learn the tree structure and the associated parameters.
2
Setting
We commence with the game theoretic setting. There are n players indexed by a position in [n] ,
{1, 2, . . . , n}. These players can be visualized as nodes of a tree-structured graph T with undirected
edges E. We denote the set of neighbors of node i by Ni , i.e., (i, j) ? E ?? j ? Ni ? i ? Nj , and
abbreviate (i, j) ? E as ij ? T without introducing ambiguity. Each player i has a finite discrete
set of strategies Yi . A strategy
or label configuration is an n-dimensional vector of the form
Qprofile
n
y = (y1 , y2 , . . . , yn ) ? Y = i=1 Yi . We denote the parametric potential function associated with
the tree by f (y; x, T, ?), where y is a strategy profile, ? the set of parameters, and x ? X is a context
[14]. We obtain an (n ? 1)-dimensional vector y?i = (y1 , . . . , yi?1 , yi+1 , . . . , yn ) by considering
the strategies of all players other than i. Thus, we may equivalently write y = (yi , y?i ). Moreover,
we use yNi to denote the strategy profile pertaining to the neighbors of node i. We can extract from
f (y; x, T, ?) individual payoff (or cost) functions fi (yi , yNi ; x, T, ?), i ? [n], which merely include
all the terms that pertain to the strategy of the ith player yi .
The choice of a particular equilibrium (local optimum) in a context results from a stochastic process.
Starting with an initial configuration y at time t = 0 (e.g., chosen at random), the game proceeds
in an iterative fashion: during each subsequent iteration t = 1, 2, . . ., a player pt ? [n] is chosen
uniformly at random. The player pt then computes the best response candidate set
Zpt = arg max fpt (z, yNpt ; x, T, ?),
z?Ypt
and switches to a strategy within this set uniformly at random if their current strategy does not already
belong to this set, i.e., player changes their strategy only if a better option presents itself. The game
finishes when a locally optimal configuration y? ? Y has been reached, i.e., when no player can
improve their payoff unilaterally. Since many locally optimal configurations could have been reached
in the given context x, the stochastic process induces a distribution over the strategy profiles.
We assume that our training data S = {(x1 , y 1 ), . . . , (xM , y M )} is generated by some distribution
over contexts and the induced conditional distribution over strategy profiles with respect to some tree
structured potential function. Our objective is to learn both the underlying tree structure T and the
parameters ? using a max-margin setting. Specifically, given S, we are interested in finding T and ?
such that
m
? m ? [M ], i ? [n], yi ? Yi , f (y m ; xm , T, ?) ? f (y?i
, yi ; xm , T, ?) + e(y, y m ),
where e(y, y m ) is a non-negative loss (e.g. Hamming loss), which is 0 if and only if y = y m .
Note that the maximum margin framework does not make an explicit use of the assumed induced
distribution over equilibria.
The setting here is superficially similar to relaxations of structured prediction tasks such as pseudolikelihood [15] or decomposed learning [16]. These methods are, however, designed to provide
computationally efficient approximations of the original structured prediction task by using fewer
constraints during learning. Instead, we are specifically interested in modeling the observations as
locally optimal solutions with respect to the potential function.
We only state the results of our theorems in the main text, and defer all the proofs to the Supplementary.
3
Learning Tree Structured Potential Games
We first show that it is NP-hard to learn a tree structured potential game in a discriminative maxmargin setting. Previous hardness results are available about learning structured prediction models
under global constraints and arbitrary graphs [15], and under global constraints and tree structured
models [17], also in a max-margin setting.
Theorem 1. Given a set of training examples S = {(xm , y m )}M
m=1 and a family of potential
functions of the form
X
X
X
f (y; x, T, ?) =
?ij (yi , yj ) +
?i (yi ) +
xi (yi ),
i
ij?T
2
i
it is NP-hard to decide whether there exists a tree T and parameters ? (up to model equivalence) such
that the following holds:
m
? m, i, yi , f (y m ; xm , T, ?) ? f (y?i
, yi ; xm , T, ?) + e(y, y m ).
3.1
Dual decomposition algorithm
The remainder of this section concerns with developing an approximate method for learning the
potential function by appeal to dual decomposition. Dual decomposition methods are typically
employed to solve inference tasks over combinatorial structures (e.g., [12; 13]). In contrast, we
decompose the problem on two levels. On one hand, we break the problem into independent local
neighborhood choices and use dual variables to reconcile these choices across the players so as to
obtain a single tree-structured model. On the other hand, we ensure that initially disjoint parameters
mediating the interactions between a player and its neighbors are in agreement across the edges in the
resulting structure. The two constraints ensure that there is a single tree-structured global potential
function.
For each node i, let Ni be the set of neighbors of i represented in terms of indicator variables such
that Nij = 1 if i selects j as a neighbor. Nij can be chosen independently from Nji but the two
will be enforced to agree at the solution. We will use Ni as a set of neighbors and as a set of
indicator variables interchangeably. Similarly, we decompose the parameters into node potentials
?i ? ?(yi ; x) = ?i (yi ; x) and edge potentials ?ij ? ?(yi , yj ; x) = ?i,j (yi , yj ; x) where again ?ij may
be chosen separately from ?ji but will be encouraged to agree. The set of parameters associated with
each player then consists of locally controllable parameters ?i = {?i , ?i? } and Ni , where Ni selects
the relevant subset of interaction terms:
X
f (y; x, Ni , ?i ) = ?i (yi ; x) +
Nij ?i,j (yi , yj ; x)
j6=i
Given a training set S = {(x1 , y 1 ), . . . , (xM , y M )}, the goal is to learn the set of neighbors N =
{N1 , . . . , Nn }, and weights ? = {?1 , . . . , ?n } so as to minimize
n
M
1
C XX
m
||?||2 +
max f (y?i
, yi ; xm , Ni , ?i ) ? f (y m ; xm , Ni , ?i ) + e(yi , yim )
2
M n i=1 m=1 yi
{z
}
|
(1)
,Rmi (Ni ,?i )
subject toP N forming a tree and ? agreeing across the players.
Let Ri (Ni , ?i ) =
C/(M n) m Rmi (Ni , ?i ). We force the neighbor choices to agree with a global tree structure
represented by indicators N 0 . Similarly, we enforce parameters ?i to agree across neighbors. The
resulting Lagrangian can be written as
n
X
1
2
i=1
|
||?i ||2 + Ri (Ni , ?i ) +
X
X
0
(?ij Nij + ?ij ? ?ij ) + ?
?ij Nij
+ G(N 0 )
j6=i
i,j6=i
{z
}
L(?i ,Ni ;?,?)
|
{z
G(N 0 ,?)
}
where G(N 0 ) = 0 if N 0 forms a tree and ? otherwise, and ?ij = ??ji . For the dual decomposition
algorithm, we must be able to solve min?i L(?i , Ni ; ?, ?) to obtain ??i and minNi L(?i , Ni ; ?, ?) to
get Ni? . The former is a QP while the latter is more challenging though may permit efficient solutions
via additional relaxations, exploiting combinatorial properties in restricted cases (sub-modularity), or
even brute force for smaller problems. G(N 0 , ?) corresponds to a minimum weighted spanning tree,
and thus can be efficiently solved using any standard algorithm like Bor?uvka?s, Kruskal?s or Prim?s.
?
The basic dual decomposition alternatively solves ??i , Ni? , and N 0 , resulting in updates of the dual
variables based on disagreements. While the method has been successful for enforcing structural
constraints (e.g., parsing), it is less appropriate for constraints involving continuous variables. To
address this, we employ the alternating direction method of multipliers (ADMM) [18; 19; 20] for
parameter agreements. Specifically, we encourage ?i? and ??i to agree with their mean ui? , by
introducing an additional term to the Lagrangian L
?
LA (?i , Ni ; ui? , ?, ?) = L(?i , Ni ; ?, ?) + ||?i? ? ui? ||2
2
3
where ui? is updated as an independent parameter.
There are many ways to schedule the updates. We employ a two-phase algorithm that learns the
structure of the game tree and the parameters separately. The algorithm is motivated by the following
theorem. Since the result applies broadly to the dual decomposition paradigm, we state the theorem
in a slightly more generic form than that required for our purpose. The theorem applies to our setting
with
X
f (N 0 ) = ?G(N 0 ), A = [n], and gi (?i , Ni ) =
?ij Nij ? L(?i , Ni ; ?, ?).
j6=i
We now set up the conditions of the theorem. Consider the following combinatorial problem
(
)
X
Opt = max f (z) +
g? (z? ) ,
z
??A
where f (z) specifies global constraints on admissible z, and g? (z? ) represent local terms guiding the
assignment of values to different subsets of variables z? = {zj }j?? . Let the problem be minimized
with respect to the dual coefficients {?i,? (zi )} by following a dual decomposition approach. Suppose
we can find a global assignment z? and dual coefficients such that this assignment nearly attains the
local maxima for all ? ? A, i.e.,
X
X
g? (?
z? ) +
?j,? (?
zj ) ? max g? (z? ) +
?j,? (zj ) ? .
j??
z?
j??
Assume further, without loss of generality,1 that the assignment attains the max for the global
constraint. Then, we have the following result.
Theorem 2. If there exists an assignment z? and associated dual coefficients such that the assignment
obtains
-maximum ofeach term in the decomposition, for some > 0, then the objective value for
z? ? Opt ? |A|, Opt .
The theorem implies that if a global structure nearly attains the optima for the local neighborhoods,
then we might as well shift our focus to finding the global structure rather than optimize for the
parameters corresponding to the exact local optima. The result guarantees that the value of such a
global structure cannot be too far from that of the optimal global structure.
We outline our two-phase approach in Algorithm 1. The first phase concerns only with iteratively
finding a globally consistent structure. It is possible that at the conclusion of this phase, the local
structures do not fully agree (the relaxation is not tight). For this reason, the procedure runs for a
specified maximum number of iterations and selects the global tree corresponding to an iteration that
is least inconsistent with the local neighborhoods. Note that this phase does not precisely solve the
original problem we posed earlier. Instead, the structure is obtained without constraining parameters
to agree. In this sense, the first phase does not consider strictly potential games as the interactions
between players can remain intrinsic to the players themselves.
The second phase simply optimizes the parameters for the already specified global tree. This step
realizes a potential game as the parameters and the structure will be in agreement. We note that such
parameters could be optimized directly for the selected tree without the need of dual decomposition.
However, Algorithm 1 remains suitable in a distributed setting since each player is required to solve
only local problems during the entire execution of the algorithm.
3.2
Scaling the algorithm
As already noted, Algorithm 1 exhaustively enumerates all neighborhoods for each local optimization
problem. This makes the algorithm computationally prohibitive in realistic settings. We now outline
an approximation procedure that restricts the candidate neighborhood assignments. Specifically, for a
local optimization at any node i, we may restrict the possible local neighborhoods at any iteration t to
only those configurations that are at most h Hamming distance away from the best local configuration
for i in iteration t-1. That is, we update each local max-structure incrementally, still guided by the
1
We can adjust the bound with a term that depends on the difference between the value of the optimal global
structure and the value of the global structure under consideration if these values do not coincide.
4
overall tree within the same dual decomposition framework. Note that we recover Algorithm 1 as a
special case when h = n. A small h corresponds to searching over a much smaller space compared to
the brute force algorithm. For instance, if we take h = 1, then the total complexity of the approximate
algorithm reduces to O(n2 ? M axIter) since in each iteration we need to solve n local problems
each having O(n) candidate neighborhoods.
Algorithm 1 Learning tree structured potential games
1: procedure L EARN T REE P OTENTIAL G AME
2: Input: parameters ?, ?, M axIter, and > 0.
3:
4: Phase 1: Learn Tree Structure
5:
Initialize t = 1, ?ij = 0, ?ij = 0, M inGap = ?.
6:
repeat
7:
Find N 0 = argmin G(N , ?) using a minimum spanning tree algorithm
N
8:
9:
10:
11:
12:
for each i ? [n] do
for each Ni do
Compute ?i?t+1 = min L(?i , Ni ; ?, 0)
?i
Find Ni? = argmin L(??t+1
, Ni ; ?, 0)
i
Ni
X
?
0
Compute gap: Gap =
I(Nij
6= Nij
).
i,j
13:
if Gap < M inGap then
M inGap = Gap, Global = N 0
0
?
Update ? ?i, j 6= i: ?ij = ?ij + ?t (Nij
? Nij )
t?t+1
until M inGap = 0 or t > M axIter.
Set N 0? = Global.
14:
15:
16:
17:
18:
19: Phase 2: Learn Parameters
20:
Set N = N 0?
21:
Compute ?i?t+1 = min L(?i , Ni ; 0, ?)
?i
25:
repeat
?t+1
?t+1
Compute u ?i, j 6= i: ut+1
+ ?ji
)/2
ij = (?ij
?t+1
t+1
Update ? ?i, j 6= i: ?ij = ?ij + ?(?ij ? uij )
Compute ?i?t+1 = min LA (?i , Ni ; ui? , 0, ?)
26:
27:
t?
Xt + 1
?t+1
?t+1
until
||?ij
? ?ji
||2 <
28:
?t+1
?t+1
?
?
Set ?ij
, ?ji
= (?ij
+ ?ji
)/2
22:
23:
24:
?i
i,j6=i
29: Output: N 0? , ??
4
Experimental Results
We now describe the results of our experiments on both synthetic and real data to demonstrate the
efficacy of our algorithm. We found the algorithm to perform well for a wide range of C and ?
across different data. We report below the results of our experiments with the following setting
of parameters: ? = 1, ?t = 0.005 (for all t), C = 10, = 0.1, and M axIter = 100. For each
local optimization problem, the configurations were constrained to share the slack variable in order
to reduce the total number of optimization variables. Moreover, we used a scaled 0-1 loss [15],
e(y, y m ) = 1{y 6= y m }/n for each local optimization. We set h = 1 for the approximate method.
We conducted different sets of experiments to underscore the different aspects of our approach.
Our experiments with toy synthetic data highlight recovery of an underlying true structure under
controlled conditions (pertaining to the data generation process). The results on a real, but toy dataset,
Supreme Court vindicate the applicability of the exhaustive approach to unraveling the interactions
5
latent in real datasets. Finally, we address the scalability issues inherent in the exhaustive search, by
demonstrating the approximate version on the larger Congressional Votes real dataset.
4.1
Synthetic Dataset
We will now describe how the brute force method recovered the true structure on a synthetic dataset.
For this, data were assumed to come from the underlying model
X
X
f (y; x, ?) =
?ij (yi , yj ) +
xi ?i (yi ),
i
ij?E
where x represents the context that varies. The parameters were set as follows. We designed a n-node
degenerate or pathological tree, n = 6, with edges between node i and i + 1, i ? {1, 2, . . . , n ? 1}.
On each edge (i, j) ? E, we sampled ?ij (yi , yj ), yi , yj ? {0, 1} uniformly at random from [?1, 1]
independently of the other edges. For each node i, we also sampled ?i (yi ), yi ? {0, 1} independently
from the same range. Each training example pair (xm , ym ) was sampled in two steps. First, each
xmj , j ? [n] was set uniformly at random in the range [?10, 10], independently of each other. The
associated ym was then generated according to the stochastic process described in Section 2. Briefly,
starting with ym ? {0, 1}n sampled uniformly at random, we successively updated the configuration
by changing a randomly chosen coordinate of ym , and accepting the move only if the associated
score was higher. Since there are 2n possible configurations of binary vectors, we were guaranteed
that, in finite time, this procedure ended in a locally stable configuration. Once this locally stable
configuration was reached, we checked if the score of this configuration exceeded all the other
configurations with Hamming distance one by at least 1/n. If yes, then we included the pair (xm , ym )
in our synthetic data set, otherwise we discarded the pair. Starting with 100 examples, this procedure
resulted in a total of 78 stable configurations that scored higher than each configuration one Hamming
distance away by at least 1/n. These configurations formed our synthetic data set. We were able
to exactly recover the tree structure at the end of the Phase 1 of our algorithm using the training
0
data. Fig. 1 shows the evolution of the global tree structure (i.e. N in the iterations that resulted in
decrease of Gap). Note how the algorithm corrects for incorrect edges, starting from a star tree till it
recovers the pathological tree structure. Fig. 2 elucidates the synergy between the global tree and
local neighborhoods toward recovering the correct structure.
2
3
2
1
5
4
6
3
2
1
5
4
6
3
2
1
5
4
6
3
1
5
4
6
Figure 1: Recovery on synthetic data. Evolution of the tree structure is shown from left to right.
Each incorrect edge is indicated by coloring one of the end nodes in red. After first iteration, only the
edge (1, 2) is identified correctly. At termination, all edges in the underlying structure are recovered.
We show in Fig. 3 the evolution of the tree when the observations were falsely treated as globally
optimal points. Clearly, structured prediction failed to recover the underlying tree structure.
4.2
Real Dataset 1: Supreme Court Rulings
For both real datasets, we assumed the following decomposition:
X
X
?i (yi ).
f (y; ?) =
?ij (yi , yj ) +
i
ij?E
For our first real dataset,2 we considered the rulings of a Supreme Court bench comprising Justices
Alito (A), Breyer (B), Ginsburg (G), Kennedy (K), Roberts (R), Scalia (S), and Thomas (T ), during
2
Publicly available at http://scdb.wustl.edu/.
6
1
5
6
4
2
3
3
2
6
1
5
2
1
4
3
6
4
5
Figure 2: Global-Local Synergy. (Center & Right) Spanning trees formed from two separate local
neighborhoods (in different iterations). (Left) The common global tree structure. The global tree
structure reappears during the execution of the algorithm. On first occurrence, the global tree is
misaligned from chain 2-3-4 of the local neighborhood tree at node 5, as indicated by tree in the
center. The algorithm takes corrective action, and on the next occurrence, node 5 moves to the desired
position, as seen from tree on the right. The algorithm proceeds to correct the positioning of node 6.
2
3
5
1
5
4
6
3
5
1
2
4
6
3
5
1
2
6
4
3
1
2
4
6
Figure 3: Evolution of structured prediction. Structured prediction fails to recover true structure.
S
T
Scalia
Thomas
Kennedy
Alito
Roberts
K
A
R
Breyer
G
B
K
B
A
T
R
G
S
Ginsburg
Conservatives
Liberals
Figure 4: (Left) Tree recovered from Supreme Court data. The tree is consistent with widely known
ideology of the justices: Justice Kennedy (K) is considered largely moderate, while the others espouse
a more conservative or liberal jurisprudence. The thickness of an edge indicates the strength of
interaction in terms of (scaled) l2 -norm of the edge parameters. (Right) Enforcing global constraints
(structured prediction) resulted in a qualitatively incorrect structure.
the year 2013. Justices Alito, Roberts, Scalia, and Thomas are known to be conservatives, while
Justices Breyer and Ginsburg belong to the liberal side of the Court. Justice Kennedy generally takes
a moderate stand on most issues. On every case under their jurisdiction, each Justice chose an integer
from the set {1, 2, . . . , 8}. We considered all the rulings of this bench that had at least one ?dissent".
For our purposes, we created a dataset from those rulings that did not register a value 6, 7, 8 from any
of the Justices, since these values seem to have a complex interpretation instead of a simple yes/no.
For all other values, we used the interpretation by [21]: dissent value 2 was treated as 0 (no), and
others with 1 (yes). Fig. 4 shows that we were able to recover the known ideology of the Justices
by correctly treating the rulings as local optimal, whereas structured prediction failed to identify a
qualitatively correct structure.
7
Figure 5: (Congressional Votes.) The recovered tree is consistent with the expected voting pattern
that, in general, Democrats and Republicans vote along their respective party principles.
4.3
Real Dataset 2: Congressional Voting Records
We also experimented with a dataset3 obtained by compiling the votes on all the bills of the 110th
United States Congress (Session 2). The US Congress records the voting proceedings of the legislative
branch of the US federal government [11]. The U.S. Senate consists of 100 senators: each of the
50 U.S. states is represented by two senators. We compiled all the votes of the first 30 senators (in
data order) over this period on bills without unanimity. Each vote takes one of the two values: +1 or
-1, to denote whether the vote was in favor or against the proposed bill. We treated vote values -1
with 0 to create a binary dataset. Fig. 5 shows how the approximate algorithm is able to recover a
qualitatively correct structure that Democrats and Republicans typically vote along their respective
party ideologies (note that there might be more than one qualitatively correct structure). Specifically,
we obtain a structure where no Democrat is sandwiched between two Republicans, or vice-versa.
Discussion
A primary goal of this work is to argue that complex strategic interactions are better modeled as
locally optimal solutions instead of globally optimal assignments (as done, for instance, in structured
prediction). We believe this local versus global distinction has not been accorded due significance in
the literature, and we hope our work fosters more research in that direction.
The work opens up several interesting avenues. All the results presented in this paper are qualitative
in nature, primarily because quantitative evaluation is non-trivial in our setting since a strategic game
may have multiple equilibria (local optima). The incremental method proposed in this paper does
not come with any certificate of optimality, unlike most dual decomposition settings. We assumed
the dynamics of the underlying game follow a stochastic process, whereas players typically take
deterministic turns in real game settings. From a statistical learning perspective, it will be interesting
to estimate the generalization bounds in terms of the number of local equibria samples. Learning
across (repeated) games and exploring sub-modular potential functions are other directions.
Acknowledgments
Jean Honorio provided the Congressional Votes dataset for our experiments. We would also like to
thank the anonymous reviewers for their helpful comments.
3
Publicly available at http://www.senate.gov/.
8
References
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and interdependent output variables, JMLR, 6(2), pp. 1453-1484, 2005.
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segmenting and labeling sequence data, ICML, 2001.
[4] J. K. Bradley and C. Guestrin. Learning tree conditional random fields, ICML, 2010.
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[8] Y. Song, S. H. Y. Wong, and K.-W. Lee. Optimal gateway selection in multi-domain wireless
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Games from Behavioral Data, JMLR, 16, pp. 1157-1210, 2015.
[12] A. M. Rush and M. Collins. A Tutorial on Dual Decomposition and Lagrangian Relaxation for
Inference in Natural Language Processing, JAIR, 45, pp. 305-362, 2012.
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Programming Relaxations for Natural Language Processing, EMNLP, 2010.
[14] M. Hoefer and A. Skopalik. Social Context in Potential Games, Internet and Network Economics,
pp. 364-377, 2012.
[15] D. Sontag, O. Meshi, T. Jaakkola, and A. Globerson. More data means less inference: A
pseudo-max approach to structured learning, NIPS, 2010.
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2012.
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Dual Decomposition for MAP Inference, NIPS, 2010.
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9
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5,695 | 6,153 | Estimating Nonlinear Neural Response Functions
using GP Priors and Kronecker Methods
Cristina Savin
IST Austria
Klosterneuburg, AT 3400
csavin@ist.ac.at
Gasper Tka?cik
IST Austria
Klosterneuburg, AT 3400
tkacik@ist.ac.at
Abstract
Jointly characterizing neural responses in terms of several external variables
promises novel insights into circuit function, but remains computationally prohibitive in practice. Here we use gaussian process (GP) priors and exploit recent
advances in fast GP inference and learning based on Kronecker methods, to efficiently estimate multidimensional nonlinear tuning functions. Our estimator
requires considerably less data than traditional methods and further provides principled uncertainty estimates. We apply these tools to hippocampal recordings during
open field exploration and use them to characterize the joint dependence of CA1
responses on the position of the animal and several other variables, including the
animal?s speed, direction of motion, and network oscillations. Our results provide
an unprecedentedly detailed quantification of the tuning of hippocampal neurons.
The model?s generality suggests that our approach can be used to estimate neural
response properties in other brain regions.
1
Introduction
An important facet of neural data analysis concerns characterizing the tuning properties of neurons,
defined as the average firing rate of a cell conditioned on the value of some external variables, for
instance the orientation of an image patch in a V1 cell, or the position of the animal within an
environment for hippocampal cells. As experiments become more complex and more naturalistic, the
number of variables that modulate neural responses increases. These include not only experimentally
targeted inputs but also variables that are no longer under the experimenter?s control but which can
be (to a certain extent) measured, either external (the behavior of the animal) or internal (attentional
level, network oscillations, etc). Characterizing these complex dependencies is very difficult, yet it
could provide important insights into neural circuits computation and function.
Traditional estimates of a cell?s tuning properties often manipulate one variable at the time or consider
simple dependencies between inputs and the neural responses e.g. Generalized Linear Models,
GLM [1, 2]). There is comparatively little work that allows for complex input-output functional
relationships on multidimensional input spaces [3?5]. The reasons for this are twofold. On one hand,
dealing with complex nonlinearities is computationally challenging, on the other hand, constraints
on experimental duration lead to a potentially very sparse sampling of the stimulus space, requiring
additional assumptions for a sensible interpolation. This problem is further exacerbated in experiments
in awake animals where the sampling of the stimulus space is driven by the animal behavior. The
few solutions for nonlinear tuning properties rely on spline-based approximation of one-dimensional
functions (for position on a linear track) [6] or assume a log-Gaussian Cox process generative model
as a way to enforce smoothness of 2D functional maps [3?5]. These methods are usually restricted to
at most two input dimensions (but see [4]).
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Here we take advantage of recent advances in scaling GP inference and learning using Kronecker
methods [7] to extend the approach in [3] to the multidimensional setting,
keeping the
while
comd+1
2
putational and memory requirements almost linear in dataset size N , O dN d and O dN d ,
respectively (for d dimensions) [8]. Our formulation requires a discretization of the input space,1
but allows for a flexible selection of the kernels specifying different assumptions about the nature
of the functional dependencies we are looking for in the data, with hyperparameters inferred by
maximizing marginal likelihood. We deal with the non-gaussian likelihood in the traditional way by
using a Laplace approximation of the posterior [8]. The critical ingredient for our approach is the
particular form of the covariance matrix that decomposes into a Kronecker product over covariances
corresponding to individual input dimensions, dramatically simplifying computations. The focus
here is not on the methods per se but rather on their previously unacknowledged utility for estimating
multidimensional nonlinear tuning functions.
The inferred tuning functions are probabilistic. The estimator is adaptive, in the sense that it relies
strongly on the prior in regions of the input space where data is scarce, but can flexibly capture
complex input-output relations where enough data is available. It naturally comes equipped with error
bars which can be used for instance for detecting shifts in receptive field properties due to learning.
Using artificial data we show that inference and learning in our model can robustly recover the
underlying structure of neural responses even in the experimentally realistic setting where the
sampling of the input space is sparse and strongly non-uniform (due to stereotyped animal behavior).
We further argue for the utility of spectral mixture kernels as a powerful tool for detecting complex
functional relationships beyond simple smoothing/interpolation. We go beyond artificial data that
follows the assumptions of the model exactly, and show robust estimation of tuning properties in
several experimental recordings. For illustration purposes we focus here on data from the CA1 region
of the hippocampus of rats, during an open field exploration task. We characterize several 3D tuning
functions as a function of the animal?s position but also additional internal (the overall activity in
the network at the time) or external variables (speed or direction of motion, time within experiment)
and use these to derive new insights into the distribution of spatial and non-spatial information at the
level of CA1 principal cell activity.
2
Methods
Generative model
Given data in the form of spike count ? input pairs D = {y (i) , x(i) }i=1:N , we model neural activity
as an inhomogeneous Poisson process with input-dependent firing rate ? (as in [3], see. Fig. 1a):
Y
1
P(y|x) =
Poisson y (i) ; ?(x)(i) ,
where Poisson (y; ?) = ?y e?? .
(1)
y!
i
The inputs x are defined on a d-dimensional lattice and the spike counts are measured within a time
window ?t for which the input is roughly constant (25.6ms, given by the frequency of positional
tracking).2 We formalize assumptions about neural tuning as a GP prior f ? GP(?, k? ), with
f = log ?(x), with a constant mean ?i = ? (for the overall scale of neural responses) and a
covariance function k(?, ?) with hyperparameters ?. This covariance function defines our assumptions
about what kind of functional dependencies are expected in the data (smoothness, periodicity, etc.).
The exponential linking f to ? provides a mathematically convenient way to enforce positivity of the
mean firing while keeping the posterior log-concave in f , justifiying the use of Laplace methods for
approximating the posterior (see also [3]).
For computational tractability we restrict our model to the class of product kernels k(x, x0 ) =
Q
0
d kd (xd , xd ) for which the covariance matrix decomposes as a Kronecker product K = K1 ?
K2 ? . . . Kd , allowing for efficient computation of determinants, matrix multiplications and eigendecomposition in terms of the individual factors Ki (see Suppl.Info. and [7]).
The individual kernels can be tailored to the specific application, allowing for a flexible characterization of individual input dimensions (inputs need not live in the same space, e.g. space-time, or can be
1
In practice many input dimensions are discrete to begin with (e.g. measurements of an animal?s position), so
this is a weak requirement. The coarseness of the discretization depends on the application.
2
Input noise is ignored here, but could be explicitly incorporated in the generative model [9].
2
a
b
f (x) ? GP (?? (x), k? (?, ?))
Poisson
x
y
D-dimensional
input
neural
response
true field
trajectory
estimate histogram
c
estimate GP
rate (Hz)
10
ground truth
estimate
0
d
data subgroups, 6min each
time
Figure 1: Model overview and estimator validation. a) Generative model: spike counts arise as
Poisson draws with an input dependent mean, f (x) with an exponential linkage function. b) A GP
prior specifies the assumptions concerning the properties of this function (smoothness, periodicity,
etc). c) Place field estimates from artificial data; left to right: the position of the animal modelled as
a bounded random walk, ground-truth, traditional estimate (without smoothing), posterior mean of
the inferred functional. d) Vertical slice through the posterior with shaded area showing the 2 ? sd
confidence region. d) Estimates of place field selectivity in an example CA1 recording during open
field exploration in a cross-shaped box; separate estimates for 6min subsets.
periodic, e.g. the phase of theta oscillations). Here we use a classic squared-exponential (SE kernel
2
(x?x0 )
for simple interpolation/smoothing tasks, kd (x, x0 ) = ?2d exp 2?2 , with parameters ? = {?, ?}
d
specifying the output variance and lengthscale [9]. For tasks involving extrapolation or discovering
complex patterns we use spectral mixture (SM) kernels, as a powerful and mathematically tractable
route towards automated kernel design [10]. SMs are stationary kernels defined as a linear mixture of
basis functions in the spectral domain:
0
kd (x, x ) =
Q
X
q=1
wq exp ?2? 2 (x ? x0 )2 vq cos(2?(x ? x0 )?q )
(2)
with parameters ? = {w, ?, v} defining the weights, spectral means and variances for each of the
mixture components. Assuming Q is large enough, such a spectral mixture can approximate any
arbitrary kernel (the same way Gaussian mixtures can be used to approximate an arbitrary density).
Moreover, many traditional kernels can be recovered as special cases; for instance the SE kernel
corresponds to a single component spectral density with zero mean (see also [10]).
Inference and learning
We sketch the main steps of the derivation here and provide the details in the Suppl. Info. Our goal is
to find the hyperparameters ? = {?, ?} that maximize P (?|y) ? P (y|?) ? P(?). We follow common
practice in using a point estimate ?? = argmax? P (?|y) for the hyperparameters, and leave a fully
probabilistic treatment to future work (e.g. using [11]). We use ?? to infer a predictive distribution
P(f ? |D, x? , ?? ) for a set of test inputs x? . Because of the Poisson observation noise these quantities
do not have simple closed form solutions and some approximations are required. As it is customary
[9], we use the Laplace method to approximate the log posterior log P(f |D) = log P(y|f ) + log P(f )
with its second-order Taylor expansion around the maximum ?f . This results in a multivariate Gaussian
?1
approximate posterior, with mean ?f and covariance H + K ?1
, where H = ??? log P(y|f ) |?f
is the Hessian of the log likelihood, and K is the covariance matrix. Substituting the approximate
posterior, we obtain the Laplace approximate marginal likelihood of the form:
log (y|?) = log P(y|?f ) ? 0.5z0 K ?1 z ? 0.5 log |I + KH|
(3)
3
with z = K ?1 (?f ? ?). The approximate predictive distribution for ?? is a multivariate Gaussian with
?1
mean k? ? log P(y|?f ) and covariance k?? ? k0? H ?1 + K
k? , where k? and k?? correspond
to the test-data and test-test covariances, respectively [8]. Lastly the predicted tuning function for
an individual test point ?? ) = exp(f ? ), is log-normal with closed-form expressions for mean and
variance (see Suppl. Info.).
Standard
methods for implementing
these computations using the Cholesky decomposition require
O N 3 computations and O N 2 memory, restricting their use to a few hundred data points. The
efficient implementation proposed here relies on the Kronecker structure of the covariance matrix
(which makes eigenvalue decomposition and matrix vector products very fast, see Suppl.Info.),
with linear conjugate gradients optimization and a lower bound on the marginal likelihood
for
d+1
hyperparameter learning. The predictive distribution can be efficiently evaluated in O dN d
(with a hidden constant given by the number of Newton steps needed for convergence, cf. [8]) Our
implementation is based on the gpml library [9] and the code is available online. A more detailed
description of the algorithmic details is provided in the Suppl. Info. In practice, this means that
it takes minutes on a laptop to estimate a 3D field for a 30min dataset (2-5min depending on the
coarseness of the grid), with a traditional 2D field estimated in 20-30sec.
3
Results
Estimator validation
We first validated our implementation on artificial data with known statistics.3 We defined a circular
arena with 1m diameter and simulated the animal?s behavior as a random walk with reflective bounds
(Fig. 1 b, left panel). This random process would eventually uniformly cover the space, but for
short sessions it yields occupancy maps similar to those seen in real data. We calibrated diffusion
parameters to roughly match CA1 statistics (average speed 5cm/sec, peak firing 5-10Hz, 10-30min
long sessions). Inferring the underlying place field was already robust with 10min sessions, with the
posterior mean f ? close to the ground truth (SE kernel, see Fig. 1 c). In comparison, the traditional
histogram-based estimates is quite poor (Fig. 1 b, left panel), though it can potentially be improved
by gaussian smoothing at the right spatial scale (although not without caveats, see Suppl. Info.).
It is more difficult to quantify the effects of the various approximations on real data where the
assumptions of the model are not matched exactly. Our approach was to check the robustness of the
GP-based estimates on subsets of the data constructed by combining every 5th data point (see left
panel in Fig. 1 d). This partitioning was designed to ensure that subsets are as statistically similar as
possible, sharing slow fluctuations in responses (e.g. due to variations in attentional levels, or changes
in behavior). An example cell?s response is shown in Fig. 1 d. Our analysis revealed robust field
estimation in most cells, provided they were reasonably active during the session (with mean firing
rates >0.1Hz; we discarded the non-responsive cells from subsequent analyses).
stereotyped behaviour
extrapolation
true field
estimate histogram estimate GP (kSM)
trajectory
estimate histogram estimate GP (kSM)
Figure 2: Spectral mixture kernels for modelling complex structure. We use artificial data with
hexagonal grid structure mimicking MEC responses. Extrapolation task: the animal?s position is
restricted to the orange delimited region of the environment. Stereotyped behavior: the simulated
animal performs a bounded random walk within an annulus . In both cases, we recover the full field,
beyond these borders (GP estimate) using a spectral mixture kernel (kSM).
3
Here we show a 2D example for simplicity; we obtained very similar results with 3D artificial inputs.
4
Spectral mixture kernels for complex functional dependencies
Place field estimation is relatively easy in a traditional open field exploration session (30min). The
main challenge is getting robust estimates on the time scale of a few minutes (e.g. in order to be
able to detect changes due to learning), which we have seen a GP-based estimator can do well.
A much more difficult problem is detecting tuning properties in a cheeseboard memory task [12].
What distinguishes this setup is that fact that the animal quickly discovers the location of the wells
containing rewards, after which its running patterns become highly stereotypical, close to the shortest
path that traverses the reward locations. While it is hard to figure out place field selectivity for
locations that the animal never visits, GP-based estimators may have an advantage compared to
traditional methods when functional dependencies are structured, as is the case for grid cells in the
medial enthorinal cortex (MEC) [13, 14]. When tuning properties are complex and structured we can
exploit the expressive power of spectral mixture kernels (SM) to make the most of very limited data.
We simulated two versions of this scenario. First, we defined an extrapolation task in which the
animal?s behaviour is restricted to a subregion of the environment (marked by orange lines in the
2nd panel of Fig. 2) but we want to infer the spatial selectivity outside these borders. The second
scenario attempts to mimic the animal running patterns in a cheeseboard maze (after learning) by
restricting the trajectory within a ring (random walk with reflective boundaries in both cases). Using a
5 component spectral mixture kernel we were able to fully reconstruct the hexagonal lattice structure
of the true field despite the size of the observed region covering only about 2 times the length scale of
the periodic pattern. In contrast, traditional methods (including GP-based inference with standard
SE kernels) would fail completely at such extrapolation. While such complex patterns of spatial
dependence are restricted to MEC (and the estimator is probably best suited for ventral MEC, where
grids have a small length scale [15]) it is conceivable that such extrapolation may also be useful in
the temporal domain, or more generally for cortical responses in neurons which have so far eluded a
simple functional characterization.
Spatial and non-spatial modulation of CA1 responses
To explore the multidimensional characterization of principal cell responses in CA1 we constructed
several 3D estimators where the input combines the position of the animal within a 2D environment
with an additional non-spatial variable.4 The first non-spatial variable we considered is the network
PNneurons
state, quantified as the population spike count, k =
yi (naturally a discrete variable
i=1
between 0 and some kmax ). This quantity provides a computationally convenient proxy for network
oscillations and has been recently used in a series of studies on the statistics of population activity
in the retina and cortex [16?19]. Second, we considered the animal?s speed and direction of motion
(with a coarse discretization), motivated by past work on non-spatial modulation of place fields on
linear tracks [20]. Third, we also considered input variable t measuring time within a session (SE
kernel; 3-5 min windows), as a way to examine the stability of spatial tuning over time. For all
analyses, positional information was discretized on a 32 ? 32 grid, corresponding to a spacing of
2.5cm, comparable to the binning resolution used in traditional place field estimates. The animal
speed (estimated from the positional information with 250ms temporal smoothing) varied between 0
and about 25cm/sec, with a very skewed distribution (not shown). Small to medium variations in the
coarseness of the discretization did not qualitatively affect the results although the choice of prior
becomes more important on the tail of the speed distribution, where data is scarce.
The resulting 3D tuning functions are shown in Fig. 3 for a few example neurons. First, network state
modulates the place field selectivity in most CA1 neurons in our recordings. The typical modulation
pattern is a monotonic increase in firing with k (Fig. 3, a, top), although we also found k-dependent
flickering in a minority of the cells (Fig. 3a, middle), and very rarely k invariance (Fig. 3a, bottom).
Rate remapping is also the dominant pattern of speed-dependent modulation in our data set (Fig. 3b).
In terms of place field stability over time, about half the cells were stable during a 30min session in a
familiar environment, with occasionally higher firing rates at the very beginning of the trial (Fig. 3c,
top), while the rest showed fluctuation in representations (Fig. 3c, bottom). Results shown for 5min
windows, but results very similar for 3min.
4
We chose to estimate multiple 3D fields rather than jointly conditioning on all variables mainly for simplicity;
this strategy has the added bonus of providing sanity checks for the quality of the different estimates.
5
a
3D estimate GP
traditional
place field
k=26
network state
k=0
time
c
5min
time
speed
k
cell1
e
30min
5min
d
3rd
dimension
>10cm/s
30min
spatial marginals
cell2
cell3
cell4
cell5
f
firing rate (Hz)
3
10
nonspatial modulation
0cm/s
60
network state
0
k=0
6
k=26
speed
0cm/sec
5
0
5min
30min
spatial information
MI(y,x)
temporal instability,
MI(y,t)
0.6
0.4
1
0
0
familiar
10
0
>10cm/sec
time
2
novel
speed
b
0.2
0
1
2
familiar
3
0
0
0.2
0.4 0.6
familiar
Figure 3: Estimating 3D response dependences in CA1 cells. a) Conditional place fields when
constraining the network state, defined by the average population activity k. c) Conditional place
fields as a function of the time within a 30min session, used to assess the stability of the representation.
In all cases, the rightmost field corresponds to the traditional place field ignoring the 3rd dimension.
d) Sanity check: marginal statistics of the place field selectivity obtained independently from the 3D
fields in 5 example cells. e) Population summary of the degree of modulation of spatial selectivity
by non-spatial variables; see text for details. f) Within comparison of cell properties during the
exploration of a familiar vs. a novel environment.
As a sanity check of our 3D estimators? quality, we independently computed the traditional place
field by marginalizing out the 3rd dimension for each of our 3D estimates. We used the empirical
distribution as a prior for the non-spatial dimensions, and an uniform prior for space. Reassuringly
we find that the estimates computed after marginalization are very close to the simple 2D place field
map in all but 2 cells, which we exclude from the next analysis (examples in Fig. 3d). This provides
additional confidence in the robustness of the estimator in the multidimensional case.
Since we have a closed form expression for the map between stimulus dimensions and neural
responses, we can estimate the mutual information between neural activity and various input variables
as a way to dissect their contribution to coding. First, we visualize the modulation of spatial
selectivity by the non-spatial variable as the spatial information conditioned on the 3rd variable,
normalized by the marginal spatial information, MI(x,y|z)
MI(x,y) , with z generically denoting any of the
non-spatial variables (approximate closed form expression given f and Poisson observation noise).
We see monotonic increases in spatial information with k (Fig. 3e, top), and speed (Fig. 3e, top)
at the level of the population, and a weak decrease in spatial information over time (possibly due
to higher speeds at the beginning of the session, combined with heightened attention/motivation
levels). In terms of the division of spatial vs. non-spatial information across cells, we found that
space selective cells have weaker k-modulation (Spearman corr(MI(y, x), MI(y, k) = ?0.17). This
however does not exclude the possibility that theta-coupled cells have additional spatial information
at the fine temporal scale. Additionally, there is little correlation between the coding of position and
speed (corr(MI(y, x), MI(y, speed) = ?0.03), suggesting that the encoding of the two is relatively
orthogonal at the level of the population. Somewhat unexpectedly, we found a cell?s temporal stability
to be largely independent of its spatial selectivity corr(MI(y, x), MI(y, t) = ?0.04).
6
Motivated by recent observations that the overall excitability of cells may be predictive of both their
spatial selectivity and of the rigidity of their representation [21], we compared the overall firing rate
of the cells with their spatial and non-spatial selectivity. We found relatively strong dependencies,
with positive correlations between firing rate and spatial information (cc = 0.21), network influence
(cc = 0.43) and the cell?s stability (cc = 0.38). When comparing these quantities in the same cells
as the animal visits a familiar or a novel environment (93 cells, 20min in each environment) we
found additional nontrivial dependences between spatial and non-spatial tuning. Although the overall
firing rates of the cells are remarkably preserved across conditions (reflecting general cell excitability,
cc = 0.66), the subpopulation of cells with strong spatial selectivity is largely non-overlapping
across environments (corr(MIfam (y, x), MInov (y, x) = 0.07). Moreover, the temporal stability of
the representation is also environment specific (corr(MIfam (y, t), MInov (y, t) = ?0.04). Overall,
these results paint a complex picture of hippocampal coding, the implications of which need further
empirical and theoretical investigation.
Lastly, we studied the dependence of CA1 responses on the animal?s direction of motion. Although
directional selectivity is well documented on a linear track [20] it remains unclear if a similar
behavior occurs in a 2D environment. The main challenge comes from the poor sampling of the
position?direction-of-motion input space, something which our methods can handle readily. To
construct directionally selective place field estimates in 2D we took inspiration from recent analyses
of 2D phase procession [22] conditioning the responses on the main direction of motion within the
place field. Specifically, we used our estimation of a traditional 2D place field to define a region of
interest (ROI) that covers 90% of the field for each cell (Fig. 4. We isolated all trajectory segments
that traverse this ROI and classified them based on the primary direction of motion along the cardinal
orientations. We then computed place field estimates for each direction, with data outside the ROI
shared across conditions. To avoid artefacts due to the stereotypical pattern of running along the
box borders, we restricted this analysis to cells with fields in the central part of the environment (10
cells). A set of representative examples for the resulting directional fields are shown in Fig. 4d. We
found the fields to be largely invariant to direction of motion in our setup, with small displacements
in peak firing possibly due to differences between the perceived vs. the camera-based measurements
of position (see also [22]). Overall, these results suggest that, in contrast to linear track behavior,
CA1 responses are largely invariant to the direction of motion in an open field exploration task.
a
c
traditional field
d
directional fields (GP)
GP
histogram
cell1
b
cell6
Figure 4: Directional selectivity in CA1 cells. a) Cell specific ROI that covers the classic place
field (example corresponding to cell 6). b) Classification of the traversals of the region of interest
as a function of the primary direction of motion along the cardinal directions. Out of ROI data
shared across conditions. c) Traditional place field estimates for example CA1 cells and d) their
corresponding direction-specific tuning.
7
4
Discussion
Strong constraints on experiment duration, poor sampling of the stimulus space and additional sources
of variability that are not under direct experimental control make the estimation of tuning properties
during awake behavior particularly challenging. Here we have shown that recent advances on fast GP
inference based on Kronecker methods allow for a robust characterization of multidimensional nonlinear tuning functions, which was inaccessible to traditional methods. Furthermore, our estimators
inherit all the advantages of a probabilistic approach, including a principled way of dealing with the
non-uniform sampling of the input space and natural uncertainty estimates.
Our methods can robustly estimate place fields with one order of magnitude fewer data points.
Furthermore, they allow for more than two-dimensional inputs. While one could imagine it would
suffice to estimate separate place fields conditioned on each value of the non-spatial dimension, z,
the joint estimator has the advantage that it allows for smoothing across z values, borrowing strength
from well-sampled regions of the z space to make better estimates for poorly sampled z values.
Several related algorithms have been proposed in the literature [3?5], which vary primarily in how
they handle the tradeoff between kernel flexibility and the computational time required for inference
and learning (see Table 1). At one extreme, [3] strongly restricts the nature of the covariance matrix to
nearest-neighbour interactions on a 2D grid (resulting in a band-diagonal inverse covariance matrix)
which allows them to exploit sparse matrix techniques to estimate the posterior mean in linear time.
At the other extreme,
[4, 5] allow for an arbitrary covariance structure, but are computationally
prohibitive, O N 3 . Our proposal sits between these extremes in that it achieves close-to-linear
computational and memory costs without significantly restricting the flexibility of the covariance
structure (for a better intuition of the effect of different covariances, see also Fig. S1). In particular, it
can be combined with powerful spectral mixture kernels to extract complex functional dependencies
that go beyond simple smoothing. This opens the door to a variety of previously inaccessible
tasks such as extrapolation. Moreover, it allows for an agnostic exploration of the neural responses
functional space, which could be used to discover novel tuning properties in cells for which coding is
poorly understood.
When applied to CA1 data, our multidimensional estimators revealed a complex picture of the
modulation of neural responses by spatial and non-spatial inputs in the hippocampus. First we
confirmed linear track results concerning the speed and oscillatory modulation of spatial tuning.
Furthermore, we revealed additional insights into the interaction between the representation of space
and these non-spatial dimensions, which go beyond the capabilities of traditional methods. Most
notably we found 1) a mostly orthogonal representation of speed and position, that 2) place field
stability cannot be easily explained in terms of cell excitability or spatial selectivity, although 3) it is
environment specific. Lastly, while we showed 2D place field maps to be direction-invariant in an
open field exploration task, more interesting directional dependencies may be revealed in other 2D
tasks, where the direction of motion is behavioraly more relevant (e.g. cheeseboard). Importantly,
there is nothing hippocampus-specific in the methodology. Hence fast GP inference using Kronecker
methods, combined with expressive kernels, may provide a general-purpose tool for characterizing
neural responses across brain regions.
Table 1: Summary comparison of different estimators.
Algorithm
Kernel function
Computing cost
Memory cost
Rad et al. 2010 [3]
sparse banded inverse covariance
SE, any in principle
O (N )
O (N )
Park et al. 2014 [4]
Savin & Tkacik
SE and SM, works for
any tensor-product
O N 3
O dN
d+1
d
O N 2
2
O dN d
Data
size
105
< 103
105
Acknowledgments
We thank Jozsef Csicsvari for kindly sharing the CA1 data. This work was supported by the
People Programme (Marie Curie Actions) of the European Union?s Seventh Framework Programme
(FP7/2007-2013) under REA grant agreement no. 291734.
8
References
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9
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5,696 | 6,154 | A Simple Practical Accelerated Method for Finite
Sums
Aaron Defazio
Ambiata, Sydney Australia
Abstract
We describe a novel optimization method for finite sums (such as empirical risk
minimization problems) building on the recently introduced SAGA method. Our
method achieves an accelerated convergence rate on strongly convex smooth problems. Our method has only one parameter (a step size), and is radically simpler
than other accelerated methods for finite sums. Additionally it can be applied
when the terms are non-smooth, yielding a method applicable in many areas where
operator splitting methods would traditionally be applied.
Introduction
A large body of recent developments in optimization have focused on minimization of convex finite
sums of the form:
n
1X
f (x) =
fi (x),
n i=1
a very general class of problems including the empirical risk minimization (ERM) framework as a
special case. Any function h can be written in this form by setting f1 (x) = h(x) and fi = 0 for
i 6= 1, however when each fi is sufficiently regular in a way that can be made precise, it is possible to
optimize such sums more efficiently than by treating them as black box functions.
In most cases recently developed methods such as SAG [Schmidt et al., 2013] can find an -minimum
faster than either stochastic gradient descent or accelerated black-box approaches, both in theory and
in practice. We call this class of methods fast incremental gradient methods (FIG).
FIG methods are randomized methods similar to SGD, however unlike SGD they are able to achieve
linear convergence rates under Lipschitz-smooth and strong convexity conditions [Mairal, 2014,
Defazio et al., 2014b, Johnson and Zhang, 2013, Kone?cn? and Richt?rik, 2013]. The linear rate in the
first wave of FIG methods directly depended on the condition number L/? of the problem, whereas
recently several methods have been developed that depend on the square-root of the condition number
[Lan and Zhou, 2015, Lin et al., 2015, Shalev-Shwartz and Zhang, 2013c, Nitanda, 2014], at least
when n is not too large. Analogous to the black-box case, these methods are known as accelerated
methods.
In this work we develop another accelerated method, which is conceptually simpler and requires
less tuning than existing accelerated methods. The method we give is a primal approach, however it
makes use of a proximal operator oracle for each fi instead of a gradient oracle, unlike other primal
approaches. The proximal operator is also used by dual methods such as some variants of SDCA
[Shalev-Shwartz and Zhang, 2013a].
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
Algorithm 1
Pick some starting point x0 and step size ?. Initialize each gi0 = fi0 (x0 ), where fi0 (x0 ) is any
gradient/subgradient at x0 .
Then at step k + 1:
1. Pick index j from 1 to n uniformly at random.
2. Update x:
"
#
n
X
1
zjk = xk + ? gjk ?
gk ,
n i=1 i
xk+1 = prox?j zjk .
3. Update the gradient table: Set gjk+1 = ?1 zjk ? xk+1 , and leave the rest of the entries
unchanged (gik+1 = gik for i 6= j).
1
Algorithm
Our algorithm?s main step makes use of the proximal operator for a randomly chosen fi . For
convenience, we define:
1
2
prox?i (x) = argminy ?fi (y) + kx ? yk .
2
This proximal operator can be computed efficiently or in closed form in many cases, see Section 4 for
details. Like SAGA, we also maintain a table of gradients gi , one for each function fi . We denote the
state of gi at the end of step k by gik . The iterate (our guess at the solution) at the end of step k is
denoted xk . The starting iterate x0 may be chosen arbitrarily.
Pn
The full algorithm is given as Algorithm 1. The sum of gradients n1 i=1 gik can be cached and
updated efficiently at each step, and in most cases instead of storing a full vector for each gi , only a
single real value needs to be stored. This is the case for linear regression or binary classification with
logistic loss or hinge loss, in precisely the same way as for standard SAGA. A discussion of further
implementation details is given in Section 4.
With step size
q
(n ? 1)2 + 4n L
?
1 ? n1
,
2Ln
2L
the expected convergence rate in terms of squared distance to the solution is given by:
k
2
??
?+L
x0 ? x?
2 ,
E
xk ? x?
? 1 ?
1 + ??
?
?
=
?
when each fi : Rd ? R is L-smooth and ?-strongly convex. See Nesterov [1998] for definitions of
these conditions. Using big-O notation, the number of steps required to reduce the distance to the
solution by a factor is:
s
!
!
nL
1
k=O
+ n log
,
?
as ? 0. This rate matches the lower bound known for this problem [Lan and Zhou, 2015] under the
gradient oracle. We conjecture that this rate is optimal under the proximal operator oracle as well.
Unlike other accelerated approaches though, we have only a single tunable parameter (the step size
?), and the algorithm doesn?t need knowledge of L or ? except for their appearance in the step size.
Compared to the O ((L/? + n) log (1/)) rate for SAGA and other non-accelerated FIG methods,
accelerated FIG methods are significantly faster when n is small compared to L/?, however for
n ? L/? the performance is essentially the same. All known FIG methods hit a kind of wall at
n ? L/?, where they decrease the error at each step by no more than 1 ? n1 . Indeed, when n ? L/?
the problem is so well conditioned so as to be easy for any FIG method to solve it efficiently. This is
sometimes called the big data setting [Defazio et al., 2014b].
2
Our convergence rate can also
be compared
to that of
optimal first-order black box methods, which
p
have rates of the form k = O
L/? log (1/) per epoch equivalent. We are able to achieve
?
a n speedup on a per-epoch basis, for n not too large. Of course, all of the mentioned rates are
significantly better than the O ((L/?) log (1/)) rate of gradient descent.
For non-smooth but strongly convex problems, we prove a 1/-type rate under a standard iterate
averaging scheme. This rate does not require the use of decreasing step sizes, so our algorithm
requires less tuning than other primal approaches on non-smooth problems.
2
Relation to other approaches
Our method is most closely related to the SAGA method. To make the relation clear, we may write
our method?s main step as:
"
#
n
1X k
k+1
k
0 k+1
k
x
= x ? ? fj (x
) ? gj +
g ,
n i=1 i
whereas SAGA has a step of the form:
"
x
k+1
k
=x ??
fj0 (xk )
?
gjk
#
n
1X k
g .
+
n i=1 i
The difference is the point at which the gradient of fj is evaluated at. The proximal operator has the
effect of evaluating the gradient at xk+1 instead of xk . While a small difference on the surface, this
change has profound effects. It allows the method to be applied directly to non-smooth problems
using fixed step sizes, a property not shared by SAGA or other primal FIG methods. Additionally, it
allows for much larger step sizes to be used, which is why the method is able to achieve an accelerated
rate.
It is also illustrative to look at how the methods behave at n = 1. SAGA degenerates into regular
gradient descent, whereas our method becomes the proximal-point method [Rockafellar, 1976]:
xk+1 = prox?f (xk ).
The proximal point method has quite remarkable properties. For strongly convex problems, it
converges for any ? > 0 at a linear rate. The downside being the inherent difficulty of evaluating
the proximal operator. For the n = 2 case, if each term is an indicator function for a convex set, our
algorithm matches Dykstra?s projection algorithm if we take ? = 2 and use cyclic instead of random
steps.
Accelerated incremental gradient methods
Several acceleration schemes have been recently developed as extensions of non-accelerated FIG
methods. The earliest approach developed was the ASDCA algorithm [Shalev-Shwartz and Zhang,
2013b,c]. The general approach of applying the proximal-point method as the outer-loop of a doubleloop scheme has been dubbed the Catalyst algorithm Lin et al. [2015]. It can be applied to accelerate
any FIG method. Recently a very interesting primal-dual approach has been proposed by Lan and
Zhou [2015]. All of the prior accelerated methods are significantly more complex than the approach
we propose, and have more complex proofs.
3
3.1
Theory
Proximal operator bounds
In this section we rehash some simple bounds from proximal operator theory that we will use in
this work. Define the short-hand p?f (x) = prox?f (x), and let g?f (x) = ?1 (x ? p?f (x)), so that
p?f (x) = x ? ?g?f (x). Note that g?f (x) is a subgradient of f at the point p?f (x). This relation is
known as the optimality condition of the proximal operator. Note that proofs for the following two
propositions are in the supplementary material.
3
Notation
xk
x?
?
p?f (x)
Description
Current iterate at step k
Solution
Step size
Short-hand in results for generic f
prox?i (x)
gik
gi?
vi
Proximal operator of ?fi at x
j
zjk
Additional relation
xk ? R d
x? ? R d
p?f (x)
n = prox?f (x)
= argminy ?fi (y) +
A stored subgradient of fi as seen at step k
A subgradient of fi at x?
vi = x? + ?gi?
Chosen componentindex (random variable)
Pn
zjk = xk + ? gjk ? n1 i=1 gik
1
2
2
kx ? yk
o
Pn
?
i=1 gi = 0
x? = prox?i (vi )
xk+1
= prox?j zjk
j
Table 1: Notation quick reference
Proposition 1. (Strengthening firm non-expansiveness under strong convexity) For any x, y ? Rd ,
and any convex function f : Rd ? R with strong convexity constant ? ? 0,
2
hx ? y, p?f (x) ? p?f (y)i ? (1 + ??) kp?f (x) ? p?f (y)k .
In operator theory this property is known as (1 + ??)-cocoerciveness of p?f .
Proposition 2. (Moreau decomposition) For any x ? Rd , and any convex function f : Rd ? R
with Fenchel conjugate f ? :
p?f (x) = x ? ?p ?1 f ? (x/?).
(1)
Recall our definition of g?f (x) = ?1 (x ? p?f (x)) also. After combining, the following relation thus
holds between the proximal operator of the conjugate f ? and g?f :
p ?1 f ? (x/?) =
1
(x ? p?f (x)) = g?f (x).
?
(2)
Theorem 3. For any x, y ? Rd , and any convex L-smooth function f : Rd ? R:
1
hg?f (x) ? g?f (y), x ? yi ? ? 1 +
L?
2
kg?f (x) ? g?f (y)k ,
Proof. We will apply cocoerciveness of the proximal operator of f ? as it appears in the decomposition.
Note that L-smoothness of f implies 1/L-strong convexity of f ? . In particular we apply it to the
points ?1 x and ?1 y:
Pulling
3.2
2
1
1
1
? 1+
p 1 ? ( x) ? p ?1 f ? ( y)
.
L?
? f ?
?
1
1
1
1
p ?1 f ? ( x) ? p ?1 f ? ( y), x ? y
?
?
?
?
1
?
from the right side of the inner product out, and plugging in Equation 2, gives the result.
Notation
Let x? be the unique minimizer (due to strong convexity) of f . In addition to the notation used in the
description of the algorithm, we also fix a set of subgradients gj? , one for each of fj at x? , chosen
Pn
such that j=1 gj? = 0. We also define vj = x? + ?gj? . Note that at the solution x? , we want to
apply a proximal step for component j of the form:
x? = prox?j x? + ?gj? = prox?j (vj ) .
4
Lemma 4. (Technical lemma needed by main proof) Under Algorithm 1, taking the expectation over
the random choice of j, conditioning on xk and each gik , allows us to bound the following inner
product at step k:
* "
#
"
#
+
n
n
1X k
1X k
k
?
k
?
k
?
E ? gj ?
g ? ?gj , x ? x + ? gj ?
g ? ?gj
n i=1 i
n i=1 i
n
? ?2
1 X
g k ? gi?
2 .
n i=1 i
The proof is in the supplementary material.
3.3
Main result
Theorem 5. (single step Lyapunov descent) We define the Lyapunov function T k of our algorithm
(Point-SAGA) at step k as:
n
c X
gik ? gi?
2 +
xk ? x?
2 ,
Tk =
n i=1
q
(n?1)2 +4n L
?
1? 1
? 2Ln , the expectation of T k+1 , over the
for c = 1/?L. Then using step size ? =
2Ln
k
k
random choice of j, conditioning on x and each gi , is:
??
,
E T k+1 ? (1 ? ?) T k for ? =
1 + ??
when each fi : Rd ? R is L-smooth and ?-strongly convex and 0 < ? < L. This is the same
Lyapunov function as used by Hofmann et al. [2015].
Proof. Term 1 of T k+1 is straight-forward to simplify:
n
n
c X
1 c X
k+1
?
2
gik ? gi?
2 + c E
g k+1 ? gj?
2 .
E
gi ? gi = 1 ?
j
n i=1
n n i=1
n
For term 2 of T k+1 we start by applying cocoerciveness (Theorem 1):
2
(1 + ??)E
xk+1 ? x?
2
(1 + ??)E
prox?j (zjk ) ? prox?j (vj )
? E prox?j (zjk ) ? prox?j (vj ), zjk ? vj
= E xk+1 ? x? , zjk ? vj .
=
Now we add and subtract xk :
= E xk+1 ? xk + xk ? x? , zjk ? vj
= E xk ? x? , zjk ? vj + E xk+1 ? xk , zjk ? vj
2
=
xk ? x?
+ E xk+1 ? xk , zjk ? vj ,
where we have pulled out the quadratic term by using E[zjk ? vj ] = xk ? x? (we can take the
expectation
since the left
hand side of the inner product doesn?t depend on j). We now expand
E xk+1 ? xk , zjk ? vj further:
E xk+1 ? xk , zjk ? vj
= E xk+1 ? ?gj? + ?gj? ? xk , zjk ? vj
*
"
#
n
1X k
k+1
k
k
g ? ?gj? + ?gj? ? xk ,
= E x ? ?gj + ? gj ?
n i=1 i
"
#
+
n
1X k
k
?
k
?
x ? x + ? gj ?
g ? ?gj .
(3)
n i=1 i
5
We further split the left side of the inner product to give two separate inner products:
"
+
#
#
* "
n
n
1X k
1X k
?
k
?
k
?
k
= E ? gj ?
g ? ?gj , x ? x + ? gj ?
g ? ?gj
n i=1 i
n i=1 i
*
"
#
+
n
1X k
k+1
?
k
?
k
?
+ E ?gj ? ?gj , x ? x + ? gj ?
g ? ?gj .
n i=1 i
(4)
The first inner product in Equation 4 is the quantity we bounded in Lemma 4 by
2
Pn
? 2 n1 i=1
gik ? gi?
. The second inner product in Equation 4, can be simplified using Theorem 3 (note the right side of the inner product is equal to zjk ? vj ):
2
k+1
1
?
k
2
E
gjk+1 ? gj?
.
??E gj ? gj , zj ? vj ? ?? 1 +
L?
k+1
2
Combing these gives the following bound on (1 + ??)E
x
? x?
:
n
X
k
k+1
2
k
1
21
?
2
?
2
?
2
2
+?
? x ?x
?x
(1+??)E x
E
gjk+1 ? gj?
.
gi ? gi ?? 1 +
n i=1
L?
??
1
Define ? = 1+??
. Now we multiply the above inequality through by ?
= 1 ? ?, where ? = 1+??
and combine with the rest of the Lyapunov function, giving:
n
c 1 X
gik ? gi?
2
E T k+1 ? T k + ?? 2 ?
n n i
c
2
2
??
? ?? 2 ?
+
E
gjk+1 ? gj?
? ?E
xk ? x?
.
n
L
We want an ? convergence rate, so we pull out the required terms:
n
c 1 X
gik ? gi?
2
E T k+1 ? ?T k + ?? 2 + ?c ?
n n i
c
2
??
+
? ?? 2 ?
E
gjk+1 ? gj?
.
n
L
q
(n?1)2 +4n L
?
1? 1
Now to complete the proof we note that c = 1/?L and ? =
? 2Ln ensure that both
2Ln
k+1
k
terms inside the round brackets are non-positive, giving ET
? ?T . These constants were found
by equating the equations in the brackets to zero, and solving with respect to the two unknowns, ?
and c. It is easy to verify that ? is always positive, as a consequence of the condition number L/?
always being at least 1.
Corollary 6. (Smooth case) Chaining Theorem 5 gives a convergence rate for Point-SAGA at step k
under the constants given in Theorem 5 of:
2
k ?+L
x0 ? x?
2 ,
E
xk ? x?
? (1 ? ?)
?
if each fi : Rd ? R is L-smooth and ?-strongly convex.
Theorem
case) Suppose each fi : Rd ? R is ?-strongly convex,
gi0 ? gi?
? B
0 7. ?(Non-smooth
?
and
x ? x
? R. Then after k iterations of Point-SAGA with step size ? = R/B n:
?
?
k
2
n (1 + ? (R/B n))
? ? x?
? 2
E
x
RB,
?k
Pk
where x
?k = k1 E t=1 xt . The proof of this theorem is included in the supplementary material.
6
4
Implementation
Care must be taken for efficient implementation, particularly in the sparse gradient case. We discuss
the key points below. A fast Cython implementation is available on the author?s website incorporating
these techniques.
Proximal operators For the most common binary classification and regression methods, implementing the proximal operator is straight-forward. We include details of the computation
of the proximal operators for the hinge, square and logistic losses in the supplementary
material. The logistic loss does not have a closed form proximal operator, however it may
be computed very efficiently in practice using Newton?s method on a 1D subproblem. For
problems of a non-trivial dimensionality the cost of the dot products in the main step is
much greater than the cost of the proximal operator evaluation. We also detail how to handle
a quadratic regularizer within each term?s prox operator, which has a closed form in terms
of the unregularized prox operator.
Initialization Instead of setting gi0 = fi0 (x0 ) before commencing the algorithm, we recommend
using gi0 = 0 instead. This avoids the cost of a initial pass over the data. In practical effect
this is similar to the SDCA initialization of each dual variable to 0.
5
Experiments
We tested our algorithm which we call Point-SAGA against SAGA [Defazio et al., 2014a], SDCA
[Shalev-Shwartz and Zhang, 2013a], Pegasos/SGD [Shalev-Shwartz et al., 2011] and the catalyst
acceleration scheme [Lin et al., 2015]. SDCA was chosen as the inner algorithm for the catalyst
scheme as it doesn?t require a step-size, making it the most practical of the variants. Catalyst applied
to SDCA is essentially the same algorithm as proposed in Shalev-Shwartz and Zhang [2013c]. A
single inner epoch was used for each SDCA invocation. Accelerated MISO as well as the primal-dual
FIG method [Lan and Zhou, 2015] were excluded as we wanted to test on sparse problems and
they are not designed to take advantage of sparsity. The step-size parameter for each method (? for
catalyst-SDCA) was chosen using a grid search of powers of 2. The step size that gives the lowest
error at the final epoch is used for each method.
We selected a set of commonly used datasets from the LIBSVM repository [Chang and Lin, 2011].
The pre-scaled versions were used when available. Logistic regression with L2 regularization was
applied to each problem. The L2 regularization constant for each problem was set by hand to ensure
f was not in the big data regime n ? L/?; as noted above, all the methods perform essentially the
same when n ? L/?. The constant used is noted beneath each plot. Open source code to exactly
replicate the experimental results is available at https://github.com/adefazio/point-saga.
Algorithm scaling with respect to n The key property that distinguishes accelerated FIG methods
from their non-accelerated counterparts is their performance scaling with respect to the dataset size.
For large datasets on well-conditioned problems we expect from the theory to see little difference
between the methods. To this end, we ran experiments including versions of the datasets subsampled
randomly without replacement in 10% and 5% increments, in order to show the scaling with n
empirically. The same amount of regularization was used for each subset.
Figure 1 shows the function value sub-optimality for each dataset-subset combination. We see that
in general accelerated methods dominate the performance of their non-accelerated counter-parts.
Both SDCA and SAGA are much slower on some datasets comparatively than others. For example,
SDCA is very slow on the 5 and 10% COVTYPE datasets, whereas both SAGA and SDCA are much
slower than the accelerated methods on the AUSTRALIAN dataset. These differences reflect known
properties of the two methods. SAGA is able to adapt to inherent strong convexity while SDCA can
be faster on very well-conditioned problems.
There is no clear winner between the two accelerated methods, each gives excellent results on each
problem. The Pegasos (stochastic gradient descent) algorithm with its slower than linear rate is a
clear loser on each problem, almost appearing as an almost horizontal line on the log scale of these
plots.
Non-smooth problems We also tested the RCV1 dataset on the hinge loss. In general we did not
expect an accelerated rate for this problem, and indeed we observe that Point-SAGA is roughly as
fast as SDCA across the different dataset sizes.
7
10?2
10?3
10?4
10?5
10?6
10?7
10?8
0
5
10
Epoch
15
20
100
10?1
10?2
10?3
10?4
10?5
10?6
10?7
10?8
10?9
Function Suboptimality
Function Suboptimality
Function Suboptimality
100
10?1
0
5
10
Epoch
15
20
101
100
10?1
10?2
10?3
10?4
10?5
10?6
10?7
10?8
10?9
10?10
10?11
10?12
0
5
10
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15
20
(a) COVTYPE ? = 2 ? 10?6 : 5%, 10%, 100% subsets
100
10?1
10?2
10?3
10?4
0
5
10
15
Epoch
20
25
10?1
10?2
10?3
10?4
30
100
Function Suboptimality
100
Function Suboptimality
Function Suboptimality
101
0
5
10
15
Epoch
20
25
10?1
10?2
10?3
10?4
10?5
10?6
10?7
10?8
30
0
5
10
15
Epoch
20
25
30
5
10
15
Epoch
20
25
30
Function Suboptimality
100
10?1
10?2
10?3
10
?4
0
5
10
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20
25
30
0
5
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20
25
30
100
10?1
10?2
10?3
10?4
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10?7
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0
101 (c) MUSHROOMS ? = 10?4 : 5%, 10%, 100% subsets
100
10
10
10 1
10
10
10 2
10
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3
10
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10
4
10
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10
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1510
20 525 30 35 40
0
5 10 15 20 25 30 35 40
0
5 10
Epoch
Epoch
10 6
(d)
5 ? 10?5 : 5%,
0 RCV1 with hinge
5 loss, ? =10
1510%, 100% subsets
20
Point-SAGA
Epoch SDCA
0
0
Function Suboptimality
5
100
10?1
10?2
10?3
10?4
10?5
10?6
10?7
10?8
10?9
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Function Suboptimality
Function Suboptimality
0
Function Suboptimality
100
10?1
10?2
10?3
10?4
10?5
10?6
10?7
10?8
10?9
10?10
Function Suboptimality
Function Suboptimality
(b) AUSTRALIAN ? = 10?4 : 5%, 10%, 100% subsets
?1
?2
?3
?4
?5
?1
?2
?3
?4
?5
Pegasos
SAGA
Catalyst-SDCA
Figure 1: Experimental results
8
15
20 25
Epoch
30
35
40
References
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9
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5,697 | 6,155 | Active Learning with Oracle Epiphany
Tzu-Kuo Huang ?
Uber Advanced Technologies Group
Pittsburgh, PA 15201
Ara Vartanian
University of Wisconsin?Madison
Madison, WI 53706
Saleema Amershi
Microsoft Research
Redmond, WA 98052
Lihong Li
Microsoft Research
Redmond, WA 98052
Xiaojin Zhu
University of Wisconsin?Madison
Madison, WI 53706
Abstract
We present a theoretical analysis of active learning with more realistic interactions
with human oracles. Previous empirical studies have shown oracles abstaining on
difficult queries until accumulating enough information to make label decisions.
We formalize this phenomenon with an ?oracle epiphany model? and analyze active
learning query complexity under such oracles for both the realizable and the agnostic cases. Our analysis shows that active learning is possible with oracle epiphany,
but incurs an additional cost depending on when the epiphany happens. Our results
suggest new, principled active learning approaches with realistic oracles.
1
Introduction
There is currently a wide gap between theory and practice of active learning with oracle interaction.
Theoretical active learning assumes an omniscient oracle. Given a query x, the oracle simply
answers its label y by drawing from the conditional distribution p(y | x). This oracle model is
motivated largely by its convenience for analysis. However, there is mounting empirical evidence
from psychology and human-computer interaction research that humans behave in far more complex
ways. The oracle may abstain on some queries [Donmez and Carbonell, 2008] (note this is distinct
from classifier abstention [Zhang and Chaudhuri, 2014, El-Yaniv and Wiener, 2010]), or their answers
can be influenced by the identity and order of previous queries [Newell and Ruths, 2016, Sarkar et al.,
2016, Kulesza et al., 2014] and by incentives [Shah and Zhou, 2015]. Theoretical active learning
has yet to account for such richness in human behaviors, which are critical to designing principled
algorithms to effectively learn from human annotators.
This paper takes a step toward bridging this gap. Specifically, we formalize and analyze the phenomenon of ?oracle epiphany.? Consider active learning from a human oracle to build a webpage
classifier on basketball sport vs. others. It is well-known in practice that no matter how simple
the task looks, the oracle can encounter difficult queries. The oracle may easily answer webpage
queries that are obviously about basketball or obviously not about the sport, until she encounters
a webpage on basketball jerseys. Here, the oracle cannot immediately decide how to label (?Does
this jersey webpage qualify as a webpage about basketball??). One solution is to allow the oracle
to abstain by answering with a special I-don?t-know label [Donmez and Carbonell, 2008]. More
interestingly, Kulesza et al. [2014] demonstrated that with proper user interface support, the oracle
may temporarily abstain on similar queries but then have an ?epiphany?: she may suddenly decide
how to label all basketball apparel-related webpages. Empirical evidence in [Kulesza et al., 2014]
suggests that epiphany may be induced by the accumulative effect of seeing multiple similar queries.
If a future basketball-jersey webpage query arrives, the oracle will no longer abstain but will answer
?
Part of this work was done while the author was with Microsoft Research.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
with the label she determined during epiphany. In this way, the oracle improves herself on the subset
of the input space that corresponds to basketball apparel-related webpages.
Empirical evidence also suggests that oracle abstention, and subsequent epiphany, may happen
separately on different subsets of the input space. When building a cooking vs. others text classifier,
Kulesza et al. [2014] observed oracle epiphany on a subset of cooking supplies documents, and
separately on the subset of culinary service documents; on gardening vs. others, they observed
separate oracle epiphany on plant information and on local garden documents; on travel vs. others,
they observed separate oracle epiphany on photography, rental cars, and medical tourism documents.
Our contributions are three-fold: (i) We formalize oracle epiphany in Section 2; (ii) We analyze
EPICAL, a variant of the CAL algorithm [Cohn et al., 1994], for realizable active learning with
oracle epiphany in Section 3. (iii) We analyze Oracular-EPICAL, a variant of the Oracular-CAL
algorithm [Hsu, 2010, Huang et al., 2015], for agnostic active learning in Section 4. Our query
complexity bounds show that active learning is possible with oracle epiphany, although we may
incur a penalty waiting for epiphany to happen. This is verified with simulations in Section 5, which
highlights the nuanced dependency between query complexity and epiphany parameters.
2 Problem Setting
As in standard active learning, we are given a hypothesis class H ? YX for some input space X
and a binary label set Y , {?1, 1}. There is an unknown distribution ? over X ? Y, from which
examples are drawn IID. The marginal distribution over X is ?X . Define the expected classification
error, or risk, of a classifier h ? H to be err(h) , E(x,y)?? [1 (h(x) 6= y)]. As usual, the active
learning goal is as follows: given any fixed , ? ? (0, 1), we seek an active learning algorithm which,
with probability at least 1 ? ?, returns a hypothesis with classification error at most after sending a
?small? number of queries to the oracle. What is unique here is an ?oracle epiphany model.?
The input space consists of two disjoint sets X = K ? U. The oracle knows the label for items in K
(for ?known?) but initially does not know the labels in U (for ?unknown?). The oracle will abstain if
a query comes from U (unless epiphany happens, see below). Furthermore, U is partitioned into K
disjoint subsets U = U1 ? U2 ? . . . ? UK . These correspond to the photograph/rental cars/medical
tourism subsets in the travel task earlier. The active learner does not know the partitions nor K.
When the active learner submits a query x ? X to the oracle, the learner will receive one of three
outcomes in Y+ , {?1, 1, ?}, where ? indicates I-don?t-know abstention.
Importantly, we assume that epiphany is modeled as K Markov chains: Whenever a unique x ? Uk
is queried on some unknown region k ? {1, . . . , K} which did not experience epiphany yet, the
oracle has a probability ? ? [0, 1] of epiphany on that region. If epiphany happens, the oracle
then understands how to label everything in Uk . In effect, the state of Uk is flipped from unknown
to known. Epiphany is irrevocable: Uk will stay known from now on and the oracle will answer
accordingly for all future x therein. Thus the oracle will only answer ? if Uk remains unknown. The
requirement for a unique x is to prevent a trivial active learning algorithm which repeatedly queries
the same ? item in an attempt to induce oracle epiphany. This requirement does not pose difficulty
for analysis if ?X is continuous on X, since all queries will be unique with probability one.
Therefore, our oracle epiphany model is parameterized by (?, K, U1 , . . . , UK ). All our analyses
below will be based on this epiphany model. Of course, the model is only an approximation to real
human oracle behaviors; In Section 6 we will discuss more sophisticated epiphany models for future
work.
3
The Realizable Case
In this section, we study the realizable active learning case, where we assume there exists some
h? ? H such that the label of an example x ? X is y = h? (x). It follows that err(h? ) = 0. Although
the realizability assumption is strong, the analysis is insightful on the role of epiphany. We will
show that the worst-case query complexity has an additional 1/? dependence. We also discuss nice
cases where this 1/? can be avoided depending on U?s interaction with the disagreement region.
Furthermore, our analysis focuses on the K = 1 case; that is, the oracle has only one unknown region
U = U1 . This case is the simplest but captures the essence of the algorithm we propose in this section.
2
For convenience, we will drop the superscript and write U. In the next section, we will eliminate both
assumptions, and present and analyze an algorithm for the agnostic case with an arbitrary K ? 1.
We modify the standard CAL algorithm [Cohn et al., 1994] to accommodate oracle epiphany. The
modified algorithm, which we call EPICAL for ?epiphany CAL,? is given in Alg. 1. Like CAL,
EPICAL receives a stream of unlabeled items; It maintains a version space; If the unlabeled item falls
into the disagreement region of the version space the oracle is queried. The essential difference to
CAL is that if the oracle answers ?, no update to the version space happens. The stopping criterion
ensures that the true risk of any hypothesis in the version space is at most , with high probability.
Algorithm 1 EPICAL
Input: , ?, oracle, X, H
Version space V ? H
Disagreement region D ? {x ? X | ?h, h0 ? V, h(x) 6= h0 (x)}
for t = 1, 2, 3, . . . do
Sample an unlabeled example from the marginal distribution restricted to D: xt ? ?X|D
Query oracle with xt to get yt
if yt 6= ? then
V ? {h ? V | h(xt ) = yt }
D ? {x ? X | ?h, h0 ? V, h(x) 6= h0 (x)}
end if
if ?X (D) ? then
Return any h ? V
end if
end for
Our analysis is based on the following observation: before oracle epiphany and ignoring all queries
that result in ?, EPICAL behaves exactly the same as CAL on an induced active-learning problem.
The induced problem has input space K, but with a projected hypothesis space we detail below. Hence,
standard CAL analysis bounds the number of queries to find a good hypothesis in the induced problem.
Now consider the sequence of probabilities of getting a ? label in each step of EPICAL. If these
probabilities tend to be small, EPICAL will terminate with an-risk hypothesis without even having
to wait for epiphany. If these probabilities tend to be large, we may often hit the unknown region U.
But the number of such steps is bounded because epiphany will happen with high probability.
? , K, and
Formally, we define the induced active-learning problem as follows. The input space is X
the output space is still Y. The sampling distribution is ?
?X (x) , ?X (x)1 (x ? K) /?X (K). The
? ? YX? | ?h ? H, ?x ? X
?
? H
? , {h
? : h(x)
hypothesis space is the projection of H onto X:
= h(x)}.
? ? be the projected target hypothesis. Let ? be the
Clearly, the induced problem is still realizable; let h
disagreement coefficient [Hanneke, 2014] for the original problem without unknown regions. The
induced problem potentially has a different disagreement coefficient:
??H
? ? (x) 6= h(x),
?
? 0 ) 6= h
? ? (x0 ) ? r .
? s.t. h
?? , sup r?1 ? Ex???X 1 ?h
Ex0 ???X 1 h(x
r>0
Let m
? be the number of queries required for the CAL algorithm to find a hypothesis of /2 risk with
probability 1 ? ?/4 in the induced problem. It is known [Hanneke, 2014, Theorem 5.1] that
2
4 2
?
?
?
?
ln
ln .
m
? ? M , ? dim(H) ln ? + ln
?
where dim(?) is the VC dimension. Similarly, let mCAL be the number of queries required for
CAL to find a hypothesis of risk with probability
1 ? ?/4 in the original problem, and we have
mCAL ? MCAL , ? dim(H) ln ? + ln 4? ln 1 ln 1 . Furthermore, define m? , |{t | yt = ?}|
to be the number of queries in EPICAL for which the oracle returns ?. We define Ut to be U for an
iteration t before epiphany, and ? after that. We define Dt to be the disagreement region D at iteration
t. Finally, define the unknown fraction within disagreement as ?t , ?X (Dt ? Ut )/?X (Dt ). We are
now ready to state the main result of this section.
? ? H with
Theorem 1. Given any and ?, EPICAL will, with probability at least 1 ? ?, return an h
3
4
?
?
err(h) ? , after making at most MCAL + M + ln queries.
?
3
?
Remark The bound above consists of three terms. The first is the standard CAL query complexity
bound with an omniscient oracle. The other two are the price we pay when the oracle is imperfect. The
second term is the query complexity for finding a low-risk hypothesis in the induced active-learning
problem. In situations where ?X (U) = /2 and ? 1, it is hard to induce epiphany, but it suffices to
? with /2 risk in the induced problem (which implies at most risk under
find a hypothesis from H
? is unavoidable in some cases. The third term is roughly
the original distribution ?X ); it indicates M
the extra query complexity required to induce epiphany. It is unavoidable in the worst case: when
U = X, one has to wait for oracle epiphany to start collecting labeled examples to infer h? ; the
average number of steps until epiphany is on the order of 1/?. Finally, note that not all three terms
contribute simultaneously to the query complexity of EPICAL. As we will see in the analysis and in
the experiments, usually one or two of them will dominate, depending on how U interacts with the
disagreement region. Summing them up simplifies our exposition, without changing the order of the
worst-case bounds.
Our analysis starts with the definition of the following two events. Lemmas 2 and 3 show that they
hold with high probability when running EPICAL; the proofs are delegated to Appendix A. Define:
1 4
2 4
E? , m? ? ln
and E? , |{t | ?t > 1/2}| ? ln
.
? ?
? ?
Lemma 2. Pr{E? } ? 1 ? ?/4 .
Lemma 3. Pr{E? } ? 1 ? ?/4.
Lemma 4. Assume event E? holds. Then, the number of queries from K before oracle epiphany or
before EPICAL terminates, whichever happens first, is at most m
? + ?2 ln 4? .
Proof. (sketch) Denote the quantity by m. Before epiphany, V and D in EPICAL behave in exactly
the same way as in CAL on K. It takes m
? queries to get to /2 accuracy in K by the definition of
m.
? If m ? m,
? then m < m
? + ?2 ln 4? trivially, and we are done. Otherwise, it must be the case that
?t > 1/2 for every step after V reaches /2 accuracy on K. Suppose not. Then there is a step t where
?t ? 1/2. Note V reaching /2 accuracy on K implies ?X (Dt ) ? ?X (Dt ? Ut ) ? /2. Together with
?t = ?X (Dt ? Ut )/?X (Dt ) ? 1/2, we have ?X (Dt ) < . But this would have triggered termination
of EPICAL at step t, a contradiction. Since we assume E? holds, we have m ? m
? + ?2 ln 4? .
Proof of Theorem 1. We will prove the query complexity bound, assuming (i) events E? and E? hold;
? and MCAL successfully upper bound the corresponding query complexity of standard
and (ii) M
CAL. By Lemmas 2 and 3 and a union bound, the above holds with probability at least 1 ? ?.
Suppose epiphany happens before EPICAL terminates. By event E? and Lemma 4, the total number
of queried examples before epiphany is at most m
? + ?3 ln 4? . After epiphany, the total number of
queries is no more than that of running CAL from scratch; this number is at most MCAL . Therefore,
? + MCAL + 3 ln 4 .
the total query complexity is at most M
?
?
Suppose epiphany does not happen before EPICAL terminates. In this case, the number of queries in
the unknown region is at most ?1 ln 4? (event E? ), and the number of queries in the known region is at
? + 3 ln 4 .
most m
? + ?2 ln 4? (Lemma 4). Thus, the total number of queries is at most M
?
?
4
The Agnostic Case
In the agnostic setting the best hypothesis, h? , arg minh err(h), has a nonzero error. We want
an active learning algorithm that, for a given accuracy > 0, returns a hypothesis h with small
regret reg(h, h? ) , err(h) ? err(h? ) ? while making a small number of queries. Among existing
agnostic active learning algorithms we choose to adapt the Oracular-CAL algorithm, first proposed
by Hsu [2010] and later improved by Huang et al. [2015]. Oracular-CAL makes no assumption on H
or ?, and can be implemented solely with an empirical risk minimization (ERM) subroutine, which is
often well approximated by convex optimization over a surrogate loss in practice. This is a significant
advantage over several existing agnostic algorithms, which either explicitly maintain a version space,
as done in A2 [Balcan et al., 2006], or require a constrained ERM routine [Dasgupta et al., 2007] that
may not be well approximated efficiently in practice. IWAL [Beygelzimer et al., 2010] and Active
4
Algorithm 2 Oracular-EPICAL
?
1: Set c1 , 4 and c2 , 2 6 + 9. Let ?0 , 1 and ?t ,
12
t
ln
32t|H| ln t
?
, t ? 1.
2: Initialize labeled data Z0 ? ?, the version space V1 ? H, and the ERM h1 as any h ? H.
3: for t = 1, 2, . . . do
i.i.d.
Observe new example xt , where (xt , yt ) ? ?.
if xt ? Dt , {x | x ? X, ?(h, h0 ) ? V2t s.t. h(x) 6= h0 (x)} then
Query oracle
with xt .
Zt?1 ? {(xt , yt )}, oracle returns yt .
Zt ?
Zt?1 ,
oracle returns ?.
ut ? 1 (oracle returns ?) .
else
Zt ? Zt?1 ? {(xt , ht (xt ))}. // update the labeled data with the current ERM?s prediction
ut ? 0.
end if
Pt
err(h, Zt ) , 1t i=1 1 (xi ? Di ) (1 ? ui )1 (h(xi ) 6= yi ) + 1 (xi ?
/ Di ) 1 (h(xi ) 6= hi (xi )) .
ht+1 ? arg minh?H err(h, Zt ).
Pt
bt ? 1t i=1 ui .
p
?t ? c1 ?t err(ht+1 , Zt ) + c2 (?t + bt ).
Vt+1 ? {h ? H | err(h, Zt ) ? err(ht+1 , Zt ) ? ?t }.
end for
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
Cover [Huang et al., 2015] are agnostic algorithms that are implementable with an ERM routine,
both using importance weights to correct for querying bias. But in the presence of ??s, choosing
proper importance weights becomes challenging. Moreover, the improved Oracular-CAL [Huang
et al., 2015] we use2 has stronger guarantees than IWAL, and in fact, the best known worst-case
guarantees among efficient, agnostic active learning algorithms.
Our proposed algorithm, Oracular-EPICAL, is given in Alg. 2. Note t here counts unlabeled data,
while in Alg. 1 it counts queries. Roughly speaking, Oracular-EPICAL also has an additive factor
of O(K/?) compared to Oracular-CAL?s query complexity. It keeps a growing set Z of labeled
examples. If the unlabeled example xt falls in the disagreement region, the algorithm queries its label:
when the oracle returns a label yt , the algorithm adds xt and yt to Z; when the oracle returns ?, no
update to Z happens. If xt is outside the disagreement region, the algorithm adds xt and the label
predicted by the current ERM hypothesis ht (xt ) to Z. Alg. 2 keeps an indicator ut , which records
whether ? was returned on xt , and it always updates the ERM and the version space after every new
xt . For simplicity we assume a finite H; this can be extended to H with finite VC dimension.
The critical modification we make here to accommodate oracle abstention is that the threshold ?t
defining the version space additively depends on the average number of ??s received up to round
t. This allows us to show that Oracular-EPICAL retains the favorable bias guarantee of OracularCAL: with high probability, all of the imputed labels are consistent with the classifications of h? ,
so imputation never pushes the algorithm away from h? . Oracular-EPICAL only uses the version
space in the disagreement test. With the same technique used by Oracular-CAL, summarized in
Appendix B, the algorithm is able to perform the test solely with an ERM routine.
We now state Oracular-EPICAL?s general theoretical guarantees, which hold for any oracle model,
and then specialize them for the epiphany model in Section 2. We start with a consistency result:
p
Theorem 5 (Consistency Guarantee). Pick any 0 < ? < 1/e and let ??t := c1 ?t err(h? ) + c2 (?t +
bt ). With probability at least 1 ? ?, the following holds for all t ? 1,
err(h) ? err(h? ) ? 4??t
err(h , Zt ) ? err(ht+1 , Zt ) ? ?t .
?
2
for all h ? Vt+1 ,
and
(1)
(2)
This improved version of Oracular-CAL defines the version space using a tighter threshold than the one
used by Hsu [2010], and has the same worst-case guarantees as Active Cover [Huang et al., 2015].
5
All hypotheses in the current version space, including the current ERM, have controlled expected
regrets. Compared with Oracular-CAL?s consistency guarantee, this is worse by an additive factor of
O(bt ), the average number of ??s over t examples. Importantly, h? always remains in the version
space, as implied by (2). This guarantees that all predicted labels used by the algorithm are consistent
with h? , since the entire version space makes the same prediction. The query complexity bound is:
Pt
Theorem 6 (Query Complexity Bound). Let Qt , i=1 1 (xi ? Di ) denote the total number of
queries Alg. 2 makes after observing t examples. Under the conditions of Theorem 5, with probability
at least 1 ? ? the following holds: ?t > 0, Qt is bounded by
q
4? err(h? )t + ? ? O
t err(h? ) ln(t|H|/?) ln2 t + ln(t|H|/?) ln t + tbt ln t + 8 ln(8t2 ln t/?) ,
where ? denotes the disagreement coefficient [Hanneke, 2014].
Again, this result is worse than Oracular-CAL?s query complexity [Huang et al., 2015] by an additive
factor. The magnitude of this factor is less trivial than it seems: since the algorithm increases the
threshold by bt , it includes more hypotheses in the version space, which may cause the algorithm
to query a lot more. However, our analysis shows that the number of queries only increases by
O(tbt ln t), i.e., ln t times the total number of ??s received over t examples.
The full proofs of both theorems are in Appendix C. Here we provide the key ingredient. Consider an
imaginary dataset Zt? where all the labels queried by the algorithm but not returned by the oracle are
imputed, and define the error on this imputed data:
t
err(h, Zt? ) ,
1X
1 (xi ? Di ) 1 (h(xi ) 6= yi ) + 1 (xi ?
/ Di ) 1 (h(xi ) 6= hi (xi )) .
t i=1
(3)
Note that the version space Vt and therefore the disagreement region Dt are still defined in terms
of err(h, Zt ), not err(h, Zt? ). Also define the empirical regrets between two hypotheses h and h0 :
reg(h, h0 , Zt ) , err(h, Zt ) ? err(h0 , Zt ) and reg(h, h0 , Zt? ) on Zt? in the same way. The empirical
error and regret on Zt? are not observable, but can be easily bounded by observable quantities:
err(h, Zt ) ? err(h, Zt? ) ? err(h, Zt ) + bt ,
0
0
|reg(h, h , Zt ) ? reg(h, h
, Zt? )|
(4)
? bt ,
(5)
Pt
where bt = i=1 ui /t is also observable. Using a martingale analysis resembling Huang et al.
[2015]?s for Oracular-CAL, we prove concentration of the empirical regret reg(h, h? , Zt? ) to its
expectation. For every h ? Vt+1 , the algorithm controls its empirical regret on Zt , which bounds
reg(h, h? , Zt? ) by the above. This leads to a bound on the expected regret of h. The query complexity
analysis follows the standard framework of Hsu [2010] and Huang et al. [2015].
Next, we specialize the guarantees to the oracle epiphany model in Section 2:
e ,
Corollary 7. Assume the epiphany model in Section 2. Fix > 0, ? > 0. Let d? , ln(|H|/(?)), K
?
?
K ln(K/?) and e , err(h ). With probability at least 1 ? ?,
following holds:
The ERM
the
??
e
hypothesis ht +1 satisfies err(ht +1 ) ? e? ? , where t = O de2 + 1 d? + K , and the total
number of queries made
round
? up to
t is ?
? ?
e
e
d?e
K
??O
+ ln e2 + 1 d? +
+ ?
e
K
?
?
e
?
?
+1 d+
?
e
K
?
.
The proof is in Appendix D. This corollary reveals how the epiphany parameters K and ? affect query
e = 0 recovers the result for a perfect oracle, showing that the (unlabeled) sample
complexity. Setting K
e
complexity t worsens by an additive factor of K/(?)
in both realizable and agnostic settings. For
e
e
query complexity, in the realizable setting the bound becomes ? ? O ln d?+ K/?
/ d?+ K/?
.
e ? )/(?) . In both cases,
In the agnostic setting, the leading term in our bound is ? ? O (e? /)2 d?+ (Ke
e
our bounds are worse by roughly an additive factor of O(K/?)
than bounds for perfect oracles.
As for the effect of U, the above corollary is a worst-case result: it uses an upper bound on tbt
that holds even for U = X. For certain U ?s the upper bound can be much tighter. For example, if
U ? Dt = ? for sufficiently large t, then tbt will be O(1) for all ?, with or without epiphany.
6
5
Experiments
To complement our theoretical results, we present two simulated experiments on active learning
with oracle epiphany: learning a 1D threshold classifier and handwritten digit recognition (OCR).
Specifically, we will highlight query complexity dependency on the epiphany parameter? and on U.
EPICAL on 1D Threshold Classifiers. Take ?X to be the uniform distribution over the interval
X = [0, 1]. Our hypothesis space is the set of threshold classifiers H = {ha : a ? [0, 1]} where
ha (x) = 1 (x ? a). We choose h? = h 12 and set the target classification error at = 0.05.
0.0
0.5
?
(a) U = [0.4, 0.6]
1.0
0
Excess queries
20 40 60
EPICAL
Passive
0
0
EPICAL
Passive
80
Queries
100 200 300 400
Queries
100 200 300 400
We illustrate epiphany with a single unknown region K = 1, U = U1 . However, we contrast two
shapes of U: in one set of experiments we set U = [0.4, 0.6] which contains the decision boundary
0.5. In this case, the active learner EPICAL must induce oracle epiphany in order to achieve risk.
In another set of experiments U = [0.7, 0.9], where we expect the learner to be able to ?bypass? the
need for epiphany. Intuitively, this latter U could soon be excluded from the disagreement region.
For both U, we systematically vary the oracle epiphany parameter ? ? {2?6 , 2?5 , . . . , 20 }. A small
? means epiphany is less likely per query, thus we expect the learner to spend more queries trying
to induce epiphany in the case of U = [0.4, 0.6]. In contrast, ? may not matter much in the case of
U = [0.7, 0.9] since epiphany may not be required. Note that ? = 20 = 1 reverts back to the standard
active learning oracle, since epiphany always happens immediately. We run each combination of ?, U
0.0
0.5
?
(b) U = [0.7, 0.9]
1.0
0
20
40
1/?
60
80
(c) U = [0.4, 0.6]
Figure 1: EPICAL results on 1D threshold classifiers
for 10, 000 trials. The results are shown in Figure 1. As expected, (a) shows a clear dependency on
?. This indicates that epiphany is necessary in the case U = [0.4, 0.6] for learning to be successful.
In contrast, the dependence on ? vanishes in (b) when U is shifted sufficiently away from the target
threshold (and thus from later disagreement regions). The oracle need not reach epiphany for learning
to happen. Note (b) does not contradict with EPICAL query complexity analysis since Theorem 1 is
a worst case bound that must hold true for all U.
To further clarify the role of ?, note EPICAL query complexity bound predicts an additive term
? and MCAL ). This term
of O(1/?) on top of the standard CAL query complexities (i.e., both M
represents ?excess queries? needed to induce epiphany. In Figure 1(c) we plot this excess against ?1
for U = [0.4, 0.6]. Excess is computed as the number of EPICAL queries minus the average number
of queries for ? = 1. Indeed, we see a near linear relationship between excess queries and 1/?.
Finally, as a baseline we compare EPICAL to passive learning. In passive learning x1 , x2 , . . . are
chosen randomly according to ?X instead of adaptively. Note passive learning here is also subject
to oracle epiphany. That is, the labels yt are produced by the same oracle epiphany model, some of
them can be ? initially. Our passive learning simply maintains a version space. If it encounters ? it
does not update the version space. All EPICAL results are better than passive learning.
Oracular-EPICAL on OCR. We consider the binary classification task of 5 vs. other digits on
MNIST [LeCun et al., 1998]. This allows us to design the unknown regions {Uk } as certain other
digits, making the experiments more interpretable. Furthermore, we can control how confusable the
U digits are to ?5? to observe the influence on oracle epiphany.
Although Alg. 2 is efficiently implementable with an ERM routine, it still requires two calls to a
supervised learning algorithm on every new example. To scale it up, we implement an approximate
version of Alg. 2 that uses online optimization in place of the ERM. More details are in Appendix E.
While being efficient in practice, this online algorithm may not retain Alg. 2?s theoretical guarantees.
7
14000
1800
Oracular?EPICAL
Passive
12000
1400
Queries
Queries
10000
8000
6000
1200
1000
4000
800
2000
600
0
Oracular?EPICAL
Passive
1600
0
1e?4 1e?3 1e?2 1e?1
?
400
1
(a) U =?3?
0
1e?4 1e?3 1e?2 1e?1
?
1
(b) U =?1?
Figure 2: Oracular-EPICAL results on OCR.
We use epiphany parameters ? ? {1, 10?1 , 10?2 , 10?3 , 10?4 , 0}, K = 1, and U is either ?3? or ?1?.
By using ? = 1 and ? = 0, we include the boundary cases where the oracle is perfect or never has an
epiphany. The two different U?s correspond to two contrasting scenarios: ?3? is among the ?nearest?
digits to ?5? as measured by the binary classification error between ?5? and every other single digit,
while ?1? is the farthest. The two U?s are about the same size, each covering roughly 10% of the
data. More details and experimental results with other choices of U can be found in Appendix E.
For each combination of ? and U, we perform 100 random trials. In each trial, we run both the
online version of Alg. 2 and online passive logistic regression (also subject to oracle epiphany) over
a randomly permuted training set of 60, 000 examples, and check the error of the online ERM on
the 10, 000 testing examples every 10 queries from 200 up to our query budget of 13, 000. In each
trial we record the smallest number of queries for achieving a test error of 4%. Fig. 2(a) and Fig. 2(b)
show the median of this number over the 100 random trials, with error bars being the 25th and 75th
quantiles. The effect of ? on query complexity is dramatic for the near U = ?3? but subdued for
the far U = ?1?. In particular, for U = ?3? small ??s force active learning to query as many labels
as passive learning. The flattening at 13, 000 at the end means no algorithm could achieve a 4%
test error within our query budget. For U = ?1?, active learning is always much better than passive
regardless of ?. Again, this illustrates that both ? and U affect the query complexity. As performance
references, passive learning on the entire labeled training data achieves a test error of 2.6%, while
predicting the majority class (non-5) has a test error of 8.9%.
6
Discussions
Our analysis reveals a worst case O(1/?) term in query complexity due to the wait for epiphany, and
we hypothesize ?(K/?) to be the tight lower bound. This immediately raises the question: can we
decouple active learning queries from epiphany induction? What if the learner can quickly induce
epiphany by showing the oracle a screenful of unlabeled items at a time, without the oracle labeling
them? This possibility is hinted in empirical studies. For example, Kulesza et al. [2014] observed
epiphanies resulting from seeing items. Then there is a tradeoff between two learner actions toward
the oracle: asking a query (getting a label or small contribution toward epiphany), or showing several
items (not getting labels but potentially large contribution toward epiphany). One must formalize
the cost and benefit of this tradeoff. Of course, real human behaviors are even richer. Epiphanies
may be reversible on certain queries, where the oracle begins to have doubts on her previous labeling.
Extending our model under more relaxed assumptions is an interesting open question for future
research.
Acknowledgments
This work is supported in part by NSF grants IIS-0953219, IIS-1623605, DGE-1545481, CCF1423237, and by the University of Wisconsin-Madison Graduate School with funding from the
Wisconsin Alumni Research Foundation.
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9
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5,698 | 6,156 | ?-risk: a New Surrogate Risk for Learning
from Weakly Labeled Data
Valentina Zantedeschi?
R?mi Emonet
Marc Sebban
firstname.lastname@univ-st-etienne.fr
Univ Lyon, UJM-Saint-Etienne, CNRS, Institut d Optique Graduate School,
Laboratoire Hubert Curien UMR 5516, F-42023, SAINT-ETIENNE, France
Abstract
During the past few years, the machine learning community has paid attention to
developing new methods for learning from weakly labeled data. This field covers
different settings like semi-supervised learning, learning with label proportions,
multi-instance learning, noise-tolerant learning, etc. This paper presents a generic
framework to deal with these weakly labeled scenarios. We introduce the ?-risk as
a generalized formulation of the standard empirical risk based on surrogate marginbased loss functions. This risk allows us to express the reliability on the labels and
to derive different kinds of learning algorithms. We specifically focus on SVMs
and propose a soft margin ?-SVM algorithm which behaves better that the state of
the art.
1
Introduction
The growing amount of data available nowadays allowed us to increase the confidence in the models
induced by machine learning methods. On the other hand, it also caused several issues, especially in
supervised classification, regarding the availability of labels and their reliability. Because it may be
expensive and tricky to assign a reliable and unique label to each training instance, the data at our
disposal for the application at hand are often weakly labeled. Learning from weak supervision has
received important attention over the past few years [14, 12]. This research field includes different
settings: only a fraction of the labels are known (Semi-Supervised learning [22]); we can access only
the proportions of the classes (Learning with Label Proportions [19] and Multi-Instance Learning [8]);
the labels are uncertain or noisy (Noise-Tolerant Learning [1, 18, 16]); different discording labels are
given to the same instance by different experts (Multi-Expert Learning [21]); labels are completely
unknown (Unsupervised Learning [11]). As a consequence of this statement of fact, the data provided
in all these situations cannot be fully exploited using supervised techniques, at the risk of drastically
reducing the performance of the learned models. To address this issue, numerous machine learning
methods have been developed to deal with each of the previous specific situations. However, all
these weakly labeled learning tasks share common features mainly relying on the confidence in
the labels, opening the door to the development of generic frameworks. Unfortunately, only a few
attempts have tried to address several settings with the same approach. The most interesting one has
been presented in [14] where the authors propose W ELL SVM which is dedicated to deal with three
different weakly labeled learning scenarios: semi-supervised learning, multi-instance learning and
clustering. However, W ELL SVM focuses specifically on Support Vector Machines and it requires to
?
http://vzantedeschi.com/
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
derive a new optimization problem for each new task. Even though W ELL SVM constitutes a step
further towards general models, it stopped in midstream constraining the learner to use SVMs.
This paper aims to bridge this gap by presenting a generic framework for learning from weakly labeled
data. Our approach is based on the derivation of the ?-risk , a new surrogate empirical risk defined as
a strict generalization of the standard empirical risk relying on surrogate margin-based loss functions.
The main interesting property of the ?-risk comes from its ability to exploit the information given
by the weakly supervised setting and encoded as a ? matrix reflecting the supervision on the labels.
Moreover, the instance-specific weights ? let one integrate in classical methods the side information
provided by the setting. This is the peculiarity w.r.t. [18, 16]: in both papers, the proposed losses are
defined using class-dependent weights (fixed to 1/2 for the first paper, and dependent on the class
noise rate for the latter) while in our approach the used weights are provided for each instance, which
gives a more flexible formulation. Making use of this ?-risk , we design a generic algorithm devoted
to address different kinds of aforementioned weakly labeled settings. To allow a comparison with
the state of the art, we instantiate it with a learner that takes the form of an SVM algorithm. In this
context, we derive a soft margin ?-SVM algorithm and show that it outperforms W ELL SVM.
The remainder of this paper is organized as follows: in Section 2, we define the empirical surrogate
?-risk and show under which conditions it can be used to learn without explicitly accessing the
labels; we also show how to instantiate ? according to the weakly labeled learning setting at hand; in
Section 3, we present our generic iterative algorithm for learning with weakly labeled data and in
Section 4 we exploit our new framework to derive a novel formulation of the Support Vector Machine
problem, the ?-SVM ; finally, we report experiments in semi-supervised learning and learning with
label noise, conducted on classical datasets from the UCI repository [15], in order to compare our
algorithm with the state of the art approaches.
2
From Classical Surrogate Losses and Surrogate Risks to the ?-risk
In this section, we first provide reminders about surrogate losses and then exploit the characteristics of
the popular loss functions to introduce the empirical surrogate ?-risk . The ?-risk formulation allows
us to tackle the problem of learning with weakly labeled data. We show under which conditions it
can be used instead of the standard empirical surrogate risk (defined in a fully supervised context).
Those conditions give insight on how to design algorithms that learn from weak supervision. We
restrain our study to the context of binary classification.
2.1
Preliminaries
In statistical learning, a common approach for choosing the optimal hypothesis h? from a hypothesis
class H is to select the classifier that minimizes the expected risk over the joint space Z = X ? Y ,
where X is the feature space and Y the label space, expressed as
Z
R` (h) =
`(yh(x))p(x, y)dxdy
X?Y
with ` : H ? Z ? R+ a margin-based loss function.
Since the true distribution of the data p(x, y) is usually unknown, machine learning algorithms
typically minimize the empirical version of the risk, computed over a finite set S composed of m
instances (xi , yi ) i.i.d. drawn from a distribution over X ? {?1, 1}:
m
R` (S, h) =
1 X
`(yi h(xi )).
m i=1
The most natural loss function is the so-called 0-1 loss. As this function is not convex, not differentiable and has zero gradient, other loss functions are commonly employed instead. These losses,
such as the logistic loss (e.g., for the logistic regression [6]), the exponential loss (e.g., for boosting
techniques [10]) and the hinge loss (e.g., for the SVM [7]), are convex and smooth relaxations of
the 0-1 loss. Theoretical studies on the characteristics and behavior of such surrogate losses can
be found in [17, 2, 20]. In particular, [17] showed that each commonly used surrogate loss can be
2
characterized by a permissible function ? (see below) and rewritten as F? (x)
F? (x) =
?? (?x) ? a?
b?
where ?? (x) = supa (xa ? ?(a)) is the Legendre conjugate of ? (for more details, see [4]), a? =
??(0) = ??(1) ? 0 and b? = ??( 12 ) ? a? > 0. As presented by the authors of [13] and [17],
a permissible function is a function f : [0, 1] ? R? , symmetric about ? 12 , differentiable on
]0, 1[ and strictly convex. For instance, the permissible function ?log related to the logistic loss
F? (x) = log(1 + exp?x ) is:
?log (x) = x log(x) + (1 ? x) log(1 ? x)
and a? = 0 and b? = log(2).
As detailed in [17], considering a surrogate loss F? , the empirical surrogate risk of an hypothesis
h : X ? R w.r.t. S can be expressed as:
R? (S, h) =
m
m
b? X
1 X
D? yi , ??1
(h(x
))
=
F? (yi h(xi ))
i
?
m i=1
m i=1
with D? the Bregman Divergence
D? (x, y) = ?(x) ? ?(y) ? (x ? y)?? (y).
In order to evaluate such risk R? (S, h), it is mandatory to provide the labels y for all the instances. In
addition, it is not possible to take into account eventual uncertainties on the given labels. Consequently,
R? is defined in a totally supervised context, where the labels y are known and considered to be
true. In order to face the numerous situations where training data may be weakly labeled, we claim
that there is a need to fill the gap by defining a new empirical surrogate risk that can deal with such
settings. In the following section, we propose a generalization of the empirical surrogate risk, called
the empirical surrogate ??risk, which can be employed in the context of weakly labeled data instead
of the standard one under some linear conditions on the margin.
2.2
The Empirical Surrogate ?-risk
Before defining the empirical surrogate ?-risk for any loss F? and hypothesis h ? H, let us rewrite
the definition of R? introducing a new set of variables named ?, and that can be laid out as a 2?m
matrix.
Lemma 2.1. For any S, ? and h, and for any non-negative real coefficients ?i-1 and ?i+1 defined
for each instance xi ? S such that ?i-1 + ?i+1 = 1, the empirical surrogate risk R? (S, h) can be
rewritten as
R? (S, h) = R? (S, h, ?)
where
R? (S, h, ?) =
m
m
b? X X ?
1 X -yi
?i F? (?h(xi )) +
? (?yi h(xi )).
m i=1 ??
m i=1 i
{-1,+1}
The coefficient ?i+1 (resp. ?i-1 ) for an instance xi can be interpreted here as the degree of confidence
in (or the probability of) the label +1 (resp. -1) assigned to xi .
3
Proof.
m
b? X
R? (S, h) =
F? (yi h(xi ))
m i=1
m
b? X yi
?i F? (yi h(xi )) + ?i-yi F? (yi h(xi ))
m i=1
m
b? X
yi h(xi )
=
?iyi F? (yi h(xi )) + ?i-yi F? (?yi h(xi )) ?
m i=1
b?
=
=
m
m
b? X X ?
1 X -yi
?i F? (?h(xi )) +
? (?yi h(xi )).
m i=1 ??
m i=1 i
(1)
(2)
(3)
{-1,+1}
Eq. (1) is because ?i-1 + ?i+1 = 1; Eq. (2) is due to the fact that ?? (?x) = ?? (x) ? x (see the sup?? (?x)?a?
?? (x)?a? ?x
plementary material) for any permissible function ?, so that F? (x) =
=
=
b?
b?
x
F? (?x) ? b? .
From Eq. (3), and considering that the sample S is composed by the finite set of features X and labels
Y, we can write that
m
R? (S, h) = R? (S, h, ?) = R?? (X , h) ?
where
R?? (X , h) =
1 X -yi
? yi h(xi )
m i=1 i
(4)
m
b? X X ?
? F? (?h(xi ))
m i=1 ?? i
{-1,+1}
-1
+1 -1
].
|?0 , ..., ?m
is the empirical surrogate ?-risk for a matrix ? = [?0+1 , ..., ?m
It is worth noticing that R? (S, h, ?) is expressed in the form of a sum of two terms: the second one
takes into account the labels of the data, while the first one, the ?-risk, focuses on the loss suffered by
h over X without explicitly needing the labels Y.
The empirical ?-risk is a generalization of the empirical risk: setting ?iyi = 1 (and thus ?i?yi = 0)
for each instance, the second term vanishes and we retrieve the classical formulation of the empirical
risk. Additionally, as developed in Section 2.3, the introduction of ? makes it possible to inject some
side-information about the labels. For this reason, we claim that the ?-risk is suited to deal with
classification in the context of weakly labeled data.
Let us now focus on the conditions allowing the empirical ?-risk (i) to be a surrogate of the 0-1
loss-based empirical risk and (ii) to be sufficient to learn with a weak supervision on the labels.
From (4), we deduce:
m
R?? (X , h) = R? (S, h, ?) +
m
1 X -yi
1 X -yi
?i yi h(xi ) ? R0/1 (S, h) +
? yi h(xi )
m i=1
m i=1 i
(5)
where R0/1 (S, h) the empirical risk related to the 0-1 loss and Eq. (5) is because b? F? (x) ? F0/1 (x)
(for any surrogate loss).
It is possible to ensure that the ?-risk is both a convex upper-bound of the 0-1 loss based risk and a
relaxation as tight as the traditional risk (i.e., that we have R0/1 (S, h) ? R?? (X , h) ? R? (S, h)) is
Pm
to force the following constraint: i=1 ?i-yi yi h(xi ) = 0.
Pm
Unfortunately, the constraint i=1 ?i-yi yi h(xi ) = 0 still depends on the vector y of labels, which is
not always provided and most likely uncertain or inaccurate in a weakly labeled data setting. We will
show in Section 3 that this issue can be overcome by means of an iterative 2-step learning procedure,
that first learns a classifier minimizing the ?-risk , possibly violating the constraint, and then learns a
new matrix ? that fulfills the constraint.
4
2.3
Instantiating ? for Different Weakly Supervised Settings
The ?-risk can be used as the basis for handling different learning settings, including weakly labeled
learning. This can be achieved by fixing the ? values, choosing their initial values or putting a prior
on them. We have already seen that, fully supervised learning can be obtained by fixing all ? values
to 1 for the assigned class and to 0 for the opposite class. The current section provides guidance on
how ? could be instantiated to handle various weakly labeled settings.
In a semi-supervised setting, as detailed in the experimental section, we propose to initialize the
? of unlabeled points to 0.5 and then to automatically refine them in an iterative process. Going
further, and if we are ready to integrate spatial or topological information in the process, the ?
values of each unlabeled point could be initialized using a density estimation procedure (e.g., by
considering the label proportions of the k nearest labeled neighbors). In the context of multi-expert
learning, the experts? votes for each instance i can simply be averaged to produce the ?i values (or
their initialization, or a prior). The case of learning with label proportions is especially useful for
privacy-preserving data processing: the training points are grouped into bags and, for each bag, the
proportion of labels are given. One way of handling such supervision is to initialize, for each bag,
all the ? with the same value that corresponds to the provided proportion of labels. Noise-tolerant
learning aims at learning in the presence of label noise, where labels are given but can be wrong. For
any point that can be possibly noisy, a direct approach is to use lower ? values (instead of 1 in the
supervised case) and refine them as in the semi-supervised setting. ? can also be initialized using the
label proportion of the k nearest labeled example (as done in the experimental section). The case of
Multiple Instance Learning (MIL) is trickier: in a typical MIL setting, instances are grouped in bags
and the supervision is given as a single label per bag that is positive if the bag contains at least one
positive instance (negative bags contain only negative instances). A straightforward solution would
be to recast the MIL supervision as a ?learning with label proportion? (e.g., considering exactly one
positive instance in each bag). It is not fully satisfying and a more promising solution would be to
consider, within each bag, the set of ? +1 variables and put a sparsity-inducing prior on them. This
approach would be a less-constrained version of the relaxation proposed in WellSVM [14] (where it
is supposed that there is exactly one positive instance per positive bag) and could be achieved by a l1
penalty or using a Dirichlet prior (with low ? to promote sparsity).
3
An Iterative Algorithm for Weakly-labeled Learning
As explained in Section 2, a sufficient condition for guaranteeing that the ?-risk is a convex
upper-bound
of the 0-1 loss based risk and it is not worse than the traditional risk is to fix
Pm -yi
?
y
h(x
i
i ) = 0. However, the previous constraint depends on the labels. We overcome
i=1 i
this problem by (i) iteratively learning a classifier minimizing the ?-risk and most likely violating the
constraint and then (ii) learning a new matrix ? that fulfills it. The algorithm is generic. It can be used
in different weakly labeled settings and can be instantiated with different losses and regularizations,
as we will do in the next Section with SVMs.
As the process is iterative, let t ? be the estimation of ? at iteration t. At each iteration, our algorithm
consists in two steps. We first learn an hypothesis h for the following problem P1 :
t
?
ht+1 = P1 (X , t ?) = arg min cR? (X , h) + N (h)
h
which boils down to minimizing the N -regularized empirical surrogate ?-risk over the training
sample X of size m, where N , for instance, can take the form of a L1 or a L2 norm.
Then, we find the optimal ? of the following problem P2 for the points of X :
t+1
? = P2 (X , ht+1 ) = arg min R?? (X , ht+1 )
?
s.t.
m
X
?i-yi (?yi ht+1 (xi )) = 0
i=1
?i-1 + ?i+1 = 1, ?i-1 ? 0, ?i+1 ? 0 ?i = 1..m .
For this step, a vector of labels is required. We choose to re-estimate it at each iteration according
to the current value of ?: we affect to an instance the most probable label, i.e. the ? corresponding
5
to the biggest ? ? . The matrix ? has to be initialized at the beginning of the algorithm according to
the problem setting (see Section 2.3). While some stabilization criterion does not exceed a given
threshold , the two steps are repeated.
4
Soft-margin ?-SVM
A major advantage of the empirical surrogate ?-risk is that it can be plugged in numerous learning
settings without radically modifying the original formulations. As an example, in this section we
derive a new version of the Support Vector Machine problem, using the empirical surrogate ?-risk ,
that takes into account the knowledge provided for each training instance (through the matrix ?).
The soft-margin ?-SVM optimization problem is a direct generalization of a standard soft-margin
SVM and is defined as follows:
m
X
1
2
arg min k?k2 + c
?i-1 ?i-1 + ?i+1 ?i+1
2
?
i=1
s.t. ?(?T ?(xi ) + b) ? 1 ? ?i? ?i = 1..m, ? ? {?1, 1}
?i? ? 0 ?i = 1..m, ? ? {?1, 1}
where ? ? X 0 is the vector defining the margin hyperplane and b its offset, ? : X ? X 0 a mapping
function and c ? R a tuned hyper-parameter. In the rest of the paper, we will refer to K : X ?X ? R
as the kernel function corresponding to ?, i.e. K(xi , xj ) = ?(xi )?(xj ).
The corresponding Lagrangian dual problem is given by (the complete derivation is provided in the
supplementary material):
max ?
?
m
m
m
X
X
1X X X X ? ? 0
?i ??j ? K(xi , xj ) +
?i?
2 i=1 ?? j=1 0
i=1 ??
{-1,+1}
? ?
{-1,+1}
{-1,+1}
s.t. 0 ? ?i? ? c?i? ?i = 1..m, ? ? {?1, 1}
m
X
X
?i? ? = 0 ?i = 1..m, ? ? {?1, 1}
i=1 ??
{-1,+1}
which is concave w.r.t. ? as for the standard SVM.
The ?-SVM formulation differs from the SVM one in two points: first, the number of Lagrangian
multipliers is doubled, because we consider both positive and negative labels for each instance;
second, the upper-bounds for ? are not the same for all instances but depend on the given matrix
?. Like the coefficient c in the classical formulation of SVM, those upper-bounds play the role of
trade-off between under-fitting and over-fitting: the smaller they are, the more robust to outliers the
learner is but the less it adapts to the data. It is then logical that the upper-bound for an instance
i depends on ?i? because it reflects the reliability on the label ? for that instance: if the label ? is
unlikely, its corresponding ?i? will be constrained to be null (and its adversary will have more chance
to be selected as a support vector, as ?i? + ?i? ? = 1). Also, those points for which no label is more
probable than the other (?i? ? 0.5) will have less importance in the learning process compared to
those for which a label is almost certain. In order to fully exploit the advantages of our formulation, c
has to be finite and bigger than 0. As a matter of fact, when c ? ? or c ? 0, the constraints become
exactly those of the original formulation.
5
Experimental Results
In the first part of this section, we present some experimental results obtained by adapting the iterative
algorithm presented in Section 3 for semi-supervised learning and combining it with the previously
derived ?-SVM . Note that some approaches based on SVMs have been already presented in the
literature to address the problem of semi-supervised learning. Among them, TransductiveSVM [5]
6
iteratively learns a separator with the labeled instances, classifies a subset of the unlabeled instances
and adds it to the training set. On the other hand, WellSVM [14] combines the classical SVM with a
label generation strategy that allows one to learn the optimal separator, even when the training sample
is not completely labeled, by convexly relaxing the original Mixed-Integer Programming problem. In
[14], WellSVM has been shown to be very effective and better than TransductiveSVM and the state of
the art. For this reason, we compare in this section ?-SVM to WellSVM. In the second subsection, we
present some preliminary results in the noise-tolerant learning setting, showing how ?-SVM behaves
when facing data with label noise.
Iterative ?-SVM for semi-supervised learning
5.1
We compare our method?s performances to those of WellSVM, that has been proved, in [14], to
performs in average better than the state of the art semi-supervised learning methods based on SVM
and the standard SVM as well. In a semi-supervised context, a set Xl of labeled instances of size ml
and a set Xu of unlabeled instances of size mu are provided. The matrix ? is initialized as follows:
?i = 1..ml and ?? in {?1, 1}, 0 ? ?i = 1 if ? = yi , 0 otherwise,
?i = ml +1..mu and ?? in {?1, 1}, 0 ? ?i = 0.5
and we learn an optimal separator:
t
?
t
?
ht+1 = P1 (Xl ? Xu , t ?) = arg min c1 R? (Xl , h) + c2 R? (Xu , h) + N (h).
h
Here c1 and c2 are balance constants between the labeled and unlabeled set: when the number of
unlabeled instances become greater than the number of labeled instances, we need to reduce the
importance of the unlabeled set in the learning procedure because there exists the risk that the labeled
set will be ignored. We consider the provided labels to be correct, so we keep the corresponding
t+1
). The
l ? fixed during the iterations of the algorithm and estimate u ? by optimizing P2 (Xu , h
iterative algorithm with ?-SVM is implemented in Python using Cvxopt (for optimizing ?-SVM )
and Cvxpy 2 with its Ecos solver [9].
For each dataset, we show in Figure 1 the accuracy of the two methods with an increasing proportion
of labeled data. The different approaches are compared on the same kernel, either the linear or the
gaussian, the one that gives higher overall accuracy. As a matter of fact, the choice of the kernel
depends on the geometry of the data, not on the learning method.
For each proportion of labeled data, we perform a 4-fold cross-validation and we show the average
accuracy over 10 iterations. Concerning the hyper-parameters of the different methods, we fix c2
l
of ?-SVM to c1 m
m , c1 of WellSVM to 1 as explained in [14] and all the other hyper-parameters
(c1 for ?-SVM and c2 for WellSVM) are tuned by cross-validation through grid search. As for
the stopping criteria, we fix of ?-SVM to 10?5 + 10?3 khkF and of WellSVM to 10?3 and the
maximal number of iterations to 20 for both methods. When using the gaussian kernel, the ? in
2
K(xi , xj ) = exp(?kxi ? xj k2 /?) is fixed to the mean distance between instances.
Our method performs better than WellSVM, with few exceptions, and is more efficient in terms
of CPU time: for the Australian dataset, the biggest dataset in number of features and instances,
WellSVM is in average 30 times slower than our algorithm (without particular optimization efforts).
5.2
Preliminary results under label-noise
We quickly tackle another setting of the weakly labeled data field: the noise-tolerant learning, the
task of learning from data that have noisy or uncertain labels. It has been shown in [3] that SVM
learning is extremely sensitive to outliers, especially the ones lying next to the boundary. We study,
the sensitivity of ?-SVM to label noise artificially introduced on the Ionosphere dataset. We consider
two initialization strategies for ?: the standard on where ? yi = 1 and ? ?yi = 0 and the 4-nn one
where ? ? is set to the proportion of neighboring instances with label ?. In Figure 2, we draw the mean
accuracy over 4 repetitions w.r.t. an increasing percentage (as a proportion of the smallest dataset) of
two kinds of noise: the symmetric noise, introduced by swapping the labels of instances belonging
to different classes, and the asymmetric noise, introduced by gradually changing the labels of the
2
http://cvxopt.org/ and http://www.cvxpy.org/
7
0.85
0.8
0.75
5
10
15
0.8
0.65
0.8
0.75
0.6
0.75
0.7
0.55
0.7
20
5
(a) Ionosphere,
gaussian kernel.
10
15
0.7
0.65
0.65
0.6
10
15
10
15
20
20
5
10
15
20
(d) Australian,
gaussian kernel.
WellSVM
betaSVM
5
(f) Sonar,
linear kernel.
(e) Pima,
linear kernel.
15
0.65
0.6
0.55
0.5
5
20
10
(c) Liver,
linear kernel.
(b) Heart-statlog,
linear kernel.
0.75
5
5
20
10
15
20
(g) Splice,
gaussian kernel.
Figure 1: Comparison of the mean accuracies of WellSVM and ?-SVM versus the percentage of labeled data on
different UCI datasets.
0.85
0.85
0.8
0.8
standard
4-nn
0.75
0.75
10
20
30
40
(a) Symmetric Noise.
10
50
20
30
40
50
(b) Asymmetric Noise.
Figure 2: Comparison of the mean accuracy versus the percentage of noise of iterative ?-SVM with different
initializations of ?. The standard curve refers to the initialization of ? yi = 1 and ? ?yi = 0 and the 4-nn to the
initialization of ? ? to the proportion of neighboring instances with label ?.
instances of one class. These preliminary results are encouraging and show that locally estimating
the conditional class density to initialize the ? matrix improves the robustness of our method to label
noise.
6
Conclusion
This paper focuses on the problem of learning from weakly labeled data. We introduced the ?risk which generalizes the standard empirical risk while allowing the integration of weak supervision.
From the expression of the ?-risk , we derived a generic algorithm for weakly labeled data and
specialized it in an SVM-like context. The resulting ?-SVM algorithm has been applied in two
different weakly labeled settings, namely semi-supervised learning and learning with label noise,
showing the advantages of the approach.
The perspectives of this work are numerous and of two main kinds: covering new weakly labeled
settings and studying theoretical guarantees. As proposed in Section 2.3, the ?-risk can be used in
various weakly labeled scenarios. This requires to use different strategies for the initialization and the
refinement of ?, and also to propose proper priors for these parameters. Generalizing the proposed
?-risk to a multi-class setting is a natural extension as ? is already a matrix of class probabilities.
Another broad direction involves deriving robustness and convergence bounds for the algorithms
built on the ?-risk .
7
Acknowledgments
We thank the reviewers for their valuable remarks. We also thank the ANR projects SOLSTICE
(ANR-13-BS02-01) and LIVES (ANR-15-CE230026-03).
8
References
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9
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5,699 | 6,157 | Double Thompson Sampling for Dueling Bandits
Huasen Wu
University of California, Davis
hswu@ucdavis.edu
Xin Liu
University of California, Davis
xinliu@ucdavis.edu
Abstract
In this paper, we propose a Double Thompson Sampling (D-TS) algorithm for
dueling bandit problems. As its name suggests, D-TS selects both the first and the
second candidates according to Thompson Sampling. Specifically, D-TS maintains
a posterior distribution for the preference matrix, and chooses the pair of arms for
comparison according to two sets of samples independently drawn from the posterior distribution. This simple algorithm applies to general Copeland dueling bandits,
including Condorcet dueling bandits as a special case. For general Copeland dueling bandits, we show that D-TS achieves O(K 2 log T ) regret. Moreover, using
a back substitution argument, we refine the regret to O(K log T + K 2 log log T )
in Condorcet dueling bandits and most practical Copeland dueling bandits. In
addition, we propose an enhancement of D-TS, referred to as D-TS+ , to reduce the
regret in practice by carefully breaking ties. Experiments based on both synthetic
and real-world data demonstrate that D-TS and D-TS+ significantly improve the
overall performance, in terms of regret and robustness.
1
Introduction
The dueling bandit problem [1] is a variant of the classical multi-armed bandit (MAB) problem, where
the feedback comes in the form of pairwise comparison. This model has attracted much attention as
it can be applied in many systems such as information retrieval (IR) [2, 3], where user preferences
are easier to obtain and typically more stable. Most earlier work [1, 4, 5] focuses on Condorcet
dueling bandits, where there exists an arm, referred to as the Condorcet winner, that beats all other
arms. Recent work [6, 7] turns to a more general and practical case of a Copeland winner(s), which is
the arm (or arms) that beats the most other arms. Existing algorithms are mainly generalized from
traditional MAB algorithms along two lines: 1) UCB (Upper Confidence Bound)-type algorithms,
such as RUCB [4] and CCB [6]; and, 2) MED (Minimum Empirical Divergence)-type algorithms,
such as RMED [5] and CW-RMED/ECW-RMED [7].
In traditional MAB, an alternative effective solution is Thompson Sampling (TS) [8]. Its principle is
to choose the optimal action that maximizes the expected reward according to the randomly drawn
belief. TS has been successfully applied in traditional MAB [9, 10, 11, 12] and other online learning
problems [13, 14]. In particular, empirical studies in [9] show that TS not only achieves lower regret
than other algorithms in practice, but is also more robust as a randomized algorithm.
In the wake of the success of TS in these online learning problems, a natural question is whether
and how TS can be applied to dueling bandits to further improve the performance. However, it is
challenging to apply the standard TS framework to dueling bandits, because not all comparisons
provide information about the system statistics. Specifically, a good learning algorithm for dueling
bandits will eventually compare the winner against itself. However, comparing one arm against itself
does not provide any statistical information, which is critical in TS to update the posterior distribution.
Thus, TS needs to be adjusted so that 1) comparing the winners against themselves is allowed, but, 2)
trapping in comparing a non-winner arm against itself is avoided.
In this paper, we propose a Double Thompson Sampling (D-TS) algorithm for dueling bandits,
including both Condorcet dueling bandits and general Copeland dueling bandits. As its name
suggests, D-TS typically selects both the first and the second candidates according to samples
independently drawn from the posterior distribution. D-TS also utilizes the idea of confidence
bounds to eliminate the likely non-winner arms, and thus avoids trapping in suboptimal comparisons.
Compared to prior studies on dueling bandits, D-TS has both practical and theoretical advantages.
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
First, the double sampling structure of D-TS better suits the nature of dueling bandits. Launching two
independent rounds of sampling provides us the opportunity to select the same arm in both rounds
and thus to compare the winners against themselves. This double sampling structure also leads to
more extensive utilization of TS (e.g., compared to RCS [3]), and significantly reduces the regret.
In addition, this simple framework applies to general Copeland dueling bandits and achieves lower
regret than existing algorithms such as CCB [6]. Moreover, as a randomized algorithm, D-TS is more
robust in practice.
Second, this double sampling structure enables us to obtain theoretical bounds for the regret of D-TS.
As noted in traditional MAB literature [10, 15], theoretical analysis of TS is usually more difficult than
UCB-type algorithms. The analysis in dueling bandits is even more challenging because the selection
of arms involves more factors and the two selected arms may be correlated. To address this issue, our
D-TS algorithm draws the two sets of samples independently. Because their distributions are fully
captured by historic comparison results, when the first candidate is fixed, the comparison between it
and all other arms is similar to traditional MAB and thus we can borrow ideas from traditional MAB.
Using the properties of TS and confidence bounds, we show that D-TS achieves O(K 2 log T ) regret
for a general K-armed Copeland dueling bandit. More interestingly, the property that the sample
distribution only depends on historic comparing results (but not t) enables us to refine the regret
using a back substitution argument, where we show that D-TS achieves O(K log T + K 2 log log T )
in Condorcet dueling bandits and many practical Copeland dueling bandits.
Based on the analysis, we further refine the tie-breaking criterion in D-TS and propose its enhancement
called D-TS+ . D-TS+ achieves the same theoretical regret bound as D-TS, but performs better in
practice especially when there are multiple winners.
In summary, the main contributions of this paper are as follows:
? We propose a D-TS algorithm and its enhancement D-TS+ for general Copeland dueling
bandits. The double sampling structure suits the nature of dueling bandits and leads to more
extensive usage of TS, which significantly reduces the regret.
? We obtain theoretical regret bounds for D-TS and D-TS+ . For general Copeland dueling
bandits, we show that D-TS and D-TS+ achieve O(K 2 log T ) regret. In Condorcet dueling
bandits and most practical Copeland dueling bandits, we further refine the regret bound to
O(K log T + K 2 log log T ) using a back substitution argument.
? We evaluate the D-TS and D-TS+ algorithms through experiments based on both synthetic
and real-world data. The results show that D-TS and D-TS+ significantly improve the
overall performance, in terms of regret and robustness, compared to existing algorithms.
2
Related Work
Early dueling bandit algorithms study finite-horizon settings, using the ?explore-then-exploit? approaches, such as IF [1], BTM [16], and SAVAGE [17]. For infinite horizon settings, recent work
has generalized the traditional MAB algorithms to dueling bandits along two lines. First, RUCB
[4] and CCB [6] are generalizations of UCB for Condorcet and general Copeland dueling bandits,
respectively. In addition, [18] reduces dueling bandits to traditional MAB, which is then solved
by UCB-type algorithms, called MutiSBM and Sparring. Second, [5] and [7] extend the MED
algorithm to dueling bandits, where they present the lower bound on the regret and propose the
corresponding optimal algorithms, including RMED for Condorcet dueling bandits [5], CW-RMED
and its computationally efficient version ECW-RMED for general Copeland dueling bandits [7].
Different from such existing work, we study algorithms for dueling bandits from the perspective of
TS, which typically achieves lower regret and is more robust in practice.
Dated back to 1933, TS [8] is one of the earliest algorithms for exploration/exploitation tradeoff.
Nowadays, it has been applied in many variants of MAB [11, 12, 13] and other more complex
problems, e.g., [14], due to its simplicity, good performance, and robustness [9]. Theoretical analysis
of TS is much more difficult. Only recently, [10] proposes a logarithmic bound for the standard
frequentist expected regret, whose constant factor is further improved in [15]. Moreover [19, 20]
derive the bounds for its Bayesian expected regret through information-theoretic analysis.
TS has been preliminarily considered for dueling bandits [3, 21]. In particular, recent work [3]
proposes a Relative Confidence Sampling (RCS) algorithm that combines TS with RUCB [4] for
Condorcet dueling bandits. Under RCS, the first arm is selected by TS while the second arm is
selected according to their RUCB. Empirical studies demonstrate the performance improvement of
using RCS in practice, but no theoretical bounds on the regret are provided.
2
3
System Model
We consider a dueling bandit problem with K (K ? 2) arms, denoted by A = {1, 2, . . . , K}. At
(2)
each time-slot t > 0, a pair of arms (a(1)
t , at ) is displayed to a user and a noisy comparison outcome
wt is obtained, where wt = 1 if the user prefers a(1)
to a(2)
t
t , and wt = 2 otherwise. We assume the
user preference is stationary over time and the distribution of comparison outcomes is characterized
by the preference matrix P = [pij ]K?K , where pij is the probability that the user prefers arm i to
arm j, i.e., pij = P{i j}, i, j = 1, 2, . . . , K. We assume that the displaying order does not affect
the preference, and hence, pij + pji = 1 and pii = 1/2. We say that arm i beats arm j if pij > 1/2.
We study the general Copeland dueling bandits, where the Copeland winner is defined as the arm
(or arms) that maximizes
the number of other arms it beats [6, 7]. Specifically, the Copeland
P
score is defined as j6=i 1(pij > 1/2), and the normalized Copeland score is defined as ?i =
P
1
?
j6=i 1(pij > 1/2), where 1(?) is the indicator function. Let ? be the highest normalized
K?1
?
Copeland score, i.e., ? = max1?i?K ?i . Then the Copeland winner is defined as the arm (or arms)
with the highest normalized Copeland score, i.e., C ? = {i : 1 ? i ? K, ?i = ? ? }. Note that the
Condorcet winner is a special case of Copeland winner with ? ? = 1.
A dueling bandit algorithm ? decides which pair of arms to compare depending on the historic obser(2)
vations. Specifically, define a filtration Ht?1 as the history before t, i.e., Ht?1 = {a(1)
? , a? , w? , ? =
(1)
(2)
1, 2, . . . , t ? 1}. Then a dueling bandit algorithm ? is a function that maps Ht?1 to (at , at ), i.e.,
(1)
(2)
(at , at ) = ?(Ht?1 ). The performance of a dueling bandit algorithm ? is measured by its expected
cumulative regret, which is defined as
R? (T ) = ? ? T ?
T
1X
E ?a(1) + ?a(2) .
t
t
2 t=1
(1)
The objective of ? is then to minimize R? (T ). As pointed out in [6], the results can be adapted to
other regret definitions because the above definition bounds the number of suboptimal comparisons.
4
Double Thompson Sampling
4.1
D-TS Algorithm
We present the D-TS algorithm for Copeland dueling bandits, as described in Algorithm 1 (time
index t is omitted in pseudo codes for brevity). As its name suggests, the basic idea of D-TS is to
select both the first and the second candidates by TS. For each pair (i, j) with i 6= j, we assume a
beta prior distribution for its preference probability pij . These distributions are updated according to
the comparison results Bij (t ? 1) and Bji (t ? 1), where Bij (t ? 1) (resp. Bji (t ? 1)) is the number
of time-slots when arm i (resp. j) beats arm j (resp. i) before t. D-TS selects the two candidates by
sampling from the posterior distributions.
Specifically, at each time-slot t, the D-TS algorithm consists of two phases that select the first and
the second candidates, respectively. When choosing the first candidate a(1)
t , we first use the RUCB
[4] of pij to eliminate the arms that are unlikely to be the Copeland winner, resulting in a candidate
(1)
set Ct (Lines 4 to 6). The algorithm then samples ?ij
(t) from the posterior beta distribution, and the
(1)
first candidate at is chosen by ?majority voting?, i.e., the arm within Ct that beats the most arms
(1)
according to ?ij
(t) will be selected (Lines 7 to 11). The ties are broken randomly here for simplicity
and will be refined later in Section 4.3. A similar idea is applied to select the second candidate a(2)
t ,
where new samples ?(2)(1) (t) are generated and the arm with the largest ?(2)(1) (t) among all arms with
iat
iat
lia(1) ? 1/2 is selected as the second candidate (Lines 13 to 14).
t
The double sampling structure of D-TS is designed based on the nature of dueling bandits, i.e., at each
time-slot, two arms are needed for comparison. Unlike RCS [3], D-TS selects both candidates using
TS. This leads to more extensive utilization of TS and thus achieves much lower regret. Moreover,
the two sets of samples are independently distributed, following the same posterior that is only
determined by the comparison statistics Bij (t ? 1) and Bji (t ? 1). This property enables us to obtain
an O(K 2 log T ) regret bound and further refine it by a back substitution argument, as discussed later.
We also note that RUCB-based elimination (Lines 4 to 6) and RLCB (Relative Lower Confidence
Bound)-based elimination (Line 14) are essential in D-TS. Without these eliminations, the algorithm
3
Algorithm 1 D-TS for Copeland Dueling Bandits
1: Init: B ? 0K?K ; // Bij is the number of time-slots that the user prefers arm i to j.
2: for t = 1 to T do
3:
// Phase 1: Choose the first candidate a(1)
q
q
Bij
Bij
? log t
? log t
,
l
=
4:
U := [uij ], L := [lij ], where uij = Bij +B
+
?
ij
Bij +Bji
Bij +Bji
Bij +Bji , if
ji
i 6= j, and uii = lii = 1/2, ?i; // x0 := 1 for any x.
P
1
??i ? K?1
j6=i 1(uij > 1/2); // Upper bound of the normalized Copeland score.
?
C ? {i : ?i = maxj ??j };
for i, j = 1, . . . , K with i < j do
(1)
Sample ?ij ? Beta(Bij + 1, Bji + 1);
5:
6:
7:
8:
(1)
(1)
9:
10:
11:
?ji ? 1 ? ?ij ;
end for
P
(1)
a(1) ? arg max j6=i 1(?ij > 1/2); // Choosing from C to eliminate likely non-winner
12:
13:
arms; Ties are broken randomly.
// Phase 2: Choose the second candidate a(2)
(2)
(2)
Sample ?ia(1) ? Beta(Bia(1) + 1, Ba(1) i + 1) for all i 6= a(1) , and let ?a(1) a(1) = 1/2;
14:
a(2) ? arg max ?ia(1) ; // Choosing only from uncertain pairs.
i?C
(2)
i:lia(1) ?1/2
15:
// Compare and Update
16:
Compare pair (a(1) , a(2) ) and observe the result w;
17:
Update B: Ba(1) a(2) ? Ba(1) a(2) + 1 if w = 1, or Ba(2) a(1) ? Ba(2) a(1) + 1 if w = 2;
18: end for
may trap in suboptimal comparisons. Consider one extreme case in Condorcet dueling bandits1 :
assume arm 1 is the Condorcet winner with p1j = 0.501 for all j > 1, and arm 2 is not the Condorcet
winner, but with p2j = 1 for all j > 2. Then for a larger K (e.g., K > 4), without RUCB-based
elimination, the algorithm may trap in a(1)
= 2 for a long time, because arm 2 is likely to receive
t
higher score than arm 1. This issue can be addressed by RUCB-based elimination as follows: when
chosen as the first candidate, arm 2 has a great probability to compare with arm 1; after sufficient
comparisons with arm 1, arm 2 will have u21 (t) < 1/2 with high probability; then arm 2 is likely
to be eliminated because arm 1 has ??1 (t) = 1 > ??2 (t) with high probability. Similarly, RLCB-based
elimination (Line 14, where we restrict to the arms with lia(1) (t) ? 1/2) is important especially
t
for non-Condorcet dueling bandits. Specifically, lia(1) (t) > 1/2 indicates that arm i beats a(1)
with
t
t
high probability. Thus, comparing a(1)
and arm i brings little information gain and thus should be
t
eliminated to minimize the regret.
4.2
Regret Analysis
Before conducting the regret analysis, we first introduce certain notations that will be used later.
Gap to 1/2: In dueling bandits, an important benchmark for pij is 1/2, and thus we let ?ij be the
gap between pij and 1/2, i.e., ?ij = |pij ? 1/2|.
(1)
(2)
Number of Comparisons: Under D-TS, (i, j) can be compared in the form of (at , at ) = (i, j)
(1) (2)
and (at , at ) = (j, i). We consider these two cases separately and define the following counters:
Pt
Pt
(1)
(1)
(2)
(2)
(1)
(2)
Nij (t) = ? =1 1(a? = i, a? = j) and Nij (t) = ? =1 1(a? = j, a? = i). Then the total
(1)
(2)
(1)
(2)
number of comparisons is Nij (t) = Nij (t) + Nij (t) for i 6= j, and Nii (t) = Nii (t) = Nii (t)
for i = j.
4.2.1
O(K 2 log T ) Regret
To obtain theoretical bounds for the regret of D-TS, we make the following assumption:
1
A Borda winner may be more appropriate in this special case [22], and we mainly use it to illustrate the
dilemma.
4
Assumption 1: The preference probability pij 6= 1/2 for any i 6= j.
Under Assumption 1, we present the first result for D-TS in general Copeland dueling bandits:
Proposition 1. When applying D-TS with ? > 0.5 in a Copeland dueling bandit with a preference
matrix P = [pij ]K?K satisfying Assumption 1, its regret is bounded as:
X
log T
4? log T
K2
+
(1
+
)
(2)
RD-TS (T ) ?
+ O( 2 ),
2
?ij
D(pij ||1/2)
i6=j:pij <1/2
where > 0 is an arbitrary constant, and D(p||q) = p log
p
q
+ (1 ? p) log
1?p
1?q
is the KL divergence.
The summation operation in Eq. (2) is conducted over all pairs (i, j) with pij < 1/2. Thus,
Proposition 1 states that D-TS achieves O(K 2 log T ) regret in Copeland dueling bandits. To the best
of our knowledge, this is the first theoretical bound for TS in dueling bandits. The scaling behavior
of this bound with respect to T is order optimal, since a lower bound ?(log T ) has been shown in [7].
The refinement of the scaling behavior with respect to K will be discussed later.
Proving Proposition 1 needs to bound the number of comparisons for all pairs (i, j) with i ?
/ C?
(1)
?
or j ?
/ C . When fixing the first candidate as at = i, the selection of the second candidate a(2)
t
is similar to a traditional K-armed bandit problem with expected utilities pji (j = 1, 2, . . . , K).
However, the analysis is more complex here since different arms are eliminated differently depending
on the value of pji . We prove Proposition 1 through Lemmas 1 to 3, which bound the number of
comparisons for all suboptimal pairs (i, j) under different scenarios, i.e., pji < 1/2, pji > 1/2, and
pji = 1/2 (j = i ?
/ C ? ), respectively.
Lemma 1. Under D-TS, for an arbitrary constant > 0 and one pair (i, j) with pji < 1/2, we have
log T
1
(1)
E[Nij (T )] ? (1 + )
+ O( 2 ).
(3)
D(pji ||1/2)
Proof. We can prove this lemma by viewing the comparison between the first candidate arm i and its
inferiors as a traditional MAB. In fact, it may be even simpler than that in [15] because under D-TS,
arm j with pji < 1/2 is competing with arm i with pii = 1/2, which is known and fixed. Then we
(1)
can bound E[Nij (T )] using the techniques in [15]. Details can be found in Appendix B.1.
Lemma 2. Under D-TS with ? > 0.5, for one pair (i, j) with pji > 1/2, we have
4? log T
(1)
E[Nij (T )] ?
+ O(1).
?2ji
(1)
(4)
(2)
Proof. We note that when at = i, arm j can be selected as at only when its RLCB lji (t) ? 1/2.
(1)
T
Then we can bound E[Nij (T )] by O( 4??log
) similarly to the analysis of traditional UCB algorithms
2
ji
[23]. Details can be found in Appendix B.2.
Lemma 3. Under D-TS, for any arm i ?
/ C ? , we have
X
1
1
1
E[Nii (T )] ? O(K) +
? 2 + 2
+ 4 = O(K).
?ki
?ki D(1/2||pki ) ?ki
(5)
k:pki >1/2
Before proving Lemma 3, we present an important property for ??? (t) := max1?i?K ??i (t). Recall
that ? ? is the maximum normalized Copeland score. Using the concentration property of RUCB
(Lemma 6 in Appendix A), the following lemma shows that ??? (t) is indeed a UCB of ? ? .
log t
2?
Lemma 4. For any ? > 0.5 and t > 0, P{??? (t) ? ? ? } ? 1 ? K
+ 1 t? ?+1/2 .
log(?+1/2)
Return to the proof of Lemma 3. To prove Lemma 3, we consider the cases of ??? (t) < ? ? and
??? (t) ? ? ? . The former case ??? (t) < ? ? can be bounded by Lemma 4. For the latter case, we
(2)
note that when ??? (t) ? ? ? , the event (a(1)
t , at ) = (i, i) occurs only if: a) there exists at least one
(2)
k ? K with pki > 1/2, such that lki (t) ? 1/2; and b) ?ki (t) ? 1/2 for all k with lki (t) ? 1/2. In
(1) (2)
(1) (2)
this case, we can bound the probability of (at , at ) = (i, i) by that of (at , at ) = (i, k), for k
with pki > 1/2 but lki (t) ? 1/2, where the coefficient decays exponentially. Then we can bound
E[Nii (T )] by O(1) similar to [15]. Details of proof can be found in Appendix B.4.
The conclusion of Proposition 1 then follows by combining Lemmas 1 to 3.
5
4.2.2
Regret Bound Refinement
In this section, we refine the regret bound for D-TS and reduce its scaling factor with respect to the
number of arms K.
We sort the arms for each i ?
/ C ? in the descending order of pji , and let (?i(1) , ?i(2) , . . . , ?i(K) )
be a permutation of (1, 2, . . . , K), such that p?i(1) ,i ? p?i(2) ,i ? . . . ? p?i(K) ,i . In addition, for a
PK
Copeland winner i? ? C, let LC = j=1 1(pji? > 1/2) be the number of arms that beat arm i? . To
refine the regret, we introduce an additional no-tie assumption:
Assumption 2: For each arm i ?
/ C ? , p?i(LC +1) ,i > p?i(j) ,i for all j > LC + 1.
We present a refined regret bound for D-TS as follows:
Theorem 1. When applying D-TS with ? > 0.5 in a Copeland dueling bandit with a preference
matrix P = [pij ]K?K satisfying Assumptions 1 and 2, its regret is bounded as:
RD-TS (T )
?
X
i?C ?
X
j:pji >1/2
+?(1 + )2
4? log T
+
?2ji
X
K
X
i?C
/ ? j=LC +2
X
j:pji <1/2
(1 + )
X LX
C +1
log T
4? log T
+
D(pji ||1/2)
?2?i(j) ,i
? j=1
i?C
/
K2
log log T
+ O(K 3 ) + O( 2 ),
D(p?i(j) ,i ||p?i(LC +1) ,i )
(6)
where ? > 2 and > 0 are constants, and D(?||?) is the KL-divergence.
In (6), the first term corresponds to the regret when the first candidate a(1)
is a winner, and is
t
O(K|C ? | log T ). The second term corresponds to the comparisons between a non-winner arm
and its first LC + 1 superiors, which is bounded by O(K(LC + 1) log T ). The remaining terms
correspond to the comparisons between a non-winner arm and the remaining arms, and is bounded by
O K 2 log log T . As demonstrated in [6], LC is relatively small compared to K, and can be viewed
as a constant. Thus, the total regret RD-TS (T ) is bounded as RD-TS (T ) = O(K log T + K 2 log log T ).
In particular, this asymptotic trend can be easily seen for Condorcet dueling bandits where LC = 0.
Comparing Eq. (6) with Eq. (2), we can see the difference is the third and fourth terms in (6), which
refine the regret of comparing a suboptimal arm and its last (K ? LC ? 1) inferiors into O(log log T ).
Thus, to prove Theorem 1, it suffices to show the following additional lemma:
Lemma 5. Under Assumptions 1 and 2, for any suboptimal arm i ?
/ C ? and j > LC + 1, we have
(1)
E[Ni?i(j) (T )] ?
?(1 + )2 log log T
1
+ O(K) + O( 2 ),
D(p?i(j) ,i ||p?i(LC +1) ,i )
(7)
where ? > 2 and > 0 are constants.
Proof. We prove this lemma using a back substitution argument. The intuition is that when fixing the
(1)
first candidate as a(1)
= i, the comparison between at and the other arms is similar to a traditional
t
P
MAB with expected utilities pji (1 ? j ? K). Let Ni(1) (T ) = Tt=1 1(a(1)
= i) be the number
t
of time-slots when this type of MAB is played. Using the fact that the distribution of the samples
(1)
(1)
only depends on the historic comparison results (but not t), we can show E[Ni,?
(T )|Ni (T )] =
i(j)
(1)
(1)
(1)
O(log Ni (T )), which holds for any Ni (T ). We have shown that E[Ni (T )] = O(K log T ) for
(1)
any i 6= C ? when proving Proposition 1. Then, substituting the bound of E[Ni (T )] back and
(1)
(1)
(1)
using the concavity of the log(?) function, we have E[Ni,?i(j) (T )] = E E[Ni,?i(j) (T )|Ni (T )] ?
(1)
O(log E[Ni (T )]) = O(log log T + log K). Details can be found in Appendix C.1
4.3
Further Improvement: D-TS+
D-TS is a TS framework for dueling bandits, and its performance can be improved by refining certain
components of it. In this section, we propose an enhanced version of D-TS, referred to as D-TS+ ,
that carefully breaks the ties to reduce the regret.
Note that by randomly breaking the ties (Line 11 in Algorithm 1), D-TS tends to explore all potential
winners. This may be desirable in certain applications such as restaurant recommendation, where
6
users may not want to stick to a single winner. However, because of this, the regret of D-TS scales
with the number of winners |C ? | as shown in Theorem 1. To further reduce the regret, we can break
the ties according to estimated regret.
(1)
Specifically, with samples ?ij
(t), the normalized Copeland score for each arm i can be estiP
(1)
1
?
mated as ?i (t) = K?1 j6=i 1(?ij (t) > 1/2). Then the maximum normalized Copeland score is
??? (t) = maxi ??i (t), and the loss of comparing arm i and arm j is r?ij (t) = ??? (t) ? 1 ??i (t) + ??j (t) .
2
T
For pij 6= 1/2, we need about ?( D(plog
) time-slots to distinguish it from 1/2 [5]. Thus, when
ij ||1/2)
choosing i as the first candidate, the regret of comparing it with all other arms can be estimated by
(1)
? (1) (t) = P (1)
R
r? (t)/D(?ij (t)||1/2). We propose the following D-TS+ algorithm that
i
j:? (t)6=1/2 ij
ij
? (1) (t).
breaks the ties to minimize R
i
D-TS+ : Implement the same operations as D-TS, except for the selection of the first candidate
(Line 11 in Algorithm 1) is replaced by the following two steps:
A(1) ? {i ? C : ?i = max
i?C
(1)
a
X
(1)
1(?ij > 1/2)};
j6=i
? (1) ;
? arg min R
i
i?A(1)
D-TS+ only changes the tie-breaking criterion in selecting the first candidate. Thus, the regret bound
of D-TS directly applies to D-TS+ :
Corollary 1. The regret of D-TS+ , RD-TS+ (T ), satisfies inequality (6) under Assumptions 1 and 2.
Corollary 1 provides an upper bound for the regret of D-TS+ . In practice, however, D-TS+ performs
better than D-TS in the scenarios with multiple winners, as we can see in Section 5 and Appendix D.
Our conjecture is that with this regret-minimization criterion, the D-TS+ algorithm tends to focus
on one of the winners (if there is no tie in terms of expected regret), and thus reduces the first term
in (6) from O(K|C ? | log T ) to O(K log T ). The proof of this conjecture requires properties for the
evolution of the statistics for all arms and the majority voting results based on the Thompson samples,
and is complex. This is left as part of our future work.
In the above D-TS+ algorithm, we only consider the regret of choosing i as the first candidate. From
Theorem 1, we know that comparing other arms with their superiors will also result in ?(log T )
regret. Thus, although the current D-TS+ algorithm performs well in most practical scenarios, one
? (1) (t).
may further improve its performance by taking these additional comparisons into account in R
i
5
Experiments
To evaluate the proposed D-TS and D-TS+ algorithms, we run experiments based on synthetic and
real-world data. Here we present the results for experiments based on the Microsoft Learning to Rank
(MSLR) dataset [24], which provides the relevance for queries and ranked documents. Based on this
dataset, [6] derives a preference matrix for 136 rankers, where each ranker is a function that maps a
user?s query to a document ranking and can be viewed as one arm in dueling bandits. We use the
two 5-armed submatrices in [6], one for Condorcet dueling bandit and the other for non-Condorcet
dueling bandit. More experiments and discussions can be found in Appendix D 2 .
We compare D-TS and D-TS+ with the following algorithms: BTM [16], SAVAGE [17], Sparring
[18], RUCB [4], RCS [3], CCB [6], SCB [6], RMED1 [5], and ECW-RMED [7]. For BTM, we set
the relaxed factor ? = 1.3 as [16]. For algorithms using RUCB and RLCB, including D-TS and
D-TS+ , we set the scale factor ? = 0.51. For RMED1, we use the same settings as [5], and for
ECW-RMED, we use the same setting as [7]. For the ?explore-then-exploit? algorithms, BTM and
SAVAGE, each point is obtained by resetting the time horizon as the corresponding value. The results
are averaged over 500 independent experiments, where in each experiment, the arms are randomly
shuffled to prevent algorithms from exploiting special structures of the preference matrix.
In Condorcet dueling bandits, our D-TS and D-TS+ algorithms achieve almost the same performance
and both perform much better than existing algorithms, as shown in Fig. 1(a). In particular, compared
with RCS, we can see that the full utilization of TS in D-TS and D-TS+ significantly reduces the
2
Source codes are available at https://github.com/HuasenWu/DuelingBandits.
7
500
15
B
RU
/CC
CB
ED
CW
-RM
D1/E
E
/RM
S
RC
#10 4
BTM
SAVAGE
RUCB
RCS
Sparring
CCB
SCB
RMED1
ECW-RMED
D-TS
10
Regret
Regret
1000
BTM
SAVAGE
RUCB
RCS
Sparring
CCB
SCB
RMED1
ECW-RMED
D-TS
+
S
D-T
D-TS+
-T
S/D
5
D-TS+
S
D-T
ECW-RMED
0
10 2
10 4
Time t
(a) K = 5, Condorcet
10 6
0
10 4
10 6
19
3.0
9%
D-TS+
150
100
50
29
D-TS +
10 5
ECW-RMED
D-TS
200
CCB
Normalized STD of regret
1500
.37
% 5%
1%
.6 30.5
27
29
.12
%
13
10 7
Time t
(b) K = 5, non-Condorcet
Figure 1: Regret in MSLR dataset. In (b), there are 3 Copeland
winners with normalized Copeland score ? ? = 3/4.
0
.16
%
K = 5, Condorcet
K = 5, non-Condorcet
Dataset
Figure 2: Standard deviation
(STD) of regret for T = 106 (normalized by RECW?RMED (T )).
regret. Compared with RMED1 and ECW-RMED, our D-TS and D-TS+ algorithms also perform
better. [5] has shown that RMED1 is optimal in Condorcet dueling bandits, not only in the sense of
asymptotic order, but also the coefficients in the regret bound. The simulation results show that D-TS
and D-TS+ not only achieve the similar slope as RMED1/ECW-RMED, but also converge faster to
the asymptotic regime and thus achieve much lower regret. This inspires us to further refine the regret
bounds for D-TS and D-TS+ in the future.
In non-Condorcet dueling bandits, as shown in Fig. 1(b), D-TS and D-TS+ significantly reduce the
regret compared to the UCB-type algorithm, CCB (e.g., the regret of D-TS+ is less than 10% of that
of CCB). Compared with ECW-RMED, D-TS achieves higher regret, mainly because it randomly
explores all Copeland winners due to the random tie-breaking rule. With a regret-minimization
tie-breaking rule, D-TS+ further reduces the regret, and outperforms ECW-RMED in this dataset.
Moreover, as randomized algorithms, D-TS and D-TS+ are more robust to the preference probabilities.
As shown in Fig. 2, D-TS and D-TS+ have much smaller regret STD than that of ECW-RMED in the
non-Condorcet dataset, where certain preference probabilities (for different arms) are close to 1/2.
In particular, the STD of regret for ECW-RMED is almost 200% of its mean value, while it is only
13.16% for D-TS+ . In addition, as shown in Appendix D.2.3, D-TS and D-TS+ are also robust to
delayed feedback, which is typically batched and provided periodically in practice.
Overall, D-TS and D-TS+ significantly outperform all existing algorithms, with the exception of
ECW-RMED. Compared to ECW-RMED, D-TS+ achieves much lower regret in the Condorcet case,
lower or comparable regret in the non-Condorcet case, and much more robustness in terms of regret
STD and delayed feedback. Thus, the simplicity, good performance, and robustness of D-TS and
D-TS+ make them good algorithms in practice.
6
Conclusions and Future Work
In this paper, we study TS algorithms for dueling bandits. We propose a D-TS algorithm and its
enhanced version D-TS+ for general Copeland dueling bandits, including Condorcet dueling bandits
as a special case. Our study reveals desirable properties of D-TS and D-TS+ from both theoretical
and practical perspectives. Theoretically, we show that the regret of D-TS and D-TS+ is bounded by
O(K 2 log T ) in general Copeland dueling bandits, and can be refined to O(K log T + K 2 log log T )
in Condorcet dueling bandits and most practical Copeland dueling bandits. Practically, experimental
results demonstrate that these simple algorithms achieve significantly better overall-performance than
existing algorithms, i.e., D-TS and D-TS+ typically achieve much lower regret in practice and are
robust to many practical factors, such as preference matrix and feedback delay.
Although logarithmic regret bounds have been obtained for D-TS and D-TS+ , our analysis relies
heavily on the properties of RUCB/RLCB and the regret bounds are likely loose. In fact, we see
from experiments that RUCB-based elimination seldom occurs under most practical settings. We
will further refine the regret bounds by investigating the properties of TS-based majority-voting.
Moreover, results from recent work such as [7] may be leveraged to improve TS algorithms. Last, it
is also an interesting future direction to study D-TS type algorithms for dueling bandits with other
definition of winners.
Acknowledgements: This research was supported in part by NSF Grants CCF-1423542, CNS1457060, and CNS-1547461. The authors would like to thank Prof. R. Srikant (UIUC), Prof. Shipra
Agrawal (Columbia University), Masrour Zoghi (University of Amsterdam), and Dr. Junpei
Komiyama (University of Tokyo) for their helpful discussions and suggestions.
8
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[24] Microsoft Research, Microsoft Learning to Rank Datasets. http://research.microsoft.com/enus/projects/mslr/, 2010.
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