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+Sharper analysis of sparsely activated wide neural
+networks with trainable biases
+Hongru Yang ∗
+Ziyu Jiang †
+Ruizhe Zhang ‡
+Zhangyang Wang §
+Yingbin Liang ¶
+Abstract
+This work studies training one-hidden-layer overparameterized ReLU networks via gradient
+descent in the neural tangent kernel (NTK) regime, where, differently from the previous works,
+the networks’ biases are trainable and are initialized to some constant rather than zero. The
+tantalizing benefit of such initialization is that the neural network will provably have sparse
+activation pattern before, during and after training, which can enable fast training procedures
+and, therefore, reduce the training cost. The first set of results of this work characterize the
+convergence of the network’s gradient descent dynamics. Surprisingly, we show that the net-
+work after sparsification can achieve as fast convergence as the original network. Further, the
+required width is provided to ensure gradient descent can drive the training error towards zero
+at a linear rate. The contribution over previous work is that not only the bias is allowed to
+be updated by gradient descent under our setting but also a finer analysis is given such that
+the required width to ensure the network’s closeness to its NTK is improved. Secondly, the
+networks’ generalization bound after training is provided. A width-sparsity dependence is pre-
+sented which yields sparsity-dependent localized Rademacher complexity and a generalization
+bound matching previous analysis (up to logarithmic factors). To our knowledge, this is the
+first sparsity-dependent generalization result via localized Rademacher complexity. As a by-
+product, if the bias initialization is chosen to be zero, the width requirement improves the
+previous bound for the shallow networks’ generalization. Lastly, since the generalization bound
+has dependence on the smallest eigenvalue of the limiting NTK and the bounds from previous
+works yield vacuous generalization, this work further studies the least eigenvalue of the limiting
+NTK. Surprisingly, while it is not shown that trainable biases are necessary, trainable bias helps
+to identify a nice data-dependent region where a much finer analysis of the NTK’s smallest
+eigenvalue can be conducted, which leads to a much sharper lower bound than the previously
+known worst-case bound and, consequently, a non-vacuous generalization bound. Experimental
+evaluation is provided to evaluate our results.
+1
+Introduction
+The literature of sparse neural networks can be dated back to the early work of LeCun et al. (1989)
+where they showed that a fully-trained neural network can be pruned to preserve generalization.
+∗Department of Computer Science, The University of Texas at Austin; e-mail: hy6385@utexas.edu
+†Department of Computer Science, Texas A&M University; e-mail: jiangziyu@tamu.edu
+‡Department of Computer Science, The University of Texas at Austin; e-mail: ruizhe@utexas.edu
+§Department
+of
+Electrical
+and
+Computer
+Engineering,
+The
+University
+of
+Texas
+at
+Austin;
+e-mail:
+atlaswang@utexas.edu
+¶Department of Electrical and Computer Engineering, The Ohio State University; e-mail: liang.889@osu.edu
+1
+arXiv:2301.00327v1  [cs.LG]  1 Jan 2023
+
+Recently, training sparse neural networks has been receiving increasing attention since the discovery
+of the lottery ticket hypothesis (Frankle and Carbin, 2018). In their work, they showed that if we
+repeatedly train and prune a neural network and then rewind the weights to the initialization,
+we are able to find a sparse neural network that can be trained to match the performance of its
+dense counterpart. However, this method is more of a proof of concept and is computationally
+expensive for any practical purposes. Nonetheless, this inspires further interest in the machine
+learning community to develop efficient methods to find the sparse pattern at the initialization
+such that the performance of the sparse network matches the dense network after training (Lee
+et al., 2018; Wang et al., 2019; Tanaka et al., 2020; Liu and Zenke, 2020; Chen et al., 2021; He
+et al., 2017; Liu et al., 2021b).
+On the other hand, instead of trying to find some desired sparsity patterns at the initialization,
+another line of research has been focusing on inducing the sparsity pattern naturally and then
+creatively utilizing such sparse structure via high-dimensional geometric data structures as well as
+sketching or even quantum algorithms to speedup per-step gradient descent training (Song et al.,
+2021a,b; Hu et al., 2022; Gao et al., 2022). In this line of theoretical studies, the sparsity is induced
+by shifted ReLU which is the same as initializing the bias of the network’s linear layer to some
+large constant instead of zero and holding the bias fixed throughout the entire training. By the
+concentration of Gaussian, at the initialization, the total number of activated neurons (i.e., ReLU
+will output some non-zero value) will be sublinear in the total number m of neurons, as long as the
+bias is initialized to be C√log m for some appropriate constant C. We call this sparsity-inducing
+initialization. If the network is in the NTK regime, each neuron weight will exhibit microscopic
+change after training, and thus the sparsity can be preserved throughout the entire training process.
+Therefore, during the entire training process, only a sublinear number of the neuron weights need
+to be updated, which can significantly speedup the training process.
+The focus of this work is along the above line of theoretical studies of sparsely trained overpa-
+rameterized neural networks and address the two main research limitations in the aforementioned
+studies. (1) The bias parameters used in the previous works are not trainable, contrary to what
+people are doing in practice. (2) The previous works only provided the convergence guarantee,
+while lacking the generalization performance which is of the central interest in deep learning
+theory. Thus, our study will fill the above important gaps, by providing a comprehensive study of
+training one-hidden-layer sparsely activated neural networks in the NTK regime with (a) trainable
+biases incorporated in the analysis; (b) finer analysis of the convergence; and (c) first general-
+ization bound for such sparsely activated neural networks after training with sharp bound on the
+restricted smallest eigenvalue of the limiting NTK. We further elaborate our technical contributions
+are follows:
+1. Convergence. Surprisingly, Theorem 3.1 shows that the network after sparsification can
+achieve as fast convergence as the original network. It further provides the required width
+to ensure that gradient descent can drive the training error towards zero at a linear rate.
+Our convergence result contains two novel ingredients compared to the existing study. (1)
+Our analysis handles trainable bias, and shows that even though the biases are allowed to
+be updated from its initialization, the network’s activation remains sparse during the entire
+training. This relies on our development of a new result showing that the change of bias is
+also diminishing with a O(1/√m) dependence on the network width m. (2) A finer analysis is
+provided such that the required network width to ensure the convergence can be much smaller,
+with an improvement upon the previous result by a factor of �Θ(n8/3) under appropriate bias
+2
+
+initialization, where n is the sample size. This relies on our novel development of (1) a better
+characterization of the activation flipping probability via an analysis of the Gaussian anti-
+concentration based on the location of the strip and (2) a finer analysis of the initial training
+error.
+2. Generalization. Theorem 3.8 studies the generalization of the network after gradient descent
+training where we characterize how the network width should depend on activation sparsity,
+which lead to a sparsity-dependent localized Rademacher complexity and a generalization
+bound matching previous analysis (up to logarithmic factors).
+To our knowledge, this is
+the first sparsity-dependent generalization result via localized Rademacher complexity. In
+addition, compared with previous works, our result yields a better width’s dependence by a
+factor of n10. This relies on (1) the usage of symmetric initialization and (2) a finer analysis
+of the weight matrix change in Frobenius norm in Lemma 3.13.
+3. Restricted Smallest Eigenvalue. Theorem 3.8 shows that the generalization bound heav-
+ily depends on the smallest eigenvalue λmin of the limiting NTK. However, the previously
+known worst-case lower bounds on λmin under data separation have a 1/n2 explicit depen-
+dence in (Oymak and Soltanolkotabi, 2020; Song et al., 2021a), making the generalization
+bound vacuous. Instead, our Theorem 3.11 establishes a much sharper lower bound restricted
+to a data-dependent region, which is sample-size-independent. This hence yields a desirable
+generalization bound that vanishes as fast as O(1/√n), given that the label vector is in this
+region, which can be done with simple label-shifting.
+1.1
+Further Related Works
+Besides the works mentioned in the introduction, another work related to ours is (Liao and Kyril-
+lidis, 2022) where they also considered training a one-hidden-layer neural network with sparse
+activation and studied its convergence. However, different from our work, their sparsity is induced
+by sampling a random mask at each step of gradient descent whereas our sparsity is induced by
+non-zero initialization of the bias terms. Also, their network has no bias term, and they only focus
+on studying the training convergence but not generalization. We discuss additional related works
+here.
+Training Overparameterized Neural Networks. Over the past few years, a tremendous
+amount of efforts have been made to study training overparameterized neural networks. A series
+of works have shown that if the neural network is wide enough (polynomial in depth, number
+of samples, etc), gradient descent can drive the training error towards zero in a fast rate either
+explicitly (Du et al., 2018, 2019; Ji and Telgarsky, 2019) or implicitly (Allen-Zhu et al., 2019; Zou
+and Gu, 2019; Zou et al., 2020) using the neural tangent kernel (NTK) (Jacot et al., 2018). Further,
+under some conditions, the networks can generalize (Cao and Gu, 2019). Under the NTK regime,
+the trained neural network can be well-approximated by its first order Taylor approximation from
+the initialization and Liu et al. (2020) showed that this transition to linearity phenomenon is a
+result from a diminishing Hessian 2-norm with respect to width. Later on, Frei and Gu (2021)
+and Liu et al. (2022) showed that closeness to initialization is sufficient but not necessary for
+gradient descent to achieve fast convergence as long as the non-linear system satisfies some variants
+of the Polyak-�Lojasiewicz condition. On the other hand, although NTK offers good convergence
+explanation, it contradicts the practice since (1) the neural networks need to be unrealistically wide
+and (2) the neuron weights merely change from the initialization. As Chizat et al. (2019) pointed
+3
+
+out, this “lazy training” regime can be explained by a mere effect of scaling. Other works have
+considered the mean-field limit (Chizat and Bach, 2018; Mei et al., 2019; Chen et al., 2020), feature
+learning (Allen-Zhu and Li, 2020, 2022; Shi et al., 2021; Telgarsky, 2022) which allow the weights
+to travel far away from the initialization.
+Sparse Neural Networks in Practice. Besides finding a fixed sparse mask at the initial-
+ization as we mentioned in introduction, on the other hand, dynamic sparse training allows the
+sparse mask to be updated during training, e.g., (Mocanu et al., 2018; Mostafa and Wang, 2019;
+Evci et al., 2020; Jayakumar et al., 2020; Liu et al., 2021a,c,d).
+2
+Preliminaries
+Notations. We use ∥·∥2 to denote vector or matrix 2-norm and ∥·∥F to denote the Frobenius norm
+of a matrix. When the subscript of ∥·∥ is unspecified, it is default to be the 2-norm. For matrices
+A ∈ Rm×n1 and B ∈ Rm×n2, we use [A, B] to denote the row concatenation of A, B and thus [A, B]
+is a m × (n1 + n2) matrix. For matrix X ∈ Rm×n, the row-wise vectorization of X is denoted by
+⃗X = [x1, x2, . . . , xm]⊤ where xi is the i-th row of X. For a given integer n ∈ N, we use [n] to
+denote the set {0, . . . , n}, i.e., the set of integers from 0 to n. For a set S, we use S to denote the
+complement of S. We use N(µ, σ2) to denote the Gaussian distribution with mean µ and standard
+deviation σ. In addition, we use �O, �Θ, �Ω to suppress (poly-)logarithmic factors in O, Θ, Ω.
+2.1
+Problem Formulation
+Let the training set to be (X, y) where X = (x1, x2, . . . , xn) ∈ Rd×n denotes the feature matrix
+consisting of n d-dimensional vectors, and y = (y1, y2, . . . , yn) ∈ Rn consists of the corresponding
+n response variables. We assume ∥xi∥2 ≤ 1 and yi = O(1) for all i ∈ [n]. We use one-hidden-layer
+neural network and consider the regression problem with the square loss function:
+f(x; W, b) :=
+1
+√m
+m
+�
+r=1
+arσ(⟨wr, x⟩ − br),
+L(W, b) := 1
+2
+n
+�
+i=1
+(f(xi; W, b) − yi)2,
+where W ∈ Rm×d with its r-th row being wr, b ∈ Rm is a vector with br being the bias of r-th
+neuron, ar is the second layer weight, and σ(·) denotes the ReLU activation function. We initialize
+the neural network by Wr,i ∼ N(0, 1) and ar ∼ Uniform({±1}) and br = B for some value B ≥ 0
+of choice, for all r ∈ [m], i ∈ [d]. We train only the parameters W and b (i.e., the linear layer ar
+for r ∈ [m] is not trained) via gradient descent, the update of which are given by
+wr(t + 1) = wr(t) − η∂L(W(t), b(t))
+∂wr
+,
+br(t + 1) = br(t) − η∂L(W(t), b(t))
+∂br
+.
+By the chain rule, we have
+∂L
+∂wr = ∂L
+∂f
+∂f
+∂wr . The gradient of the loss with respect to the network is
+∂L
+∂f = �n
+i=1(f(xi; W, b) − yi) and the network gradients with respect to weights and bias are
+∂f(x; W, b)
+∂wr
+=
+1
+√marxI(w⊤
+r x ≥ br),
+∂f(x; W, b)
+∂br
+= − 1
+√marI(w⊤
+r x ≥ br),
+4
+
+where I(·) is the indicator function. We further define H as the NTK matrix of this network with
+Hi,j(W, b) :=
+�∂f(xi; W, b)
+∂W
+, ∂f(xj; W, b)
+∂W
+�
++
+�∂f(xi; W, b)
+∂b
+, ∂f(xj; W, b)
+∂b
+�
+= 1
+m
+m
+�
+r=1
+(⟨xi, xj⟩ + 1)I(w⊤
+r xi ≥ br, w⊤
+r xj ≥ br)
+(2.1)
+and the infinite-width version H∞(B) of the NTK matrix H is given by
+H∞
+ij (B) :=
+E
+w∼N(0,I)
+�
+(⟨xi, xj⟩ + 1)I(w⊤xi ≥ B, w⊤xj ≥ B)
+�
+.
+Let λ(B) := λmin(H∞(B)). We define Ir,i(W, b) := I(w⊤
+r xi ≥ br) and the matrix Z(W, b) as
+Z(W, b) :=
+1
+√m
+�
+��
+I1,1(W, b)a1[x⊤
+1 , −1]⊤
+. . .
+I1,n(W, b)a1[x⊤
+n , −1]⊤
+...
+...
+...
+Im,1(W, b)am[x⊤
+1 , −1]⊤
+. . .
+Im,n(W, b)am[x⊤
+n , −1]⊤
+�
+�� ∈ Rm(d+1)×n.
+Note that H(W, b) = Z(W, b)⊤Z(W, b). Hence, the gradient descent step can be written as
+⃗
+[W, b](t + 1) =
+⃗
+[W, b](t) − ηZ(t)(f(t) − y)
+where [W, b](t) ∈ Rm×(d+1) denotes the row-wise concatenation of W(t) and b(t) at the t-th step of
+gradient descent, and Z(t) := Z(W(t), b(t)).
+3
+Main Theory
+3.1
+Convergence and Sparsity
+We present the convergence of gradient descent for the sparsely activated neural networks. Com-
+pared to the existing convergence result in (Song et al., 2021a), our study handles the trainable bias
+with constant initialization in the convergence analysis (which is the first of such a type). Also, our
+bound is sharper and yields a much smaller bound on the width of neural networks to guarantee
+the convergence.
+Theorem 3.1 (Convergence). Let the learning rate η ≤ O( λ(B) exp(B2)
+n2
+), and the bias initialization
+B ∈ [0, √0.5 log m]. Assume λ(B) = λ0 exp(−B2/2) for some λ0 > 0 independent of B. Then, if
+the network width satisfies m ≥ �Ω
+�
+λ−4
+0 n4 exp(B2)
+�
+, over the randomness in the initialization,
+P
+�
+∀t : L(W(t), b(t)) ≤ (1 − ηλ(B)/4)tL(W(0), b(0))
+�
+≥ 1 − δ − e−Ω(n).
+This theorem show that the training loss decreases linearly, and its rate depends on the smallest
+eigenvalue of the NTK. The assumption on λ(B) in Theorem 3.1 can be justified by (Song et al.,
+2021a, Theorem F.1) which shows that under some mild conditions, the NTK’s least eigenvalue
+λ(B) is positive and has an exp(−B2/2) dependence. This further implies that the network after
+sparsification can achieve as fast convergence as the original network.
+5
+
+Remark 3.2. Theorem 3.1 establishes a much sharper bound on the width of the neural network
+than previous work to guarantee the linear convergence.
+To elaborate, our bound only requires
+m ≥ �Ω
+�
+λ−4
+0 n4 exp(B2)
+�
+, as opposed to the bound m ≥ �Ω(λ−4
+0 n4B2 exp(2B2)) in (Song et al.,
+2021a, Lemma D.9). If we take B = √0.25 log m (as allowed by the theorem), then our lower
+bound yields a polynomial improvement by a factor of �Θ(n/λ0)8/3, which implies that the neural
+network width can be much smaller to achieve the same linear convergence.
+Key ideas in the proof of Theorem 3.1. The proof mainly consists of developing a novel
+bound on activation flipping probability and a novel upper bound on initial error, as we elaborate
+below.
+Like previous works, in order to prove convergence, we need to show that the NTK during
+training is close to its initialization. Inspecting the expression of NTK in Equation (2.1), observe
+that the training will affect the NTK by changing the output of each indicator function. We say
+that the r-th neuron flips its activation with respect to input xi at the k-th step of gradient descent
+if I(wr(k)⊤xi − br(k) > 0) ̸= I(wr(k − 1)⊤xi − br(k − 1) > 0) for all r ∈ [m]. The central idea
+is that for each neuron, as long as the weight and bias movement Rw, Rb from its initialization is
+small, then the probability of activation flipping (with respect to random initialization) should not
+be large. We first present the bound on the probability that a given neuron flips its activation.
+Lemma 3.3 (Bound on Activation flipping probability). Let B ≥ 0 and Rw, Rb ≤ min{1/B, 1}. Let
+�
+W = ( �w1, . . . , �wm) be vectors generated i.i.d. from N(0, I) and �b = (�b1, . . . ,�bm) = (B, . . . , B), and
+weights W = (w1, . . . , wm) and biases b = (b1, . . . , bm) that satisfy for any r ∈ [m], ∥ �wr − wr∥2 ≤
+Rw and |�br − br| ≤ Rb. Define the event
+Ai,r = {∃wr, br : ∥ �wr − wr∥2 ≤ Rw, |br − �br| ≤ Rb, I(x⊤
+i �wr ≥ �br) ̸= I(x⊤
+i wr ≥ br)}.
+Then, for some constant c,
+P [Ai,r] ≤ c(Rw + Rb) exp(−B2/2).
+(Song et al., 2021a, Claim C.11) presents a O(min{R, exp(−B2/2)}) bound on P[Ai,r]. The
+reason that their bound involving the min operation is because P[Ai,r] can be bounded by the
+standard Gaussian tail bound and Gaussian anti-concentration bound separately and then, take
+the one that is smaller. On the other hand, our bound replaces the min operation by the product
+which creates a more convenient (and tighter) interpolation between the two bounds. Later, we will
+show that the maximum movement of neuron weights and biases, Rw and Rb, both have a O(1/√m)
+dependence on the network width, and thus our bound offers a exp(−B2/2) improvement where
+exp(−B2/2) can be as small as 1/m1/4 when we take B = √0.5 log m.
+Proof idea of Lemma 3.3.
+First notice that P[Ai,r] = Px∼N(0,1)[|x − B| ≤ Rw + Rb].
+Thus, here we are trying to solve a fine-grained Gaussian anti-concentration problem with the
+strip centered at B. The problem with the standard Gaussian anti-concentration bound is that
+it only provides a worst case bound and, thus, is location-oblivious.
+Centered in our proof is
+a nice Gaussian anti-concentration bound based on the location of the strip, which we describe
+as follows: Let’s first assume B > Rw + Rb. A simple probability argument yields a bound of
+2(Rw + Rb)
+1
+√
+2π exp(−(B − Rw − Rb)2). Since later in the Appendix we can show that Rw and
+Rb have a O(1/√m) dependence (Lemma A.9 bounds the movement for gradient descent and
+Lemma A.10 for gradient flow) and we only take B = O(√log m), by making m sufficiently large,
+6
+
+we can safely assume that Rw and Rb is sufficiently small. Thus, the probability can be bounded by
+O((Rw + Rb) exp(−B2/2)). However, when B < Rw + Rb the above bound no longer holds. But a
+closer look tells us that in this case B is close to zero, and thus (Rw +Rb)
+1
+√
+2π exp(−B2/2) ≈ Rw+Rb
+√
+2π
+which yields roughly the same bound as the standard Gaussian anti-concentration.
+Next, our proof of Theorem 3.1 develops the following initial error bound.
+Lemma 3.4 (Initial error upper bound). Let B > 0 be the initialization value of the biases and all
+the weights be initialized from standard Gaussian. Let δ ∈ (0, 1) be the failure probability. Then,
+with probability at least 1 − δ over the randomness in the initialization, we have
+L(W(0), b(0)) = O
+�
+n + n
+�
+exp(−B2/2) + 1/m
+�
+log3(2mn/δ)
+�
+.
+(Song et al., 2021a, Claim D.1) gives a rough estimate of the initial error with O(n(1 +
+B2) log2(n/δ) log(m/δ)) bound.
+When we set B = C√log m for some constant C, our bound
+improves the previous result by a polylogarithmic factor. The previous bound is not tight in the
+following two senses: (1) the bias will only decrease the magnitude of the neuron activation instead
+of increasing and (2) when the bias is initialized as B, only roughly O(exp(−B2/2)) · m neurons
+will activate. Thus, we can improve the B2 dependence to exp(−B2/2).
+By combining the above two improved results, we can prove our convergence result with im-
+proved lower bound of m as in Remark 3.2. We provide the complete proof in Appendix A.
+Lastly, since the total movement of each neuron’s bias has a O(1/√m) dependence (shown in
+Lemma A.9), combining with the number of activated neurons at the initialization, we can show
+that during the entire training, the number of activated neurons is small.
+Lemma 3.5 (Number of Activated Neurons per Iteration). Assume the parameter settings in
+Theorem 3.1. With probability at least 1 − e−Ω(n) over the random initialization, we have
+|Son(i, t)| = O(m · exp(−B2/2))
+for all 0 ≤ t ≤ T and i ∈ [n], where Son(i, t) = {r ∈ [m] : wr(t)⊤xi ≥ br(t)}.
+3.2
+Generalization and Restricted Least Eigenvalue
+In this section, we present the sparsity-dependent generalization of our neural networks after gra-
+dient descent training. However, for technical reasons stated in Section 3.3, we use symmetric
+initialization defined below. Further, we adopt the setting in (Arora et al., 2019) and use a non-
+degenerate data distribution to make sure the infinite-width NTK is positive definite.
+Definition 3.6 (Symmetric Initialization). For a one-hidden layer neural network with 2m neu-
+rons, the network is initialized as the following:
+1. For r ∈ [m], independently initialize wr ∼ N(0, I) and ar ∼ Uniform({−1, 1}).
+2. For r ∈ {m + 1, . . . , 2m}, let wr = wr−m and ar = −ar−m.
+Definition 3.7 ((λ0, δ, n)-non-degenerate distribution, (Arora et al., 2019)). A distribution D over
+Rd × R is (λ0, δ, n)-non-degenerate, if for n i.i.d. samples {(xi, yi)}n
+i=1 from D, with probability
+1 − δ we have λmin(H∞(B)) ≥ λ0 > 0.
+7
+
+Theorem 3.8. Fix a failure probability δ ∈ (0, 1) and an accuracy parameter ϵ ∈ (0, 1). Suppose
+the training data S = {(xi, yi)}n
+i=1 are i.i.d. samples from a (λ, δ, n)-non-degenerate distribution D
+defined in Definition 3.7. Assume the one-hidden layer neural network is initialized by symmetric
+initialization in Definition 3.6. Further, assume the parameter settings in Theorem 3.1 except we
+let m ≥ �Ω
+�
+λ(B)−6n6 exp(−B2)
+�
+. Consider any loss function ℓ : R×R → [0, 1] that is 1-Lipschitz in
+its first argument. Then with probability at least 1 − 2δ − e−Ω(n) over the randomness in symmetric
+initialization of W(0) ∈ Rm×d and a ∈ Rm and the training samples, the two layer neural network
+f(W(t), b(t), a) trained by gradient descent for t ≥ Ω(
+1
+ηλ(B) log n log(1/δ)
+ϵ
+) iterations has empirical
+Rademacher complexity (see its formal definition in Definition C.1 in Appendix) bounded as
+RS(F) ≤
+�
+y⊤(H∞(B))−1y · 8 exp(−B2/2)
+n
++ �O
+�exp(−B2/4)
+n1/2
+�
+and the population loss LD(f) = E(x,y)∼D[ℓ(f(x), y)] can be upper bounded as
+LD(f(W(t), b(t), a)) ≤
+�
+y⊤(H∞(B))−1y · 32 exp(−B2/2)
+n
++ �O
+� 1
+n1/2
+�
+.
+(3.1)
+To show good generalization, we need a larger width: the second term in the Rademacher
+complexity bound is diminishing with m and to make this term O(1/√n), the width needs to
+have (n/λ(B))6 dependence as opposed to (n/λ(B))4 for convergence. Now, at the first glance of
+our generalization result, it seems we can make the Rademacher complexity arbitrarily small by
+increasing B. Recall from the discussion of Theorem 3.1 that the smallest eigenvalue of H∞(B)
+also has an exp(−B2/2) dependence. Thus, in the worst case, the exp(−B2/2) factor gets canceled
+and sparsity will not hurt the network’s generalization.
+Before we present the proof, we make a corollary of Theorem 3.8 for the zero-initialized bias
+case.
+Corollary 3.9. Take the same setting as in Theorem 3.8 except now the biases are initialized as
+zero, i.e., B = 0. Then, if we let m ≥ �Ω(λ(0)−6n6), the empirical Rademacher complexity and
+population loss are both bounded by
+RS(F), LD(f(W(t), b(t), a)) ≤
+�
+y⊤(H∞(0))−1y · 32
+n
++ �O
+� 1
+n1/2
+�
+.
+Corollary 3.9 requires the network width m ≥ �Ω((n/λ(0))6) which significantly improves upon
+the previous result in (Song and Yang, 2019, Theorem G.7) m ≥ �Ω(n16 poly(1/λ(0))) (including
+the dependence on the rescaling factor κ) which is a much wider network.
+Generalization Bound via Least Eigenvalue. Note that in Theorem 3.8, the worst case of
+the first term in the generalization bound in Equation (3.1) is given by �O(
+�
+1/(λ(B) · n)). Hence,
+the least eigenvalue λ(B) of the NTK matrix can significantly affect the generalization bound.
+Previous works (Oymak and Soltanolkotabi, 2020; Song et al., 2021a) established lower bounds
+on λ(B) with an explicit 1/n2 dependence on n under the δ data separation assumption (see
+Theorem 3.11), which clearly makes a vacuous generalization bound of �O(n). This thus motivates
+us to provide a tighter bound (desirably independent on n) on the least eigenvalue of the infinite-
+width NTK in order to make the generalization bound in Theorem 3.8 valid and useful. However,
+it turns out that there are major difficulties in proving a better lower bound in the general case
+8
+
+and thus, we are only able to present a better lower bound when we restrict the domain to some
+(data-dependent) regions.
+Definition 3.10 (Data-dependent Region). Let pij = Pw∼N(0,I)[w⊤xi ≥ B, w⊤xj ≥ B] for i ̸= j.
+Define the (data-dependent) region R = {a ∈ Rn : �
+i̸=j aiajpij ≥ mini′̸=j′ pi′j′ �
+i̸=j aiaj}.
+Notice that R is non-empty for any input data-set since Rn
++ ⊂ R where Rn
++ denotes the set of
+vectors with non-negative entries, and R = Rn if pij = pi′j′ for all i ̸= i′, j ̸= j′.
+Theorem 3.11 (Restricted Least Eigenvalue). Let X = (x1, . . . , xn) be points in Rd with ∥xi∥2 = 1
+for all i ∈ [n] and w ∼ N(0, Id). Suppose that there exists δ ∈ [0,
+√
+2] such that
+min
+i̸=j∈[n](∥xi − xj∥2 , ∥xi + xj∥2) ≥ δ.
+Let B ≥ 0. Consider the minimal eigenvalue of H∞ over the data-dependent region R defined
+above, i.e., let λ := min∥a∥2=1, a∈R a⊤H∞a. Then, λ ≥ max(0, λ′) where
+λ′ ≥ max
+�
+1
+2 −
+B
+√
+2π,
+� 1
+B − 1
+B3
+� e−B2/2
+√
+2π
+�
+− e−B2/(2−δ2/2)
+π − arctan
+�
+δ√
+1−δ2/4
+1−δ2/2
+�
+2π
+.
+(3.2)
+To demonstrate the usefulness of our result, if we take the bias initialization B = 0 in Equa-
+tion (3.2), this bound yields 1/(2π) · arctan((δ
+�
+1 − δ2/4)/(1 − δ2/2)) ≈ δ/(2π), when δ is close
+to 0 whereas (Song et al., 2021a) yields a bound of δ/n2. On the other hand, if the data has
+maximal separation, i.e., δ =
+√
+2, we get a max
+�
+1
+2 −
+B
+√
+2π,
+� 1
+B −
+1
+B3
+� e−B2/2
+√
+2π
+�
+lower bound, whereas
+(Song et al., 2021a) yields a bound of exp(−B2/2)
+√
+2/n2. Connecting to our convergence result
+in Theorem 3.1, if f(t) − y ∈ R, then the error can be reduced at a much faster rate than the
+(pessimistic) rate with 1/n2 dependence in the previous studies as long as the error vector lies in
+the region.
+Remark 3.12. The lower bound on the restricted smallest eigenvalue λ in Theorem 3.11 is in-
+dependent on n, which makes that the worst case generalization bound in Theorem 3.8 be O(1)
+under constant data separation margin (note that this is optimal since the loss is bounded). Such a
+lower bound is much sharper than the previous results with a 1/n2 explicit dependence which yields
+vacuous generalization. This improvement relies on a fact that the label vector should lie in the
+region R, which can be justified by a simple label-shifting strategy as follows. Since Rn
++ ⊂ R, the
+condition can be easily achieved by training the neural network on the shifted labels y + C (with
+appropriate broadcast) where C is a constant such that mini yi + C ≥ 0.
+Careful readers may notice that in the proof of Theorem 3.11 in Appendix B, the restricted
+least eigenvalue on Rn
++ is always positive even if the data separation is zero. However, we would like
+to point out that the generalization bound in Theorem 3.8 is meaningful only when the training is
+successful: when the data separation is zero, the limiting NTK is no longer positive definite and
+the training loss cannot be minimized toward zero.
+3.3
+Key Ideas in the Proof of Theorem 3.8
+Since each neuron weight and bias move little from their initialization, a natural approach is to
+bound the generalization via localized Rademacher complexity. After that, we can apply appro-
+priate concentration bounds to derive generalization. The main effort of our proof is devoted to
+9
+
+bounding the weight movement to bound the localized Rademacher complexity. If we directly take
+the setting in Theorem 3.1 and compute the network’s localized Rademacher complexity, we will
+encounter a non-diminishing (with the number of samples n) term which can be as large as O(√n)
+since the network outputs non-zero values at the initialization. Arora et al. (2019) and Song and
+Yang (2019) resolved this issue by initializing the neural network weights instead by N(0, κ2I) to
+force the neural network output something close to zero at the initialization. The magnitude of
+κ is chosen to balance different terms in the Rademacher complexity bound in the end. Similar
+approach can also be adapted to our case by initializing the weights by N(0, κ2I) and the biases
+by κB. However, the drawback of such an approach is that the effect of κ to all the previously
+established results for convergence need to be carefully tracked or derived. In particular, in order
+to guarantee convergence, the neural network’s width needs to have a polynomial dependence on
+1/κ where 1/κ has a polynomial dependence on n and 1/λ, which means their network width needs
+to be larger to compensate for the initialization scaling. We resolve this issue by symmetric ini-
+tialization Definition 3.6 which yields no effect (up to constant factors) on previously established
+convergence results, see (Munteanu et al., 2022). Symmetric initialization allows us to organically
+combine the results derived for convergence to be reused for generalization, which leads to a more
+succinct analysis. Further, we replace the ℓ1-ℓ2 norm upper bound by finer inequalities in various
+places in the original analysis. All these improvements lead to the following upper bound of the
+weight matrix change in Frobenius norm. Further, combining our sparsity-inducing initialization,
+we present our sparsity-dependent Frobenius norm bound on the weight matrix change.
+Lemma 3.13. Assume the one-hidden layer neural network is initialized by symmetric initialization
+in Definition 3.6. Further, assume the parameter settings in Theorem 3.1. Then with probability
+at least 1 − δ − e−Ω(n) over the random initialization, we have for all t ≥ 0,
+∥[W, b](t) − [W, b](0)∥F ≤
+�
+y⊤(H∞)−1y + O
+�
+n
+λ
+�exp(−B2/2) log(n/δ)
+m
+�1/4�
++ O
+�
+n
+�
+R exp(−B2/2)
+λ
+�
++ n
+λ2 · O
+�
+exp(−B2/4)
+�
+log(n2/δ)
+m
++ R exp(−B2/2)
+�
+where R = Rw + Rb denote the maximum magnitude of neuron weight and bias change.
+By Lemma A.9 and Lemma A.11 in the Appendix, we have R = �O(
+n
+λ√m). Plugging in and
+setting B = 0, we get ∥[W, b](t) − [W, b](0)∥F ≤
+�
+y⊤(H∞)−1y + �O(
+n
+λm1/4 +
+n3/2
+λ3/2m1/4 +
+n
+λ2√m +
+n2
+λ3√m). On the other hand, taking κ = 1, (Song and Yang, 2019, Lemma G.6) yields a bound
+of ∥W(t) − W(0)∥F ≤
+�
+y⊤(H∞)−1y + �O( n
+λ + n7/2 poly(1/λ)
+m1/4
+). Notice that the �O( n
+λ) term has no
+dependence on 1/m and is removed by symmetric initialization in our analysis and we improve the
+upper bound’s dependence on n by a factor of n2.
+We defer the full proof of Theorem 3.8 and Lemma 3.13 to Appendix C.
+3.4
+Key Ideas in the Proof of Theorem 3.11
+In this section, we analyze the smallest eigenvalue λ := λmin(H∞) of the limiting NTK H∞ with
+δ data separation. We first note that H∞ ⪰ Ew∼N(0,I)
+�
+I(Xw ≥ B)I(Xw ≥ B)⊤�
+and for a fixed
+vector a, we are interested in the lower bound of Ew∼N(0,I)[|a⊤I(Xw ≥ B)|2]. In previous works,
+Oymak and Soltanolkotabi (2020) showed a lower bound Ω(δ/n2) for zero-initialized bias, and later
+10
+
+Song et al. (2021a) generalized this result to a lower bound Ω(e−B2/2δ/n2) for non-zero initialized
+bias.
+Both lower bounds have a dependence of 1/n2.
+Their approach is by using an intricate
+Markov’s inequality argument and then proving an lower bound of P[|a⊤I(Xw ≥ B)| ≥ c ∥a∥∞].
+The lower bound is proved by only considering the contribution from the largest coordinate of a
+and treating all other values as noise. It is non-surprising that the lower bound has a factor of 1/n
+since a can have identical entries. On the other hand, the diagonal entries can give a exp(−B2/2)
+upper bound and thus there is a 1/n2 gap between the two. Now, we give some evidence suggesting
+the 1/n2 dependence may not be tight in some cases. Consider the following scenario: Assume
+n ≪ d and the data set is orthonormal. For a fixed a, we have
+a⊤
+E
+w∼N(0,I)
+�
+I(Xw ≥ B)I(Xw ≥ B)⊤�
+a
+= �
+i,j∈[n] aiaj P[w⊤xi ≥ B, w⊤xj ≥ B] = p0 ∥a∥2
+2 + p1
+�
+i̸=j aiaj
+= p0 − p1 + p1 (�
+i ai)2 > p0 − p1
+where p0, p1 ∈ [0, 1] are defined such that due to the spherical symmetry of the standard Gaussian
+we are able to let p0 = P[w⊤xi ≥ B], ∀i ∈ [n] and p1 = P[w⊤xi ≥ B, w⊤xj ≥ B], ���i, j ∈ [n], i ̸= j.
+Notice that p0 > p1. Since this is true for all a ∈ Rn, we get a lower bound of p0 − p1 with no
+explicit dependence on n and this holds for all n ≤ d. When d is large and n = d/2, this bound is
+better than previous bound by a factor of Θ(1/d2). However, it turns out that the product terms
+with i ̸= j above creates major difficulties in analyzing the general case. Due to such technical
+difficulties, we are only able to prove a better lower bound by utilizing the extra constant factor in
+the NTK thanks to the trainable bias, when we restrict the domain to some data-dependent region.
+We defer the proof of Theorem 3.11 to Appendix B.
+4
+Experiments
+In this section, we study how the activation sparsity patterns of multi-layer neural networks change
+during training when the bias parameters are initialized as non-zero.
+Settings. We train a 6-layer multi-layer perceptron (MLP) of width 1024 with trainable bias
+terms on MNIST image classification (LeCun et al., 2010).
+The biases of the fully-connected
+layers are initialized as 0, −0.5 and −1.
+For the weights in the linear layer, we use Kaiming
+Initialization (He et al., 2015) which is sampled from an appropriately scaled Gaussian distribution.
+The traditional MLP architecture only has linear layers with ReLU activation. However, we found
+out that using the sparsity-inducing initialization, the magnitude of the activation will decrease
+geometrically layer-by-layer, which leads to vanishing gradients and that the network cannot be
+trained. Thus, we made a slight modification to the MLP architecture to include an extra Batch
+Normalization after ReLU to normalize the activation. Our MLP implementation is based on (Zhu
+et al., 2021). We train the neural network by stochastic gradient descent with a small learning
+rate 5e-3 to make sure the training is in the NTK regime. The sparsity is measured as the total
+number of activated neurons (i.e., ReLU outputs some positive values) divided by total number of
+neurons, averaged over every SGD batch. We plot how the sparsity patterns changes for different
+layers during training.
+Observation and Implication. As demonstrated at Figure 1, when we initialize the bias with
+three different values, the sparsity patterns are stable across all layers during training: when the
+bias is initialized as 0 and −0.5, the sparsity change is within 2.5%; and when the bias is initialized
+11
+
+0
+10k
+20k
+30k
+40k
+Iterations
+0.600
+0.625
+0.650
+0.675
+0.700
+0.725
+0.750
+0.775
+Sparsity
+layer 0
+layer 1
+layer 2
+layer 3
+layer 4
+layer 5
+(a) Init Bias as 0
+0
+10k
+20k
+30k
+40k
+Iterations
+0.65
+0.70
+0.75
+0.80
+0.85
+Sparsity
+layer 0
+layer 1
+layer 2
+layer 3
+layer 4
+layer 5
+(b) Init Bias as -0.5
+0
+10k
+20k
+30k
+40k
+Iterations
+0.65
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+Sparsity
+layer 0
+layer 1
+layer 2
+layer 3
+layer 4
+layer 5
+(c) Init Bias as -1.0
+Figure 1: Sparsity pattern on different layers across different training iterations for three different
+bias initialization. The x and y axis denote the iteration number and sparsity level, respectively.
+The models can achieve 97.9%, 97.7% and 97.3% accuracy after training, respectively. Note that,
+in Figure (a), the lines of layers 1-5 overlap together except layer 0.
+as −1.0, the sparsity change is within 10%. Meanwhile, by increasing the initialization magnitude
+for bias, the sparsity level increases with only marginal accuracy dropping. This implies that our
+theory can be extended to the multi-layer setting (with some extra care for coping with vanishing
+gradient) and multi-layer neural networks can also benefit from the sparsity-inducing initialization
+and enjoy reduction of computational cost. Another interesting observation is that the input layer
+(layer 0) has a different sparsity pattern from other layers while all the rest layers behave similarly.
+5
+Discussion
+In this work, we study training one-hidden-layer overparameterized ReLU networks in the NTK
+regime with its biases being trainable and initialized as some constants rather than zero. We showed
+sparsity-dependent results on convergence, restricted least eigenvalue and generalization. A future
+direction is to generalize our analysis to multi-layer neural networks. In practice, label shifting
+is unnecessary for achieving good generalization. An open problem is whether it is possible to
+improve the dependence on the sample size of the lower bound of the infinite-width NTK’s least
+eigenvalue, or even whether a lower bound purely dependent on the data separation is possible so
+that the generalization bound is no longer vacuous for all labels.
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+16
+
+A
+Convergence
+Notation simplification. Since the smallest eigenvalue of the limiting NTK appeared in this
+proof all has dependence on the bias initialization parameter B, for the ease of notation of our
+proof, we suppress its dependence on B and use λ to denote λ := λ(B) = λmin(H∞(B)).
+A.1
+Difference between limit NTK and sampled NTK
+Lemma A.1. For a given bias vector b ∈ Rm with br ≥ 0, ∀r ∈ [m], the limit NTK H∞ and the
+sampled NTK H are given as
+H∞
+ij :=
+E
+w∼N(0,I)
+�
+(⟨xi, xj⟩ + 1)I(w⊤
+r xi ≥ br, w⊤
+r xj ≥ br)
+�
+,
+Hij := 1
+m
+m
+�
+r=1
+(⟨xi, xj⟩ + 1)I(w⊤
+r xi ≥ br, w⊤
+r xj ≥ br).
+Let’s define λ := λmin(H∞) and assume λ > 0. If the network width m = Ω(λ−1n · log(n/δ)), then
+P
+�
+λmin(H) ≥ 3
+4λ
+�
+≥ 1 − δ.
+Proof. Let Hr := 1
+m �
+X(wr)⊤ �
+X(wr), where �
+X(wr) ∈ R(d+1)×n is defined as
+�
+X(wr) := [I(w⊤
+r x1 ≥ b) · (x1, 1), . . . , I(w⊤
+r xn ≥ b) · (xn, 1)],
+where (xi, 1) denotes appending the vector xi by 1. Hence Hr ⪰ 0. Since for each entry Hij we
+have
+(Hr)ij = 1
+m(⟨xi, xj⟩ + 1)I(w⊤
+r xi ≥ br, w⊤
+r xj ≥ br) ≤ 1
+m(⟨xi, xj⟩ + 1) ≤ 2
+m,
+and naively, we can upper bound ∥Hr∥2 by:
+∥Hr∥2 ≤ ∥Hr∥F ≤
+�
+n2 4
+m2 = 2n
+m .
+Then H = �m
+r=1 Hr and E[H] = H∞. Hence, by the Matrix Chernoff Bound in Lemma D.2 and
+choosing m = Ω(λ−1n · log(n/δ)), we can show that
+P
+��
+λmin(H) ≤ 3
+4λ
+�
+≤ n · exp
+�
+− 1
+16λ/(4n/m)
+�
+= n · exp
+�
+− λm
+64n
+�
+≤ δ.
+Lemma A.2. Assume m = nO(1) and exp(B2/2) = O(√m) where we recall that B is the initializa-
+tion value of the biases. With probability at least 1−δ, we have ∥H(0) − H∞∥F ≤ 4n exp(−B2/4)
+�
+log(n2/δ)
+m
+.
+17
+
+Proof. First, we have E[((⟨xi, xj⟩ + 1)Ir,i(0)Ir,j(0))2] ≤ 4 exp(−B2/2). Then, by Bernstein’s in-
+equality in Lemma D.1, with probability at least 1 − δ/n2,
+|Hij(0) − H∞
+ij | ≤ 2 exp(−B2/4)
+�
+2log(n2/δ)
+m
++ 2 2
+m log(n2/δ) ≤ 4 exp(−B2/4)
+�
+log(n2/δ)
+m
+.
+By a union bound, the above holds for all i, j ∈ [n] with probability at least 1 − δ, which implies
+∥H(0) − H∞∥F ≤ 4n exp(−B2/4)
+�
+log(n2/δ)
+m
+.
+A.2
+Bounding the number of flipped neurons
+Definition A.3 (No-flipping set). For each i ∈ [n], let Si ⊂ [m] denote the set of neurons that are
+never flipped during the entire training process,
+Si := {r ∈ [m] : ∀t ∈ [T] sign(⟨wr(t), xi⟩ − br(t)) = sign(⟨wr(0), xi⟩ − br(0))}.
+Thus, the flipping set is Si for i ∈ [n].
+Lemma A.4 (Bound on flipping probability). Let B ≥ 0 and Rw, Rb ≤ min{1/B, 1}. Let �
+W =
+( �w1, . . . , �wm) be vectors generated i.i.d.
+from N(0, I) and �b = (�b1, . . . ,�bm) = (B, . . . , B), and
+weights W = (w1, . . . , wm) and biases b = (b1, . . . , bm) that satisfy for any r ∈ [m], ∥ �wr − wr∥2 ≤
+Rw and |�br − br| ≤ Rb. Define the event
+Ai,r = {∃wr, br : ∥ �wr − wr∥2 ≤ Rw, |br − �br| ≤ Rb, I(x⊤
+i �wr ≥ �br) ̸= I(x⊤
+i wr ≥ br)}.
+Then,
+P [Ai,r] ≤ c(Rw + Rb) exp(−B2/2)
+for some constant c.
+Proof. Notice that the event Ai,r happens if and only if | �w⊤
+r xi − �br| < Rw + Rb. First, if B > 1,
+then by Lemma D.3, we have
+P [Ai,r] ≤ (Rw + Rb)
+1
+√
+2π exp(−(B − Rw − Rb)2/2) ≤ c1(Rw + Rb) exp(−B2/2)
+for some constant c1. If 0 ≤ B < 1, then the above analysis doesn’t hold since it is possible that
+B − Rw − Rb ≤ 0. In this case, the probability is at most P[Ai,r] ≤ 2(Rw + Rb)
+1
+√
+2π exp(−02/2) =
+2(Rw+Rb)
+√
+2π
+. However, since 0 ≤ B < 1 in this case, we have exp(−12/2) ≤ exp(−B2/2) ≤ exp(−02/2).
+Therefore, P[Ai,r] ≤ c2(Rw + Rb) exp(−B2/2) for c2 = 2 exp(1/2)
+√
+2π
+. Take c = max{c1, c2} finishes the
+proof.
+Corollary A.5. Let B > 0 and Rw, Rb ≤ min{1/B, 1}. Assume that ∥wr(t) − wr(0)∥2 ≤ Rw and
+18
+
+|br(t) − br(0)| ≤ Rb for all t ∈ [T]. For i ∈ [n], the flipping set Si satisfies that
+P[r ∈ Si] ≤ c(Rw + Rb) exp(−B2/2)
+for some constant c, which implies
+P[∀i ∈ [n] : |Si| ≤ 2mc(Rw + Rb) exp(−B2/2)] ≥ 1 − n · exp
+�
+−2
+3mc(Rw + Rb) exp(−B2/2)
+�
+.
+Proof. The proof is by observing that P[r ∈ Si] ≤ P[Ai,r]. Then, by Bernstein’s inequality,
+P[|Si| > t] ≤ exp
+�
+−
+t2/2
+mc(Rw + Rb) exp(−B2/2) + t/3
+�
+.
+Take t = 2mc(Rw + Rb) exp(−B2/2) and a union bound over [n], we have
+P[∀i ∈ [n] : |Si| ≤ 2mc(Rw + Rb) exp(−B2/2)] ≥ 1 − n · exp
+�
+−2
+3mc(Rw + Rb) exp(−B2/2)
+�
+.
+A.3
+Bounding NTK if perturbing weights and biases
+Lemma A.6. Assume λ > 0. Let B > 0 and Rb, Rw ≤ min{1/B, 1}. Let �
+W = ( �w1, . . . , �wm) be
+vectors generated i.i.d. from N(0, I) and �b = (�b1, . . . ,�bm) = (B, . . . , B). For any set of weights
+W = (w1, . . . , wm) and biases b = (b1, . . . , bm) that satisfy for any r ∈ [m], ∥ �wr − wr∥2 ≤ Rw and
+|�br − br| ≤ Rb, we define the matrix H(W, b) ∈ Rn×n by
+Hij(W, b) = 1
+m
+m
+�
+r=1
+(⟨xi, xj⟩ + 1)I(w⊤
+r xi ≥ br, w⊤
+r xj ≥ br).
+It satisfies that for some small positive constant c,
+1. With probability at least 1 − n2 exp
+�
+− 2
+3cm(Rw + Rb) exp(−B2/2)
+�
+, we have
+���H(�
+W,�b) − H(W, b)
+���
+F ≤ n · 8c(Rw + Rb) exp(−B2/2),
+���Z(�
+W,�b) − Z(W, b)
+���
+F ≤
+�
+n · 8c(Rw + Rb) exp(−B2/2).
+2. With probability at least 1 − δ − n2 exp
+�
+− 2
+3cm(Rw + Rb) exp(−B2/2)
+�
+,
+λmin(H(W, b)) > 0.75λ − n · 8c(Rw + Rb) exp(−B2/2).
+Proof. We have
+���Z(W, b) − Z(�
+W,�b)
+���
+2
+F =
+�
+i∈[n]
+�
+� 2
+m
+�
+r∈[m]
+�
+I(w⊤
+r xi ≥ br) − I( �w⊤
+r xi ≥ �br)
+�2
+�
+�
+19
+
+=
+�
+i∈[n]
+�
+� 2
+m
+�
+r∈[m]
+tr,i
+�
+�
+and
+���H(W, b) − H(�
+W,�b)
+���
+2
+F
+=
+�
+i∈[n], j∈[n]
+(Hij(W, b) − Hij(�
+W,�b))2
+≤ 4
+m2
+�
+i∈[n], j∈[n]
+�
+� �
+r∈[m]
+|I(w⊤
+r xi ≥ br, w⊤
+r xj ≥ br) − I( �w⊤
+r xi ≥ �br, �w⊤
+r xj ≥ �br)|
+�
+�
+2
+= 4
+m2
+�
+i,j∈[n]
+�
+� �
+r∈[m]
+sr,i,j
+�
+�
+2
+,
+where we define
+sr,i,j := |I(w⊤
+r xi ≥ br, w⊤
+r xj ≥ br) − I( �w⊤
+r xi ≥ �br, �w⊤
+r xj ≥ �br)|,
+tr,i := (I(w⊤
+r xi ≥ br) − I( �w⊤
+r xi ≥ �br))2.
+Notice that tr,i = 1 only if the event Ai,r happens (recall the definition of Ai,r in Lemma A.4) and
+sr,i,j = 1 only if the event Ai,r or Aj,r happens. Thus,
+�
+r∈[m]
+tr,i ≤
+�
+r∈[m]
+I(Ai,r),
+�
+r∈[m]
+sr,i,j ≤
+�
+r∈[m]
+I(Ai,r) + I(Aj,r).
+By Lemma A.4, we have
+E
+�wr[sr,i,j] ≤ E
+�wr[s2
+r,i,j] ≤ P
+�wr[Ai,r] + P
+�wr[Aj,r] ≤ 2c(Rw + Rb) exp(−B2/2).
+Define si,j = �m
+r=1 I(Ai,r) + I(Aj,r). By Bernstein’s inequality in Lemma D.1,
+P
+�
+si,j ≥ m · 2c(Rw + Rb) exp(−B2/2) + mt
+�
+≤ exp
+�
+−
+m2t2/2
+m · 2c(Rw + Rb) exp(−B2/2) + mt/3
+�
+,
+∀t ≥ 0.
+Let t = 2c(Rw + Rb) exp(−B2/2). We get
+P[si,j ≥ m · 4c(Rw + Rb) exp(−B2/2)] ≤ exp
+�
+−2
+3cm(Rw + Rb) exp(−B2/2)
+�
+.
+Thus, we obtain with probability at least 1 − n2 exp
+�
+− 2
+3cm(Rw + Rb) exp(−B2/2)
+�
+,
+���H(�
+W,�b) − H(W, b)
+���
+F ≤ n · 8c(Rw + Rb) exp(−B2/2),
+���Z(�
+W,�b) − Z(W, b)
+���
+F ≤
+�
+n · 8c(Rw + Rb) exp(−B2/2).
+20
+
+For the second result, by Lemma A.1, P[λmin(H(�
+W,�b)) ≥ 0.75λ] ≥ 1 − δ. Hence, with probability
+at least 1 − δ − n2 exp
+�
+− 2
+3cm(Rw + Rb) exp(−B2/2)
+�
+,
+λmin(H(W, b)) ≥ λmin(H(�
+W,�b)) −
+���H(W, b) − H(�
+W,�b)
+���
+≥ λmin(H(�
+W,�b)) −
+���H(W, b) − H(�
+W,�b)
+���
+F
+≥ 0.75λ − n · 8c(Rw + Rb) exp(−B2/2).
+A.4
+Total movement of weights and biases
+Definition A.7 (NTK at time t). For t ≥ 0, let H(t) be an n × n matrix with (i, j)-th entry
+Hij(t) :=
+�∂f(xi; θ(t))
+∂θ(t)
+, ∂f(xj; θ(t))
+∂θ(t)
+�
+= 1
+m
+m
+�
+r=1
+(⟨xi, xj⟩ + 1)I(wr(t)⊤xi ≥ br(t), wr(t)⊤xj ≥ br(t)).
+We follow the proof strategy from (Du et al., 2018). Now we derive the total movement of weights
+and biases. Let f(t) = f(X; θ(t)) where fi(t) = f(xi; θ(t)). The dynamics of each prediction is
+given by
+d
+dtfi(t) =
+�∂f(xi; θ(t))
+∂θ(t)
+, dθ(t)
+dt
+�
+=
+n
+�
+j=1
+(yj − fj(t))
+�∂f(xi; θ(t))
+∂θ(t)
+, ∂f(xj; θ(t))
+∂θ(t)
+�
+=
+n
+�
+j=1
+(yj − fj(t))Hij(t),
+which implies
+d
+dtf(t) = H(t)(y − f(t)).
+(A.1)
+Lemma A.8 (Gradient Bounds). For any 0 ≤ s ≤ t, we have
+����
+∂L(W(s), b(s))
+∂wr(s)
+����
+2
+≤
+� n
+m ∥f(s) − y∥2 ,
+����
+∂L(W(s), b(s))
+∂br(s)
+����
+2
+≤
+� n
+m ∥f(s) − y∥2 .
+Proof. We have:
+����
+∂L(W(s), b(s))
+∂wr(s)
+����
+2
+=
+�����
+1
+√m
+n
+�
+i=1
+(f(xi; W(s), b(s)) − yi)arxiI(wr(s)⊤xi ≥ br)
+�����
+2
+≤
+1
+√m
+n
+�
+i=1
+|f(xi; W(s), b(s)) − yi|
+≤
+� n
+m ∥f(s) − y∥2 ,
+where the first inequality follows from triangle inequality, and the second inequality follows from
+Cauchy-Schwarz inequality.
+21
+
+Similarly, we also have:
+����
+∂L(W(s), b(s))
+∂br(s)
+����
+2
+=
+�����
+1
+√m
+n
+�
+i=1
+(f(xi; W(s), b(s)) − yi)arI(wr(s)⊤xi ≥ br)
+�����
+2
+≤
+1
+√m
+n
+�
+i=1
+|f(xi; W(s), b(s)) − yi|
+≤
+� n
+m ∥f(s) − y∥2 .
+A.4.1
+Gradient Descent
+Lemma A.9. Assume λ > 0. Assume ∥y − f(k)∥2
+2 ≤ (1 − ηλ/4)k ∥y − f(0)∥2
+2 holds for all k′ ≤ k.
+Then for every r ∈ [m],
+∥wr(k + 1) − wr(0)∥2 ≤ 8√n ∥y − f(0)∥2
+√mλ
+:= Dw,
+|br(k + 1) − br(0)| ≤ 8√n ∥y − f(0)∥2
+√mλ
+:= Db.
+Proof.
+∥wr(k + 1) − wr(0)∥2 ≤ η
+k
+�
+k′=0
+����
+∂L(W(k′))
+∂wr(k′)
+����
+2
+≤ η
+k
+�
+k′=0
+� n
+m
+��y − f(k′)
+��
+2
+≤ η
+k
+�
+k′=0
+� n
+m(1 − ηλ/4)k′/2 ∥y − f(0)∥2
+≤ η
+k
+�
+k′=0
+� n
+m(1 − ηλ/8)k′ ∥y − f(0)∥2
+≤ η
+∞
+�
+k′=0
+� n
+m(1 − ηλ/8)k′ ∥y − f(0)∥2
+≤ 8√n
+√mλ ∥y − f(0)∥2 ,
+where the first inequality is by Triangle inequality, the second inequality is by Lemma A.8, the
+third inequality is by our assumption and the fourth inequality is by (1−x)1/2 ≤ 1−x/2 for x ≥ 0.
+The proof for b is similar.
+22
+
+A.4.2
+Gradient Flow
+Lemma A.10. Suppose for 0 ≤ s ≤ t, λmin(H(s)) ≥
+λ0
+2
+> 0.
+Then we have ∥y − f(t)∥2
+2 ≤
+exp(−λ0t) ∥y − f(0)∥2
+2 and for any r ∈ [m], ∥wr(t) − wr(0)∥2 ≤
+√n∥y−f(0)∥2
+√mλ0
+and |br(t) − br(0)| ≤
+√n∥y−f(0)∥2
+√mλ0
+.
+Proof. By the dynamics of prediction in Equation (A.1), we have
+d
+dt ∥y − f(t)∥2
+2 = −2(y − f(t))⊤H(t)(y − f(t))
+≤ −λ0 ∥y − f(t)∥2
+2 ,
+which implies
+∥y − f(t)∥2
+2 ≤ exp(−λ0t) ∥y − f(t)∥2
+2 .
+Now we bound the gradient norm of the weights
+����
+d
+dswr(s)
+����
+2
+=
+�����
+n
+�
+i=1
+(yi − fi(s)) 1
+√marxiI(wr(s)⊤xi ≥ b(s))
+�����
+2
+≤
+1
+√m
+n
+�
+i=1
+|yifi(s)| ≤
+√n
+√m ∥y − f(s)∥2 ≤
+√n
+√m exp(−λ0s) ∥y − f(0)∥2 .
+Integrating the gradient, the change of weight can be bounded as
+∥wr(t) − wr(0)∥2 ≤
+� t
+0
+����
+d
+dswr(s)
+�����
+2
+ds ≤
+√n ∥y − f(0)∥2
+√mλ0
+.
+For bias, we have
+����
+d
+dsbr(s)
+����
+2
+=
+�����
+n
+�
+i=1
+(yi − fi(s)) 1
+√marI(wr(s)⊤xi ≥ b(s))
+�����
+2
+≤
+1
+√m
+n
+�
+i=1
+|yi − fi(s)| ≤
+√n
+√m ∥y − f(s)∥2 ≤
+√n
+√m exp(−λ0s) ∥y − f(0)∥2 .
+Now, the change of bias can be bounded as
+∥br(t) − br(0)∥2 ≤
+� t
+0
+����
+d
+dswr(s)
+����
+2
+ds ≤
+√n ∥y − f(0)∥2
+√mλ0
+.
+A.5
+Gradient Descent Convergence Analysis
+A.5.1
+Upper bound of the initial error
+Lemma A.11 (Initial error upper bound). Let B > 0 be the initialization value of the biases and
+all the weights be initialized from standard Gaussian. Let δ ∈ (0, 1) be the failure probability. Then,
+23
+
+with probability at least 1 − δ, we have
+∥f(0)∥2
+2 = O(n(exp(−B2/2) + 1/m) log3(mn/δ)),
+∥f(0) − y∥2
+2 = O
+�
+n + n
+�
+exp(−B2/2) + 1/m
+�
+log3(2mn/δ)
+�
+.
+Proof. Since we are only analyzing the initialization stage, for notation ease, we omit the depen-
+dence on time without any confusion. We compute
+∥y − f∥2
+2 =
+n
+�
+i=1
+(yi − f(xi))2
+=
+n
+�
+i=1
+�
+yi −
+1
+√m
+m
+�
+r=1
+arσ(w⊤
+r xi − B)
+�2
+=
+n
+�
+i=1
+�
+�y2
+i − 2 yi
+√m
+m
+�
+r=1
+arσ(w⊤
+r xi − B) + 1
+m
+� m
+�
+r=1
+arσ(w⊤
+r xi − B)
+�2�
+� .
+Since w⊤
+r xi ∼ N(0, 1) for all r ∈ [m] and i ∈ [n], by Gaussian tail bound and a union bound over
+r, i, we have
+P[∀i ∈ [n], j ∈ [m] : w⊤
+r xi ≤
+�
+2 log(2mn/δ)] ≥ 1 − δ/2.
+Let E1 denote this event. Conditioning on the event E1, let
+zi,r :=
+1
+√m · ar · min
+�
+σ(w⊤
+r xi − B),
+�
+2 log(2mn/δ)
+�
+.
+Notice that zi,r ̸= 0 with probability at most exp(−B2/2). Thus,
+E
+ar,wr[z2
+i,r] ≤ exp(−B2/2) 1
+m2 log(2mn/δ).
+By randomness in ar, we know E[zi,r] = 0. Now apply Bernstein’s inequality in Lemma D.1, we
+have for all t > 0,
+P
+������
+m
+�
+r=1
+zi,r
+����� > t
+�
+≤ exp
+�
+− min
+�
+t2/2
+4 exp(−B2/2) log(2mn/δ),
+√mt/2
+2
+�
+2 log(2mn/δ)
+��
+.
+Thus, by a union bound, with probability at least 1 − δ/2, for all i ∈ [n],
+�����
+m
+�
+r=1
+zi,r
+����� ≤
+�
+2 log(2mn/δ) exp(−B2/2)2 log(2n/δ) + 2
+�
+2 log(2mn/δ)
+m
+log(2n/δ)
+≤
+�
+2 exp(−B2/4) + 2
+�
+2/m
+�
+log3/2(2mn/δ).
+24
+
+Let E2 denote this event. Thus, conditioning on the events E1, E2, with probability 1 − δ,
+∥f(0)∥2
+2 =
+n
+�
+i=1
+� m
+�
+r=1
+zi,r
+�2
+= O(n(exp(−B2/2) + 1/m) log3(mn/δ))
+and
+∥y − f(0)∥2
+2
+=
+n
+�
+i=1
+y2
+i − 2
+n
+�
+i=1
+yi
+m
+�
+r=1
+zi,r +
+n
+�
+i=1
+� m
+�
+r=1
+zi,r
+�2
+≤
+n
+�
+i=1
+y2
+i + 2
+n
+�
+i=1
+|yi|
+�
+2 exp(−B2/4) + 2
+�
+2/m
+�
+log3/2(2mn/δ)
++
+n
+�
+i=1
+��
+2 exp(−B2/4) + 2
+�
+2/m
+�
+log3/2(2mn/δ)
+�2
+= O
+�
+n + n
+�
+exp(−B2/2) + 1/m
+�
+log3(2mn/δ)
+�
+,
+where we assume yi = O(1) for all i ∈ [n].
+A.5.2
+Error Decomposition
+We follow the proof outline in (Song and Yang, 2019; Song et al., 2021a) and we generalize it to
+networks with trainable b. Let us define matrix H⊥ similar to H except only considering flipped
+neurons by
+H⊥
+ij (k) := 1
+m
+�
+r∈Si
+(⟨xi, xj⟩ + 1)I(wr(k)⊤xi ≥ br(k), wr(k)⊤xj ≥ br(k))
+and vector v1, v2 by
+v1,i :=
+1
+√m
+�
+r∈Si
+ar(σ(⟨wr(k + 1), xi⟩ − br(k + 1)) − σ(⟨wr(k), xi⟩ − br(k))),
+v2,i :=
+1
+√m
+�
+r∈Si
+ar(σ(⟨wr(k + 1), xi⟩ − br(k + 1)) − σ(⟨wr(k), xi⟩ − br(k))).
+Now we give out our error update.
+Claim A.12.
+∥y − f(k + 1)∥2
+2 = ∥y − f(k)∥2
+2 + B1 + B2 + B3 + B4,
+where
+B1 := −2η(y − f(k))⊤H(k)(y − f(k)),
+B2 := 2η(y − f(k))⊤H⊥(k)(y − f(k)),
+B3 := −2(y − f(k))⊤v2,
+25
+
+B4 := ∥f(k + 1) − f(k)∥2
+2 .
+Proof. First we can write
+v1,i =
+1
+√m
+�
+r∈Si
+ar
+�
+σ
+��
+wr(k) − η ∂L
+∂wr
+, xi
+�
+−
+�
+br(k) − η ∂L
+∂br
+��
+− σ(⟨wr(k), xi⟩ − br(k))
+�
+=
+1
+√m
+�
+r∈Si
+ar
+��
+−η ∂L
+∂wr
+, xi
+�
++ η ∂L
+∂br
+�
+I(⟨wr(k), xi⟩ − br(k) ≥ 0)
+=
+1
+√m
+�
+r∈Si
+ar
+�
+�η 1
+√m
+n
+�
+j=1
+(yj − fj(k))ar(⟨xj, xi⟩ + 1)I(wr(k)⊤xj ≥ br(k))
+�
+� I(⟨wr(k), xi⟩ − br(k) ≥ 0)
+= η
+n
+�
+j=1
+(yj − fj(k))(Hij(k) − H⊥
+ij (k))
+which means
+v1 = η(H(k) − H⊥(k))(y − f(k)).
+Now we compute
+∥y − f(k + 1)∥2
+2 = ∥y − f(k) − (f(k + 1) − f(k))∥2
+2
+= ∥y − f(k)∥2
+2 − 2(y − f(k))⊤(f(k + 1) − f(k)) + ∥f(k + 1) − f(k)∥2
+2 .
+Since f(k + 1) − f(k) = v1 + v2, we can write the cross product term as
+(y − f(k))⊤(f(k + 1) − f(k))
+= (y − f(k))⊤(v1 + v2)
+= (y − f(k))⊤v1 + (y − f(k))⊤v2
+= η(y − f(k))⊤H(k)(y − f(k))
+− η(y − f(k))⊤H⊥(k)(y − f(k)) + (y − f(k))⊤v2.
+A.5.3
+Bounding the decrease of the error
+Lemma A.13. Assume λ > 0. Assume we choose Rw, Rb, B where Rw, Rb ≤ min{1/B, 1} such
+that 8cn(Rw + Rb) exp(−B2/2) ≤ λ/8. Denote δ0 = δ + n2 exp(− 2
+3cm(Rw + Rb) exp(−B2/2)).
+Then,
+P[B1 ≤ −η5λ ∥y − f(k)∥2
+2 /8] ≥ 1 − δ0.
+Proof. By Lemma A.6 and our assumption,
+λmin(H(W)) > 0.75λ − n · 8c(Rw + Rb) exp(−B2/2) ≥ 5λ/8
+26
+
+with probability at least 1 − δ0. Thus,
+(y − f(k))⊤H(k)(y − f(k)) ≥ ∥y − f(k)∥2
+2 5λ/8.
+A.5.4
+Bounding the effect of flipped neurons
+Here we bound the term B2, B3. First, we introduce a fact.
+Fact A.14.
+���H⊥(k)
+���
+2
+F ≤ 4n
+m2
+n
+�
+i=1
+|Si|2.
+Proof.
+���H⊥(k)
+���
+2
+F =
+�
+i,j∈[n]
+�
+� 1
+m
+�
+r∈Si
+(x⊤
+i xj + 1)I(wr(k)⊤xi ≥ br(k), wr(k)⊤xj ≥ br(k))
+�
+�
+2
+≤
+�
+i,j∈[n]
+� 1
+m2|Si|
+�2
+≤ 4n
+m2
+n
+�
+i=1
+|Si|2.
+Lemma A.15. Denote δ0 = n exp(− 2
+3cm(Rw + Rb) exp(−B2/2)). Then,
+P[B2 ≤ 8ηnc(Rw + Rb) exp(−B2/2) · ∥y − f(k)∥2
+2] ≥ 1 − δ0.
+Proof. First, we have
+B2 ≤ 2η ∥y − f(k)∥2
+2
+���H⊥(k)
+���
+2 .
+Then, by Fact A.14,
+���H⊥(k)
+���
+2
+2 ≤
+���H⊥(k)
+���
+2
+F ≤ 4n
+m2
+n
+�
+i=1
+|Si|2.
+By Corollary A.5, we have
+P[∀i ∈ [n] : |Si| ≤ 2mc(Rw + Rb) exp(−B2/2)] ≥ 1 − δ0.
+Thus, with probability at least 1 − δ0,
+���H⊥(k)
+���
+2 ≤ 4nc(Rw + Rb) exp(−B2/2).
+27
+
+Lemma A.16. Denote δ0 = n exp(− 2
+3cm(Rw + Rb) exp(−B2/2)). Then,
+P[B3 ≤ 4cηn(Rw + Rb) exp(−B2/2) ∥y − f(k)∥2
+2] ≥ 1 − δ0.
+Proof. By Cauchy-Schwarz inequality, we have B3 ≤ 2 ∥y − f(k)∥2 ∥v2∥2. We have
+∥v2∥2
+2 ≤
+n
+�
+i=1
+�
+� η
+√m
+�
+r∈Si
+����
+� ∂L
+∂wr
+, xi
+����� +
+����
+∂L
+∂br
+����
+�
+�
+2
+≤
+n
+�
+i=1
+η2
+m max
+i∈[n]
+�����
+� ∂L
+∂wr
+, xi
+����� +
+����
+∂L
+∂br
+����
+�2
+|Si|2
+≤ nη2
+m
+�
+2
+� n
+m ∥f(k) − y∥2 2mc(Rw + Rb) exp(−B2/2)
+�2
+= 16c2η2n2 ∥y − f(k)∥2
+2 (Rw + Rb)2 exp(−B2),
+where the last inequality is by Lemma A.8 and Corollary A.5 which holds with probability at least
+1 − δ0.
+A.5.5
+Bounding the network update
+Lemma A.17.
+B4 ≤ C2
+2η2n2 ∥y − f(k)∥2
+2 exp(−B2).
+for some constant C2.
+Proof. Recall that the definition that Son(i, t) = {r ∈ [m] : wr(t)⊤xi ≥ br(t)}, i.e., the set of
+neurons that activates for input xi at the t-th step of gradient descent.
+∥f(k + 1) − f(k)∥2
+2 ≤
+n
+�
+i=1
+�
+� η
+√m
+�
+r:r∈Son(i,k+1)∪Son(i,k)
+����
+� ∂L
+∂wr
+, xi
+����� +
+����
+∂L
+∂br
+����
+�
+�
+2
+≤ nη2
+m (|Son(i, k + 1)| + |Son(i, k)|)2 max
+i∈[n]
+�����
+� ∂L
+∂wr
+, xi
+����� +
+����
+∂L
+∂br
+����
+�2
+≤ nη2
+m
+�
+C2m exp(−B2/2) ·
+� n
+m ∥y − f(k)∥2
+�2
+≤ C2
+2η2n2 ∥y − f(k)∥2
+2 exp(−B2).
+where the third inequality is by Lemma A.19 for some C2.
+A.5.6
+Putting it all together
+Theorem A.18 (Convergence). Assume λ > 0. Let η ≤ λ exp(B2)
+5C2
+2n2 , B ∈ [0, √0.5 log m] and
+m ≥ �Ω
+�
+λ−4n4 �
+1 +
+�
+exp(−B2/2) + 1/m
+�
+log3(2mn/δ)
+�
+exp(−B2)
+�
+.
+28
+
+Assume λ = λ0 exp(−B2/2) for some constant λ0. Then,
+P
+�
+∀t : ∥y − f(t)∥2
+2 ≤ (1 − ηλ/4)t ∥y − f(0)∥2
+2
+�
+≥ 1 − δ − e−Ω(n).
+Proof. From Lemma A.13, Lemma A.15, Lemma A.16 and Lemma A.17, we know with probability
+at least 1 − 2n2 exp(− 2
+3cm(Rw + Rb) exp(−B2/2)) − δ, we have
+∥y − f(k + 1)∥2
+2 ≤ ∥y − f(k)∥2
+2 (1 − 5ηλ/8 + 12ηnc(Rw + Rb) exp(−B2/2) + C2
+2η2n2 ∥y − f(k)∥2
+2 exp(−B2)).
+By Lemma A.9, we need
+Dw = 8√n ∥y − f(0)∥2
+√mλ
+≤ Rw,
+Db = 8√n ∥y − f(0)∥2
+√mλ
+≤ Rb.
+By Lemma A.11, we have
+P[∥f(0) − y∥2
+2 = O
+�
+n + n
+�
+exp(−B2/2) + 1/m
+�
+log3(2mn/δ)
+�
+] ≥ 1 − δ.
+Let R = min{Rw, Rb}, D = max{Dw, Db}. Combine the results we have
+R > Ω(λ−1m−1/2n
+�
+1 + (exp(−B2/2) + 1/m) log3(2mn/δ)).
+Lemma A.13 requires
+8cn(Rw + Rb) exp(−B2/2) ≤ λ/8
+⇒ R ≤ λ exp(B2/2)
+128cn
+.
+which implies a lower bound on m
+m ≥ Ω
+�
+λ−4n4 �
+1 +
+�
+exp(−B2/2) + 1/m
+�
+log3(2mn/δ)
+�
+exp(−B2)
+�
+.
+Lemma A.1 further requires a lower bound of m = Ω(λ−1n · log(n/δ)) which can be ignored.
+Lemma A.6 further requires R < min{1/B, 1} which implies
+B <
+128cn
+λ exp(B2/2),
+m ≥ �Ω
+�
+λ−4n4 �
+1 +
+�
+exp(−B2/2) + 1/m
+�
+log3(2mn/δ)
+�
+exp(−B2)
+�
+.
+From Theorem F.1 in (Song et al., 2021a) we know that λ = λ0 exp(−B2/2) for some λ0 with
+no dependence on B and λ exp(B2/2) ≤ 1. Thus, by our constraint on m and B, this is always
+satisfied.
+Finally, to require
+12ηnc(Rw + Rb) exp(−B2/2) + C2
+2η2n2 exp(−B2) ≤ ηλ/4,
+29
+
+we need η ≤ λ exp(B2)
+5C2
+2n2 . By our choice of m, B, we have
+2n2 exp(−2
+3cm(Rw + Rb) exp(−B2/2)) = e−Ω(n).
+A.6
+Bounding the Number of Activated Neurons per Iteration
+First we define the set of activated neurons at iteration t for training point xi to be
+Son(i, t) = {r ∈ [m] : wr(t)⊤xi ≥ br(t)}.
+Lemma A.19 (Number of Activated Neurons at Initialization). Assume the choice of m in The-
+orem A.18. With probability at least 1 − e−Ω(n) over the random initialization, we have
+|Son(i, t)| = O(m · exp(−B2/2)),
+for all 0 ≤ t ≤ T and i ∈ [n]. And As a by-product,
+∥Z(0)∥2
+F ≤ 8n exp(−B2/2).
+Proof. First we bound the number of activated neuron at the initialization. We have P[w⊤
+r xi ≥
+B] ≤ exp(−B2/2). By Bernstein’s inequality,
+P[|Son(i, 0)| ≥ m exp(−B2/2) + t] ≤ exp
+�
+−
+t2
+m exp(−B2/2) + t/3
+�
+.
+Take t = m exp(−B2/2) we have
+P[|Son(i, 0)| ≥ 2m exp(−B2/2)] ≤ exp
+�
+−m exp(−B2/2)/4
+�
+.
+By a union bound over i ∈ [n], we have
+P[∀i ∈ [n] : |Son(i, 0)| ≤ 2m exp(−B2/2)] ≥ 1 − n exp
+�
+−m exp(−B2/2)/4
+�
+.
+Notice that
+∥Z(0)∥2
+F ≤ 4
+m
+m
+�
+r=1
+n
+�
+i=1
+Ir,i(0) ≤ 8n exp(−B2/2).
+Lemma A.20 (Number of Activated Neurons per Iteration). Assume the parameter settings in
+Theorem A.18. With probability at least 1 − e−Ω(n) over the random initialization, we have
+|Son(i, t)| = O(m · exp(−B2/2))
+for all 0 ≤ t ≤ T and i ∈ [n].
+30
+
+Proof. By Corollary A.5 and Theorem A.18, we have
+P[∀i ∈ [n] : |Si| ≤ 4mc exp(−B2/2)] ≥ 1 − e−Ω(n).
+Recall Si is the set of flipped neurons during the entire training process. Notice that |Son(i, t)| ≤
+|Son(i, 0)| + |Si|. Thus, by Lemma A.19
+P[∀i ∈ [n] : |Son(i, t)| = O(m exp(−B2/2))] ≥ 1 − e−Ω(n).
+B
+Bounding the Restricted Smallest Eigenvalue with Data Sepa-
+ration
+Theorem B.1. Let X = (x1, . . . , xn) be points in Rd with ∥xi∥2 = 1 for all i ∈ [n] and w ∼
+N(0, Id). Suppose that there exists δ ∈ [0,
+√
+2] such that
+min
+i̸=j∈[n](∥xi − xj∥2 , ∥xi + xj∥2) ≥ δ.
+Let B ≥ 0. Recall the limit NTK matrix H∞ defined as
+H∞
+ij :=
+E
+w∼N(0,I)
+�
+(⟨xi, xj⟩ + 1)I(w⊤xi ≥ B, w⊤xj ≥ B)
+�
+.
+Define p0 = P[w⊤x1 ≥ B] and pij = P[w⊤xi ≥ B, w⊤xj ≥ B] for i ̸= j.
+Define the (data-
+dependent) region R = {a ∈ Rn : �
+i̸=j aiajpij ≥ mini′̸=j′ pi′j′ �
+i̸=j aiaj} and let λ := min∥a∥2=1, a∈R a⊤H∞a.
+Then, λ ≥ max(0, λ′) where
+λ′ ≥ p0 − min
+i̸=j pij
+≥ max
+�
+1
+2 −
+B
+√
+2π,
+� 1
+B − 1
+B3
+� e−B2/2
+√
+2π
+�
+− e−B2/(2−δ2/2)
+π − arctan
+�
+δ√
+1−δ2/4
+1−δ2/2
+�
+2π
+.
+Proof. Define ∆ := maxi̸=j | ⟨xi, xj⟩ |. Then by our assumption,
+1 − ∆ = 1 − max
+i̸=j | ⟨xi, xj⟩ | = mini̸=j(∥xi − xj∥2
+2 , ∥xi + xj∥2
+2)
+2
+≥ δ2/2
+⇒ ∆ ≤ 1 − δ2/2.
+Further, we define
+Z(w) := [x1I(w⊤x1 ≥ B), x2I(w⊤x2 ≥ B), . . . , xnI(w⊤xn ≥ B)] ∈ Rd×n.
+Notice that H∞ = Ew∼N(0,I)
+�
+Z(w)⊤Z(w) + I(Xw ≥ B)I(Xw ≥ B)⊤�
+. We need to lower bound
+min
+∥a∥2=1,a∈R a⊤H∞a =
+min
+∥a∥2=1,a∈R a⊤
+E
+w∼N(0,I)
+�
+Z(w)⊤Z(w)
+�
+a
+31
+
++ a⊤
+E
+w∼N(0,I)
+�
+I(Xw ≥ B)I(Xw ≥ B)⊤�
+a
+≥
+min
+∥a∥2=1,a∈R a⊤
+E
+w∼N(0,I)
+�
+I(Xw ≥ B)I(Xw ≥ B)⊤�
+a.
+Now, for a fixed a,
+a⊤
+E
+w∼N(0,I)
+�
+I(Xw ≥ B)I(Xw ≥ B)⊤�
+a =
+n
+�
+i=1
+a2
+i P[w⊤xi ≥ B] +
+�
+i̸=j
+aiaj P[w⊤xi ≥ B, w⊤xj ≥ B]
+= p0 ∥a∥2
+2 +
+�
+i̸=j
+aiajpij,
+where the last equality is by P[w⊤x1 ≥ B] = . . . = P[w⊤xn ≥ B] = p0 which is due to spherical
+symmetry of standard Gaussian. Notice that maxi̸=j pij ≤ p0. Since a ∈ R,
+E
+w∼N(0,I)
+�
+(a⊤I(Xw ≥ B))2�
+≥ (p0 − min
+i̸=j pij) ∥a∥2
+2 + (min
+i̸=j pij) ∥a∥2
+2 + (min
+i̸=j pij)
+�
+i̸=j
+aiaj
+= (p0 − min
+i̸=j pij) ∥a∥2
+2 + (min
+i̸=j pij)
+��
+i
+ai
+�2
+.
+Thus,
+λ ≥
+min
+∥a∥2=1,a∈R
+E
+w∼N(0,I)
+�
+(a⊤I(Xw ≥ B))2�
+≥
+min
+∥a∥2=1,a∈R(p0 − min
+i̸=j pij) ∥a∥2
+2 +
+min
+∥a∥2=1,a∈R(min
+i̸=j pij)
+��
+i
+ai
+�2
+≥ p0 − min
+i̸=j pij.
+Now we need to upper bound
+min
+i̸=j pij ≤ max
+i̸=j pij.
+We divide into two cases: B = 0 and B > 0.
+Consider two fixed examples x1, x2.
+Then, let
+v = (I − x1x⊤
+1 )x2/
+��(I − x1x⊤
+1 )x2
+�� and c = | ⟨x1, x2⟩ | 1.
+Case 1: B = 0. First, let us define the region A0 as
+A0 =
+�
+(g1, g2) ∈ R2 : g1 ≥ 0, g1 ≥ −
+√
+1 − c2
+c
+g2
+�
+.
+Then,
+P[w⊤x1 ≥ 0, w⊤x2 ≥ 0] = P[w⊤x1 ≥ 0, w⊤(cx1 +
+�
+1 − c2v) ≥ 0]
+= P[g1 ≥ 0, cg1 +
+�
+1 − c2g2 ≥ 0]
+1Here we force c to be positive. Since we are dealing with standard Gaussian, the probability is exactly the same
+if c < 0 by symmetry and therefore, we force c > 0.
+32
+
+= P[A0]
+=
+π − arctan
+� √
+1−c2
+|c|
+�
+2π
+≤
+π − arctan
+� √
+1−∆2
+|∆|
+�
+2π
+,
+where we define g1 := w⊤x1 and g2 := w⊤v and the second equality is by the fact that since x1 and
+v are orthonormal, g1 and g2 are two independent standard Gaussian random variables; the last
+inequality is by arctan is a monotonically increasing function and
+√
+1−c2
+|c|
+is a decreasing function in
+|c| and |c| ≤ ∆. Thus,
+min
+i̸=j pij ≤ max
+i̸=j pij ≤
+π − arctan
+� √
+1−∆2
+|∆|
+�
+2π
+.
+Case 2: B > 0. First, let us define the region
+A =
+�
+(g1, g2) ∈ R2 : g1 ≥ B, g1 ≥ B
+c −
+√
+1 − c2
+c
+g2
+�
+.
+Then, following the same steps as in case 1, we have
+P[w⊤x1 ≥ B, w⊤x2 ≥ B] = P[g1 ≥ B, cg1 +
+�
+1 − c2g2 ≥ B] = P[A].
+Let B1 = B and B2 = B
+�
+1−c
+1+c. Further, notice that A = A0 + (B1, B2). Then,
+P[A] =
+��
+(g1,g2)∈A
+1
+2π exp
+�
+−g2
+1 + g2
+2
+2
+�
+dg1 dg2
+=
+��
+(g1,g2)∈A0
+1
+2π exp
+�
+−(g1 + B1)2 + (g2 + B2)2
+2
+�
+dg1 dg2
+= e−(B2
+1+B2
+2)/2
+��
+(g1,g2)∈A0
+1
+2π exp {−B1g1 − B2g2} exp
+�
+−g2
+1 + g2
+2
+2
+�
+dg1 dg2.
+Now, B1g1 + B2g2 = Bg1 + B
+�
+1−c
+1+cg2 ≥ 0 always holds if and only if g1 ≥ −
+�
+1−c
+1+cg2. Define the
+region A+ to be
+A+ =
+�
+(g1, g2) ∈ R2 : g1 ≥ 0, g1 ≥ −
+�
+1 − c
+1 + cg2
+�
+.
+Observe that
+�
+1 − c
+1 + c ≤
+√
+1 − c2
+c
+=
+�
+(1 − c)(1 + c)
+c
+⇔ c ≤ 1 + c.
+33
+
+Thus, A0 ⊂ A+. Therefore,
+P[A] ≤ e−(B2
+1+B2
+2)/2
+��
+(g1,g2)∈A0
+1
+2π exp
+�
+−g2
+1 + g2
+2
+2
+�
+dg1 dg2
+= e−(B2
+1+B2
+2)/2 P[A0]
+= e−(B2
+1+B2
+2)/2 π − arctan
+� √
+1−c2
+|c|
+�
+2π
+≤ e−B2/(1+∆) π − arctan
+� √
+1−∆2
+|∆|
+�
+2π
+.
+Finally, we need to lower bound p0. This can be done in two ways: when B is small, we apply
+Gaussian anti-concentration bound and when B is large, we apply Gaussian tail bounds. Thus,
+p0 = P[w⊤x1 ≥ B] ≥ max
+�
+1
+2 −
+B
+√
+2π,
+� 1
+B − 1
+B3
+� e−B2/2
+√
+2π
+�
+.
+Combining the lower bound of p0 and upper bound on maxi̸=j pij we have
+λ ≥ p0 − min
+i̸=j pij ≥ max
+�
+1
+2 −
+B
+√
+2π,
+� 1
+B − 1
+B3
+� e−B2/2
+√
+2π
+�
+− e−B2/(1+∆) π − arctan
+� √
+1−∆2
+|∆|
+�
+2π
+.
+Applying ∆ ≤ 1 − δ2/2 and noticing that H∞ is positive semi-definite gives our final result.
+C
+Generalization
+C.1
+Rademacher Complexity
+In this section, we would like to compute the Rademacher Complexity of our network. Rademacher
+complexity is often used to bound the deviation from empirical risk and true risk (see, e.g. (Shalev-
+Shwartz and Ben-David, 2014).)
+Definition C.1 (Empirical Rademacher Complexity). Given n samples S, the empirical Rademacher
+complexity of a function class F, where f : Rd → R for f ∈ F, is defined as
+RS(F) = 1
+n E
+ϵ
+�
+sup
+f∈F
+n
+�
+i=1
+ϵif(xi)
+�
+where ϵ = (ϵ1, . . . , ϵn)⊤ and ϵi is an i.i.d Rademacher random variable.
+Theorem C.2 ((Shalev-Shwartz and Ben-David, 2014)). Suppose the loss function ℓ(·, ·) is bounded
+in [0, c] and is ρ-Lipschitz in the first argument. Then with probability at least 1 − δ over sample S
+of size n:
+sup
+f∈F
+LD(f) − LS(f) ≤ 2ρRS(F) + 3c
+�
+log(2/δ)
+2n
+.
+34
+
+In order to get meaningful generalization bound via Rademacher complexity, previous results,
+such as (Arora et al., 2019; Song and Yang, 2019), multiply the neural network by a scaling factor
+κ to make sure the neural network output something small at the initialization, which requires at
+least modifying all the previous lemmas we already established. We avoid repeating our arguments
+by utilizing symmetric initialization to force the neural network to output exactly zero for any
+inputs at the initialization. 2
+Definition C.3 (Symmetric Initialization). For a one-hidden layer neural network with 2m neu-
+rons, the network is initialized as the following
+1. For r ∈ [m], initialize wr ∼ N(0, I) and ar ∼ Uniform({−1, 1}).
+2. For r ∈ {m + 1, . . . , 2m}, let wr = wr−m and ar = −ar−m.
+It is not hard to see that all of our previously established lemmas hold including expectation
+and concentration. The only effect this symmetric initialization brings is to worse the concentration
+by a constant factor of 2 which can be easily addressed. For detailed analysis, see (Munteanu et al.,
+2022).
+In order to state our final theorem, we need to use Definition 3.7. Now we can state our theorem
+for generalization.
+Theorem C.4. Fix a failure probability δ ∈ (0, 1) and an accuracy parameter ϵ ∈ (0, 1). Suppose
+the training data S = {(xi, yi)}n
+i=1 are i.i.d. samples from a (λ, δ, n)-non-degenerate distribution
+D. Assume the settings in Theorem A.18 except now we let
+m ≥ �Ω
+�
+λ−4n6 �
+1 +
+�
+exp(−B2/2) + 1/m
+�
+log3(2mn/δ)
+�
+exp(−B2)
+�
+.
+Consider any loss function ℓ : R × R → [0, 1] that is 1-Lipschitz in its first argument. Then with
+probability at least 1 − 2δ − e−Ω(n) over the symmetric initialization of W(0) ∈ Rm×d and a ∈ Rm
+and the training samples, the two layer neural network f(W(k), b(k), a) trained by gradient descent
+for k ≥ Ω( 1
+ηλ log n log(1/δ)
+ϵ
+) iterations has population loss LD(f) = E(x,y)∼D[ℓ(f(x), y)] upper bounded
+as
+LD(f(W(k), b(k), a)) ≤
+�
+y⊤(H∞)−1y · 32 exp(−B2/2)
+n
++ �O
+� 1
+n1/2
+�
+.
+Proof. First, we need to bound LS. After training, we have ∥f(k) − y∥2 ≤ ϵ < 1, and thus
+LS(f(W(k), b(k), a)) = 1
+n
+n
+�
+i=1
+[ℓ(fi(k), yi) − ℓ(yi, yi)]
+≤ 1
+n
+n
+�
+i=1
+|fi(k) − yi|
+≤
+1
+√n ∥f(k) − y∥2
+2While preparing the manuscript, the authors notice that this can be alternatively solved by reparameterized the
+neural network by f(x; W)−f(x; W0) and thus minimizing the following objective L = 1
+2
+�n
+i=1(f(xi; W)−f(xi; W0)−
+yi)2. The corresponding generalization is the same since Rademacher complexity is invariant to translation. However,
+since the symmetric initialization is widely adopted in theory literature, we go with symmetric initialization here.
+35
+
+≤
+1
+√n.
+By Theorem C.2, we know that
+LD(f(W(k), b(k), a)) ≤ LS(f(W(k), b(k), a)) + 2RS(F) + �O(n−1/2)
+≤ 2RS(F) + �O(n−1/2).
+Then, by Theorem C.5, we get that for sufficiently large m,
+RS(F) ≤
+�
+y⊤(H∞)−1y · 8 exp(−B2/2)
+n
++ �O
+�exp(−B2/4)
+n1/2
+�
+≤
+�
+y⊤(H∞)−1y · 8 exp(−B2/2)
+n
++ �O
+� 1
+n1/2
+�
+,
+where the last step follows from B > 0.
+Therefore, we conclude that:
+LD(f(W(k), b(k), a)) ≤
+�
+y⊤(H∞)−1y · 32 exp(−B2/2)
+n
++ �O
+� 1
+n1/2
+�
+.
+Theorem C.5. Fix a failure probability δ ∈ (0, 1). Suppose the training data S = {(xi, yi)}n
+i=1 are
+i.i.d. samples from a (λ, δ, n)-non-degenerate distribution D. Assume the settings in Theorem A.18
+except now we let
+m ≥ �Ω
+�
+λ−6n6 �
+1 +
+�
+exp(−B2/2) + 1/m
+�
+log3(2mn/δ)
+�
+exp(−B2)
+�
+.
+Denote the set of one-hidden-layer neural networks trained by gradient descent as F. Then with
+probability at least 1 − 2δ − e−Ω(n) over the randomness in the symmetric initialization and the
+training data, the set F has empirical Rademacher complexity bounded as
+RS(F) ≤
+�
+y⊤(H∞)−1y · 8 exp(−B2/2)
+n
++ �O
+�exp(−B2/4)
+n1/2
+�
+.
+Note that the only extra requirement we make on m is the (n/λ)6 dependence instead of (n/λ)4
+which is needed for convergence. The dependence of m on n is significantly better than previous
+work (Song and Yang, 2019) where the dependence is n14. We take advantage of our initialization
+and new analysis to improve the dependence on n.
+Proof. Let Rw (Rb) denotes the maximum distance moved any any neuron weight (bias), the same
+role as Dw (Db) in Lemma A.9. From Lemma A.9 and Lemma A.11, and we have
+max(Rw, Rb) ≤ O
+�
+�n
+�
+1 + (exp(−B2/2) + 1/m) log3(2mn/δ)
+√mλ
+�
+� .
+36
+
+The rest of the proof depends on the results from Lemma C.6 and Lemma C.8.
+Let R :=
+∥[W, b](k) − [W, b](0)∥F . By Lemma C.6 we have
+RS(FRw,Rb,R) ≤ R
+�
+8 exp(−B2/2)
+n
++ 4c(Rw + Rb)2√m exp(−B2/2)
+≤ R
+�
+8 exp(−B2/2)
+n
++ O
+�n2(1 + (exp(−B2/2) + 1/m) log3(2mn/δ)) exp(−B2/2)
+√mλ2
+�
+.
+Lemma C.8 gives that
+R ≤
+�
+y⊤(H∞)−1y + O
+�
+n
+λ
+�exp(−B2/2) log(n/δ)
+m
+�1/4�
++ O
+�
+n
+�
+(Rw + Rb) exp(−B2/2)
+λ
+�
++ n
+λ2 · O
+�
+exp(−B2/4)
+�
+log(n2/δ)
+m
++ (Rw + Rb) exp(−B2/2)
+�
+.
+Combining the above results and using the choice of m, R, B in Theorem A.18 gives us
+R(F) ≤
+�
+y⊤(H∞)−1y · 8 exp(−B2/2)
+n
++ O
+��
+n exp(−B2/2)
+λ
+�exp(−B2/2) log(n/δ)
+m
+�1/4�
++ O
+��
+n(Rw + Rb)
+λ exp(B2/2)
+�
++
+√n
+λ2 · O
+�
+exp(−B2/2)
+�
+log(n2/δ)
+m
++ (Rw + Rb) exp(−3B2/4)
+�
++ O
+�n2(1 + (exp(−B2/2) + 1/m) log3(2mn/δ)) exp(−B2/2)
+√mλ2
+�
+.
+Now, we analyze the terms one by one by plugging in the bound of m and Rw, Rb and show
+that they can be bounded by �O(exp(−B2/4)/n1/2). For the second term, we have
+O
+��
+n exp(−B2/2)
+λ
+�exp(−B2/2) log(n/δ)
+m
+�1/4�
+= O
+�√
+λ exp(−B2/8) log1/4(n/δ)
+n
+�
+.
+For the third term, we have
+O
+��
+n(Rw + Rb)
+λ exp(B2/2)
+�
+= O
+�
+√n
+λ exp(B2/2)
+√n(1 + (exp(−B2/2) + 1/m) log3(2mn/δ))1/4
+m1/4λ1/2
+�
+= O
+�
+n
+exp(B2/2)n6/4 exp(−B2/4)
+�
+= O
+�exp(−B2/4)
+n1/2
+�
+.
+For the fourth term, we have
+√n
+λ2 · O
+�
+exp(−B2/2)
+�
+log(n2/δ)
+m
++ (Rw + Rb) exp(−3B2/4)
+�
+37
+
+= O
+�
+λ
+�
+log(n/δ)
+n2.5
+�
++ O
+�exp(−B2/4)
+n1.5
+�
+.
+For the last term, we have
+O
+�n2(1 + (exp(−B2/2) + 1/m) log3(2mn/δ)) exp(−B2/2)
+√mλ2
+�
+= O
+�
+�λ
+�
+1 + (exp(−B2/2) + 1/m) log3(2mn/δ)
+n
+�
+� .
+Recall our discussion on λ in Section 3.4 that λ = λ0 exp(−B2/2) ≤ 1 for some λ0 independent
+of B. Putting them together, we get the desired upper bound for R(F), and the theorem is then
+proved.
+Lemma C.6. Assume the choice of Rw, Rb, m in Theorem A.18. Given R > 0, with probability at
+least 1 − e−Ω(n) over the random initialization of W(0), a, the following function class
+FRw,Rb,R = {f(W, a, b) : ∥W − W(0)∥2,∞ ≤ Rw, ∥b − b(0)∥∞ ≤ Rb,
+���
+⃗
+[W, b] − [W(0), b(0)]
+��� ≤ R}
+has empirical Rademacher complexity bounded as
+RS(FRw,Rb,R) ≤ R
+�
+8 exp(−B2/2)
+n
++ 4c(Rw + Rb)2√m exp(−B2/2).
+Proof. We need to upper bound RS(FRw,Rb,R). Define the events
+Ar,i = {|wr(0)⊤xi − br(0)| ≤ Rw + Rb}, i ∈ [n], r ∈ [m]
+and a shorthand I(wr(0)⊤xi − B ≥ 0) = Ir,i(0). Then,
+n
+�
+i=1
+ϵi
+m
+�
+r=1
+arσ(w⊤
+r xi − br) −
+n
+�
+i=1
+ϵi
+m
+�
+r=1
+arIr,i(0)(w⊤
+r xi − br)
+=
+n
+�
+i=1
+m
+�
+r=1
+ϵiar
+�
+σ(w⊤
+r xi − br) − Ir,i(0)(w⊤
+r xi − br)
+�
+=
+n
+�
+i=1
+m
+�
+r=1
+I(Ar,i)ϵiar
+�
+σ(w⊤
+r xi − br) − Ir,i(0)(w⊤
+r xi − br)
+�
+=
+n
+�
+i=1
+m
+�
+r=1
+I(Ar,i)ϵiar
+�
+σ(w⊤
+r xi − br) − Ir,i(0)(wr(0)⊤xi − br(0)) − Ir,i(0)((wr − wr(0))⊤xi − (br − br(0)))
+�
+=
+n
+�
+i=1
+m
+�
+r=1
+I(Ar,i)ϵiar
+�
+σ(w⊤
+r xi − br) − σ(wr(0)⊤xi − br(0)) − Ir,i(0)((wr − wr(0))⊤xi − (br − br(0)))
+�
+≤
+n
+�
+i=1
+m
+�
+r=1
+I(Ar,i)2(Rw + Rb),
+38
+
+where the second equality is due to the fact that σ(w⊤
+r xi − br) = Ir,i(0)(w⊤
+r xi − br) if r /∈ Ar,i.
+Thus, the Rademacher complexity can be bounded as
+RS(FRw,Rb,R)
+= 1
+n E
+ϵ
+�
+�����
+sup
+∥W−W(0)∥2,∞≤Rw, ∥b−b(0)∥∞≤Rb,
+���
+⃗
+[W,b]−[W(0),b(0)]
+���≤R
+n
+�
+i=1
+ϵi
+m
+�
+r=1
+ar
+√mσ(w⊤
+r xi − br)
+�
+�����
+≤ 1
+n E
+ϵ
+�
+�����
+sup
+∥W−W(0)∥2,∞≤Rw, ∥b−b(0)∥∞≤Rb,
+���
+⃗
+[W,b]−[W(0),b(0)]
+���≤R
+n
+�
+i=1
+ϵi
+m
+�
+r=1
+ar
+√mIr,i(0)(w⊤
+r xi − br)
+�
+�����
++ 2(Rw + Rb)
+n√m
+n
+�
+i=1
+m
+�
+r=1
+I(Ar,i)
+= 1
+n E
+ϵ
+�
+��
+sup
+���
+⃗
+[W,b]−[W(0),b(0)]
+���≤R
+⃗
+[W, b]
+⊤Z(0)ϵ
+�
+�� + 2(Rw + Rb)
+n√m
+n
+�
+i=1
+m
+�
+r=1
+I(Ar,i)
+= 1
+n E
+ϵ
+�
+��
+sup
+���
+⃗
+[W,b]−[W(0),b(0)]
+���≤R
+⃗
+[W, b] − [W(0), b(0)]
+⊤Z(0)ϵ
+�
+�� + 2(Rw + Rb)
+n√m
+n
+�
+i=1
+m
+�
+r=1
+I(Ar,i)
+≤ 1
+n E
+ϵ [R ∥Z(0)ϵ∥2] + 2(Rw + Rb)
+n√m
+n
+�
+i=1
+m
+�
+r=1
+I(Ar,i)
+≤ R
+n
+�
+E
+ϵ [∥Z(0)ϵ∥2
+2] + 2(Rw + Rb)
+n√m
+n
+�
+i=1
+m
+�
+r=1
+I(Ar,i)
+= R
+n ∥Z(0)∥F + 2(Rw + Rb)
+n√m
+n
+�
+i=1
+m
+�
+r=1
+I(Ar,i),
+where we recall the definition of the matrix
+Z(0) =
+1
+√m
+�
+��
+I1,1(0)a1[x⊤
+1 , −1]⊤
+. . .
+I1,n(0)a1[x⊤
+n , −1]⊤
+...
+...
+Im,1(0)am[x⊤
+1 , −1]⊤
+. . .
+Im,n(0)am[x⊤
+n , −1]⊤
+�
+�� ∈ Rm(d+1)×n.
+By Lemma A.19, we have ∥Z(0)∥F ≤
+�
+8n exp(−B2/2) and by Corollary A.5, we have
+P
+�
+∀i ∈ [n] :
+m
+�
+r=1
+I(Ar,i) ≤ 2mc(Rw + Rb) exp(−B2/2)
+�
+≥ 1 − e−Ω(n).
+Thus, with probability at least 1 − e−Ω(n), we have
+RS(FRw,Rb,R) ≤ R
+�
+8 exp(−B2/2)
+n
++ 4c(Rw + Rb)2√m exp(−B2/2).
+39
+
+C.2
+Analysis of Radius
+Theorem C.7. Assume the parameter settings in Theorem A.18. With probability at least 1 − δ −
+e−Ω(n) over the initialization we have
+f(k) − y = −(I − ηH∞)ky ± e(k),
+where
+∥e(k)∥2 = k(1 − ηλ/4)(k−1)/2ηn3/2 · O
+�
+exp(−B2/4)
+�
+log(n2/δ)
+m
++ (Rw + Rb) exp(−B2/2)
+�
+.
+Proof. Before we start, we assume all the events needed in Theorem A.18 succeed, which happens
+with probability at least 1 − δ − e−Ω(n).
+Recall the no-flipping set Si in Definition A.3. We have
+fi(k + 1) − fi(k) =
+1
+√m
+m
+�
+r=1
+ar[σ(wr(k + 1)⊤xi − br(k + 1)) − σ(wr(k)⊤xi − br(k))]
+=
+1
+√m
+�
+r∈Si
+ar[σ(wr(k + 1)⊤xi − br(k + 1)) − σ(wr(k)⊤xi − br(k))]
++
+1
+√m
+�
+r∈Si
+ar[σ(wr(k + 1)⊤xi − br(k + 1)) − σ(wr(k)⊤xi − br(k))]
+�
+��
+�
+ϵi(k)
+.
+(C.1)
+Now, to upper bound the second term ϵi(k),
+|ϵi(k)| =
+������
+1
+√m
+�
+r∈Si
+ar[σ(wr(k + 1)⊤xi − br(k + 1)) − σ(wr(k)⊤xi − br(k))]
+������
+≤
+1
+√m
+�
+r∈Si
+|wr(k + 1)⊤xi − br(k + 1) − (wr(k)⊤xi − br(k))|
+≤
+1
+√m
+�
+r∈Si
+∥wr(k + 1) − wr(k)∥2 + |br(k + 1) − br(k)|
+=
+1
+√m
+�
+r∈Si
+������
+η
+√mar
+n
+�
+j=1
+(fj(k) − yj)Ir,j(k)xj
+������
+2
++
+������
+η
+√mar
+n
+�
+j=1
+(fj(k) − yj)Ir,j(k)
+������
+≤ 2η
+m
+�
+r∈Si
+n
+�
+j=1
+|fj(k) − yj|
+≤ 2η√n|Si|
+m
+∥f(k) − y∥2
+⇒ ∥ϵ∥2 =
+�
+�
+�
+�
+n
+�
+i=1
+4η2n|Si|2
+m2
+∥f(k) − y∥2
+2 ≤ ηnO((Rw + Rb) exp(−B2/2)) ∥f(k) − y∥2
+(C.2)
+40
+
+where we apply Corollary A.5 in the last inequality. To bound the first term,
+1
+√m
+�
+r∈Si
+ar[σ(wr(k + 1)⊤xi − br(k + 1)) − σ(wr(k)⊤xi − br(k))]
+=
+1
+√m
+�
+r∈Si
+arIr,i(k)
+�
+(wr(k + 1) − wr(k))⊤xi − (br(k + 1) �� br(k))
+�
+=
+1
+√m
+�
+r∈Si
+arIr,i(k)
+�
+�
+�
+�
+�− η
+√mar
+n
+�
+j=1
+(fj(k) − yj)Ir,j(k)xj
+�
+�
+⊤
+xi −
+η
+√mar
+n
+�
+j=1
+(fj(k) − yj)Ir,j(k)
+�
+�
+�
+=
+1
+√m
+�
+r∈Si
+arIr,i(k)
+�
+�− η
+√mar
+n
+�
+j=1
+(fj(k) − yj)Ir,j(k)(x⊤
+j xi + 1)
+�
+�
+= −η
+n
+�
+j=1
+(fj(k) − yj) 1
+m
+�
+r∈Si
+Ir,i(k)Ir,j(k)(x⊤
+j xi + 1)
+= −η
+n
+�
+j=1
+(fj(k) − yj)Hij(k) + η
+n
+�
+j=1
+(fj(k) − yj) 1
+m
+�
+r∈Si
+Ir,i(k)Ir,j(k)(x⊤
+j xi + 1)
+�
+��
+�
+ϵ′
+i(k)
+(C.3)
+where we can upper bound |ϵ′
+i(k)| as
+|ϵ′
+i(k)| ≤ 2η
+m |Si|
+n
+�
+j=1
+|fj(k) − yj| ≤ 2η√n|Si|
+m
+∥f(k) − y∥2
+⇒
+��ϵ′��
+2 =
+�
+�
+�
+�
+n
+�
+i=1
+4η2n|Si|2
+m2
+∥f(k) − y∥2
+2 ≤ ηnO((Rw + Rb) exp(−B2/2)) ∥f(k) − y∥2 .
+(C.4)
+Combining Equation (C.1), Equation (C.2), Equation (C.3) and Equation (C.4), we have
+fi(k + 1) − fi(k) = −η
+n
+�
+j=1
+(fj(k) − yj)Hij(k) + ϵi(k) + ϵ′
+i(k)
+⇒ f(k + 1) − f(k) = −ηH(k)(f(k) − y) + ϵ(k) + ϵ′(k)
+= −ηH∞(f(k) − y) + η(H∞ − H(k))(f(k) − y) + ϵ(k) + ϵ′(k)
+�
+��
+�
+ζ(k)
+⇒ f(k) − y = (I − ηH∞)k(f(0) − y) +
+k−1
+�
+t=0
+(I − ηH∞)tζ(k − 1 − t)
+= −(I − ηH∞)ky + (I − ηH∞)kf(0) +
+k−1
+�
+t=0
+(I − ηH∞)tζ(k − 1 − t)
+�
+��
+�
+e(k)
+.
+Now the rest of the proof bounds the magnitude of e(k). From Lemma A.2 and Lemma A.6, we
+41
+
+have
+∥H∞ − H(k)∥2 ≤ ∥H(0) − H∞∥2 + ∥H(0) − H(k)∥2
+= O
+�
+n exp(−B2/4)
+�
+log(n2/δ)
+m
+�
++ O(n(Rw + Rb) exp(−B2/2)).
+Thus, we can bound ζ(k) as
+∥ζ(k)∥2 ≤ η ∥H∞ − H(k)∥2 ∥f(k) − y∥2 + ∥ϵ(k)∥2 +
+��ϵ′(k)
+��
+2
+= O
+�
+ηn
+�
+exp(−B2/4)
+�
+log(n2/δ)
+m
++ (Rw + Rb) exp(−B2/2)
+��
+∥f(k) − y∥2 .
+Notice that ∥H∞∥2 ≤ Tr(H∞) ≤ n since H∞ is symmetric.
+By Theorem A.18, we pick η =
+O(λ/n2) ≪ 1/ ∥H∞∥2 and, with probability at least 1 − δ − e−Ω(n) over the random initialization,
+we have ∥f(k) − y∥2 ≤ (1 − ηλ/4)k/2 ∥f(0) − y∥2.
+Since we are using symmetric initialization, we have (I − ηH∞)kf(0) = 0.
+Thus,
+∥e(k)∥2 =
+�����
+k−1
+�
+t=0
+(I − ηH∞)tζ(k − 1 − t)
+�����
+2
+≤
+k−1
+�
+t=0
+∥I − ηH∞∥t
+2 ∥ζ(k − 1 − t)∥2
+≤
+k−1
+�
+t=0
+(1 − ηλ)tηnO
+�
+exp(−B2/4)
+�
+log(n2/δ)
+m
++ (Rw + Rb) exp(−B2/2)
+�
+∥f(k − 1 − t) − y∥2
+≤
+k−1
+�
+t=0
+(1 − ηλ)tηnO
+�
+exp(−B2/4)
+�
+log(n2/δ)
+m
++ (Rw + Rb) exp(−B2/2)
+�
+· (1 − ηλ/4)(k−1−t)/2 ∥f(0) − y∥2
+≤ k(1 − ηλ/4)(k−1)/2ηnO
+�
+exp(−B2/4)
+�
+log(n2/δ)
+m
++ (Rw + Rb) exp(−B2/2)
+�
+∥f(0) − y∥2
+≤ k(1 − ηλ/4)(k−1)/2ηn3/2O
+� �
+exp(−B2/4)
+�
+log(n2/δ)
+m
++ (Rw + Rb) exp(−B2/2)
+�
+·
+��
+1 + (exp(−B2/2) + 1/m) log3(2mn/δ)
+� �
+= k(1 − ηλ/8)k−1ηn3/2O
+�
+exp(−B2/4)
+�
+log(n2/δ)
+m
++ (Rw + Rb) exp(−B2/2)
+�
+.
+Lemma C.8. Assume the parameter settings in Theorem A.18. Then with probability at least
+42
+
+1 − δ − e−Ω(n) over the random initialization, we have for all k ≥ 0,
+∥[W, b](k) − [W, b](0)∥F ≤
+�
+y⊤(H∞)−1y + O
+�
+n
+λ
+�exp(−B2/2) log(n/δ)
+m
+�1/4�
++ O
+�
+n
+�
+R exp(−B2/2)
+λ
+�
++ n
+λ2 · O
+�
+exp(−B2/4)
+�
+log(n2/δ)
+m
++ R exp(−B2/2)
+�
+where R = Rw + Rb.
+Proof. Before we start, we assume all the events needed in Theorem A.18 succeed, which happens
+with probability at least 1 − δ − e−Ω(n).
+⃗
+[W, b](K) −
+⃗
+[W, b](0)
+=
+K−1
+�
+k=0
+⃗
+[W, b](k + 1) −
+⃗
+[W, b](k)
+= −
+K−1
+�
+k=0
+Z(k)(u(k) − y)
+=
+K−1
+�
+k=0
+ηZ(k)((I − ηH∞)ky − e(k))
+=
+K−1
+�
+k=0
+ηZ(k)(I − ηH∞)ky −
+K−1
+�
+k=0
+ηZ(k)e(k)
+=
+K−1
+�
+k=0
+ηZ(0)(I − ηH∞)ky
+�
+��
+�
+T1
++
+K−1
+�
+k=0
+η(Z(k) − Z(0))(I − ηH∞)ky
+�
+��
+�
+T2
+−
+K−1
+�
+k=0
+ηZ(k)e(k)
+�
+��
+�
+T3
+.
+(C.5)
+Now, by Lemma A.6, we have ∥Z(k) − Z(0)∥F ≤ O(
+�
+nR exp(−B2/2)) which implies
+∥T2∥2 =
+�����
+K−1
+�
+k=0
+η(Z(k) − Z(0))(I − ηH∞)ky
+�����
+2
+≤
+K−1
+�
+k=0
+η · O(
+�
+nR exp(−B2/2)) ∥I − ηH∞∥k
+2 ∥y∥2
+≤ η · O(
+�
+nR exp(−B2/2))
+K−1
+�
+k=0
+(1 − ηλ)k√n
+= O
+�
+n
+�
+R exp(−B2/2)
+λ
+�
+.
+(C.6)
+43
+
+By ∥Z(k)∥2 ≤ ∥Z(k)∥F ≤
+√
+2n, we get
+∥T3∥2 =
+�����
+K−1
+�
+k=0
+ηZ(k)e(k)
+�����
+2
+≤
+K−1
+�
+k=0
+η
+√
+2n
+�
+k(1 − ηλ/8)k−1ηn3/2O
+�
+exp(−B2/4)
+�
+log(n2/δ)
+m
++ R exp(−B2/2)
+� �
+= n
+λ2 · O
+�
+exp(−B2/4)
+�
+log(n2/δ)
+m
++ R exp(−B2/2)
+�
+.
+(C.7)
+Define T = η �K−1
+k=0 (I−ηH∞)k. By Lemma A.2, we know ∥H(0) − H∞∥2 ≤ O(n exp(−B2/4)
+�
+log(n/δ)
+m
+)
+and this implies
+∥T1∥2
+2 =
+�����
+K−1
+�
+k=0
+ηZ(0)(I − ηH∞)ky
+�����
+2
+2
+= ∥Z(0)Ty∥2
+2
+= y⊤TZ(0)⊤Z(0)Ty
+= y⊤TH(0)Ty
+≤ y⊤TH∞Ty + ∥H(0) − H∞∥2 ∥T∥2
+2 ∥y∥2
+2
+≤ y⊤TH∞Ty + O
+�
+n exp(−B2/4)
+�
+log(n/δ)
+m
+� �
+η
+K−1
+�
+k=0
+(1 − ηλ)k
+�2
+n
+= y⊤TH∞Ty + O
+�
+n2 exp(−B2/4)
+λ2
+�
+log(n/δ)
+m
+�
+.
+Let H∞ = UΣU ⊤ be the eigendecomposition. Then
+T = U
+�
+η
+K−1
+�
+k=0
+(I − ηΣ)k
+�
+U ⊤ = U((I − (I − ηΣ)K)Σ−1)U ⊤
+⇒ TH∞T = U((I − (I − ηΣ)K)Σ−1)2ΣU ⊤ = U(I − (I − ηΣ)K)2Σ−1U ⊤ ⪯ UΣ−1U ⊤ = (H∞)−1.
+Thus,
+∥T1∥2
+2 =
+�����
+K−1
+�
+k=0
+ηZ(0)(I − ηH∞)ky
+�����
+2
+≤
+�
+�
+�
+�y⊤(H∞)−1y + O
+�
+n2 exp(−B2/4)
+λ2
+�
+log(n/δ)
+m
+�
+≤
+�
+y⊤(H∞)−1y + O
+�
+n
+λ
+�exp(−B2/2) log(n/δ)
+m
+�1/4�
+.
+(C.8)
+Finally, plugging in the bounds in Equation (C.5), Equation (C.8), Equation (C.6), and Equa-
+44
+
+tion (C.7), we have
+∥[W, b](K) − [W, b](0)∥F
+=
+���
+⃗
+[W, b](K) −
+⃗
+[W, b](0)
+���
+2
+≤
+�
+y⊤(H∞)−1y + O
+�
+n
+λ
+�exp(−B2/2) log(n/δ)
+m
+�1/4�
++ O
+�
+n
+�
+R exp(−B2/2)
+λ
+�
++ n
+λ2 · O
+�
+exp(−B2/4)
+�
+log(n2/δ)
+m
++ R exp(−B2/2)
+�
+.
+D
+Probability
+Lemma D.1 (Bernstein’s Inequality). Assume Z1, . . . , Zn are n i.i.d.
+random variables with
+E[Zi] = 0 and |Zi| ≤ M for all i ∈ [n] almost surely. Let Z = �n
+i=1 Zi. Then, for all t > 0,
+P[Z > t] ≤ exp
+�
+−
+t2/2
+�n
+j=1 E[Z2
+j ] + Mt/3
+�
+≤ exp
+�
+− min
+�
+t2
+2 �n
+j=1 E[Z2
+j ],
+t
+2M
+��
+which implies with probability at least 1 − δ,
+Z ≤
+�
+�
+�
+�2
+n
+�
+j=1
+E[Z2
+j ] log 1
+δ + 2M log 1
+δ .
+Lemma D.2 (Matrix Chernoff Bound, (Tropp et al., 2015)). Let X1, . . . , Xm ∈ Rn×n be m in-
+dependent random Hermitian matrices. Assume that 0 ⪯ Xi ⪯ L · I for some L > 0 and for all
+i ∈ [m]. Let X := �m
+i=1 Xi. Then, for ϵ ∈ (0, 1], we have
+P [λmin(X) ≤ ϵλmin(E[X])] ≤ n · exp(−(1 − ϵ)2λmin(E[X])/(2L)).
+Lemma D.3 ((Li and Shao, 2001, Theorem 3.1) with Improved Upper Bound for Gaussian)). Let
+b > 0 and r > 0. Then,
+exp(−b2/2)
+P
+w∼N(0,1)[|w| ≤ r] ≤
+P
+w∼N(0,1)[|x − b| ≤ r] ≤ 2r ·
+1
+√
+2π exp(−(max{b − r, 0})2/2).
+Proof. To prove the upper bound, we have
+P
+w∼N(0,1)[|x − b| ≤ r] =
+� b+r
+b−r
+1
+√
+2π exp(−x2/2) dx ≤ 2r ·
+1
+√
+2π exp(−(max{b − r, 0})2/2).
+45
+
+Lemma D.4 (Anti-concentration of Gaussian). Let Z ∼ N(0, σ2). Then for t > 0,
+P[|Z| ≤ t] ≤
+2t
+√
+2πσ.
+E
+The Benefit of Constant Initialization of Biases
+In short, the benefit of constant initialization of biases lies in inducing sparsity in activation and thus
+reducing the per step training cost. This is the main motivation of our work on studying sparsity
+from a deep learning theory perspective. Since our convergence shows that sparsity doesn’t change
+convergence rate, the total training cost is also reduced.
+To address the width’s dependence on B, our argument goes like follows. In practice, people
+set up neural network models by first picking a neural network of some pre-chosen size and then
+choose other hyper-parameters such as learning rate, initialization scale, etc. In our case, the hyper-
+parameter is the bias initialization. Thus, the network width is picked before B. Let’s say we want
+to apply our theoretical result to guide our practice. Since we usually don’t know the exact data
+separation and the minimum eigenvalue of the NTK, we don’t have a good estimate on the exact
+width needed for the network to converge and generalize. We may pick a network with width that
+is much larger than needed (e.g. we pick a network of width Ω(n12) whereas only Ω(n4) is needed;
+this is possible because the smallest eigenvalue of NTK can range from [Ω(1/n2), O(1)]). Also, it
+is an empirical observation that the neural networks used in practice are very overparameterized
+and there is always room for sparsification.
+If the network width is very large, then per step
+gradient descent is very costly since the cost scales linearly with width and can be improved to
+scale linearly with the number of active neurons if done smartly. If the bias is initialized to zero
+(as people usually do in practice), then the number of active neurons is O(m). However, since
+we can sparsify the neural network activation by non-zero bias initialization, the number of active
+neurons can scale sub-linearly in m. Thus, if the neural network width we choose at the beginning
+is much larger than needed, then we are indeed able to obtain total training cost reduction by this
+initialization. The above is an informal description of the result proven in (Song et al., 2021a)
+and the message is sparsity can help reduce the per step training cost. If the network width is
+pre-chosen, then the lower bound on network width m ≥ �Ω(λ−4
+0 n4 exp(B2)) in Theorem 3.1 can be
+translated into an upper bound on bias initialization: B ≤ �O(
+�
+log λ4
+0m
+n4 ) if m ≥ �Ω(λ−4
+0 n4). This
+would be a more appropriate interpretation of our result. Note that this is different from how
+Theorem 3.1 is presented: first pick B and then choose m; since m is picked later, m can always
+satisfy B ≤ √0.5 log m and m ≥ �Ω(λ−4
+0 n4 exp(B2)). Of course, we don’t know the best (largest)
+possible B that works but as long as we can get some B to work, we can get computational gain
+from sparsity.
+In summary, sparsity can reduce the per step training cost since we don’t know the exact width
+needed for the network to converge and generalize. Our result should be interpreted as an upper
+bound on B since the width is always chosen before B in practice.
+46
+