diff --git a/-9FST4oBgHgl3EQfczg_/content/tmp_files/2301.13804v1.pdf.txt b/-9FST4oBgHgl3EQfczg_/content/tmp_files/2301.13804v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ebe624f375f655f3dff4408cde3b743377742c71 --- /dev/null +++ b/-9FST4oBgHgl3EQfczg_/content/tmp_files/2301.13804v1.pdf.txt @@ -0,0 +1,1120 @@ +arXiv:2301.13804v1 [cs.GT] 31 Jan 2023 +Fairness in the Assignment Problem with Uncertain Priorities∗ +Zeyu Shen† +Zhiyi Wang∗ +Xingyu Zhu∗ +Brandon Fain∗ +Kamesh Munagala∗ +Abstract +In the assignment problem, a set of items must be allocated to unit-demand agents who ex- +press ordinal preferences (rankings) over the items. In the assignment problem with priorities, +agents with higher priority are entitled to their preferred goods with respect to lower priority +agents. A priority can be naturally represented as a ranking and an uncertain priority as a +distribution over rankings. For example, this models the problem of assigning student appli- +cants to university seats or job applicants to job openings when the admitting body is uncertain +about the true priority over applicants. This uncertainty can express the possibility of bias in +the generation of the priority ranking. We believe we are the first to explicitly formulate and +study the assignment problem with uncertain priorities. We introduce two natural notions of +fairness in this problem: stochastic envy-freeness (SEF) and likelihood envy-freeness (LEF). We +show that SEF and LEF are incompatible and that LEF is incompatible with ordinal efficiency. +We describe two algorithms, Cycle Elimination (CE) and Unit-Time Eating (UTE) that satisfy +ordinal efficiency (a form of ex-ante Pareto optimality) and SEF; the well known random serial +dictatorship algorithm satisfies LEF and the weaker efficiency guarantee of ex-post Pareto op- +timality. We also show that CE satisfies a relaxation of LEF that we term 1-LEF which applies +only to certain comparisons of priority, while UTE satisfies a version of proportional allocations +with ranks. We conclude by demonstrating how a mediator can model a problem of school +admission in the face of bias as an assignment problem with uncertain priority. +1 +Introduction +Consider a motivating example of the assignment problem where a number of university admission +slots (the items) must be assigned to student applicants (the agents). The university slots could be +at a single university or several. Applicants might have preferences over different universities, or +might have preferences over different slots at the same university (for example, some slots might be +associated with merit-based financial aid, or include admission to particular academic programs). +Applicants are unit-demand, meaning they only need to be assigned a single slot (and derive no +benefit from being assigned multiple). +Most university systems employ some form of priority-based admissions; this can be expressed +through a ranking over applicants. For example, a priority might rank applicants by standardized +exam scores, or perhaps by some more complex holistic assessment. Given any deterministic priority +(a ranking), one might naturally solve the assignment problem using the serial dictatorship rule, +so that students choose their most preferred remaining university slot one at a time in order of +∗This work is supported by NSF grant CCF-2113798. +†Computer +Science +Department, +Duke +University, +Durham, +NC +27708-0129. +Email: +{zeyu.shen,zhiyi.wang,xingyu.zhu}@duke.edu, {btfain,kamesh}@cs.duke.edu. +1 + +their standardized exam score. Indeed, systems roughly like this are employed in several countries +around the world such as the Indian Institutes of Technology [13]. +Despite the appeal of such a simple and ostensibly fair system, there is reason to suspect that +any scoring or ranking system is based on imperfect noisy signals of the true underlying priority +(whatever that might be). For example, an applicant A scoring 1 point higher on a standardized +exam or holistic assessment than another applicant B is not, in general, 100% more likely to be a +better student than B. Even more worryingly, studies show that standardized exam performance is +closely related to demographic factors such as race and income [8], leading to uncertainty based on +social bias and inequality in addition to random noise like whether one had a good breakfast the day +of an exam. More holistic assessments are further vulnerable to the well documented phenomenon +of implicit bias against historically marginalized groups [5]. Ignoring these uncertainties may result +in arbitrary decisions (deterministically preferring one applicant over another when the comparison +is unclear and noisy) and systemic discrimination against historically marginalized groups. +Previous work has attempted to solve the second problem of bias without explicitly modeling +an uncertain priority by adapting the so-called “Rooney Rule” [15, 7]. There are variations, but +roughly speaking these methods reserve a number of “minority” spots and prioritize this many +“minority” applicants in some serial dictatorship assignment. This approach can lead to fairness +gerrymandering [14] by which structured subgroups remain disadvantaged. In particular, Rooney +Rule style approaches are predicated on a single binary distinction of the applicant population into +“majority” (or privileged) and “minority” (or disadvantaged) applicants. But in reality, applicant +identity is multidimensional (race, gender, income, disability, first language, etc.) and bias can +compound along intersections. In fact, it is perfectly plausible that the vast majority of applicants +are disadvantaged (that is, suffer from bias leading to underestimation of their priority) along +one or more dimensions of identity, though not all to the same extent. +In addition to group +identity, there may sometimes be uncertainties related to the priority of individual applicants, +unique circumstances that merit accounting. +For these reasons, we consider the more general problem that takes as input an uncertain +priority, expressed as a probability distribution over rankings of applicants. The generality of the +input to our algorithms ensures that a decision maker can fully model the complexity of uncertainty +and bias inherent in the creation of a priority. This modeling problem is outside the scope of this +paper, though we do provide an example for our experiments in Section 6. Rather, our emphasis +is on the question of characterizing fairness and efficiency given a random priority, and providing +algorithms to compute random assignments that satisfy these desiderata. +1.1 +Contributions +We study an extension of the random assignment problem [6, 18, 2] in which a decision maker must +allocate a number of items to unit-demand agents in a way that is consistent with an uncertain +priority represented as a distribution over rankings of the agents. To the best of our knowledge, +we are the first to characterize this more general problem. +In general we want to compute a random assignment that is simultaneously efficient with respect +to agent preferences over the items and fair with respect to the agent priorities. Ordinal efficiency +(OE) [6] generalizes the concept of Pareto efficiency to the case of a random assignment. Our main +contribution is to characterize two alternative notions of fairness for the random assignment problem +with uncertain priorities in Section 3. +The first notion, which we call stochastic envy-freeness +(SEF), guarantees that any agent whose priority first-order stochastically dominates another agent’s +2 + +priority should prefer their own (random) assignment to that of the other agent. The second notion, +which we call likelihood envy-freeness (LEF), guarantees that the likelihood (over the random +assignment) that an agent prefers the assignment of another should be at most the likelihood (over +the uncertain priority) that the latter agent has higher priority than the former. +We introduce additional notions that helps more finely distinguish between algorithms that +satisfy one of the above notions. The first is a relaxation of LEF called 1-LEF that holds only when +an agent has higher priority than another with probability 1. The next is ranked proportionality +(PROP), where the allocation of any agent should stochastically dominate the allocation where +she gets her i-th preferred item with probability pi if she herself is ranked at position i with that +probability. +Formal definitions are provided in Section 3. We provide illustrative examples of these concepts +as well as justification for why multiple definitions of fairness might be appropriate in Section 3.3. +In Section 4 we show that it is impossible to guarantee OE and LEF simultaneously. +We +also show that it is impossible to guarantee SEF and LEF simultaneously. Given this, we focus +on achieving OE and SEF. In Section 5 we describe two algorithms: Unit-time Eating (UTE) and +Cycle Elimination (CE). We show that both of these algorithms satisfy OE and SEF. To more finely +distinguish between these algorithms, we show that CE also satisfies the relaxed 1-LEF property, +while UTE satisfies PROP. We also show that any algorithm achieving OE cannot achieve PROP +and 1-LEF simultaneously, so that we cannot achieve a super-set of the properties achieved by +these algorithms. +It is straightforward to observe that the well known Random Serial Dictatorship (RSD) that +samples a priority from Σ and then uses the serial dictatorship satisfies LEF, PROP, and is ex-post +Pareto efficient, though it does not satisfy OE [6]. We obtain a nearly complete characterization of +achievable subsets of our efficiency and fairness properties, as shown in Table 1. +Algorithm +OE +SEF +LEF +1-LEF +PROP +RSD +✓ +✓ +✓ +UTE (new) +✓ +✓ +✓ +CE (new) +✓ +✓ +✓ +Table 1: Summary of fairness properties achieved. +In Section 6 we return to a consideration of our motivating application of biased school ad- +missions. We provide a practical example modeling an uncertain priority in the presence of bias +and compare our CE and UTE algorithms with previous approaches to address bias using “Rooney +Rule” style approaches [15, 7]. +1.2 +Related Work +Random Assignment. +There is a large body of work studying the problem of random assign- +ment with no priority (or, in our framework, when the priority is uniform). The work of [1] proposed +a random serial dictatorship mechanism, which draws an ordering of agents uniformly at random +and let them choose items in that order, and showed that this mechanism is ex-post efficient. The +work of [23] observed that though random serial dictatorship is fair, it is not efficient when the +agents are endowed with Von Neumann-Morgenstern preferences over lotteries. The work of [6] +3 + +introduced a notion of efficiency that is stronger than ex-post efficiency, namely ordinal efficiency, +and showed that random serial dictatorship is not ordinally efficient. They proposed the probabilis- +tic serial rule that is ordinally efficient. Moreover, probabilistic serial is (stochastically) envy-free +while random serial dictatorship is not. The work of [2] studied the relationship between ex-post +efficiency and ordinal efficiency, showing that a lottery induces an ordinally efficient random as- +signment if and only if each subset of the full support of the lottery is undominated (in a specific +sense). +Subsequent works investigated natural extensions of the canonical setup. The work of [18] con- +sidered the problem of random assignment in the case where agents can opt out, and characterised +probabilistic serial by ordinal efficiency, envy-freeness, strategyproofness, and equal treatment of +equals in this setting. The work of [10] studied the notion of rank efficiency, which maximises the +number of agents matched to their first choices. +Fair Ranking. +The assignment problem with priority is closely related to the subset selection +problem that has been studied extensively as a problem in fair ranking [15, 16, 19, 7, 9, 17, 12] +where the goal is to optimize some latent measure of utility for the algorithm designer subject to +group fairness constraints on the resulting ranking. Recent work considers explicitly modeling the +uncertainty from bias when estimating a ranking based on observed utilities [22], similar to our +approach in modeling an uncertain priority. Our work differs from the fair ranking literature in that +we study a more general assignment problem in which agents may not all have the same preferences +over items. Of course, one can always translate a given ranking into an assignment by employing +the serial dictatorship rule, but this need not be ordinally efficient [6]. Instead, we formulate our +desiderata more explicitly in the wider context of the assignment problem itself. +Two-sided matching. +School choice problems are often studied in the context of two-sided +matching, where applicants have preferences over schools and schools have preferences over ap- +plicants. +For example, the deferred acceptance algorithm (and its extensions) calculates stable +matchings and has been extensively studied and deployed in the real world [11, 20, 21, 4, 3]. Our +problem is different in two ways. First, the “items” in our problem (eg., school seats) share a single +common priority over applicants, so the notion of stability simply means no applicant of lower +priority is assigned an item preferred by an agent of higher priority. However, our setting is more +complex in the second sense: The shared priority is uncertain, and the assignment will be random, +requiring an extension of existing fairness properties and algorithms. +2 +Preliminaries +We are given n unit demand agents A = {1, 2, . . . , n} and a set of m items I. We assume without +loss of generality that m ≥ n (if not, one can create additional “dummy” items that are least +preferred by all agents). We write a ≻i b to denote that agent i prefers item a to item b. Each +agent has ordinal preferences represented as a total order over I, that is, for every agent i we have +a permutation πi : I → {1, . . . , n} such that πi(a) < πi(b) if and only if a ≻i b.1 +1In general, results extend trivially to the case where agents may have objective indifferences between items, +meaning that if any agent is indifferent between two items then all agents are indifferent between those items. +However, our results do not necessarily extend straightforwardly if agents have subjective indifferences, see [6]. +4 + +A simple priority over agents is a permutation σ : A → {1, . . . , n} where σ(i) < σ(j) means that +i has higher priority than j. A random priority is a probability distribution over simple priorities +which we denote as Σ = {(σk, ρk)} where each σk is a simple priority, ρk ≥ 0, and � +k ρk = 1. +A simple assignment is a matching f : A → I. A lottery is a probability distribution over simple +assignments which we denote as L = {(fk, pk)} where each fk is a simple assignment, pk ≥ 0, and +� +k pk = 1. +Following [6], we call a probability distribution over [m] itself a random allocation to an agent. +It is important to note that agents have ordinal preferences over deterministic items which only +induces a partial order over random allocations. That is, given πi, it may be unclear whether i +would prefer one random allocation to another. We denote by P = {pij} a random assignment, +the n by m matrix where Pi, the i-th row, is agent i’s random allocation, and where � +i pij = 1 +for all columns j. In general, a random assignment P can be induced by one or more lotteries, the +existence of which is guaranteed by the Birkhoff-von Neumann Theorem, but a particular lottery +induces a unique random assignment P. +In the assignment problem with uncertain priorities we are given a random priority Σ and agent +preferences {πi} and we must compute a random assignment. +3 +Desiderata +In this section we introduce the normative properties that an algorithm for the random assignment +with uncertain priorities problem should satisfy. Broadly speaking, these desiderata require that +the algorithm be efficient with respect to agent preferences and fair with respect to agent priorities. +3.1 +Efficiency +A simple assignment f is Pareto efficient (or Pareto optimal) if it is not dominated by any other +simple assignment, which simply means that there is no alternative such that no agent is worse off +and at least one agent is better off. +Definition 1 (Pareto Efficiency). A simple assignment f is Pareto efficient if for all simple as- +signments g one of the following holds: (i) ∃i ∈ A such that f(i) ≻i g(i), or (ii) g(i) ⊁ f(i) for all +i ∈ [n]. +A lottery L is ex-post Pareto efficient if every simple assignment in the support of L (i.e., every +simple assignment fk with pk > 0) is Pareto efficient. +A stronger efficiency property for a random assignment is ordinal efficiency (OE) [6]. To define +ordinal efficiency we must first define the notion of stochastic dominance. +Definition 2 (Stochastic Domination). A probability distribution X stochastically dominates an- +other distribution Y under permutation π (denoted X ≻sd +π Y ) if for all t ∈ {1, . . . , n} it holds that +�t +r=1 Xπ−1(r) ≥ �t +r=1 Yπ−1(r), where π−1 is the inverse permutation. A random assignment P is +stochastically dominated by a random assignment Q ̸= P if the random allocation induced by Q +stochastically dominates the random allocation induced by P under preferences πi for every agent +i ∈ [n]. +Note that this implies the following: If random assignment Q stochastically dominates random +assignment P, then every agent prefers Q to P under any Von Neumann-Morgenstern utility func- +tion consistent with their ordinal preferences. Now we can define ordinal efficiency, following [6]. +5 + +Definition 3 (Ordinal Efficiency, OE). We say that a random assignment P is ordinally efficient +if it is not stochastically dominated by any other random assignment. +At a high level, a random assignment is ordinally efficient if there is no other random assignment +that is better for all agents and all utility functions consistent with their ordinal preferences. The +property is not trivial: Some natural algorithms such as random serial dictatorship are Pareto +efficient but not ordinally efficient. +3.2 +Fairness +We define fairness in terms of envy. We say that one agent envies another if the former prefers +the item assigned to the latter. Envy of a lower priority agent constitutes a justified complaint +against an assignment; ideally we would like to compute an envy-free assignment with respect to +the priority. +Definition 4 (Envy-Freeness). We say that a simple assignment f is envy-free with respect to a +simple priority σ if for all i, j ∈ [n], σ(i) < σ(j) =⇒ f(i) ≻i f(j). +However, it is immediately evident that it is impossible to compute a single simple assignment +that is envy-free in this sense for every simple priority in the support of a random priority (for +example, if there are two agents with uncertain priority who both prefer the same item). Instead, +we need to compute a random assignment so that each agent is fairly treated ex-ante (for example, +so that each agent has a fair probability of receiving the preferred good). +There are two natural ways to generalize the concept of envy to a random assignment with a +random priority. One is to imagine that one agent envies another if the random allocation of the +latter stochastically dominates that of the former under the former’s ordinal preferences. Envy of +this type forms a justified complaint if the envying agent also stochastically dominates the envied +agent in terms of the random priority. More formally, +Definition 5 (Stochastic Envy-Freeness, SEF). Consider a random assignment P generated under +a random priority Σ. Let Si be the probability distribution over [n] induced by Σ for agent i, that +is, for r ∈ [n], Sir = � +k:σk(i)=r ρk. +Let σ∗ be the identity permutation, i.e., σ∗(i) = i. +P is +stochastically envy-free (SEF) with respect to Σ if for all i, j ∈ [n], Si ≻sd +σ∗ Sj =⇒ Pi ≻sd +πi Pj. +Loosely speaking, the implication of stochastic envy-freeness can be read as “if agent i probably +has higher priority than j then i should prefer their random allocation to j’s under all utility +functions consistent with i’s ordinal preferences.” +A second way to generalize envy is by considering the likelihood of envy (in the simple sense) +with respect to a lottery inducing a given random assignment. Envy of this type is justified if the +likelihood of agent i envying another agent j is greater than the likelihood over the random priority +that i has lower priority than agent j. We call a random assignment likelihood envy-free if there is +a lottery which induces it and has no envy of this kind. +Definition 6 (Likelihood Envy-Freeness, LEF). A random assignment P satisfies likelihood envy- +freeness (LEF) under Σ if P can be induced by a lottery L such that for all i, j ∈ [n], Prσ∼Σ[σ(i) < +σ(j)] ≤ Prf∼L[f(i) ≻i f(j)]. +In other words, LEF means that an agent i who is ℓ-likely to have higher priority than another +agent j should be at least ℓ-likely to prefer their assigned item to j’s. +6 + +We say an algorithm satisfies OE (resp. SEF, LEF) if it always produces random assignment +that satisfies OE (resp. SEF, LEF). As we show in Section 4, it is not possible to guarantee SEF +and LEF simultaneously. +3.3 +Relationship between LEF and SEF +The relationship between SEF and LEF is subtle; neither implies the other and it is not immediately +evident which is the “better” or more “natural” fairness property. We present two examples to +illustrate that an assignment satisfying only one of SEF and LEF might still be unfair, so that both +properties are useful competing notions of fairness, and neither is strictly stronger than the other. +We first present an example which shows that an assignment that satisfies SEF can be unfair. +Consider n = 2 agents and m = 2 items which we label a, b for clarity. Both agents prefer a to b, +and the random priority is simply Σ = {(σ, 1)} with σ(1) < σ(2), i.e. agent 1 has higher priority +than agent 2 with probability 1. In this setup, allocating 1 +2 unit of a and b to both agent yields +an assignment that satisfies SEF. However, this assignment is clearly unfair, because even though +agent 1 has higher priority than agent 2, they are getting the same assignment. Notice that this +assignment does not satisfy LEF. In this instance, LEF could be used to characterize how much +one agent is prioritized over the other. +The next example shows that an assignment that only satisfies LEF can also be unfair. Consider +n = 2 agents and m = 100 items which we label i1, . . . , i100 for clarity. The preferences of both +agents are i1 ≻ · · · ≻ i100. The random priority is given by Σ = {(σ1, 1 +2), (σ2, 1 +2)} with σ1(1) < σ1(2) +and σ2(2) < σ2(1). In other words, both agents have the same priority. In this setup, allocating +1 +2 unit of i1 and 1 +2 unit of i100 to agent 1 and 1 +2 unit of i99 and 1 +2 unit of i100 to agent 2 yields +an assignment that satisfies LEF. Notice that this assignment can be induced by a lottery L = +{(f1, 1 +2), (f2, 1 +2)} where f1(1) = i1, f1(2) = i100, f2(1) = i100, f2(2) = i99. However, this assignment +is clearly unfair, because even though the two agents have the same priority, agent 1 gets a strictly +better assignment than agent 2. +This shows that LEF alone has limitations as well, and the +appropriate concept here is SEF. +The above examples show that SEF and LEF provide reasonable competing notions of fairness. +When combined with the efficiency notion of OE, we will show in Section 4 that LEF and OE +are incompatible. If OE is replaced by the weaker notion of Pareto-efficiency, then it is easy to +check that random serial dictatorship (RSD), which simply samples a priority of agents from the +distribution and allocates each agent their favorite remaining item in this priority order, satisfies +LEF2 and pareto efficiency. Thus, in our work, we will focus on the more non-trivial part of finding +algorithms that satisfy SEF and OE. +3.4 +Additional Fairness Criteria +As we show in Section 5, there can be multiple algorithms that satisfy the same subset of the +fairness criteria. We therefore consider two additional notions to more finely distinguish between +them. +The first criterion is the following relaxation of LEF: If agent i with probability 1 has higher +priority than another agent j then agent i should certainly (again, with probability 1) not envy j. +2To see why RSD satisfies LEF, suppose the random priority is given by Σ = {(σk, ρk)}, then the random +assignment produced by RSD can be induced by the lottery L = {(fk, ρk)}, where fk is the deterministic assignment +produced by letting agents successively choose an item based on the order given by σk. +7 + +Definition 7 (1-LEF). A random assignment P under random priority Σ satisfies 1-LEF if there +exists some lottery L which induces P such that for all agents i ̸= j ∈ [n], if Prσ∼Σ[σ(i) < σ(j)] = 1, +then Prf∼L[f(i) ≻i f(j)] = 1. +The next criterion is called Ranked Proportionality (PROP), which captures stochastic dom- +inance over an allocation that matches the probability an agent gets her ith ranked item to the +probability of she being ranked at position i. Note that if all rankings of agents were equally likely, +this captures stochastic dominance to an allocation that assigns every item to every agent uniformly +at random. +Definition 8 (PROP). Given a random priority Σ = {(σk, ρk)}, we define the baseline allocation +P i for agent i by P iπ−1 +i +(r) = Sir = � +k:σk(i)=r ρk for all r ∈ [n]. In other words, if an agent i ranks +the r-th in the random priorities with probability p, then we add p fraction of the r-th preferred item +of agent i to her baseline allocation. For an allocation to satisfy ranked proportionality (PROP), +it should stochastically dominate this baseline for each agent. +4 +Impossibility Results +In this part, we present several impossibility results. We note that these are existential hardness +results, not computational. We begin by observing that LEF is incompatible with OE. +Theorem 1. LEF is incompatible with OE. +Proof. We present an instance in which no random assignment can satisfy both LEF and OE. There +are n = 4 agents and m = 4 items which we label a, b, c, d for clarity. Agent preferences are given +by +π1, π3 : a ≻ b ≻ c ≻ d, +π2, π4 : b ≻ a ≻ c ≻ d +Moreover, we consider the priority Σ = {(σ1, 1 +2), (σ2, 1 +2)} where +σ1(4) < σ1(2) < σ1(3) < σ1(1), +σ2(3) < σ2(1) < σ2(4) < σ2(2). +In other words, with probability 1 +2 under σ1, agent 4 has the highest priority, then agent 2, then +agent 3, finally agent 1. Similarly for σ2 with probability 1 +2. Assume for contradiction that there +exists a random assignment P = [pij], together with a lottery L which induces P, satisfying LEF +and OE. By definition of LEF, we note that +Pr +f∼L[f(3) ≻3 f(1)] ≥ Pr +σ∼Σ[σ(3) < σ(1)] = 1, +so it must be that Prf∼L[f(3) ≻3 f(1)] = 1. Thus, we must have p1a = 0, because otherwise there +would exist a simple assignment in the lottery in which agent 1 is assigned with a and agent 3 is +assigned with some less preferred item under π3. By the same reasoning, we note that p2b = 0. +Also by definition of LEF, observe that +Pr +f∼L[f(2) ≻2 f(3)] ≥ Pr +σ∼Σ[σ(2) < σ(3)] = 1 +2. +8 + +This implies p3a < 1, as otherwise we have f(3) = a for all f ∼ L; combined with the fact that +p2b = 0, we would have f(3) ≻2 f(2) for all f ∼ L, which contradicts Prf∼L[f(2) ≻2 f(3)] ≥ 1 +2. +Since p1a = 0, p3a < 1, and � +i pia = 1, it follows that p2a + p4a > 0. Similarly, we have +p1b +p3b > 0. Without loss of generality, we assume that p2a > 0 and p1b > 0 (if p4a > 0 or p3b > 0, +the proof proceeds similarly). Let pmin = min(p2a, p1b); define random assignment Q = [qij] by +qij = + + + + + +pij if i /∈ {1, 2} and j /∈ {a, b} +pij + pmin if (i, j) = (1, a) or (2, b) +pij − pmin if (i, j) = (1, b) or (2, a) +We can see that Q stochastically dominates P. In particular, all that is different in Q is that +agent 1 swaps agent 2 some of agent 2’s allocated probability mass on item a in exchange for an +equivalent amount of agent 1’s probability mass on item b. Since a ≻1 b and b ≻2 a and nothing else +changes, agents 1 and 2 prefer Q, and nothing has changed for agents 3 and 4. This contradicts with +the fact that P satisfies OE. Thus, we can conclude that no random assignment in this instance +satisfies LEF and OE. +Theorem 1 can be interpreted as a fundamental tradeoff between efficiency and fairness con- +ceived as LEF. Next, we show that LEF and SEF are two fundamentally different notions of fairness +that are incompatible with one another. As we will see later in Section 5, each of LEF and SEF +independently can be guaranteed. Thus, neither notion of fairness is subsumed by the other. +Theorem 2. LEF is incompatible with SEF. +Proof. We present an instance in which no random assignment can satisfy both LEF and SEF. +There are n = 5 agents and m = 5 items which we label a, b, c, d, e for clarity. Preferences are given +by +π1, π3 : a ≻ b ≻ c ≻ d ≻ e, +π2, π4 : b ≻ a ≻ c ≻ d ≻ e, +π5 : a ≻ c ≻ b ≻ d ≻ e. +We consider the priority Σ = {(σ1, 1 +2), (σ2, 1 +2)} defined by +σ1(3) < σ1(5) < σ1(1) < σ1(4) < σ1(2), +σ2(4) < σ2(5) < σ2(2) < σ2(3) < σ2(1). +In other words, with probability 1 +2 under σ1, agent 3 has the highest priority, then agents 5, 1, +4, and finally 2. Similarly for σ2. +Assume for contradiction that there exists a random assignment P = [pij], together with a +lottery L which induces P, that satisfies LEF and SEF. Since agent 3 always has higher priority +than agent 1 and agent 3 prefers a over all other items, LEF implies that p1a = 0. Similarly, since +agent 4 always has higher priority than agent 2 prefers b over all other itmes, LEF implies that +p2b = 0. +Recall that Si is the probability density over [n] induced by Σ for agent i and σ∗ is the identity +permutation. Since S1 ≻sd +σ∗ S2 by construction and P satisfies (SEF) by assumption, we have +P1 ≻sd +π1 P2. Combined with the fact that p1a = 0, we must have p2a = 0. Similarly, p1b = 0. +9 + +We next show p1c = p2c = 1 +2. First, observe that LEF guarantees +Pr +f∼L[f(4) ≻4 f(2)] ≥ Pr +σ∼Σ[σ(4) < σ(2)] = 1, +Thus, since e is the least preferred item by agent 4, we must have p4e = 0. Also by LEF, we have +Pr +f∼L[f(1) ≻1 f(4)] ≥ Pr +σ∼Σ[σ(1) < σ(4)] = 1 +2, +i.e. Prf∼L[f(1) ≻1 f(4)] ≥ 1 +2. On the other hand, since p4e = 0, the worst item that agent 4 can +get under π4 is d, so +Pr +f∼L[f(1) ≻1 f(4)] ≤ p1a + p1b + p1c = p1c, +since we earlier found that p1a = p1b = 0. +Recall Prf∼L[f(1) ≻1 f(4)] ≥ +1 +2, we get p1c ≥ +1 +2. +Similarly, we have p2c ≥ 1 +2. Since � +i pic = 1, it must be the case that p1c = p2c = 1 +2. We deduce +that for any f ∼ L, either f(1) = c or f(2) = c, because on one hand, for any fixed f, we should +have f(1) ̸= f(2), while on the other hand, p1c + p2c = 1. +Observe that p5a ≤ 1 +2. This follows directly from LEF, because +Pr +f∼L[f(3) ≻3 f(5)] ≥ Pr +σ∼Σ[σ(3) < σ(5)] = 1 +2; +if p5a > 1 +2, we would have Prf∼L[f(3) ≻3 f(5)] < 1 − p5a = 1 +2, leading to contradiction. What’s +more, we have p5c = 0, since we already have p1c + p2c = 1. +On one hand, we should have Prf∼L[f(5) ≻5 f(1) and f(5) ≻5 f(2)] = 1, since σi(5) < σi(1) and +σi(5) < σi(2) for i ∈ {1, 2}; but on the other hand, we have Prf∼L[f(5) ≻5 f(1) and f(5) ≻5 f(2)] ≤ +p5a ≤ 1 +2, because for any f ∼ L, either f(1) = c or f(2) = c, so f(5) ≻5 f(1) and f(5) ≻5 f(2) if +and only if f(5) = a. This leads to contradiction. Thus, LEF and SEF are incompatible. +We finally show that OE, 1-LEF, and PROP are simultaneously incompatible. This will inform +the design of algorithms in Section 5. +Lemma 1. There is an instance where no allocation simultaneously satisfies OE, 1-LEF, and +PROP. +Proof. We use the instance in the proof of Theorem 1. Since agent 3 is ranked higher than agent +1 with probability 1 and since their preferences over items is identical, 1-LEF implies agent 1 is +allocated item a with probability 0. Now PROP implies agent 1 receives item b with probability at +least 1/2. By a similar reasoning, agent 2 must get item a with probability at least 1/2. But any +such allocation cannot be OE, completing the proof. +5 +Algorithms +As we have seen, LEF is a very strong notion of fairness which is incompatible with both OE and +SEF. In the following, we present two algorithms – cycle elimination (CE) and unit time eating +(UTE) – that satisfy both OE and SEF. In addition, we show that CE satisfies 1-LEF and UTE +satisfies PROP. Given Lemma 1, we cannot design an algorithm that achieves OE and both these +properties. +10 + +Therefore, both CE and UTE are reasonable fair allocation algorithms in that they satisfy +efficiency (OE) and envy-freeness (SEF). The choice of which to implement depends on whether +we care more about a form of proportionality in the resulting allocation (UTE satisfies PROP) or +whether we care about additional envy-freeness in a deterministic sense (CE satisfies 1-LEF). +5.1 +Cycle Elimination algorithm +We first introduce a Cycle Elimination algorithm (CE), which works by constructing a directed +graph based on the random priority, and allocate items based on this graph. +To begin with, we introduce the Probabilistic Serial rule [6], a continuous algorithm which +works as follows. Initially, each agent i goes to their favorite item j and starts “eating” it (that is, +increasing pij) at unit speed. It is possible that several agents eat the same item at the same time. +Whenever an item is fully eaten, each of the agents eating it goes to their favorite remaining item +not fully allocated (that is, � +i pij < 1) and starts eating it in the same way. This process continues +until all items are consumed, or all the agents are full (that is, � +j pij = 1). We use PS(A, I) to +denote the assignment produced by running Probabilistic Serial rule on the set of agents A and +items I. +We construct a graph from Σ, which we call a Stochastic-Dominance graph (SD-graph), as +follows: Start with a graph with n vertices, where the i-th vertex corresponds to the i-th agent. +For any pair of distinct agents i and j, if Si ≻sd +σ∗ Sj, then we draw a directed edge from i to j. The +algorithm is now formally stated in Algorithm 1. +Algorithm 1: Cycle Elimination, Eliminate(A, I, G) +Input: Set of agents A, set of items I, SD-graph G; +Let �G be the condensation3of G; +Let � +A be the set of agents that belong to a strongly connected component whose in-degree +in �G is zero; +if A = � +A then +Output PS(A, I); +else +A′ ← A \ � +A; I′ ← I \ PS( � +A; I); G′ ← G \ � +A; +Output PS( � +A, I) + Eliminate(A′, I′, G′); +end +Analysis. +Our main result is the following theorem. +Theorem 3. The Cycle Elimination algorithm satisfies OE, SEF, and 1-LEF. It runs in O(n3 + +nm + n|Σ|) time. +Proof. Theorem 1 in [6] states that any simultaneous eating algorithm where each agent always +eats from her favorite remaining item satisfies OE. Hence, CE satisfies OE. +3Condensation of a graph is a directed acyclic graph formed by contracting each strongly connected component +to a single vertex. +11 + +To show SEF, fix two agents i and j, and assume Si ≻sd +σ∗ Sj. Let P be the random assignment +produced by CE. We show that Pi ≻sd +πi Pj. Since Si ≻sd +σ∗ Sj, there exists an edge from i to j +in the SD-graph. Thus, i and j either belong to the same strongly connected component, or the +strongly connected component of i has higher topological order than that of j’s. Either way, we +have Pi ≻sd +πi Pj, from which we can conclude that CE satisfies SEF. +To show 1-LEF, fix two agents i and j, and assume Prσ∼Σ[σ(i) < σ(j)] = 1. +Let P be +the random assignment produced by CE. We show that, for any lottery L inducing P, we have +Prf∼L[f(i) ≻i f(j)] = 1. We use proof by contradiction. Assume that there exists a lottery L0 +which induces P such that Prf∼L0[f(i) ≻i f(j)] < 1. This implies that there exists two items a +and b such that a ≻i b, pib > 0 and pja > 0. On the other hand, since Prσ∼Σ[σ(i) < σ(j)] = 1, +so we have Si ≻sd +σ∗ Sj; thus, the strongly connected component that agent i belongs to must have +higher topological order than that of j’s. By CE, agent j could start eating only when agent i is +completely full. Thus, pja > 0 implies that there is still item a remaining when agent i finishes +eating; this leads to contradiction, because a ≻i b implies that i could eat a instead of b. Therefore, +we must have Prf∼L[f(i) ≻i f(j)] = 1 for all lotteries L which induces P, from which we can +conclude that CE satisfies 1-LEF. +To show the running time, preprocessing Σ in order to compute the stochastic dominance +relation between agents takes O(n|Σ| + n3) time. Constructing the SD-graph by the stochastic +dominance relation between agents takes O(n2) time, as there are +�n +2 +� +pair of agents. Given the +SD-graph, running CE takes O(nm) time. This is because we only need to consider at most m +time points: the time at which each item is eaten up. We divide this process into m time intervals. +During each time interval, each agent keeps eating the same item, so it simply takes O(n) time to +keep track of the state of each agent, and the running time over m intervals is O(nm). Hence, the +total running time is O(n3 + nm + n|Σ|). +5.2 +Unit-time Eating Algorithm +We next introduce the Unit-time Eating Algorithm (UTE). Recall that Σ = {(σk, ρk)}. Essentially, +the algorithm works by dividing the time into n units, each of duration one; in time unit t, the t-th +ranked agent in σk eats their favorite item among those left over at rate ρk for all k. The procedure +is formally stated in Algorithm 2. +Algorithm 2: Unit-time Eating Algorithm +for t = 1, . . . , n do +The t-th ranked agent in each σk eats their favorite item among those left over at rate +ρk for all (σk, ρk) ∈ Σ; +end +Analysis. +We show the following theorem. +Theorem 4. The Unit-time Eating Algorithm satisfies OE, SEF, and PROP. Further, it runs in +O(n2|Σ| + nm) time. +Proof. By Theorem 1 in [6], we have UTE satisfies OE. +12 + +To show SEF, fix two agents i and j; assume that Si ≻sd +σ∗ Sj. Let P be the random assignment +produced by UTE, we show that Pi ≻sd +πi Pj. Let tk be the time when item π−1 +i +(k) has been eaten +up. Fix some k ∈ [m]; because Si ≻sd +σ∗ Sj, we have +⌊tk⌋ +� +t=1 +Sit ≥ +⌊tk⌋ +� +t=1 +Sjt +and +⌈tk⌉ +� +t=1 +Sit ≥ +⌈tk⌉ +� +t=1 +Sjt +Combining these gives +� +tk − ⌊tk⌋ +� +Si⌈tk⌉ + +⌊tk⌋ +� +t=1 +Sit ≥ +� +tk − ⌊tk⌋ +� +Sj⌈tk⌉ + +⌊tk⌋ +� +t=1 +Sjt. +(1) +Observe that +k +� +r=1 +Piπ−1 +i +(r) = +� +tk − ⌊tk⌋ +� +Si⌈tk⌉ + +⌊tk⌋ +� +t=1 +Sit, +k +� +r=1 +Pjπ−1 +i +(r) ≤ +� +tk − ⌊tk⌋ +� +Sj⌈tk⌉ + +⌊tk⌋ +� +t=1 +Sjt, +which gives �k +r=1 Piπ−1 +i +(r) ≥ �k +r=1 Pjπ−1 +i +(r). Because this holds for all k ∈ [m], we conclude that +Pi ≻sd +πi Pj. Hence, UTE satisfies SEF. +To show PROP, fix some agent i. Suppose the allocation produced by UTE for this agent is Pi, +and the baseline allocation for this agent is P i. We will show that Pi ≻sd +πi P i. Let tk be the time +when item π−1 +i +(k) has been eaten. Fix some k ∈ [m]. Clearly, we have tk ≥ k, because in order to +eat up π−1 +i +(k), we have to eat up π−1(r) for all r < k. We observe that +k +� +r=1 +Piπ−1 +i +(r) = +� +tk − ⌊tk⌋ +� +Si⌈tk⌉ + +⌊tk⌋ +� +t=1 +Sit. +Combined with tk ≥ k, we have +k +� +r=1 +Piπ−1 +i +(r) ≥ +k +� +t=1 +Sit = +k +� +r=1 +P iπ−1 +i +(r). +Because this holds for all k ∈ [m], we conclude that that Pi ≻sd +πi P i, and hence UTE satisfies PROP. +To show running time, preprocessing Σ to obtain the eating speed of each agent in each unit +time interval takes O(n2|Σ|) time. Then, running UTE takes O(nm) time, as we similarly only +need to consider at most m time points and keeping track of the state of each agent in each time +interval takes O(n) time. Therefore, the total running time is O(n2|Σ| + nm). +6 +Generating Random Priorities and Empirical Results +In this section, we will demonstrate how one could obtain random priorities in practical settings +using an example of school admission under implicit bias. In several environments based on such +13 + +generative model, we will compare our proposed algorithms, namely Cycle Elimination (CE) and +Unit-time Eating (UTE), with other common bias mitigating allocation algorithms such as “the +Rooney Rule” [7]. We empirically demonstrate that all existing algorithms induce stochastic envy. +To show this, using the same notation as in Definition 5, we say a pair of agents i and j form a +stochastic envy pair if Zi ≻sd +σ∗ Zj but Pi ⊁sd +πi Pj, and we will count the number of stochastic envy +pairs produced by each algorithm. +6.1 +Random Priority in School Admission +Consider a group of N students, including n disadvantaged students with indices {1, . . . , n} and +N − n advantaged students with indices {n + 1, . . . , N}. +Suppose that they are competing for +admission priorities of ℓ schools with capacities c1, . . . , cℓ ∈ N that �ℓ +i=1 ci = N, in which process +disadvantaged students are subjected to implicit bias on their capability. We will quantify the effect +of implicit bias in the experiments. This is equivalent to allocating N items to N agents, where +items correspond to seats, and the agents’ preferences are their school choices. +Denote the j-th seat of school i as sij, then the set of seats is S ≜ �ℓ +i=1{si1, si2, . . . , sici}. For +any ordinal preference �π : [ℓ] → [ℓ] over the schools, it induces an ordinal preference π : S → [N] +over the seats such that for any sij ∈ S, π(sij) = j + � +k:�π(k)<�π(i) ck. In other words, if a student +prefers school a to school b, then all seats of school a are preferred over the seats of school b. For the +seats in the same school, smaller indices are preferred. In the following, we describe how random +priorities over students are generated. +For each student, we assign a “capability score” xi that is drawn from the same distribution +D, and students with higher capability score should have higher priority. Moreover, assume every +student from the disadvantaged group is subjected to a multiplicative implicit bias bi, which is +independently sampled from some distribution B. A disadvantaged student with capability score +xi is perceived to have a biased score �xi ≜ bixi. We will also consider additive bias �xi ≜ xi + bi in +our experiments. The admission committee make decisions based on the perceived scores (which +are biased for disadvantaged students and equal to the true scores for advantaged students). +For each experiment, we fix a set of unbiased capability scores {xi}N +i=1 for the students, where +xi +iid +∼ D. Then, we take n bias parameters {bi}n +i=1 independently from B. The perceived scores of +the students are {�x1, . . . , �xn, xn+1, . . . , xN}, where �xi = bixi. Now imagine we are the admission +committee. We know B, D, and the perceived scores of the students. The goal is to approximately +recover the underlying true scores of the students. To do this, we compute a posterior distribution +for the bias factor of each disadvantaged student given B, the biased score of this student, and +D. Concretely, the density of the posterior distribution for the bias factor of the ith disadvantaged +student, which we denote by bi, is fbi(b) = +fB(b)fD(�xi/b) +� ∞ +0 +fB(u)fD(�xi/u)du. Given {bi}n +i=1, we independently +draw q sets of bias parameters for disadvantaged students, where we denote the jth set of bias +parameters as {b(j) +i }n +i=1, i.e. b(j) +i +iid +∼ bi. Let the ordinal relationship induced by {b(j) +i }n +i=1 be σ(j). +We consider the random priority {(σ(j), 1 +q)}q +j=1. We denote the random priority induced by q sets +of bias parameters as Σ(q). +6.2 +Algorithms for Comparison +To compare with CE and UTE, we consider four alternative solutions to the allocation problem +under implicit bias. Fix a set of biased scores {�x1, . . . , �xn, xn+1, . . . , xN}, let �σ denote its induced +14 + +ℓ +β +N +RN +R +RR +CE +UTE +0.2 +3.4 +0 +3.4 +10 +0 +0 +1 +0.5 +1.2 +0 +1.2 +10 +0 +0 +0.8 +0.6 +0 +0.6 +10 +0 +0 +0.2 +14.3 +42.8 +2.6 +3.8 +0 +0 +2 +0.5 +14.5 +42.8 +1.0 +3.8 +0 +0 +0.8 +19.7 +42.8 +0.6 +3.8 +0 +0 +0.2 +88.8 +175.7 +1.6 +2.5 +0 +0 +3 +0.5 +98.9 +175.7 +0.7 +2.5 +0 +0 +0.8 +103.5 +175.7 +0.4 +2.5 +0 +0 +ℓ +β +N +RN +R +RR +CE +UTE +0.2 +7.0 +0 +15.4 +25.4 +0 +0 +1 +0.5 +7.3 +0 +6.2 +36.9 +0 +0 +0.8 +7.8 +0 +2.8 +42.2 +0 +0 +0.2 +38.2 +36.2 +11.2 +16.8 +0 +0 +2 +0.5 +37.0 +38.3 +4.5 +22.9 +0 +0 +0.8 +37.4 +39.6 +2.2 +25.6 +0 +0 +0.2 +156.2 +183.3 +7.3 +9.8 +0 +0 +3 +0.5 +141.4 +200.5 +3.4 +12.6 +0 +0 +0.8 +127.6 +205.5 +1.9 +15.5 +0 +0 +Figure 1: Number of stochastic envy pairs under multiplicative bias (left) and additive bias (right). +ordinal relationship. For a deterministic priority σ over the students and ordinal preferences Π ≜ +{πi}i∈[N] of students over the seats, let GS(σ, Π) denote the deterministic assignment produced by +the Gale-Shapley algorithm [11] which produces a stable matching between students and seats. +The algorithms that we compare with are as follows: +1. Naive Stable Matching (N) takes deterministic priority �σ and returns the assignment PN(�σ, Π) ≜ +GS(�σ, Π). +2. Random Naive Stable Matching (RN) takes the random priority Σ(q) = {(σi, pi)}q +i=1 and +outputs a lottery based on (N), namely {(PN(σi, Π), pi)}q +i=1. +3. Rooney Stable Matching (R) takes in the deterministic priority �σ as input. Using the Rooney +constraint in Theorem 3.3 of [7], it creates a new priority �σR. +We present this formally +in Algorithm 3. +Using �σR, Rooney Stable Matching returns the assignment PR(�σR, Π) ≜ +GS(�σR, Π). +4. Random Rooney Stable Matching (RR) takes the random priority Σ(q) = {(σi, pi)}q +i=1 and +outputs a lottery based on (R), namely {(PR(σi, Π), pi)}q +i=1. +6.3 +Prevalence of Stochastic Envy +We now demonstrate that with random priority induced by the generative model described in +Section 6.1, stochastic envy exists for the bias mitigating algorithms N, RN, R, RR. +15 + +Algorithm 3: Proportional Rooney-rule-like Constraint [7] +Let A, B be the ordered sub-sequences of disadvantaged and advantaged candidates in �σ +respectively, i.e. p < q ⇐⇒ �σ(A[p]) < �σ(A[q]); +i, j ← 0; +while i + j < N do +if ⌊ +i +i+j ⌋ < n +N or �σ(i) < �σ(n + j) then +�σR(A[i]) = i + j and i ← i + 1; +else +�σR(B[j]) = i + j and j ← j + 1; +end +end +return �σR +We consider an admission problem with ℓ schools each with ⌊ N +ℓ+1⌋ seats. Every student i ∈ [N] +has a uniformly random preference order over the ℓ schools. There is also a ”dummy school” with +N − ℓ⌊ N +ℓ+1⌋ seats representing no admission. Every student prefers seats in the dummy school the +least. For seats in the same school, all students have the same preference order. This represents +the situation in which schools may distribute educational resources to students based on their rank +when admitted. +We take N = 35 and n = 10, and experiment with ℓ = 1, 2, 3. For each choice of k, we experiment +with β = 0.2, 0.5, 0.8. For multiplicative bias, we take D = Exponential(1) and B = Exponential(β); +for additive bias, we take D = Uniform(0, 2) and B = Uniform(0, β). Figure 1 presents the number +of stochastic envy pairs for each algorithm averaged over 100 experiments. For each experiment, +the random priority is computed with 1000 sets of bias parameters. +Except for RN in the one school setting, stochastic envy exists in all other scenarios for N, +RN, R, RR. While empirically Rooney-rule-like constraints do significantly reduce the number of +stochastic envy pairs compared to applying no mitigation mechanism at all, we still need CE or +UTE to obtain guaranteed SEF. +7 +Conclusion +We conclude with some open questions. First, even though SEF and LEF are incompatible, we +do not know whether they can be compatible under certain natural generative assumptions on the +random priorities and agent preferences. Second, it is known [6] that OE (and hence CE and UTE) +is incompatible with strategyproofness under natural assumptions. However, it is interesting to +explore whether SEF alone is also incompatible with strategy-proofness. 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Journal of Economic +Theory, 52:123–135, 1990. +18 + diff --git a/-9FST4oBgHgl3EQfczg_/content/tmp_files/load_file.txt b/-9FST4oBgHgl3EQfczg_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c15675fb24c74cde5165a4c8378a1ae83047d6bb --- /dev/null +++ b/-9FST4oBgHgl3EQfczg_/content/tmp_files/load_file.txt @@ -0,0 +1,677 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf,len=676 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='13804v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='GT] 31 Jan 2023 Fairness in the Assignment Problem with Uncertain Priorities∗ Zeyu Shen† Zhiyi Wang∗ Xingyu Zhu∗ Brandon Fain∗ Kamesh Munagala∗ Abstract In the assignment problem, a set of items must be allocated to unit-demand agents who ex- press ordinal preferences (rankings) over the items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In the assignment problem with priorities, agents with higher priority are entitled to their preferred goods with respect to lower priority agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' A priority can be naturally represented as a ranking and an uncertain priority as a distribution over rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For example, this models the problem of assigning student appli- cants to university seats or job applicants to job openings when the admitting body is uncertain about the true priority over applicants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' This uncertainty can express the possibility of bias in the generation of the priority ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We believe we are the first to explicitly formulate and study the assignment problem with uncertain priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We introduce two natural notions of fairness in this problem: stochastic envy-freeness (SEF) and likelihood envy-freeness (LEF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We show that SEF and LEF are incompatible and that LEF is incompatible with ordinal efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We describe two algorithms, Cycle Elimination (CE) and Unit-Time Eating (UTE) that satisfy ordinal efficiency (a form of ex-ante Pareto optimality) and SEF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' the well known random serial dictatorship algorithm satisfies LEF and the weaker efficiency guarantee of ex-post Pareto op- timality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We also show that CE satisfies a relaxation of LEF that we term 1-LEF which applies only to certain comparisons of priority, while UTE satisfies a version of proportional allocations with ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We conclude by demonstrating how a mediator can model a problem of school admission in the face of bias as an assignment problem with uncertain priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 1 Introduction Consider a motivating example of the assignment problem where a number of university admission slots (the items) must be assigned to student applicants (the agents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The university slots could be at a single university or several.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Applicants might have preferences over different universities, or might have preferences over different slots at the same university (for example, some slots might be associated with merit-based financial aid, or include admission to particular academic programs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Applicants are unit-demand, meaning they only need to be assigned a single slot (and derive no benefit from being assigned multiple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Most university systems employ some form of priority-based admissions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' this can be expressed through a ranking over applicants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For example, a priority might rank applicants by standardized exam scores, or perhaps by some more complex holistic assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Given any deterministic priority (a ranking), one might naturally solve the assignment problem using the serial dictatorship rule, so that students choose their most preferred remaining university slot one at a time in order of ∗This work is supported by NSF grant CCF-2113798.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' †Computer Science Department, Duke University, Durham, NC 27708-0129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Email: {zeyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='shen,zhiyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='wang,xingyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='zhu}@duke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='edu, {btfain,kamesh}@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='duke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 1 their standardized exam score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Indeed, systems roughly like this are employed in several countries around the world such as the Indian Institutes of Technology [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Despite the appeal of such a simple and ostensibly fair system, there is reason to suspect that any scoring or ranking system is based on imperfect noisy signals of the true underlying priority (whatever that might be).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For example, an applicant A scoring 1 point higher on a standardized exam or holistic assessment than another applicant B is not, in general, 100% more likely to be a better student than B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Even more worryingly, studies show that standardized exam performance is closely related to demographic factors such as race and income [8], leading to uncertainty based on social bias and inequality in addition to random noise like whether one had a good breakfast the day of an exam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' More holistic assessments are further vulnerable to the well documented phenomenon of implicit bias against historically marginalized groups [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Ignoring these uncertainties may result in arbitrary decisions (deterministically preferring one applicant over another when the comparison is unclear and noisy) and systemic discrimination against historically marginalized groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Previous work has attempted to solve the second problem of bias without explicitly modeling an uncertain priority by adapting the so-called “Rooney Rule” [15, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' There are variations, but roughly speaking these methods reserve a number of “minority” spots and prioritize this many “minority” applicants in some serial dictatorship assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' This approach can lead to fairness gerrymandering [14] by which structured subgroups remain disadvantaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In particular, Rooney Rule style approaches are predicated on a single binary distinction of the applicant population into “majority” (or privileged) and “minority” (or disadvantaged) applicants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' But in reality, applicant identity is multidimensional (race, gender, income, disability, first language, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=') and bias can compound along intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In fact, it is perfectly plausible that the vast majority of applicants are disadvantaged (that is, suffer from bias leading to underestimation of their priority) along one or more dimensions of identity, though not all to the same extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In addition to group identity, there may sometimes be uncertainties related to the priority of individual applicants, unique circumstances that merit accounting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For these reasons, we consider the more general problem that takes as input an uncertain priority, expressed as a probability distribution over rankings of applicants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The generality of the input to our algorithms ensures that a decision maker can fully model the complexity of uncertainty and bias inherent in the creation of a priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' This modeling problem is outside the scope of this paper, though we do provide an example for our experiments in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Rather, our emphasis is on the question of characterizing fairness and efficiency given a random priority, and providing algorithms to compute random assignments that satisfy these desiderata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='1 Contributions We study an extension of the random assignment problem [6, 18, 2] in which a decision maker must allocate a number of items to unit-demand agents in a way that is consistent with an uncertain priority represented as a distribution over rankings of the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' To the best of our knowledge, we are the first to characterize this more general problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In general we want to compute a random assignment that is simultaneously efficient with respect to agent preferences over the items and fair with respect to the agent priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Ordinal efficiency (OE) [6] generalizes the concept of Pareto efficiency to the case of a random assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Our main contribution is to characterize two alternative notions of fairness for the random assignment problem with uncertain priorities in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The first notion, which we call stochastic envy-freeness (SEF), guarantees that any agent whose priority first-order stochastically dominates another agent’s 2 priority should prefer their own (random) assignment to that of the other agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The second notion, which we call likelihood envy-freeness (LEF), guarantees that the likelihood (over the random assignment) that an agent prefers the assignment of another should be at most the likelihood (over the uncertain priority) that the latter agent has higher priority than the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We introduce additional notions that helps more finely distinguish between algorithms that satisfy one of the above notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The first is a relaxation of LEF called 1-LEF that holds only when an agent has higher priority than another with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The next is ranked proportionality (PROP), where the allocation of any agent should stochastically dominate the allocation where she gets her i-th preferred item with probability pi if she herself is ranked at position i with that probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Formal definitions are provided in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We provide illustrative examples of these concepts as well as justification for why multiple definitions of fairness might be appropriate in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In Section 4 we show that it is impossible to guarantee OE and LEF simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We also show that it is impossible to guarantee SEF and LEF simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Given this, we focus on achieving OE and SEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In Section 5 we describe two algorithms: Unit-time Eating (UTE) and Cycle Elimination (CE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We show that both of these algorithms satisfy OE and SEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' To more finely distinguish between these algorithms, we show that CE also satisfies the relaxed 1-LEF property, while UTE satisfies PROP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We also show that any algorithm achieving OE cannot achieve PROP and 1-LEF simultaneously, so that we cannot achieve a super-set of the properties achieved by these algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' It is straightforward to observe that the well known Random Serial Dictatorship (RSD) that samples a priority from Σ and then uses the serial dictatorship satisfies LEF, PROP, and is ex-post Pareto efficient, though it does not satisfy OE [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We obtain a nearly complete characterization of achievable subsets of our efficiency and fairness properties, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Algorithm OE SEF LEF 1-LEF PROP RSD ✓ ✓ ✓ UTE (new) ✓ ✓ ✓ CE (new) ✓ ✓ ✓ Table 1: Summary of fairness properties achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In Section 6 we return to a consideration of our motivating application of biased school ad- missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We provide a practical example modeling an uncertain priority in the presence of bias and compare our CE and UTE algorithms with previous approaches to address bias using “Rooney Rule” style approaches [15, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='2 Related Work Random Assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' There is a large body of work studying the problem of random assign- ment with no priority (or, in our framework, when the priority is uniform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The work of [1] proposed a random serial dictatorship mechanism, which draws an ordering of agents uniformly at random and let them choose items in that order, and showed that this mechanism is ex-post efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The work of [23] observed that though random serial dictatorship is fair, it is not efficient when the agents are endowed with Von Neumann-Morgenstern preferences over lotteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The work of [6] 3 introduced a notion of efficiency that is stronger than ex-post efficiency, namely ordinal efficiency, and showed that random serial dictatorship is not ordinally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' They proposed the probabilis- tic serial rule that is ordinally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Moreover, probabilistic serial is (stochastically) envy-free while random serial dictatorship is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The work of [2] studied the relationship between ex-post efficiency and ordinal efficiency, showing that a lottery induces an ordinally efficient random as- signment if and only if each subset of the full support of the lottery is undominated (in a specific sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Subsequent works investigated natural extensions of the canonical setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The work of [18] con- sidered the problem of random assignment in the case where agents can opt out, and characterised probabilistic serial by ordinal efficiency, envy-freeness, strategyproofness, and equal treatment of equals in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The work of [10] studied the notion of rank efficiency, which maximises the number of agents matched to their first choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Fair Ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The assignment problem with priority is closely related to the subset selection problem that has been studied extensively as a problem in fair ranking [15, 16, 19, 7, 9, 17, 12] where the goal is to optimize some latent measure of utility for the algorithm designer subject to group fairness constraints on the resulting ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Recent work considers explicitly modeling the uncertainty from bias when estimating a ranking based on observed utilities [22], similar to our approach in modeling an uncertain priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Our work differs from the fair ranking literature in that we study a more general assignment problem in which agents may not all have the same preferences over items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Of course, one can always translate a given ranking into an assignment by employing the serial dictatorship rule, but this need not be ordinally efficient [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Instead, we formulate our desiderata more explicitly in the wider context of the assignment problem itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Two-sided matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' School choice problems are often studied in the context of two-sided matching, where applicants have preferences over schools and schools have preferences over ap- plicants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For example, the deferred acceptance algorithm (and its extensions) calculates stable matchings and has been extensively studied and deployed in the real world [11, 20, 21, 4, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Our problem is different in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' First, the “items” in our problem (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=', school seats) share a single common priority over applicants, so the notion of stability simply means no applicant of lower priority is assigned an item preferred by an agent of higher priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' However, our setting is more complex in the second sense: The shared priority is uncertain, and the assignment will be random, requiring an extension of existing fairness properties and algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 2 Preliminaries We are given n unit demand agents A = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' , n} and a set of m items I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We assume without loss of generality that m ≥ n (if not, one can create additional “dummy” items that are least preferred by all agents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We write a ≻i b to denote that agent i prefers item a to item b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Each agent has ordinal preferences represented as a total order over I, that is, for every agent i we have a permutation πi : I → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' , n} such that πi(a) < πi(b) if and only if a ≻i b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='1 1In general, results extend trivially to the case where agents may have objective indifferences between items, meaning that if any agent is indifferent between two items then all agents are indifferent between those items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' However, our results do not necessarily extend straightforwardly if agents have subjective indifferences, see [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 4 A simple priority over agents is a permutation σ : A → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' , n} where σ(i) < σ(j) means that i has higher priority than j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' A random priority is a probability distribution over simple priorities which we denote as Σ = {(σk, ρk)} where each σk is a simple priority, ρk ≥ 0, and � k ρk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' A simple assignment is a matching f : A → I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' A lottery is a probability distribution over simple assignments which we denote as L = {(fk, pk)} where each fk is a simple assignment, pk ≥ 0, and � k pk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Following [6], we call a probability distribution over [m] itself a random allocation to an agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' It is important to note that agents have ordinal preferences over deterministic items which only induces a partial order over random allocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' That is, given πi, it may be unclear whether i would prefer one random allocation to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We denote by P = {pij} a random assignment, the n by m matrix where Pi, the i-th row, is agent i’s random allocation, and where � i pij = 1 for all columns j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In general, a random assignment P can be induced by one or more lotteries, the existence of which is guaranteed by the Birkhoff-von Neumann Theorem, but a particular lottery induces a unique random assignment P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In the assignment problem with uncertain priorities we are given a random priority Σ and agent preferences {πi} and we must compute a random assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 3 Desiderata In this section we introduce the normative properties that an algorithm for the random assignment with uncertain priorities problem should satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Broadly speaking, these desiderata require that the algorithm be efficient with respect to agent preferences and fair with respect to agent priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='1 Efficiency A simple assignment f is Pareto efficient (or Pareto optimal) if it is not dominated by any other simple assignment, which simply means that there is no alternative such that no agent is worse off and at least one agent is better off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Definition 1 (Pareto Efficiency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' A simple assignment f is Pareto efficient if for all simple as- signments g one of the following holds: (i) ∃i ∈ A such that f(i) ≻i g(i), or (ii) g(i) ⊁ f(i) for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' A lottery L is ex-post Pareto efficient if every simple assignment in the support of L (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=', every simple assignment fk with pk > 0) is Pareto efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' A stronger efficiency property for a random assignment is ordinal efficiency (OE) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' To define ordinal efficiency we must first define the notion of stochastic dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Definition 2 (Stochastic Domination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' A probability distribution X stochastically dominates an- other distribution Y under permutation π (denoted X ≻sd π Y ) if for all t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' , n} it holds that �t r=1 Xπ−1(r) ≥ �t r=1 Yπ−1(r), where π−1 is the inverse permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' A random assignment P is stochastically dominated by a random assignment Q ̸= P if the random allocation induced by Q stochastically dominates the random allocation induced by P under preferences πi for every agent i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Note that this implies the following: If random assignment Q stochastically dominates random assignment P, then every agent prefers Q to P under any Von Neumann-Morgenstern utility func- tion consistent with their ordinal preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Now we can define ordinal efficiency, following [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 5 Definition 3 (Ordinal Efficiency, OE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We say that a random assignment P is ordinally efficient if it is not stochastically dominated by any other random assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' At a high level, a random assignment is ordinally efficient if there is no other random assignment that is better for all agents and all utility functions consistent with their ordinal preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The property is not trivial: Some natural algorithms such as random serial dictatorship are Pareto efficient but not ordinally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='2 Fairness We define fairness in terms of envy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We say that one agent envies another if the former prefers the item assigned to the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Envy of a lower priority agent constitutes a justified complaint against an assignment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' ideally we would like to compute an envy-free assignment with respect to the priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Definition 4 (Envy-Freeness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We say that a simple assignment f is envy-free with respect to a simple priority σ if for all i, j ∈ [n], σ(i) < σ(j) =⇒ f(i) ≻i f(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' However, it is immediately evident that it is impossible to compute a single simple assignment that is envy-free in this sense for every simple priority in the support of a random priority (for example, if there are two agents with uncertain priority who both prefer the same item).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Instead, we need to compute a random assignment so that each agent is fairly treated ex-ante (for example, so that each agent has a fair probability of receiving the preferred good).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' There are two natural ways to generalize the concept of envy to a random assignment with a random priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' One is to imagine that one agent envies another if the random allocation of the latter stochastically dominates that of the former under the former’s ordinal preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Envy of this type forms a justified complaint if the envying agent also stochastically dominates the envied agent in terms of the random priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' More formally, Definition 5 (Stochastic Envy-Freeness, SEF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Consider a random assignment P generated under a random priority Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Let Si be the probability distribution over [n] induced by Σ for agent i, that is, for r ∈ [n], Sir = � k:σk(i)=r ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Let σ∗ be the identity permutation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=', σ∗(i) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' P is stochastically envy-free (SEF) with respect to Σ if for all i, j ∈ [n], Si ≻sd σ∗ Sj =⇒ Pi ≻sd πi Pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Loosely speaking, the implication of stochastic envy-freeness can be read as “if agent i probably has higher priority than j then i should prefer their random allocation to j’s under all utility functions consistent with i’s ordinal preferences.” A second way to generalize envy is by considering the likelihood of envy (in the simple sense) with respect to a lottery inducing a given random assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Envy of this type is justified if the likelihood of agent i envying another agent j is greater than the likelihood over the random priority that i has lower priority than agent j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We call a random assignment likelihood envy-free if there is a lottery which induces it and has no envy of this kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Definition 6 (Likelihood Envy-Freeness, LEF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' A random assignment P satisfies likelihood envy- freeness (LEF) under Σ if P can be induced by a lottery L such that for all i, j ∈ [n], Prσ∼Σ[σ(i) < σ(j)] ≤ Prf∼L[f(i) ≻i f(j)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In other words, LEF means that an agent i who is ℓ-likely to have higher priority than another agent j should be at least ℓ-likely to prefer their assigned item to j’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 6 We say an algorithm satisfies OE (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' SEF, LEF) if it always produces random assignment that satisfies OE (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' SEF, LEF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' As we show in Section 4, it is not possible to guarantee SEF and LEF simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='3 Relationship between LEF and SEF The relationship between SEF and LEF is subtle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' neither implies the other and it is not immediately evident which is the “better” or more “natural” fairness property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We present two examples to illustrate that an assignment satisfying only one of SEF and LEF might still be unfair, so that both properties are useful competing notions of fairness, and neither is strictly stronger than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We first present an example which shows that an assignment that satisfies SEF can be unfair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Consider n = 2 agents and m = 2 items which we label a, b for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Both agents prefer a to b, and the random priority is simply Σ = {(σ, 1)} with σ(1) < σ(2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' agent 1 has higher priority than agent 2 with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In this setup, allocating 1 2 unit of a and b to both agent yields an assignment that satisfies SEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' However, this assignment is clearly unfair, because even though agent 1 has higher priority than agent 2, they are getting the same assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Notice that this assignment does not satisfy LEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In this instance, LEF could be used to characterize how much one agent is prioritized over the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The next example shows that an assignment that only satisfies LEF can also be unfair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Consider n = 2 agents and m = 100 items which we label i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' , i100 for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The preferences of both agents are i1 ≻ · · · ≻ i100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The random priority is given by Σ = {(σ1, 1 2), (σ2, 1 2)} with σ1(1) < σ1(2) and σ2(2) < σ2(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In other words, both agents have the same priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In this setup, allocating 1 2 unit of i1 and 1 2 unit of i100 to agent 1 and 1 2 unit of i99 and 1 2 unit of i100 to agent 2 yields an assignment that satisfies LEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Notice that this assignment can be induced by a lottery L = {(f1, 1 2), (f2, 1 2)} where f1(1) = i1, f1(2) = i100, f2(1) = i100, f2(2) = i99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' However, this assignment is clearly unfair, because even though the two agents have the same priority, agent 1 gets a strictly better assignment than agent 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' This shows that LEF alone has limitations as well, and the appropriate concept here is SEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The above examples show that SEF and LEF provide reasonable competing notions of fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' When combined with the efficiency notion of OE, we will show in Section 4 that LEF and OE are incompatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' If OE is replaced by the weaker notion of Pareto-efficiency, then it is easy to check that random serial dictatorship (RSD), which simply samples a priority of agents from the distribution and allocates each agent their favorite remaining item in this priority order, satisfies LEF2 and pareto efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Thus, in our work, we will focus on the more non-trivial part of finding algorithms that satisfy SEF and OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='4 Additional Fairness Criteria As we show in Section 5, there can be multiple algorithms that satisfy the same subset of the fairness criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We therefore consider two additional notions to more finely distinguish between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The first criterion is the following relaxation of LEF: If agent i with probability 1 has higher priority than another agent j then agent i should certainly (again, with probability 1) not envy j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 2To see why RSD satisfies LEF, suppose the random priority is given by Σ = {(σk, ρk)}, then the random assignment produced by RSD can be induced by the lottery L = {(fk, ρk)}, where fk is the deterministic assignment produced by letting agents successively choose an item based on the order given by σk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 7 Definition 7 (1-LEF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' A random assignment P under random priority Σ satisfies 1-LEF if there exists some lottery L which induces P such that for all agents i ̸= j ∈ [n], if Prσ∼Σ[σ(i) < σ(j)] = 1, then Prf∼L[f(i) ≻i f(j)] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The next criterion is called Ranked Proportionality (PROP), which captures stochastic dom- inance over an allocation that matches the probability an agent gets her ith ranked item to the probability of she being ranked at position i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Note that if all rankings of agents were equally likely, this captures stochastic dominance to an allocation that assigns every item to every agent uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Definition 8 (PROP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Given a random priority Σ = {(σk, ρk)}, we define the baseline allocation P i for agent i by P iπ−1 i (r) = Sir = � k:σk(i)=r ρk for all r ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In other words, if an agent i ranks the r-th in the random priorities with probability p, then we add p fraction of the r-th preferred item of agent i to her baseline allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For an allocation to satisfy ranked proportionality (PROP), it should stochastically dominate this baseline for each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 4 Impossibility Results In this part, we present several impossibility results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We note that these are existential hardness results, not computational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We begin by observing that LEF is incompatible with OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' LEF is incompatible with OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We present an instance in which no random assignment can satisfy both LEF and OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' There are n = 4 agents and m = 4 items which we label a, b, c, d for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Agent preferences are given by π1, π3 : a ≻ b ≻ c ≻ d, π2, π4 : b ≻ a ≻ c ≻ d Moreover, we consider the priority Σ = {(σ1, 1 2), (σ2, 1 2)} where σ1(4) < σ1(2) < σ1(3) < σ1(1), σ2(3) < σ2(1) < σ2(4) < σ2(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In other words, with probability 1 2 under σ1, agent 4 has the highest priority, then agent 2, then agent 3, finally agent 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Similarly for σ2 with probability 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Assume for contradiction that there exists a random assignment P = [pij], together with a lottery L which induces P, satisfying LEF and OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' By definition of LEF, we note that Pr f∼L[f(3) ≻3 f(1)] ≥ Pr σ∼Σ[σ(3) < σ(1)] = 1, so it must be that Prf∼L[f(3) ≻3 f(1)] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Thus, we must have p1a = 0, because otherwise there would exist a simple assignment in the lottery in which agent 1 is assigned with a and agent 3 is assigned with some less preferred item under π3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' By the same reasoning, we note that p2b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Also by definition of LEF, observe that Pr f∼L[f(2) ≻2 f(3)] ≥ Pr σ∼Σ[σ(2) < σ(3)] = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 8 This implies p3a < 1, as otherwise we have f(3) = a for all f ∼ L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' combined with the fact that p2b = 0, we would have f(3) ≻2 f(2) for all f ∼ L, which contradicts Prf∼L[f(2) ≻2 f(3)] ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Since p1a = 0, p3a < 1, and � i pia = 1, it follows that p2a + p4a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Similarly, we have p1b +p3b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Without loss of generality, we assume that p2a > 0 and p1b > 0 (if p4a > 0 or p3b > 0, the proof proceeds similarly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Let pmin = min(p2a, p1b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' define random assignment Q = [qij] by qij = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 pij if i /∈ {1, 2} and j /∈ {a, b} pij + pmin if (i, j) = (1, a) or (2, b) pij − pmin if (i, j) = (1, b) or (2, a) We can see that Q stochastically dominates P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In particular, all that is different in Q is that agent 1 swaps agent 2 some of agent 2’s allocated probability mass on item a in exchange for an equivalent amount of agent 1’s probability mass on item b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Since a ≻1 b and b ≻2 a and nothing else changes, agents 1 and 2 prefer Q, and nothing has changed for agents 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' This contradicts with the fact that P satisfies OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Thus, we can conclude that no random assignment in this instance satisfies LEF and OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Theorem 1 can be interpreted as a fundamental tradeoff between efficiency and fairness con- ceived as LEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Next, we show that LEF and SEF are two fundamentally different notions of fairness that are incompatible with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' As we will see later in Section 5, each of LEF and SEF independently can be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Thus, neither notion of fairness is subsumed by the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' LEF is incompatible with SEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We present an instance in which no random assignment can satisfy both LEF and SEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' There are n = 5 agents and m = 5 items which we label a, b, c, d, e for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Preferences are given by π1, π3 : a ≻ b ≻ c ≻ d ≻ e, π2, π4 : b ≻ a ≻ c ≻ d ≻ e, π5 : a ≻ c ≻ b ≻ d ≻ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We consider the priority Σ = {(σ1, 1 2), (σ2, 1 2)} defined by σ1(3) < σ1(5) < σ1(1) < σ1(4) < σ1(2), σ2(4) < σ2(5) < σ2(2) < σ2(3) < σ2(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In other words, with probability 1 2 under σ1, agent 3 has the highest priority, then agents 5, 1, 4, and finally 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Similarly for σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Assume for contradiction that there exists a random assignment P = [pij], together with a lottery L which induces P, that satisfies LEF and SEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Since agent 3 always has higher priority than agent 1 and agent 3 prefers a over all other items, LEF implies that p1a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Similarly, since agent 4 always has higher priority than agent 2 prefers b over all other itmes, LEF implies that p2b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Recall that Si is the probability density over [n] induced by Σ for agent i and σ∗ is the identity permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Since S1 ≻sd σ∗ S2 by construction and P satisfies (SEF) by assumption, we have P1 ≻sd π1 P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Combined with the fact that p1a = 0, we must have p2a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Similarly, p1b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 9 We next show p1c = p2c = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' First, observe that LEF guarantees Pr f∼L[f(4) ≻4 f(2)] ≥ Pr σ∼Σ[σ(4) < σ(2)] = 1, Thus, since e is the least preferred item by agent 4, we must have p4e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Also by LEF, we have Pr f∼L[f(1) ≻1 f(4)] ≥ Pr σ∼Σ[σ(1) < σ(4)] = 1 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Prf∼L[f(1) ≻1 f(4)] ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' On the other hand, since p4e = 0, the worst item that agent 4 can get under π4 is d, so Pr f∼L[f(1) ≻1 f(4)] ≤ p1a + p1b + p1c = p1c, since we earlier found that p1a = p1b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Recall Prf∼L[f(1) ≻1 f(4)] ≥ 1 2, we get p1c ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Similarly, we have p2c ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Since � i pic = 1, it must be the case that p1c = p2c = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We deduce that for any f ∼ L, either f(1) = c or f(2) = c, because on one hand, for any fixed f, we should have f(1) ̸= f(2), while on the other hand, p1c + p2c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Observe that p5a ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' This follows directly from LEF, because Pr f∼L[f(3) ≻3 f(5)] ≥ Pr σ∼Σ[σ(3) < σ(5)] = 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' if p5a > 1 2, we would have Prf∼L[f(3) ≻3 f(5)] < 1 − p5a = 1 2, leading to contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' What’s more, we have p5c = 0, since we already have p1c + p2c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' On one hand, we should have Prf∼L[f(5) ≻5 f(1) and f(5) ≻5 f(2)] = 1, since σi(5) < σi(1) and σi(5) < σi(2) for i ∈ {1, 2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' but on the other hand, we have Prf∼L[f(5) ≻5 f(1) and f(5) ≻5 f(2)] ≤ p5a ≤ 1 2, because for any f ∼ L, either f(1) = c or f(2) = c, so f(5) ≻5 f(1) and f(5) ≻5 f(2) if and only if f(5) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' This leads to contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Thus, LEF and SEF are incompatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We finally show that OE, 1-LEF, and PROP are simultaneously incompatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' This will inform the design of algorithms in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' There is an instance where no allocation simultaneously satisfies OE, 1-LEF, and PROP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We use the instance in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Since agent 3 is ranked higher than agent 1 with probability 1 and since their preferences over items is identical, 1-LEF implies agent 1 is allocated item a with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Now PROP implies agent 1 receives item b with probability at least 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' By a similar reasoning, agent 2 must get item a with probability at least 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' But any such allocation cannot be OE, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 5 Algorithms As we have seen, LEF is a very strong notion of fairness which is incompatible with both OE and SEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In the following, we present two algorithms – cycle elimination (CE) and unit time eating (UTE) – that satisfy both OE and SEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In addition, we show that CE satisfies 1-LEF and UTE satisfies PROP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Given Lemma 1, we cannot design an algorithm that achieves OE and both these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 10 Therefore, both CE and UTE are reasonable fair allocation algorithms in that they satisfy efficiency (OE) and envy-freeness (SEF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The choice of which to implement depends on whether we care more about a form of proportionality in the resulting allocation (UTE satisfies PROP) or whether we care about additional envy-freeness in a deterministic sense (CE satisfies 1-LEF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='1 Cycle Elimination algorithm We first introduce a Cycle Elimination algorithm (CE), which works by constructing a directed graph based on the random priority, and allocate items based on this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' To begin with, we introduce the Probabilistic Serial rule [6], a continuous algorithm which works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Initially, each agent i goes to their favorite item j and starts “eating” it (that is, increasing pij) at unit speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' It is possible that several agents eat the same item at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Whenever an item is fully eaten, each of the agents eating it goes to their favorite remaining item not fully allocated (that is, � i pij < 1) and starts eating it in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' This process continues until all items are consumed, or all the agents are full (that is, � j pij = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We use PS(A, I) to denote the assignment produced by running Probabilistic Serial rule on the set of agents A and items I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We construct a graph from Σ, which we call a Stochastic-Dominance graph (SD-graph), as follows: Start with a graph with n vertices, where the i-th vertex corresponds to the i-th agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For any pair of distinct agents i and j, if Si ≻sd σ∗ Sj, then we draw a directed edge from i to j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The algorithm is now formally stated in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Algorithm 1: Cycle Elimination, Eliminate(A, I, G) Input: Set of agents A, set of items I, SD-graph G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Let �G be the condensation3of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Let � A be the set of agents that belong to a strongly connected component whose in-degree in �G is zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' if A = � A then Output PS(A, I);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' else A′ ← A \\ � A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' I′ ← I \\ PS( � A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' I);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' G′ ← G \\ � A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Output PS( � A, I) + Eliminate(A′, I′, G′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' end Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Our main result is the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The Cycle Elimination algorithm satisfies OE, SEF, and 1-LEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' It runs in O(n3 + nm + n|Σ|) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Theorem 1 in [6] states that any simultaneous eating algorithm where each agent always eats from her favorite remaining item satisfies OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Hence, CE satisfies OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 3Condensation of a graph is a directed acyclic graph formed by contracting each strongly connected component to a single vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 11 To show SEF, fix two agents i and j, and assume Si ≻sd σ∗ Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Let P be the random assignment produced by CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We show that Pi ≻sd πi Pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Since Si ≻sd σ∗ Sj, there exists an edge from i to j in the SD-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Thus, i and j either belong to the same strongly connected component, or the strongly connected component of i has higher topological order than that of j’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Either way, we have Pi ≻sd πi Pj, from which we can conclude that CE satisfies SEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' To show 1-LEF, fix two agents i and j, and assume Prσ∼Σ[σ(i) < σ(j)] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Let P be the random assignment produced by CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We show that, for any lottery L inducing P, we have Prf∼L[f(i) ≻i f(j)] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We use proof by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Assume that there exists a lottery L0 which induces P such that Prf∼L0[f(i) ≻i f(j)] < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' This implies that there exists two items a and b such that a ≻i b, pib > 0 and pja > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' On the other hand, since Prσ∼Σ[σ(i) < σ(j)] = 1, so we have Si ≻sd σ∗ Sj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' thus, the strongly connected component that agent i belongs to must have higher topological order than that of j’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' By CE, agent j could start eating only when agent i is completely full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Thus, pja > 0 implies that there is still item a remaining when agent i finishes eating;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' this leads to contradiction, because a ≻i b implies that i could eat a instead of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Therefore, we must have Prf∼L[f(i) ≻i f(j)] = 1 for all lotteries L which induces P, from which we can conclude that CE satisfies 1-LEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' To show the running time, preprocessing Σ in order to compute the stochastic dominance relation between agents takes O(n|Σ| + n3) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Constructing the SD-graph by the stochastic dominance relation between agents takes O(n2) time, as there are �n 2 � pair of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Given the SD-graph, running CE takes O(nm) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' This is because we only need to consider at most m time points: the time at which each item is eaten up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We divide this process into m time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' During each time interval, each agent keeps eating the same item, so it simply takes O(n) time to keep track of the state of each agent, and the running time over m intervals is O(nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Hence, the total running time is O(n3 + nm + n|Σ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='2 Unit-time Eating Algorithm We next introduce the Unit-time Eating Algorithm (UTE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Recall that Σ = {(σk, ρk)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Essentially, the algorithm works by dividing the time into n units, each of duration one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' in time unit t, the t-th ranked agent in σk eats their favorite item among those left over at rate ρk for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The procedure is formally stated in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Algorithm 2: Unit-time Eating Algorithm for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' , n do The t-th ranked agent in each σk eats their favorite item among those left over at rate ρk for all (σk, ρk) ∈ Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' end Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We show the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The Unit-time Eating Algorithm satisfies OE, SEF, and PROP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Further, it runs in O(n2|Σ| + nm) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' By Theorem 1 in [6], we have UTE satisfies OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 12 To show SEF, fix two agents i and j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' assume that Si ≻sd σ∗ Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Let P be the random assignment produced by UTE, we show that Pi ≻sd πi Pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Let tk be the time when item π−1 i (k) has been eaten up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Fix some k ∈ [m];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' because Si ≻sd σ∗ Sj, we have ⌊tk⌋ � t=1 Sit ≥ ⌊tk⌋ � t=1 Sjt and ⌈tk⌉ � t=1 Sit ≥ ⌈tk⌉ � t=1 Sjt Combining these gives � tk − ⌊tk⌋ � Si⌈tk⌉ + ⌊tk⌋ � t=1 Sit ≥ � tk − ⌊tk⌋ � Sj⌈tk⌉ + ⌊tk⌋ � t=1 Sjt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' (1) Observe that k � r=1 Piπ−1 i (r) = � tk − ⌊tk⌋ � Si⌈tk⌉ + ⌊tk⌋ � t=1 Sit, k � r=1 Pjπ−1 i (r) ≤ � tk − ⌊tk⌋ � Sj⌈tk⌉ + ⌊tk⌋ � t=1 Sjt, which gives �k r=1 Piπ−1 i (r) ≥ �k r=1 Pjπ−1 i (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Because this holds for all k ∈ [m], we conclude that Pi ≻sd πi Pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Hence, UTE satisfies SEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' To show PROP, fix some agent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Suppose the allocation produced by UTE for this agent is Pi, and the baseline allocation for this agent is P i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We will show that Pi ≻sd πi P i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Let tk be the time when item π−1 i (k) has been eaten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Fix some k ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Clearly, we have tk ≥ k, because in order to eat up π−1 i (k), we have to eat up π−1(r) for all r < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We observe that k � r=1 Piπ−1 i (r) = � tk − ⌊tk⌋ � Si⌈tk⌉ + ⌊tk⌋ � t=1 Sit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Combined with tk ≥ k, we have k � r=1 Piπ−1 i (r) ≥ k � t=1 Sit = k � r=1 P iπ−1 i (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Because this holds for all k ∈ [m], we conclude that that Pi ≻sd πi P i, and hence UTE satisfies PROP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' To show running time, preprocessing Σ to obtain the eating speed of each agent in each unit time interval takes O(n2|Σ|) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Then, running UTE takes O(nm) time, as we similarly only need to consider at most m time points and keeping track of the state of each agent in each time interval takes O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Therefore, the total running time is O(n2|Σ| + nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 6 Generating Random Priorities and Empirical Results In this section, we will demonstrate how one could obtain random priorities in practical settings using an example of school admission under implicit bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In several environments based on such 13 generative model, we will compare our proposed algorithms, namely Cycle Elimination (CE) and Unit-time Eating (UTE), with other common bias mitigating allocation algorithms such as “the Rooney Rule” [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We empirically demonstrate that all existing algorithms induce stochastic envy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' To show this, using the same notation as in Definition 5, we say a pair of agents i and j form a stochastic envy pair if Zi ≻sd σ∗ Zj but Pi ⊁sd πi Pj, and we will count the number of stochastic envy pairs produced by each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='1 Random Priority in School Admission Consider a group of N students, including n disadvantaged students with indices {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' , n} and N − n advantaged students with indices {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Suppose that they are competing for admission priorities of ℓ schools with capacities c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' , cℓ ∈ N that �ℓ i=1 ci = N, in which process disadvantaged students are subjected to implicit bias on their capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We will quantify the effect of implicit bias in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' This is equivalent to allocating N items to N agents, where items correspond to seats, and the agents’ preferences are their school choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Denote the j-th seat of school i as sij, then the set of seats is S ≜ �ℓ i=1{si1, si2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' , sici}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For any ordinal preference �π : [ℓ] → [ℓ] over the schools, it induces an ordinal preference π : S → [N] over the seats such that for any sij ∈ S, π(sij) = j + � k:�π(k)<�π(i) ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In other words, if a student prefers school a to school b, then all seats of school a are preferred over the seats of school b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For the seats in the same school, smaller indices are preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In the following, we describe how random priorities over students are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For each student, we assign a “capability score” xi that is drawn from the same distribution D, and students with higher capability score should have higher priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Moreover, assume every student from the disadvantaged group is subjected to a multiplicative implicit bias bi, which is independently sampled from some distribution B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' A disadvantaged student with capability score xi is perceived to have a biased score �xi ≜ bixi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We will also consider additive bias �xi ≜ xi + bi in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The admission committee make decisions based on the perceived scores (which are biased for disadvantaged students and equal to the true scores for advantaged students).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For each experiment, we fix a set of unbiased capability scores {xi}N i=1 for the students, where xi iid ∼ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Then, we take n bias parameters {bi}n i=1 independently from B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The perceived scores of the students are {�x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' , �xn, xn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' , xN}, where �xi = bixi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Now imagine we are the admission committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We know B, D, and the perceived scores of the students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The goal is to approximately recover the underlying true scores of the students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' To do this, we compute a posterior distribution for the bias factor of each disadvantaged student given B, the biased score of this student, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Concretely, the density of the posterior distribution for the bias factor of the ith disadvantaged student, which we denote by bi, is fbi(b) = fB(b)fD(�xi/b) � ∞ 0 fB(u)fD(�xi/u)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Given {bi}n i=1, we independently draw q sets of bias parameters for disadvantaged students, where we denote the jth set of bias parameters as {b(j) i }n i=1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' b(j) i iid ∼ bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Let the ordinal relationship induced by {b(j) i }n i=1 be σ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We consider the random priority {(σ(j), 1 q)}q j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We denote the random priority induced by q sets of bias parameters as Σ(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='2 Algorithms for Comparison To compare with CE and UTE, we consider four alternative solutions to the allocation problem under implicit bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Fix a set of biased scores {�x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' , �xn, xn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' , xN}, let �σ denote its induced 14 ℓ β N RN R RR CE UTE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='4 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='4 10 0 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='2 10 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='6 10 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='3 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='8 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='8 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='8 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='8 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='5 0 0 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='5 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='9 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='8 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='5 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='5 0 0 ℓ β N RN R RR CE UTE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='2 7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='8 0 0 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='5 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='4 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='6 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='8 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='6 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='9 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='5 0 0 Figure 1: Number of stochastic envy pairs under multiplicative bias (left) and additive bias (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' ordinal relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For a deterministic priority σ over the students and ordinal preferences Π ≜ {πi}i∈[N] of students over the seats, let GS(σ, Π) denote the deterministic assignment produced by the Gale-Shapley algorithm [11] which produces a stable matching between students and seats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' The algorithms that we compare with are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Naive Stable Matching (N) takes deterministic priority �σ and returns the assignment PN(�σ, Π) ≜ GS(�σ, Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Random Naive Stable Matching (RN) takes the random priority Σ(q) = {(σi, pi)}q i=1 and outputs a lottery based on (N), namely {(PN(σi, Π), pi)}q i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Rooney Stable Matching (R) takes in the deterministic priority �σ as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Using the Rooney constraint in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='3 of [7], it creates a new priority �σR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We present this formally in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Using �σR, Rooney Stable Matching returns the assignment PR(�σR, Π) ≜ GS(�σR, Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Random Rooney Stable Matching (RR) takes the random priority Σ(q) = {(σi, pi)}q i=1 and outputs a lottery based on (R), namely {(PR(σi, Π), pi)}q i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='3 Prevalence of Stochastic Envy We now demonstrate that with random priority induced by the generative model described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='1, stochastic envy exists for the bias mitigating algorithms N, RN, R, RR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 15 Algorithm 3: Proportional Rooney-rule-like Constraint [7] Let A, B be the ordered sub-sequences of disadvantaged and advantaged candidates in �σ respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' p < q ⇐⇒ �σ(A[p]) < �σ(A[q]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' i, j ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' while i + j < N do if ⌊ i i+j ⌋ < n N or �σ(i) < �σ(n + j) then �σR(A[i]) = i + j and i ← i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' else �σR(B[j]) = i + j and j ← j + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' end end return �σR We consider an admission problem with ℓ schools each with ⌊ N ℓ+1⌋ seats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Every student i ∈ [N] has a uniformly random preference order over the ℓ schools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' There is also a ”dummy school” with N − ℓ⌊ N ℓ+1⌋ seats representing no admission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Every student prefers seats in the dummy school the least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For seats in the same school, all students have the same preference order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' This represents the situation in which schools may distribute educational resources to students based on their rank when admitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' We take N = 35 and n = 10, and experiment with ℓ = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For each choice of k, we experiment with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For multiplicative bias, we take D = Exponential(1) and B = Exponential(β);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' for additive bias, we take D = Uniform(0, 2) and B = Uniform(0, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Figure 1 presents the number of stochastic envy pairs for each algorithm averaged over 100 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' For each experiment, the random priority is computed with 1000 sets of bias parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Except for RN in the one school setting, stochastic envy exists in all other scenarios for N, RN, R, RR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' While empirically Rooney-rule-like constraints do significantly reduce the number of stochastic envy pairs compared to applying no mitigation mechanism at all, we still need CE or UTE to obtain guaranteed SEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 7 Conclusion We conclude with some open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' First, even though SEF and LEF are incompatible, we do not know whether they can be compatible under certain natural generative assumptions on the random priorities and agent preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Second, it is known [6] that OE (and hence CE and UTE) is incompatible with strategyproofness under natural assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' However, it is interesting to explore whether SEF alone is also incompatible with strategy-proofness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Finally, can our framework be extended to the scenario where the agent preferences are random as well, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' each agent reports a distribution over preferences instead of a deterministic preference?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 16 References [1] Atila Abdulkadiroglu and Tayfun S¨onmez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Random serial dictatorship and the core from random endowments in house allocation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Econometrica, 66:689–701, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' [2] Atila Abdulkadiroglu and Tayfun S¨onmez.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' In M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Ranzato, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Beygelzimer, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Dauphin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Liang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Wortman Vaughan, editors, Advances in Neural Information Processing Systems, volume 34, pages 11896–11908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' [23] Lin Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' On a conjecture by gale about one-sided matching problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' Journal of Economic Theory, 52:123–135, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FST4oBgHgl3EQfczg_/content/2301.13804v1.pdf'} diff --git a/.gitattributes b/.gitattributes index fda89d47278810f75bb260542bad9d01733529ac..72fb9d0eff182a8954301913e727b06149e056c3 100644 --- a/.gitattributes +++ b/.gitattributes @@ -196,3 +196,8 @@ SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf filter=lfs diff=lfs merge=lfs -tex wdE3T4oBgHgl3EQfOgk_/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text YNFQT4oBgHgl3EQfdzaO/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text ytE1T4oBgHgl3EQfQwPe/content/2301.03045v1.pdf filter=lfs diff=lfs merge=lfs -text +TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf filter=lfs diff=lfs merge=lfs -text +ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf filter=lfs diff=lfs merge=lfs -text +ytE1T4oBgHgl3EQfQwPe/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf filter=lfs diff=lfs merge=lfs -text +I9FAT4oBgHgl3EQfux7Z/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text diff --git a/39FST4oBgHgl3EQfZTjC/content/tmp_files/2301.13791v1.pdf.txt b/39FST4oBgHgl3EQfZTjC/content/tmp_files/2301.13791v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ca2c58489115f14d494fef1386bb1858a5c8781a --- /dev/null +++ b/39FST4oBgHgl3EQfZTjC/content/tmp_files/2301.13791v1.pdf.txt @@ -0,0 +1,8492 @@ +Improved Algorithms for Multi-period Multi-class Packing Problems +with Bandit Feedback +Wonyoung Kim 1 Garud Iyengar 1 Assaf Zeevi 1 +Abstract +We consider the linear contextual multi-class +multi-period packing problem (LMMP) where +the goal is to pack items such that the total vector +of consumption is below a given budget vector +and the total value is as large as possible. We +consider the setting where the reward and the con- +sumption vector associated with each action is +a class-dependent linear function of the context, +and the decision-maker receives bandit feedback. +LMMP includes linear contextual bandits with +knapsacks and online revenue management as spe- +cial cases. We establish a new more efficient esti- +mator which guarantees a faster convergence rate, +and consequently, a lower regret in such problems. +We propose a bandit policy that is a closed-form +function of said estimated parameters. When the +contexts are non-degenerate, the regret of the pro- +posed policy is sublinear in the context dimen- +sion, the number of classes, and the time hori- +zon T when the budget grows at least as +√ +T. We +also resolve an open problem posed in Agrawal & +Devanur (2016), and extend the result to a multi- +class setting. Our numerical experiments clearly +demonstrate that the performance of our policy is +superior to other benchmarks in the literature. +1. Introduction +In the multi-period packing problem (MPP) the decision- +maker “packs” the arrivals so that the total consumption +across a set a resources is below a given budget vector +and the reward is maximized. A variant of the packing +problem, where items consume multiple resources and the +decisions must be made sequentially with bandit feedback +for a fixed time horizon, is known as bandits with knapsacks +(Agrawal & Devanur, 2014a; Badanidiyuru et al., 2018; +Immorlica et al., 2019). MPPs also arise in online revenue +1Columbia University, New York, NY, USA. Correspondence +to: <>. +Preliminary work. Under review by the International Conference +on Machine Learning (ICML). Do not distribute. +management (Besbes & Zeevi, 2012; Ferreira et al., 2018). +MPPs in the literature assume that all arrivals belong to a +single class. However, in several application domains (e.g., +operations, healthcare, and e-commerce), the arrivals are +heterogeneous, and personalizing decisions to each distinct +population or class is of paramount importance. In this +paper we consider a class of linear multi-class multi-period +packing problems (LMMP). At each round, there is a single +arrival that belongs to one of J classes, and the decision- +maker observes the d-dimensional context and the cost for +K different available actions. The outcome of selecting an +action is a random sample of the reward and a consumption +vector for m resources with an expected value that is a class- +dependent linear function of the d-dimensional contexts. +The goal of the problem is to minimize the cumulative regret +over a time horizon T while ensuring that the total resource +consumed is at most B. +The LMMP problem is a generalization of several prob- +lems including linear contextual bandits with knapsacks +(LinCBwK) introduced by Agrawal & Devanur (2016). +They proposed an online mirror descent-based algorithm +that achieves ˜O(OPT/B · d +√ +T) regret when the budget +B for each of the m resources is Ω( +√ +dT 3/4), where OPT +is the reward obtained by the oracle policy. Although the +regret bound is meaningful for B ≥ Ω(d +√ +T), establishing +the regret bound for smaller budget values was left as an +open problem. Chu et al. (2011) established a regret bound +sublinear in d for the linear contextual bandit setting, which +is a special case of LinCBwK with no budget constraints. +Thus, the following question remained open: “Is there an +algorithm for LinCBwK that achieves sublinear dependence +on d with budget B = Ω( +√ +T)?” +We propose a novel algorithm and an improved estimation +strategy that settles this open problem and generalizes the +result to the more general class of LMMP. The proposed +algorithm achieves ˜O(OPT/B +√ +JdT) regret with budget +B = Ω( +√ +JdT) under non-degenerate contexts. While re- +gret of the existing algorithms grows linearly in the number +of classes J, our estimator is able to pool data from differ- +ent classes and avoids linear dependence on J. To reiterate, +the improved regret bound results from the novel estimator +which yields faster convergence rates. +arXiv:2301.13791v1 [stat.ML] 31 Jan 2023 + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Our main contributions are summarized as follows: +• We propose a new problem class – linear multi-class +multi-period packing problems (LMMP). This problem +generalizes a variety of problems including LinCBwK +and online revenue management problems to the multi- +class setting. +• We propose a novel estimator that uses contexts for +all actions (including the contexts in skipped rounds) +and yields O( +� +Jd/n) convergence rate for J classes, +context dimension d, and n admitted arrivals (Theorem +4.2). +• We propose a novel AMF (Allocate to the Maximum +First) algorithm which achieves ˜O(OPT/B +√ +JdT) +regret with budget B = Ω( +√ +JdT) where OPT is +the reward obtained by oracle policy (Theorem 5.1). +For the single class setting with J = 1, we improve +the existing bound by +√ +d and show that the bound +is valid when B = Ω( +√ +dT), and thus resolving an +open problem posed in Agrawal & Devanur (2016) +regarding LinCBwK. +• We evaluate our proposed algorithm on a suite of syn- +thetic experiments and demonstrate its superior perfor- +mance. +All proofs omitted from the front matter can be found in the +Appendix. +2. Related Works +There are two streams of work that are relevant for +LMMP. In online revenue management literature, Gallego & +Van Ryzin (1994) introduced the dynamic pricing problem +where the demand is a known function of price (action). Bes- +bes & Zeevi (2009) and Besbes & Zeevi (2012) extended the +problem under unknown demands with multiple resource +constraints. Ferreira et al. (2018) proposed a Thompson +sampling-based algorithm and extended it to contextual ban- +dits with knapsacks. When the expected demand is a linear +function of the price vector, the dynamic pricing problem +is a special case of linear contextual bandits with knap- +sack (LinCBwK) proposed by Agrawal & Devanur (2016). +The LinCBwk is a common generalization of bandits with +knapsacks (Badanidiyuru et al., 2018; Immorlica et al., 2019; +Li et al., 2021) and online stochastic packing problems (Feld- +man et al., 2010; Agrawal & Devanur, 2014b; Devanur et al., +2011). Recently, Sankararaman & Slivkins (2021) proved a +logarithmic regret bound for LinCBwK when there exists a +problem-dependent gap between the reward of the optimal +action and the other actions. Instead of the gap assump- +tion, we require non-degeneracy of the stochastic contexts +(see Assumption 3 for a precise definition) to obtain a re- +gret bound sublinear in d and extends to the case when the +contexts are generated from J different class. +Amani et al. (2019) proposed a variant of LinCBwK where +the selected action must satisfy a single constraint with high +probability in all rounds, i.e., LinCBwK with anytime con- +straints. Moradipari et al. (2021) and Pacchiano et al. (2021) +proposed a Thompson sampling-based algorithm and an +upper confidence bound-based algorithm, respectively, for +LinCBwK with a single anytime constraint. Liu et al. (2021) +highlighted the difference between global and anytime con- +straints, and proposed an pessimistic-optimistic algorithm +for the anytime constraints. We focus on the global con- +straints; however, we note that the extension to the anytime +constraints is straightforward with minor modifications. +2.1. Notation +Let R+ denote the set of positive real numbers. For two +real numbers a, b ∈ R, we write a ∧ b := min{a, b} and +a ∨ b := max{a, b}. For a natural number N, let [N] = +{1, . . . , N}. +3. Linear Multi-period Packing Problem +Let [J] denote the set of classes with arrival probabilities +p = {pj}j∈[J], where pmin := minj∈[J] pj > 0. In each +round t ∈ [T], the covariates {x(j) +k,t ∈ [0, 1]d : k ∈ [K]} +and costs {c(j) +k,t ∈ [0, 1] : k ∈ [K]} are drawn from a class- +specific distribution Fj. We assume that the class arrival +probabilities p are known to the decision-maker; however, +the distributions {Fj}j∈[J] are not known. +At time t ∈ [T], the decision-maker observes an arrival of +the form (jt, {x(jt) +k,t , c(jt) +k,t : k ∈ [K]}), where jt ∈ [J] is +the arrived class. Upon observing the arrival, the decision- +maker can either take one of K different actions or skip the +arrival. When the arrival is skipped, the decision-maker does +not obtain any rewards or consume any resources. When +the decision-maker chooses an action at ∈ [K], the reward +and consumption of the resource are given by +E +� +r(jt) +at,t +��� Ht +� += +� +θ(jt) +⋆ +�⊤ +x(jt) +at,t ∈ [−1, 1], +E +� +b(jt) +at,t +��� Ht +� += +� +W (jt) +⋆ +�⊤ +x(jt) +at,t ∈ [0, 1]m, +for some unknown class-specific parameters θ(j) +⋆ +∈ [0, 1]d +and W (j) +⋆ +∈ [0, 1]d×m. The sigma algebra Ht is generated +by the class-specific variables {js, x(js) +k,s , c(js) +k,s , : s ∈ [t], k ∈ +[K]}, actions {as : s ∈ At}, consumption vectors {b(js) +as,s : +s ∈ At−1} and rewards {r(js) +as,t : s ∈ At−1}, where At +is the rounds admitted by the decision-maker until round +t. The process terminates at the horizon T or runs out of + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +budget B ∈ Rm ++ for some resources r ∈ [m]. The problem +reduces to LinCBwK when the number of class is J = 1 +and the costs are c(j) +k,t = 0 +Let ρ = B/T ∈ Rm ++ denote per-period budget for m re- +sources. Without loss of generality, one can assume that +ρ(r) = ρ for all r ∈ [m], By rescaling W (j) +⋆ . We assume +that ρ is known to the decision-maker and B is possibly un- +known at first, but known at the end of the round. This case +happens when the total budget B is difficult to count in the +early rounds. When ρ is not available, the decision-maker +requires B and T to compute ρ. However, this assumption +is more practical than in Agrawal & Devanur (2016) where +B and OPT must be known to the decision-maker. When +OPT is unknown, they estimate OPT with +√ +T number +of rounds, which requires the knowledge of T and budget +B = Ω( +√ +dT +3 +4 ). Instead of estimating OPT, we use ρ to +avoid the required budget B = Ω( +√ +dT +3 +4 ). +We benchmark the performance of the decision-maker’s pol- +icy relative to that of an oracle who knows the distributions +{Fj : j ∈ [J]} and the parameters {θ(j) +⋆ , W (j) +⋆ +: j ∈ [J]}, +but does not know the arrivals {(jt, x(j) +k,t, c(j) +k,t) : t ∈ [T]} +a-priori. In this case, the optimal static policy for the oracle +{π⋆(j) +k +: j ∈ [J], k ∈ [K]} is the solution to the following +optimization problem: +max +π(j) +k +J +� +j=1 +K +� +k=1 +pjπ(j) +k E(xk,ck)∼Fj +�� +θ(j) +⋆ +�⊤ +xk − ck +� +s.t. +j +� +j=1 +K +� +k=1 +pjπ(j) +k Exk∼Fj +�� +W (j) +⋆ +�⊤ +xk +� +≤ ρ, +K +� +k=1 +π(j) +k +≤ 1, ∀j ∈ [J], +π(j) +k +≥ 0, ∀j ∈ [J], ∀k ∈ [K], +(1) +Let π⋆ denote the optimal oracle policy. Then the expected +reward obtained by the oracle is +OPT := +T +J +� +j=1 +K +� +k=1 +pjπ⋆(j) +k +E(xk,ck)∼Fj +�� +θ(j) +⋆ +�⊤ +xk − ck +� +. +Let π := {π(j) +k,t : j ∈ [J], k ∈ [K], t ∈ [T]} denote the +adapted (randomized) control policy of the decision-maker, +i.e. she chooses action k ∈ [K] when the arrival at time +t ∈ [T] belongs to class j ∈ [J]. Note that �K +k=1 π(j) +k,t ≤ 1 +in order to allow the decision-maker to skip an arrival and +save the inventory for later use. Our goal is to compute a +policy that minimizes the cumulative regret Rπ +T defined as +Rπ +T := OPT − E +� T +� +t=1 +Rπ +t +� +, +where Rπ +t := �K +k=1 π(jt) +k,t E +�� +θ(jt) +⋆ +�⊤ +x(jt) +k,t − c(jt) +k,t +� +is the +expected reward obtained by policy π at time t. +For the LMMP problem, we assume the following regularity +conditions on the stochastic processes. +Assumption 1. (Sub-Gaussian and bounded errors) For +each t ∈ [T], the error of the reward ηk,t = r(jt) +k,t − +� +θ(jt) +⋆ +�⊤ +x(jt) +k,t is conditionally zero-mean σr-sub-Gaussian +for a fixed constant σr ≥ 0, i.e. E [exp (vηk,t)| Ht] ≤ +exp +� +v2σ2 +r +2 +� +for all v ∈ R. For the consumption vectors, +E +� +v⊤{b(jt) +k,t − (W (jt) +⋆ +)⊤x(jt) +k,t } +��� Ht +� +≤ exp( ∥v∥2 +2σ2 +b +2 +) for +all v ∈ Rm. +Assumption 2. (Independently distributed contexts and +costs) The set of contexts {x(j) +k,t : k ∈ [K]} and {c(j) +k,t : +k ∈ [K]} are generated independently over t ∈ [T]. The +contexts and cost in the same round and class can be corre- +lated with each other. +Assumption 3. (Positive definiteness of average covari- +ances) For each t ∈ [T] and j ∈ [J], there exists α > 0, +such that +λmin +� +E +� +1 +K +K +� +k=1 +x(j) +k,t +� +x(j) +k,t +�⊤ +�� +≥ α. +Assumptions 1 and 2 are standard in stochastic contex- +tual bandits with knapsacks literature (Agrawal & Devanur, +2016; Sankararaman & Slivkins, 2021; Sivakumar et al., +2022). In the multi-class case, Assumption 2 implies that all +the contexts are drawn independently over time steps, but +their distribution may vary depending on the class. Assump- +tion 3 implies that the density of the covariate distribution is +non-degenerate. Recent contextual bandit literature (without +constraints) exploits Assumption 3 to improve the depen- +dency of d on the regret bound (Bastani & Bayati, 2020; +Kim et al., 2021; Bastani et al., 2021; Oh et al., 2021). The +contexts with independent Gaussian perturbation used in +Kannan et al. (2018); Sivakumar et al. (2020; 2022) satisfy +the Assumption 3. +4. Proposed Method +In this section, we present our proposed estimator for the +parameters {θ(j) +⋆ , W (j) +⋆ +: j ∈ [J]} and the proposed bandit +policy. +4.1. Proposed Estimator +In sequential decision-making problems with contexts, the +decision-maker observes the contexts for all actions, but +the reward for only selected actions, i.e. the rewards for +unselected actions remain missing. Kim & Paik (2019); + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Dimakopoulou et al. (2019); Kim et al. (2021) use doubly +robust (DR) method to handle the missing rewards for the +linear contextual bandits. However, extensions to LinCBwK +or LMMP problem have not explored yet. +We adapt the DR method to the LMMP problem. For each +n ∈ N, let τ(n) be the round when the n-th admission +happens (recall the bandit policy allows for skipping some +arrivals). Clearly, n ≤ τ(n) < τ(n + 1) holds. Let +Θ⋆ := +� +� +� +� +θ(1) +⋆ +... +θ(J) +⋆ +� +� +� +� , W⋆ := +� +� +� +� +W (1) +⋆ +... +W (J) +⋆ +� +� +� +� , ˜Xk,n := +� +� +� +� +� +� +0d +... +x +(jτ(n)) +k,τ(n) +0d +� +� +� +� +� +� +denote the stacked parameter vectors, and zero padded con- +texts where x +(jτ(n)) +k,τ(n) is located after the j − 1 of 0d vectors. +Then the score for the ridge estimator for Θ⋆ at round τ(n) +is: +n +� +ν=1 +� +r +(jτ(ν)) +aτ(ν),τ(ν) − Θ⊤ ˜Xaτ(ν),ν +� +˜Xaτ(ν),ν += +n +� +ν=1 +K +� +k=1 +I +� +aτ(ν) = k +� � +r +(jτ(ν)) +k,τ(ν) − Θ⊤ ˜Xk,ν +� +˜Xk,ν, +where Θ ∈ RJ·d. Dividing the score by the probability +π +(jτ(ν)) +k,τ(ν) gives the inverse probability weighted (IPW) score, +n +� +ν=1 +K +� +k=1 +I +� +aτ(ν) = k +� +π +(jτ(ν)) +k,τ(ν) +� +r +(jτ(ν)) +k,τ(ν) − Θ⊤ ˜Xk,ν +� +˜Xk,ν. +To obtain the DR score, Bang & Robins (2005); Kim et al. +(2021) proposed to subtract the nuisance tangent space gen- +erated by an imputed estimator ˇΘ: +n +� +ν=1 +K +� +k=1 +I +� +aτ(ν) = k +� +π +(jτ(ν)) +k,τ(ν) +� +˜X⊤ +k,ν ˇΘ − ˜X⊤ +k,νΘ +� +˜Xk,ν, +from the IPW score. Then the following DR score +n +� +ν=1 +K +� +k=1 +� +r +DR(jτ(ν)) +k,τ(ν) +− ˜X⊤ +k,νΘ +� +˜Xk,ν, +(2) +is obtained where +rDR( ˇΘ) +k,ν +:=I +� +aτ(ν)=k +� +π +(jτ(ν)) +k,τ(ν) +r +(jτ(ν)) +k,τ(ν) + +� +� +�1−I +� +aτ(ν)=k +� +π +(jτ(ν)) +k,τ(ν) +� +� +� +˜X⊤ +k,ν ˇΘ. +(3) +The score (2) has a similar form with the score equation +for the ridge estimator. The difference with the ridge es- +timator is that it uses contexts for all actions k ∈ [K] +with the pseudo-reward rDR( ˇΘ) +k,ν +which is unbiased, i.e., +E[rDR( ˇΘ) +k,ν +] = E[r +(jτ(ν)) +k,τ(ν) ], for any given ˇΘ ∈ RJ·d. Adding +the ℓ2 regularization norm and solving (2) leads to the DR +estimator: +� n +� +ν=1 +K +� +k=1 +˜Xk,ν ˜X⊤ +k,ν+IJ·d +�−1� n +� +ν=1 +K +� +k=1 +˜Xk,νrDR( ˇΘ) +k,τ(ν) +� +. +The main advantage of the DR estimator is that it uses +contexts from all K actions. However, in our policy, some +π +(jτ(ν)) +k,τ(ν) can be zero, and therefore, the pseudo-reward (3) is +not defined. To handle this problem, we propose to introduce +a random variable. After taking an action at round τ(ν) +and observing the selected action aτ(ν), the decision-maker +samples hν from the distribution: +φk,ν:=P +� +hν = k| Hτ(n) +� += +� +� +� +1− +16(K−1) log( dJ +δ ) +λmin(Fν) +k=aτ(ν) +16 log( dJ +δ ) +λmin(Fν) +k̸=aτ(ν) +(4) +where Fν := �ν,K +i,k=1 ˜Xk,i ˜X⊤ +k,i+16d(K −1) log +� dJ +δ +� +IJ·d +is the Gram matrix of contexts from ν admitted rounds +and δ ∈ (0, 1) is the confidence level. We would like to +emphasize that hν is sampled after observing the actions +aτ(ν) and does not affect the policies until round τ(ν). +Sampling the random variables hν after choosing actions +is motivated by bootstrap methods (Efron & Tibshirani, +1994) and resampling methods (Good, 2006). To obtain the +unbiased pseudo-rewards similar to (3), we resample the +action with another non-zero probabilities. The probabil- +ities {φk,ν : k ∈ [K]} is designed to control the level of +exploration and exploitation for future rounds based on the +ratio of confidence level to the number of admitted rounds. +When the minimum eigenvalue of Fν is small compared +to log(1/δ), the distribution of hν is less concentrated on +aτ(ν) and tends to explore other actions. As ν increases, the +probabilities {φk,ν : k ∈ [K]} concentrates on aτ(ν), and +the decision-maker tends to exploit. +Since we obtain non-zero probabilities {φk,ν : k ∈ [K], ν ∈ +[n]}, we define novel unbiased pseudo-rewards: +˜rk,ν :=I (hν =k) +φk,ν +r +(jτ(ν)) +k,τ(ν) + +� +1− I (hν =k) +φk,ν +� +˜X⊤ +k,νˇΘn, (5) +where the imputation estimator ˇΘn is an IPW estimator with + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +new probabilities: +ˇΘt :=A−1 +n +� � +ν∈Ψn +K +� +k=1 +I (hν =k) +φk,ν +˜Xk,νr +(jτ(ν)) +k,τ(ν) ++ +� +ν /∈Ψn +˜Xaτ(ν),νr +(jτ(ν)) +aτ(ν),τ(ν) +� +, +An := +� +ν∈Ψn +K +� +k=1 +I (hν =k) +φk,ν +˜Xk,ν ˜X⊤ +k,ν ++ +� +ν /∈Ψn +˜Xaτ(ν),ν ˜X⊤ +aτ(ν),ν + IJ·d, +Ψn := +� +ν ∈ [n] : hν = aτ(ν) +� +. +The set Ψn is introduced because we cannot observe +I(hν=k) +φk,ν +r +(jτ(ν)) +k,τ(ν) in case of hν ̸= aτ(ν). In other words, we +use the pseudo-rewards in (5) only at the rounds that satisfy +hν = aτ(ν). Then our estimator with n admitted samples is +defined as +�Θn :=V −1 +n +� +� +� +� +ν∈Ψn +K +� +k=1 +˜Xk,ν˜rk,ν + +� +ν /∈Ψn +˜Xaτ(ν),νr +(jτ(ν)) +aτ(ν),τ(ν) +� +� +� +Vn := +� +ν∈Ψn +K +� +k=1 +˜Xk,ν ˜X⊤ +k,ν + +� +ν /∈Ψn +˜Xaτ(ν),ν ˜X⊤ +aτ(ν),ν +IJ·d. +(6) +Analogous to the construction of (6), we can also define the +estimator for the resource consumption parameters {W (j) +⋆ +: +j ∈ [J]}, +� +Wn :=V −1 +n +� � +ν∈Ψn +K +� +k=1 +˜Xk,ν ˜b⊤ +k,ν+ +� +ν /∈Ψn +˜Xaτ(ν),νb +(jτ(ν))⊤ +aτ(ν)τ(ν) +� +, +(7) +where the pseudo-consumption vectors and the imputation +estimator are +˜bk,ν := I (hν =k) +φk,ν +b +(jτ(ν)) +aτ(ν),τ(ν)+ +� +1− I (hν =k) +φk,ν +� +ˇ +W⊤ +n ˜Xk,ν, +ˇ +Wn := A−1 +n +� � +ν∈Ψn +K +� +k=1 +I (hν =k) +φk,ν +˜Xk,ν +� +b +(jτ(ν)) +k,ν +�⊤ ++ +� +ν /∈Ψn +˜Xaτ(ν),ν +� +b +(jτ(ν)) +aτ(ν),τ(ν) +�⊤ +� +. +The two estimators use the novel Gram matrix Vn defined +in (6) consist of contexts from all K actions. Now, we +present estimation error bounds normalized by the novel +Gram matrix Vn. +Theorem 4.1. (Self-normalized bound for the estimator) +Suppose Assumptions 1 and 2 hold. For each t ∈ [T], +denote nt the number of admitted arrivals until round t and +Ψnt := {ν ∈ [nt] : hν = aτ(ν)}, where hν is defined in +(4). Suppose Fnt := �nt +ν=1 +�K +k=1 ˜Xk,ν ˜X⊤ +k,ν + 16d(K − +1) log Jd +δ IJ·d satisfies +λmin(Fnt)≥12Kd +� nt +� +ν=1 +48(K−1) log +� Jd +δ +� +λmin(Fν) ++2 log Jd +δ +� +, +(8) +for δ ∈ (0, 1). For each r ∈ [m] , let � +Wnt,r and W⋆,r +be the r-th column of � +Wnt and W⋆, respectively. Denote +βσ(δ) := 8 +√ +Jd + 96σ +� +Jd log 4 +δ. Then with probability +at least 1 − 4(m + 1)δ, +����Θnt − Θ∗��� +Vnt +≤βσr(δ), +max +r∈[m] +���� +Wnt,r − W⋆,r +��� +Vnt +≤βσb(δ). +(9) +The widely used self-normalized bound in Abbasi-Yadkori +et al. (2011) uses the Gram matrix consisting of selected +contexts only, while our bounds are normalized by Vnt +This change in the Gram matrix enables us to develop a +fast convergence rate. The condition (8) is required for +the eigenvalues of the Gram matrix Fnt to be large so that +the probability φaτ(ν),ν is large and the estimators use the +pseudo rewards and pseudo consumption vectors for most +of the rounds. We show in Lemma 5.3 that the condition (8) +requires at most rounds logarithmic in T, and does not affect +the main order of the regret bound. +Using the novel estimators, we define the estimates for +utility and resource consumption. Denote C(j) +t +:= {s ∈ [t] : +js = j} and +�u(j) +k,t := +���C(j) +t +��� +−1 � +s∈C(j) +t +�� +�θ(j) +t−1 +�⊤ +x(j) +k,s − c(j) +k,s +� +, +�b(j) +k,t := +���C(j) +t +��� +−1 � +s∈C(j) +t +� +� +W (j) +t−1 +�⊤ +x(j) +k,s. +(10) +The estimates (10) use the average of contexts in the same +class to estimate the expected value over the context dis- +tribution. In this way, the decision-maker effectively uses +previous contexts in all rounds including the skipped rounds. +Next, we establish the convergence rate for the estimators +�u(j) +k,t and �b(j) +k,t. +Theorem 4.2. (Convergence rate for the estimates) Suppose +Assumptions 1-3 hold. Denote the expected utility u⋆(j) +k +:= +E(xk,ck)∼Fj +�� +θ(j) +⋆ +�⊤ +xk − ck +� +and consumption b⋆(j) +k +:= +Exk∼Fj +�� +W (j) +⋆ +�⊤ +xk +� +. Set γt,σ(δ) := +16√ +J log(JKT ) +√ +t ++ +4 +√ +2βσ(δ) +√nt +, where nt is the number of admitted arrivals until + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +round t and βσ(δ) is defined in Theorem 4.1. Suppose +t ≥ 8dα−1p−1 +min log JT, δ ∈ (0, T −1) and Fnt satisfies (8). +Then with probability at least 1 − 4(m + 1)δ − 7T −1, +� +� +� +� +J +� +j=1 +pj max +k∈[K] +���u⋆(j) +k +− �u(j) +k,t+1 +��� +2 +≤ γt,σr(δ), +� +� +� +� +J +� +j=1 +pj max +k∈[K] +���b⋆(j) +k +− �b(j) +k,t+1 +��� +2 +∞ ≤ γt,σb(δ). +(11) +The convergence rate of the estimates is O( +√ +Jdn−1/2 +t +). In +deriving the fast rate, the novel Gram matrix Vnt plays a +significant role. To prove Theorem 4.2, we bound the sum +of squared maximum prediction error as follows: +1 +nt +� +s∈Ψnt +max +k∈[K] +�� +θ(j) +⋆ −�θ(j) +t +�⊤ +x(j) +k,s +�2 += 1 +nt +� +s∈Ψnt +max +k∈[K] +� +θ(j) +⋆ −�θ(j) +t +� � +x(j) +k,sx(j) +k,s +�⊤� +θ(j) +⋆ −�θ(j) +t +� +≤ 1 +nt +� +s∈Ψnt +� +θ(j) +⋆ −�θ(j) +t +� � K +� +k=1 +x(j) +k,sx(j) +k,s +�⊤� +θ(j) +⋆ −�θ(j) +t +� +≤ 1 +nt +���θ(j) +⋆ −�θ(j) +t +��� +2 +Vnt +. +Such a bound is not available if the Gram matrix is con- +structed using only contexts corresponding to selected ac- +tions. In this way, we obtain faster convergence rate for the +estimates for utility and consumption vectors. +4.2. Proposed Algorithm +Let (K + 1)-th action denote skipping the arrival and +π(j) +K+1,t := P (Skip the round t| Ht) denote the probabil- +ity of skipping the arrival. Since the decision-maker must +choose an action or skip the round, we have �K+1 +k=1 π(j) +k,t = 1. +When the decision-maker skips round t, we set x(j) +K+1,t := 0, +c(j) +K+1,t := 0, and b(j) +K+1,t := 0. In round t, the randomized +bandit policy is given by the optimal solution of the follow- +ing optimization problem: +max +π(jt) +k,t +K+1 +� +k=1 +π(jt) +k,t +� +�u(jt) +k,t + γt−1,σr(δ) +√pjt +I (k ∈ [K]) +� +, +s.t. +K+1 +� +k=1 +π(jt) +k,t +� +�b(jt) +k,t − γt−1,σb(δ) +√pjt +1m +� +≤ ρt ∨ 0, +K+1 +� +k=1 +π(jt) +k,t = 1, +π(jt) +k,t ≥ 0, +∀k ∈ [K + 1], +(12) +Algorithm 1 Allocate to the Maximum First algorithm +(AMF) +INPUT: confidence lengths γθ, γb > 0, confidence level +δ ∈ (0, 1). +Initialize F0 := 16d(K − 1) log Jd +δ IJ·d, ρ1 := ρ, �Θ0 := +0J·d, � +W0 := 0J·d×m +for t = 1 to T do +Observe arrival (jt, {x(jt) +k,t , c(jt) +k,t }k∈[K]). +if Ft−1 does not satisfy (8) then +Take action at = arg maxk∈[K] ρ∥�b(jt) +k,t ∥−1 +∞ . +else +Compute �u(jt) +k,t and �b(jt) +k,t with �θ(jt) +t−1 and � +W(jt) +t−1. +Compute ˜u(jt) +k,t := �u(jt) +k,t + +γθ +√nt and ˜b(jt) +k,t := �b(jt) +k,t − +γb +√nt 1m. +Take action at with the policy �π(jt) +1,t , . . . , �π(jt) +K+1,t de- +fined in (13). +end if +if at ∈ [K] then +Observe r(jt) +at,t and b(jt) +at,t, then estimate �Θt and � +Wt +as in (6) and (7), respectively. +Update Ft = Ft−1 + �K +k=1 ˜Xk,t ˜X⊤ +k,t. +end if +Update available resource ρt+1 = ρt + ρ − b(jt) +at,t. +if �t +s=1 b(js) +as,s ≥ Tρ then +Exit +end if +end for +where ρt := tρ − �t−1 +s=1 b(js) +as,s is the difference between +the used resources and planned budget until round t. The +algorithm is optimistic in that it uses upper confidence bound +(UCB) in rewards and lower confidence bound (LCB) in +consumption while it regulates the consumption to be less +than tρ with ρt. In this way, the problem (12) balances +between admitting the arrivals and saving the resources for +later use. Next, we show that the optimal solution (12) is +available in a closed-form. +Lemma 4.3. (Optimal policy for bandit) Let ˜u(jt) +k,t +:= +�u(jt) +k,t ++ p−1/2 +jt +γt−1,σr(δ)I (k ∈ [K]) and ˜b(jt) +k,t (r) +:= +�b(jt) +k,t (r) − p−1/2 +jt +γt−1,σb(δ), for r ∈ [m]. For i ∈ [K + 1], +let ˜u(jt) +k⟨i⟩,t be an sequence of ordered variables of ˜u(jt) +k,t in +decreasing order, i.e. ˜u(jt) +k⟨1⟩,t ≥ ˜u(jt) +k⟨2⟩,t ≥ · · · ≥ ˜u(jt) +k⟨K+1⟩t. +When there is a tie between ˜u(jt) +k⟨i⟩,t and ˜u(jt) +k⟨i+1⟩,t, the index +k⟨i⟩ with the higher value for +� +� min +r∈[m] +ρt(r) ∨ 0 − �i−1 +h=1 �π(jt) +k⟨h⟩,t˜b(jt) +k⟨h⟩,t(r) +˜b(jt) +k⟨h⟩,t(r) +� +� + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +goes first. Then the policy defined as, +�π(jt) +k⟨1⟩,t = +� +� min +r∈[m] +ρt(r)∨0 +˜b(jt) +k⟨1⟩,t(r) +� +� ∧ 1, +�π(jt) +k⟨i⟩,t = +� +�min +r∈[m] +ρt(r)∨0−�i−1 +h=1�π(jt) +k⟨h⟩,t˜b(jt) +k⟨h⟩,t(r) +˜b(jt) +k⟨i⟩,t(r) +� +� +∧ +� +1 − +i−1 +� +h=i +�π(jt) +k⟨h⟩,t +� +, ∀i ∈ [2, K + 1], +(13) +is the optimal solution to (12). +Since the objective function of (12) is linear, we can obtain +the maximum value by permuting the objective coefficients +in decreasing order and allocating the greatest possible prob- +ability value in decreasing order of the objective coefficients. +Note that �π(jt) +k,t is automatically set to zero when the utility +is negative. This is because of the probability of skipping +the arrival, �π(jt) +K+1,t = 1−�l−1 +h=1 �π(jt) +k⟨h⟩,t, when ˜u(jt) +K+1,t is the +l-th largest weighted utility function and all the remaining +probability is allocated to �π(jt) +K+1,t. Therefor, the probabili- +ties for actions k with ˜u(jt) +k,t < ˜u(jt) +K+1,t := 0 are all zero. +Our proposed algorithm, Allocate to the Maximum First +(AMF) is presented in Algorithm 1. The algorithm first ex- +plores with the least consumption action until the eigenvalue +condition for the estimator (8) holds. In each round of ex- +ploration, the Gram matrix of all actions is added to Fnt, +and any choice of action increases the eigenvalue of Fnt. +Once the condition (8) holds, the algorithm solves the prob- +lem (12) by computing the closed-form policy (13). The +computational complexity of our algorithm is discussed in +Appendix A.4. +5. Regret Analysis +In this section, we present our regret bound and regret anal- +ysis for the AMF algorithm. +Theorem 5.1. (Regret bound of AMF) Suppose Assumptions +1-3 hold. Let Mα,p,T := 1152α−2p−2 +min log T + 96α−1p−1 +min +and Cσ(δ) := 8 +√ +2(8 + 96σ +� +log 4 +δ ). Suppose T and +ρ satisfies T ≥ 8dα−1p−1 +min log JdT, and ρ ≥ +� +Jd/T. +Setting γθ = 16√J log JKT + 4 +√ +2βσr(δ) and γb = +16√J log JKT + 4 +√ +2βσb(δ), the regret bound of AMF is +R�π +T ≤ +� +2+ OPT +ρT +�� +4d log JdT +αpmin ++2dMα,p,T log Jd +δ +15 ++ +� +96 +� +log JKT +3Cσr∨σb(δ) +�� +JdT log T +10mT 3δ +� +. +For δ ∈ (0, m−1T −3), the regret bound is +R�π +T = O +�OPT +Tρ +� +JdT log mJKT log T +� +. +(14) +The regret bound (14) holds when the hyperparameter δ = +m−1T −3, which requires the knowledge of T. However, +in practice, selecting another value of δ does not affect the +performance of the algorithm. We provide the discussion on +the sensitivity to the hyperparameter choice in Section 6.3. +Setting B = Tρ, the main term of the regret bound is +˜O(OPT/B +√ +JdT) for B = Ω( +√ +JdT). The sublinear +dependence of the regret bound on J, d, and T is a direct +consequence of the improved ˜O( +� +Jd/nt) convergence rate +for the parameter estimates. Agrawal & Devanur (2016) +establish a regret bound Rπ +T = ˜O(OPT/B · d +√ +T) for the +LinCBwK when B = Ω( +√ +dT 3/4). Our bound for LMMP +(which subsumes LinCBwK as a special case) is improved +by a +√ +d factor, and is valid under budget constraints that +relaxed from Ω( +√ +dT +3 +4 ) to Ω( +√ +dT +1 +2 ). +For the proof of the regret bound, we first present the lower +bound of the reward obtained by our algorithm. +Lemma 5.2. Let ˜u(j) +k,t and �b(j) +k,t be the estimates defined in +(10). Denote �π the policy of AMF. Define the good events, +Et := +� +˜u(j) +k,t and �b(j) +k,t satisfies (11). +� +, +Mt := {Fnt satisfies (8).} , +(15) +and Gt := Et ∩ Mt−1. Let τ be the stopping time for the +algorithm and ξ := inft∈[T ] {Mt−1 ∩ {ρt > 0}} be the +starting time after the exploration for condition (8). Then, +the total reward +E +� T +� +t=1 +R�π +t +� +≥ OPT +T +E [τ − ξ]− +� +2+ OPT +ρT +� T +� +t=1 +P(Gc +t ) +−2 +� +1+ OPT +ρT +�� +� +� +�TE +� T +� +t=1 +γt−1,σr∨σb(δ)2I (at ∈ [K]) +� +. +The lower bound consists of three main terms. The first +term OP T +T +E [τ − ξ] relates to the time span for which the +algorithm uses the optimal policy (13). The second term +(2+ OP T +ρT ) �T +t=1P (Gc +t ) is the sum of the probability of bad +events Mc +t−1 over which the minimum eigenvalue of the +Gram matrix Fnt is not large enough for the fast conver- +gence rate, and the event Ec +t over which the estimator goes +out of the confidence interval. And, the third term con- +sists of the sum of confidence lengths for the reward and +consumption. +The following result bounds τ, ξ and the sum of bad events +{Mc +t : t ∈ [T]}. + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Lemma 5.3. Suppose Assumptions 1-3 holds and ρ > +� +Jd/T Let Mα,p,T and γt,σ(δ) denote the variables +defined in Theorem 5.1 and Theorem 4.2, respectively. +Then, for any δ ∈ (0, 1/T 2), the starting time ξ := +inft∈[T ]{Mt−1 ∩ {ρt > 0}} and the stopping time τ of +the AMF algorithm is bounded as +E [ξ] ≤ 1+dMα,p,T log +� Jd +δ +� ++T 2δ +ρ ++ 1, +E[T −τ]≤ 4(m + 1)Tδ + 7 + 2γ1,σb(δ) +ρ +, +and for Mt defined as in (15), +T +� +t=1 +P +� +Mc +t−1 +� +≤ T 2δ + dMα,p,T log +�Jd +δ +� +. +The regret bound follows from bounding the probability of +Ec +t with Theorem 4.2 and showing that the sum of square +of γt,σ(δ) is O(Jd log T). The bound holds because the +summation of γt,σ(δ)2 = ˜O( Jd +nt ) over the rounds that at ∈ +[K] happens is �nT +n=1 O(Jd/n) = O(Jd log T). +6. Numerical results +We report the cumulative regrets for given budgets. For the +computation of the regret, we use the following settings. For +each round t ∈ [T] there exists the optimal action whose +reward is 1 with consumption ρ, while the reward of other +actions is less than 1 and the consumption is possibly greater +than ρ. In this case, we can compute the instantaneous regret +by subtracting the reward of a selected action from 1. Detail +settings of the parameter and contexts are in Appendix A.1. +6.1. Regret R�π +T as a function of d +Figure 1 plots log(R�π +T ) vs. log(d) for a single-class (J = +1) LMMP for T = {5000, 20000} and the budget B = +√ +dT, where our ˜O( OP T +B +√ +JdT) regret bound implies that +log(R�π +T ) is constant over d. The regression line on the +plot is nearly flat and the slope of the best fit line is 0.136 +(resp. 0.008) for T = 5000 (resp. T = 20000). The weak +increase in T = 5000 is captured by the O(d log JdmT) +term in our bound, which diminishes for large T. +6.2. Comparison of AMF with OCO +In order to compare AMF with OCO (Agrawal & Devanur, +2016), we set the costs c(1) +k,t = 0 and J = 1. The hyperpa- +rameters for AMF were set to γθ = 1, γb = 1 and δ = 0.01. +Figure 2(a) (resp. (b)) plots the cumulative regret of the +two algorithms with budget B = +√ +dT +3 +4 (resp. B = +√ +dT). +Note that OCO requires a minimum budget B = +√ +dT +3 +4 +whereas AMF requires a lower minimum budget of B = +Figure 1. Logarithm of cumulative regret of the proposed AMF +algorithm on various dimension d when the per-period budget is +ρ = +� +d/T. The gray (resp. black) line is the best fit line on the +points when T = 5000 (resp. T = 20000). +(a) Regret comparison with budget B = +√ +dT 3/4 +(b) Regret comparison with budget B = +√ +dT +Figure 2. Regret of AMF and OCO algorithms for K = 20 and +m = 20. The line and shade represent the average and standard +deviation based on 20 independent experiments. Additional results +on different K and m are in Section A.2. +√ +dT. The regret lines cross because AMF is allowed to skip +arrivals whereas OCO does not skip arrivals. The sudden +bend points at the end of the round in OCO show that it +runs out of budget and has regret = 1. In all cases, our +algorithm performs better and the performance gap increases +as d increases. Note that regret plot for OCO never flattens +out for most cases, where the regret of AMF flattens as t +increases. This is because our new estimator, that uses +contexts from all actions with unbiased pseudo-rewards (5) +for unselected actions, has significantly faster convergence +rate as compared with the estimator used in OCO. +6.3. Sensitivity Analysis +Our proposed AMF algorithm has three hyperparameters: γθ, +γb and δ. The choice of hyperparameters is not sensitive +because the effect of γθ and γb diminishes fast by n−1/2 +t +term and our policy finds the order of the utilities rather than +their absolute values. For δ, which controls the sampling +probabilities (4) in estimators and the exploration rounds +in (8), it also has small effect. This is because the minimum +eigenvalue of Fnt increases in Ω(nt)-rate and reduces the +effect of log 1 +δ terms in (4) and (8). Therefore, our algorithm +guarantees the similar performance for other hyperparame- +ters than specified in Theorem 5.1. For details including the +numerical results and specific recommendations for choos- +ing the hyperparameters, see Appendix A.3. + +7.5 +7.0 +6.5 +. +6.0 +5.5 - +O +5.0 +1.0 +1.5 +2.0 +2.5 +3.0 +log d2000 +OCO +1500 +AMF +1000 - +500 - +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision points3500 +OCo +3000 +AMF +2500 +2000 +1500 - +1000- +500 +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision pointsImproved Algorithms for Multi-period Packing Problems with Bandit Feedback +References +Abbasi-Yadkori, Y., P´al, D., and Szepesv´ari, C. Improved +algorithms for linear stochastic bandits. In Advances in +Neural Information Processing Systems, pp. 2312–2320, +2011. +Agrawal, S. and Devanur, N. Linear contextual bandits with +knapsacks. Advances in Neural Information Processing +Systems, 29, 2016. +Agrawal, S. and Devanur, N. R. +Bandits with concave +rewards and convex knapsacks. 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User-friendly tail bounds for sums of random +matrices. Foundations of computational mathematics, 12 +(4):389–434, 2012. +Tropp, J. A. An introduction to matrix concentration inequal- +ities. Foundations and Trends® in Machine Learning, 8 +(1-2):1–230, 2015. + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +(a) Regret comparison under K = 10 and m = 10 +(b) Regret comparison under K = 20 and m = 10 +(c) Regret comparison under K = 10 and m = 20 +(d) Regret comparison under K = 20 and m = 20 +Figure 3. Regret comparison of AMF and OCO algorithms under B = dT 3/4. The line and shade represent the average and standard +deviation based on 20 repeated experiments. +A. Supplementary for Experiments +A.1. Settings of Parameters and Contexts for Regret Computation +For numerical experiments, we devise a setting where explicit regret computation is available. We set J = 1 and c(1) +k,t = 0 for +OCO to be compatible with the setting. For x ∈ R+, let ⌈x⌉ be the smallest integer greater than equal to x. For parameters, +we set θ⋆ = (−1, · · · , −1, ⌈d/2⌉−1, · · · , ⌈d/2⌉−1) and +W⋆ = +� +� +� +� +� +� +� +� +� +� +ρ⌈d/2⌉−1 +· · · +ρ⌈d/2⌉−1 +... +· · · +... +ρ⌈d/2⌉−1 +· · · +ρ⌈d/2⌉−1 +ρ +· · · +ρ +... +... +... +ρ +· · · +ρ +� +� +� +� +� +� +� +� +� +� +, +where the ⌈d/2⌉−1 and ρ⌈d/2⌉−1 terms are in the first ⌈d/2⌉ entries. +For contexts, we set the optimal action as +(0, · · · , 0, 1, · · · , 1), and for other actions, we set (U0,0.05, · · · , U0,0.05, U0,−0.05, · · · , U0,−0.05),where Ua,b the uniform +random variable supports on [a, b]. Then we have the optimal arm with reward 1 and consumption ρ, while other arms have +reward less than 1 and consumption more than ρ. +A.2. Additional Results on Regret Comparison. +Figure 3 (a)-(d) show the regret comparison of AMF and OCO on different terms of K = 10, 20, m = 10, 20, and B = dT 3/4. +Similar to the results in Figure 2(a), our algorithm has less regret than OCO in all cases, especially at the end of the rounds. +The crossing line occurs when our algorithm skips in the middle round when ρt < 0 while OCO does not skip until the +inventory runs out. +Figure 4 (a)-(d) show the regret of AMF and OCO algorithm on various K = 10, 20 and m = 10, 20 with budget B = +√ +dT. +Even in the smaller budget, our algorithm AMF does not run out the inventory and gains more reward than OCO. The gap of +the performance tends to be larger than B = +√ +dT +3 +4 case. + +1000 +d=10, K=20,m=20 +OCO +AMF +800 +600 +400 +200 : +0 · +0 +2000 +4000 +6000 +8000 +10000 +Decision points1000 +d=20, K=20, m=20 +OCO +AMF +800 +600 +400 +200 : +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision points1000 +d=10,K=10, m=10 +OCO +AMF +800 +600 +400 +200 +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision points1000 +d=20,K=10,m=10 +OCO +AMF +800 +600 - +400 - +200 - +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision points1000 +d=10,K=20, m=10 +OCO +AMF +800 +600 +400 +200 : +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision points1000 +d=20,K=20,m=10 +OCO +AMF +800 +600 +400 +200 - +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision points1000 +d=10,K=10, m=20 +OCO +AMF +800 +600 +400 +200 : +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision points1000 +d=20,K=10, m=20 +OCO +AMF +800 +600 +400 +200 : +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision pointsImproved Algorithms for Multi-period Packing Problems with Bandit Feedback +(a) Regret comparison under K = 10 and m = 10 +(b) Regret comparison under K = 20 and m = 10 +(c) Regret comparison under K = 10 and m = 20 +(d) Regret comparison under K = 20 and m = 20 +Figure 4. Regret comparison of AMF and OCO algorithms under B = +√ +dT. The line and shade represent the average and standard +deviation based on 20 repeated experiments. +(a) On various γθ +(b) On various γb +(c) On various δ +Figure 5. The reward and inventory of AMF on various hyperparameters γθ, γb and δ. The solid (resp. dashed) line represents the reward +(resp. inventory). The line and shade represent the average and standard deviation based on 10 repeated experiments, respectively. +A.3. Sensitivity Analysis +In this experiment, we present the sensitivity of our algorithm to various hyperparameters. The number of classes is J = 3 +with a uniform prior p = (1/3, 1/3, 1/3)⊤ and every d = 5 elements of K = 10 contexts are generated from the uniform +distribution on [ kj +KJ − 1, kj +KJ + 1] for k ∈ [K] and j ∈ [J]. The costs are generated from the uniform distribution on +[ k(J−j+1)−1 +KJ +, k(J−j+1)+1 +KJ +] for k ∈ [K] and j ∈ [J]. Each element of θ(j) +⋆ +and W (j) +⋆ +is generated from U0,1 and fixed +throughout the experiment. The generated rewards and consumption vectors are not truncated to one to impose more +variability, because our algorithm does not show apparent sensitivity on bounded rewards and consumption vectors. The +budget is ρ = dT −1/2 with a time horizon of T = 2000. +In our algorithm, there are three hyperparameters: (i) a confidence bound for the reward γθ, (ii) a confidence bound for +the consumption γb and (iii) confidence level δ which affects the minimum eigenvalue condition (8). Figure 5(a) and 5(b) +show the reward and inventory of our algorithm on various γθ ∈ {0.01, 0.1, 1} and γb ∈ {0.01, 0.1, 1}. Outside of the +hyperparameter regions, the variability of the reward and the inventory of the algorithm are hardly visible. The algorithm +consumes the budget earlier than previous experiments because the consumption vector is not bounded to 1. As γθ and +γb increase the algorithm is more optimistic and admits the arrival more often, which leads to faster consumption of the +resource. Increasing γθ has small effect on the inventory because the algorithm automatically skips when ρt < 0, i.e., when +the consumption is too fast. However, when γb increases, the LCB of consumption is small and the algorithm uses more + +2500 +d=10,K=10,m=10 +OCO +AMF +2000 +1500 - +1000 - +500 - +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision points2500 +d=20,K=10,m=10 +OCO +AMF +2000 +1500 - +1000 - +500 - +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision points2500 +d=10,K=20,m=10 +OCO +AMF +2000 +1500 - +1000 - +500 - +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision points2500 +d=20,K=20,m=10 +OCO +AMF +2000 +1500 - +1000 - +500 - +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision points2500 +d=10,K=10,m=20 +OCO +AMF +2000 +1500 - +1000 - +500 - +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision points2500 +d=20, K=10,m=20 +OCO +AMF +2000 +1500 - +1000 - +500 - +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision points2500 +d=10,K=20,m=20 +OCO +AMF +2000 +1500 - +1000 - +500 - +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision points2500 +d=20, K=20, m=20 +OCO +AMF +2000 +1500 - +1000 - +500 - +0 +0 +2000 +4000 +6000 +8000 +10000 +Decision points700 +600 +500 +400 +300 +200 +Ye=0.01 +100 +Ye=0.10 +0 +Ye=1.00 +0 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +Decision points700 +600 +500 +400 +300 +200 +Yb=0.01 +100 +Yb=0.10 +0 +Yb=1.00 +0 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +Decision points700 +600 +500 +400 +300 +200 +6=1e-01 +§=1e-04 +100 +6=1e-07 +0 +0 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +Decision pointsImproved Algorithms for Multi-period Packing Problems with Bandit Feedback +resource than tρ. For the specific value of the hyperparameters we recommend to use grid search on γθ × γb ∈ [0, 1]2 to +maximize the reward. +Figure 5(c) shows the reward and inventory of AMF on various δ ∈ {10−1, 10−4, 10−7}. When δ ≥ 10−1 (resp. δ ≤ 10−7) +the reward and inventories are same with δ = 10−1 (resp. δ = 10−7). As δ decreases, the threshold for condition (8) +increases and the algorithm explores more with minimum possible consumption. This results in the slower consumption +of the resource. However, we recommend using δ = 0.1, which is greater than the specified value in Theorem 5.1 for the +algorithm to start using its policy in earlier rounds. +A.4. Computational Complexity of AMF +The computational complexity of our algorithm is ˜O(d3mKT + Jd3T) where the main order occurs from updating the +estimators and computing the eigenvalues of J symmetric positive-definite matrix Fnt. Note that Computing estimators +does not depend on J because the algorithm updates only jt-th variables for each t ∈ [T]. +B. Missing Proofs +B.1. Proof of Theorem 4.1 +Proof. Because the construction of �Θt and � +Wt is the same, the bound for the � +Wt follows immediately from the bound for +�Θt by replacing {r +(jτ(ν)) +aτ(ν),τ(ν) : ν ∈ [nt]} with m entries of {b +(jτ(ν)) +aτ(ν),τ(ν) : ν ∈ [nt]}. Thus, it is sufficient to prove the bound +for �Θt. +Step 1. Estimation error decomposition: +Let us fix t ∈ [T] throughout the proof. For each ν ∈ [nt] and k ∈ [K], denote +Xk,ν := ˜Xk,ν ˜X⊤ +k,ν. Then we can write +Vnt := +� +ν∈Ψnt +K +� +k=1 +Xk,ν + +� +ν /∈Ψnt +Xaτ(ν),ν + IJ·d, +Ant := +� +ν∈Ψnt +K +� +k=1 +I (hν = k) +φk,ν +Xk,ν + +� +ν /∈Ψnt +Xk,ν + IJ·d. +Denote the errors ˜ηk,ν := ˜rk,ν − ˜X⊤ +k,νΘ⋆ and ηk,ν := r +(jτ(ν)) +k,τ(ν) − ˜X⊤ +k,νΘ⋆. By the definition of the estimator �Θnt, +����Θnt − Θ∗��� +Vnt += +������ +V −1/2 +nt +� +� +�−Θ∗ + +� +ν∈Ψnt +K +� +k=1 +˜ηk,ν ˜Xk,ν + +� +ν /∈Ψnt +ηk,ν ˜Xaτ(ν),ν +� +� +� +������ +2 +≤ λmax +� +V −1/2 +nt +� +∥Θ∗∥2 + +������ +V −1/2 +nt +� +� +� +� +ν∈Ψnt +K +� +k=1 +˜ηk,ν ˜Xk,ν + +� +ν /∈Ψnt +ηk,ν ˜Xaτ(ν),ν +� +� +� +������ +2 +≤ +√ +Jd + +������ +V −1/2 +nt +� +� +� +� +ν∈Ψnt +K +� +k=1 +˜ηk,ν ˜Xk,ν + +� +ν /∈Ψnt +ηk,ν ˜Xaτ(ν),ν +� +� +� +������ +2 +, +(16) +where and the last inequality holds because +���θ(j) +⋆ +��� +2 ≤ +√ +d. Plugging in ˜rk,ν defined in (5), +˜ηk,ν ˜Xk,ν = +� +1 − I (hν = k) +φk,ν +� +˜Xk,ν ˜X⊤ +k,ν +�ˇΘt − Θ∗� ++ I (hν = k) +φk,ν +ηk,ν ˜Xk,ν, + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +and the term � +ν∈Ψnt +�K +k=1 ˜ηk,ν ˜Xk,ν is decomposed as, +� +ν∈Ψnt +K +� +k=1 +˜ηk,ν ˜Xk,ν = +� +ν∈Ψnt +K +� +k=1 +�� +1 − I (hν = k) +φk,ν +� +Xk,ν +�ˇΘt − Θ∗� ++ I (hν = k) +φk,ν +ηk,ν ˜Xk,ν +� +. +(17) +By definition of the IPW estimator ˇΘt, +� +ν∈Ψnt +K +� +k=1 +� +1 − I (hν = k) +φk,ν +� +Xk,ν +�ˇΘt − Θ∗� += +� +� +� +� +ν∈Ψnt +K +� +k=1 +� +1 − I (hν = k) +φk,ν +� +Xk,ν +� +� +� A−1 +nt +� +�−Θ∗ + +� +ν∈Ψnt +K +� +k=1 +I (hν = k) +φk,ν +ηk,ν ˜Xk,ν + +� +ν /∈Ψnt +ηaτ(ν),ν ˜Xaτ(ν),ν +� +� += (Vnt − Ant) A−1 +nt +� +�−Θ∗ + +� +ν∈Ψnt +K +� +k=1 +I (hν = k) +φk,ν +ηk,ν ˜Xk,ν + +� +ν /∈Ψnt +ηaτ(ν),ν ˜Xaτ(ν),ν +� +� +:= (Vnt − Ant) A−1 +nt (−Θ∗ + Snt) , +(18) +where +Snt := +� +ν∈Ψnt +K +� +k=1 +I (hν = k) +φk,ν +ηk,ν ˜Xk,ν + +� +ν /∈Ψnt +ηaτ(ν),ν ˜Xaτ(ν),ν, +then, +����Θnt − Θ∗��� +Vnt +≤ +(16) +√ +Jd + +������ +V −1/2 +nt +� +� +� +� +ν∈Ψnt +K +� +k=1 +˜ηk,ν ˜Xk,ν + +� +ν /∈Ψnt +ηk,ν ˜Xaτ(ν),ν +� +� +� +������ +2 += +(17),(18) +√ +Jd + +���V −1/2 +nt +� +(Vnt − Ant) A−1 +nt (−Θ∗ + Snt) + Snt +���� +2 . +By triangular inequality, +����Θt − Θ∗��� +Vnt +≤ +√ +Jd + +���V −1/2 +nt +� +(Vnt − Ant) A−1 +nt (−Θ∗ + Snt) + Snt +���� +2 +≤ +√ +Jd + +���V −1/2 +nt +(Vnt − Ant) A−1 +nt (−Θ∗ + Snt) +��� +2 + ∥Snt∥V −1 +nt +≤ +√ +Jd + +��� +� +V 1/2 +nt A−1 +nt V 1/2 +nt +− IJ·d +� � +−V −1/2 +nt +Θ∗ + V −1/2 +t +Snt +���� +2 + ∥Snt∥V −1 +nt +≤ +√ +Jd + +���V 1/2 +nt A−1 +nt V 1/2 +nt +− IJ·d +��� +2 +���−V −1/2 +nt +Θ∗ + V −1/2 +t +Snt +��� +2 + ∥Snt∥V −1 +nt +≤ +√ +Jd + +���V 1/2 +nt A−1 +nt V 1/2 +nt +− IJ·d +��� +2 +�√ +Jd + ∥Snt∥V −1 +nt +� ++ ∥Snt∥V −1 +nt += +����V 1/2 +nt A−1 +nt V 1/2 +nt +− IJ·d +��� +2 + 1 +� �√ +Jd + ∥Snt∥V −1 +nt +� +. +(19) +Step 2. Bounding the ∥ · ∥2 of the matrix in (19) +We claim that +V 1/2 +nt A−1 +nt V 1/2 +nt +⪰ 1 +8IJ·d +(20) + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Define Fnt := �nt +ν=1 +�K +k=1 Xk,ν + 16Kd log( Jd +δ )IJ·d. Then we have Vnt ⪯ Fnt and V 1/2 +nt A−1 +nt V 1/2 +nt +⪰ F −1/2 +nt +AntF −1/2 +nt +. +Now we decompose the matrix Ant as +F −1/2 +nt +AntF −1/2 +nt +=F −1/2 +nt +� nt +� +ν=1 +K +� +k=1 +I (hν = k) +φk,ν +Xk,ν + IJ·d +� +F −1/2 +nt ++ F −1/2 +nt +� +� � +ν /∈Ψnt +� +Xaτ(ν),ν − +K +� +k=1 +I (hν = k) +φk,ν +Xk,ν +�� +� F −1/2 +nt +. +(21) +For each ν ∈ [nt], the matrix �K +k=1 +I(hν=k) +φk,ν +F −1/2 +nt +Xk,νF −1/2 +nt +symmetric positive definite and +λmax +� +8 log Jd +δ +K +� +k=1 +I (hν = k) +φk,ν +F −1/2 +nt +Xk,νF −1/2 +nt +� +≤8 log Jd +δ +K +� +k=1 +I (hν = k) +φk,ν +λmax +� +F −1 +nt +� +≤8 log Jd +δ +λmin(Fν) +16 log +� Jd +δ +�λmax +� +F −1 +nt +� +≤1 +2 +λmin(Fν) +λmin(Fnt) +≤1 +2. +(22) +With the filtration F0 := Ht and Fn := F0 ∪ {hν : ν ∈ [n]}, we use Lemma C.3 to have with probability at least 1 − δ, +8 log Jd +δ F −1/2 +nt +� nt +� +ν=1 +K +� +k=1 +I (hν = k) +φk,ν +Xk,ν + IJ·d +� +F −1/2 +nt +⪰ 8 log Jd +δ F −1/2 +nt +� nt +� +ν=1 +K +� +k=1 +Xk,ν + IJ·d +� +F −1/2 +nt += 4 log Jd +δ IJ·d − log Jd +δ IJ·d += 3 log Jd +δ IJ·d, +which implies +F −1/2 +nt +� nt +� +ν=1 +K +� +k=1 +I (hν = k) +φk,ν +Xk,ν + IJ·d +� +F −1/2 +nt +⪰ 3 +8IJ·d, +(23) +and the first term in (21) is bounded as +F −1/2 +nt +AntF −1/2 +nt +⪰ 3 +8IJ·d + F −1/2 +nt +� +� � +ν /∈Ψnt +� +Xaτ(ν),ν − +K +� +k=1 +I (hν = k) +φk,ν +Xk,ν +�� +� F −1/2 +nt +. +(24) +To bound the other term, observe that for ν /∈ Ψnt, +E +� K +� +k=1 +I (hν = k) +φk,ν +Xk,ν +����� Ht +� += +� +i̸=aτ(ν) +K +� +k=1 +φi,ν +I (i = k) +φk,ν +Xk,ν = +� +k̸=aτ(ν) +Xk,ν. +Because (22) holds for ν /∈ Ψnt, we can use Lemma C.3, to have with probability at least 1 − δ +8 log Jd +δ F −1/2 +nt +� +� � +ν /∈Ψnt +K +� +k=1 +I (hν = k) +φk,ν +Xk,ν +� +� F −1/2 +nt +⪯ 12 log Jd +δ F −1/2 +nt +� +� � +ν /∈Ψnt +� +k̸=aτ(ν) +Xk,ν +� +� F −1/2 +nt ++ log Jd +δ IJ·d. +Rearranging the terms, +F −1/2 +nt +� +� � +ν /∈Ψnt +K +� +k=1 +I (hν = k) +φk,ν +Xk,ν +� +� F −1/2 +nt +⪯ 3 +2F −1/2 +nt +� +� � +ν /∈Ψnt +� +k̸=aτ(ν) +Xk,ν +� +� F −1/2 +nt ++ 1 +8IJ·d. + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Thus the second term in (24) is bounded as, +F −1/2 +nt +� +� � +ν /∈Ψnt +� +Xaτ(ν),ν − +K +� +k=1 +I (hν = k) +φk,ν +Xk,ν +�� +� F −1/2 +nt +⪰ F −1/2 +nt +� +� � +ν /∈Ψnt +� +� +�Xaτ(ν),ν − 3 +2 +� +ν /∈Ψt +� +k̸=aτ(ν) +Xk,ν +� +� +� +� +� F −1/2 +nt +− 1 +8IJ·d +⪰ −3 +2F −1/2 +nt +� +� � +ν /∈Ψnt +K +� +k=1 +Xk,ν +� +� F −1/2 +nt +− 1 +8IJ·d +⪰ − +�3dK +2 +��Ψc +nt +�� λmax +� +F −1 +nt +� ++ 1 +8 +� +IJ·d, +(25) +where the last inequality holds by λmax(Xk,ν) ≤ d. By Lemma C.3, with probability at least 1 − δ, +1 +2 +��Ψc +nt +�� = 1 +2 +nt +� +ν=1 +I +� +hν ̸= aτ(ν) +� +≤ 3 +2 +nt +� +ν=1 +� +k̸=aτ(ν) +φk,ν + log 1 +δ , +which implies +3dK +2 +��Ψc +nt +�� λmax +� +F −1 +nt +� +≤ +3Kd +2λmin(Fnt) +� +� +�3 +nt +� +ν=1 +� +k̸=aτ(ν) +φk,ν + 2 log 1 +δ +� +� +� += +3Kd +2λmin(Fnt) +� nt +� +ν=1 +48 (K − 1) log +� Jd +δ +� +λmin(Fν) ++ 2 log 1 +δ +� +Because the assumption (8), +λmin(Fnt) ≥ 12Kd +� nt +� +ν=1 +48 (K − 1) log +� Jd +δ +� +λmin(Fν) ++ 2 log Jd +δ +� +, +implies +3dK +2 +��Ψc +nt +�� λmax +� +F −1 +nt +� +≤ +3Kd +2λmin(Fnt) +� nt +� +ν=1 +48 (K − 1) log +� Jd +δ +� +λmin(Fν) ++ 2 log 1 +δ +� +≤ 1 +8, +(26) +plugging in (25), with probability at least 1 − 2δ, +F −1/2 +nt +� +� � +ν /∈Ψnt +� +Xaτ(ν),ν − +K +� +k=1 +I (hν = k) +φk,ν +Xk,ν +�� +� F −1/2 +nt +⪰ −1 +4IJ·d. +With (24), +F −1/2 +nt +AntF −1/2 +nt +⪰ 1 +8IJ·d, +which proves (20) and the claim implies +���V 1/2 +nt A−1 +nt V 1/2 +nt +− IJ·d +��� +2 ≤ 7. +Step 3. Bounding the self-normalized vector-valued martingale Snt +Let F0 be a sigma algebra generated by contexts +{x(js) +k,s : k ∈ [K], s ∈ [t]}, and Ψt. Define filtration as Fν := σ(F0 ∪ Hτ(ν+1)). Then Sν is a RJ·d-valued martingale + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +because +E [Sν − Sν−1| Fν−1] = E +� +I (ν ∈ Ψnt) +K +� +k=1 +I (hν = k) +φk,ν +ηk,ν ˜Xk,ν + I (ν /∈ Ψnt) ηaτ(ν),τ(ν) ˜Xk,ν +����� Fν−1 +� += E +� +I (ν ∈ Ψnt) +K +� +k=1 +I +� +aτ(ν) = k +� +φk,ν +ηk,ν ˜Xk,ν + I (ν /∈ Ψnt) ηaτ(ν),τ(ν) ˜Xk,ν +����� Fν−1 +� += E +�� +I (ν ∈ Ψnt) +φaτ(ν),ν ++ I (ν /∈ Ψnt) +� +ηaτ(ν),ν ˜Xk,ν +����� Fν−1 +� += E +�� +I (ν ∈ Ψnt) +φaτ(ν),ν ++ I (ν /∈ Ψnt) +� +ηaτ(ν),ν ˜Xk,ν +����� Hτ(ν) +� += 0, +where the second equality holds by definition of Ψnt and the fourth inequality holds because the distribution of {x(js) +k,s : k ∈ +[K], s ∈ (τ(ν), t]} is independent of Hτ(ν) by Assumption 2. By Assumption 1, for any λ ∈ R, +E +� +exp +� +λ +� +I (ν ∈ Ψnt) +φaτ(ν),ν ++I (ν /∈ Ψnt) +� +ηk,aτ(ν) +������ Fν−1 +� +≤ E +� +�exp +� +�λ2σ2 +2 +� +I (ν ∈ Ψnt) +φaτ(ν),ν ++I (ν /∈ Ψnt) +�2� +� +������ +Fν−1 +� +� +≤ exp +� +2λ2σ2 +r +� +, +Thus, +� +I(ν∈Ψnt) +φaτ(ν),ν + I (ν /∈ Ψnt) +� +ηk,aτ(ν) is 2σr-sub-Gaussian. Because +∥Snt∥V −1 +t += +������ +� +ν∈Ψnt +K +� +k=1 +I (hν = k) +φk,ν +ηk,ν ˜Xk,ν + +� +ν /∈Ψnt +ηaτ(ν),ν ˜Xaτ(ν),ν +������ +V −1 +nt += +����� +nt +� +ν=1 +� +I (ν ∈ Ψnt) +φaτ(ν),ν ++ I (ν /∈ Ψnt) +� +ηaτ(ν),ν ˜Xk,ν +����� +V −1 +nt += +����� +nt +� +ν=1 +� +I (ν ∈ Ψnt) +φaτ(ν),ν ++ I (ν /∈ Ψnt) +� +ηaτ(ν),νV −1/2 +nt +˜Xk,ν +����� +2 +, +by Lemma C.6, with probability at least 1 − δ, +∥St∥V −1 +t +≤ +����� +nt +� +ν=1 +� +I (ν ∈ Ψnt) +φaτ(ν),ν ++ I (ν /∈ Ψt) +� +ηaτ(ν),νV −1/2 +nt +˜Xk,ν +����� +2 +, +≤12σr +� +� +� +� +nt +� +ν=1 +���V −1/2 +nt +˜Xk,ν +��� +2 +2 log 4 +δ +≤12σr +� +Jd log 4 +δ , +where the last inequality holds because +nt +� +ν=1 +���V −1/2 +nt +˜Xk,ν +��� +2 +2 = +nt +� +ν=1 +˜X⊤ +k,νV −1 +nt +˜Xk,ν = Tr +� nt +� +ν=1 +˜X⊤ +k,νV −1 +nt +˜Xk,ν +� += Tr +� nt +� +ν=1 +˜Xk,ν ˜X⊤ +k,νV −1 +nt +� +≤ Tr +� +VntV −1 +nt +� += Jd. +With (19), the proof is completed + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +B.2. Proof of Theorem 4.2 +Proof. Similar to the proof of Theorem 4.1, the bound for consumption vector immediately follows from the bound for the +utilities. Therefore we provide the proof for the utility bound. +Step 1. Decomposition: +For each k ∈ [K] and j ∈ [J], +���u⋆(j) +k +− ˜u(j) +k,t+1 +��� ≤ +�����E +� +x(j) +k +�⊤ +θ(j) +⋆ +− +� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +� +x(j) +k,s +�⊤ �θ(j) +t +������ ++ +�����E +� +c(j) +k +� +− +� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) c(j) +k,s +������ +≤ +�����E +� +x(j) +k +�⊤ +θ(j) +⋆ +− +� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +� +x(j) +k,s +�⊤ +θ(j) +⋆ +������ ++ +�����E +� +c(j) +k +� +− +� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) c(j) +k,s +������ ++ +����� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +� +�θ(j) +t +− θ(j) +⋆ +�⊤ +x(j) +k,s +����� +����� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +� +E +� +x(j) +k +�⊤ +θ(j) +⋆ +− +� +x(j) +k,s +�⊤ +θ(j) +⋆ +������ ++ +����� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +� +E +� +c(j) +k +� +− c(j) +k,s +������ ++ +����� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +� +�θ(j) +t +− θ(j) +⋆ +�⊤ +x(j) +k,s +����� . +Taking maximum over k ∈ [K] gives the decomposition, +max +k∈[K] +���u⋆(j) +k +− ˜u(j) +k,t+1 +��� ≤ max +k∈[K] +����� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +� +E +� +x(j) +k +�⊤ +θ(j) +⋆ +− +� +x(j) +k,s +�⊤ +θ(j) +⋆ +������ ++ max +k∈[K] +����� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +� +E +� +c(j) +k +� +− c(j) +k,s +������ ++ max +k∈[K] +����� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +� +�θ(j) +t +− θ(j) +⋆ +�⊤ +x(j) +k,s +����� . +(27) +Step 2. +Bounding the difference between expectation and empirical distribution: +The random variables +�� +x(j) +k,s +�⊤ +θ(j) +⋆ +: s ∈ [t] +� +and {c(j) +k,s : s ∈ [t]} are IID by Assumption 2. Using Lemma C.1, +����� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +� +E +� +x(j) +k +�⊤ +θ(j) +⋆ +− +� +x(j) +k,s +�⊤ +θ(j) +⋆ +������ += +1 +�t+1 +s=1 I (js = j) +����� +t+1 +� +s=1 +I (js = j) +� +E +� +x(j) +k +�⊤ +θ(j) +⋆ +− +� +x(j) +k,s +�⊤ +θ(j) +⋆ +������ +≤ +4 +��t+1 +s=1 I (js = j) +� +log JKT, + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +and +����� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +� +E +� +c(j) +k +� +− c(j) +k,s +������ ≤ +4 +��t+1 +s=1 I (js = j) +� +log JKT +with probability at least 1 − 4(JKT)−1. By Lemma C.3, with probability at least 1 − (JT)−1, +t+1 +� +s=1 +I (js = j) ≥ 1 +2pj (t + 1) − 2 log JT ≥ 1 +4pj (t + 1) , +(28) +where the last inequality holds by the assumption t ≥ 8dα−1p−1 +min log JT. Summing up the probability bounds, with +probability at least 1 − 5T −1, +max +k∈[K] +����� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +� +E +� +x(j) +k +�⊤ +θ(j) +⋆ +− +� +x(j) +k,s +�⊤ +θ(j) +⋆ +������ ≤ +4 +��t+1 +s=1 I (js = j) +� +log JKT +≤ +8 +� +pj (t + 1) +� +log JKT, +max +k∈[K] +����� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +� +E +� +c(j) +k +� +− c(j) +k,s +������ ≤ +8 +� +pj (t + 1) +� +log JKT. +Plugging in the decomposition (27), for each j ∈ [J], +max +k∈[K] +���u⋆(j) +k +− ˜u(j) +k,t+1 +��� ≤ +16 +� +pj (t + 1) +� +log JKT ++ max +k∈[K] +����� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +� +�θ(j) +t−1 − θ(j) +⋆ +�⊤ +x(j) +k,s +����� . +Taking square and summing up over j ∈ [J] gives +J +� +j=1 +pj max +k∈[K] +���u⋆(j) +k +− ˜u(j) +k,t+1 +��� +2 +≤16J log JKT +t + 1 ++ +J +� +j=1 +pj max +k∈[K] +����� +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +� +�θ(j) +t +− θ(j) +⋆ +�⊤ +x(j) +k,s +����� +2 +, +(29) +Step 3. Bounding the prediction error: +By Cauchy-Schwartz inequality and (28), +J +� +j=1 +max +k∈[K] pj +1 +��t+1 +s=1 I (js = j) +�2 +����� +t+1 +� +s=1 +I (js = j) +� +�θ(j) +t +− θ(j) +⋆ +�⊤ +x(j) +k,s +����� +2 +≤ +J +� +j=1 +max +k∈[K] pj +1 +�t+1 +s=1 I (js = j) +t+1 +� +s=1 +I (js = j) +�� +�θ(j) +t +− θ(j) +⋆ +�⊤ +x(j) +k,s +�2 +≤ +J +� +j=1 +max +k∈[K] +4pj +pj (t + 1) +t+1 +� +s=1 +I (js = j) +�� +�θ(j) +t +− θ(j) +⋆ +�⊤ +x(j) +k,s +�2 += +4 +(t + 1) +J +� +j=1 +max +k∈[K] +� +�θ(j) +t +− θ(j) +⋆ +�⊤ +�t+1 +� +s=1 +I (js = j) x(j) +k,s +� +x(j) +k,s +�⊤ +� � +�θ(j) +t +− θ(j) +⋆ +� +≤ +4 +(t + 1) +J +� +j=1 +� +�θ(j) +t +− θ(j) +⋆ +�⊤ +�t+1 +� +s=1 +K +� +k=1 +I (js = j) x(j) +k,s +� +x(j) +k,s +�⊤ +� � +�θ(j) +t +− θ(j) +⋆ +� += +4 +(t + 1) +� +�Θt − Θ⋆�⊤ +�t+1 +� +s=1 +K +� +k=1 +˜Xk,s ˜X⊤ +k,s +� � +�Θt − Θ⋆� +, +(30) + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +where Θ⋆ := (θ(1) +⋆ , . . . , θ(J) +⋆ +)T ∈ RJ·d and +˜Xk,s := +� +� +� +� +� +0d +... +x(js) +k,s +0d +� +� +� +� +� ∈ RJ·d, +where the context x(js) +k,s is located after js − 1 of 0d vectors. We claim that +1 +t + 1 +t+1 +� +s=1 +K +� +k=1 +˜Xk,s ˜X⊤ +k,s ⪯ 2E +� +˜Xk,1 ˜X⊤ +k,1 +� +⪯ +7 +|Ψnt| +� +s∈Ψnt +K +� +k=1 +˜Xk,s ˜X⊤ +k,s, +(31) +with probability at least 1 − 2T −1. The matrix Xs := �K +k=1 ˜Xk,s ˜X⊤ +k,sis symmetric nonnegative definite which satisfies +λmax +� +1 +2dK Xs +� +≤ 1 +2. +By Lemma C.3, with probability at least 1 − T −1, +1 +2Kd +t+1 +� +s=1 +Xs ⪯ +3 +4Kd +t+1 +� +s=1 +E [Xs] + (log JdT) IJ·d, +which implies +1 +t + 1 +t+1 +� +s=1 +Xs ⪯ +3 +2 (t + 1) +t+1 +� +s=1 +E [Xs] + 2dK +t + 1 (log JdT) IJ·d. +(32) +By Assumption 3, for s ∈ [t + 1], +λmin(E [Xs]) =λmin +� +� +� +� +� +� +� +� +� +� +p1Exk∼F1 +��K +k=1 xkx⊤ +k +� +0 +0 +0 +... +0 +0 +0 +pJExk∼FJ +��K +k=1 xkx⊤ +k +� +� +� +� +� +� +� +� +� +� +� +≥λmin +� +� +� +� +� +� +p1KαId +0 +0 +0 +... +0 +0 +0 +pJKαId +� +� +� +� +� +� +≥Kpminα. +For t ≥ 8dα−1p−1 +min log JdT , +t+1 +� +s=1 +E [Xs] ⪰ +t+1 +� +s=1 +λmin(E [Xs]) IJ·d ⪰ (t + 1) KpminαIJ·d ⪰ 4dK +� +log Jd +δ +� +Plugging in (32) proves the first inequality of (31), +1 +t + 1 +t+1 +� +s=1 +Xs ⪯ +2 +(t + 1) +t+1 +� +s=1 +E [Xs] = 2E [X1] , +where the equality holds because EXs = EX1 for all s ∈ [T]. To prove the second inequality, +E [X1] = |Ψnt|−1 � +ν∈Ψnt +E +� +Xτ(ν) +� +, + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +and by Lemma C.3, with probability at least 1 − T −1, +1 +2Kd +� +ν∈Ψnt +Xτ(ν) ⪰ +1 +4Kd +� +ν∈Ψnt +E +� +Xτ(ν) +� +− (log JdT) IJ·d. +Rearranging the terms, +� +ν∈Ψnt +E +� +Xτ(ν) +� +⪯ 2 +� +ν∈Ψnt +Xτ(ν) + 4Kd (log JdT) IJ·d +(33) +By definition of Fnt, +� +ν∈Ψnt +Xτ(ν) =Fnt − +� +ν /∈Ψnt +Xτ(ν) − 16d(K − 1) log Jd +δ IJ·d +⪰Fnt − +� +Kd +��Ψc +nt +�� + 16d(K − 1) log Jd +δ +� +IJ·d. +(34) +Because the condition (8) holds, we can use (26) to have, +Kd +��Ψc +nt +�� ≤ +1 +12λmax +� +F −1 +nt +� = λmin(Fnt) +12 +, +(35) +and +Fnt − +� +Kd +��Ψc +nt +�� + 16d(K − 1) log Jd +δ +� +IJ·d ⪰Fnt − +� 1 +12λmin(Fnt) + 16d(K − 1) log Jd +δ +� +IJ·d +⪰ +�11 +12λmin(Fnt) − 16d(K − 1) log Jd +δ +� +IJ·d +⪰ +�11 +1224Kd log Jd +δ IJ·d − 16d(K − 1) log Jd +δ +� +IJ·d +⪰ +� +6dK log Jd +δ +� +IJ·d +⪰ {6dK log JdT} IJ·d, +(36) +where the third inequality holds by condition (8) and the last inequality holds by δ < T −1. Collecting the bounds (34) +and (36) +� +ν∈Ψnt +Xτ(ν) ⪰ Fnt − +� +Kd +��Ψc +nt +�� + 16d(K − 1) log Jd +δ +� +IJ·d ⪰ {6dK log JdT} IJ·d. +Plugging in (33), +� +ν∈Ψnt +E +� +Xτ(ν) +� +⪯ 2 +� +ν∈Ψnt +Xτ(ν) + 4Kd (log JdT) IJ·d ⪯ 7 +2 +� +ν∈Ψnt +Xτ(ν), +proves the second inequality in claim (20). From (30), +J +� +j=1 +max +k∈[K] pj +1 +��t+1 +s=1 I (js = j) +�2 +����� +t+1 +� +s=1 +I (js = j) +� +�θ(j) +t +− θ(j) +⋆ +�⊤ +x(j) +k,s +����� +2 +≤ +4 +(t + 1) +� +�Θt − Θ⋆�⊤ +�t+1 +� +s=1 +K +� +k=1 +˜Xk,s ˜X⊤ +k,s +� � +�Θt − Θ⋆� +≤ +28 +|Ψnt| +� +�Θt − Θ⋆�⊤ +� +� +� +� +s∈Ψnt +K +� +k=1 +˜Xk,s ˜X⊤ +k,s +� +� +� +� +�Θt − Θ⋆� +≤ +28 +|Ψnt| +� +�Θt − Θ⋆�⊤ +{Vnt} +� +�Θt − Θ⋆� += +28 +|Ψnt| +����Θt − Θ⋆��� +2 +Vnt +(37) + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +On bounding the normalizing matrix, the novel Gram matrix Vnt plays a crucial role. To obtain an upper bound for (37), we +need a matrix whose eigenvalue is greater than that of: +� +ν∈Ψnt +Xτ(ν) = +� +ν∈Ψnt +K +� +k=1 +˜Xk,τ(ν) ˜X⊤ +k,τ(ν), +(38) +However, with � +ν∈Ψnt ˜Xaτ(ν),τ(ν) ˜X⊤ +aτ(ν),τ(ν) , a Gram matrix consist of only selected contexts, we cannot bound the +matrix (38). Instead, by using a Gram matrix Vt, we can bound (38) as, +� +ν∈Ψnt +Xτ(ν) = +� +ν∈Ψnt +K +� +k=1 +˜Xk,τ(ν) ˜X⊤ +k,τ(ν) +⪯ +� +ν∈Ψnt +K +� +k=1 +˜Xk,τ(ν) ˜X⊤ +k,τ(ν) + +� +ν /∈Ψnt +˜Xaτ(ν),τ(ν) ˜X⊤ +aτ(ν),τ(ν) +⪯Vnt, +and prove the bound (37) to relate the prediction error to the self-normalized bound. From (37), by Theorem 4.1 +J +� +j=1 +max +k∈[K] pj +1 +��t+1 +s=1 I (js = j) +�2 +����� +t+1 +� +s=1 +I (js = j) +� +�θ(j) +t +− θ(j) +⋆ +�⊤ +x(j) +k,s +����� +2 +≤ 28 +|Ψnt| +����Θt − Θ⋆��� +2 +Vnt +≤ 28 +|Ψnt|βσr(δ)2, +with probability at least 1 − 4(m + 1)δ. Because |Ψnt| + +��Ψc +nt +�� = nt and (35) holds, +|Ψnt| ≥ nt − +��Ψc +nt +�� ≥ nt − λmin(Fnt) +12Kd +≥ nt − ntKd +12Kd = 11 +12nt. +Thus, +J +� +j=1 +max +k∈[K] pj +1 +��t+1 +s=1 I (js = j) +�2 +����� +t+1 +� +s=1 +I (js = j) +� +�θ(j) +t +− θ(j) +⋆ +�⊤ +x(j) +k,s +����� +2 +≤ 28 +|Ψnt|βσr(δ)2 +≤12 +11 · 28 +nt +βσr(δ)2 +≤32 +nt +βσr(δ)2. +From (29), +� +� +� +� +J +� +j=1 +pj max +k∈[K] +���u⋆(j) +k +− ˜u(j) +k,t+1 +��� +2 +≤ 16√J log JKT +√ +t ++ 4 +√ +2βσr(δ) +√nt +and the proof is completed. +B.3. Proof of Lemma 4.3 +Proof. Suppose a feasible policy ˜π(j) +k,t for the optimization problem (1) satisfies +K+1 +� +k=1 +˜π(jt) +k,t ˜u(jt) +k,t > +K+1 +� +k=1 +�π(jt) +k,t ˜u(jt) +k,t , +which is equivalent to +K+1 +� +l=1 +˜π(jt) +k⟨l⟩,t˜u(jt) +k⟨l⟩,t > +K+1 +� +l=1 +�π(jt) +k⟨l⟩,t˜u(jt) +k⟨l⟩,t. +(39) + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Without loss of generality we assume �u(jt) +k⟨l⟩,t ≥ 0 (Because �K+1 +l=1 ˜π(jt) +k⟨l⟩,t = �K+1 +l=1 �π(jt) +k⟨l⟩,t = 1, we can subtract �u(jt) +k⟨K+1⟩,t +on both side of (39)). By the constraints on the resources, +˜π(jt) +k⟨1⟩,t ≤ +� +� min +r∈[m] +ρt(r) +b(jt) +k⟨1⟩,t(r) +� +� ∧ 1 = �π(jt) +k⟨1⟩,t +Suppose ˜π(jt) +k⟨1⟩,t < �π(jt) +k⟨1⟩,t. Because �K+1 +l=1 ˜π(jt) +k⟨l⟩,t = �K+1 +l=1 �π(jt) +k⟨l⟩,t = 1, by Lemma C.2, +K+1 +� +l=1 +˜π(jt) +k⟨l⟩,t˜u(jt) +k⟨l⟩,t ≤ +K+1 +� +l=1 +�π(jt) +k⟨l⟩,t˜u(jt) +k⟨l⟩,t, +which contradicts with (39). Thus we have ˜π(jt) +k⟨1⟩,t = �π(jt) +k⟨1⟩,t and +K+1 +� +l=2 +˜π(jt) +k⟨l⟩,t˜u(jt) +k⟨l⟩,t > +K+1 +� +l=2 +�π(jt) +k⟨l⟩,t˜u(jt) +k⟨l⟩,t. +(40) +Again, by the constraints on the resources,˜π(jt) +k⟨2⟩,t ≤ �π(jt) +k⟨2⟩,t. Suppose ˜π(jt) +k⟨2⟩,t < �π(jt) +k⟨2⟩,t. Because �K+1 +l=2 ˜π(jt) +k⟨l⟩,t = +�K+1 +l=2 �π(jt) +k⟨l⟩,t, by Lemma C.2, +K+1 +� +l=2 +˜π(jt) +k⟨l⟩,t˜u(jt) +k⟨l⟩,t ≤ +K+1 +� +l=2 +�π(jt) +k⟨l⟩,t˜u(jt) +k⟨l⟩,t. +which contradicts with (40). Thus we have ˜π(jt) +k⟨2⟩,t = �π(jt) +k⟨2⟩,t Recursively, we have ˜π(jt) +k⟨l⟩,t = �π(jt) +k⟨l⟩,t for all l ∈ [K + 1]. Thus +there exist no feasible solution ˜π(j) +k,t such that (39) holds and the proof is completed. +B.4. Proof of Lemma 5.2 +Proof. For each t ∈ [T], denote the good events Gt := Et ∩ Mt−1. +Step 1. Bounds for the estimates ˜u(jt) +k,t and ˜b(jt) +k,t : +For each t ∈ [T] and k ∈ [K], +˜u(jt) +k,t = ˜u(jt) +k,t − u⋆(jt) +k ++ u⋆(jt) +k +=γt−1,σr(δ) +√pjt ++ �u(jt) +k,t − u⋆(jt) +k ++ u⋆(jt) +k +≥ +γt−1,σr(δ) − √pjt maxk∈[K] +����u(jt) +k,t − u⋆(jt) +k +��� +√pjt ++ u⋆(jt) +k +. +Under the event Gt, +√pjt max +k∈[K] +���u⋆(jt) +k +− �u(jt) +k,t +��� = +� +pjt max +k∈[K] +���u⋆(jt) +k +− �u(jt) +k,t +��� +2 +≤ +� +� +� +� +J +� +j=1 +pj max +k∈[K] +���u⋆(j) +k +− �u(j) +k,t +��� +2 +≤γt−1,σr(δ), +which implies +˜u(jt) +k,t ≥ u⋆(jt) +k +. +(41) +Similarly, +˜b(jt) +k,t ≤ b⋆(jt) +k +. +(42) + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Another useful bound for ˜u(jt) +k,t is +E +�� +t∈U +K +� +k=1 +�π(jt) +k,t +���˜u(jt) +k,t − u⋆(jt) +k +��� I (Gt) +� +≤ 2γt−1,σr(δ) +� +E [I (at ∈ [K])]. +(43) +This bound is proved by the tower property of conditional expectation and Cauchy-Schwartz inequality, +E +� K +� +k=1 +�π(jt) +k,t +���˜u(jt) +k,t − u⋆(jt) +k +��� I (Gt) +� +=E +� +max +k∈[K] +���˜u(jt) +k,t − u⋆(jt) +k +��� +K +� +k=1 +�π(jt) +k,t I (Gt) +� +=E +� +max +k∈[K] +���˜u(jt) +k,t − u⋆(jt) +k +��� I (at ∈ [K]) I (Gt) +� +=E +� +� +J +� +j=1 +pj max +k∈[K] +���˜u(j) +k,t − u⋆(j) +k +��� I (at ∈ [K]) I (Gt) +� +� +≤E +� +� +� +� +� +� +J +� +j=1 +pj max +k∈[K] +���˜u(j) +k,t − u⋆(j) +k +��� +2 +� +� +� +� +J +� +j=1 +pjI (at ∈ [K])I (Gt) +� +� +By definition of ˜u(j) +k,t and triangular inequality for ℓ2-norm, +� +� +� +� +J +� +j=1 +pj max +k∈[K] +���˜u(j) +k,t − u⋆(j) +k +��� +2 +I (Gt) = +� +� +� +� +J +� +j=1 +pj max +k∈[K] +�����u(j) +k,t − u⋆(j) +k ++ γt−1,σr(δ) +√pj +���� +2 +I (Gt) +≤ +� +� +� +� +� +� +J +� +j=1 +pj max +k∈[K] +����u(j) +k,t − u⋆(j) +k +��� +2 ++ +� +� +� +� +J +� +j=1 +pj +�γt−1,σr(δ) +√pj +�2 +� +� I (Gt) +≤2γt−1,σr(δ)I (Gt) +≤2γt−1,σr(δ). +Then by Jensen’s inequality, +E +� K +� +k=1 +�π(jt) +k,t +���˜u(jt) +k,t − u⋆(jt) +k +��� I (Gt) +� +≤E +� +� +� +� +� +� +J +� +j=1 +pj max +k∈[K] +���˜u(j) +k,t − u⋆(j) +k +��� +2 +� +� +� +� +J +� +j=1 +pjI (at ∈ [K])I (Gt) +� +� +≤2γt−1,σr(δ)E +� +� +� +� +� +� +J +� +j=1 +pjI (at ∈ [K]) +� +� +≤2γt−1,σr(δ) +� +� +� +� +�E +� +� +J +� +j=1 +pjI (at ∈ [K]) +� +� +=2γt−1,σr(δ) +� +E [I (at ∈ [K])], +which proves (43). Similarly, +E +� K +� +k=1 +�π(jt) +k,t +���˜b(jt) +k,t − b⋆(jt) +k +��� +∞ I (Gt) +� +≤ 2γt−1,σb(δ) +� +E [I (at ∈ [K])] +(44) +Step 2. Reward decomposition: +Let τ be the stopping time of the algorithm and let U := {t ∈ [τ] : ρt > 0}. Then for +t /∈ U, the allocated resource is ρt ∨ 0 = 0 and the algorithm skips the round. Thus, +E +� T +� +t=1 +R�π +t +� +=E +�� +t∈U +R�π +t +� +. + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Then, the reward is decomposed as +E +�� +t∈U +R�π +t +� +=E +�� +t∈U +R�π +t I (Gt) +� ++ E +�� +t∈U +R�π +t I (Gc +t ) +� +≥E +�� +t∈U +K +� +k=1 +�π(jt) +k,t u⋆(jt) +k +I (Gt) +� +− +T +� +t=1 +P (Gc +t ) +≥E +�� +t∈U +K +� +k=1 +�π(jt) +k,t ˜u(jt) +k,t I (Gt) +� +− E +�� +t∈U +K +� +k=1 +�π(jt) +k,t +���˜u(jt) +k,t − u⋆(jt) +k +��� I (Gt) +� +− +T +� +t=1 +P (Gc +t ) +≥E +�� +t∈U +K +� +k=1 +�π(jt) +k,t ˜u(jt) +k,t I (Gt) +� +− +T +� +t=1 +E +� K +� +k=1 +�π(jt) +k,t +���˜u(jt) +k,t − u⋆(jt) +k +��� I (Gt) +� +− +T +� +t=1 +P (Gc +t ) . +By the bound (43), +T +� +t=1 +E +� K +� +k=1 +�π(jt) +k,t +���˜u(jt) +k,t − u⋆(jt) +k +��� I (Gt) +� +≤2 +T +� +t=1 +γt−1,σr(δ) +� +E [I (at ∈ [K])] +≤2 +� +� +� +�T +T +� +t=1 +γt−1,σr(δ)2E [I (at ∈ [K])] +=2 +� +� +� +�TE +� T +� +t=1 +γt−1,σr(δ)2I (at ∈ [K]) +� +where the last ineqaulity holds by Cauchy-Schwartz inequality. Thus, the reward is decomposed as +E +� T +� +t=1 +R�π +t +� +=E +�� +t∈U +R�π +t +� +≥E +�� +t∈U +K +� +k=1 +�π(jt) +k,t ˜u(jt) +k,t I (Gt) +� +− 2 +� +� +� +�TE +� T +� +t=1 +γt−1,σr(δ)2I (at ∈ [K]) +� +− +T +� +t=1 +P (Gc +t ) +(45) +Step 3. A lower bound for ρt: +Denote u1 < u2 < . . . < u|U| the indexes in U. For s /∈ U, we have ρs = 0m and +b(js) +as,s = 0m. Thus for ν ∈ [|U| − 1], +ρuν+1 = uν+1ρ − +uν+1−1 +� +s=1 +b(js) +as,s = uν+1ρ − +uν +� +s=1 +b(js) +as,s. +(46) +By the resource constrain at round uν, +K +� +k=1 +�π(juν ) +k,uν ˜b(juν ) +k,uν ≤uνρ − +uν−1 +� +s=1 +b(js) +as,s +=uνρ + b(juν ) +auν ,uν − +uν +� +s=1 +b(js) +as,s. +Plugging in (46), +ρuν+1 ≥ (uν+1 − uν) ρ − b(juν ) +auν ,uν + +K +� +k=1 +�π(juν ) +k,uν ˜b(juν ) +k,uν +≥ (uν+1 − uν) ρ − b(juν ) +auν ,uν + +K +� +k=1 +�π(juν ) +k,uν b⋆(juν ) +k ++ +K +� +k=1 +�π(juν ) +k,uν +� +˜b(juν ) +k,uν − b⋆(juν ) +k +� +. + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Taking conditional expectation on both sides gives +E +� +ρuν+1 +�� juν+1 +� +≥E +� +uν+1 − uν| juν+1 +� +ρ + E +� +−b(juν ) +auν ,uν + +K +� +k=1 +�π(juν ) +k,uν b⋆(juν ) +k +����� juν+1 +� ++ E +� K +� +k=1 +�π(juν ) +k,uν +� +˜b(juν ) +k,uν − b⋆(juν ) +k +������ juν+1 +� +=E +� +uν+1 − uν| juν+1 +� +ρ + E +� K +� +k=1 +�π(juν ) +k,uν +� +˜b(juν ) +k,uν − b⋆(juν ) +k +�� +≥E +� +uν+1 − uν| juν+1 +� +ρ + E +� K +� +k=1 +�π(juν ) +k,uν +� +˜b(juν ) +k,uν − b⋆(juν ) +k +� +I (Guν) +� +− P +� +Gc +uν +� +1m, +where the equality holds by Assumption 1 and +E +�� +−b(juν ) +auν ,uν + +K +� +k=1 +�π(juν ) +k,uν b⋆(juν ) +k +�� +=E +�� +−b⋆(juν ) +auν ++ +K +� +k=1 +�π(juν ) +k,uν b⋆(juν ) +k +�� +=E +�� +− +K +� +k=1 +�π(juν ) +k,uν b⋆(juν ) +k ++ +K +� +k=1 +�π(juν ) +k,uν b⋆(juν ) +k +�� +=0. +For the last term, by the bound (44), +E +� K +� +k=1 +�π(juν ) +k,uν +� +˜b(juν ) +k,uν − b⋆(juν ) +k +� +I (Guν) +� +≥ − E +� K +� +k=1 +�π(juν ) +k,uν +���˜b(juν ) +k,uν − b⋆(juν ) +k +��� +∞ I (Guν) +� +1m +≥ − E +� +2γuν−1,σb(δ) +� +E [I (auν ∈ [K])| uν] +� +1m. +Thus we obtain a lower bound, +E +� +ρuν+1 +�� juν+1 +� +≥E +� +uν+1 − uν| juν+1 +� +ρ − P +� +Gc +uν +� +1m +− 2E +� +γuν−1,σb(δ) +� +E [I (auν ∈ [K])| uν] +� +1m. +(47) +Step 4. An upper bound for OPT +In the optimization problem (1), all constraints are linear with respect to the variable +and there exist a feasible solution. Thus the problem satisfies the Slater’s condition and strong duality (Boyd et al., 2004). +Then, +OPT +T += max +π(j) +k +min +λ∈Rm ++ +min +µ(j)≥0 min +ν(j) +k +≥0 +L +� +π(j) +k , λ, µ(j), ν(j) +k +� +, +where L is the Lagrangian function: +L +� +π(j) +k , λ, µ(j), ν(j) +k +� +:= +J +� +j=1 +K +� +k=1 +pjπ(j) +k u⋆(j) +k ++ +� +�ρ − +J +� +j=1 +K +� +k=1 +pjπ(j) +k b⋆(j) +k +� +� +⊤ +λ ++ +J +� +j=1 +µ(j) +� +1 − +K +� +k=1 +π(j) +k,1 +� ++ +J +� +j=1 +K +� +k=1 +ν(j) +k π(j) +k,1. + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Minimizing over µ(j) and ν(j) +k +gives +min +µ(j) +t +≥0 +min +ν(j) +k,t≥0 +L +� +π(j) +k , λ, µ(j), ν(j) +k +� += +� +� +� +�J +j=1 +�K +k=1 pjπ(j) +k u⋆(j) +k ++ +� +ρ − �J +j=1 +�K +k=1 pjπ(j) +k b⋆(j) +k +�⊤ +λ +�K +k=1 π(j) +k +≤ 1, π(j) +k +≥ 0 +−∞ +o.w. +, +which implies +OPT +T += max +π(j) +k +min +λ∈Rm ++ +min +µ(j) +t +≥0 +min +ν(j) +k,t≥0 +L +� +π(j) +k , λ, µ(j), ν(j) +k +� +≤ +max +�K +k=1 π(j) +k +≤1,π(j) +k +≥0 +min +λ∈Rm ++ +J +� +j=1 +K +� +k=1 +pjπ(j) +k u⋆(j) +k ++ +� +�ρ − +J +� +j=1 +K +� +k=1 +pjπ(j) +k b⋆(j) +k +� +� +⊤ +λ += +max +�K +k=1 π(j) +k +≤1,π(j) +k +≥0 +min +λ∈Rm ++ +J +� +j=1 +pj +� +� +� +K +� +k=1 +π(j) +k u⋆(j) +k ++ +� +ρ − +K +� +k=1 +π(j) +k b⋆(j) +k +�⊤ +λ +� +� +� +≤ min +λ∈Rm ++ +max +�K +k=1 π(j) +k +≤1,π(j) +k +≥0 +J +� +j=1 +pj +� +� +� +K +� +k=1 +π(j) +k u⋆(j) +k ++ +� +ρ − +K +� +k=1 +π(j) +k b⋆(j) +k +�⊤ +λ +� +� +� +≤ min +λ∈Rm ++ +J +� +j=1 +pj +max +�K +k=1 π(j) +k +≤1,π(j) +k +≥0 +� +� +� +K +� +k=1 +π(j) +k u⋆(j) +k ++ +� +ρ − +K +� +k=1 +π(j) +k b⋆(j) +k +�⊤ +λ +� +� +� . +Let {¯π(j) +k +: j ∈ [J], k ∈ [K]} be the maximizer. If ρ − �K +k=1 ¯π(j) +k b⋆(j) +k +is negative for some element and j ∈ [J], then the +optimal value becomes −∞. Thus +OPT +T +≤ min +λ∈Rm ++ +J +� +j=1 +pj +max +�K +k=1 π(j) +k +≤1,π(j) +k +≥0 +� +� +� +K +� +k=1 +π(j) +k u⋆(j) +k ++ +� +ρ − +K +� +k=1 +π(j) +k b⋆(j) +k +�⊤ +λ +� +� +� += min +λ∈Rm ++ +J +� +j=1 +pj +max +�K +k=1 π(j) +k +≤1,π(j) +k +≥0,ρ−�K +k=1 π(j) +k +b⋆(j) +k +≥0 +� +� +� +K +� +k=1 +π(j) +k u⋆(j) +k ++ +� +ρ − +K +� +k=1 +π(j) +k b⋆(j) +k +�⊤ +λ +� +� +� += +J +� +j=1 +pj +max +�K +k=1 π(j) +k +≤1,π(j) +k +≥0,ρ−�K +k=1 π(j) +k +b⋆(j) +k +≥0 +� K +� +k=1 +π(j) +k u⋆(j) +k +� +. +For each j ∈ [J] and v ∈ Rm ++, let ˜π(j) +k,v be the solution to the optimization problem: +max +π(j) +k,v +K +� +k=1 +π(j) +k,vu⋆(j) +k +s.t. +K +� +k=1 +π(j) +k,vb⋆(j) +k +≤ v. +(48) +Then, +OPT +T +≤ +J +� +j=1 +pj +max +�K +k=1 π(j) +k +≤1,π(j) +k +≥0,ρ−�K +k=1 π(j) +k +b⋆(j) +k +≥0 +� K +� +k=1 +π(j) +k u⋆(j) +k +� += +J +� +j=1 +pj +K +� +k=1 +˜π(j) +k,ρu⋆(j) +k +. + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +For each ν ∈ [|U| − 1], +E +� +(uν+1 − uν) OPT +T +� +≤E +� +�(uν+1 − uν) +J +� +j=1 +pj +K +� +k=1 +˜π(j) +k,ρu⋆(j) +k +� +� +=E +� +(uν+1 − uν) +K +� +k=1 +˜π +(juν+1) +k,ρ +u +⋆(juν+1) +k +� +In (48), all constraints are linear with respect to the variable and there exist a feasible solution. Thus the problem satisfies +the Slater’s condition and strong duality (Boyd et al., 2004). The dual problem of (48) is +min +λ(j) +v ∈Rm ++ +v⊤λ(j) +v +s.t. +� +b⋆(j) +k +�⊤ +λ(j) +v +≥ u⋆(j) +k +, +∀k ∈ [K]. +(49) +Let ˜λ(j) +v +be the solution to (49). By strong duality, for each ν ∈ [|U| − 1], +E +� +(uν+1 − uν) +K +� +k=1 +˜π +(juν+1) +k,ρ +u +⋆(juν+1) +k +� += E +� +(uν+1 − uν) ρ⊤˜λ +(juν+1) +ρ +� += E +� +E +� +(uν+1 − uν) ρ| juν+1 +�⊤ ˜λ +(juν+1) +ρ +� += E +�� +P +� +Gc +uν +� ++ 2E +�� +E [I (auν ∈ [K])| uν]γuν−1,λ(σb) +�� +1⊤ +m˜λ +(juν+1) +ρ +� ++ E +�� +E +� +(uν+1 − uν) ρ| juν+1 +� +− P +� +Gc +uν +� +− 2E +�� +E [I (auν ∈ [K])| uν]γuν−1,σb(δ) +�� +1⊤ +m˜λ +(juν+1) +ρ +� +. +(50) +For the first term, we observe the dual problem of (1), +min +λ∈Rm ++ +ρ1⊤ +mλ +s.t.λ⊤b⋆(j) +k +≥ u⋆(j) +k +, ∀j ∈ [J], ∀k ∈ [K]. +(51) +Comparing to the dual problem (49), when v = ρ1m and j = juν+1, (51) has more constraints than (49) with same objective +function. Denote λ⋆ be the solution to (51). Then, +ρ1⊤ +m˜λ +(juν+1) +ρ1m +≤ ρ1⊤ +mλ⋆ = OPT +T +, +where the last equality holds by strong duality for the oracle problem (1). Thus the first term in (50) is bounded as +E +�� +P +� +Gc +uν +� ++ 2E +�� +E [I (auν ∈ [K])| uν]γuν−1,λ(σb) +�� +1⊤ +m˜λ +(juν+1) +ρ1m +� +≤ +� +P +� +Gc +uν +� ++ 2E +�� +E [I (auν ∈ [K])| uν]γuν−1,λ(σb) +�� OPT +ρT . +For the second term in (50), we observe that ˜λ(juν +1) +E[ ρuν+1|juν+1] is a feasible solution to (49) when v = ρ1m and j = juν+1. +Thus +E +�� +E +� +(uν+1 − uν) ρ| juν+1 +� +−2E +�� +E [I (auν ∈[K])| uν]γuν−1,λ(σb) +� +−P +� +Gc +uν +�� +1⊤ +m˜λ +(juν+1) +ρ1m +� +≤E +��� +E +� +(uν+1 − uν) ρ| juν+1 +� +−2E +�� +E [I (auν ∈[K])| uν]γuν−1,λ(σb) +� +−P +� +Gc +uν +�� +∨ 0 +� +1⊤ +m˜λ +(juν+1) +ρ1m +� +≤E +��� +E +� +(uν+1 − uν) ρ| juν+1 +� +−2E +�� +E [I (auν ∈[K])| uν]γuν−1,λ(σb) +� +−P +� +Gc +uν +�� +∨ 0 +� +1⊤ +m˜λ(juν +1) +E[ ρuν+1|juν+1] +� +≤E +�� +E +� +ρuν+1 +�� juν+1 +� +∨ 0m +�⊤ ˜λ(juν +1) +E[ ρuν+1|juν+1] +� +, + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +where the last inequality holds by (47). Because uν+1 ∈ U, we have ρuν+1 > 0 and +E +�� +E +� +ρuν+1 +�� juν+1 +� +∨ 0m +�⊤ ˜λ(juν +1) +E[ ρuν+1|juν+1] +� +≤E +� +E +� +ρuν+1 ∨ 0m +�� juν+1 +�⊤ ˜λ(juν +1) +E[ ρuν+1|juν+1] +� +=E +� +E +� +ρuν+1 +�� juν+1 +�⊤ ˜λ(juν +1) +E[ ρuν+1|juν+1] +� +. +Collecting the bounds, we have +E +� +(uν+1 − uν) OPT +T +� +≤E +� +E +� +ρuν+1 +�� juν+1 +�⊤ ˜λ(juν +1) +E[ ρuν+1|juν+1] +� ++ +� +2E +�� +E [I (auν ∈[K])| uν]γuν−1,σb(δ) +� ++ P +� +Gc +uν +�� OPT +ρT . +Similar to Step 4, by strong duality, +E +� +ρuν+1 +�� juν+1 +�⊤ ˜λ(juν +1) +E[ ρuν+1|juν+1] += +max +�K +k=1 π +(juν +1) +k +≤1,π +(juν +1) +k +≥0 +min +λ∈Rm ++ +K +� +k=1 +π(juν +1) +k +u⋆(juν +1) +k ++ +� +E +� +ρuν+1 +�� juν+1 +� +− +K +� +k=1 +π(juν +1) +k +b⋆(juν +1) +k +�⊤ +λ +≤ min +λ∈Rm ++ +max +�K +k=1 π +(juν +1) +k +≤1,π +(juν +1) +k +≥0 +K +� +k=1 +π(juν +1) +k +u⋆(juν +1) +k ++ +� +E +� +ρuν+1 +�� juν+1 +� +− +K +� +k=1 +π(juν +1) +k +b⋆(juν +1) +k +�⊤ +λ +≤ min +λ∈Rm ++ +E +� +� +max +�K +k=1 π +(juν +1) +k +≤1,π +(juν +1) +k +≥0 +K +� +k=1 +π(juν +1) +k +u⋆(juν +1) +k ++ +� +ρuν+1 − +K +� +k=1 +π(juν +1) +k +b⋆(juν +1) +k +�⊤ +λ +������ +juν+1 +� +� +≤ E +� +� +max +�K +k=1 π +(juν +1) +k +≤1,π +(juν +1) +k +≥0,ρuν+1−�K +k=1 π +(juν +1) +k +b +⋆(juν +1) +k +≥0 +K +� +k=1 +π(juν +1) +k +u⋆(juν +1) +k +������ +juν+1 +� +� += +K +� +k=1 +˜π +(juν+1) +k,ρuν+1 u⋆(juν +1) +k +. +Thus we have +E +� +(uν+1 − uν) OPT +T +� +≤E +� K +� +k=1 +˜π +(juν+1) +k,ρuν+1 u⋆(juν +1) +k +� ++ +� +2E +�� +E [I (auν ∈[K])| uν]γuν−1,σb(δ) +� ++ P +� +Gc +uν +�� OPT +ρT . +Under the event Guν+1, the policy ˜π +(juν+1) +k,ρuν+1 is a feasible solution to the bandit problem (12), +E +� K +� +k=1 +˜π +(juν+1) +k,ρuν+1 u⋆(juν +1) +k +� +≤E +� K +� +k=1 +˜π +(juν+1) +k,ρuν+1 u⋆(juν +1) +k +I +� +Guν+1 +� +� ++ P +� +Gc +uν+1 +� +≤E +� K +� +k=1 +˜π +(juν+1) +k,ρuν+1 ˜u +(juν+1) +k,uν+1 I +� +Guν+1 +� +� ++ P +� +Gc +uν+1 +� +≤E +� K +� +k=1 +�π +(juν+1) +k,uν+1 ˜u +(juν+1) +k,uν+1 I +� +Guν+1 +� +� ++ P +� +Gc +uν+1 +� +. + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Thus, for each ν ∈ [|U| − 1], +E +� +(uν+1 − uν) OPT +T +� +≤E +� K +� +k=1 +�π +(juν+1) +k,uν+1 ˜u +(juν+1) +k,uν+1 I +� +Guν+1 +� +� ++ P +� +Gc +uν+1 +� ++ +� +2E +�� +E [I (auν ∈[K])| uν]γuν−1,σb(δ) +� ++ P +� +Gc +uν +�� OPT +ρT . +Summing up over ν, +E +� +� +|U|−1 +� +ν=1 +(uν+1 − uν) OPT +T +� +� ≤E +�� +t∈U +K +� +k=1 +�π(jt) +k,t ˜u(jt) +k,t I (Gt) +� ++ +� +1 + OPT +ρT +� +T +� +t=1 +P (Gc +t ) ++ +� +� +|U|−1 +� +ν=1 +2E +�� +E [I (auν ∈[K])| uν]γuν−1,σb(δ) +� +� +� OPT +ρT +≤E +�� +t∈U +K +� +k=1 +�π(jt) +k,t ˜u(jt) +k,t I (Gt) +� ++ +� +1 + OPT +ρT +� +T +� +t=1 +P (Gc +t ) ++ 2 +� T +� +t=1 +E +�� +E [I (at ∈[K])]γt−1,σb(δ) +�� +OPT +ρT +≤E +�� +t∈U +K +� +k=1 +�π(jt) +k,t ˜u(jt) +k,t I (Gt) +� ++ +� +1 + OPT +ρT +� +T +� +t=1 +P (Gc +t ) ++ 2 +� +� +� +�TE +� T +� +t=1 +γt−1,σb(δ)2I (at ∈ [K]) +� +OPT +ρT , +where the last inequality holds by Cauchy-Schwartz inequality, By (45), +E +� +� +|U|−1 +� +ν=1 +(uν+1 − uν) OPT +T +� +� ≤E +�� +t∈U +K +� +k=1 +�π(jt) +k,t ˜u(jt) +k,t I (Gt) +� ++ +� +1 + OPT +ρT +� +T +� +t=1 +P (Gc +t ) ++ 2 +� +� +� +�TE +� T +� +t=1 +γt−1,σb(δ)2I (at ∈ [K]) +� +OPT +ρT +≤E +� T +� +t=1 +R�π +t +� ++ +� +2 + OPT +ρT +� +T +� +t=1 +P (Gc +t ) ++ 2 +� +� +� +�TE +� T +� +t=1 +γt−1,σb(δ)2I (at ∈ [K]) +� +OPT +ρT +2 +� +� +� +�TE +� T +� +t=1 +γt−1,σr(δ)2I (at ∈ [K]) +� +≤E +� T +� +t=1 +R�π +t +� ++ +� +2 + OPT +ρT +� +T +� +t=1 +P (Gc +t ) +2 +� +1 + OPT +ρT +� � +� +� +�TE +� T +� +t=1 +γt−1,σb∨σr(δ)2I (at ∈ [K]) +� + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Because the last choice of the algorithm happens at round τ, we have ρτ > 0 and u|U| = τ. And by definition, u1 = ξ. Thus +E +� +� +|U|−1 +� +ν=1 +(uν+1 − uν) OPT +T +� +� = E +�� +u|U| − u1 +� OPT +T +� += OPT +T +E [τ − ξ] . +Rearranging the terms +E +� T +� +t=1 +R�π +t +� +≥ E +� +� +|U|−1 +� +ν=1 +(uν+1 − uν) OPT +T +� +� − +� +2 + OPT +ρT +� +T +� +t=1 +P (Gc +t ) +− 2 +� +1 + OPT +ρT +� � +� +� +�TE +� T +� +t=1 +γt−1,σb∨σr(δ)2I (at ∈ [K]) +� +≥ OPT +T +E [τ − ξ] − +� +2 + OPT +ρT +� +T +� +t=1 +P (Gc +t ) +− 2 +� +1 + OPT +ρT +� � +� +� +�TE +� T +� +t=1 +γt−1,σb∨σr(δ)2I (at ∈ [K]) +� +, +completes the proof. +B.5. Proof of Lemma 5.3 +Proof. Let us fix δ ∈ (0, T −2) throughout the proof. +Step 1. Bounding the minimum eigenvalue of {Fν : ν ∈ [nT ]}: +By Lemma C.3, with probability at least 1 − Tδ, +1 +2KdFν = +1 +2Kd +ν +� +u=1 +˜Xk,τ(u) ˜X⊤ +k,τ(u) + 8K − 1 +K +log Jd +δ +⪰ +1 +4Kd +ν +� +u=1 +E +� +˜Xk,τ(u) ˜X⊤ +k,τ(u) +��� Hτ(u)−1 +� ++ 8K − 1 +K +log Jd +δ − log Jd +δ +⪰ +1 +4Kd +ν +� +u=1 +E +� +˜Xk,τ(u) ˜X⊤ +k,τ(u) +��� Hτ(u)−1 +� +, +for all ν ∈ [nT ]. By Assumption 2 and 3, +λmin +� +E +� +˜Xk,τ(u) ˜X⊤ +k,τ(u) +��� Hτ(u)−1 +�� +=λmin +� +� +� +� +� +p1EXk∼F1 +��K +k=1 XkX⊤ +k +� +0 +0 +0 +... +0 +0 +0 +pJExk∼FJ +��K +k=1 XkX⊤ +k +� +� +� +� +� +� +≥pmin min +j∈[J] λmin +� +EXk∼Fj +� K +� +k=1 +XkX⊤ +k +�� +≥pminKα. +Thus, with probability at least 1 − Tδ, +λmin(Fν) ≥1 +2λmin +� ν +� +u=1 +E +� +˜Xk,u ˜X⊤ +k,u +��� Hu−1 +�� +≥1 +2 +ν +� +u=1 +λmin +� +E +� +˜Xk,u ˜X⊤ +k,u +��� Hu−1 +�� +≥pminKαν +2 +, + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +for all ν ∈ [nT ]. +Step 2. Bounding the probability of Mt: +Under the event proved in Step 1, the event Mt is implied by +pminKαnt +2 +≥ 12Kd +� nt +� +ν=1 +96 (K − 1) log +� Jd +δ +� +αKpminν ++ 2 log Jd +δ +� +, +(52) +for all t ∈ [T]. The left hand side is bounded as +12Kd +� nt +� +ν=1 +96 (K − 1) log +� Jd +δ +� +αKpminν ++ 2 log Jd +δ +� +≤12 · 96Kd log +� Jd +δ +� +log nt +αpmin ++ 24Kd log Jd +δ +≤12 · 96Kd log +� Jd +δ +� +log T +αpmin ++ 24Kd log Jd +δ . +Plugging in (52) and rearranging the terms, +nt ≥ 96d log +�Jd +δ +� �24 log T +α2p2 +min ++ +1 +αpmin +� +, +implies the event Mt for all t ∈ [T] with probability at least 1 − Tδ. In other words, +P (Mc +t) ≤ P +� +nt < dMα,p,T log +�Jd +δ +�� ++ Tδ, +for all t ∈ [T], where Mα,p,T := 96 +� +24 log T +α2p2 +min + +1 +αpmin +� +. +Step 3. Bounding ξ: +Let ˜t = inft∈[T ]{Mt happens} be the first round that Mt happens. After round ˜t, the algorithm +skips the rounds until ρt > 0 holds and then pulls an action according to the policy. Thus, for the round ξ − 1, +(ξ − 1) ρ − +ξ−2 +� +s=1 +b(js) +as,s = (ξ − 1) ρ − +˜t +� +s=1 +b(js) +as,s ≤ 0. +Rearraging the terms, and taking expectation, +E [ξ] ≤ 1 + ρ−1E +� +� +˜t +� +s=1 +b(js) +as,s +� +� ≤ 1 + ρ−1E +�˜t +� +. +(53) +Now we need an upper bound for ˜t. For t ∈ [˜t − 1], the event Mt does not happen and the algorithm admits the arrival for +t ∈ [˜t]. Thus, nt = t for all t ∈ [˜t]. For t = ˜t − 1, the event M˜t−1 does not happen and +λmin +� +Fn˜t−1 +� +≤ 12Kd +�n˜t−1 +� +ν=1 +48 (K − 1) log +� Jd +δ +� +λmin(Fν) ++ 2 log Jd +δ +� +. +By the fact proved in Step 1, with probability at least 1 − Tδ, +pminKαn˜t−1 +2 +≤ 12Kd +�n˜t−1 +� +ν=1 +96 (K − 1) log +� Jd +δ +� +pminKαν ++ 2 log Jd +δ +� +. +Plugging in n˜t−1 = ˜t − 1 and rearranging the terms, +˜t − 1 ≤ 24d +pminα +� +� +� +˜t−1 +� +ν=1 +96 (K − 1) log +� Jd +δ +� +pminKαν ++ 2 log Jd +δ +� +� +� +≤ 24d +pminα +�96 log T +pminα + 2 log Jd +δ +� +. + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Then with probability at least 1 − Tδ, +˜t ≤1 + 96d log +�Jd +δ +� �24 log T +α2p2 +min ++ +1 +αpmin +� +:=1 + Mα,p,T d log +�Jd +δ +� +. +Thus, +E +�˜t +� +=E +� +˜tI +� +˜t < 1 + Mα,p,T d log +�Jd +δ +��� ++ E +� +˜tI +� +˜t ≥ 1 + Mα,p,T d log +�Jd +δ +��� +≤1 + dMα,p,T log +�Jd +δ +� ++ TP +� +˜t ≥ 1 + Mα,p,T d log +�Jd +δ +�� +≤1 + dMα,p,T log +�Jd +δ +� ++ T 2δ +Plugging in (53), +E [ξ] ≤1 + ρ−1E +�˜t +� +≤1 + 1 + dMα,p,T log +� Jd +δ +� ++ T 2δ +ρ +. +Step 4. Proving the lower bound for τ: +Let τ be the stopping time of the algorithm. Because the algorithm admits +arrival at round τ, we have ρτ > 0. From the resource constraint in the bandit problem (12), +K +� +k=1 +�π(jτ ) +k,τ +� +�b(jτ ) +k,τ − γτ−1,σb(δ) +√pjτ +1m +� +:= +K +� +k=1 +�π(jτ ) +k,τ ˜b(jτ ) +k,τ :≤ τρ − +τ−1 +� +s=1 +b(js) +as,s +Because algorithm stops at round τ, there exists an r ∈ [m] such that �τ +s=1 b(js) +as,s(r) ≥ Tρ(r). Rearranging the terms, +τρ ≥ +τ−1 +� +s=1 +b(js) +as,s(r) + +K +� +k=1 +�π(jτ ) +k,τ ˜b(jτ ) +k,τ (r) +≥Tρ − b(jτ ) +aτ ,τ(r) + +K +� +k=1 +�π(jτ ) +k,τ ˜b(jτ ) +k,τ (r) +=Tρ − b(jτ ) +aτ ,τ(r) + +K +� +k=1 +�π(jτ ) +k,τ b⋆(jτ ) +k,τ +(r) + +K +� +k=1 +�π(jτ ) +k,τ +� +˜b(jτ ) +k,τ (r) − b⋆(jτ ) +k,τ +(r) +� +≥Tρ − b(jτ ) +aτ ,τ(r) + +K +� +k=1 +�π(jτ ) +k,τ b⋆(jτ ) +k +(r) − +K +� +k=1 +�π(jτ ) +k,τ +���˜b(jτ ) +k,τ − b⋆(jτ ) +k +��� +∞ . +Taking expectation on both side, +E [τρ] ≥Tρ + E +� +−b(jτ ) +aτ ,τ(r) + +K +� +k=1 +�π(jτ ) +k,τ b⋆(jτ ) +k,τ +(r) +� +− E +� K +� +k=1 +�π(jτ ) +k,τ +���˜b(jτ ) +k,τ − b⋆(jτ ) +k +��� +∞ +� +=Tρ − E +� K +� +k=1 +�π(jτ ) +k,τ +���˜b(jτ ) +k,τ − b⋆(jτ ) +k +��� +∞ +� +=Tρ − E +� K +� +k=1 +�π(jτ ) +k,τ +���˜b(jτ ) +k,τ − b⋆(jτ ) +k +��� +∞ I (Eτ ∩ Mτ−1) +� +− E +� K +� +k=1 +�π(jτ ) +k,τ +���˜b(jτ ) +k,τ − b⋆(jτ ) +k +��� +∞ I +� +Ec +τ ∪ Mc +τ−1 +� +� +≥Tρ − 2E +� +γτ−1,σb(δ) +� +E [I (at ∈ [K])| τ] +� +− E +� K +� +k=1 +�π(jτ ) +k,τ +���˜b(jτ ) +k,τ − b⋆(jτ ) +k +��� +∞ I +� +Ec +τ ∪ Mc +τ−1 +� +� +, +(54) + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +where the last inequality holds by (44). +Because ˜b(jτ ) +k,τ (r) ≤ Tρ almost surely, +E +� K +� +k=1 +�π(jτ ) +k,τ +���˜b(jτ ) +k,τ − b⋆(jτ ) +k +��� +∞ I +� +Ec +τ ∪ Mc +τ−1 +� +� +≤TρP +� +Ec +τ ∪ Mc +τ−1 +� +=TρP (Ec +τ) +≤Tρ +� +4(m + 1)δ + 7T −1� +=7ρ + 4(m + 1)Tδ, +where the equality holds because the algorithm takes action according to the policy at round τ and the last inequality holds +by Theorem 4.2. from (54), +E [τρ] ≥Tρ − 7ρ + 4(m + 1)Tρδ − 2E +� +γτ−1,σb(δ) +� +E [I (at ∈ [K])| τ] +� +≥Tρ − 7ρ + 4(m + 1)Tρδ − 2γ1,σb(δ) +Rearranging the terms, +E [T − τ] ≤4(m + 1)Tδ + 7 + 2γτ−1,σb(δ) +ρ +. +Step 5. Proving a bound for the sum of probabilities +Because the algorithm admits the arrival when Mt−1 does not +happen, +Mc +t−1 = Mc +t−1 ∩ {at ∈ [K]} . +Then +P +� +Mc +t−1 +� +=P +� +Mc +t−1 ∩ {at ∈ [K]} +� +=P +� +Mc +t−1 ∩ {at ∈ [K]} ∩ +� +nt−1 ≥ Mα,p,T d log +�Jd +δ +��� ++ P +� +Mc +t−1 ∩ {at ∈ [K]} ∩ +� +nt−1 < Mα,p,T d log +�Jd +δ +��� +≤P +� +Mc +t−1 ∩ +� +nt−1 ≥ Mα,p,T d log +�Jd +δ +��� ++ P +� +{at ∈ [K]} ∩ +� +nt−1 < Mα,p,T d log +�Jd +δ +��� +≤Tδ + P +� +{at ∈ [K]} ∩ +� +nt−1 < Mα,p,T d log +�Jd +δ +��� +, +where the last inequality holds by the fact proved in Step 2. Summing over t ∈ [T], +T +� +t=1 +P +� +Mc +t−1 +� +≤T 2δ + +T +� +t=1 +P +� +{at ∈ [K]} ∩ +� +nt−1 < Mα,p,T d log +�Jd +δ +��� +=T 2δ + E +� T +� +t=1 +I (at ∈ [K]) I +� +nt−1 < Mα,p,T d log +�Jd +δ +��� +. +Set µ := Mα,p,T d log +� Jd +δ +� +and suppose +T +� +t=1 +I (at ∈ [K]) I (nt−1 < µ) > µ. +(55) + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Let τ(1) < τ(2) < · · · < τ(|A|) be the ordered admitted round in A := {t ∈ [T] : at ∈ [K]}. By definition, nτ(ν) = ν +for ν ∈ [|A|]. By (55), the event{at ∈ [K]} happens at least µ + 1 times over the horizon [T] and |A| > µ. For any +ν ∈ (µ, |A|],the number of admitted round is nτ(ν) > µ and +T −1 +� +t=ε +I (nt−1 < µ) I (at ∈ [K]) = +|A| +� +ν=1 +I +� +nτ(ν)−1 < µ +� +I +� +aτ(ν) ∈ [K] +� +≤ +|A| +� +ν=1 +I +� +nτ(ν)−1 < µ +� +I +� +nτ(ν) = nτ(ν)−1 + 1 +� += +|A| +� +ν=1 +I +� +nτ(ν)−1 < µ +� +I +� +ν = nτ(ν)−1 + 1 +� +≤ +|A| +� +ν=1 +I (ν − 1 < µ) , += +|A| +� +ν=1 +I (ν < µ + 1) +=µ, +which contradicts with (55). Thus +E +� T +� +t=1 +I (at ∈ [K]) I +� +nt−1 < Mα,p,T d log +�Jd +δ +��� +≤ µ := Mα,p,T d log +�Jd +δ +� +, +which proves, +T +� +t=1 +P +� +Mc +t−1 +� +≤ T 2δ + Mα,p,T d log +�Jd +δ +� +. +B.6. Proof of Theorem 5.1 +Proof. From Lemma 5.2, rearranging the terms, +R�π +T :=OPT − E +� T +� +t=1 +R�π +t +� +≤OPT +T +{T − E [τ − ξ]} ++ +� +2 + OPT +ρT +� +T +� +t=1 +P +� +Mc +t−1 ∪ Ec +t +� ++ 2 +� +� +� +�TE +� T +� +t=1 +γt−1,σr(δ)2I (at ∈ [K]) +� ++ 2 +� +� +� +�TE +� T +� +t=1 +γt−1,σb(δ)2I (at ∈ [K]) +� +OPT +ρT . + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +By Lemma 5.3, +E [ξ] ≤ 1 + 1+dMα,p,T log +� Jd +δ +� ++T 2δ +ρ +, +E [T − τ] ≤ 4(m + 1)Tδ + 7 + 2γτ−1,σb(δ) +ρ +. +By definition of γt,σ(δ), +E [T − τ] ≤4(m + 1)Tδ + 7 + 32√J log JKT + 8 +√ +2βσb(δ) +ρ +=4(m + 1)Tδ + 7 + 32√J log JKT + Cσ(δ) +√ +Jd +ρ +. +This implies +OPT +T +{T − E [τ − ξ]} +≤ OPT +Tρ +� +ρ + 4(m + 1)Tδ + 8 + 32 +� +J log +�JK +δ +� ++Cσb(δ) +√ +Jd + dMα,p,T log +�Jd +δ +� ++T 2δ +� +≤ OPT +Tρ +� +ρ + 8 + +� +5mT + T 2� +δ + 32 +� +J log +�JK +δ +� ++Cσb(δ) +√ +Jd + dMα,p,T log +�Jd +δ +�� +. +[Step 3. Bounding the sum of probability] Because T ≥ 8dα−1p−1 +min log JdT, by Theorem 4.2 and Lemma 5.3, +T +� +t=1 +P +� +Mc +t−1 ∪ Ec +t +� += +T +� +t=1 +� +P +� +Mc +t−1 +� ++ P (Mt−1 ∩ Ec +t ) +� +≤T 3δ + dMα,p,T log +�Jd +δ +� ++ +T +� +t=1 +P (Mt−1 ∩ Ec +t ) +≤T 3δ + dMα,p,T log +�Jd +δ +� ++ 8dα−1p−1 +min log JdT ++ +T +� +t=8dα−1p−1 +min log JdT +P (Mt−1 ∩ Ec +t ) +≤T 3δ + dMα,p,T log +�Jd +δ +� ++ 8dα−1p−1 +min log JdT + 4(m + 1)Tδ + 7. +By definition of γt,σ(δ) and βσ(δ), +E +� T +� +t=1 +γt−1,σr(δ)2I (at ∈ [K]) +� +=E +� +� +T +� +t=1 +� +16√J log JKT +√t − 1 ++ 4 +√ +2βσr(δ) +√nt−1 +�2 +I (at ∈ [K]) +� +� +≤E +� T +� +t=1 +� +16√J log JKT + 4 +√ +2βσr(δ) +�2 +nt−1 +I (at ∈ [K]) +� +≤E +� T +� +t=1 +� +16√J log JKT + 4 +√ +2βσr(δ) +�2 +nt−1 +I (nt = nt−1 + 1) +� +≤ +� +16 +� +J log JKT + 4 +√ +2βσr(δ) +�2 +log T, + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +where the first inequality holds by nt ≤ t almost surely. Thus by definition of βσ(δ) := 8 +√ +Jd + 96σ +� +Jd log 4 +δ , +2 +� +� +� +�TE +� T +� +t=1 +γt−1,σr(δ)2I (at ∈ [K]) +� +≤ +� +32 +� +J log JKT + 4 +√ +6βσr(δ) +� � +T log T +≤ +� +32 +� +J log JKT + Cσr(δ) +√ +Jd +� � +T log T, +where Cσ(δ) := 8 +√ +2 · +� +8 + 96σ +� +log 4 +δ +� +. Similarly, +2 +� +� +� +�TE +� T +� +t=1 +γt−1,σb(δ)2I (at ∈ [K]) +� +OPT +ρT +≤ +� +32 +� +J log JKT + Cσb(δ) +√ +Jd +� � +T log T OPT +ρT +Collecting the bounds, +R�π +T ≤ OPT +Tρ +� +ρ + 8 + +� +5mT + T 2� +δ + 32 +� +J log JKT +Cσb(δ) +√ +Jd + dMα,p,T log +�Jd +δ +�� ++ +� +2 + OPT +ρT +� � +T 3δ + dMα,p,T log +�Jd +δ +� ++ 4dα−1p−1 +min log JdT + 4(m + 1)Tδ + 7 +� ++ 2 +� +1 + OPT +ρT +� � +32 +� +J log JKT + Cσb∨σr(δ) +√ +Jd +� � +T log T +≤ +� +2 + OPT +ρT +� � � +96 +� +J log JKT + 3Cσr∨σr(δ) +√ +Jd +� � +T log T + 2dMα,p,T log +�Jd +δ +� ++ 4dα−1p−1 +min log JdT + 15 + 10mT 3δ +� +, +Plugging in δ = m−1T −3 proves (14). +C. Technical lemmas +Lemma C.1. (Azuma-Hoeffding’s inequality) Azuma (1967) If a super-martingale (Yt; t ≥ 0) corresponding to filtration +Ft, satisfies |Yt − Yt−1| ≤ ct for some constant ct, for all t = 1, . . . , T, then for any a ≥ 0, +P (YT − Y0 ≥ a) ≤ e +− +a2 +2 �T +t=1 c2 +t . +Thus with probability at least 1 − δ, +YT − Y0 ≤ +� +� +� +�2 log 1 +δ +T +� +t=1 +c2 +t. +Lemma C.2. For a sequence u1 ≥ u2 ≥ · · · ≥ un ≥ 0 and nonnegative real sequences {pi}i∈[n] and {qi}i∈[n] such that +�n +i=1 pi = �n +i=1 qi, if p1 > q1 then +n +� +i=1 +piui ≥ +n +� +i=1 +qiui. +Proof. When n = 1, p1u1 ≥ q1u1, for any u1 ≥ 0. Suppose for any sequence u1 ≥ u2 ≥ · · · ≥ un−1 ≥ 0 and nonnegative +real sequences {pi}i∈[n−1] and {qi}i∈[n−1] such that �n−1 +i=1 pi = �n−1 +i=1 qi, +p1 > q1 =⇒ +n−1 +� +i=1 +piui ≥ +n−1 +� +i=1 +qiui. + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +For a sequence u1 ≥ u2 ≥ · · · ≥ un ≥ 0 and nonnegative real sequences {pi}i∈[n] and {qi}i∈[n] such that �n +i=1 pi = +�n +i=1 qi, and p1 > q1, there exist k ∈ [n]\{1} such that pk < qk. In case of k = n, define a sequence +˜qi = qi, +∀i ∈ [n − 2] +˜qn−1 = qn−1 − pn + qn ≥ 0. +Then �n−1 +i=1 ˜qi = �n−1 +i=1 pi and +n +� +i=1 +piui = +n−1 +� +i=1 +piui + pnun +≥ +n−1 +� +i=1 +˜qiui + pnun += +n−1 +� +i=1 +qiui + (−pn + qn) un−1 + pnun +≥ +n−1 +� +i=1 +qiui + (−pn + qn) un + pnun += +n +� +i=1 +qiui. +In case of k ̸= n, denote a sequence +˜qi = qi, +∀i ∈ [n − 1]\{k} +˜qk = qk − pk + qn. +Then �n−1 +i=1 ˜qi = � +j̸=k pi and +n +� +i=1 +piui = +� +i̸=k +piui + pkuk +≥ +n−1 +� +i=1 +˜qiui + pkuk +≥ +n−1 +� +i=1 +qiui − pkuk + qnuk + pkuk += +n−1 +� +i=1 +qkuk + qnuk +≥ +n +� +i=1 +qkuk. +By induction, the proof is complete. +Lemma C.3. Let {Xτ : τ ∈ [t]} be a Rd×d-valued stochastic process adapted to the filtration {Fτ : τ ∈ [t]}, i.e., Xτ +is Fτ-measurable for τ ∈ [t]. Suppose Xτ is a positive definite symmetric matrices such thatλmax(Xτ) ≤ 1 +2.Then with +probability at least 1 − δ, +t +� +τ=1 +Xτ ⪰ 1 +2 +t +� +τ=1 +E [Xτ| Fτ−1] − log d +δ Id. + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +In addition, with probability at least 1 − δ, +t +� +τ=1 +Xτ ⪯ 3 +2 +t +� +τ=1 +E [Xτ| Fτ−1] + log d +δ Id. +Proof. This proof is an adapted version of Lemma 12.2 in Lattimore & Szepesv´ari (2020) for matrix stochastic process +using the argument of Tropp (2012). For the lower bound, It is sufficient to prove that +λmax +� +− +t +� +τ=1 +Xτ + 1 +2 +t +� +τ=1 +E [Xτ| Fτ−1] +� +≤ log d +δ , +with probability at least 1 − δ. By the spectral mapping theorem, +exp +� +λmax +� +− +t +� +τ=1 +Xτ + 1 +2 +t +� +τ=1 +E [Xτ| Fτ−1] +�� +≤λmax +� +exp +� +− +t +� +τ=1 +Xτ + 1 +2 +t +� +τ=1 +E [Xτ| Fτ−1] +�� +≤Tr +� +exp +� +− +t +� +τ=1 +Xτ + 1 +2 +t +� +τ=1 +E [Xτ| Fτ−1] +�� +. +Taking expectation on both side gives, +E exp +� +λmax +� +− +t +� +τ=1 +Xτ + 1 +2 +t +� +τ=1 +E [Xτ| Fτ−1] +�� +≤ETr +� +exp +� +− +t +� +τ=1 +Xτ + 1 +2 +t +� +τ=1 +E [Xτ| Fτ−1] +�� +=ETr +� +E +� +exp +� +− +t−1 +� +τ=1 +Xτ + 1 +2 +t +� +τ=1 +E [Xτ| Fτ−1] + log exp (−Xt) +������ Ft−1 +�� +≤ETr +� +exp +� +− +t−1 +� +τ=1 +Xτ + 1 +2 +t +� +τ=1 +E [Xτ| Fτ−1] + log E [exp (−Xt)| Ft−1] +�� +. +The last inequality holds due to Lieb’s theorem Tropp (2015). Because ex ≤ 1+ 1 +2xfor all x ∈ [−1/2, 0], and the eigenvalue +of −Xt lies in [−1/2, 0], we have +E [exp (−Xt)| Ft−1] ⪯ I − 1 +2E [Xt| Ft−1] ⪯ exp +� +−1 +2E [Xt| Ft−1] +� +, +by the spectral mapping theorem. Thus we have +E exp +� +λmax +� +− +t +� +τ=1 +Xτ + 1 +2 +t +� +τ=1 +E [Xτ| Fτ−1] +�� +≤ ETr +� +exp +� +− +t−1 +� +τ=1 +Xτ + 1 +2 +t +� +τ=1 +E [Xτ| Fτ−1] + log exp +� +−1 +2E [Xt| Ft−1] +��� += ETr +� +exp +� +− +t−1 +� +τ=1 +Xτ + 1 +2 +t +� +τ=1 +E [Xτ| Fτ−1] − 1 +2E [Xt| Ft−1] +�� += ETr +� +exp +� +− +t−1 +� +τ=1 +Xτ + 1 +2 +t−1 +� +τ=1 +E [Xτ| Fτ−1] +�� +≤ ... +≤ ETr (exp (O)) = d + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Now my Markov’s inequality, +P +� +λmax +� +− +t +� +τ=1 +Xτ + 1 +2 +t +� +τ=1 +E [Xτ| Fτ−1] +� +> log d +δ +� +≤ E exp +� +λmax +� +− +t +� +τ=1 +Xτ + 1 +2 +t +� +τ=1 +E [Xτ| Fτ−1] +�� +δ +d +≤ δ. +For the upper bound, we prove +λmax +� +t +� +τ=1 +Xτ − 3 +2 +t +� +τ=1 +E [Xτ| Fτ−1] +� +≤ log d +δ , +in a similar way using the fact that ex ≤ 1 + (3/2)x on x ∈ [0, 1/2]. +Lemma C.4. Suppose a random variable X satisfies E[X] = 0, and let Y be an σ-sub-Gaussian random variable. If +|X| ≤ |Y | almost surely, then X is 6σ-sub-Gaussian. +Proof. Because |X| ≤ |Y | +E +� X2 +6σ2 +� +≤E +� Y 2 +6σ2 +� +. +=1 + E +�� ∞ +0 +I (|Y | ≥ x) x +3σ2 e +x2 +6σ2 dx +� +≤1 + +� ∞ +0 +P (|Y | ≥ x) x +3σ2 e +x2 +6σ2 dx. +Because +P (|Y | ≥ x) =P (Y ≥ x) + P (−Y ≤ x) +≤2e− x2 +2σ2 , +we have +E +� X2 +6σ2 +� +≤1 + +� ∞ +0 +2x +3σ2 e− x2 +3σ2 dx +≤2. +Now for any λ ∈ R, +E [exp (λX)] =E +� ∞ +� +n=0 +(λX)n +n! +� +=1 + E +� ∞ +� +n=2 +(λX)n +n! +� +≤1 + E +� +λ2X2 +2 +∞ +� +n=2 +|λX|n−2 +(n − 2)! +� +≤1 + λ2 +2 E +� +X2 exp (|λX|) +� +. + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +Because 6σ2λ2 + +X2 +12σ2 ≥ |λX| , +E [exp (λX)] ≤1 + λ2 +2 exp +� +6σ2λ2� +E +� +X2 exp +� X2 +12σ2 +�� +=1 + 6σ2λ2 exp +� +6σ2λ2� +E +� X2 +12σ2 exp +� X2 +12σ2 +�� +≤1 + 6σ2λ2 exp +� +6σ2λ2� +E +� +exp +� X2 +6σ2 +�� +≤1 + 12σ2λ2 exp +� +6σ2λ2� +≤ +� +1 + 12σ2λ2� +exp +� +6σ2λ2� +≤ exp +�36 +2 σ2λ2 +� +. +Thus X is 6σ-sub-Gaussian. +Lemma C.5. (Lee et al., 2016, Lemma 2.3) Let {Nt} be a martingale on a Hilbert space (H, ∥·∥H). Then there exists a +R2-valued martingale {Pt} such that for any time t ≥ 0, ∥Pt∥2 = ∥Nt∥H and ∥Pt+1 − Pt∥2 = ∥Nt+1 − Nt∥H. +Lemma C.6. (A dimension-free bound for vector-valued martingales.) Let {Fs}t +s=0 be a filtration and {ηs}t +s=1 be a +real-valued stochastic process such that ηs is Fτ-measurable. Let {Xs}t +s=1 be an Rd-valued stochastic process where Xs +is F0-measurable. Assume that {ηs}t +s=1 are σ-sub-Gaussian as in Assumption 1. Then with probability at least 1 − δ, +����� +t +� +s=1 +ηsXs +����� +2 +≤ 12σ +� +� +� +� +t +� +s=1 +∥Xs∥2 +2 +� +log 4t2 +δ . +(56) +Proof. Fix a t ≥ 1. For each s = 1, . . . , t, we have E [ηs| Fs−1] = 0 and Xs is F0-measurable. Thus the stochastic process, +� u +� +s=1 +ηsXs +�t +u=1 +(57) +is a (Rd, ∥·∥2)-martingale. Since (Rd, ∥·∥2) is a Hilbert space, by Lemma C.5, there exists an R2-martingale {Mu}t +u=1 +such that +����� +u +� +s=1 +ηsXs +����� +2 += ∥Mu∥2 , ∥ηuXu∥2 = ∥Mu − Mu−1∥2 , +(58) +and M0 = 0. Set Mu = (M1(u), M2(u))⊤. Then for each i = 1, 2, and u ≥ 2, +|Mi(u) − Mi(u − 1)| ≤ ∥Mu − Mu−1∥2 += ∥ηuXu∥2 += |ηu| ∥Xu∥2 , +almost surely. By Lemma C.4, Mi(u) − Mi(u − 1) is 6σ-sub-Gaussian. By Lemma C.1, for x > 0, +P (|Mi(t)| > x) =P +������ +t +� +u=1 +Mi(u) − Mi(u − 1) +����� > x +� +≤2 exp +� +− +x2 +72tσ2 �t +s=1 ∥Xs∥2 +2 +� +, +for each i = 1, 2. Thus, with probability 1 − δ/2, +Mi(t)2 ≤ 72 +� +t +� +s=1 +∥Xs∥2 +2 +� +σ2 log 4 +δ . + +Improved Algorithms for Multi-period Packing Problems with Bandit Feedback +In summary, with probability at least 1 − δ/2, +����� +t +� +τ=1 +ηsXs +����� +2 += +� +M1(t)2 + M2(t)2 ≤ 6σ +� +� +� +� +t +� +s=1 +∥Xs∥2 +2 +� +2 log 4t2 +δ . + diff --git a/39FST4oBgHgl3EQfZTjC/content/tmp_files/load_file.txt b/39FST4oBgHgl3EQfZTjC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a2d0674f885fe9af744d39c1cf37c0eb3e693d4e --- /dev/null +++ b/39FST4oBgHgl3EQfZTjC/content/tmp_files/load_file.txt @@ -0,0 +1,1451 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf,len=1450 +page_content='Improved Algorithms for Multi-period Multi-class Packing Problems with Bandit Feedback Wonyoung Kim 1 Garud Iyengar 1 Assaf Zeevi 1 Abstract We consider the linear contextual multi-class multi-period packing problem (LMMP) where the goal is to pack items such that the total vector of consumption is below a given budget vector and the total value is as large as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We consider the setting where the reward and the con- sumption vector associated with each action is a class-dependent linear function of the context, and the decision-maker receives bandit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' LMMP includes linear contextual bandits with knapsacks and online revenue management as spe- cial cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We establish a new more efficient esti- mator which guarantees a faster convergence rate, and consequently, a lower regret in such problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We propose a bandit policy that is a closed-form function of said estimated parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' When the contexts are non-degenerate, the regret of the pro- posed policy is sublinear in the context dimen- sion, the number of classes, and the time hori- zon T when the budget grows at least as √ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We also resolve an open problem posed in Agrawal & Devanur (2016), and extend the result to a multi- class setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Our numerical experiments clearly demonstrate that the performance of our policy is superior to other benchmarks in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Introduction In the multi-period packing problem (MPP) the decision- maker “packs” the arrivals so that the total consumption across a set a resources is below a given budget vector and the reward is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' A variant of the packing problem, where items consume multiple resources and the decisions must be made sequentially with bandit feedback for a fixed time horizon, is known as bandits with knapsacks (Agrawal & Devanur, 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Badanidiyuru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Immorlica et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' MPPs also arise in online revenue 1Columbia University, New York, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Correspondence to: <>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Preliminary work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Under review by the International Conference on Machine Learning (ICML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Do not distribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' management (Besbes & Zeevi, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Ferreira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' MPPs in the literature assume that all arrivals belong to a single class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' However, in several application domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', operations, healthcare, and e-commerce), the arrivals are heterogeneous, and personalizing decisions to each distinct population or class is of paramount importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In this paper we consider a class of linear multi-class multi-period packing problems (LMMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' At each round, there is a single arrival that belongs to one of J classes, and the decision- maker observes the d-dimensional context and the cost for K different available actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The outcome of selecting an action is a random sample of the reward and a consumption vector for m resources with an expected value that is a class- dependent linear function of the d-dimensional contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The goal of the problem is to minimize the cumulative regret over a time horizon T while ensuring that the total resource consumed is at most B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The LMMP problem is a generalization of several prob- lems including linear contextual bandits with knapsacks (LinCBwK) introduced by Agrawal & Devanur (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' They proposed an online mirror descent-based algorithm that achieves ˜O(OPT/B · d √ T) regret when the budget B for each of the m resources is Ω( √ dT 3/4), where OPT is the reward obtained by the oracle policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Although the regret bound is meaningful for B ≥ Ω(d √ T), establishing the regret bound for smaller budget values was left as an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (2011) established a regret bound sublinear in d for the linear contextual bandit setting, which is a special case of LinCBwK with no budget constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus, the following question remained open: “Is there an algorithm for LinCBwK that achieves sublinear dependence on d with budget B = Ω( √ T)?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We propose a novel algorithm and an improved estimation strategy that settles this open problem and generalizes the result to the more general class of LMMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The proposed algorithm achieves ˜O(OPT/B √ JdT) regret with budget B = Ω( √ JdT) under non-degenerate contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' While re- gret of the existing algorithms grows linearly in the number of classes J, our estimator is able to pool data from differ- ent classes and avoids linear dependence on J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' To reiterate, the improved regret bound results from the novel estimator which yields faster convergence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='13791v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ML] 31 Jan 2023 Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Our main contributions are summarized as follows: We propose a new problem class – linear multi-class multi-period packing problems (LMMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' This problem generalizes a variety of problems including LinCBwK and online revenue management problems to the multi- class setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We propose a novel estimator that uses contexts for all actions (including the contexts in skipped rounds) and yields O( � Jd/n) convergence rate for J classes, context dimension d, and n admitted arrivals (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We propose a novel AMF (Allocate to the Maximum First) algorithm which achieves ˜O(OPT/B √ JdT) regret with budget B = Ω( √ JdT) where OPT is the reward obtained by oracle policy (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For the single class setting with J = 1, we improve the existing bound by √ d and show that the bound is valid when B = Ω( √ dT), and thus resolving an open problem posed in Agrawal & Devanur (2016) regarding LinCBwK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We evaluate our proposed algorithm on a suite of syn- thetic experiments and demonstrate its superior perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' All proofs omitted from the front matter can be found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Related Works There are two streams of work that are relevant for LMMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In online revenue management literature, Gallego & Van Ryzin (1994) introduced the dynamic pricing problem where the demand is a known function of price (action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Bes- bes & Zeevi (2009) and Besbes & Zeevi (2012) extended the problem under unknown demands with multiple resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Ferreira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (2018) proposed a Thompson sampling-based algorithm and extended it to contextual ban- dits with knapsacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' When the expected demand is a linear function of the price vector, the dynamic pricing problem is a special case of linear contextual bandits with knap- sack (LinCBwK) proposed by Agrawal & Devanur (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The LinCBwk is a common generalization of bandits with knapsacks (Badanidiyuru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Immorlica et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', 2021) and online stochastic packing problems (Feld- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Agrawal & Devanur, 2014b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Devanur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Recently, Sankararaman & Slivkins (2021) proved a logarithmic regret bound for LinCBwK when there exists a problem-dependent gap between the reward of the optimal action and the other actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Instead of the gap assump- tion, we require non-degeneracy of the stochastic contexts (see Assumption 3 for a precise definition) to obtain a re- gret bound sublinear in d and extends to the case when the contexts are generated from J different class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Amani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (2019) proposed a variant of LinCBwK where the selected action must satisfy a single constraint with high probability in all rounds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', LinCBwK with anytime con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Moradipari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (2021) and Pacchiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (2021) proposed a Thompson sampling-based algorithm and an upper confidence bound-based algorithm, respectively, for LinCBwK with a single anytime constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (2021) highlighted the difference between global and anytime con- straints, and proposed an pessimistic-optimistic algorithm for the anytime constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We focus on the global con- straints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' however, we note that the extension to the anytime constraints is straightforward with minor modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Notation Let R+ denote the set of positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For two real numbers a, b ∈ R, we write a ∧ b := min{a, b} and a ∨ b := max{a, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For a natural number N, let [N] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Linear Multi-period Packing Problem Let [J] denote the set of classes with arrival probabilities p = {pj}j∈[J], where pmin := minj∈[J] pj > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In each round t ∈ [T], the covariates {x(j) k,t ∈ [0, 1]d : k ∈ [K]} and costs {c(j) k,t ∈ [0, 1] : k ∈ [K]} are drawn from a class- specific distribution Fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We assume that the class arrival probabilities p are known to the decision-maker;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' however, the distributions {Fj}j∈[J] are not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' At time t ∈ [T], the decision-maker observes an arrival of the form (jt, {x(jt) k,t , c(jt) k,t : k ∈ [K]}), where jt ∈ [J] is the arrived class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Upon observing the arrival, the decision- maker can either take one of K different actions or skip the arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' When the arrival is skipped, the decision-maker does not obtain any rewards or consume any resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' When the decision-maker chooses an action at ∈ [K], the reward and consumption of the resource are given by E � r(jt) at,t ��� Ht � = � θ(jt) ⋆ �⊤ x(jt) at,t ∈ [−1, 1], E � b(jt) at,t ��� Ht � = � W (jt) ⋆ �⊤ x(jt) at,t ∈ [0, 1]m, for some unknown class-specific parameters θ(j) ⋆ ∈ [0, 1]d and W (j) ⋆ ∈ [0, 1]d×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The sigma algebra Ht is generated by the class-specific variables {js, x(js) k,s , c(js) k,s , : s ∈ [t], k ∈ [K]}, actions {as : s ∈ At}, consumption vectors {b(js) as,s : s ∈ At−1} and rewards {r(js) as,t : s ∈ At−1}, where At is the rounds admitted by the decision-maker until round t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The process terminates at the horizon T or runs out of Improved Algorithms for Multi-period Packing Problems with Bandit Feedback budget B ∈ Rm + for some resources r ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The problem reduces to LinCBwK when the number of class is J = 1 and the costs are c(j) k,t = 0 Let ρ = B/T ∈ Rm + denote per-period budget for m re- sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Without loss of generality, one can assume that ρ(r) = ρ for all r ∈ [m], By rescaling W (j) ⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We assume that ρ is known to the decision-maker and B is possibly un- known at first, but known at the end of the round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' This case happens when the total budget B is difficult to count in the early rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' When ρ is not available, the decision-maker requires B and T to compute ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' However, this assumption is more practical than in Agrawal & Devanur (2016) where B and OPT must be known to the decision-maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' When OPT is unknown, they estimate OPT with √ T number of rounds, which requires the knowledge of T and budget B = Ω( √ dT 3 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Instead of estimating OPT, we use ρ to avoid the required budget B = Ω( √ dT 3 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We benchmark the performance of the decision-maker’s pol- icy relative to that of an oracle who knows the distributions {Fj : j ∈ [J]} and the parameters {θ(j) ⋆ , W (j) ⋆ : j ∈ [J]}, but does not know the arrivals {(jt, x(j) k,t, c(j) k,t) : t ∈ [T]} a-priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In this case, the optimal static policy for the oracle {π⋆(j) k : j ∈ [J], k ∈ [K]} is the solution to the following optimization problem: max π(j) k J � j=1 K � k=1 pjπ(j) k E(xk,ck)∼Fj �� θ(j) ⋆ �⊤ xk − ck � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' j � j=1 K � k=1 pjπ(j) k Exk∼Fj �� W (j) ⋆ �⊤ xk � ≤ ρ, K � k=1 π(j) k ≤ 1, ∀j ∈ [J], π(j) k ≥ 0, ∀j ∈ [J], ∀k ∈ [K], (1) Let π⋆ denote the optimal oracle policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then the expected reward obtained by the oracle is OPT := T J � j=1 K � k=1 pjπ⋆(j) k E(xk,ck)∼Fj �� θ(j) ⋆ �⊤ xk − ck � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Let π := {π(j) k,t : j ∈ [J], k ∈ [K], t ∈ [T]} denote the adapted (randomized) control policy of the decision-maker, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' she chooses action k ∈ [K] when the arrival at time t ∈ [T] belongs to class j ∈ [J].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Note that �K k=1 π(j) k,t ≤ 1 in order to allow the decision-maker to skip an arrival and save the inventory for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Our goal is to compute a policy that minimizes the cumulative regret Rπ T defined as Rπ T := OPT − E � T � t=1 Rπ t � , where Rπ t := �K k=1 π(jt) k,t E �� θ(jt) ⋆ �⊤ x(jt) k,t − c(jt) k,t � is the expected reward obtained by policy π at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For the LMMP problem, we assume the following regularity conditions on the stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (Sub-Gaussian and bounded errors) For each t ∈ [T], the error of the reward ηk,t = r(jt) k,t − � θ(jt) ⋆ �⊤ x(jt) k,t is conditionally zero-mean σr-sub-Gaussian for a fixed constant σr ≥ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' E [exp (vηk,t)| Ht] ≤ exp � v2σ2 r 2 � for all v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For the consumption vectors, E � v⊤{b(jt) k,t − (W (jt) ⋆ )⊤x(jt) k,t } ��� Ht � ≤ exp( ∥v∥2 2σ2 b 2 ) for all v ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (Independently distributed contexts and costs) The set of contexts {x(j) k,t : k ∈ [K]} and {c(j) k,t : k ∈ [K]} are generated independently over t ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The contexts and cost in the same round and class can be corre- lated with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (Positive definiteness of average covari- ances) For each t ∈ [T] and j ∈ [J], there exists α > 0, such that λmin � E � 1 K K � k=1 x(j) k,t � x(j) k,t �⊤ �� ≥ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Assumptions 1 and 2 are standard in stochastic contex- tual bandits with knapsacks literature (Agrawal & Devanur, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Sankararaman & Slivkins, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Sivakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In the multi-class case, Assumption 2 implies that all the contexts are drawn independently over time steps, but their distribution may vary depending on the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Assump- tion 3 implies that the density of the covariate distribution is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Recent contextual bandit literature (without constraints) exploits Assumption 3 to improve the depen- dency of d on the regret bound (Bastani & Bayati, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Bastani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The contexts with independent Gaussian perturbation used in Kannan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Sivakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 2022) satisfy the Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Proposed Method In this section, we present our proposed estimator for the parameters {θ(j) ⋆ , W (j) ⋆ : j ∈ [J]} and the proposed bandit policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Proposed Estimator In sequential decision-making problems with contexts, the decision-maker observes the contexts for all actions, but the reward for only selected actions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' the rewards for unselected actions remain missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Kim & Paik (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Dimakopoulou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (2021) use doubly robust (DR) method to handle the missing rewards for the linear contextual bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' However, extensions to LinCBwK or LMMP problem have not explored yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We adapt the DR method to the LMMP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For each n ∈ N, let τ(n) be the round when the n-th admission happens (recall the bandit policy allows for skipping some arrivals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Clearly, n ≤ τ(n) < τ(n + 1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Let Θ⋆ := � � � � θ(1) ⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' θ(J) ⋆ � � � � , W⋆ := � � � � W (1) ⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' W (J) ⋆ � � � � , ˜Xk,n := � � � � � � 0d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' x (jτ(n)) k,τ(n) 0d � � � � � � denote the stacked parameter vectors, and zero padded con- texts where x (jτ(n)) k,τ(n) is located after the j − 1 of 0d vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then the score for the ridge estimator for Θ⋆ at round τ(n) is: n � ν=1 � r (jτ(ν)) aτ(ν),τ(ν) − Θ⊤ ˜Xaτ(ν),ν � ˜Xaτ(ν),ν = n � ν=1 K � k=1 I � aτ(ν) = k � � r (jτ(ν)) k,τ(ν) − Θ⊤ ˜Xk,ν � ˜Xk,ν, where Θ ∈ RJ·d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Dividing the score by the probability π (jτ(ν)) k,τ(ν) gives the inverse probability weighted (IPW) score, n � ν=1 K � k=1 I � aτ(ν) = k � π (jτ(ν)) k,τ(ν) � r (jτ(ν)) k,τ(ν) − Θ⊤ ˜Xk,ν � ˜Xk,ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' To obtain the DR score, Bang & Robins (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (2021) proposed to subtract the nuisance tangent space gen- erated by an imputed estimator ˇΘ: n � ν=1 K � k=1 I � aτ(ν) = k � π (jτ(ν)) k,τ(ν) � ˜X⊤ k,ν ˇΘ − ˜X⊤ k,νΘ � ˜Xk,ν, from the IPW score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then the following DR score n � ν=1 K � k=1 � r DR(jτ(ν)) k,τ(ν) − ˜X⊤ k,νΘ � ˜Xk,ν, (2) is obtained where rDR( ˇΘ) k,ν :=I � aτ(ν)=k � π (jτ(ν)) k,τ(ν) r (jτ(ν)) k,τ(ν) + � � �1−I � aτ(ν)=k � π (jτ(ν)) k,τ(ν) � � � ˜X⊤ k,ν ˇΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (3) The score (2) has a similar form with the score equation for the ridge estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The difference with the ridge es- timator is that it uses contexts for all actions k ∈ [K] with the pseudo-reward rDR( ˇΘ) k,ν which is unbiased, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', E[rDR( ˇΘ) k,ν ] = E[r (jτ(ν)) k,τ(ν) ], for any given ˇΘ ∈ RJ·d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Adding the ℓ2 regularization norm and solving (2) leads to the DR estimator: � n � ν=1 K � k=1 ˜Xk,ν ˜X⊤ k,ν+IJ·d �−1� n � ν=1 K � k=1 ˜Xk,νrDR( ˇΘ) k,τ(ν) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The main advantage of the DR estimator is that it uses contexts from all K actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' However, in our policy, some π (jτ(ν)) k,τ(ν) can be zero, and therefore, the pseudo-reward (3) is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' To handle this problem, we propose to introduce a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' After taking an action at round τ(ν) and observing the selected action aτ(ν), the decision-maker samples hν from the distribution: φk,ν:=P � hν = k| Hτ(n) � = � � � 1− 16(K−1) log( dJ δ ) λmin(Fν) k=aτ(ν) 16 log( dJ δ ) λmin(Fν) k̸=aτ(ν) (4) where Fν := �ν,K i,k=1 ˜Xk,i ˜X⊤ k,i+16d(K −1) log � dJ δ � IJ·d is the Gram matrix of contexts from ν admitted rounds and δ ∈ (0, 1) is the confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We would like to emphasize that hν is sampled after observing the actions aτ(ν) and does not affect the policies until round τ(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Sampling the random variables hν after choosing actions is motivated by bootstrap methods (Efron & Tibshirani, 1994) and resampling methods (Good, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' To obtain the unbiased pseudo-rewards similar to (3), we resample the action with another non-zero probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The probabil- ities {φk,ν : k ∈ [K]} is designed to control the level of exploration and exploitation for future rounds based on the ratio of confidence level to the number of admitted rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' When the minimum eigenvalue of Fν is small compared to log(1/δ), the distribution of hν is less concentrated on aτ(ν) and tends to explore other actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' As ν increases, the probabilities {φk,ν : k ∈ [K]} concentrates on aτ(ν), and the decision-maker tends to exploit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Since we obtain non-zero probabilities {φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν : k ∈ [K],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' ν ∈ [n]},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' we define novel unbiased pseudo-rewards: ˜rk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν :=I (hν =k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν r (jτ(ν)) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ(ν) + � 1− I (hν =k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � ˜X⊤ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='νˇΘn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (5) where the imputation estimator ˇΘn is an IPW estimator with Improved Algorithms for Multi-period Packing Problems with Bandit Feedback new probabilities: ˇΘt :=A−1 n � � ν∈Ψn K � k=1 I (hν =k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='νr (jτ(ν)) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ(ν) + � ν /∈Ψn ˜Xaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='νr (jτ(ν)) aτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ(ν) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' An := � ν∈Ψn K � k=1 I (hν =k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜X⊤ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + � ν /∈Ψn ˜Xaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜X⊤ aτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + IJ·d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Ψn := � ν ∈ [n] : hν = aτ(ν) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The set Ψn is introduced because we cannot observe I(hν=k) φk,ν r (jτ(ν)) k,τ(ν) in case of hν ̸= aτ(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In other words, we use the pseudo-rewards in (5) only at the rounds that satisfy hν = aτ(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then our estimator with n admitted samples is defined as �Θn :=V −1 n � � � � ν∈Ψn K � k=1 ˜Xk,ν˜rk,ν + � ν /∈Ψn ˜Xaτ(ν),νr (jτ(ν)) aτ(ν),τ(ν) � � � Vn := � ν∈Ψn K � k=1 ˜Xk,ν ˜X⊤ k,ν + � ν /∈Ψn ˜Xaτ(ν),ν ˜X⊤ aτ(ν),ν +IJ·d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (6) Analogous to the construction of (6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' we can also define the estimator for the resource consumption parameters {W (j) ⋆ : j ∈ [J]},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' � Wn :=V −1 n � � ν∈Ψn K � k=1 ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜b⊤ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν+ � ν /∈Ψn ˜Xaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='νb (jτ(ν))⊤ aτ(ν)τ(ν) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (7) where the pseudo-consumption vectors and the imputation estimator are ˜bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν := I (hν =k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν b (jτ(ν)) aτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ(ν)+ � 1− I (hν =k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � ˇ W⊤ n ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' ˇ Wn := A−1 n � � ν∈Ψn K � k=1 I (hν =k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � b (jτ(ν)) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν �⊤ + � ν /∈Ψn ˜Xaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � b (jτ(ν)) aτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ(ν) �⊤ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The two estimators use the novel Gram matrix Vn defined in (6) consist of contexts from all K actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Now, we present estimation error bounds normalized by the novel Gram matrix Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (Self-normalized bound for the estimator) Suppose Assumptions 1 and 2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For each t ∈ [T], denote nt the number of admitted arrivals until round t and Ψnt := {ν ∈ [nt] : hν = aτ(ν)}, where hν is defined in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Suppose Fnt := �nt ν=1 �K k=1 ˜Xk,ν ˜X⊤ k,ν + 16d(K − 1) log Jd δ IJ·d satisfies λmin(Fnt)≥12Kd � nt � ν=1 48(K−1) log � Jd δ � λmin(Fν) +2 log Jd δ � , (8) for δ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For each r ∈ [m] , let � Wnt,r and W⋆,r be the r-th column of � Wnt and W⋆, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Denote βσ(δ) := 8 √ Jd + 96σ � Jd log 4 δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then with probability at least 1 − 4(m + 1)δ, ����Θnt − Θ∗��� Vnt ≤βσr(δ), max r∈[m] ���� Wnt,r − W⋆,r ��� Vnt ≤βσb(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (9) The widely used self-normalized bound in Abbasi-Yadkori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (2011) uses the Gram matrix consisting of selected contexts only, while our bounds are normalized by Vnt This change in the Gram matrix enables us to develop a fast convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The condition (8) is required for the eigenvalues of the Gram matrix Fnt to be large so that the probability φaτ(ν),ν is large and the estimators use the pseudo rewards and pseudo consumption vectors for most of the rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We show in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3 that the condition (8) requires at most rounds logarithmic in T, and does not affect the main order of the regret bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Using the novel estimators, we define the estimates for utility and resource consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Denote C(j) t := {s ∈ [t] : js = j} and �u(j) k,t := ���C(j) t ��� −1 � s∈C(j) t �� �θ(j) t−1 �⊤ x(j) k,s − c(j) k,s � , �b(j) k,t := ���C(j) t ��� −1 � s∈C(j) t � � W (j) t−1 �⊤ x(j) k,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (10) The estimates (10) use the average of contexts in the same class to estimate the expected value over the context dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In this way, the decision-maker effectively uses previous contexts in all rounds including the skipped rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Next, we establish the convergence rate for the estimators �u(j) k,t and �b(j) k,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (Convergence rate for the estimates) Suppose Assumptions 1-3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Denote the expected utility u⋆(j) k := E(xk,ck)∼Fj �� θ(j) ⋆ �⊤ xk − ck � and consumption b⋆(j) k := Exk∼Fj �� W (j) ⋆ �⊤ xk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Set γt,σ(δ) := 16√ J log(JKT ) √ t + 4 √ 2βσ(δ) √nt , where nt is the number of admitted arrivals until Improved Algorithms for Multi-period Packing Problems with Bandit Feedback round t and βσ(δ) is defined in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Suppose t ≥ 8dα−1p−1 min log JT, δ ∈ (0, T −1) and Fnt satisfies (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then with probability at least 1 − 4(m + 1)δ − 7T −1, � � � � J � j=1 pj max k∈[K] ���u⋆(j) k − �u(j) k,t+1 ��� 2 ≤ γt,σr(δ), � � � � J � j=1 pj max k∈[K] ���b⋆(j) k − �b(j) k,t+1 ��� 2 ∞ ≤ γt,σb(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (11) The convergence rate of the estimates is O( √ Jdn−1/2 t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In deriving the fast rate, the novel Gram matrix Vnt plays a significant role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' To prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2, we bound the sum of squared maximum prediction error as follows: 1 nt � s∈Ψnt max k∈[K] �� θ(j) ⋆ −�θ(j) t �⊤ x(j) k,s �2 = 1 nt � s∈Ψnt max k∈[K] � θ(j) ⋆ −�θ(j) t � � x(j) k,sx(j) k,s �⊤� θ(j) ⋆ −�θ(j) t � ≤ 1 nt � s∈Ψnt � θ(j) ⋆ −�θ(j) t � � K � k=1 x(j) k,sx(j) k,s �⊤� θ(j) ⋆ −�θ(j) t � ≤ 1 nt ���θ(j) ⋆ −�θ(j) t ��� 2 Vnt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Such a bound is not available if the Gram matrix is con- structed using only contexts corresponding to selected ac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In this way, we obtain faster convergence rate for the estimates for utility and consumption vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Proposed Algorithm Let (K + 1)-th action denote skipping the arrival and π(j) K+1,t := P (Skip the round t| Ht) denote the probabil- ity of skipping the arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Since the decision-maker must choose an action or skip the round, we have �K+1 k=1 π(j) k,t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' When the decision-maker skips round t, we set x(j) K+1,t := 0, c(j) K+1,t := 0, and b(j) K+1,t := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In round t, the randomized bandit policy is given by the optimal solution of the follow- ing optimization problem: max π(jt) k,t K+1 � k=1 π(jt) k,t � �u(jt) k,t + γt−1,σr(δ) √pjt I (k ∈ [K]) � , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' K+1 � k=1 π(jt) k,t � �b(jt) k,t − γt−1,σb(δ) √pjt 1m � ≤ ρt ∨ 0, K+1 � k=1 π(jt) k,t = 1, π(jt) k,t ≥ 0, ∀k ∈ [K + 1], (12) Algorithm 1 Allocate to the Maximum First algorithm (AMF) INPUT: confidence lengths γθ, γb > 0, confidence level δ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Initialize F0 := 16d(K − 1) log Jd δ IJ·d, ρ1 := ρ, �Θ0 := 0J·d, � W0 := 0J·d×m for t = 1 to T do Observe arrival (jt, {x(jt) k,t , c(jt) k,t }k∈[K]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' if Ft−1 does not satisfy (8) then Take action at = arg maxk∈[K] ρ∥�b(jt) k,t ∥−1 ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' else Compute �u(jt) k,t and �b(jt) k,t with �θ(jt) t−1 and � W(jt) t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Compute ˜u(jt) k,t := �u(jt) k,t + γθ √nt and ˜b(jt) k,t := �b(jt) k,t − γb √nt 1m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Take action at with the policy �π(jt) 1,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' , �π(jt) K+1,t de- fined in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' end if if at ∈ [K] then Observe r(jt) at,t and b(jt) at,t, then estimate �Θt and � Wt as in (6) and (7), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Update Ft = Ft−1 + �K k=1 ˜Xk,t ˜X⊤ k,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' end if Update available resource ρt+1 = ρt + ρ − b(jt) at,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' if �t s=1 b(js) as,s ≥ Tρ then Exit end if end for where ρt := tρ − �t−1 s=1 b(js) as,s is the difference between the used resources and planned budget until round t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The algorithm is optimistic in that it uses upper confidence bound (UCB) in rewards and lower confidence bound (LCB) in consumption while it regulates the consumption to be less than tρ with ρt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In this way, the problem (12) balances between admitting the arrivals and saving the resources for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Next, we show that the optimal solution (12) is available in a closed-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (Optimal policy for bandit) Let ˜u(jt) k,t := �u(jt) k,t + p−1/2 jt γt−1,σr(δ)I (k ∈ [K]) and ˜b(jt) k,t (r) := �b(jt) k,t (r) − p−1/2 jt γt−1,σb(δ), for r ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For i ∈ [K + 1], let ˜u(jt) k⟨i⟩,t be an sequence of ordered variables of ˜u(jt) k,t in decreasing order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' ˜u(jt) k⟨1⟩,t ≥ ˜u(jt) k⟨2⟩,t ≥ · · · ≥ ˜u(jt) k⟨K+1⟩t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' When there is a tie between ˜u(jt) k⟨i⟩,t and ˜u(jt) k⟨i+1⟩,t, the index k⟨i⟩ with the higher value for � � min r∈[m] ρt(r) ∨ 0 − �i−1 h=1 �π(jt) k⟨h⟩,t˜b(jt) k⟨h⟩,t(r) ˜b(jt) k⟨h⟩,t(r) � � Improved Algorithms for Multi-period Packing Problems with Bandit Feedback goes first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then the policy defined as, �π(jt) k⟨1⟩,t = � � min r∈[m] ρt(r)∨0 ˜b(jt) k⟨1⟩,t(r) � � ∧ 1, �π(jt) k⟨i⟩,t = � �min r∈[m] ρt(r)∨0−�i−1 h=1�π(jt) k⟨h⟩,t˜b(jt) k⟨h⟩,t(r) ˜b(jt) k⟨i⟩,t(r) � � ∧ � 1 − i−1 � h=i �π(jt) k⟨h⟩,t � , ∀i ∈ [2, K + 1], (13) is the optimal solution to (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Since the objective function of (12) is linear, we can obtain the maximum value by permuting the objective coefficients in decreasing order and allocating the greatest possible prob- ability value in decreasing order of the objective coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Note that �π(jt) k,t is automatically set to zero when the utility is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' This is because of the probability of skipping the arrival, �π(jt) K+1,t = 1−�l−1 h=1 �π(jt) k⟨h⟩,t, when ˜u(jt) K+1,t is the l-th largest weighted utility function and all the remaining probability is allocated to �π(jt) K+1,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Therefor, the probabili- ties for actions k with ˜u(jt) k,t < ˜u(jt) K+1,t := 0 are all zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Our proposed algorithm, Allocate to the Maximum First (AMF) is presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The algorithm first ex- plores with the least consumption action until the eigenvalue condition for the estimator (8) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In each round of ex- ploration, the Gram matrix of all actions is added to Fnt, and any choice of action increases the eigenvalue of Fnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Once the condition (8) holds, the algorithm solves the prob- lem (12) by computing the closed-form policy (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The computational complexity of our algorithm is discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Regret Analysis In this section, we present our regret bound and regret anal- ysis for the AMF algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (Regret bound of AMF) Suppose Assumptions 1-3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Let Mα,p,T := 1152α−2p−2 min log T + 96α−1p−1 min and Cσ(δ) := 8 √ 2(8 + 96σ � log 4 δ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Suppose T and ρ satisfies T ≥ 8dα−1p−1 min log JdT, and ρ ≥ � Jd/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Setting γθ = 16√J log JKT + 4 √ 2βσr(δ) and γb = 16√J log JKT + 4 √ 2βσb(δ), the regret bound of AMF is R�π T ≤ � 2+ OPT ρT �� 4d log JdT αpmin +2dMα,p,T log Jd δ +15 + � 96 � log JKT +3Cσr∨σb(δ) �� JdT log T +10mT 3δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For δ ∈ (0, m−1T −3), the regret bound is R�π T = O �OPT Tρ � JdT log mJKT log T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (14) The regret bound (14) holds when the hyperparameter δ = m−1T −3, which requires the knowledge of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' However, in practice, selecting another value of δ does not affect the performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We provide the discussion on the sensitivity to the hyperparameter choice in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Setting B = Tρ, the main term of the regret bound is ˜O(OPT/B √ JdT) for B = Ω( √ JdT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The sublinear dependence of the regret bound on J, d, and T is a direct consequence of the improved ˜O( � Jd/nt) convergence rate for the parameter estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Agrawal & Devanur (2016) establish a regret bound Rπ T = ˜O(OPT/B · d √ T) for the LinCBwK when B = Ω( √ dT 3/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Our bound for LMMP (which subsumes LinCBwK as a special case) is improved by a √ d factor, and is valid under budget constraints that relaxed from Ω( √ dT 3 4 ) to Ω( √ dT 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For the proof of the regret bound, we first present the lower bound of the reward obtained by our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Let ˜u(j) k,t and �b(j) k,t be the estimates defined in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Denote �π the policy of AMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Define the good events, Et := � ˜u(j) k,t and �b(j) k,t satisfies (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' � , Mt := {Fnt satisfies (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='} , (15) and Gt := Et ∩ Mt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Let τ be the stopping time for the algorithm and ξ := inft∈[T ] {Mt−1 ∩ {ρt > 0}} be the starting time after the exploration for condition (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then, the total reward E � T � t=1 R�π t � ≥ OPT T E [τ − ξ]− � 2+ OPT ρT � T � t=1 P(Gc t ) −2 � 1+ OPT ρT �� � � �TE � T � t=1 γt−1,σr∨σb(δ)2I (at ∈ [K]) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The lower bound consists of three main terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The first term OP T T E [τ − ξ] relates to the time span for which the algorithm uses the optimal policy (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The second term (2+ OP T ρT ) �T t=1P (Gc t ) is the sum of the probability of bad events Mc t−1 over which the minimum eigenvalue of the Gram matrix Fnt is not large enough for the fast conver- gence rate, and the event Ec t over which the estimator goes out of the confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' And, the third term con- sists of the sum of confidence lengths for the reward and consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The following result bounds τ, ξ and the sum of bad events {Mc t : t ∈ [T]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Suppose Assumptions 1-3 holds and ρ > � Jd/T Let Mα,p,T and γt,σ(δ) denote the variables defined in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then, for any δ ∈ (0, 1/T 2), the starting time ξ := inft∈[T ]{Mt−1 ∩ {ρt > 0}} and the stopping time τ of the AMF algorithm is bounded as E [ξ] ≤ 1+dMα,p,T log � Jd δ � +T 2δ ρ + 1, E[T −τ]≤ 4(m + 1)Tδ + 7 + 2γ1,σb(δ) ρ , and for Mt defined as in (15), T � t=1 P � Mc t−1 � ≤ T 2δ + dMα,p,T log �Jd δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The regret bound follows from bounding the probability of Ec t with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2 and showing that the sum of square of γt,σ(δ) is O(Jd log T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The bound holds because the summation of γt,σ(δ)2 = ˜O( Jd nt ) over the rounds that at ∈ [K] happens is �nT n=1 O(Jd/n) = O(Jd log T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Numerical results We report the cumulative regrets for given budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For the computation of the regret, we use the following settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For each round t ∈ [T] there exists the optimal action whose reward is 1 with consumption ρ, while the reward of other actions is less than 1 and the consumption is possibly greater than ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In this case, we can compute the instantaneous regret by subtracting the reward of a selected action from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Detail settings of the parameter and contexts are in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Regret R�π T as a function of d Figure 1 plots log(R�π T ) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' log(d) for a single-class (J = 1) LMMP for T = {5000, 20000} and the budget B = √ dT, where our ˜O( OP T B √ JdT) regret bound implies that log(R�π T ) is constant over d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The regression line on the plot is nearly flat and the slope of the best fit line is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='136 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='008) for T = 5000 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' T = 20000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The weak increase in T = 5000 is captured by the O(d log JdmT) term in our bound, which diminishes for large T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Comparison of AMF with OCO In order to compare AMF with OCO (Agrawal & Devanur, 2016), we set the costs c(1) k,t = 0 and J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The hyperpa- rameters for AMF were set to γθ = 1, γb = 1 and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Figure 2(a) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (b)) plots the cumulative regret of the two algorithms with budget B = √ dT 3 4 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' B = √ dT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Note that OCO requires a minimum budget B = √ dT 3 4 whereas AMF requires a lower minimum budget of B = Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Logarithm of cumulative regret of the proposed AMF algorithm on various dimension d when the per-period budget is ρ = � d/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The gray (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' black) line is the best fit line on the points when T = 5000 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' T = 20000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (a) Regret comparison with budget B = √ dT 3/4 (b) Regret comparison with budget B = √ dT Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Regret of AMF and OCO algorithms for K = 20 and m = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The line and shade represent the average and standard deviation based on 20 independent experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Additional results on different K and m are in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' √ dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The regret lines cross because AMF is allowed to skip arrivals whereas OCO does not skip arrivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The sudden bend points at the end of the round in OCO show that it runs out of budget and has regret = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In all cases, our algorithm performs better and the performance gap increases as d increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Note that regret plot for OCO never flattens out for most cases, where the regret of AMF flattens as t increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' This is because our new estimator, that uses contexts from all actions with unbiased pseudo-rewards (5) for unselected actions, has significantly faster convergence rate as compared with the estimator used in OCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Sensitivity Analysis Our proposed AMF algorithm has three hyperparameters: γθ, γb and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The choice of hyperparameters is not sensitive because the effect of γθ and γb diminishes fast by n−1/2 t term and our policy finds the order of the utilities rather than their absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For δ, which controls the sampling probabilities (4) in estimators and the exploration rounds in (8), it also has small effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' This is because the minimum eigenvalue of Fnt increases in Ω(nt)-rate and reduces the effect of log 1 δ terms in (4) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Therefore, our algorithm guarantees the similar performance for other hyperparame- ters than specified in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For details including the numerical results and specific recommendations for choos- ing the hyperparameters, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='5 - O 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='0 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Foundations of computational mathematics, 12 (4):389–434, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Tropp, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' An introduction to matrix concentration inequal- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Foundations and Trends® in Machine Learning, 8 (1-2):1–230, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback (a) Regret comparison under K = 10 and m = 10 (b) Regret comparison under K = 20 and m = 10 (c) Regret comparison under K = 10 and m = 20 (d) Regret comparison under K = 20 and m = 20 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Regret comparison of AMF and OCO algorithms under B = dT 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The line and shade represent the average and standard deviation based on 20 repeated experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Supplementary for Experiments A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Settings of Parameters and Contexts for Regret Computation For numerical experiments, we devise a setting where explicit regret computation is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We set J = 1 and c(1) k,t = 0 for OCO to be compatible with the setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For x ∈ R+, let ⌈x⌉ be the smallest integer greater than equal to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For parameters, we set θ⋆ = (−1, · · · , −1, ⌈d/2⌉−1, · · · , ⌈d/2⌉−1) and W⋆ = � � � � � � � � � � ρ⌈d/2⌉−1 · · ρ⌈d/2⌉−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' ρ⌈d/2⌉−1 · · ρ⌈d/2⌉−1 ρ · · ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' ρ · · ρ � � � � � � � � � � , where the ⌈d/2⌉−1 and ρ⌈d/2⌉−1 terms are in the first ⌈d/2⌉ entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For contexts, we set the optimal action as (0, · · · , 0, 1, · · · , 1), and for other actions, we set (U0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='05, · · · , U0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='05, U0,−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='05, · · · , U0,−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='05),where Ua,b the uniform random variable supports on [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then we have the optimal arm with reward 1 and consumption ρ, while other arms have reward less than 1 and consumption more than ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Additional Results on Regret Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Figure 3 (a)-(d) show the regret comparison of AMF and OCO on different terms of K = 10, 20, m = 10, 20, and B = dT 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Similar to the results in Figure 2(a), our algorithm has less regret than OCO in all cases, especially at the end of the rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The crossing line occurs when our algorithm skips in the middle round when ρt < 0 while OCO does not skip until the inventory runs out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Figure 4 (a)-(d) show the regret of AMF and OCO algorithm on various K = 10, 20 and m = 10, 20 with budget B = √ dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Even in the smaller budget, our algorithm AMF does not run out the inventory and gains more reward than OCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The gap of the performance tends to be larger than B = √ dT 3 4 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 1000 d=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' K=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='m=20 OCO AMF 800 600 400 200 : 0 · 0 2000 4000 6000 8000 10000 Decision points1000 d=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' K=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' m=20 OCO AMF 800 600 400 200 : 0 0 2000 4000 6000 8000 10000 Decision points1000 d=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='K=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' m=10 OCO AMF 800 600 400 200 0 0 2000 4000 6000 8000 10000 Decision points1000 d=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='K=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='m=10 OCO AMF 800 600 - 400 - 200 - 0 0 2000 4000 6000 8000 10000 Decision points1000 d=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='K=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' m=10 OCO AMF 800 600 400 200 : 0 0 2000 4000 6000 8000 10000 Decision points1000 d=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='K=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='m=10 OCO AMF 800 600 400 200 - 0 0 2000 4000 6000 8000 10000 Decision points1000 d=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='K=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' m=20 OCO AMF 800 600 400 200 : 0 0 2000 4000 6000 8000 10000 Decision points1000 d=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='K=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' m=20 OCO AMF 800 600 400 200 : 0 0 2000 4000 6000 8000 10000 Decision pointsImproved Algorithms for Multi-period Packing Problems with Bandit Feedback (a) Regret comparison under K = 10 and m = 10 (b) Regret comparison under K = 20 and m = 10 (c) Regret comparison under K = 10 and m = 20 (d) Regret comparison under K = 20 and m = 20 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Regret comparison of AMF and OCO algorithms under B = √ dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The line and shade represent the average and standard deviation based on 20 repeated experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (a) On various γθ (b) On various γb (c) On various δ Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The reward and inventory of AMF on various hyperparameters γθ, γb and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The solid (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' dashed) line represents the reward (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' inventory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The line and shade represent the average and standard deviation based on 10 repeated experiments, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Sensitivity Analysis In this experiment, we present the sensitivity of our algorithm to various hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The number of classes is J = 3 with a uniform prior p = (1/3, 1/3, 1/3)⊤ and every d = 5 elements of K = 10 contexts are generated from the uniform distribution on [ kj KJ − 1, kj KJ + 1] for k ∈ [K] and j ∈ [J].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The costs are generated from the uniform distribution on [ k(J−j+1)−1 KJ , k(J−j+1)+1 KJ ] for k ∈ [K] and j ∈ [J].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Each element of θ(j) ⋆ and W (j) ⋆ is generated from U0,1 and fixed throughout the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The generated rewards and consumption vectors are not truncated to one to impose more variability, because our algorithm does not show apparent sensitivity on bounded rewards and consumption vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The budget is ρ = dT −1/2 with a time horizon of T = 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In our algorithm, there are three hyperparameters: (i) a confidence bound for the reward γθ, (ii) a confidence bound for the consumption γb and (iii) confidence level δ which affects the minimum eigenvalue condition (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Figure 5(a) and 5(b) show the reward and inventory of our algorithm on various γθ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1, 1} and γb ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Outside of the hyperparameter regions, the variability of the reward and the inventory of the algorithm are hardly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The algorithm consumes the budget earlier than previous experiments because the consumption vector is not bounded to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' As γθ and γb increase the algorithm is more optimistic and admits the arrival more often, which leads to faster consumption of the resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Increasing γθ has small effect on the inventory because the algorithm automatically skips when ρt < 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', when the consumption is too fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' when γb increases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' the LCB of consumption is small and the algorithm uses more 2500 d=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='K=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='m=10 OCO AMF 2000 1500 - 1000 - 500 - 0 0 2000 4000 6000 8000 10000 Decision points2500 d=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='K=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='m=10 OCO AMF 2000 1500 - 1000 - 500 - 0 0 2000 4000 6000 8000 10000 Decision points2500 d=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='K=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='m=10 OCO AMF 2000 1500 - 1000 - 500 - 0 0 2000 4000 6000 8000 10000 Decision points2500 d=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='K=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='m=10 OCO AMF 2000 1500 - 1000 - 500 - 0 0 2000 4000 6000 8000 10000 Decision points2500 d=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='K=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='m=20 OCO AMF 2000 1500 - 1000 - 500 - 0 0 2000 4000 6000 8000 10000 Decision points2500 d=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' K=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='m=20 OCO AMF 2000 1500 - 1000 - 500 - 0 0 2000 4000 6000 8000 10000 Decision points2500 d=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='K=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='m=20 OCO AMF 2000 1500 - 1000 - 500 - 0 0 2000 4000 6000 8000 10000 Decision points2500 d=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' K=20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' m=20 OCO AMF 2000 1500 - 1000 - 500 - 0 0 2000 4000 6000 8000 10000 Decision points700 600 500 400 300 200 Ye=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='01 100 Ye=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='10 0 Ye=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='00 0 250 500 750 1000 1250 1500 1750 2000 Decision points700 600 500 400 300 200 Yb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='01 100 Yb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='10 0 Yb=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='00 0 250 500 750 1000 1250 1500 1750 2000 Decision points700 600 500 400 300 200 6=1e-01 §=1e-04 100 6=1e-07 0 0 250 500 750 1000 1250 1500 1750 2000 Decision pointsImproved Algorithms for Multi-period Packing Problems with Bandit Feedback resource than tρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For the specific value of the hyperparameters we recommend to use grid search on γθ × γb ∈ [0, 1]2 to maximize the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Figure 5(c) shows the reward and inventory of AMF on various δ ∈ {10−1, 10−4, 10−7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' When δ ≥ 10−1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' δ ≤ 10−7) the reward and inventories are same with δ = 10−1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' δ = 10−7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' As δ decreases, the threshold for condition (8) increases and the algorithm explores more with minimum possible consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' This results in the slower consumption of the resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' However, we recommend using δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1, which is greater than the specified value in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1 for the algorithm to start using its policy in earlier rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Computational Complexity of AMF The computational complexity of our algorithm is ˜O(d3mKT + Jd3T) where the main order occurs from updating the estimators and computing the eigenvalues of J symmetric positive-definite matrix Fnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Note that Computing estimators does not depend on J because the algorithm updates only jt-th variables for each t ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Missing Proofs B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Because the construction of �Θt and � Wt is the same, the bound for the � Wt follows immediately from the bound for �Θt by replacing {r (jτ(ν)) aτ(ν),τ(ν) : ν ∈ [nt]} with m entries of {b (jτ(ν)) aτ(ν),τ(ν) : ν ∈ [nt]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus, it is sufficient to prove the bound for �Θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Estimation error decomposition: Let us fix t ∈ [T] throughout the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For each ν ∈ [nt] and k ∈ [K], denote Xk,ν := ˜Xk,ν ˜X⊤ k,ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then we can write Vnt := � ν∈Ψnt K � k=1 Xk,ν + � ν /∈Ψnt Xaτ(ν),ν + IJ·d, Ant := � ν∈Ψnt K � k=1 I (hν = k) φk,ν Xk,ν + � ν /∈Ψnt Xk,ν + IJ·d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Denote the errors ˜ηk,ν := ˜rk,ν − ˜X⊤ k,νΘ⋆ and ηk,ν := r (jτ(ν)) k,τ(ν) − ˜X⊤ k,νΘ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By the definition of the estimator �Θnt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' ����Θnt − Θ∗��� Vnt = ������ V −1/2 nt � � �−Θ∗ + � ν∈Ψnt K � k=1 ˜ηk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + � ν /∈Ψnt ηk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � � � ������ 2 ≤ λmax � V −1/2 nt � ∥Θ∗∥2 + ������ V −1/2 nt � � � � ν∈Ψnt K � k=1 ˜ηk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + � ν /∈Ψnt ηk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � � � ������ 2 ≤ √ Jd + ������ V −1/2 nt � � � � ν∈Ψnt K � k=1 ˜ηk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + � ν /∈Ψnt ηk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � � � ������ 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (16) where and the last inequality holds because ���θ(j) ⋆ ��� 2 ≤ √ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Plugging in ˜rk,ν defined in (5), ˜ηk,ν ˜Xk,ν = � 1 − I (hν = k) φk,ν � ˜Xk,ν ˜X⊤ k,ν �ˇΘt − Θ∗� + I (hν = k) φk,ν ηk,ν ˜Xk,ν, Improved Algorithms for Multi-period Packing Problems with Bandit Feedback and the term � ν∈Ψnt �K k=1 ˜ηk,ν ˜Xk,ν is decomposed as, � ν∈Ψnt K � k=1 ˜ηk,ν ˜Xk,ν = � ν∈Ψnt K � k=1 �� 1 − I (hν = k) φk,ν � Xk,ν �ˇΘt − Θ∗� + I (hν = k) φk,ν ηk,ν ˜Xk,ν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (17) By definition of the IPW estimator ˇΘt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' � ν∈Ψnt K � k=1 � 1 − I (hν = k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν �ˇΘt − Θ∗� = � � � � ν∈Ψnt K � k=1 � 1 − I (hν = k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � � � A−1 nt � �−Θ∗ + � ν∈Ψnt K � k=1 I (hν = k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ηk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + � ν /∈Ψnt ηaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � � = (Vnt − Ant) A−1 nt � �−Θ∗ + � ν∈Ψnt K � k=1 I (hν = k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ηk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + � ν /∈Ψnt ηaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � � := (Vnt − Ant) A−1 nt (−Θ∗ + Snt) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (18) where Snt := � ν∈Ψnt K � k=1 I (hν = k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ηk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + � ν /∈Ψnt ηaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' ����Θnt − Θ∗��� Vnt ≤ (16) √ Jd + ������ V −1/2 nt � � � � ν∈Ψnt K � k=1 ˜ηk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + � ν /∈Ψnt ηk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � � � ������ 2 = (17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='(18) √ Jd + ���V −1/2 nt � (Vnt − Ant) A−1 nt (−Θ∗ + Snt) + Snt ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By triangular inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='����Θt − Θ∗��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='Vnt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='Jd + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='���V −1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='(Vnt − Ant) A−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt (−Θ∗ + Snt) + Snt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='Jd + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='���V −1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='(Vnt − Ant) A−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt (−Θ∗ + Snt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2 + ∥Snt∥V −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='Jd + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='V 1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt A−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt V 1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='− IJ·d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='−V −1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='Θ∗ + V −1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='Snt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2 + ∥Snt∥V −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='Jd + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='���V 1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt A−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt V 1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='− IJ·d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='���−V −1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='Θ∗ + V −1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='Snt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2 + ∥Snt∥V −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='Jd + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='���V 1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt A−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt V 1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='− IJ·d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='Jd + ∥Snt∥V −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='+ ∥Snt∥V −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='����V 1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt A−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt V 1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='− IJ·d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='� �√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='Jd + ∥Snt∥V −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='nt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (19) Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Bounding the ∥ · ∥2 of the matrix in (19) We claim that V 1/2 nt A−1 nt V 1/2 nt ⪰ 1 8IJ·d (20) Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Define Fnt := �nt ν=1 �K k=1 Xk,ν + 16Kd log( Jd δ )IJ·d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then we have Vnt ⪯ Fnt and V 1/2 nt A−1 nt V 1/2 nt ⪰ F −1/2 nt AntF −1/2 nt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Now we decompose the matrix Ant as F −1/2 nt AntF −1/2 nt =F −1/2 nt � nt � ν=1 K � k=1 I (hν = k) φk,ν Xk,ν + IJ·d � F −1/2 nt + F −1/2 nt � � � ν /∈Ψnt � Xaτ(ν),ν − K � k=1 I (hν = k) φk,ν Xk,ν �� � F −1/2 nt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (21) For each ν ∈ [nt], the matrix �K k=1 I(hν=k) φk,ν F −1/2 nt Xk,νF −1/2 nt symmetric positive definite and λmax � 8 log Jd δ K � k=1 I (hν = k) φk,ν F −1/2 nt Xk,νF −1/2 nt � ≤8 log Jd δ K � k=1 I (hν = k) φk,ν λmax � F −1 nt � ≤8 log Jd δ λmin(Fν) 16 log � Jd δ �λmax � F −1 nt � ≤1 2 λmin(Fν) λmin(Fnt) ≤1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (22) With the filtration F0 := Ht and Fn := F0 ∪ {hν : ν ∈ [n]}, we use Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3 to have with probability at least 1 − δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 8 log Jd δ F −1/2 nt � nt � ν=1 K � k=1 I (hν = k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + IJ·d � F −1/2 nt ⪰ 8 log Jd δ F −1/2 nt � nt � ν=1 K � k=1 Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + IJ·d � F −1/2 nt = 4 log Jd δ IJ·d − log Jd δ IJ·d = 3 log Jd δ IJ·d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' which implies F −1/2 nt � nt � ν=1 K � k=1 I (hν = k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + IJ·d � F −1/2 nt ⪰ 3 8IJ·d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (23) and the first term in (21) is bounded as F −1/2 nt AntF −1/2 nt ⪰ 3 8IJ·d + F −1/2 nt � � � ν /∈Ψnt � Xaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν − K � k=1 I (hν = k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν �� � F −1/2 nt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (24) To bound the other term, observe that for ν /∈ Ψnt, E � K � k=1 I (hν = k) φk,ν Xk,ν ����� Ht � = � i̸=aτ(ν) K � k=1 φi,ν I (i = k) φk,ν Xk,ν = � k̸=aτ(ν) Xk,ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Because (22) holds for ν /∈ Ψnt, we can use Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3, to have with probability at least 1 − δ 8 log Jd δ F −1/2 nt � � � ν /∈Ψnt K � k=1 I (hν = k) φk,ν Xk,ν � � F −1/2 nt ⪯ 12 log Jd δ F −1/2 nt � � � ν /∈Ψnt � k̸=aτ(ν) Xk,ν � � F −1/2 nt + log Jd δ IJ·d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Rearranging the terms, F −1/2 nt � � � ν /∈Ψnt K � k=1 I (hν = k) φk,ν Xk,ν � � F −1/2 nt ⪯ 3 2F −1/2 nt � � � ν /∈Ψnt � k̸=aτ(ν) Xk,ν � � F −1/2 nt + 1 8IJ·d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Thus the second term in (24) is bounded as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' F −1/2 nt � � � ν /∈Ψnt � Xaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν − K � k=1 I (hν = k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν �� � F −1/2 nt ⪰ F −1/2 nt � � � ν /∈Ψnt � � �Xaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν − 3 2 � ν /∈Ψt � k̸=aτ(ν) Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � � � � � F −1/2 nt − 1 8IJ·d ⪰ −3 2F −1/2 nt � � � ν /∈Ψnt K � k=1 Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � � F −1/2 nt − 1 8IJ·d ⪰ − �3dK 2 ��Ψc nt �� λmax � F −1 nt � + 1 8 � IJ·d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (25) where the last inequality holds by λmax(Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν) ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' with probability at least 1 − δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 1 2 ��Ψc nt �� = 1 2 nt � ν=1 I � hν ̸= aτ(ν) � ≤ 3 2 nt � ν=1 � k̸=aτ(ν) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + log 1 δ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' which implies 3dK 2 ��Ψc nt �� λmax � F −1 nt � ≤ 3Kd 2λmin(Fnt) � � �3 nt � ν=1 � k̸=aτ(ν) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + 2 log 1 δ � � � = 3Kd 2λmin(Fnt) � nt � ν=1 48 (K − 1) log � Jd δ � λmin(Fν) + 2 log 1 δ � Because the assumption (8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' λmin(Fnt) ≥ 12Kd � nt � ν=1 48 (K − 1) log � Jd δ � λmin(Fν) + 2 log Jd δ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' implies 3dK 2 ��Ψc nt �� λmax � F −1 nt � ≤ 3Kd 2λmin(Fnt) � nt � ν=1 48 (K − 1) log � Jd δ � λmin(Fν) + 2 log 1 δ � ≤ 1 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (26) plugging in (25),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' with probability at least 1 − 2δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' F −1/2 nt � � � ν /∈Ψnt � Xaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν − K � k=1 I (hν = k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν �� � F −1/2 nt ⪰ −1 4IJ·d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' With (24), F −1/2 nt AntF −1/2 nt ⪰ 1 8IJ·d, which proves (20) and the claim implies ���V 1/2 nt A−1 nt V 1/2 nt − IJ·d ��� 2 ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Bounding the self-normalized vector-valued martingale Snt Let F0 be a sigma algebra generated by contexts {x(js) k,s : k ∈ [K], s ∈ [t]}, and Ψt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Define filtration as Fν := σ(F0 ∪ Hτ(ν+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then Sν is a RJ·d-valued martingale Improved Algorithms for Multi-period Packing Problems with Bandit Feedback because E [Sν − Sν−1| Fν−1] = E � I (ν ∈ Ψnt) K � k=1 I (hν = k) φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ηk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + I (ν /∈ Ψnt) ηaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ(ν) ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ����� Fν−1 � = E � I (ν ∈ Ψnt) K � k=1 I � aτ(ν) = k � φk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ηk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + I (ν /∈ Ψnt) ηaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ(ν) ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ����� Fν−1 � = E �� I (ν ∈ Ψnt) φaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + I (ν /∈ Ψnt) � ηaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ����� Fν−1 � = E �� I (ν ∈ Ψnt) φaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + I (ν /∈ Ψnt) � ηaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ����� Hτ(ν) � = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' where the second equality holds by definition of Ψnt and the fourth inequality holds because the distribution of {x(js) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s : k ∈ [K],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' s ∈ (τ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' t]} is independent of Hτ(ν) by Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By Assumption 1, for any λ ∈ R, E � exp � λ � I (ν ∈ Ψnt) φaτ(ν),ν +I (ν /∈ Ψnt) � ηk,aτ(ν) ������ Fν−1 � ≤ E � �exp � �λ2σ2 2 � I (ν ∈ Ψnt) φaτ(ν),ν +I (ν /∈ Ψnt) �2� � ������ Fν−1 � � ≤ exp � 2λ2σ2 r � , Thus, � I(ν∈Ψnt) φaτ(ν),ν + I (ν /∈ Ψnt) � ηk,aτ(ν) is 2σr-sub-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Because ∥Snt∥V −1 t = ������ � ν∈Ψnt K � k=1 I (hν = k) φk,ν ηk,ν ˜Xk,ν + � ν /∈Ψnt ηaτ(ν),ν ˜Xaτ(ν),ν ������ V −1 nt = ����� nt � ν=1 � I (ν ∈ Ψnt) φaτ(ν),ν + I (ν /∈ Ψnt) � ηaτ(ν),ν ˜Xk,ν ����� V −1 nt = ����� nt � ν=1 � I (ν ∈ Ψnt) φaτ(ν),ν + I (ν /∈ Ψnt) � ηaτ(ν),νV −1/2 nt ˜Xk,ν ����� 2 , by Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' with probability at least 1 − δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' ∥St∥V −1 t ≤ ����� nt � ν=1 � I (ν ∈ Ψnt) φaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν + I (ν /∈ Ψt) � ηaτ(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='νV −1/2 nt ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ����� 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' ≤12σr � � � � nt � ν=1 ���V −1/2 nt ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ��� 2 2 log 4 δ ≤12σr � Jd log 4 δ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' where the last inequality holds because nt � ν=1 ���V −1/2 nt ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ��� 2 2 = nt � ν=1 ˜X⊤ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='νV −1 nt ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν = Tr � nt � ν=1 ˜X⊤ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='νV −1 nt ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν � = Tr � nt � ν=1 ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ν ˜X⊤ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='νV −1 nt � ≤ Tr � VntV −1 nt � = Jd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' With (19), the proof is completed Improved Algorithms for Multi-period Packing Problems with Bandit Feedback B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Similar to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1, the bound for consumption vector immediately follows from the bound for the utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Therefore we provide the proof for the utility bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Decomposition: For each k ∈ [K] and j ∈ [J],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' ���u⋆(j) k − ˜u(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t+1 ��� ≤ �����E � x(j) k �⊤ θ(j) ⋆ − � 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) � x(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s �⊤ �θ(j) t ������ + �����E � c(j) k � − � 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) c(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s ������ ≤ �����E � x(j) k �⊤ θ(j) ⋆ − � 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) � x(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s �⊤ θ(j) ⋆ ������ + �����E � c(j) k � − � 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) c(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s ������ + ����� 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) � �θ(j) t − θ(j) ⋆ �⊤ x(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s ����� ����� 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) � E � x(j) k �⊤ θ(j) ⋆ − � x(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s �⊤ θ(j) ⋆ ������ + ����� 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) � E � c(j) k � − c(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s ������ + ����� 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) � �θ(j) t − θ(j) ⋆ �⊤ x(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Taking maximum over k ∈ [K] gives the decomposition, max k∈[K] ���u⋆(j) k − ˜u(j) k,t+1 ��� ≤ max k∈[K] ����� 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) � E � x(j) k �⊤ θ(j) ⋆ − � x(j) k,s �⊤ θ(j) ⋆ ������ + max k∈[K] ����� 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) � E � c(j) k � − c(j) k,s ������ + max k∈[K] ����� 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) � �θ(j) t − θ(j) ⋆ �⊤ x(j) k,s ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (27) Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Bounding the difference between expectation and empirical distribution: The random variables �� x(j) k,s �⊤ θ(j) ⋆ : s ∈ [t] � and {c(j) k,s : s ∈ [t]} are IID by Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Using Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' ����� 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) � E � x(j) k �⊤ θ(j) ⋆ − � x(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s �⊤ θ(j) ⋆ ������ = 1 �t+1 s=1 I (js = j) ����� t+1 � s=1 I (js = j) � E � x(j) k �⊤ θ(j) ⋆ − � x(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s �⊤ θ(j) ⋆ ������ ≤ 4 ��t+1 s=1 I (js = j) � log JKT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback and ����� 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) � E � c(j) k � − c(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s ������ ≤ 4 ��t+1 s=1 I (js = j) � log JKT with probability at least 1 − 4(JKT)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3, with probability at least 1 − (JT)−1, t+1 � s=1 I (js = j) ≥ 1 2pj (t + 1) − 2 log JT ≥ 1 4pj (t + 1) , (28) where the last inequality holds by the assumption t ≥ 8dα−1p−1 min log JT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Summing up the probability bounds, with probability at least 1 − 5T −1, max k∈[K] ����� 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) � E � x(j) k �⊤ θ(j) ⋆ − � x(j) k,s �⊤ θ(j) ⋆ ������ ≤ 4 ��t+1 s=1 I (js = j) � log JKT ≤ 8 � pj (t + 1) � log JKT, max k∈[K] ����� 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) � E � c(j) k � − c(j) k,s ������ ≤ 8 � pj (t + 1) � log JKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Plugging in the decomposition (27), for each j ∈ [J], max k∈[K] ���u⋆(j) k − ˜u(j) k,t+1 ��� ≤ 16 � pj (t + 1) � log JKT + max k∈[K] ����� 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) � �θ(j) t−1 − θ(j) ⋆ �⊤ x(j) k,s ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Taking square and summing up over j ∈ [J] gives J � j=1 pj max k∈[K] ���u⋆(j) k − ˜u(j) k,t+1 ��� 2 ≤16J log JKT t + 1 + J � j=1 pj max k∈[K] ����� 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) � �θ(j) t − θ(j) ⋆ �⊤ x(j) k,s ����� 2 , (29) Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Bounding the prediction error: By Cauchy-Schwartz inequality and (28),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' J � j=1 max k∈[K] pj 1 ��t+1 s=1 I (js = j) �2 ����� t+1 � s=1 I (js = j) � �θ(j) t − θ(j) ⋆ �⊤ x(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s ����� 2 ≤ J � j=1 max k∈[K] pj 1 �t+1 s=1 I (js = j) t+1 � s=1 I (js = j) �� �θ(j) t − θ(j) ⋆ �⊤ x(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s �2 ≤ J � j=1 max k∈[K] 4pj pj (t + 1) t+1 � s=1 I (js = j) �� �θ(j) t − θ(j) ⋆ �⊤ x(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s �2 = 4 (t + 1) J � j=1 max k∈[K] � �θ(j) t − θ(j) ⋆ �⊤ �t+1 � s=1 I (js = j) x(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s � x(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s �⊤ � � �θ(j) t − θ(j) ⋆ � ≤ 4 (t + 1) J � j=1 � �θ(j) t − θ(j) ⋆ �⊤ �t+1 � s=1 K � k=1 I (js = j) x(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s � x(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s �⊤ � � �θ(j) t − θ(j) ⋆ � = 4 (t + 1) � �Θt − Θ⋆�⊤ �t+1 � s=1 K � k=1 ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s ˜X⊤ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s � � �Θt − Θ⋆� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (30) Improved Algorithms for Multi-period Packing Problems with Bandit Feedback where Θ⋆ := (θ(1) ⋆ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' , θ(J) ⋆ )T ∈ RJ·d and ˜Xk,s := � � � � � 0d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' x(js) k,s 0d � � � � � ∈ RJ·d, where the context x(js) k,s is located after js − 1 of 0d vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' We claim that 1 t + 1 t+1 � s=1 K � k=1 ˜Xk,s ˜X⊤ k,s ⪯ 2E � ˜Xk,1 ˜X⊤ k,1 � ⪯ 7 |Ψnt| � s∈Ψnt K � k=1 ˜Xk,s ˜X⊤ k,s, (31) with probability at least 1 − 2T −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The matrix Xs := �K k=1 ˜Xk,s ˜X⊤ k,sis symmetric nonnegative definite which satisfies λmax � 1 2dK Xs � ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3, with probability at least 1 − T −1, 1 2Kd t+1 � s=1 Xs ⪯ 3 4Kd t+1 � s=1 E [Xs] + (log JdT) IJ·d, which implies 1 t + 1 t+1 � s=1 Xs ⪯ 3 2 (t + 1) t+1 � s=1 E [Xs] + 2dK t + 1 (log JdT) IJ·d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (32) By Assumption 3, for s ∈ [t + 1], λmin(E [Xs]) =λmin � � � � � � � � � � p1Exk∼F1 ��K k=1 xkx⊤ k � 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 0 0 0 pJExk∼FJ ��K k=1 xkx⊤ k � � � � � � � � � � � ≥λmin � � � � � � p1KαId 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 0 0 0 pJKαId � � � � � � ≥Kpminα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For t ≥ 8dα−1p−1 min log JdT , t+1 � s=1 E [Xs] ⪰ t+1 � s=1 λmin(E [Xs]) IJ·d ⪰ (t + 1) KpminαIJ·d ⪰ 4dK � log Jd δ � Plugging in (32) proves the first inequality of (31), 1 t + 1 t+1 � s=1 Xs ⪯ 2 (t + 1) t+1 � s=1 E [Xs] = 2E [X1] , where the equality holds because EXs = EX1 for all s ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' To prove the second inequality, E [X1] = |Ψnt|−1 � ν∈Ψnt E � Xτ(ν) � , Improved Algorithms for Multi-period Packing Problems with Bandit Feedback and by Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3, with probability at least 1 − T −1, 1 2Kd � ν∈Ψnt Xτ(ν) ⪰ 1 4Kd � ν∈Ψnt E � Xτ(ν) � − (log JdT) IJ·d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Rearranging the terms, � ν∈Ψnt E � Xτ(ν) � ⪯ 2 � ν∈Ψnt Xτ(ν) + 4Kd (log JdT) IJ·d (33) By definition of Fnt, � ν∈Ψnt Xτ(ν) =Fnt − � ν /∈Ψnt Xτ(ν) − 16d(K − 1) log Jd δ IJ·d ⪰Fnt − � Kd ��Ψc nt �� + 16d(K − 1) log Jd δ � IJ·d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (34) Because the condition (8) holds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' we can use (26) to have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Kd ��Ψc nt �� ≤ 1 12λmax � F −1 nt � = λmin(Fnt) 12 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (35) and Fnt − � Kd ��Ψc nt �� + 16d(K − 1) log Jd δ � IJ·d ⪰Fnt − � 1 12λmin(Fnt) + 16d(K − 1) log Jd δ � IJ·d ⪰ �11 12λmin(Fnt) − 16d(K − 1) log Jd δ � IJ·d ⪰ �11 1224Kd log Jd δ IJ·d − 16d(K − 1) log Jd δ � IJ·d ⪰ � 6dK log Jd δ � IJ·d ⪰ {6dK log JdT} IJ·d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (36) where the third inequality holds by condition (8) and the last inequality holds by δ < T −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Collecting the bounds (34) and (36) � ν∈Ψnt Xτ(ν) ⪰ Fnt − � Kd ��Ψc nt �� + 16d(K − 1) log Jd δ � IJ·d ⪰ {6dK log JdT} IJ·d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Plugging in (33), � ν∈Ψnt E � Xτ(ν) � ⪯ 2 � ν∈Ψnt Xτ(ν) + 4Kd (log JdT) IJ·d ⪯ 7 2 � ν∈Ψnt Xτ(ν), proves the second inequality in claim (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' From (30),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' J � j=1 max k∈[K] pj 1 ��t+1 s=1 I (js = j) �2 ����� t+1 � s=1 I (js = j) � �θ(j) t − θ(j) ⋆ �⊤ x(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s ����� 2 ≤ 4 (t + 1) � �Θt − Θ⋆�⊤ �t+1 � s=1 K � k=1 ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s ˜X⊤ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s � � �Θt − Θ⋆� ≤ 28 |Ψnt| � �Θt − Θ⋆�⊤ � � � � s∈Ψnt K � k=1 ˜Xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s ˜X⊤ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='s � � � � �Θt − Θ⋆� ≤ 28 |Ψnt| � �Θt − Θ⋆�⊤ {Vnt} � �Θt − Θ⋆� = 28 |Ψnt| ����Θt − Θ⋆��� 2 Vnt (37) Improved Algorithms for Multi-period Packing Problems with Bandit Feedback On bounding the normalizing matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' the novel Gram matrix Vnt plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' To obtain an upper bound for (37), we need a matrix whose eigenvalue is greater than that of: � ν∈Ψnt Xτ(ν) = � ν∈Ψnt K � k=1 ˜Xk,τ(ν) ˜X⊤ k,τ(ν), (38) However, with � ν∈Ψnt ˜Xaτ(ν),τ(ν) ˜X⊤ aτ(ν),τ(ν) , a Gram matrix consist of only selected contexts, we cannot bound the matrix (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Instead, by using a Gram matrix Vt, we can bound (38) as, � ν∈Ψnt Xτ(ν) = � ν∈Ψnt K � k=1 ˜Xk,τ(ν) ˜X⊤ k,τ(ν) ⪯ � ν∈Ψnt K � k=1 ˜Xk,τ(ν) ˜X⊤ k,τ(ν) + � ν /∈Ψnt ˜Xaτ(ν),τ(ν) ˜X⊤ aτ(ν),τ(ν) ⪯Vnt, and prove the bound (37) to relate the prediction error to the self-normalized bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' From (37), by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1 J � j=1 max k∈[K] pj 1 ��t+1 s=1 I (js = j) �2 ����� t+1 � s=1 I (js = j) � �θ(j) t − θ(j) ⋆ �⊤ x(j) k,s ����� 2 ≤ 28 |Ψnt| ����Θt − Θ⋆��� 2 Vnt ≤ 28 |Ψnt|βσr(δ)2, with probability at least 1 − 4(m + 1)δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Because |Ψnt| + ��Ψc nt �� = nt and (35) holds, |Ψnt| ≥ nt − ��Ψc nt �� ≥ nt − λmin(Fnt) 12Kd ≥ nt − ntKd 12Kd = 11 12nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus, J � j=1 max k∈[K] pj 1 ��t+1 s=1 I (js = j) �2 ����� t+1 � s=1 I (js = j) � �θ(j) t − θ(j) ⋆ �⊤ x(j) k,s ����� 2 ≤ 28 |Ψnt|βσr(δ)2 ≤12 11 · 28 nt βσr(δ)2 ≤32 nt βσr(δ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' From (29), � � � � J � j=1 pj max k∈[K] ���u⋆(j) k − ˜u(j) k,t+1 ��� 2 ≤ 16√J log JKT √ t + 4 √ 2βσr(δ) √nt and the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Suppose a feasible policy ˜π(j) k,t for the optimization problem (1) satisfies K+1 � k=1 ˜π(jt) k,t ˜u(jt) k,t > K+1 � k=1 �π(jt) k,t ˜u(jt) k,t , which is equivalent to K+1 � l=1 ˜π(jt) k⟨l⟩,t˜u(jt) k⟨l⟩,t > K+1 � l=1 �π(jt) k⟨l⟩,t˜u(jt) k⟨l⟩,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (39) Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Without loss of generality we assume �u(jt) k⟨l⟩,t ≥ 0 (Because �K+1 l=1 ˜π(jt) k⟨l⟩,t = �K+1 l=1 �π(jt) k⟨l⟩,t = 1, we can subtract �u(jt) k⟨K+1⟩,t on both side of (39)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By the constraints on the resources, ˜π(jt) k⟨1⟩,t ≤ � � min r∈[m] ρt(r) b(jt) k⟨1⟩,t(r) � � ∧ 1 = �π(jt) k⟨1⟩,t Suppose ˜π(jt) k⟨1⟩,t < �π(jt) k⟨1⟩,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Because �K+1 l=1 ˜π(jt) k⟨l⟩,t = �K+1 l=1 �π(jt) k⟨l⟩,t = 1, by Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2, K+1 � l=1 ˜π(jt) k⟨l⟩,t˜u(jt) k⟨l⟩,t ≤ K+1 � l=1 �π(jt) k⟨l⟩,t˜u(jt) k⟨l⟩,t, which contradicts with (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus we have ˜π(jt) k⟨1⟩,t = �π(jt) k⟨1⟩,t and K+1 � l=2 ˜π(jt) k⟨l⟩,t˜u(jt) k⟨l⟩,t > K+1 � l=2 �π(jt) k⟨l⟩,t˜u(jt) k⟨l⟩,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (40) Again, by the constraints on the resources,˜π(jt) k⟨2⟩,t ≤ �π(jt) k⟨2⟩,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Suppose ˜π(jt) k⟨2⟩,t < �π(jt) k⟨2⟩,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Because �K+1 l=2 ˜π(jt) k⟨l⟩,t = �K+1 l=2 �π(jt) k⟨l⟩,t, by Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2, K+1 � l=2 ˜π(jt) k⟨l⟩,t˜u(jt) k⟨l⟩,t ≤ K+1 � l=2 �π(jt) k⟨l⟩,t˜u(jt) k⟨l⟩,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' which contradicts with (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus we have ˜π(jt) k⟨2⟩,t = �π(jt) k⟨2⟩,t Recursively, we have ˜π(jt) k⟨l⟩,t = �π(jt) k⟨l⟩,t for all l ∈ [K + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus there exist no feasible solution ˜π(j) k,t such that (39) holds and the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For each t ∈ [T], denote the good events Gt := Et ∩ Mt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Bounds for the estimates ˜u(jt) k,t and ˜b(jt) k,t : For each t ∈ [T] and k ∈ [K], ˜u(jt) k,t = ˜u(jt) k,t − u⋆(jt) k + u⋆(jt) k =γt−1,σr(δ) √pjt + �u(jt) k,t − u⋆(jt) k + u⋆(jt) k ≥ γt−1,σr(δ) − √pjt maxk∈[K] ����u(jt) k,t − u⋆(jt) k ��� √pjt + u⋆(jt) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Under the event Gt, √pjt max k∈[K] ���u⋆(jt) k − �u(jt) k,t ��� = � pjt max k∈[K] ���u⋆(jt) k − �u(jt) k,t ��� 2 ≤ � � � � J � j=1 pj max k∈[K] ���u⋆(j) k − �u(j) k,t ��� 2 ≤γt−1,σr(δ), which implies ˜u(jt) k,t ≥ u⋆(jt) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (41) Similarly, ˜b(jt) k,t ≤ b⋆(jt) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (42) Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Another useful bound for ˜u(jt) k,t is E �� t∈U K � k=1 �π(jt) k,t ���˜u(jt) k,t − u⋆(jt) k ��� I (Gt) � ≤ 2γt−1,σr(δ) � E [I (at ∈ [K])].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (43) This bound is proved by the tower property of conditional expectation and Cauchy-Schwartz inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' E � K � k=1 �π(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t ���˜u(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t − u⋆(jt) k ��� I (Gt) � =E � max k∈[K] ���˜u(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t − u⋆(jt) k ��� K � k=1 �π(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t I (Gt) � =E � max k∈[K] ���˜u(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t − u⋆(jt) k ��� I (at ∈ [K]) I (Gt) � =E � � J � j=1 pj max k∈[K] ���˜u(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t − u⋆(j) k ��� I (at ∈ [K]) I (Gt) � � ≤E � � � � � � J � j=1 pj max k∈[K] ���˜u(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t − u⋆(j) k ��� 2 � � � � J � j=1 pjI (at ∈ [K])I (Gt) � � By definition of ˜u(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t and triangular inequality for ℓ2-norm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' � � � � J � j=1 pj max k∈[K] ���˜u(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t − u⋆(j) k ��� 2 I (Gt) = � � � � J � j=1 pj max k∈[K] �����u(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t − u⋆(j) k + γt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='σr(δ) √pj ���� 2 I (Gt) ≤ � � � � � � J � j=1 pj max k∈[K] ����u(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t − u⋆(j) k ��� 2 + � � � � J � j=1 pj �γt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='σr(δ) √pj �2 � � I (Gt) ≤2γt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='σr(δ)I (Gt) ≤2γt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='σr(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then by Jensen’s inequality, E � K � k=1 �π(jt) k,t ���˜u(jt) k,t − u⋆(jt) k ��� I (Gt) � ≤E � � � � � � J � j=1 pj max k∈[K] ���˜u(j) k,t − u⋆(j) k ��� 2 � � � � J � j=1 pjI (at ∈ [K])I (Gt) � � ≤2γt−1,σr(δ)E � � � � � � J � j=1 pjI (at ∈ [K]) � � ≤2γt−1,σr(δ) � � � � �E � � J � j=1 pjI (at ∈ [K]) � � =2γt−1,σr(δ) � E [I (at ∈ [K])], which proves (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Similarly, E � K � k=1 �π(jt) k,t ���˜b(jt) k,t − b⋆(jt) k ��� ∞ I (Gt) � ≤ 2γt−1,σb(δ) � E [I (at ∈ [K])] (44) Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Reward decomposition: Let τ be the stopping time of the algorithm and let U := {t ∈ [τ] : ρt > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then for t /∈ U, the allocated resource is ρt ∨ 0 = 0 and the algorithm skips the round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus, E � T � t=1 R�π t � =E �� t∈U R�π t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' the reward is decomposed as E �� t∈U R�π t � =E �� t∈U R�π t I (Gt) � + E �� t∈U R�π t I (Gc t ) � ≥E �� t∈U K � k=1 �π(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t u⋆(jt) k I (Gt) � − T � t=1 P (Gc t ) ≥E �� t∈U K � k=1 �π(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t ˜u(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t I (Gt) � − E �� t∈U K � k=1 �π(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t ���˜u(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t − u⋆(jt) k ��� I (Gt) � − T � t=1 P (Gc t ) ≥E �� t∈U K � k=1 �π(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t ˜u(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t I (Gt) � − T � t=1 E � K � k=1 �π(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t ���˜u(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t − u⋆(jt) k ��� I (Gt) � − T � t=1 P (Gc t ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By the bound (43), T � t=1 E � K � k=1 �π(jt) k,t ���˜u(jt) k,t − u⋆(jt) k ��� I (Gt) � ≤2 T � t=1 γt−1,σr(δ) � E [I (at ∈ [K])] ≤2 � � � �T T � t=1 γt−1,σr(δ)2E [I (at ∈ [K])] =2 � � � �TE � T � t=1 γt−1,σr(δ)2I (at ∈ [K]) � where the last ineqaulity holds by Cauchy-Schwartz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus, the reward is decomposed as E � T � t=1 R�π t � =E �� t∈U R�π t � ≥E �� t∈U K � k=1 �π(jt) k,t ˜u(jt) k,t I (Gt) � − 2 � � � �TE � T � t=1 γt−1,σr(δ)2I (at ∈ [K]) � − T � t=1 P (Gc t ) (45) Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' A lower bound for ρt: Denote u1 < u2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' < u|U| the indexes in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For s /∈ U, we have ρs = 0m and b(js) as,s = 0m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus for ν ∈ [|U| − 1], ρuν+1 = uν+1ρ − uν+1−1 � s=1 b(js) as,s = uν+1ρ − uν � s=1 b(js) as,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (46) By the resource constrain at round uν, K � k=1 �π(juν ) k,uν ˜b(juν ) k,uν ≤uνρ − uν−1 � s=1 b(js) as,s =uνρ + b(juν ) auν ,uν − uν � s=1 b(js) as,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Plugging in (46), ρuν+1 ≥ (uν+1 − uν) ρ − b(juν ) auν ,uν + K � k=1 �π(juν ) k,uν ˜b(juν ) k,uν ≥ (uν+1 − uν) ρ − b(juν ) auν ,uν + K � k=1 �π(juν ) k,uν b⋆(juν ) k + K � k=1 �π(juν ) k,uν � ˜b(juν ) k,uν − b⋆(juν ) k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Taking conditional expectation on both sides gives E � ρuν+1 �� juν+1 � ≥E � uν+1 − uν| juν+1 � ρ + E � −b(juν ) auν ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='uν + K � k=1 �π(juν ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='uν b⋆(juν ) k ����� juν+1 � + E � K � k=1 �π(juν ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='uν � ˜b(juν ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='uν − b⋆(juν ) k ������ juν+1 � =E � uν+1 − uν| juν+1 � ρ + E � K � k=1 �π(juν ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='uν � ˜b(juν ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='uν − b⋆(juν ) k �� ≥E � uν+1 − uν| juν+1 � ρ + E � K � k=1 �π(juν ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='uν � ˜b(juν ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='uν − b⋆(juν ) k � I (Guν) � − P � Gc uν � 1m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' where the equality holds by Assumption 1 and E �� −b(juν ) auν ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='uν + K � k=1 �π(juν ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='uν b⋆(juν ) k �� =E �� −b⋆(juν ) auν + K � k=1 �π(juν ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='uν b⋆(juν ) k �� =E �� − K � k=1 �π(juν ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='uν b⋆(juν ) k + K � k=1 �π(juν ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='uν b⋆(juν ) k �� =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For the last term, by the bound (44), E � K � k=1 �π(juν ) k,uν � ˜b(juν ) k,uν − b⋆(juν ) k � I (Guν) � ≥ − E � K � k=1 �π(juν ) k,uν ���˜b(juν ) k,uν − b⋆(juν ) k ��� ∞ I (Guν) � 1m ≥ − E � 2γuν−1,σb(δ) � E [I (auν ∈ [K])| uν] � 1m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus we obtain a lower bound, E � ρuν+1 �� juν+1 � ≥E � uν+1 − uν| juν+1 � ρ − P � Gc uν � 1m − 2E � γuν−1,σb(δ) � E [I (auν ∈ [K])| uν] � 1m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (47) Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' An upper bound for OPT In the optimization problem (1), all constraints are linear with respect to the variable and there exist a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus the problem satisfies the Slater’s condition and strong duality (Boyd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then, OPT T = max π(j) k min λ∈Rm + min µ(j)≥0 min ν(j) k ≥0 L � π(j) k , λ, µ(j), ν(j) k � , where L is the Lagrangian function: L � π(j) k , λ, µ(j), ν(j) k � := J � j=1 K � k=1 pjπ(j) k u⋆(j) k + � �ρ − J � j=1 K � k=1 pjπ(j) k b⋆(j) k � � ⊤ λ + J � j=1 µ(j) � 1 − K � k=1 π(j) k,1 � + J � j=1 K � k=1 ν(j) k π(j) k,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Minimizing over µ(j) and ν(j) k gives min µ(j) t ≥0 min ν(j) k,t≥0 L � π(j) k , λ, µ(j), ν(j) k � = � � � �J j=1 �K k=1 pjπ(j) k u⋆(j) k + � ρ − �J j=1 �K k=1 pjπ(j) k b⋆(j) k �⊤ λ �K k=1 π(j) k ≤ 1, π(j) k ≥ 0 −∞ o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' which implies OPT T = max π(j) k min λ∈Rm + min µ(j) t ≥0 min ν(j) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t≥0 L � π(j) k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' µ(j),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' ν(j) k � ≤ max �K k=1 π(j) k ≤1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='π(j) k ≥0 min λ∈Rm + J � j=1 K � k=1 pjπ(j) k u⋆(j) k + � �ρ − J � j=1 K � k=1 pjπ(j) k b⋆(j) k � � ⊤ λ = max �K k=1 π(j) k ≤1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='π(j) k ≥0 min λ∈Rm + J � j=1 pj � � � K � k=1 π(j) k u⋆(j) k + � ρ − K � k=1 π(j) k b⋆(j) k �⊤ λ � � � ≤ min λ∈Rm + max �K k=1 π(j) k ≤1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='π(j) k ≥0 J � j=1 pj � � � K � k=1 π(j) k u⋆(j) k + � ρ − K � k=1 π(j) k b⋆(j) k �⊤ λ � � � ≤ min λ∈Rm + J � j=1 pj max �K k=1 π(j) k ≤1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='π(j) k ≥0 � � � K � k=1 π(j) k u⋆(j) k + � ρ − K � k=1 π(j) k b⋆(j) k �⊤ λ � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Let {¯π(j) k : j ∈ [J], k ∈ [K]} be the maximizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' If ρ − �K k=1 ¯π(j) k b⋆(j) k is negative for some element and j ∈ [J], then the optimal value becomes −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus OPT T ≤ min λ∈Rm + J � j=1 pj max �K k=1 π(j) k ≤1,π(j) k ≥0 � � � K � k=1 π(j) k u⋆(j) k + � ρ − K � k=1 π(j) k b⋆(j) k �⊤ λ � � � = min λ∈Rm + J � j=1 pj max �K k=1 π(j) k ≤1,π(j) k ≥0,ρ−�K k=1 π(j) k b⋆(j) k ≥0 � � � K � k=1 π(j) k u⋆(j) k + � ρ − K � k=1 π(j) k b⋆(j) k �⊤ λ � � � = J � j=1 pj max �K k=1 π(j) k ≤1,π(j) k ≥0,ρ−�K k=1 π(j) k b⋆(j) k ≥0 � K � k=1 π(j) k u⋆(j) k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For each j ∈ [J] and v ∈ Rm +, let ˜π(j) k,v be the solution to the optimization problem: max π(j) k,v K � k=1 π(j) k,vu⋆(j) k s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' K � k=1 π(j) k,vb⋆(j) k ≤ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (48) Then, OPT T ≤ J � j=1 pj max �K k=1 π(j) k ≤1,π(j) k ≥0,ρ−�K k=1 π(j) k b⋆(j) k ≥0 � K � k=1 π(j) k u⋆(j) k � = J � j=1 pj K � k=1 ˜π(j) k,ρu⋆(j) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback For each ν ∈ [|U| − 1], E � (uν+1 − uν) OPT T � ≤E � �(uν+1 − uν) J � j=1 pj K � k=1 ˜π(j) k,ρu⋆(j) k � � =E � (uν+1 − uν) K � k=1 ˜π (juν+1) k,ρ u ⋆(juν+1) k � In (48), all constraints are linear with respect to the variable and there exist a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus the problem satisfies the Slater’s condition and strong duality (Boyd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The dual problem of (48) is min λ(j) v ∈Rm + v⊤λ(j) v s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' � b⋆(j) k �⊤ λ(j) v ≥ u⋆(j) k , ∀k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (49) Let ˜λ(j) v be the solution to (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By strong duality, for each ν ∈ [|U| − 1], E � (uν+1 − uν) K � k=1 ˜π (juν+1) k,ρ u ⋆(juν+1) k � = E � (uν+1 − uν) ρ⊤˜λ (juν+1) ρ � = E � E � (uν+1 − uν) ρ| juν+1 �⊤ ˜λ (juν+1) ρ � = E �� P � Gc uν � + 2E �� E [I (auν ∈ [K])| uν]γuν−1,λ(σb) �� 1⊤ m˜λ (juν+1) ρ � + E �� E � (uν+1 − uν) ρ| juν+1 � − P � Gc uν � − 2E �� E [I (auν ∈ [K])| uν]γuν−1,σb(δ) �� 1⊤ m˜λ (juν+1) ρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (50) For the first term, we observe the dual problem of (1), min λ∈Rm + ρ1⊤ mλ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='λ⊤b⋆(j) k ≥ u⋆(j) k , ∀j ∈ [J], ∀k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (51) Comparing to the dual problem (49), when v = ρ1m and j = juν+1, (51) has more constraints than (49) with same objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Denote λ⋆ be the solution to (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then, ρ1⊤ m˜λ (juν+1) ρ1m ≤ ρ1⊤ mλ⋆ = OPT T , where the last equality holds by strong duality for the oracle problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus the first term in (50) is bounded as E �� P � Gc uν � + 2E �� E [I (auν ∈ [K])| uν]γuν−1,λ(σb) �� 1⊤ m˜λ (juν+1) ρ1m � ≤ � P � Gc uν � + 2E �� E [I (auν ∈ [K])| uν]γuν−1,λ(σb) �� OPT ρT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For the second term in (50), we observe that ˜λ(juν +1) E[ ρuν+1|juν+1] is a feasible solution to (49) when v = ρ1m and j = juν+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus E �� E � (uν+1 − uν) ρ| juν+1 � −2E �� E [I (auν ∈[K])| uν]γuν−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='λ(σb) � −P � Gc uν �� 1⊤ m˜λ (juν+1) ρ1m � ≤E ��� E � (uν+1 − uν) ρ| juν+1 � −2E �� E [I (auν ∈[K])| uν]γuν−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='λ(σb) � −P � Gc uν �� ∨ 0 � 1⊤ m˜λ (juν+1) ρ1m � ≤E ��� E � (uν+1 − uν) ρ| juν+1 � −2E �� E [I (auν ∈[K])| uν]γuν−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='λ(σb) � −P � Gc uν �� ∨ 0 � 1⊤ m˜λ(juν +1) E[ ρuν+1|juν+1] � ≤E �� E � ρuν+1 �� juν+1 � ∨ 0m �⊤ ˜λ(juν +1) E[ ρuν+1|juν+1] � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback where the last inequality holds by (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Because uν+1 ∈ U, we have ρuν+1 > 0 and E �� E � ρuν+1 �� juν+1 � ∨ 0m �⊤ ˜λ(juν +1) E[ ρuν+1|juν+1] � ≤E � E � ρuν+1 ∨ 0m �� juν+1 �⊤ ˜λ(juν +1) E[ ρuν+1|juν+1] � =E � E � ρuν+1 �� juν+1 �⊤ ˜λ(juν +1) E[ ρuν+1|juν+1] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Collecting the bounds, we have E � (uν+1 − uν) OPT T � ≤E � E � ρuν+1 �� juν+1 �⊤ ˜λ(juν +1) E[ ρuν+1|juν+1] � + � 2E �� E [I (auν ∈[K])| uν]γuν−1,σb(δ) � + P � Gc uν �� OPT ρT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Similar to Step 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' by strong duality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' E � ρuν+1 �� juν+1 �⊤ ˜λ(juν +1) E[ ρuν+1|juν+1] = max �K k=1 π (juν +1) k ≤1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='π (juν +1) k ≥0 min λ∈Rm + K � k=1 π(juν +1) k u⋆(juν +1) k + � E � ρuν+1 �� juν+1 � − K � k=1 π(juν +1) k b⋆(juν +1) k �⊤ λ ≤ min λ∈Rm + max �K k=1 π (juν +1) k ≤1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='π (juν +1) k ≥0 K � k=1 π(juν +1) k u⋆(juν +1) k + � E � ρuν+1 �� juν+1 � − K � k=1 π(juν +1) k b⋆(juν +1) k �⊤ λ ≤ min λ∈Rm + E � � max �K k=1 π (juν +1) k ≤1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='π (juν +1) k ≥0 K � k=1 π(juν +1) k u⋆(juν +1) k + � ρuν+1 − K � k=1 π(juν +1) k b⋆(juν +1) k �⊤ λ ������ juν+1 � � ≤ E � � max �K k=1 π (juν +1) k ≤1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='π (juν +1) k ≥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ρuν+1−�K k=1 π (juν +1) k b ⋆(juν +1) k ≥0 K � k=1 π(juν +1) k u⋆(juν +1) k ������ juν+1 � � = K � k=1 ˜π (juν+1) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='ρuν+1 u⋆(juν +1) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus we have E � (uν+1 − uν) OPT T � ≤E � K � k=1 ˜π (juν+1) k,ρuν+1 u⋆(juν +1) k � + � 2E �� E [I (auν ∈[K])| uν]γuν−1,σb(δ) � + P � Gc uν �� OPT ρT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Under the event Guν+1, the policy ˜π (juν+1) k,ρuν+1 is a feasible solution to the bandit problem (12), E � K � k=1 ˜π (juν+1) k,ρuν+1 u⋆(juν +1) k � ≤E � K � k=1 ˜π (juν+1) k,ρuν+1 u⋆(juν +1) k I � Guν+1 � � + P � Gc uν+1 � ≤E � K � k=1 ˜π (juν+1) k,ρuν+1 ˜u (juν+1) k,uν+1 I � Guν+1 � � + P � Gc uν+1 � ≤E � K � k=1 �π (juν+1) k,uν+1 ˜u (juν+1) k,uν+1 I � Guν+1 � � + P � Gc uν+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Thus, for each ν ∈ [|U| − 1], E � (uν+1 − uν) OPT T � ≤E � K � k=1 �π (juν+1) k,uν+1 ˜u (juν+1) k,uν+1 I � Guν+1 � � + P � Gc uν+1 � + � 2E �� E [I (auν ∈[K])| uν]γuν−1,σb(δ) � + P � Gc uν �� OPT ρT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Summing up over ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' E � � |U|−1 � ν=1 (uν+1 − uν) OPT T � � ≤E �� t∈U K � k=1 �π(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t ˜u(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t I (Gt) � + � 1 + OPT ρT � T � t=1 P (Gc t ) + � � |U|−1 � ν=1 2E �� E [I (auν ∈[K])| uν]γuν−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='σb(δ) � � � OPT ρT ≤E �� t∈U K � k=1 �π(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t ˜u(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t I (Gt) � + � 1 + OPT ρT � T � t=1 P (Gc t ) + 2 � T � t=1 E �� E [I (at ∈[K])]γt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='σb(δ) �� OPT ρT ≤E �� t∈U K � k=1 �π(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t ˜u(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t I (Gt) � + � 1 + OPT ρT � T � t=1 P (Gc t ) + 2 � � � �TE � T � t=1 γt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='σb(δ)2I (at ∈ [K]) � OPT ρT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' where the last inequality holds by Cauchy-Schwartz inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By (45),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' E � � |U|−1 � ν=1 (uν+1 − uν) OPT T � � ≤E �� t∈U K � k=1 �π(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t ˜u(jt) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='t I (Gt) � + � 1 + OPT ρT � T � t=1 P (Gc t ) + 2 � � � �TE � T � t=1 γt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='σb(δ)2I (at ∈ [K]) � OPT ρT ≤E � T � t=1 R�π t � + � 2 + OPT ρT � T � t=1 P (Gc t ) + 2 � � � �TE � T � t=1 γt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='σb(δ)2I (at ∈ [K]) � OPT ρT 2 � � � �TE � T � t=1 γt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='σr(δ)2I (at ∈ [K]) � ≤E � T � t=1 R�π t � + � 2 + OPT ρT � T � t=1 P (Gc t ) 2 � 1 + OPT ρT � � � � �TE � T � t=1 γt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='σb∨σr(δ)2I (at ∈ [K]) � Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Because the last choice of the algorithm happens at round τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' we have ρτ > 0 and u|U| = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' And by definition, u1 = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus E � � |U|−1 � ν=1 (uν+1 − uν) OPT T � � = E �� u|U| − u1 � OPT T � = OPT T E [τ − ξ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Rearranging the terms E � T � t=1 R�π t � ≥ E � � |U|−1 � ν=1 (uν+1 − uν) OPT T � � − � 2 + OPT ρT � T � t=1 P (Gc t ) − 2 � 1 + OPT ρT � � � � �TE � T � t=1 γt−1,σb∨σr(δ)2I (at ∈ [K]) � ≥ OPT T E [τ − ξ] − � 2 + OPT ρT � T � t=1 P (Gc t ) − 2 � 1 + OPT ρT � � � � �TE � T � t=1 γt−1,σb∨σr(δ)2I (at ∈ [K]) � , completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Let us fix δ ∈ (0, T −2) throughout the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Bounding the minimum eigenvalue of {Fν : ν ∈ [nT ]}: By Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3, with probability at least 1 − Tδ, 1 2KdFν = 1 2Kd ν � u=1 ˜Xk,τ(u) ˜X⊤ k,τ(u) + 8K − 1 K log Jd δ ⪰ 1 4Kd ν � u=1 E � ˜Xk,τ(u) ˜X⊤ k,τ(u) ��� Hτ(u)−1 � + 8K − 1 K log Jd δ − log Jd δ ⪰ 1 4Kd ν � u=1 E � ˜Xk,τ(u) ˜X⊤ k,τ(u) ��� Hτ(u)−1 � , for all ν ∈ [nT ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By Assumption 2 and 3, λmin � E � ˜Xk,τ(u) ˜X⊤ k,τ(u) ��� Hτ(u)−1 �� =λmin � � � � � p1EXk∼F1 ��K k=1 XkX⊤ k � 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 0 0 0 pJExk∼FJ ��K k=1 XkX⊤ k � � � � � � ≥pmin min j∈[J] λmin � EXk∼Fj � K � k=1 XkX⊤ k �� ≥pminKα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus, with probability at least 1 − Tδ, λmin(Fν) ≥1 2λmin � ν � u=1 E � ˜Xk,u ˜X⊤ k,u ��� Hu−1 �� ≥1 2 ν � u=1 λmin � E � ˜Xk,u ˜X⊤ k,u ��� Hu−1 �� ≥pminKαν 2 , Improved Algorithms for Multi-period Packing Problems with Bandit Feedback for all ν ∈ [nT ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Bounding the probability of Mt: Under the event proved in Step 1, the event Mt is implied by pminKαnt 2 ≥ 12Kd � nt � ν=1 96 (K − 1) log � Jd δ � αKpminν + 2 log Jd δ � , (52) for all t ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The left hand side is bounded as 12Kd � nt � ν=1 96 (K − 1) log � Jd δ � αKpminν + 2 log Jd δ � ≤12 · 96Kd log � Jd δ � log nt αpmin + 24Kd log Jd δ ≤12 · 96Kd log � Jd δ � log T αpmin + 24Kd log Jd δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Plugging in (52) and rearranging the terms, nt ≥ 96d log �Jd δ � �24 log T α2p2 min + 1 αpmin � , implies the event Mt for all t ∈ [T] with probability at least 1 − Tδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In other words, P (Mc t) ≤ P � nt < dMα,p,T log �Jd δ �� + Tδ, for all t ∈ [T], where Mα,p,T := 96 � 24 log T α2p2 min + 1 αpmin � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Bounding ξ: Let ˜t = inft∈[T ]{Mt happens} be the first round that Mt happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' After round ˜t, the algorithm skips the rounds until ρt > 0 holds and then pulls an action according to the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus, for the round ξ − 1, (ξ − 1) ρ − ξ−2 � s=1 b(js) as,s = (ξ − 1) ρ − ˜t � s=1 b(js) as,s ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Rearraging the terms, and taking expectation, E [ξ] ≤ 1 + ρ−1E � � ˜t � s=1 b(js) as,s � � ≤ 1 + ρ−1E �˜t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (53) Now we need an upper bound for ˜t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For t ∈ [˜t − 1], the event Mt does not happen and the algorithm admits the arrival for t ∈ [˜t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus, nt = t for all t ∈ [˜t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For t = ˜t − 1, the event M˜t−1 does not happen and λmin � Fn˜t−1 � ≤ 12Kd �n˜t−1 � ν=1 48 (K − 1) log � Jd δ � λmin(Fν) + 2 log Jd δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By the fact proved in Step 1, with probability at least 1 − Tδ, pminKαn˜t−1 2 ≤ 12Kd �n˜t−1 � ν=1 96 (K − 1) log � Jd δ � pminKαν + 2 log Jd δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Plugging in n˜t−1 = ˜t − 1 and rearranging the terms, ˜t − 1 ≤ 24d pminα � � � ˜t−1 � ν=1 96 (K − 1) log � Jd δ � pminKαν + 2 log Jd δ � � � ≤ 24d pminα �96 log T pminα + 2 log Jd δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Then with probability at least 1 − Tδ, ˜t ≤1 + 96d log �Jd δ � �24 log T α2p2 min + 1 αpmin � :=1 + Mα,p,T d log �Jd δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus, E �˜t � =E � ˜tI � ˜t < 1 + Mα,p,T d log �Jd δ ��� + E � ˜tI � ˜t ≥ 1 + Mα,p,T d log �Jd δ ��� ≤1 + dMα,p,T log �Jd δ � + TP � ˜t ≥ 1 + Mα,p,T d log �Jd δ �� ≤1 + dMα,p,T log �Jd δ � + T 2δ Plugging in (53), E [ξ] ≤1 + ρ−1E �˜t � ≤1 + 1 + dMα,p,T log � Jd δ � + T 2δ ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Proving the lower bound for τ: Let τ be the stopping time of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Because the algorithm admits arrival at round τ, we have ρτ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' From the resource constraint in the bandit problem (12), K � k=1 �π(jτ ) k,τ � �b(jτ ) k,τ − γτ−1,σb(δ) √pjτ 1m � := K � k=1 �π(jτ ) k,τ ˜b(jτ ) k,τ :≤ τρ − τ−1 � s=1 b(js) as,s Because algorithm stops at round τ, there exists an r ∈ [m] such that �τ s=1 b(js) as,s(r) ≥ Tρ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Rearranging the terms, τρ ≥ τ−1 � s=1 b(js) as,s(r) + K � k=1 �π(jτ ) k,τ ˜b(jτ ) k,τ (r) ≥Tρ − b(jτ ) aτ ,τ(r) + K � k=1 �π(jτ ) k,τ ˜b(jτ ) k,τ (r) =Tρ − b(jτ ) aτ ,τ(r) + K � k=1 �π(jτ ) k,τ b⋆(jτ ) k,τ (r) + K � k=1 �π(jτ ) k,τ � ˜b(jτ ) k,τ (r) − b⋆(jτ ) k,τ (r) � ≥Tρ − b(jτ ) aτ ,τ(r) + K � k=1 �π(jτ ) k,τ b⋆(jτ ) k (r) − K � k=1 �π(jτ ) k,τ ���˜b(jτ ) k,τ − b⋆(jτ ) k ��� ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Taking expectation on both side,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' E [τρ] ≥Tρ + E � −b(jτ ) aτ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ(r) + K � k=1 �π(jτ ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ b⋆(jτ ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ (r) � − E � K � k=1 �π(jτ ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ ���˜b(jτ ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ − b⋆(jτ ) k ��� ∞ � =Tρ − E � K � k=1 �π(jτ ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ ���˜b(jτ ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ − b⋆(jτ ) k ��� ∞ � =Tρ − E � K � k=1 �π(jτ ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ ���˜b(jτ ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ − b⋆(jτ ) k ��� ∞ I (Eτ ∩ Mτ−1) � − E � K � k=1 �π(jτ ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ ���˜b(jτ ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ − b⋆(jτ ) k ��� ∞ I � Ec τ ∪ Mc τ−1 � � ≥Tρ − 2E � γτ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='σb(δ) � E [I (at ∈ [K])| τ] � − E � K � k=1 �π(jτ ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ ���˜b(jτ ) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='τ − b⋆(jτ ) k ��� ∞ I � Ec τ ∪ Mc τ−1 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (54) Improved Algorithms for Multi-period Packing Problems with Bandit Feedback where the last inequality holds by (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Because ˜b(jτ ) k,τ (r) ≤ Tρ almost surely, E � K � k=1 �π(jτ ) k,τ ���˜b(jτ ) k,τ − b⋆(jτ ) k ��� ∞ I � Ec τ ∪ Mc τ−1 � � ≤TρP � Ec τ ∪ Mc τ−1 � =TρP (Ec τ) ≤Tρ � 4(m + 1)δ + 7T −1� =7ρ + 4(m + 1)Tδ, where the equality holds because the algorithm takes action according to the policy at round τ and the last inequality holds by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' from (54), E [τρ] ≥Tρ − 7ρ + 4(m + 1)Tρδ − 2E � γτ−1,σb(δ) � E [I (at ∈ [K])| τ] � ≥Tρ − 7ρ + 4(m + 1)Tρδ − 2γ1,σb(δ) Rearranging the terms, E [T − τ] ≤4(m + 1)Tδ + 7 + 2γτ−1,σb(δ) ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Proving a bound for the sum of probabilities Because the algorithm admits the arrival when Mt−1 does not happen, Mc t−1 = Mc t−1 ∩ {at ∈ [K]} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then P � Mc t−1 � =P � Mc t−1 ∩ {at ∈ [K]} � =P � Mc t−1 ∩ {at ∈ [K]} ∩ � nt−1 ≥ Mα,p,T d log �Jd δ ��� + P � Mc t−1 ∩ {at ∈ [K]} ∩ � nt−1 < Mα,p,T d log �Jd δ ��� ≤P � Mc t−1 ∩ � nt−1 ≥ Mα,p,T d log �Jd δ ��� + P � {at ∈ [K]} ∩ � nt−1 < Mα,p,T d log �Jd δ ��� ≤Tδ + P � {at ∈ [K]} ∩ � nt−1 < Mα,p,T d log �Jd δ ��� , where the last inequality holds by the fact proved in Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Summing over t ∈ [T], T � t=1 P � Mc t−1 � ≤T 2δ + T � t=1 P � {at ∈ [K]} ∩ � nt−1 < Mα,p,T d log �Jd δ ��� =T 2δ + E � T � t=1 I (at ∈ [K]) I � nt−1 < Mα,p,T d log �Jd δ ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Set µ := Mα,p,T d log � Jd δ � and suppose T � t=1 I (at ∈ [K]) I (nt−1 < µ) > µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (55) Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Let τ(1) < τ(2) < · · · < τ(|A|) be the ordered admitted round in A := {t ∈ [T] : at ∈ [K]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By definition, nτ(ν) = ν for ν ∈ [|A|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By (55), the event{at ∈ [K]} happens at least µ + 1 times over the horizon [T] and |A| > µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For any ν ∈ (µ, |A|],the number of admitted round is nτ(ν) > µ and T −1 � t=ε I (nt−1 < µ) I (at ∈ [K]) = |A| � ν=1 I � nτ(ν)−1 < µ � I � aτ(ν) ∈ [K] � ≤ |A| � ν=1 I � nτ(ν)−1 < µ � I � nτ(ν) = nτ(ν)−1 + 1 � = |A| � ν=1 I � nτ(ν)−1 < µ � I � ν = nτ(ν)−1 + 1 � ≤ |A| � ν=1 I (ν − 1 < µ) , = |A| � ν=1 I (ν < µ + 1) =µ, which contradicts with (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus E � T � t=1 I (at ∈ [K]) I � nt−1 < Mα,p,T d log �Jd δ ��� ≤ µ := Mα,p,T d log �Jd δ � , which proves, T � t=1 P � Mc t−1 � ≤ T 2δ + Mα,p,T d log �Jd δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2, rearranging the terms, R�π T :=OPT − E � T � t=1 R�π t � ≤OPT T {T − E [τ − ξ]} + � 2 + OPT ρT � T � t=1 P � Mc t−1 ∪ Ec t � + 2 � � � �TE � T � t=1 γt−1,σr(δ)2I (at ∈ [K]) � + 2 � � � �TE � T � t=1 γt−1,σb(δ)2I (at ∈ [K]) � OPT ρT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3, E [ξ] ≤ 1 + 1+dMα,p,T log � Jd δ � +T 2δ ρ , E [T − τ] ≤ 4(m + 1)Tδ + 7 + 2γτ−1,σb(δ) ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By definition of γt,σ(δ), E [T − τ] ≤4(m + 1)Tδ + 7 + 32√J log JKT + 8 √ 2βσb(δ) ρ =4(m + 1)Tδ + 7 + 32√J log JKT + Cσ(δ) √ Jd ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' This implies OPT T {T − E [τ − ξ]} ≤ OPT Tρ � ρ + 4(m + 1)Tδ + 8 + 32 � J log �JK δ � +Cσb(δ) √ Jd + dMα,p,T log �Jd δ � +T 2δ � ≤ OPT Tρ � ρ + 8 + � 5mT + T 2� δ + 32 � J log �JK δ � +Cσb(δ) √ Jd + dMα,p,T log �Jd δ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' [Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Bounding the sum of probability] Because T ≥ 8dα−1p−1 min log JdT, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3, T � t=1 P � Mc t−1 ∪ Ec t � = T � t=1 � P � Mc t−1 � + P (Mt−1 ∩ Ec t ) � ≤T 3δ + dMα,p,T log �Jd δ � + T � t=1 P (Mt−1 ∩ Ec t ) ≤T 3δ + dMα,p,T log �Jd δ � + 8dα−1p−1 min log JdT + T � t=8dα−1p−1 min log JdT P (Mt−1 ∩ Ec t ) ≤T 3δ + dMα,p,T log �Jd δ � + 8dα−1p−1 min log JdT + 4(m + 1)Tδ + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By definition of γt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='σ(δ) and βσ(δ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' E � T � t=1 γt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='σr(δ)2I (at ∈ [K]) � =E � � T � t=1 � 16√J log JKT √t − 1 + 4 √ 2βσr(δ) √nt−1 �2 I (at ∈ [K]) � � ≤E � T � t=1 � 16√J log JKT + 4 √ 2βσr(δ) �2 nt−1 I (at ∈ [K]) � ≤E � T � t=1 � 16√J log JKT + 4 √ 2βσr(δ) �2 nt−1 I (nt = nt−1 + 1) � ≤ � 16 � J log JKT + 4 √ 2βσr(δ) �2 log T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback where the first inequality holds by nt ≤ t almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus by definition of βσ(δ) := 8 √ Jd + 96σ � Jd log 4 δ , 2 � � � �TE � T � t=1 γt−1,σr(δ)2I (at ∈ [K]) � ≤ � 32 � J log JKT + 4 √ 6βσr(δ) � � T log T ≤ � 32 � J log JKT + Cσr(δ) √ Jd � � T log T, where Cσ(δ) := 8 √ 2 · � 8 + 96σ � log 4 δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' 2 � � � �TE � T � t=1 γt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='σb(δ)2I (at ∈ [K]) � OPT ρT ≤ � 32 � J log JKT + Cσb(δ) √ Jd � � T log T OPT ρT Collecting the bounds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' R�π T ≤ OPT Tρ � ρ + 8 + � 5mT + T 2� δ + 32 � J log JKT +Cσb(δ) √ Jd + dMα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='T log �Jd δ �� + � 2 + OPT ρT � � T 3δ + dMα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='T log �Jd δ � + 4dα−1p−1 min log JdT + 4(m + 1)Tδ + 7 � + 2 � 1 + OPT ρT � � 32 � J log JKT + Cσb∨σr(δ) √ Jd � � T log T ≤ � 2 + OPT ρT � � � 96 � J log JKT + 3Cσr∨σr(δ) √ Jd � � T log T + 2dMα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='T log �Jd δ � + 4dα−1p−1 min log JdT + 15 + 10mT 3δ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Plugging in δ = m−1T −3 proves (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Technical lemmas Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (Azuma-Hoeffding’s inequality) Azuma (1967) If a super-martingale (Yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' t ≥ 0) corresponding to filtration Ft, satisfies |Yt − Yt−1| ≤ ct for some constant ct, for all t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' , T, then for any a ≥ 0, P (YT − Y0 ≥ a) ≤ e − a2 2 �T t=1 c2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus with probability at least 1 − δ, YT − Y0 ≤ � � � �2 log 1 δ T � t=1 c2 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For a sequence u1 ≥ u2 ≥ · · · ≥ un ≥ 0 and nonnegative real sequences {pi}i∈[n] and {qi}i∈[n] such that �n i=1 pi = �n i=1 qi, if p1 > q1 then n � i=1 piui ≥ n � i=1 qiui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' When n = 1, p1u1 ≥ q1u1, for any u1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Suppose for any sequence u1 ≥ u2 ≥ · · · ≥ un−1 ≥ 0 and nonnegative real sequences {pi}i∈[n−1] and {qi}i∈[n−1] such that �n−1 i=1 pi = �n−1 i=1 qi, p1 > q1 =⇒ n−1 � i=1 piui ≥ n−1 � i=1 qiui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback For a sequence u1 ≥ u2 ≥ · · · ≥ un ≥ 0 and nonnegative real sequences {pi}i∈[n] and {qi}i∈[n] such that �n i=1 pi = �n i=1 qi, and p1 > q1, there exist k ∈ [n]\\{1} such that pk < qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In case of k = n, define a sequence ˜qi = qi, ∀i ∈ [n − 2] ˜qn−1 = qn−1 − pn + qn ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then �n−1 i=1 ˜qi = �n−1 i=1 pi and n � i=1 piui = n−1 � i=1 piui + pnun ≥ n−1 � i=1 ˜qiui + pnun = n−1 � i=1 qiui + (−pn + qn) un−1 + pnun ≥ n−1 � i=1 qiui + (−pn + qn) un + pnun = n � i=1 qiui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' In case of k ̸= n, denote a sequence ˜qi = qi, ∀i ∈ [n − 1]\\{k} ˜qk = qk − pk + qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then �n−1 i=1 ˜qi = � j̸=k pi and n � i=1 piui = � i̸=k piui + pkuk ≥ n−1 � i=1 ˜qiui + pkuk ≥ n−1 � i=1 qiui − pkuk + qnuk + pkuk = n−1 � i=1 qkuk + qnuk ≥ n � i=1 qkuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By induction, the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Let {Xτ : τ ∈ [t]} be a Rd×d-valued stochastic process adapted to the filtration {Fτ : τ ∈ [t]}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', Xτ is Fτ-measurable for τ ∈ [t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Suppose Xτ is a positive definite symmetric matrices such thatλmax(Xτ) ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='Then with probability at least 1 − δ, t � τ=1 Xτ ⪰ 1 2 t � τ=1 E [Xτ| Fτ−1] − log d δ Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback In addition, with probability at least 1 − δ, t � τ=1 Xτ ⪯ 3 2 t � τ=1 E [Xτ| Fτ−1] + log d δ Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' This proof is an adapted version of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='2 in Lattimore & Szepesv´ari (2020) for matrix stochastic process using the argument of Tropp (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For the lower bound, It is sufficient to prove that λmax � − t � τ=1 Xτ + 1 2 t � τ=1 E [Xτ| Fτ−1] � ≤ log d δ , with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By the spectral mapping theorem, exp � λmax � − t � τ=1 Xτ + 1 2 t � τ=1 E [Xτ| Fτ−1] �� ≤λmax � exp � − t � τ=1 Xτ + 1 2 t � τ=1 E [Xτ| Fτ−1] �� ≤Tr � exp � − t � τ=1 Xτ + 1 2 t � τ=1 E [Xτ| Fτ−1] �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Taking expectation on both side gives, E exp � λmax � − t � τ=1 Xτ + 1 2 t � τ=1 E [Xτ| Fτ−1] �� ≤ETr � exp � − t � τ=1 Xτ + 1 2 t � τ=1 E [Xτ| Fτ−1] �� =ETr � E � exp � − t−1 � τ=1 Xτ + 1 2 t � τ=1 E [Xτ| Fτ−1] + log exp (−Xt) ������ Ft−1 �� ≤ETr � exp � − t−1 � τ=1 Xτ + 1 2 t � τ=1 E [Xτ| Fτ−1] + log E [exp (−Xt)| Ft−1] �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' The last inequality holds due to Lieb’s theorem Tropp (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Because ex ≤ 1+ 1 2xfor all x ∈ [−1/2, 0], and the eigenvalue of −Xt lies in [−1/2, 0], we have E [exp (−Xt)| Ft−1] ⪯ I − 1 2E [Xt| Ft−1] ⪯ exp � −1 2E [Xt| Ft−1] � , by the spectral mapping theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus we have E exp � λmax � − t � τ=1 Xτ + 1 2 t � τ=1 E [Xτ| Fτ−1] �� ≤ ETr � exp � − t−1 � τ=1 Xτ + 1 2 t � τ=1 E [Xτ| Fτ−1] + log exp � −1 2E [Xt| Ft−1] ��� = ETr � exp � − t−1 � τ=1 Xτ + 1 2 t � τ=1 E [Xτ| Fτ−1] − 1 2E [Xt| Ft−1] �� = ETr � exp � − t−1 � τ=1 Xτ + 1 2 t−1 � τ=1 E [Xτ| Fτ−1] �� ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' ≤ ETr (exp (O)) = d Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Now my Markov’s inequality, P � λmax � − t � τ=1 Xτ + 1 2 t � τ=1 E [Xτ| Fτ−1] � > log d δ � ≤ E exp � λmax � − t � τ=1 Xτ + 1 2 t � τ=1 E [Xτ| Fτ−1] �� δ d ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For the upper bound, we prove λmax � t � τ=1 Xτ − 3 2 t � τ=1 E [Xτ| Fτ−1] � ≤ log d δ , in a similar way using the fact that ex ≤ 1 + (3/2)x on x ∈ [0, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Suppose a random variable X satisfies E[X] = 0, and let Y be an σ-sub-Gaussian random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' If |X| ≤ |Y | almost surely, then X is 6σ-sub-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Because |X| ≤ |Y | E � X2 6σ2 � ≤E � Y 2 6σ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' =1 + E �� ∞ 0 I (|Y | ≥ x) x 3σ2 e x2 6σ2 dx � ≤1 + � ∞ 0 P (|Y | ≥ x) x 3σ2 e x2 6σ2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Because P (|Y | ≥ x) =P (Y ≥ x) + P (−Y ≤ x) ≤2e− x2 2σ2 , we have E � X2 6σ2 � ≤1 + � ∞ 0 2x 3σ2 e− x2 3σ2 dx ≤2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Now for any λ ∈ R, E [exp (λX)] =E � ∞ � n=0 (λX)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' � =1 + E � ∞ � n=2 (λX)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' � ≤1 + E � λ2X2 2 ∞ � n=2 |λX|n−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' � ≤1 + λ2 2 E � X2 exp (|λX|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback Because 6σ2λ2 + X2 12σ2 ≥ |λX| , E [exp (λX)] ≤1 + λ2 2 exp � 6σ2λ2� E � X2 exp � X2 12σ2 �� =1 + 6σ2λ2 exp � 6σ2λ2� E � X2 12σ2 exp � X2 12σ2 �� ≤1 + 6σ2λ2 exp � 6σ2λ2� E � exp � X2 6σ2 �� ≤1 + 12σ2λ2 exp � 6σ2λ2� ≤ � 1 + 12σ2λ2� exp � 6σ2λ2� ≤ exp �36 2 σ2λ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus X is 6σ-sub-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=', 2016, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='3) Let {Nt} be a martingale on a Hilbert space (H, ∥·∥H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then there exists a R2-valued martingale {Pt} such that for any time t ≥ 0, ∥Pt∥2 = ∥Nt∥H and ∥Pt+1 − Pt∥2 = ∥Nt+1 − Nt∥H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (A dimension-free bound for vector-valued martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=') Let {Fs}t s=0 be a filtration and {ηs}t s=1 be a real-valued stochastic process such that ηs is Fτ-measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Let {Xs}t s=1 be an Rd-valued stochastic process where Xs is F0-measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Assume that {ηs}t s=1 are σ-sub-Gaussian as in Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then with probability at least 1 − δ, ����� t � s=1 ηsXs ����� 2 ≤ 12σ � � � � t � s=1 ∥Xs∥2 2 � log 4t2 δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' (56) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Fix a t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' For each s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' , t, we have E [ηs| Fs−1] = 0 and Xs is F0-measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus the stochastic process, � u � s=1 ηsXs �t u=1 (57) is a (Rd, ∥·∥2)-martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Since (Rd, ∥·∥2) is a Hilbert space, by Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='5, there exists an R2-martingale {Mu}t u=1 such that ����� u � s=1 ηsXs ����� 2 = ∥Mu∥2 , ∥ηuXu∥2 = ∥Mu − Mu−1∥2 , (58) and M0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Set Mu = (M1(u), M2(u))⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Then for each i = 1, 2, and u ≥ 2, |Mi(u) − Mi(u − 1)| ≤ ∥Mu − Mu−1∥2 = ∥ηuXu∥2 = |ηu| ∥Xu∥2 , almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='4, Mi(u) − Mi(u − 1) is 6σ-sub-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' By Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content='1, for x > 0, P (|Mi(t)| > x) =P ������ t � u=1 Mi(u) − Mi(u − 1) ����� > x � ≤2 exp � − x2 72tσ2 �t s=1 ∥Xs∥2 2 � , for each i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Thus, with probability 1 − δ/2, Mi(t)2 ≤ 72 � t � s=1 ∥Xs∥2 2 � σ2 log 4 δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} +page_content=' Improved Algorithms for Multi-period Packing Problems with Bandit Feedback In summary, with probability at least 1 − δ/2, ����� t � τ=1 ηsXs ����� 2 = � M1(t)2 + M2(t)2 ≤ 6σ � � � � t � s=1 ∥Xs∥2 2 � 2 log 4t2 δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FST4oBgHgl3EQfZTjC/content/2301.13791v1.pdf'} diff --git a/49E1T4oBgHgl3EQfmQS7/content/tmp_files/2301.03296v1.pdf.txt b/49E1T4oBgHgl3EQfmQS7/content/tmp_files/2301.03296v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..37faf40abde389f1497ba0ad964fb1ee0278eb0b --- /dev/null +++ b/49E1T4oBgHgl3EQfmQS7/content/tmp_files/2301.03296v1.pdf.txt @@ -0,0 +1,1074 @@ +Precise certification of a qubit space +Tomasz Białecki1, Tomasz Rybotycki2,3, Josep Batle4, Jakub Tworzydło1, and Adam Bednorz1, ∗ +1Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL02-093 Warsaw, Poland +2Systems Research Institute, Polish Academy of Sciences, 6 Newelska Street, PL01-447 Warsaw, Poland +3Center for Theoretical Physics, Polish Academy of Sciences, +Al. +Lotników 32/46, PL02-668 Warsaw, Poland +4CRISP - Centre de Recerca Independent de sa Pobla, 07420 sa Pobla, Balearic Islands, Spain +We demonstrate an implementation of the precise test of dimension on the qubit, using the +public IBM quantum computer, using the determinant dimension witness. The accuracy is below +10−3 comparing to maximal possible value of the witness in higher dimension. The test involving +minimal independent sets of preparation and measurement operations (gates) is applied both for +specific configurations and parametric ones. +The test is be robust against nonidealities such as +incoherent leakage and erroneous gate execution. Two of the IBM devices failed the test by more +than 5 standard deviations, which has no simple explanation. +I. +INTRODUCTION +Physics is an exact science, which is confirmed by pre- +cise measurements of fundamental constants and estab- +lishing definition of SI units by precise quantum experi- +ments [1–6]. Precision is also required from every com- +puter, also quantum. +Unfortunately, current quantum +technologies suffer from inevitable sources of errors, both +just from mechanical limitations and inseparable physical +environment. Of course, there are methods to mitigate +and correct the errors. Such approach relies, however, +on assumptions about the controllable space of possible +actions. +The basic building block of a quantum computer is a +qubit, a generic two-level system. Since the goal is to +manipulate accurately many qubits, it is necessary to as- +certain whether or not the qubit space is reliable, i.e. +not combined with a larger space. The most promising +implementations of qubits keep them detuned from en- +vironment and other states, except for small incoherent +disturbance. +On the other hand, the potential contri- +bution of external states can lead to systematic errors, +hard to correct. Operations on qubits, gates, realized by +microwave pulses, suffer from distortions due to nonlin- +earities of waveform generators [7], so a simple deviation +of the probability distribution from the theoretical pre- +diction is not yet a proof of extra space [8]. Therefore, +to increase the quality of classical and quantum com- +putation and communication, these systems need precise +certification, robust against imperfections of physical im- +plementations. +The dimension of the quantum space can be checked +by a dimension witness [9–14]. The construction of the +witness is based on the two-stage protocol, the initial +preparation and subsequent final measurement, which are +chosen from independent sets. The preparation must be +completed before the start of the measurement. A precise +∗Electronic address: Adam.Bednorz@fuw.edu.pl +witness must be based on equality, i.e. a quantity, which +is exactly zero up to a certain dimension, and nonzero +otherwise. Such a good witness test is the linear inde- +pendence of the specific dichotomic outcome probability +p(M|N) for the preparation N and measurement M, see +Fig. 1, tested by a suitable determinant [15–17]. It has +been been already performed on optical states [18]. It be- +longs to a family to equality-based tests, like the Sorkin +equality [19] in the three-slit experiment [20–22] testing +Born’s rule [23], benchmarking our trust in fundamental +quantum models and their actual realizations. +In this paper, we apply the test to several IBM quan- +tum device. While some results agree with the 2−level +model, taking a large statistics revealed signature of the +failure by more than 5 standard deviations. Of course it +does not immediately mean a larger space but the prob- +lem needs urgent further investigation to determine the +cause, which may be also another assumption of the test +(e.g. lack of independence of the operations). +II. +THEORY +We apply a test of the qubit space d = 2 with the +witness constructed for p(M|N) = trMN, N = N † ≥ 0, +trN = 1 and measurement 1 ≥ M = M † ≥ 0. Taking +5 preparations Nj, j = 1..5 and 4 measurements Mk, +k = 1..4. +Then the determinant W = det p, for the +5 × 5 matrix p with entries pkj = p(Mj|Nk) and p5j = 1, +must be equal to zero if all Nj and Mk are represented +in the same two-level space. In addition, it remains zero +also if all preparations and measurements contain some +constant incoherent leakage term, i.e. +N ′ +j = Nj + Ne +and M ′ +k = Mk + Me, with Ne and Me independent of +j and k and commuting with Nj and Mk. In this way, +the common leakage to higher states does not affect the +test [14]. For d = 2 we have W = 0, but d = 3 gives +maximally 27 +√ +2/64 ≃ 0.6 in the real space and ≃ 0.632 +in the complex space [16]. For d = 4 the maximum (real +and complex) is 212/37 ≃ 1.87. Even higher dimensions +are saturated by the classical maximum 3. +arXiv:2301.03296v1 [quant-ph] 9 Jan 2023 + +2 +The IBM Quantum Experience cloud computing of- +fers several devices, collections of qubits, which can be +manipulated by a user-defined set of gates (operations) +– either single qubit or two-qubit ones, also paramteric. +One can put barriers (controlling the order of operations) +or additional resets (nonunitary transition to the ground +state). The qubits are physical transmons [24], the arti- +ficial quantum states existing due to interplay of super- +conductivity (Josephson effect) and capacitance. Due to +anharmonicity one can limit the working space to two +states. The decoherence time (mostly environmental) is +long enough to perform a sequence of quantum opera- +tions and read out reliable results. +The ground state |0⟩ can be additionally assured by +a reset operation. +Gates are implemented by time- +scheduled microwave pulses prepared by waveform gen- +erators and mixers (time 30 − 70ns with sampling at +0.222ns), tuned to the drive frequency (energy difference +between qubit levels) [25] (about 4−5Ghz). The rotation +Z is not a real pulse, but an realized by an instantaneous +virtual gate VZ(θ), which adds a rotation between in- +and out-of-phase components of the next gates [26]. The +readout is performed another long microwave pulse of +frequency (different from the drive) to the resonator and +measuring the populated photons [25, 27]. +In the following, we assume the two-level description +of the qubits, expecting W = 0 up to statistical er- +ror. +Larger deviation would be an evidence that this +description is inaccurate. The states and operators will +be can be described either in a two-dimensional Hilbert +space with basis |0⟩, |1⟩ or in the Bloch sphere with +V = (v0 + v · σ)/2, with the 3-component Bloch vec- +tor v and standard Pauli matrices +σ1 = +� +0 1 +1 0 +� +, σ2 = +� +0 −i +i +0 +� +, σ3 = +� +1 +0 +0 −1 +� +. +(1) +Then the initial state |0⟩⟨0| corresponds to the vector +(0, 0, 1) while n0 = 1 and |n| ≤ 1 and 2 − |m| ≥ m0 ≥ +|m|. A microwave pulse tuned to the interlevel drive fre- +quency corresponds to parametric gates, π/2 rotations, +Sγ = Z† +γSZγ, Zθ = +� +e−iγ/2 +0 +0 +eiγ/2 +� +, +S = RX(π/2) = +√ +X = +1 +√ +2 +� +1 +−i +−i +1 +� +, +(2) +in the basis |0⟩, |1⟩ while +S = +� +� +1 0 +0 +0 0 −1 +0 1 +0 +� +� , Zγ = +� +� +cos γ − sin γ 0 +sin γ +cos γ +0 +0 +0 +1 +� +� , +(3) +on the Bloch vector, i.e. SγV S† +γ. +Physically the experiment is sequence of preparation in +the state |0⟩, two gates Sα, Sβ for the preparation, two +gates Sφ, Sθ, and the the readout pulse for the measure- +ment of the state |0⟩ again, see Fig. 2. There are 5 pairs +of angles αj, βj to be chosen independently of the 4 pairs +N +M +1 +0 +FIG. 1: Preparation and measurement scenario; the state is +prepared as N and measured by M to give an outcome of +either 1 or 0. +|0⟩ +Sα +Sβ +Sφ +Sθ +FIG. 2: The quantum circuit for the dimension test. +The +initial state |0⟩ and four gates Sγ, split into preparation and +measurement stages, are followed by the final dichotomic mea- +surement +θk, φk. Then N = SβSα|0⟩⟨0| and M = S† +φS† +θ|0⟩⟨0|SθSφ. +The actual pulse waveform of a sample sequence of gates +is depicted in Fig. 3. +III. +EXPERIMENT +In a perfect theory, we can predict a probability for +every choice of α, β, θ, φ. The experimental results can +differ for a variety of reasons. Firstly, the test is random +and we have to estimate the error due to finite statistics. +For T times the experiment is repeated, the variance of +W can be estimated as +T⟨W 2⟩ ≃ +� +kj +pkj(1 − pkj)(Adj p)2 +jk, +(4) +where Adj is the adjoint matrix (matrix of minors of p, +with crossed out a given row and column, and then trans- +0 +30 +60 +90 +119 +149 +Time (ns) +VZ(2.28) +VZ( +4.37) +VZ(7.33) +VZ( +1.57) +D0 +5.26 GHz +FIG. 3: +The actual waveform of the pulse on IBM quan- +tum computer (nairobi), with four subsequent gates Sγ, with +γ = α, β, φ, θ, consecutively. The discretization unit time is +dt = 0.222ns. Driving (level gap) frequency is denoted by D0. +The light/dark shading corresponds to in-phase/out-of-phase +amplitude component, respectively. The element VZ(ξ) is a +zero-duration virtual gate Zξ for subsequent gates SγSδ with +ξ = γ − δ [26]. + +3 +posed). Note that the identity p−1 det p = Adjp makes +no sense here as W = det p = 0 in the limit T → ∞. +Secondly, the implementation of gates may be not faith- +ful. +Our test is capable to take them into account as +long as the leakage to external states (e.g. |2⟩) is inco- +herent and does not depend on the parameters α, β, θ, φ. +Lastly, we have to assume that the pulse does not depend +on the previous ones. In other words, we can only test +the combination of assumptions, dimension of the space +and independence of operations. We have calculated W +in two ways, (i) determining p for each job and then find- +ing W (see the values for each job in Fig. 10) and finally +averaging W, (ii) averaging first p from all jobs and then +finding W. +There is no a priori best selection of preparations and +measurements but they should not lie on a single Bloch +circle. We decided to make two kinds of tests: (I) two +special configurations corresponding to either the same +Bloch vectors for preparation and measurements or max- +imal ⟨W 2⟩ for a given R; (II) a family of configurations +with one preparation vector at one of the 5 directions on +the Bloch circle. In both cases the corresponding Bloch +vectors are derived explicitly in Appendix A. The sets +of angles in the case (I) are given in the Table I, and +the corresponding Bloch vectors are visualized in Fig. +4. +We have run the test on lima and lagos, qubit 0. +The probability matrix, compared to the ideal expecta- +tion is depicted in Fig. 5. The deviation from zero and +the statistical error is given in Fig. 6. The number of +T = #jobs·#shots·#repetitions. Technically, one sends +a list of jobs to execute, each job contains up to 300 cir- +cuits, to be distributed between experiments repeated the +same number of times. Each circuit is run the number +of shots. The readout counts for each circuit is the value +returned after the job execution is accomplished. +The sets of angles in the case (II) are prepared dif- +ferently. Four preparations and measurements are fixed +while the last preparation is parameter-dependent. The +fixed angles are specified in Table II. The last prepara- +tion angles are α5 = 2πi/5 = β5 − π/2 for i = 0..4. The +corresponding Bloch vectors are depicted in Fig. 7. We +have run the test on nairobi and perth, qubit 0. +The +probability matrix, compared to the ideal expectation is +depicted in Fig. 8. The deviation from zero and the sta- +tistical error is given in Fig. 9. The deviation from the +expected 0 is more than 5 standard deviations. The data +and scripts are available at the public repository [28]. +A. +Nonidealities +There are several factors that can affect the correct- +ness of the experiment. (A) The daily calibration. The +drive frequency and the gate waveforms are corrected so +different jobs can rely on different realizations of gates. +There first order effect of calibrations is cancelled out. +Nevertheless, we made more detailed estimates on sec- +ond order effects in Appendix B. Only large, unexpected +j 1 +2’ +3’ +4’ +5’ +2” +3” +4” +5” +α 0 2π/3 2π/3 4π/3 4π/3 +0 η − π η + 5π/3 η + π/3 +β 0 π/6 −π/6 π/6 −π/6 π +0 +2π/3 +−2π/3 +k +1’ +2’ +3’ +4’ +1” +2” +3” +4” +θ 5π/3 5π/3 π/3 +π/3 +π π/2 7π/6 −π/6 +φ 7π/6 5π/6 7π/6 5π/6 0 +π +5π/3 +π/3 +TABLE I: The angles for the special two special cases, ’ and +”, with η = acos(1/3) and 1=1’=1”. +FIG. 4: The Bloch vectors for the preparations (red) and +measurements (blue) corresponding to the angles from Table +I, top ’ and bottom ”. For the case ’, the four measurement +direction are identical to four preparations. +j 1 +2 +3 +4 +α 0 η − π η + 5π/3 η + π/3 +β 0 +0 +2π/3 +−2π/3 +k 1 +2 +3 +4 +θ π π/2 7π/6 −π/6 +φ 0 +π +5π/3 +π/3 +TABLE II: The angles for the parametric case (II) for prepa- +rations and measurements 1..4 + +14 +1 +2 +3 +4 +ideal k +1 +2 +3 +4 +lima k +1 +2 +3 +4 +5 +j′ +1 +2 +3 +4 +lagos k +1 +2 +3 +4 +5 +j′′ +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +FIG. 5: Results of the test (I) with probabilities pkj for the an- +gles from Table I, for lima and lagos, compared to the ideal ex- +pectation. Lagos: 60 jobs, 32000 shots, 15 repetitions. Lima: +521’/194” jobs, 20000 shots, 5 repetitions +failures could be a problem. (B) Amplitude-dependent +leakage and distortion of the waveform. The leakage to +higher states, e.g. +|2⟩ is small, of the order 10−4 and +incoherent [8, 14], see details in Appendix C. It is pos- +sible that distortion of amplitude to the waveform de- +pends on the rotation angle (phase) but we expect this +effect to be very small, 10−3, based on the deviations ob- +served in our previous work, and so the net effect is 10−7. +(C) Memory of the waveform between successive gates. +Highly unlikely, a residual voltage amplitude can persist +up to the next gate. In principle it can be mitigated by +delay-separated gates if the effect fades out with time. +(D) Other qubits. They are usually detuned but some +crosstalk may remain. As in the case of leakage, we ex- +pect the crosstalk to be incoherent and so irrelevant for +the witness. As a sanity check we have run simulations, +using the noise models from nairobi and perth, and no +significant deviation have been found, see Appendix D. +lima′ +lima′′ +lagos′ +lagos′′ +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +×10 +4 +FIG. 6: Results of the test (I) the witness W = det p, for the +angles from Table I, for lima and lagos, with the error given +by (4). Red – W for p from each job and then averaged, blue +– p averaged from all jobs to give W. +FIG. 7: The Bloch vectors for the parametric case (II) with +fixed preparations (red) and measurements (blue) correspond- +ing to the angles from Table I, and a parametric preparation +(green). +IV. +DISCUSSION +A test of linear independence of quantum operations +reveals subtle deviations, invisible in more crude tests. +Further tests are necessary to identify the origin of the +deviations, to exclude e.g. exotic many world/copies the- +ories [29, 30]. We suggest: (i) an extreme statistics col- +lected in a relatively short time to avoid corrections due +to calibrations, (ii) a time separation between gates to ex- +clude potential overlap of the effects, (iii) a scan through +a large set of Bloch vectors to maximize the potential +deviation, (iv) run the test on a single-qubit devices to +avoid cross-talks. It is also possible to develop more so- +phisticated tests, with different assumptions, or involv- +ing different qubits. In any case, a precise diagnostics of +qubits must become a standard in quantum technologies. + +5 +1 +2 +3 +4 +j +1 +2 +3 +4 +k +theory +0 +1 +2 +3 +4 +i +j = 5 +0.0 +0.5 +1.0 +1 +2 +3 +4 +j +1 +2 +3 +4 +k +nairobi +0 +1 +2 +3 +4 +i +j = 5 +0.0 +0.5 +1.0 +1 +2 +3 +4 +j +1 +2 +3 +4 +k +perth +0 +1 +2 +3 +4 +i +j = 5 +0.0 +0.5 +1.0 +FIG. 8: Results of the test (II) with probabilities pkj for the +angles from Table II, for nairobi and perth, compared to the +theory expectation. Nairobi/perth: 115/93 jobs, both 100000 +shots and 8 repetitions +Appendix A: Bloch sphere representations +Using vectors n to represent the state N = |n⟩⟨n| = +(ˆ1 + n · σ)/2, we have SαNS† +α = Nα and S† +θMSθ = Mθ +with +nα = +� +� +cos2 α +− cos α sin α − sin α +− cos α sin α +sin2 α +− cos α +sin α +cos α +0 +� +� n, +mθ = +� +� +cos2 θ +− cos θ sin θ sin θ +− cos θ sin θ +sin2 θ +cos θ +− sin θ +− cos θ +0 +� +� n, +(A1) +For n = m = (0, 0, 1) and m0 = 1, we have Nαβ = +SβSαNS† +αS† +β with +n′ +αβ = (sin(β − α) cos β, sin(α − β) sin β, − cos(β − α)) +(A2) +while Mθφ = S† +φS† +θMSθSφ wirh +M θφ = (sin(θ − φ) cos φ, sin(φ − θ) sin φ, − cos(θ − φ)). +(A3) +0 +1 +2 +3 +4 +−2.5 +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +nairobi +×10−4 +0 +1 +2 +3 +4 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +perth +×10−4 +FIG. 9: Results of the test (II) the witness W = det p, for +the angles from Table II, for nairobi and perth, with the error +given by (4). Red – W for p from each job and then averaged, +blue – p averaged from all jobs to give W. +0 +1 +2 +3 +4 +−1.5 +−1.0 +−0.5 +0.0 +0.5 +nairobi +×10−4 +0 +1 +2 +3 +4 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +perth +×10−3 +FIG. 10: Results of the test (II) the witness W = det p, for the +angles from Table II, for nairobi and perth, for individual jobs. +Two values for nairobi are beyond the picture boundaries, +(3, 0.0023) and (4, 0.003) + +6 +1 +2 +3 +4 +j +1 +2 +3 +4 +k +nairobi-sim +0 +1 +2 +3 +4 +i +j = 5 +0.0 +0.5 +1.0 +1 +2 +3 +4 +j +1 +2 +3 +4 +k +perth-sim +0 +1 +2 +3 +4 +i +j = 5 +0.0 +0.5 +1.0 +FIG. 11: Results of the simulations of the test (II) with proba- +bilities pkj for the angles from Table II, for nairobi and perth. +0 +1 +2 +3 +4 +−2 +0 +2 +4 +nairobi-sim +×10−5 +0 +1 +2 +3 +4 +−8 +−6 +−4 +−2 +0 +2 +4 +6 +perth-sim +×10−5 +FIG. 12: Results of the simulations of the test (II) the witness +W = det p, for the angles from Table II, for nairobi and perth +noise models. Note that the two ways of calculation of W +almost coincide (the blue one covers the red one), which is +consistent with our explanation of averaged out first order +difference in Appendix B. +0 +1 +2 +3 +4 +−1.0 +−0.5 +0.0 +0.5 +nairobi-sim +×10−4 +0 +1 +2 +3 +4 +−7.5 +−5.0 +−2.5 +0.0 +2.5 +5.0 +7.5 +perth-sim +×10−4 +FIG. 13: Results of the simulations of the test (II) the witness +W = det p, for the angles from Table II, for nairobi and perth +noise models, for individual jobs +Then the probability matrix elements read +pkj = TrMkNj = (1 + n · m)/2 +(A4) +while p5j = 1. +In this way we can represent the choices used +in our experiment. +In the first choice, +prepara- +tions n′ +1 = (0, 0, −1), n′ +2 = (− +√ +3/2, 1/2, 0), n′ +3 = +(− +√ +3/4, −1/4, +√ +3/2), n′ +4 = ( +√ +3/4, −1/4, +√ +3/2), n′ +5 = +( +√ +3/2, 1/2, 0) and measurements m′ +k = n′ +k−1. In the sec- +ond choice, n′′ +1 = −n′′ +2 = (0, 0, −1), n′′ +3 = (2 +√ +2, 0, 1/3), +n′′ +4,5 = (− +√ +2/3, ∓ +� +2/3, 1/3) and measurements m′′ +1 = +(0, 0, 1), m′′ +2 = (1, 0, 0), m′′ +3,4 = (−1/2, ∓ +√ +3/2, 0). +For the parametric test we have n1 = (0, 0, 1), n2 = +(2 +√ +2, 0, 1/3), n3,4 = (− +√ +2/3, ∓ +� +2/3, 1/3) while ni +5 = +(− sin(2πi/5), − cos(2πi/5), 0). +Appendix B: Bounds on daily calibrations +Suppose that the calibration from job to job can al- +ter the matrix of probabilities. Assuming that each job +n = 1..N satisfies W (n) = 0 for probabilities p(n), we +ask if W for p = � +n p(n)/N can be nonzero. Suppose +δp(n) = p(n) − p(0) is small for some reference matrix p(0) +and |δp(n) +kj | ≤ ϵ for all kj and some small bound ϵ. Then, +in the first order of δp we have still W ≃ 0 from expand- +ing determinant in linear combinations of single columns + +7 +p(n) and the rest of columns kept equal p(0). The nonva- +nishing contribution is of the second order, when replac- +ing either of two columns by δp(n). Their length is ≤ 2ϵ. +The last row contains 0 for the replaced columns and 1 +for the rest. Subtracting 1/2 of that row from the other +rows. The moduli of remaining elements are ≤ 1/2 for +the length of the remaining 3 columns is ≤ +√ +2. From +Hadamard inequality | det A| ≤ � +j |Aj| with |Aj being +the length of the vector (column) Aj of the matrix A, +we have the upper bound |W| ≤ 80 +√ +2ϵ2 as we have 10 +choices of 2 columns out of 5. +Appendix C: Corrections from higher states +The generic Hamiltonian, in the basis states |n⟩, n = +0, 1, 2, ... (ℏ = 1) reads +H = +� +n +ωn|n⟩⟨n| + 2 cos(ωt − θ) ˆV (t) +(C1) +with energy ωn eigenstates levels and the external drive +V at frequency ω and phase shift θ (the second term). In +principle free parameters ω, θ and ˆV (t) can model a com- +pletely arbitrary evolution. We can estimate deviations +by perturbative analysis, setting ω0 = 0, ω1 = ω (reso- +nance), ω2 = 2ω+ω′ (anharmonicity, i.e. ω′ ≪ ω, in IBM +about 300Mhz compared to drive frequency ∼ 5GHz). +The state |2⟩ should give the most significant potential +contribution. We can incorporate rotation and phase into +the definition of states, |n⟩ → |n′⟩ = e−in(θ+ωt)|n⟩ so that +H′ = +� +� +2 cos(ωt − θ)V00 +(1 + e−2i(θ+ωt))V01 (e−i(θ+ωt) + e−3i(θ+ωt))V02 +(1 + e2i(ωt+θ))V10 +2 cos(ωt + θ)V11 +(1 + e−2i(θ+ωt))V12 +(e−i(θ+ωt) + e3i(ωt+θ))V20 +(1 + e2i(ωt+θ))V21 +2 cos(ωt + θ)V22 + ω′ +� +� +(C2) +Extracting the Rotating Wave Approximation (RWA) part from H′ = HRW A + ∆H, +HRW A = +� +� +0 +V01 +0 +V10 +0 +V12 +0 +V21 +ω′ +� +� , +(C3) +the correction reads +∆H = +� +� +2 cos(ωt + θ)V00 +e−2i(θ+ωt)V01 +(e−i(θ+ωt) + e−3i(θ+ωt))V02 +e2i(ωt+θ)V10 +2 cos(ωt + θ)V11 +e−2i(θ+ωt)V12 +(e−i(θ+ωt) + e3i(ωt+θ))V20 +e2i(ωt+θ)V21 +2 cos(ωt + θ)V22 +� +� +(C4) +Evolution due to RWA has the form +U(t) = T exp +� t +−∞ +HRW A(t′)dt′/i, +(C5) +where T means chronological product in Taylor expan- +sion. Then the 1st order correction to U reads +∆U = U(+∞) +� +dtU †(t)∆H(t)U(t)/i +(C6) +where the full rotation is U(+∞). 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DiVincenzo, Ex- +act rotating wave approximation, Annals of Physics 423, +168327 (2020) + diff --git a/49E1T4oBgHgl3EQfmQS7/content/tmp_files/load_file.txt b/49E1T4oBgHgl3EQfmQS7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d409c840297b610b1c7d00a76899ea7073d2dedc --- /dev/null +++ b/49E1T4oBgHgl3EQfmQS7/content/tmp_files/load_file.txt @@ -0,0 +1,512 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf,len=511 +page_content='Precise certification of a qubit space Tomasz Białecki1, Tomasz Rybotycki2,3, Josep Batle4, Jakub Tworzydło1, and Adam Bednorz1, ∗ 1Faculty of Physics, University of Warsaw, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Pasteura 5, PL02-093 Warsaw, Poland 2Systems Research Institute, Polish Academy of Sciences, 6 Newelska Street, PL01-447 Warsaw, Poland 3Center for Theoretical Physics, Polish Academy of Sciences, Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Lotników 32/46, PL02-668 Warsaw, Poland 4CRISP - Centre de Recerca Independent de sa Pobla, 07420 sa Pobla, Balearic Islands, Spain We demonstrate an implementation of the precise test of dimension on the qubit, using the public IBM quantum computer, using the determinant dimension witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The accuracy is below 10−3 comparing to maximal possible value of the witness in higher dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The test involving minimal independent sets of preparation and measurement operations (gates) is applied both for specific configurations and parametric ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The test is be robust against nonidealities such as incoherent leakage and erroneous gate execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Two of the IBM devices failed the test by more than 5 standard deviations, which has no simple explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' INTRODUCTION Physics is an exact science, which is confirmed by pre- cise measurements of fundamental constants and estab- lishing definition of SI units by precise quantum experi- ments [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Precision is also required from every com- puter, also quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Unfortunately, current quantum technologies suffer from inevitable sources of errors, both just from mechanical limitations and inseparable physical environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Of course, there are methods to mitigate and correct the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Such approach relies, however, on assumptions about the controllable space of possible actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The basic building block of a quantum computer is a qubit, a generic two-level system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Since the goal is to manipulate accurately many qubits, it is necessary to as- certain whether or not the qubit space is reliable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' not combined with a larger space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The most promising implementations of qubits keep them detuned from en- vironment and other states, except for small incoherent disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' On the other hand, the potential contri- bution of external states can lead to systematic errors, hard to correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Operations on qubits, gates, realized by microwave pulses, suffer from distortions due to nonlin- earities of waveform generators [7], so a simple deviation of the probability distribution from the theoretical pre- diction is not yet a proof of extra space [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Therefore, to increase the quality of classical and quantum com- putation and communication, these systems need precise certification, robust against imperfections of physical im- plementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The dimension of the quantum space can be checked by a dimension witness [9–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The construction of the witness is based on the two-stage protocol, the initial preparation and subsequent final measurement, which are chosen from independent sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The preparation must be completed before the start of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' A precise ∗Electronic address: Adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='Bednorz@fuw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='pl witness must be based on equality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' a quantity, which is exactly zero up to a certain dimension, and nonzero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Such a good witness test is the linear inde- pendence of the specific dichotomic outcome probability p(M|N) for the preparation N and measurement M, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 1, tested by a suitable determinant [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' It has been been already performed on optical states [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' It be- longs to a family to equality-based tests, like the Sorkin equality [19] in the three-slit experiment [20–22] testing Born’s rule [23], benchmarking our trust in fundamental quantum models and their actual realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' In this paper, we apply the test to several IBM quan- tum device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' While some results agree with the 2−level model, taking a large statistics revealed signature of the failure by more than 5 standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Of course it does not immediately mean a larger space but the prob- lem needs urgent further investigation to determine the cause, which may be also another assumption of the test (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' lack of independence of the operations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' THEORY We apply a test of the qubit space d = 2 with the witness constructed for p(M|N) = trMN, N = N † ≥ 0, trN = 1 and measurement 1 ≥ M = M † ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Taking 5 preparations Nj, j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='.5 and 4 measurements Mk, k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='.4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Then the determinant W = det p, for the 5 × 5 matrix p with entries pkj = p(Mj|Nk) and p5j = 1, must be equal to zero if all Nj and Mk are represented in the same two-level space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' In addition, it remains zero also if all preparations and measurements contain some constant incoherent leakage term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' N ′ j = Nj + Ne and M ′ k = Mk + Me, with Ne and Me independent of j and k and commuting with Nj and Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' In this way, the common leakage to higher states does not affect the test [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' For d = 2 we have W = 0, but d = 3 gives maximally 27 √ 2/64 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='6 in the real space and ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='632 in the complex space [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' For d = 4 the maximum (real and complex) is 212/37 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Even higher dimensions are saturated by the classical maximum 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='03296v1 [quant-ph] 9 Jan 2023 2 The IBM Quantum Experience cloud computing of- fers several devices, collections of qubits, which can be manipulated by a user-defined set of gates (operations) – either single qubit or two-qubit ones, also paramteric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' One can put barriers (controlling the order of operations) or additional resets (nonunitary transition to the ground state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The qubits are physical transmons [24], the arti- ficial quantum states existing due to interplay of super- conductivity (Josephson effect) and capacitance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Due to anharmonicity one can limit the working space to two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The decoherence time (mostly environmental) is long enough to perform a sequence of quantum opera- tions and read out reliable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The ground state |0⟩ can be additionally assured by a reset operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Gates are implemented by time- scheduled microwave pulses prepared by waveform gen- erators and mixers (time 30 − 70ns with sampling at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='222ns), tuned to the drive frequency (energy difference between qubit levels) [25] (about 4−5Ghz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The rotation Z is not a real pulse, but an realized by an instantaneous virtual gate VZ(θ), which adds a rotation between in- and out-of-phase components of the next gates [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The readout is performed another long microwave pulse of frequency (different from the drive) to the resonator and measuring the populated photons [25, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' In the following, we assume the two-level description of the qubits, expecting W = 0 up to statistical er- ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Larger deviation would be an evidence that this description is inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The states and operators will be can be described either in a two-dimensional Hilbert space with basis |0⟩, |1⟩ or in the Bloch sphere with V = (v0 + v · σ)/2, with the 3-component Bloch vec- tor v and standard Pauli matrices σ1 = � 0 1 1 0 � , σ2 = � 0 −i i 0 � , σ3 = � 1 0 0 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' (1) Then the initial state |0⟩⟨0| corresponds to the vector (0, 0, 1) while n0 = 1 and |n| ≤ 1 and 2 − |m| ≥ m0 ≥ |m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' A microwave pulse tuned to the interlevel drive fre- quency corresponds to parametric gates, π/2 rotations, Sγ = Z† γSZγ, Zθ = � e−iγ/2 0 0 eiγ/2 � , S = RX(π/2) = √ X = 1 √ 2 � 1 −i −i 1 � , (2) in the basis |0⟩, |1⟩ while S = � � 1 0 0 0 0 −1 0 1 0 � � , Zγ = � � cos γ − sin γ 0 sin γ cos γ 0 0 0 1 � � , (3) on the Bloch vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' SγV S† γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Physically the experiment is sequence of preparation in the state |0⟩, two gates Sα, Sβ for the preparation, two gates Sφ, Sθ, and the the readout pulse for the measure- ment of the state |0⟩ again, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' There are 5 pairs of angles αj, βj to be chosen independently of the 4 pairs N M 1 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 1: Preparation and measurement scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' the state is prepared as N and measured by M to give an outcome of either 1 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' |0⟩ Sα Sβ Sφ Sθ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 2: The quantum circuit for the dimension test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The initial state |0⟩ and four gates Sγ, split into preparation and measurement stages, are followed by the final dichotomic mea- surement θk, φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Then N = SβSα|0⟩⟨0| and M = S† φS† θ|0⟩⟨0|SθSφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The actual pulse waveform of a sample sequence of gates is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' EXPERIMENT In a perfect theory, we can predict a probability for every choice of α, β, θ, φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The experimental results can differ for a variety of reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Firstly, the test is random and we have to estimate the error due to finite statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' For T times the experiment is repeated, the variance of W can be estimated as T⟨W 2⟩ ≃ � kj pkj(1 − pkj)(Adj p)2 jk, (4) where Adj is the adjoint matrix (matrix of minors of p, with crossed out a given row and column, and then trans- 0 30 60 90 119 149 Time (ns) VZ(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='28) VZ( 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='37) VZ(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='33) VZ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='57) D0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='26 GHz FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 3: The actual waveform of the pulse on IBM quan- tum computer (nairobi), with four subsequent gates Sγ, with γ = α, β, φ, θ, consecutively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The discretization unit time is dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='222ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Driving (level gap) frequency is denoted by D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The light/dark shading corresponds to in-phase/out-of-phase amplitude component, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The element VZ(ξ) is a zero-duration virtual gate Zξ for subsequent gates SγSδ with ξ = γ − δ [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 3 posed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Note that the identity p−1 det p = Adjp makes no sense here as W = det p = 0 in the limit T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Secondly, the implementation of gates may be not faith- ful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Our test is capable to take them into account as long as the leakage to external states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' |2⟩) is inco- herent and does not depend on the parameters α, β, θ, φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Lastly, we have to assume that the pulse does not depend on the previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' In other words, we can only test the combination of assumptions, dimension of the space and independence of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' We have calculated W in two ways, (i) determining p for each job and then find- ing W (see the values for each job in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 10) and finally averaging W, (ii) averaging first p from all jobs and then finding W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' There is no a priori best selection of preparations and measurements but they should not lie on a single Bloch circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' We decided to make two kinds of tests: (I) two special configurations corresponding to either the same Bloch vectors for preparation and measurements or max- imal ⟨W 2⟩ for a given R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' (II) a family of configurations with one preparation vector at one of the 5 directions on the Bloch circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' In both cases the corresponding Bloch vectors are derived explicitly in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The sets of angles in the case (I) are given in the Table I, and the corresponding Bloch vectors are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' We have run the test on lima and lagos, qubit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The probability matrix, compared to the ideal expecta- tion is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The deviation from zero and the statistical error is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The number of T = #jobs·#shots·#repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Technically, one sends a list of jobs to execute, each job contains up to 300 cir- cuits, to be distributed between experiments repeated the same number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Each circuit is run the number of shots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The readout counts for each circuit is the value returned after the job execution is accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The sets of angles in the case (II) are prepared dif- ferently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Four preparations and measurements are fixed while the last preparation is parameter-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The fixed angles are specified in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The last prepara- tion angles are α5 = 2πi/5 = β5 − π/2 for i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='.4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The corresponding Bloch vectors are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' We have run the test on nairobi and perth, qubit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The probability matrix, compared to the ideal expectation is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The deviation from zero and the sta- tistical error is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The deviation from the expected 0 is more than 5 standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The data and scripts are available at the public repository [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Nonidealities There are several factors that can affect the correct- ness of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' (A) The daily calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The drive frequency and the gate waveforms are corrected so different jobs can rely on different realizations of gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' There first order effect of calibrations is cancelled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Nevertheless, we made more detailed estimates on sec- ond order effects in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Only large, unexpected j 1 2’ 3’ 4’ 5’ 2” 3” 4” 5” α 0 2π/3 2π/3 4π/3 4π/3 0 η − π η + 5π/3 η + π/3 β 0 π/6 −π/6 π/6 −π/6 π 0 2π/3 −2π/3 k 1’ 2’ 3’ 4’ 1” 2” 3” 4” θ 5π/3 5π/3 π/3 π/3 π π/2 7π/6 −π/6 φ 7π/6 5π/6 7π/6 5π/6 0 π 5π/3 π/3 TABLE I: The angles for the special two special cases, ’ and ”, with η = acos(1/3) and 1=1’=1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 4: The Bloch vectors for the preparations (red) and measurements (blue) corresponding to the angles from Table I, top ’ and bottom ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' For the case ’, the four measurement direction are identical to four preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' j 1 2 3 4 α 0 η − π η + 5π/3 η + π/3 β 0 0 2π/3 −2π/3 k 1 2 3 4 θ π π/2 7π/6 −π/6 φ 0 π 5π/3 π/3 TABLE II: The angles for the parametric case (II) for prepa- rations and measurements 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='.4 14 1 2 3 4 ideal k 1 2 3 4 lima k 1 2 3 4 5 j′ 1 2 3 4 lagos k 1 2 3 4 5 j′′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 5: Results of the test (I) with probabilities pkj for the an- gles from Table I, for lima and lagos, compared to the ideal ex- pectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Lagos: 60 jobs, 32000 shots, 15 repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Lima: 521’/194” jobs, 20000 shots, 5 repetitions failures could be a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' (B) Amplitude-dependent leakage and distortion of the waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The leakage to higher states, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' |2⟩ is small, of the order 10−4 and incoherent [8, 14], see details in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' It is pos- sible that distortion of amplitude to the waveform de- pends on the rotation angle (phase) but we expect this effect to be very small, 10−3, based on the deviations ob- served in our previous work, and so the net effect is 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' (C) Memory of the waveform between successive gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Highly unlikely, a residual voltage amplitude can persist up to the next gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' In principle it can be mitigated by delay-separated gates if the effect fades out with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' (D) Other qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' They are usually detuned but some crosstalk may remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' As in the case of leakage, we ex- pect the crosstalk to be incoherent and so irrelevant for the witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' As a sanity check we have run simulations, using the noise models from nairobi and perth, and no significant deviation have been found, see Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' lima′ lima′′ lagos′ lagos′′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 ×10 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 6: Results of the test (I) the witness W = det p, for the angles from Table I, for lima and lagos, with the error given by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Red – W for p from each job and then averaged, blue – p averaged from all jobs to give W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 7: The Bloch vectors for the parametric case (II) with fixed preparations (red) and measurements (blue) correspond- ing to the angles from Table I, and a parametric preparation (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' DISCUSSION A test of linear independence of quantum operations reveals subtle deviations, invisible in more crude tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Further tests are necessary to identify the origin of the deviations, to exclude e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' exotic many world/copies the- ories [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' We suggest: (i) an extreme statistics col- lected in a relatively short time to avoid corrections due to calibrations, (ii) a time separation between gates to ex- clude potential overlap of the effects, (iii) a scan through a large set of Bloch vectors to maximize the potential deviation, (iv) run the test on a single-qubit devices to avoid cross-talks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' It is also possible to develop more so- phisticated tests, with different assumptions, or involv- ing different qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' In any case, a precise diagnostics of qubits must become a standard in quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 5 1 2 3 4 j 1 2 3 4 k theory 0 1 2 3 4 i j = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 1 2 3 4 j 1 2 3 4 k nairobi 0 1 2 3 4 i j = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 1 2 3 4 j 1 2 3 4 k perth 0 1 2 3 4 i j = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 8: Results of the test (II) with probabilities pkj for the angles from Table II, for nairobi and perth, compared to the theory expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Nairobi/perth: 115/93 jobs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' both 100000 shots and 8 repetitions Appendix A: Bloch sphere representations Using vectors n to represent the state N = |n⟩⟨n| = (ˆ1 + n · σ)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' we have SαNS† α = Nα and S† θMSθ = Mθ with nα = � � cos2 α − cos α sin α − sin α − cos α sin α sin2 α − cos α sin α cos α 0 � � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' mθ = � � cos2 θ − cos θ sin θ sin θ − cos θ sin θ sin2 θ cos θ − sin θ − cos θ 0 � � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' (A1) For n = m = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 1) and m0 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' we have Nαβ = SβSαNS† αS† β with n′ αβ = (sin(β − α) cos β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' sin(α − β) sin β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' − cos(β − α)) (A2) while Mθφ = S† φS† θMSθSφ wirh M θφ = (sin(θ − φ) cos φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' sin(φ − θ) sin φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' − cos(θ − φ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' (A3) 0 1 2 3 4 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 nairobi ×10−4 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 perth ×10−4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 9: Results of the test (II) the witness W = det p, for the angles from Table II, for nairobi and perth, with the error given by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Red – W for p from each job and then averaged, blue – p averaged from all jobs to give W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 0 1 2 3 4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 nairobi ×10−4 0 1 2 3 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='00 perth ×10−3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 10: Results of the test (II) the witness W = det p, for the angles from Table II, for nairobi and perth, for individual jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Two values for nairobi are beyond the picture boundaries, (3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0023) and (4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='003) 6 1 2 3 4 j 1 2 3 4 k nairobi-sim 0 1 2 3 4 i j = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 1 2 3 4 j 1 2 3 4 k perth-sim 0 1 2 3 4 i j = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 11: Results of the simulations of the test (II) with proba- bilities pkj for the angles from Table II, for nairobi and perth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 0 1 2 3 4 −2 0 2 4 nairobi-sim ×10−5 0 1 2 3 4 −8 −6 −4 −2 0 2 4 6 perth-sim ×10−5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 12: Results of the simulations of the test (II) the witness W = det p, for the angles from Table II, for nairobi and perth noise models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Note that the two ways of calculation of W almost coincide (the blue one covers the red one), which is consistent with our explanation of averaged out first order difference in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 0 1 2 3 4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 nairobi-sim ×10−4 0 1 2 3 4 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='5 perth-sim ×10−4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 13: Results of the simulations of the test (II) the witness W = det p, for the angles from Table II, for nairobi and perth noise models, for individual jobs Then the probability matrix elements read pkj = TrMkNj = (1 + n · m)/2 (A4) while p5j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' In this way we can represent the choices used in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' In the first choice, prepara- tions n′ 1 = (0, 0, −1), n′ 2 = (− √ 3/2, 1/2, 0), n′ 3 = (− √ 3/4, −1/4, √ 3/2), n′ 4 = ( √ 3/4, −1/4, √ 3/2), n′ 5 = ( √ 3/2, 1/2, 0) and measurements m′ k = n′ k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' In the sec- ond choice, n′′ 1 = −n′′ 2 = (0, 0, −1), n′′ 3 = (2 √ 2, 0, 1/3), n′′ 4,5 = (− √ 2/3, ∓ � 2/3, 1/3) and measurements m′′ 1 = (0, 0, 1), m′′ 2 = (1, 0, 0), m′′ 3,4 = (−1/2, ∓ √ 3/2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' For the parametric test we have n1 = (0, 0, 1), n2 = (2 √ 2, 0, 1/3), n3,4 = (− √ 2/3, ∓ � 2/3, 1/3) while ni 5 = (− sin(2πi/5), − cos(2πi/5), 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Appendix B: Bounds on daily calibrations Suppose that the calibration from job to job can al- ter the matrix of probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Assuming that each job n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='.N satisfies W (n) = 0 for probabilities p(n), we ask if W for p = � n p(n)/N can be nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Suppose δp(n) = p(n) − p(0) is small for some reference matrix p(0) and |δp(n) kj | ≤ ϵ for all kj and some small bound ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Then, in the first order of δp we have still W ≃ 0 from expand- ing determinant in linear combinations of single columns 7 p(n) and the rest of columns kept equal p(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The nonva- nishing contribution is of the second order, when replac- ing either of two columns by δp(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Their length is ≤ 2ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The last row contains 0 for the replaced columns and 1 for the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Subtracting 1/2 of that row from the other rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The moduli of remaining elements are ≤ 1/2 for the length of the remaining 3 columns is ≤ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' From Hadamard inequality | det A| ≤ � j |Aj| with |Aj being the length of the vector (column) Aj of the matrix A, we have the upper bound |W| ≤ 80 √ 2ϵ2 as we have 10 choices of 2 columns out of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Appendix C: Corrections from higher states The generic Hamiltonian, in the basis states |n⟩, n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' (ℏ = 1) reads H = � n ωn|n⟩⟨n| + 2 cos(ωt − θ) ˆV (t) (C1) with energy ωn eigenstates levels and the external drive V at frequency ω and phase shift θ (the second term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' In principle free parameters ω, θ and ˆV (t) can model a com- pletely arbitrary evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' We can estimate deviations by perturbative analysis, setting ω0 = 0, ω1 = ω (reso- nance), ω2 = 2ω+ω′ (anharmonicity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' ω′ ≪ ω, in IBM about 300Mhz compared to drive frequency ∼ 5GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The state |2⟩ should give the most significant potential contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' We can incorporate rotation and phase into the definition of states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' |n⟩ → |n′⟩ = e−in(θ+ωt)|n⟩ so that H′ = � � 2 cos(ωt − θ)V00 (1 + e−2i(θ+ωt))V01 (e−i(θ+ωt) + e−3i(θ+ωt))V02 (1 + e2i(ωt+θ))V10 2 cos(ωt + θ)V11 (1 + e−2i(θ+ωt))V12 (e−i(θ+ωt) + e3i(ωt+θ))V20 (1 + e2i(ωt+θ))V21 2 cos(ωt + θ)V22 + ω′ � � (C2) Extracting the Rotating Wave Approximation (RWA) part from H′ = HRW A + ∆H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' HRW A = � � 0 V01 0 V10 0 V12 0 V21 ω′ � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' (C3) the correction reads ∆H = � � 2 cos(ωt + θ)V00 e−2i(θ+ωt)V01 (e−i(θ+ωt) + e−3i(θ+ωt))V02 e2i(ωt+θ)V10 2 cos(ωt + θ)V11 e−2i(θ+ωt)V12 (e−i(θ+ωt) + e3i(ωt+θ))V20 e2i(ωt+θ)V21 2 cos(ωt + θ)V22 � � (C4) Evolution due to RWA has the form U(t) = T exp � t −∞ HRW A(t′)dt′/i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' (C5) where T means chronological product in Taylor expan- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Then the 1st order correction to U reads ∆U = U(+∞) � dtU †(t)∆H(t)U(t)/i (C6) where the full rotation is U(+∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' All θ-dependent terms in ∆H, contain also eiωt, which exponentially damps slow-varying expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The 2nd order correction reads ∆2U = −U(+∞)× � dtU †(t)∆H(t)U(t) � t dt′U †(t′)∆H(t′)U(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' (C7) Most of components get damped exponentially, too, ex- cept when ∆H(t) contains eikωt and ∆H(t′) contains e−ikωt, k = 1, 2, 3, so kθ cancels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' The nonnegligible part of ∆2U is therefore independent of θ giving slowly Bloch-Siegert shift [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Stroboscopic corrections to RWA [32] can be neglected due to a very short sampling time, dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content='222ns, Appendix D: Simulations As a cross-check of our test we have run the identical programs on IBM simulator of a quantum computer with the noise model taken from the real devices perth and nairobi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' However, in contrast to real devices, the results are in agreement with the theory as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' 11, 12, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Eötvös, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} +page_content=' Pekár, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfmQS7/content/2301.03296v1.pdf'} 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b/5dE3T4oBgHgl3EQfQgnL/content/tmp_files/2301.04414v1.pdf.txt @@ -0,0 +1,2473 @@ +1 + +How Does Traffic Environment Quantitatively Affect +the Autonomous Driving Prediction? + +Wenbo Shao, Yanchao Xu, Jun Li, Chen Lv, Senior Member, IEEE, Weida Wang and Hong Wang☒, Senior Member, +IEEE +Abstract—An accurate trajectory prediction is crucial for safe +and efficient autonomous driving in complex traffic environments. +In recent years, artificial intelligence has shown strong capabilities +in improving prediction accuracy. However, its characteristics of +inexplicability and uncertainty make it challenging to determine +the traffic environmental effect on prediction explicitly, posing +significant challenges to safety-critical decision-making. To +address these challenges, this study proposes a trajectory +prediction framework with the epistemic uncertainty estimation +ability that outputs high uncertainty when confronting +unforeseeable or unknown scenarios. The proposed framework is +used to analyze the environmental effect on the prediction +algorithm performance. In the analysis, the traffic environment is +considered in terms of scenario features and shifts, respectively, +where features are divided into kinematic features of a target +agent, features of its surrounding traffic participants, and other +features. In addition, feature correlation and importance analyses +are performed to study the above features’ influence on the +prediction error and epistemic uncertainty. Further, a cross- +dataset case study is conducted using multiple intersection +datasets to investigate the impact of unavoidable distributional +shifts in the real world on trajectory prediction. The results +indicate that the deep ensemble-based method has advantages in +improving prediction robustness and estimating epistemic +uncertainty. The consistent conclusions are obtained by the +feature correlation and importance analyses, including the +conclusion that kinematic features of the target agent have +relatively strong effects on the prediction error and epistemic +uncertainty. Furthermore, the prediction failure caused by +distributional shifts and the potential of the deep ensemble-based +method are analyzed. + +Index Terms—Artificial intelligence, autonomous driving, +distributional shift, epistemic uncertainty, traffic environment, +trajectory prediction. +I. INTRODUCTION +RAJECTORY prediction is an indispensable part of the +autonomous driving pipeline [1]. To drive safely and +efficiently +in +complex +traffic +environments, +autonomous vehicles (AVs) are required to have the ability to +predict the future motion of surrounding traffic participants + +This work has been submitted to the IEEE for possible publication. Copyri +ght may be transferred without notice. This work was supported in part by the +National Science Foundation of China Project: 52072215and U1964203, and t +he National Key R&D Program of China:2020YFB1600303. (Corresponding +authors: Hong Wang) +Wenbo Shao, Jun Li and Hong Wang are with Tsinghua Intelligent Vehicle +Design and Safety Research Institute, School of Vehicle and Mobility, Tsingh +ua University, Beijing 100084, China. (e-mail: swb19@mails.tsinghua.edu.cn; + lijun1958@tsinghua.edu.cn; hong_wang@tsinghua.edu.cn). +(TPs), such as vehicles and pedestrians, accurately and reliably. +In recent years, with the accumulation of large-scale driving +data and rapid development of algorithms, artificial intelligence +(AI) has been widely applied to autonomous driving trajectory +prediction [2, 3], and promising results have been achieved. +However, trajectory prediction has still been challenging, +particularly in urban driving scenarios, where an agent's +movement is influenced by a combination of its historical state +and its complex interactions with the surrounding environment. +Many recent studies [4-6] have considered multiple factors +simultaneously to improve trajectory prediction algorithms, but +there has still been certain performance degradation of a +prediction model in complex traffic environments. +With the improvement in prediction accuracy, the +complexity of AI-based models has also increased gradually. +Highly elaborated models pose a great challenge to explaining +the operation and failure mechanisms of prediction algorithms, +which in turn reduces the credibility of a prediction model. In +addition, AI has its inherent uncertainty and faces many +problems, such as insufficient training data, imperfect model +architecture, and limited training process, which may lead to +functional insufficiencies of the model under specific +environmental conditions, potentially causing severe traffic +accidents [7]. The existing research mainly focuses on +improving the dataset-level accuracy of prediction algorithms +[8, 9], and little attention has been paid to the changes in +prediction +performance +under +different +environmental +conditions. However, this is not conducive to addressing the +practical challenges that a prediction algorithm confronts. +For a target agent (TA) moving in a specific scenario, a +trajectory prediction model predicts its future trajectory by +modeling time series, interaction, and other relationships based +on its historical state, surrounding TPs' features, and other +environmental features. Correspondingly, various traffic +environmental factors may have different effects on trajectory +prediction, but fewer studies have quantitatively investigated +these effects. Further, data-driven methods strongly depend on +Yanchao Xu and Weida Wang is with the School of Mechanical Engineerin +g, Beijing Institute of Technology, Beijing 100081, China. (e-mail: 31202004 +10@bit.edu.cn, wangwd0430@163.com) +Chen Lv is with the School of Mechanical and Aerospace Engineering, +Nanyang +Technological +University, +Singapore +639798 +(e-mail: +lyuchen@ntu.edu.sg). +T + +1 +Traffic environmental features and distributional shifts +Trajectory prediction with epistemic uncertainty estimation +Graph +Representation +Deep Ensemble-based prediction method +Graph +Convolution +Model +RNN-based +Trajectory +Prediction +Model +Graph Feature +How dose traffic environment affect the +autonomous driving prediction? +Distributional shifts +Surrounding traffic +participants +The target agent +Different environmental features +Scenario features analysis & research across intersection datasets +Prediction error & epistemic +uncertainty estimation +quantitative analysis +Answer +As independent +variables +qualitative analysis + +Fig. 1. Illustration of the traffic environment effect on the trajectory prediction algorithm performance. The traffic environmental +data include various TAs’ states, their surrounding TPs’ states, and other contextual information, which may affect the prediction +differently. In addition, variations in time and place can lead to distributional shifts, which may further degrade the prediction +performance. This study focuses on extracting these factors and analyzing their influence on prediction performance. + +training data, and a model trained on one dataset may not +perform well on other datasets. In real-world applications, the +operating environment of AVs may change significantly with +different factors, such as time, geography, country, and weather +conditions. This may cause certain distributional shifts, posing +additional challenges to trajectory prediction. Therefore, it is +increasingly important to study how distributional shifts [10] in +a real environment affect trajectory prediction. As shown in Fig. +1, this work focuses on the effects of both specific scenario +features and scenario shifts on the prediction algorithm. +As for the prediction algorithm performance, previous +studies have generally focused on prediction error. In recent +years, there has been an increasing interest in extracting the +uncertainty of AI-based models [11, 12], thus empowering the +models with a self-awareness ability. Epistemic uncertainty [13] +is a recurring suggestion that helps to represent the model's +confidence in its current predictions; namely, these models tend +to have greater epistemic uncertainty when they encounter +challenging environments. Therefore, the epistemic uncertainty +of a prediction model is extracted and considered a type of +performance metric in this work. As shown in Fig. 1, based on +both the prediction error and epistemic uncertainty, the effect +of the traffic environment on the prediction algorithms’ +performances can be analyzed. +The main contributions of this work can be summarized as +follows: + +A trajectory prediction framework that integrates +epistemic uncertainty estimation is proposed. The +proposed framework performs the TA’s future state +prediction and estimates the epistemic uncertainty +simultaneously; + +The potential of the proposed deep ensemble-based +trajectory prediction framework for improving the +prediction +algorithm +robustness +and +estimating +epistemic uncertainty is demonstrated; + +For the trajectory prediction task, the key features of a +traffic environment are extracted, and methods for +feature correlation analysis and feature importance +analysis are proposed to obtain the relationship between +the traffic environment and the trajectory prediction +algorithm performance; + +The distributional shifts between different intersection +datasets and the resulting trajectory prediction +degradation are investigated. The features of multiple +datasets and their prediction difficulty levels are +analyzed, and it is demonstrated that the deep ensemble +is helpful in improving the trajectory prediction +robustness against cross-dataset evaluation. +The remainder of this paper is organized as follows. Section +II presents the existing work related to this paper. Section III +introduces the proposed method. Section IV describes the +datasets and evaluation metrics used in this work, as well as +implementation details. Section V analyzes and discusses the +experimental results. Section VI concludes the paper. +II. RELATED WORK +A. Trajectory Prediction +There have been numerous studies on improving the +trajectory prediction algorithms, and according to the modeling +principles, they can be mainly divided into physics-based + +2 +methods, maneuver-based methods, and interaction-aware +methods [14]. Physics-based methods [15] consider only the +historical motion state of an object while ignoring the influence +of surrounding TPs. Therefore, they are mainly suitable for +short-term trajectory predictions. Maneuver-based methods [16] +learn prototype trajectories from the observed agent behaviors +to predict future motion, but they lack consideration of +interactions between TPs. Interaction-aware methods [17] have +shown better performance compared to the other two types of +methods through learning the interaction between a TA and +surrounding TPs. +In recent works, many methods have been used to model +interactions between agents, providing valuable information for +trajectory prediction improvement [3, 9]. For instance, social +pooling (S-pooling) [8] pools hidden states of a TA’s neighbors +within a certain spatial distance to model interactions with the +surrounding environment. Convolutional social pooling [18] +combines the convolutional and max-pooling layers to model +interactions +between +agents +in +the +occupancy +grid. +Subsequently, the grid representation is further modified to +consider only eight neighbors that have the most critical impact +on the TA [19]. In addition, recent research has focused on the +rasterized representation of scenes; the historical state of a TA +and scene context were co-encoded in a raster map [20-22], and +various information was distinguished by different channels +and colors. In addition, convolutional neural networks (CNNs) +were used to extract desired features from raster graphs. Graph +models have attracted great interest recently due to their good +performance in modeling inter-agent interactions. In graph +models, a node represents an agent, and an edge represents an +interaction between two agents. Diehl et al. [23] modeled +interactions between vehicles as a homogeneous directed graph +to achieve high computational efficiency and large model +capacity. They evaluated graph convolutional networks and +graph attention networks and introduced several adaptations for +specific scenarios. Mo et al. [2] employed a heterogeneous +edge-enhanced graph attention network to handle the +heterogeneity of TAs and TPs. The GRIP [24] represents the +input as a specific grid and uses an undirected graph to model +interactions between agents within a certain range, where fixed +graphs are considered in the graph convolution submodule. The +GRIP++ [5] improves the above-mentioned method by +adopting trainable graphs, which overcomes the shortcoming +that fixed graphs based on manually designed rules cannot +model interaction properly. In addition to the interaction +modeling, another important requirement of trajectory +prediction relates to time series processing. Recently, recurrent +neural networks (RNNs), including the long short-term memory +(LSTM) and gated recurrent unit (GRU) models, have been +widely used in modeling sequential problems, and significant +results have been achieved. Accordingly, these models have +been used as sub-modules in many trajectory prediction +algorithms [5, 6]. +Neural networks have been demonstrated to be highly +efficient in trajectory prediction for different classes of TPs. +Research on pedestrian intent modeling and motion prediction +has been conducted for decades. The Social-LSTM [8] is a +typical success case in early research in this field, which +combines S-pooling and LSTM to predict the future trajectory +of pedestrians in crowded scenes. The Social-GAN [25] uses +generative +adversarial +networks +(GANs), +sequence-to- +sequence models, and pooling mechanisms and employs the +corresponding generators and recursive discriminators to +predict pedestrians’ socially feasible future. However, the GAN +model training is difficult and may not converge and can lead +to mode collapsing and dropping. Therefore, the Social-Ways +uses the Info-GAN, which does not apply the mean square error +loss (L2 loss) to force the generated samples to be close to real +data but adds another item to consider mutual information, thus +alleviating the above-mentioned problems. Since vehicles have +higher running speeds and need to obey more road constraints +than pedestrians, predicting their future movements is a +prerequisite for realizing safe and efficient autonomous driving. +A number of studies have designed specialized networks for +vehicle trajectory prediction [26]. For instance, vehicle +trajectory prediction in highway scenarios, which are relatively +simple and where the motion pattern of a vehicle is relatively +fixed, has received early attention [18, 24, 27]. With the +collection of large-scale datasets [28, 29] and the development +of autonomous driving in urban scenes, much research has been +focused on motion prediction in complex urban environments +[4, 30-32]. TrafficPredict [33] adopted a four-dimensional +graph to model the interaction in the instance and category +layers, thus realizing the heterogeneous traffic-agent trajectory +prediction. The GRIP++ achieved joint trajectory prediction of +all observed objects while considering multiple classes of TPs, +thus greatly improving real-time prediction performance. +However, the above work focuses on the improvement in the +dataset-level accuracy while ignoring the research on the +sensitivity of the prediction algorithm to environmental factors, +which is the focus of this work. +B. Epistemic Uncertainty Estimation +Due to the rapid development of neural networks and their +application to trajectory prediction tasks, it has become +increasingly important to estimate the network confidence in its +prediction accuracy. However, the original neural network +cannot provide an estimation of its epistemic uncertainty. To +address this shortcoming, some studies have considered and +quantified the epistemic uncertainty of neural networks [11, 34, +35], which represents an indicator that can express how +confident the network is in its current prediction result. The +main epistemic uncertainty estimation methods include the +Bayesian neural network (BNN), single-pass uncertainty +estimation, and ensemble-based methods. The BNN quantifies +the epistemic uncertainty of a neural network by introducing +uncertainty into its parameters. The key challenge of these +methods is to solve the posterior distribution of network +parameters. In the early research, variational inference (VI) [36], +which uses a prespecified family of distributions [37, 38], was +widely adopted as a method with a strong theoretical basis. +However, with the rapid growth in the neural network structure +complexity, VI has faced many challenges in terms of solving +difficulty and computational complexity. To address these +limitations, the Monte Carlo (MC) dropout [39, 40] was +proposed to approximate the results obtained by sampling, +assuming that the network weights conformed to a Bernoulli +distribution. It has been theoretically demonstrated that the MC +dropout has the ability to approximate epistemic uncertainty. In + +3 +single-pass uncertainty estimation, uncertainty is obtained +through one forward propagation, which has obvious +advantages in terms of computational complexity. The deep +evidence theory is a representative method and has been widely +used in classification [41] and regression [42] tasks. However, +these methods require that the original network output has a +specific form, which limits their scalability. In addition, these +methods do not consider the uncertainty of network weights. In +view of that, some studies [43] positioned the uncertainty they +extracted as distributional uncertainty, different from epistemic +uncertainty. In deep ensemble-based methods, the training +process is adjusted to obtain multiple different models, and +epistemic uncertainty is estimated by synthesizing the +prediction results of the models. Deep ensemble [44] is a simple +and scalable uncertainty estimation method, which has attracted +extensive attention due to its excellent performance in +estimating epistemic uncertainty [45]. Currently, this method +has become a mainstream paradigm. Subsequently, to reduce +the storage and computational costs of the practical application +of deep ensemble, many improved methods have been proposed +[46, 47]. For instance, the Batch-Ensemble [46] reduces +training and testing costs by defining each weight matrix as the +Hadamard product of the shared weights of all ensemble +members and the rank-one matrix of each member, but the +uncertainty estimation performance is slightly decreased. +However, previous studies on epistemic uncertainty have +usually involved tasks such as semantic segmentation and +object detection but have lacked detailed research in the field of +trajectory prediction. This work proposes a trajectory prediction +method with epistemic uncertainty estimation, where deep +ensemble and MC dropout are used separately to estimate +epistemic uncertainty and compared on the real intersection +dataset. +C. Relationship between Prediction Performance and Traffic +Environment +Previous studies have mainly focused on enhancing the +dataset-level accuracy of trajectory prediction. However, the +actual trajectory prediction performance can be strongly +dependent on a traffic environment. Therefore, it is of great +significance to determine the relationship between the +environment +and +prediction +model +to +improve +the +interpretability of trajectory prediction algorithms and +determine their limitations. This is essential for safety-critical +autonomous driving applications. Several works focused on +modeling and complexity calculation of a traffic environment +using different methods, such as five- and six-layer scene +models [48, 49], where layer elements can have a strong +correlation with the prediction algorithm. Wang et al. [50] +proposed a method to quantify scenario complexity in traffic +but did not explore its relationship with the autonomous driving +algorithm performance. The Shapley value is a feature +attribution method that helps to measure the contribution of +input variables to model performance. Makansi et al. [51] +proposed a variant of Shapley value and analyzed the problems +that some of the existing trajectory prediction models consider +only the past trajectory of a TA and are difficult to reason about +interactions. In addition, recent studies have gradually paid +attention to the cross-dataset performance of AI algorithms in +object detection and prediction applications [52, 53]. Gesnouin +et al. [54] evaluated the impact of differences in pedestrian +poses and detection box heights in different datasets on +pedestrian crossing prediction. Gilles et al. [10] compared the +accuracy of vehicle trajectory prediction algorithms on several +datasets containing mixed scenarios. However, there has still +been a lack of comprehensive analysis of traffic environmental +factors and their changes and quantitative research on their +impact on prediction algorithms. +n this work, the research scenario is the intersection, which +is a typical and challenging urban scenario. Distributional shifts +between different intersection datasets and their effect on +trajectory prediction performance, considering both error and +epistemic uncertainty, are analyzed. +III. PROPOSED METHOD +A. +Trajectory +Prediction +with +Epistemic +Uncertainty +Estimation +1) Trajectory Prediction +Trajectory prediction is a task that estimates a TA’s future +position based on historical data on its state and context in a +scenario. Particularly, at time +0 +t  +, the historical input state S +of a TA (over previous + time steps) is represented as follows: + + + + + +  +1 +2 +0 +, +, +, +, +h +h +t +t +s +s +s + + + + + +  + +S + +(1) +where +( )t +s +is the state of a TA at t , and it is defined as +( ) +( ) +( ) +, +t +t +t +s +x +y + + +  + . +In addition, interactions between a TA and its surrounding +environment are modeled based on scene context information +C , which includes information on the states of TPs’ around the +TA and environmental conditions. +A trajectory prediction model f is trained on dataset +. +Based on the input + + +X = S,C , the trained prediction model +outputs an estimate ˆY of the real future trajectory Y of the TA +as follows: + +  + + + +ˆ +ˆ = +, +, +, +f +f +f + + + +Y +X +X +X + +(2) +where +  +  +  +1 +2 +ˆ +ˆ +ˆ +ˆ +[ +, +, +, +] +ft +s +s +s + +Y +, +ft is the predicted horizon, and ˆ +represents the trained model parameters. +In this work, the GRIP++, which is an enhanced graph-based +interaction-aware trajectory prediction method, is used as a base +model. It uses both fixed and dynamic graphs to describe the +relationship between different TPs, considering the effect of +inter-agent interactions on a TA’s motion. Furthermore, this +method employs a GRU-based encoder-decoder architecture as +a sub-module and allows joint trajectory predictions for +multiple agents, achieving good performances in terms of +prediction speed and accuracy. +2) Epistemic Uncertainty Estimation +In the previous section, a neural network-based trajectory +prediction model is presented, but the original GRIP++ can +output only deterministic prediction results. However, real- +world traffic scenarios are complex and variable, and it is +difficult to construct a training set that will effectively cover all +scenarios. In addition, deep learning-based models are +inherently uncertain and difficult to interpret, so they may not +ht + +4 +ht +Graph Convolutional Model +64 +ht +n +Trajectory Prediction Module +Predicted Trajectories +prediction +error +Random initialization, random shuffling... +epistemic +uncertainty +ADE +FDE +APE +FPE + +Fig. 2. The trajectory prediction framework with epistemic +uncertainty estimation (deep ensemble-based method). + + +be reliable enough when confronted with unknown scenarios +(e.g., scenarios unseen during training or scenarios with only +limited available information). These problems can result in +unacceptable degradation in autonomous driving performance. +In this regard, the BNN models and learns the posterior +distribution of network weights + + +ˆ +| +P + + +, which can be +used to estimate the epistemic uncertainty as follows: + + + + +  + +ˆ +, +| +, +P +| +, +f +f +X  + + + +Y +X +Y + +(3) +where the main issue is how to learn the posterior distribution +of parameters effectively. +The Bayesian approximate inference is a typical solution, +which learns an approximate distribution   +q  of  + +P +| + +. The +MC dropout has been shown to be an effective sample-based +method for approximate inference, where the network weights +are assumed to follow the Bernoulli distribution. After adding +appropriate regularization during training and turning on +dropout during testing, epistemic uncertainty can be estimated +by sampling multiple times. +In recent years, deep ensemble, as a simple, parallelizable, +and scalable method, has shown excellent uncertainty +estimation ability. In this work, a deep ensemble-based +uncertainty estimation framework for the trajectory prediction +model is proposed. Specifically, random initialization of neural +network parameters and random shuffling of a dataset are +performed because they have been proven to have enough good +performance in practice. After training, +K models of +isomorphism and different parameters are obtained. Further, by +integrating the results of + models, the final trajectory +prediction output is obtained by, + + + +1 +1 +ˆ +ˆ = +, +, +K +k +k +K +f + + + +Y +Y | X + +(4) +where ˆ +k denotes the post-training parameters of the kth model +among the ensemble models; similarly, in the MC dropout- +based method, ˆ +k indicates the model parameters for the kth +dropout during testing. +The predictive entropy is employed to quantify the epistemic +uncertainty of the proposed prediction model, where entropy +increases with uncertainty. The proposed model outputs K +continuous trajectories + + +(1) +(2) +] +ˆ +ˆ +ˆ +ˆ +[ +, +, +, +ft +k +k +k +k +s +s +s + +Y + in a prediction task, +each of which contains the predicted position at multiple future +moments. To realize a prediction-task-wise uncertainty +estimation, the predictive entropy at multiple moments is +integrated to obtain the average predictive entropy (APE) as +follows: + + + + + +( ) +( ) +( ) +1 +1 +1 +1 +ˆ +ˆ +ˆ +ˆ +A +. +PE +ln +d +f +f +t +t +t +t +t +t +i +i +f +f +s +p s +p s +s +t +t + + + + + + + + + + +  + +(5) +Assuming that +( ) +( ) +( ) +ˆ +ˆ +ˆ +, +t +t +t +s +x +y + + +  + obeys the two-dimensional +Gaussian distribution, +( ) +ˆ t +x + and +( ) +ˆ t +y + are independent of each +other, and the APE can be expressed as follows: + +  + + + + +=1 +2 +( ) +2 +( ) +1 +1 +1 +ˆ +APE += +(ln2 +1) +ln +2 +1 +1 +ˆ +ˆ +(ln2 +1) +ln +. +2 +f +f +l +t +i +f +t +t +t +i +f +t +x +x +t + + + + + + + + + + + + + + +(6) +Similarly, the final predictive entropy (FPE) is defined as, + + + + + + + + + + + +( +) +( +) +( +) +2 +2 +1 +ˆ +ˆ +FPE +ln2 +1 +ln +2 +1 +ˆ +ˆ +ln2 +1 +ln +. +2 +f +f +f +f +t +t +t +t +s +x +x + + + + + + + + + + + + + + + + + +(7) +B. Scenario Features Extraction +In prediction scenarios, a TA’s motion is related to its +historical state, interactions with surrounding TPs, and other +factors. Although the existing prediction algorithms have either +explicitly or implicitly considered different factors, their +performances may still be affected by the above-mentioned +features due to algorithm limitations. Therefore, three types of +scenario features are considered in this work: 1) dynamic +features of a TA, which include data on its historical or future +motion states; 2) features of surrounding TPs, which refer to +their states and interactions with the TA; 3) other scenario +features, which include the type, behavior pattern, compliance +with traffic rules, and current location of the TA. +1) Kinematic features of TA +The historical data on the TA motion state denote an +important input to a prediction model and have a direct impact +on the model output. In addition, the future motion state of a +TA is a key reference for evaluating the model’s prediction +accuracy. Therefore, in this work, the kinematic features of TA +are extracted to analyze their impact on the prediction algorithm +performance. +Velocity is one of the primary kinematic features, which +directly reflects the aggressiveness of TA movement. It also +represents the discrete degree of a continuous trajectory. +Considering the trajectory prediction model characteristics, +three velocity sub-features are extracted: 1) average velocity of +the historical trajectory (AVHT), which indicates the +aggressiveness of the model's input trajectory; 2) current +velocity (CV), which directly represents a TA’s current state +and has a key impact on the trajectory prediction output; 3) +average velocity of the future trajectory (AVFT), which reflects +the spatial span of the future trajectory points. +In addition, the velocity variations indicate trajectory +stationarity, which may have a significant influence on +prediction results. For instance, a sudden start of a parked +vehicle may be difficult for the model to predict timely and +accurately. Therefore, the acceleration value at each moment is +calculated to obtain the following sub-features: 1) average +acceleration of the historical trajectory (AAHT), which +K + +5 +represents the speed mutation degree of the input trajectory; 2) +average acceleration of the future trajectory (AAFT), which +reflects the overall situation of a TA's future speed mutation; 3) +maximum acceleration of the future trajectory (MAFT), +considering that a sudden speed change at any moment can lead +to severe deformation of the overall trajectory, it is necessary to +extract the fastest speed change in the future as a feature for +analysis. +Similarly, changes in the TA moving direction denote a +potentially influential factor of prediction performance. For +instance, a vehicle going straight may suddenly swerve or make +a U-turn, thus posing a serious challenge to the prediction +algorithm. Therefore, the heading change speed (HCS) is +extracted for analysis. In detail, the absolute value of the change +speed of the heading angle at each moment is calculated and +used as a basic feature, and then the analysis of the following +parameters is performed: 1) average HCS of the historical +trajectory (AHCSHT); 2) average HCS of the future trajectory +(AHCSFT), which reflects the overall curvature or volatility of +the future trajectory; 3) maximum HCS of the future trajectory +(MHCSFT), which increases when there is a sudden large +change in direction at any point in the future. +2) Features of Surrounding TPs +Convoluted interactions with other agents increase the +difficulty in trajectory prediction, and although many of the +existing prediction methods can explicitly or implicitly model +interactions, it has not been fully discussed whether the +performance of these black-box models is sensitive to actual +interactions. To examine this situation, a set of hierarchical +prediction scenario complexity metrics is proposed to analyze +the effect of a TA’s interactions with surrounding agents on the +prediction algorithm performance. +First, with a TA as a center, the prediction scenario +complexity has a positive correlation with the number of its +surrounding TPs, and a basic feature, the number of TPs within +x meters from the TA (NTPx), is defined. +In addition, the distance between the TA and its surrounding +TPs directly affects the prediction scenario complexity. +Assuming that a set of TPs within x meters of a TA i is denoted +by +  +x +N +i +, then for any +  +x +j +N +i + +, its distance from i is + + +, +j +j +i +dist s +d +s + +. The density of TPs within x meters around TA +(DTPx) is given by, + +  +1 +DTPx +j +x +N +i +d +j +e + + + +  +， +(8) +where  is a scaling factor. +Furthermore, the potential conflicts due to the movement of +surrounding TPs are analyzed. As shown in Fig. 3, for a TP +  +x +j +N +i + + within x meters around a TA i , its current state is +given by +  +  +0 +0 +[ +, +] +j +j +s +v +; then, its position after t seconds is expressed +as +  +  +t +t +j +j +s +v t + +, and the degree of conflict from TPs within x +meters around TA (DCTPx) is defined as follows, + +  + +  +T +( ) +( ) +agg +1 +DCTPx +X +t +t +j +N +i +d +j +e + + + + +  +， +(9) +where + + +( ) +( ) +( ) +, +t +t +t +j +i +j +d +dist s +s + +; T is the time horizon used for +evaluation; +( )t + + is the scaling factor for the distance at time t, +and in this study, it is set to grow faster over time to reinforce + + + + +location +, +velocity +, +j +j +jx +jy +x +y +v +v + + + + + + +location +, +velocity +, +i +i +ix +iy +x y +v +v + + + +Fig. 3. Schematic diagram of the conflict degree calculation +from TPs within x meters from a TA. +(a) Agent type +(d) Agent location +Stage-1 Stage-2 +Stage-3 +Stage-4 +Stage-5 +Stage-6 +gap +(c) Compliance with traffic rules +Red-light +running +Yellow-light +running +Obeying the +rules +For going straight +and turning left +vehicles +(b) Behavior pattern +U-turn +Going +straight +Turning +left +Turning +right + +Fig. 4. Illustration of the other extracted scenario features. + + +the focus on the short-term risk; +  +T +agg + represents the +aggregation operation, which is used to synthesize the conflicts +at T times in the future. The two basic modes used in this study +include the mean value (DCTPx_mean) and the maximum +value (DCTPx_max). +3) Other Scenario Features +In addition to the above two categories of features, the impact +of several other representative features on the prediction error +and epistemic uncertainty is studied, as shown in Fig. 4. +The considered features include: + TA type: The GRIP++ can simultaneously predict future +trajectories of multiple types of TAs, such as vehicles, +pedestrians, and cyclists. Each type of agent has its +movement pattern, which may cause different prediction +performances. Referring to the research presented in +[33], TAs can be divided into four types: small vehicles, +vehicles, pedestrians, motorcyclists, and bicyclists; + TA behavior pattern: This study mainly focused on three +basic behavior patterns of vehicles at intersections: +going straight, turning left, and turning right. In addition, +this study extracts certain corner cases, such as U-turns. +The prediction errors and epistemic uncertainty under +different behavioral patterns are compared and analyzed +in a unified manner; + +6 + TA's compliance with traffic rules: Traffic rules partially +constrain the behaviors of participants, but in real-world +scenarios, some of TAs may violate the rules, thus +affecting the trajectory prediction performance. For +instance, in a signalized intersection, the behaviors of +TAs can be classified based on their compliance with the +traffic signal into obeying the rules, yellow-light running, +and red-light running; + TA location: The whole process of a vehicle passing +through the intersection is divided according to the time +sequence into six stages: stage 1: ex-entering an +intersection; stage 2: in the gap; stage 3: in the first +crosswalk; stage 4: inside an intersection; stage 5: in the +last crosswalk; stage 6: exiting an intersection. +C. Scenario Features Analysis +To analyze the relationship between the above-mentioned +features systematically, the qualitative and quantitative analysis +methods are adopted. The main methods include feature +correlation analysis and feature importance analysis based on +random forest regression. +1) Feature Correlation Analysis +Correlation analysis is to calculate the degree of correlation +between two or more feature variables using correlation +coefficients as quantitative indicators. Typical correlation +coefficients include the Pearson correlation coefficient and +Spearman rank correlation coefficient. The Pearson correlation +coefficient requires evaluated variables to conform to the +normal distribution, but this is a strong assumption that +experimental results can hardly satisfy. In contrast, the +Spearman rank correlation coefficient does not have such strict +requirements on data as the Pearson correlation. Namely, it +requires only that observed values of the two variables are +paired rank data or rank data transformed from continuous +variable observation data. The Spearman rank coefficient can +be mainly used in the monotonic relationship evaluation. +Specifically, it is assumed that the original data ( +ix , +iy ) are +arranged in ascending order, and   +i +R x and   +i +R y are defined +as the ranking of +ix and +iy in their corresponding data, +respectively; +( ) +R x and +( ) +R y denote the mean of the ranking of +the two groups of data, and n is the number of data pairs. Then, +the Spearman rank correlation coefficient + can be defined as +follows: + +  + + + + + + +  + + + + + + +  + + + + + + +1 +2 +2 +1 +1 +2 +1 +2 +( ) +( ) +( ) +( ) +6 +1 +. +1 +n +i +i +i +n +n +i +i +i +i +n +i +i +i +R x +R x +R y +R y +R x +R x +R y +R y +R x +R y +n n + + + + + + + + + + + + +  + + + + + + +(10) +2) Feature Importance Analysis Based on Random Forest +Regression +Feature correlation analysis can be used to assess linear and +ordinal consistent correlations but cannot identify other types +of correlations. To address this limitation, this work proposes a +feature importance analysis method to evaluate the impacts of +different scenario features. +The decision tree is a non-parametric supervised learning +algorithm that can be used in solving both classification and +regression problems. It is a hierarchical tree structure mainly +composed of three types of nodes, root, internal, and leaf nodes. +Ensemble learning uses multiple models to obtain accurate +prediction results, and bootstrapping is one of its typical +application techniques that refers to the process of randomly +sampling of a sub-dataset through a given number of iterations +and variables. Random forest regression combines ensemble +learning with the decision tree framework, creating multiple +decision trees from data; then, multiple outputs are averaged to +obtain the final result, often achieving excellent performance +for regression problems. +Random forest regression can be used to evaluate feature +importance. Particularly, in this study, the extracted scenario +features are regarded as independent variables, while the +prediction error and epistemic uncertainty of the prediction +model under the corresponding conditions are regarded as +dependent variables, and a random forest regression model is +constructed. Then, the contribution of each feature to the trees +in the random forest is analyzed, as well as its importance to the +performance of trajectory prediction. The variable importance +measure is denoted as VIM, and it is assumed that there are J +features and I decision trees. Then, VIM j denotes the average +change in node split impurity of the jth feature in all decision +trees, and it is calculated by: + +  +  +1 +' +' 1 +1 +VIM +VIM = +, +VIM +I +i +j +i +j +J +I +i +j +j +i + + + + + + +(11) +where +  +VIM +i +j represents the importance of the jth feature in the +ith decision tree, and it can be obtained by calculating the +difference of Gini indices of nodes before and after branching. +D. Prediction across Different Intersection Datasets +The cross-dataset analysis aims to analyze differences in +scenarios between multiple datasets and the corresponding +prediction algorithm performance disparity. In this study, the +scenario type is limited to the intersection, and multiple +intersection datasets involving various countries are studied. +First, the scenario features are extracted to analyze +distributional shifts between different intersection datasets. +Next, +comprehensive +cross-validation +experiments +are +performed to investigate the prediction algorithm performance +in terms of distributional shifts fully. Particularly, N +intersection datasets are selected, and each of them is divided +into training and test subsets. Then, for each of the training +subsets, a trajectory prediction model is developed and trained +using the training subset first and then evaluated on the +corresponding test subset. Finally, +2 +N sets of results are +obtained. +During the analysis, the following factors are mainly studied: +1) Trajectory prediction performance degradation due to +distributional shifts: Combined with the differences in +statistical features between different datasets obtained +earlier, we analyze the impact of changes in the traffic +environment on the prediction algorithm. + + +7 +2) Effect of the deep ensemble on prediction robustness +across different datasets: The improvement in prediction +accuracy and sensitivity of the estimated epistemic +uncertainty to distributional shifts are analyzed; +3) Availability and complexity of different intersection +datasets for trajectory prediction: By synthesizing +multiple groups of results, the model performance is +evaluated using different intersection datasets as a +training set and the prediction challenge with different +intersection datasets as a test set. +IV. EXPERIMENTAL SETUP +A. Intersection Datasets +Focusing on the urban intersection scenario, multiple +trajectory datasets were used for evaluation and analysis, +involving different periods, weather, countries and regions, and +many TP types. +1) SinD [55]: The SinD dataset is a typical drone dataset +collected from a signalized intersection in Tianjin, China. These +data were recorded from a static bird’s eye view at a sampling +frequency of 10 Hz. This dataset contains about 420 minutes of +traffic recordings, including over 13,000 TPs with seven types, +including cars, trucks, buses, pedestrians, tricycles, bikes, and +motorcycles; +2) INTERACTION [29]: The INTERACTION dataset contains +12 subsets covering merging, roundabout, and intersection +scenarios, of which five intersection subsets are used in this +study: USA_Intersection_EP1 (EP1), USA_Intersection_EP2 +(EP2), USA_Intersection_MA (MA), USA_Intersection_GL +(GL) and TC_Intersection_VA (VA). They contain about 493 +minutes of recordings in total. The first four relate to +unsignalized intersections in the US, mainly involving +trajectories of vehicles, pedestrians, and bicycles recorded by +drones. The VA denotes a signalized intersection in Bulgaria, +which involves trajectories of cars, buses, trucks, motorcycles, +and bicycles recorded by traffic cameras. +B. Prediction Error Metrics +The prediction error is a preferred metric for quantifying the +performance of prediction models. Following [2, 8, 25], the +proposed model was evaluated using two error metrics: +1) Average Displacement Error (ADE): This is the mean square +error of all predicted points of a trajectory compared to the +ground truth; +2) Final Displacement Error (FDE): This is the distance +between the predicted final destination and the true final +destination at +. +C. Predictive Uncertainty Evaluation +As stated previously, the APE and FPE reflect the epistemic +uncertainty of a prediction model, which can indicate situations +where prediction models are performing poorly. Therefore, +referring to [56, 57], this study uses the error-retention curves +to evaluate the ability of the extracted epistemic uncertainty to +detect prediction errors. The curves depict the error over a +dataset as a model’s predictions are replaced by ground-truth +labels in order of decreasing uncertainty. The abscissa value of +a point represents the proportion of the retained true error (i.e., +retention fraction), while the ordinate value represents the +comprehensive error under this proportion. Similarly, the +optimal and random curves are obtained by replacing the +predictions in order of decreasing error and random order, +respectively. The area under the retention curves (AUC) is an +evaluation metric of both the robustness of prediction models +and the quality of uncertainty estimation, and an efficient +uncertainty estimation is considered to achieve a low AUC. +D. Implementation Details +For achieving fair evaluation and transferability, the datasets +were standardized to use up to 3 s of the previous data and 3 s +of the future data. The trajectory data were resampled to 2 Hz. +The single GRIP++ model in the ensemble models was +implemented in PyTorch, and its implementation details, +including input preprocessing, graph convolution, and +trajectory prediction model, mainly refer to the settings in [5]. +In the implementation of MC dropout and deep ensemble, the +value of k was set to five, which was the result of a trade-off +between uncertainty estimation quality and computational cost +[45]. In addition, during the training process of the MC dropout- +model, a regularization term was added to the loss to improve +its ability of uncertainty estimation [39], where the +regularization coefficient was set to 0.0001, and the dropout +rate was set to 0.5 +In addition, the original dataset was further processed to +obtain the labels for the subsequent analysis. The locations of +TAs were classified by combining their raw coordinate data +with the map in the Lanelet2 format [58]. +V. RESULTS AND DISCUSSION +A. Evaluation of Trajectory Prediction and Epistemic +Uncertainty Estimation +The training and test performances of the proposed model +obtained by following the train-test process in the same +intersection dataset are presented in TABLE I. Although the +MC dropout can be used to estimate epistemic uncertainty, it +increases the prediction error, which might be due to the +modifications in the original loss function. In contrast, the deep +ensemble-based +method +improves +trajectory +prediction +accuracy while estimating epistemic uncertainty. As shown in +TABLE I, the error obtained by the deep ensemble-based +method was lower than that of the single models. Thus, by +integrating the results of multiple models, deep ensemble could +effectively avoid the prediction performance degradation +caused by the deviation of a single model, which is conducive +to improving the prediction algorithm robustness. The +evaluation results of the estimated epistemic uncertainty are +shown in Fig. 5. Compared with the MC dropout, the deep +ensemble had obvious advantages in improving the model +prediction accuracy and uncertainty estimation. Therefore, in +the subsequent analysis, the epistemic uncertainty estimation +framework based on the deep ensemble was adopted. +As shown in Fig. 5, both uncertainty quantification metrics +(i.e., APE and FPE) could accurately reflect the prediction +model error (i.e., ADE or FDE), and they showed a high degree +of consistency. By comparing the left and right sides of Fig. 5, +it can be concluded that although the prediction error on the test +sets was slightly increased compared to that on the training set, +there was a small difference in the epistemic uncertainty +ft + +1 +TABLE I +TRAJECTORY PREDICTION ERROR COMPARISON1 +Dataset +ADE1/FDE1 +ADE1/FDE1 +ADE1/FDE1 +ADE1/FDE1 +ADE1/FDE1 +ADEdropout/FDEdropout +ADE/FDE +SinD +0.405/0.865 +0.404/0.865 +0.402/0.86 +0.403/0.861 +0.408/0.872 +0.477/1.021 +0.389/0.832 +VA +0.652/1.401 +0.641/1.378 +0.655/1.428 +0.657/1.421 +0.654/1.402 +0.651/1.327 +0.615/1.327 +EP0 +0.811/1.822 +0.815/1.844 +0.803/1.809 +0.809/1.814 +0.822/1.850 +1.036/2.351 +0.745/1.678 +EP1 +0.881/1.982 +0.842/1.894 +0.816/1.826 +0.853/1.915 +0.857/1.930 +1.147/2.590 +0.775/1.743 +MA +0.878/2.026 +0.857/1.979 +0.866//1.999 +0.867/2.000 +0.870/2.008 +1.083/2.546 +0.811//1.879 +GL +0.619/1.448 +0.627/1.464 +0.622/1.452 +0.625/1.458 +0.627/1.466 +0.709/1.646 +0.591/1.386 + +(a) ADE for training set +(b) FDE for training set +(c) APE for test set +(d) FPE for test set + +Fig. 5. The ADE/FDE-retention curves (top) and retention scores curves (bottom) on the training and test sets of the SinD dataset. +The optimal curve (solid green line) was obtained by replacing the model's predictions with the ground-truth labels in order of +decreasing error. Similarly, the random curve (blue dotted line) was obtained by replacing the model’s predictions with the ground- +truth labels in random order. The red and yellow solid lines correspond to the results captured in order of decreasing APE and +decreasing FPE, respectively. The retention scores corresponding to each retention fraction can be calculated by: (errorrandom – +erroruncertainty)/(errorrandom – erroroptimal). + + +estimation performance. The results indicate that the deep +ensemble-based method had good generalization ability +In the second row in Fig. 5, a consistent trend where the +retention scores first increase and then decrease with the +retention fraction can be observed. +B. Scenario Features Analysis Results +1) Feature Correlation Comparison +In general, it has been considered that scenario complexity +increases with the values of the extracted TA’s kinematic +features and the features of the TA’s surrounding TPs. +Therefore, the correlation between the model performance and +these two types of features was calculated to analyze the +influence of scenario complexity on trajectory prediction +performance. Based on the model trained on the SinD training +set, the feature correlation analysis experiment was performed +on the SinD test set, where the prediction performance was + +1 ADEk/FDEk represents the prediction error of model k among the ensemble models, ADEdropout/FDEdropout is the prediction error of the mc dropout-based +method, and ADE/FDE denotes the prediction error of the deep ensemble-based method. +represented by prediction error and epistemic uncertainty. For +the features of the TA’s surrounding TPs, four groups of +distances of x = 10, 20, 30, 50 were studied. The analysis results +are presented in Fig. 6, and based on them, the following +conclusions can be drawn: + +The distribution trends of environmental feature +correlations corresponding to the two types of prediction +errors (ADE and FDE) were highly consistent, as well as +the +distribution +trends +of +environmental +feature +correlations corresponding to the two types of epistemic +uncertainty (APE and FPE) estimates; + +The comparison of a TA’s kinematic features with the +features of its surrounding TPs shows that the former had +a strong positive correlation with the error and epistemic +uncertainty of the prediction model, while the latter had +weak correlations with the error and epistemic uncertainty +(-0.2 <  < 0.2). + +1 + +Fig. 6. Comparison of the correlation between prediction model performance and scenario features + + + +The comparison of the correlation between different +kinematic features and the prediction error shows that: + +The sorting order in terms of the correlation was: +acceleration-related features > velocity-related +features > heading change speed-related features. +This order indicates that the mutation of a TA’s +speed had a relatively large impact on the prediction +error, while the change in the TA’s movement +direction had a relatively low impact; + +The sorting order in terms of the correlation was: +features related to future trajectories > features +related to historical trajectories. This shows that the +proposed trajectory prediction model had low +adaptability to the speed and position mutations +occurring at some certain points in the future. + +The correlation between different kinematic sub-features +of a TA and the predictive uncertainty showed that the +epistemic uncertainty was highly sensitive to the velocity +and acceleration features; namely, when a TA was driving +at high speed or had a speed mutation, the model tended +to show lower confidence in the predictions. +2) Feature Importance Comparison +As mentioned above, the feature correlation analysis only +shows whether the relationship between two variables conforms +to the order consistency. Therefore, the feature importance +analysis experiment was performed based on the random forest +regression to explore whether there were other types of +correlation between the above scenario features and the +prediction model. The datasets and training settings used for the +prediction algorithm were the same as in the feature correlation +analysis. The grid-based search was employed to obtain optimal +random forest regression model, then the feature importance +analysis was performed. The results are presented in Fig. 7. +As shown in Fig. 7, the distribution trend of feature +importance was basically consistent with the feature correlation. +For instance, the features obtained from the surrounding TPs +had little effect on the error and uncertainty of the prediction +model uniformly. In contrast, the kinematic features of a TA +had a stronger influence, where velocity- and acceleration- +related features had higher importance. In random forest +regression for the ADE, the AAFT had the highest importance, + +Fig. 7. Comparison of feature importance based on the random +forest regression + +while in the random forest regression for the APE, the CV had +the strongest influence. +3) Other Scenario Features Analysis +In addition to the above-mentioned features, the impacts of +several other environmental features on the prediction +algorithm performance were explored, including the type, +behavior patterns, compliance with traffic rules, and location of +a TA. Without loss of generality, the ADE was adopted as an +error metric, and the APE was used for epistemic uncertainty +quantification. The prediction model was trained on the SinD +training set and analyzed on the SinD test set +The results indicated obvious differences in the prediction +error distribution between different TAs, as shown in Fig. 8. +Although the movement of pedestrians had high degrees of +freedom and randomness, their speed and acceleration were +generally low, so the corresponding prediction error was small. +Meanwhile, the epistemic uncertainty distribution for different +types of TAs showed high similarity with that of the prediction +error. +The results of the trajectory prediction error and epistemic +uncertainty of a vehicle under different behavior patterns are +presented in Fig. 9, where it can be seen that the trajectory +prediction performance tended to exhibit larger errors when the +TA was turning left and right compared to the going-straight +behavior. The error in the right-turn scenario was relatively +(a) ρ for ADE +(b) ρ for FDE +(a) ρ for APE +(b) ρ for FPE +(a) feature importance for ADE (b) feature importance for APE + +2 + +Fig. 8. The results of the trajectory prediction error and +epistemic uncertainty of different TA types; sv: small vehicle, +bv: large vehicle; bi: motorcyclist or bicyclist; pe: pedestrian. + +Fig. 9. The results of the trajectory prediction error and +epistemic uncertainty under different behavioral patterns. + + +Fig. 10. The results of the trajectory prediction error and +epistemic uncertainty under different traffic rule compliance. + +Fig. 11. The results of the trajectory prediction error and +epistemic uncertainty at different locations; 1: ex-entering the +intersection; 2: in the gap; 3: in the first crosswalk; 4: inside the +intersection; 5: in the last crosswalk; and 6: exiting the +intersection. + +large, and the reasons may be as follows. First, the right-turn +trajectory had a large curvature, and second, the vehicle was +less affected by traffic lights and other TPs when turning right, +compared to turning left and going straight, resulting in a higher +driving speed. In addition, although the U-Turn represents a +typical corner case, the results showed that the overall error in +this pattern was small, which may be related to the generally +low speed during the U-turning process. Furthermore, the +epistemic uncertainty distributions under different behavioral +patterns were relatively consistent with the error. +The relationship between the vehicles’ compliance with +traffic lights and the trajectory prediction performance is +presented in Fig. 10, where it can be seen that the prediction +error was larger when the vehicle ran red or yellow light than +when there was no violation of traffic lights, and +simultaneously the proposed model could output higher +epistemic uncertainty. +The impact of a TA’s location on the trajectory prediction +performance is presented in Fig. 11. When the vehicle was in +the gap area or first crosswalk before entering the intersection, +there were many possible strategic options, referring to whether +to enter the intersection and how to pass through the +intersection, which increased the prediction complexity, and +further increased the prediction error and epistemic uncertainty. +When the vehicle was inside the intersection, the prediction +model exhibited considerable error and uncertainty due to a +large number of interactions with other TPs and higher freedom +of movement. In contrast, the predominant behavior of the +vehicles before entering and after exiting the intersection was +to follow the lane, so these stages had lower prediction error +and uncertainty than the others. +C. Prediction evaluation across Different Intersection Datasets +Different intersection datasets were collected at different +times and locations, and the corresponding environmental +conditions might be relatively different, resulting in shifts in +data distribution. The distributions of velocity, acceleration, +heading, and HCS of objects in six intersection datasets are +presented in Fig. 12, where it can be seen that there were +obvious differences in the trajectory features between the +datasets. For instance, the velocity and acceleration of a portion +of trajectories in the SinD dataset were concentrated around +zero. One of the main reasons was the stopping of vehicles and +pedestrians while waiting for the green light. Furthermore, the +velocity in the SinD dataset exhibited a distinct multimodal +distribution, which could be related to the multiple movement +patterns caused by various TPs in the dataset. In contrast, the +velocity and acceleration in the GL, MA, and VA datasets +tended to be higher, reflecting more aggressive motions in these +datasets. In addition, distributional shifts could also be observed +by comparing the distribution of heading and its speed of +change in different datasets. +The aforementioned differences in data distribution between +datasets may bring application challenges of the trajectory +prediction algorithms in real-world environments. Therefore, +experiments were performed on multiple intersection datasets. +Specifically, based on the six intersection datasets mentioned +above, a set of deep ensemble-based prediction models were +trained on each dataset and evaluated on all six datasets. As +shown in Fig. 13, distributional shifts in the real-traffic + +1 + +Fig. 12. Distribution comparison of different intersection datasets. +(a) ADE1 +(b) ADE +(c) APE + +Fig. 13. The cross-dataset error and uncertainty matrix: (a) ADE1 is the error obtained by evaluating one of the ensemble models; +(b) ADE is the prediction error of the deep ensemble-based model; (c) APE also relates to the deep ensemble-based model. The +ith row and jth column of each matrix represent the results obtained by evaluating the model on the test subset at intersection j +after training it on the training subset at intersection i. + + +Fig. 14. Comprehensive performances of the prediction +algorithms on different datasets. (a) Comparison of prediction +errors of all trained models on different test subsets; (b) results +of evaluating models trained on different training subsets on all +test subsets. + +environments had a strong impact on trajectory prediction +performance. Even for the same type of scenario, for the model +trained on one intersection dataset, it was difficult to generalize +well to other intersection datasets directly. For instance, on the +test subset of the SinD dataset, the model that achieved the best +accuracy (ADE1/ADE: 0.405/0.389) was trained on the training +subset of the SinD dataset; in contrast, the prediction error +(ADE1/ADE) of the model trained on the other intersection +training subsets increased by 94.9%/8.61% - 265.0%/221.0%. +Comparing the single prediction model error (Fig. 13(a)) +with the error of the deep ensemble-based model (Fig. 13(b)), +it could be concluded that the deep ensemble-based approach +performed systematically better than a single model, achieving +significant improvements in many cases. +As presented in Fig. 13(c), the extracted epistemic +uncertainty could indicate distributional shifts between +different datasets. Particularly, the proposed model could +output higher epistemic uncertainty when the error increased +due to changes in the test scenario. +In Fig. 14, the comprehensive performances of the trajectory +prediction algorithm on specific single training or test set are +presented. As shown in Fig. 14(a), the models trained on the +training subsets of the SinD and GL datasets had better +generalization ability than the other models. This could be +because of a larger amount of data and more diverse motion +patterns of these two datasets compared to the other datasets. +Conversely, the overall error of the model based on the training +subset of VA was the highest, and the main reasons were as +follows. First, this dataset contained less data than the other +datasets. Second, this dataset was constructed by data collected +by roadside equipment, which might introduce more noise than +drone-based data collection. The comprehensive performances +of the prediction models on different intersection test subsets +are presented in Fig. 14(b). The GL and MA with higher +velocity and acceleration features showed greater prediction +difficulty, and the overall trajectory prediction error on the +SinD test subset was the smallest among all datasets. +D. Qualitative Results +The prediction results of the proposed framework in several +scenarios on different intersection datasets are presented in Fig. +15, where each row corresponds to one intersection. In Fig. 15, + +isolg +0 +2 +or +J2 +50 +52 +00.0 +20.0 +0'T0 +0'T2 +AM +0'52 +EbJ +Ebo +0E.0 +AV +2E.0 +1 2UD +926b2HC +00.0 +0'52 +0'20 +0'>2 +J'00 +J'52 +J'32 +5'00 +0 +AM +3 +Eb +Ebo +AV +2!UD +J926J6bnoit19l96 +0 +e +8 +JO +00.0 +0'52 +0'20 +AM +J'S2 +EbJ +Ebo +J'20 +AV + 2IUD +J'>2 +926b-5 +0 +A +0.0 +S.0 +0'4 +D6U +aer +AM +EbJ +8.0 +Ebo +AV +J'O +2!UD +J926J6b1 +SinD +VA +EP0 +EP1 +MA +GL + +Fig. 15. Qualitative results of trajectory prediction and uncertainty estimation on different datasets (corresponding to different +rows). The gray thick solid line represents the historical trajectory, and the black thin solid line represents the true future trajectory. +Different colors are used to denote predictions for different types of TAs: blue - small vehicle; magenta - large vehicle; yellow – +pedestrian; green - motorcycle or cyclist. In addition, the focused object in each subgraph is highlighted in red, and the current +prediction error and uncertainty estimation results of the object are presented above the subgraph. + +nipnonipno1..1 +the first column shows cases where the prediction error was +small and epistemic uncertainty was low. These cases generally +appeared when the TA exhibited obvious future intentions and +moved relatively smoothly. The second and third columns show +scenarios with large trajectory prediction errors, which were +generally accompanied by higher estimates of epistemic +uncertainty. This situation may occur when the TA was about +to enter the intersection where it was difficult to determine its +future behavior pattern, or its future motion showed a large +pattern or speed change compared to the historical trajectory. +VI. CONCLUSION +In this paper, a trajectory prediction framework that +integrates the epistemic uncertainty estimation function is +proposed, and the effects of the traffic environment and its +changes on the prediction algorithm performance are studied. A +few typical scenario features are considered, and their +influences on the prediction performance are examined by the +feature correlation and importance analyses. Further, the +distributional shifts between different intersection datasets and +the resulting performance degradation of the prediction model +are analyzed. Based on the obtained results, the following +conclusions are drawn: +(1) The extracted epistemic uncertainty is valuable for +representing the model's confidence in its current predictions +accurately, which has the potential to be used to identify +unknown scenarios where the model may be underpowered. +Compared with the MC dropout-based method, the deep +ensemble-based method performs significantly better in +estimating epistemic uncertainty and improving the trajectory +prediction robustness; +(2) Regarding the influence of different scenario features on +the trajectory prediction performance, the feature correlation +and importance analyses show similar results. Namely, there is +a positive correlation between the kinematics of a TA and the +prediction model performance. Higher velocity, acceleration, +and speed of the heading change generally pose a greater +challenge to the trajectory prediction process, while a prediction +model tends to exhibit higher epistemic uncertainty. However, +one interesting finding is that the features of surrounding TPs, +which reflect the complexity of interactions in the scenario, +show little impact on the proposed prediction model +performance. In addition, the error and uncertainty of the +prediction model vary with other abstract features, such as TA’s +type, behavior pattern, compliance with traffic rules, and +location. The conducted analyses are helpful in locating the +limitations in the prediction algorithms, thus providing +guidance for the improvement of autonomous driving functions; +(3) For the intersection scenario, different datasets show +distributional shifts due to differences in local driving habits, +road structures, and national cultures, thus posing great +challenges to the prediction algorithms. Fortunately, the +proposed framework based on the deep ensemble is beneficial +to improving the trajectory prediction robustness, and the +extracted epistemic uncertainty can respond to the reduced +confidence of the proposed model in a new environment. This +improves the self-awareness ability of autonomous driving. +Although the used basic prediction algorithm has strong +representativeness, it is still difficult to avoid the specificity of +analysis conclusions completely. However, the proposed +method is promising to analyze other prediction algorithms, +subsequent work could consider combining more types of +algorithms for systematic analysis. Moreover, this work focuses +on the extraction and analysis of epistemic uncertainty, but the +existing works in the field of multimodal trajectory forecasting +could be considered in the follow-up research to distinguish the +epistemic uncertainty from the aleatoric uncertainty better and +explore their practical significance. Furthermore, the role of +uncertainty estimation of a trajectory prediction model in an +autonomous driving decision-making mechanism needs to be +studied further to improve the robustness against the risks of +insufficient functions. + +ACKNOWLEDGMENT +This work was supported in part by the National Science +Foundation of China Project (Grant No. 52072215 and +U1964203), and the National Key R&D Program of China +under Grant NO. 2020YFB1600303. +REFERENCES +[1] K. Xu, X. Xiao, J. Miao, and Q. Luo, "Data Driven Prediction Architecture +for Autonomous Driving and its Application on Apollo Platform," in 2020 +IEEE Intelligent Vehicles Symposium (IV), 19 Oct.-13 Nov. 2020 2020, pp. +175-181, doi: 10.1109/IV47402.2020.9304810. +[2] X. Mo, Z. Huang, Y. Xing, and C. Lv, "Multi-Agent Trajectory Prediction +With Heterogeneous Edge-Enhanced Graph Attention Network," IEEE +TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, +Article Early Access 2022, doi: 10.1109/TITS.2022.3146300. +[3] A. 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His research interests include safety of +the intended functionality of autonomous driving, trajectory +prediction, +decision-making, +uncertainty +theory +and +applications. + +Yanchao Xu received B.E degree in vehicle +engineering from Hainan University in +China in 2020. He is currently pursuing the +M.S. degree in Mechanical Engineering at +Beijing Institute of Technology. He is also +one of the visiting students at IVDAS since +2020. His research interest includes +prediction, trajectory data mining, scenario +parameterization for autonomous driving. + + +Jun Li received the Ph.D. degree in +vehicle engineering from Jilin University, +Changchun, Jilin, China, in 1989. He is +currently an academician of the Chinese +Academy of Engineering, a Professor at +school of Vehicle and Mobility with +Tsinghua University, president of the +Society of Automotive Engineers of China, +director of the expert committee of China +Industry Innovation Alliance for the Intelligent and Connected +Vehicles. His research interests include internal combustion +engine, electric drive systems, electric vehicles, intelligent +vehicles and connected vehicles. + +Chen Lv (Senior Member, IEEE) +received the Ph.D. degree from the +Department of Automotive Engineering, +Tsinghua University, China, in 2016. +From 2014 to 2015, he was a Joint Ph.D. +Researcher at the EECS Department, +University of California at Berkeley, +Berkeley, CA, USA. From 2016 to 2018, +he worked as a Research Fellow at the +Advanced Vehicle Engineering Center, +Cranfield +University, +U.K. +He +is +currently an Assistant Professor with the School of Mechanical +and Aerospace Engineering, and the Cluster Director of future +mobility solutions at ERI@N, Nanyang Technological +University, Singapore. His research interests include advanced +vehicles and human–machine systems, where he has +contributed over 100 articles and obtained 12 granted patents. + + +Weida Wang received the Ph.D. degree +from Beihang University, Beijing, China, +in 2009. He is currently a Professor with +the School of Mechanical Engineering, +Beijing Institute of Technology, Beijing. +He is also the Director of the Research +Institute of Special Vehicle, Beijing +Institute of Technology. His current +research interests include electric vehicle, +automated vehicle motion planning and control, and +electromechanical transmission control. + +Hong Wang is Research Associate +Professor at Tsinghua University. She +received the Ph.D. degree in Beijing +Institute of Technology, Beijing, China, +in 2015. From the year 2015 to 2019, she +was working as a Research Associate of +Mechanical +and +Mechatronics +Engineering with the University of +Waterloo. Her research focuses on the +safety of the on-board AI algorithm, the +safe decision-making for intelligent vehicles, and the test and +evaluation of SOTIF. She becomes the IEEE member since the +year 2017. She has published over 60 papers on top +international journals. Her domestic and foreign academic part- +time includes the associate editor for IEEE Transactions on +Vehicular Technology and Intelligent Vehicles Symposium, +Guest Editor of Special Issues on Intelligent Safety of +Automotive Innovation, Young Communication Expert of +Engineering, lead Guest Editor of Special Issues on Intelligent +Safety of IEEE Intelligent Transportation Systems Magazine. + + diff --git a/5dE3T4oBgHgl3EQfQgnL/content/tmp_files/load_file.txt b/5dE3T4oBgHgl3EQfQgnL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..27952daf3359f69e4d9609a5863ee5fe70fb9728 --- /dev/null +++ b/5dE3T4oBgHgl3EQfQgnL/content/tmp_files/load_file.txt @@ -0,0 +1,1030 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf,len=1029 +page_content='1 How Does Traffic Environment Quantitatively Affect the Autonomous Driving Prediction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Wenbo Shao, Yanchao Xu, Jun Li, Chen Lv, Senior Member, IEEE, Weida Wang and Hong Wang☒, Senior Member, IEEE Abstract—An accurate trajectory prediction is crucial for safe and efficient autonomous driving in complex traffic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In recent years, artificial intelligence has shown strong capabilities in improving prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' However, its characteristics of inexplicability and uncertainty make it challenging to determine the traffic environmental effect on prediction explicitly, posing significant challenges to safety-critical decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' To address these challenges, this study proposes a trajectory prediction framework with the epistemic uncertainty estimation ability that outputs high uncertainty when confronting unforeseeable or unknown scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The proposed framework is used to analyze the environmental effect on the prediction algorithm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In the analysis, the traffic environment is considered in terms of scenario features and shifts, respectively, where features are divided into kinematic features of a target agent, features of its surrounding traffic participants, and other features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, feature correlation and importance analyses are performed to study the above features’ influence on the prediction error and epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Further, a cross- dataset case study is conducted using multiple intersection datasets to investigate the impact of unavoidable distributional shifts in the real world on trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The results indicate that the deep ensemble-based method has advantages in improving prediction robustness and estimating epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The consistent conclusions are obtained by the feature correlation and importance analyses, including the conclusion that kinematic features of the target agent have relatively strong effects on the prediction error and epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Furthermore, the prediction failure caused by distributional shifts and the potential of the deep ensemble-based method are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Index Terms—Artificial intelligence, autonomous driving, distributional shift, epistemic uncertainty, traffic environment, trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' INTRODUCTION RAJECTORY prediction is an indispensable part of the autonomous driving pipeline [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' To drive safely and efficiently in complex traffic environments, autonomous vehicles (AVs) are required to have the ability to predict the future motion of surrounding traffic participants This work has been submitted to the IEEE for possible publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Copyri ght may be transferred without notice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' This work was supported in part by the National Science Foundation of China Project: 52072215and U1964203, and t he National Key R&D Program of China:2020YFB1600303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' (Corresponding authors: Hong Wang) Wenbo Shao, Jun Li and Hong Wang are with Tsinghua Intelligent Vehicle Design and Safety Research Institute, School of Vehicle and Mobility, Tsingh ua University, Beijing 100084, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' (e-mail: swb19@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' lijun1958@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' hong_wang@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' (TPs), such as vehicles and pedestrians, accurately and reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In recent years, with the accumulation of large-scale driving data and rapid development of algorithms, artificial intelligence (AI) has been widely applied to autonomous driving trajectory prediction [2, 3], and promising results have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=" However, trajectory prediction has still been challenging, particularly in urban driving scenarios, where an agent's movement is influenced by a combination of its historical state and its complex interactions with the surrounding environment." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Many recent studies [4-6] have considered multiple factors simultaneously to improve trajectory prediction algorithms, but there has still been certain performance degradation of a prediction model in complex traffic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' With the improvement in prediction accuracy, the complexity of AI-based models has also increased gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Highly elaborated models pose a great challenge to explaining the operation and failure mechanisms of prediction algorithms, which in turn reduces the credibility of a prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, AI has its inherent uncertainty and faces many problems, such as insufficient training data, imperfect model architecture, and limited training process, which may lead to functional insufficiencies of the model under specific environmental conditions, potentially causing severe traffic accidents [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The existing research mainly focuses on improving the dataset-level accuracy of prediction algorithms [8, 9], and little attention has been paid to the changes in prediction performance under different environmental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' However, this is not conducive to addressing the practical challenges that a prediction algorithm confronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=" For a target agent (TA) moving in a specific scenario, a trajectory prediction model predicts its future trajectory by modeling time series, interaction, and other relationships based on its historical state, surrounding TPs' features, and other environmental features." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Correspondingly, various traffic environmental factors may have different effects on trajectory prediction, but fewer studies have quantitatively investigated these effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Further, data-driven methods strongly depend on Yanchao Xu and Weida Wang is with the School of Mechanical Engineerin g, Beijing Institute of Technology, Beijing 100081, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' (e-mail: 31202004 10@bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='cn, wangwd0430@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='com) Chen Lv is with the School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798 (e-mail: lyuchen@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='sg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' T 1 Traffic environmental features and distributional shifts Trajectory prediction with epistemic uncertainty estimation Graph Representation Deep Ensemble-based prediction method Graph Convolution Model RNN-based Trajectory Prediction Model Graph Feature How dose traffic environment affect the autonomous driving prediction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Distributional shifts Surrounding traffic participants The target agent Different environmental features Scenario features analysis & research across intersection datasets Prediction error & epistemic uncertainty estimation quantitative analysis Answer As independent variables qualitative analysis Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Illustration of the traffic environment effect on the trajectory prediction algorithm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The traffic environmental data include various TAs’ states, their surrounding TPs’ states, and other contextual information, which may affect the prediction differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, variations in time and place can lead to distributional shifts, which may further degrade the prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' This study focuses on extracting these factors and analyzing their influence on prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' training data, and a model trained on one dataset may not perform well on other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In real-world applications, the operating environment of AVs may change significantly with different factors, such as time, geography, country, and weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' This may cause certain distributional shifts, posing additional challenges to trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Therefore, it is increasingly important to study how distributional shifts [10] in a real environment affect trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 1, this work focuses on the effects of both specific scenario features and scenario shifts on the prediction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' As for the prediction algorithm performance, previous studies have generally focused on prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In recent years, there has been an increasing interest in extracting the uncertainty of AI-based models [11, 12], thus empowering the models with a self-awareness ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=" Epistemic uncertainty [13] is a recurring suggestion that helps to represent the model's confidence in its current predictions;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' namely, these models tend to have greater epistemic uncertainty when they encounter challenging environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Therefore, the epistemic uncertainty of a prediction model is extracted and considered a type of performance metric in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 1, based on both the prediction error and epistemic uncertainty, the effect of the traffic environment on the prediction algorithms’ performances can be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The main contributions of this work can be summarized as follows: \uf06c A trajectory prediction framework that integrates epistemic uncertainty estimation is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The proposed framework performs the TA’s future state prediction and estimates the epistemic uncertainty simultaneously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' \uf06c The potential of the proposed deep ensemble-based trajectory prediction framework for improving the prediction algorithm robustness and estimating epistemic uncertainty is demonstrated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' \uf06c For the trajectory prediction task, the key features of a traffic environment are extracted, and methods for feature correlation analysis and feature importance analysis are proposed to obtain the relationship between the traffic environment and the trajectory prediction algorithm performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' \uf06c The distributional shifts between different intersection datasets and the resulting trajectory prediction degradation are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The features of multiple datasets and their prediction difficulty levels are analyzed, and it is demonstrated that the deep ensemble is helpful in improving the trajectory prediction robustness against cross-dataset evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Section II presents the existing work related to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Section III introduces the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Section IV describes the datasets and evaluation metrics used in this work, as well as implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Section V analyzes and discusses the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Section VI concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Trajectory Prediction There have been numerous studies on improving the trajectory prediction algorithms, and according to the modeling principles, they can be mainly divided into physics-based 2 methods, maneuver-based methods, and interaction-aware methods [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Physics-based methods [15] consider only the historical motion state of an object while ignoring the influence of surrounding TPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Therefore, they are mainly suitable for short-term trajectory predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Maneuver-based methods [16] learn prototype trajectories from the observed agent behaviors to predict future motion, but they lack consideration of interactions between TPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Interaction-aware methods [17] have shown better performance compared to the other two types of methods through learning the interaction between a TA and surrounding TPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In recent works, many methods have been used to model interactions between agents, providing valuable information for trajectory prediction improvement [3, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' For instance, social pooling (S-pooling) [8] pools hidden states of a TA’s neighbors within a certain spatial distance to model interactions with the surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Convolutional social pooling [18] combines the convolutional and max-pooling layers to model interactions between agents in the occupancy grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Subsequently, the grid representation is further modified to consider only eight neighbors that have the most critical impact on the TA [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, recent research has focused on the rasterized representation of scenes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' the historical state of a TA and scene context were co-encoded in a raster map [20-22], and various information was distinguished by different channels and colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, convolutional neural networks (CNNs) were used to extract desired features from raster graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Graph models have attracted great interest recently due to their good performance in modeling inter-agent interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In graph models, a node represents an agent, and an edge represents an interaction between two agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Diehl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' [23] modeled interactions between vehicles as a homogeneous directed graph to achieve high computational efficiency and large model capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' They evaluated graph convolutional networks and graph attention networks and introduced several adaptations for specific scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' [2] employed a heterogeneous edge-enhanced graph attention network to handle the heterogeneity of TAs and TPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The GRIP [24] represents the input as a specific grid and uses an undirected graph to model interactions between agents within a certain range, where fixed graphs are considered in the graph convolution submodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The GRIP++ [5] improves the above-mentioned method by adopting trainable graphs, which overcomes the shortcoming that fixed graphs based on manually designed rules cannot model interaction properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition to the interaction modeling, another important requirement of trajectory prediction relates to time series processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Recently, recurrent neural networks (RNNs), including the long short-term memory (LSTM) and gated recurrent unit (GRU) models, have been widely used in modeling sequential problems, and significant results have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Accordingly, these models have been used as sub-modules in many trajectory prediction algorithms [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Neural networks have been demonstrated to be highly efficient in trajectory prediction for different classes of TPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Research on pedestrian intent modeling and motion prediction has been conducted for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The Social-LSTM [8] is a typical success case in early research in this field, which combines S-pooling and LSTM to predict the future trajectory of pedestrians in crowded scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The Social-GAN [25] uses generative adversarial networks (GANs), sequence-to- sequence models, and pooling mechanisms and employs the corresponding generators and recursive discriminators to predict pedestrians’ socially feasible future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' However, the GAN model training is difficult and may not converge and can lead to mode collapsing and dropping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Therefore, the Social-Ways uses the Info-GAN, which does not apply the mean square error loss (L2 loss) to force the generated samples to be close to real data but adds another item to consider mutual information, thus alleviating the above-mentioned problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Since vehicles have higher running speeds and need to obey more road constraints than pedestrians, predicting their future movements is a prerequisite for realizing safe and efficient autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' A number of studies have designed specialized networks for vehicle trajectory prediction [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' For instance, vehicle trajectory prediction in highway scenarios, which are relatively simple and where the motion pattern of a vehicle is relatively fixed, has received early attention [18, 24, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' With the collection of large-scale datasets [28, 29] and the development of autonomous driving in urban scenes, much research has been focused on motion prediction in complex urban environments [4, 30-32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' TrafficPredict [33] adopted a four-dimensional graph to model the interaction in the instance and category layers, thus realizing the heterogeneous traffic-agent trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The GRIP++ achieved joint trajectory prediction of all observed objects while considering multiple classes of TPs, thus greatly improving real-time prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' However, the above work focuses on the improvement in the dataset-level accuracy while ignoring the research on the sensitivity of the prediction algorithm to environmental factors, which is the focus of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Epistemic Uncertainty Estimation Due to the rapid development of neural networks and their application to trajectory prediction tasks, it has become increasingly important to estimate the network confidence in its prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' However, the original neural network cannot provide an estimation of its epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' To address this shortcoming, some studies have considered and quantified the epistemic uncertainty of neural networks [11, 34, 35], which represents an indicator that can express how confident the network is in its current prediction result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The main epistemic uncertainty estimation methods include the Bayesian neural network (BNN), single-pass uncertainty estimation, and ensemble-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The BNN quantifies the epistemic uncertainty of a neural network by introducing uncertainty into its parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The key challenge of these methods is to solve the posterior distribution of network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In the early research, variational inference (VI) [36], which uses a prespecified family of distributions [37, 38], was widely adopted as a method with a strong theoretical basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' However, with the rapid growth in the neural network structure complexity, VI has faced many challenges in terms of solving difficulty and computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' To address these limitations, the Monte Carlo (MC) dropout [39, 40] was proposed to approximate the results obtained by sampling, assuming that the network weights conformed to a Bernoulli distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' It has been theoretically demonstrated that the MC dropout has the ability to approximate epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In 3 single-pass uncertainty estimation, uncertainty is obtained through one forward propagation, which has obvious advantages in terms of computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The deep evidence theory is a representative method and has been widely used in classification [41] and regression [42] tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' However, these methods require that the original network output has a specific form, which limits their scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, these methods do not consider the uncertainty of network weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In view of that, some studies [43] positioned the uncertainty they extracted as distributional uncertainty, different from epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In deep ensemble-based methods, the training process is adjusted to obtain multiple different models, and epistemic uncertainty is estimated by synthesizing the prediction results of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Deep ensemble [44] is a simple and scalable uncertainty estimation method, which has attracted extensive attention due to its excellent performance in estimating epistemic uncertainty [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Currently, this method has become a mainstream paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Subsequently, to reduce the storage and computational costs of the practical application of deep ensemble, many improved methods have been proposed [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' For instance, the Batch-Ensemble [46] reduces training and testing costs by defining each weight matrix as the Hadamard product of the shared weights of all ensemble members and the rank-one matrix of each member, but the uncertainty estimation performance is slightly decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' However, previous studies on epistemic uncertainty have usually involved tasks such as semantic segmentation and object detection but have lacked detailed research in the field of trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' This work proposes a trajectory prediction method with epistemic uncertainty estimation, where deep ensemble and MC dropout are used separately to estimate epistemic uncertainty and compared on the real intersection dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Relationship between Prediction Performance and Traffic Environment Previous studies have mainly focused on enhancing the dataset-level accuracy of trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' However, the actual trajectory prediction performance can be strongly dependent on a traffic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Therefore, it is of great significance to determine the relationship between the environment and prediction model to improve the interpretability of trajectory prediction algorithms and determine their limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' This is essential for safety-critical autonomous driving applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Several works focused on modeling and complexity calculation of a traffic environment using different methods, such as five- and six-layer scene models [48, 49], where layer elements can have a strong correlation with the prediction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' [50] proposed a method to quantify scenario complexity in traffic but did not explore its relationship with the autonomous driving algorithm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The Shapley value is a feature attribution method that helps to measure the contribution of input variables to model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Makansi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' [51] proposed a variant of Shapley value and analyzed the problems that some of the existing trajectory prediction models consider only the past trajectory of a TA and are difficult to reason about interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, recent studies have gradually paid attention to the cross-dataset performance of AI algorithms in object detection and prediction applications [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Gesnouin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' [54] evaluated the impact of differences in pedestrian poses and detection box heights in different datasets on pedestrian crossing prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Gilles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' [10] compared the accuracy of vehicle trajectory prediction algorithms on several datasets containing mixed scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' However, there has still been a lack of comprehensive analysis of traffic environmental factors and their changes and quantitative research on their impact on prediction algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' n this work, the research scenario is the intersection, which is a typical and challenging urban scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Distributional shifts between different intersection datasets and their effect on trajectory prediction performance, considering both error and epistemic uncertainty, are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' PROPOSED METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Trajectory Prediction with Epistemic Uncertainty Estimation 1) Trajectory Prediction Trajectory prediction is a task that estimates a TA’s future position based on historical data on its state and context in a scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Particularly, at time 0 t \uf03d , the historical input state S of a TA (over previous time steps) is represented as follows: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 1 2 0 , , , , h h t t s s s \uf02d \uf02b \uf02b \uf0e9 \uf0f9 \uf03d \uf0eb \uf0fb S (1) where ( )t s is the state of a TA at t , and it is defined as ( ) ( ) ( ) , t t t s x y \uf0e9 \uf0f9 \uf03d \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, interactions between a TA and its surrounding environment are modeled based on scene context information C , which includes information on the states of TPs’ around the TA and environmental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' A trajectory prediction model f is trained on dataset .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Based on the input \uf05b \uf05d X = S,C , the trained prediction model outputs an estimate ˆY of the real future trajectory Y of the TA as follows: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029ˆ ˆ = , , , f f f \uf071 \uf03d \uf03d Y X X X (2) where \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 1 2 ˆ ˆ ˆ ˆ [ , , , ] ft s s s \uf03d Y , ft is the predicted horizon, and ˆ\uf071 represents the trained model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In this work, the GRIP++, which is an enhanced graph-based interaction-aware trajectory prediction method, is used as a base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' It uses both fixed and dynamic graphs to describe the relationship between different TPs, considering the effect of inter-agent interactions on a TA’s motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Furthermore, this method employs a GRU-based encoder-decoder architecture as a sub-module and allows joint trajectory predictions for multiple agents, achieving good performances in terms of prediction speed and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 2) Epistemic Uncertainty Estimation In the previous section, a neural network-based trajectory prediction model is presented, but the original GRIP++ can output only deterministic prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' However, real- world traffic scenarios are complex and variable, and it is difficult to construct a training set that will effectively cover all scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, deep learning-based models are inherently uncertain and difficult to interpret, so they may not ht 4 ht Graph Convolutional Model 64 ht n Trajectory Prediction Module Predicted Trajectories prediction error Random initialization, random shuffling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' epistemic uncertainty ADE FDE APE FPE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The trajectory prediction framework with epistemic uncertainty estimation (deep ensemble-based method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' be reliable enough when confronted with unknown scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=', scenarios unseen during training or scenarios with only limited available information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' These problems can result in unacceptable degradation in autonomous driving performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In this regard, the BNN models and learns the posterior distribution of network weights \uf028 \uf029 ˆ | P \uf071 \uf071 , which can be used to estimate the epistemic uncertainty as follows: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ˆ , | , P | , f f X \uf071 \uf071 \uf03d \uf03d\uf0f2 Y X Y (3) where the main issue is how to learn the posterior distribution of parameters effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The Bayesian approximate inference is a typical solution, which learns an approximate distribution \uf028 \uf029 q \uf071 of \uf028 \uf029 P | \uf071 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The MC dropout has been shown to be an effective sample-based method for approximate inference, where the network weights are assumed to follow the Bernoulli distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' After adding appropriate regularization during training and turning on dropout during testing, epistemic uncertainty can be estimated by sampling multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In recent years, deep ensemble, as a simple, parallelizable, and scalable method, has shown excellent uncertainty estimation ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In this work, a deep ensemble-based uncertainty estimation framework for the trajectory prediction model is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Specifically, random initialization of neural network parameters and random shuffling of a dataset are performed because they have been proven to have enough good performance in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' After training, K models of isomorphism and different parameters are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Further, by integrating the results of models, the final trajectory prediction output is obtained by, \uf028 \uf029 1 1 ˆ ˆ = , , K k k K f \uf071 \uf02d \uf03d\uf0e5 Y Y | X (4) where ˆ k\uf071 denotes the post-training parameters of the kth model among the ensemble models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' similarly, in the MC dropout- based method, ˆ k\uf071 indicates the model parameters for the kth dropout during testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The predictive entropy is employed to quantify the epistemic uncertainty of the proposed prediction model, where entropy increases with uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The proposed model outputs K continuous trajectories \uf028 \uf029 (1) (2) ] ˆ ˆ ˆ ˆ [ , , , ft k k k k s s s \uf03d Y in a prediction task, each of which contains the predicted position at multiple future moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' To realize a prediction-task-wise uncertainty estimation, the predictive entropy at multiple moments is integrated to obtain the average predictive entropy (APE) as follows: \uf028 \uf029 \uf028 \uf029 ( ) ( ) ( ) 1 1 1 1 ˆ ˆ ˆ ˆ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' PE ln d f f t t t t t t i i f f s p s p s s t t \uf03d \uf03d \uf0e9 \uf0f9 \uf03d \uf03d \uf02d \uf0eb \uf0fb \uf0e5 \uf0e5 \uf0f2 (5) Assuming that ( ) ( ) ( ) ˆ ˆ ˆ , t t t s x y \uf0e9 \uf0f9 \uf03d \uf0eb \uf0fb obeys the two-dimensional Gaussian distribution, ( ) ˆ t x and ( ) ˆ t y are independent of each other, and the APE can be expressed as follows: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 =1 2 ( ) 2 ( ) 1 1 1 ˆ APE = (ln2 1) ln 2 1 1 ˆ ˆ (ln2 1) ln .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 2 f f l t i f t t t i f t x x t \uf070 \uf070 \uf073 \uf073 \uf03d \uf02b \uf02b \uf053 \uf03d \uf02b \uf02b \uf0e5 \uf0e5 (6) Similarly, the final predictive entropy (FPE) is defined as, \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ( ) ( ) ( ) 2 2 1 ˆ ˆ FPE ln2 1 ln 2 1 ˆ ˆ ln2 1 ln .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 2 f f f f t t t t s x x \uf070 \uf070 \uf073 \uf073 \uf0e9 \uf0f9 \uf03d \uf03d \uf02b \uf02b \uf053 \uf0eb \uf0fb \uf03d \uf02b \uf02b (7) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Scenario Features Extraction In prediction scenarios, a TA’s motion is related to its historical state, interactions with surrounding TPs, and other factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Although the existing prediction algorithms have either explicitly or implicitly considered different factors, their performances may still be affected by the above-mentioned features due to algorithm limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Therefore, three types of scenario features are considered in this work: 1) dynamic features of a TA, which include data on its historical or future motion states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 2) features of surrounding TPs, which refer to their states and interactions with the TA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 3) other scenario features, which include the type, behavior pattern, compliance with traffic rules, and current location of the TA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 1) Kinematic features of TA The historical data on the TA motion state denote an important input to a prediction model and have a direct impact on the model output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, the future motion state of a TA is a key reference for evaluating the model’s prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Therefore, in this work, the kinematic features of TA are extracted to analyze their impact on the prediction algorithm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Velocity is one of the primary kinematic features, which directly reflects the aggressiveness of TA movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' It also represents the discrete degree of a continuous trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=" Considering the trajectory prediction model characteristics, three velocity sub-features are extracted: 1) average velocity of the historical trajectory (AVHT), which indicates the aggressiveness of the model's input trajectory;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 2) current velocity (CV), which directly represents a TA’s current state and has a key impact on the trajectory prediction output;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 3) average velocity of the future trajectory (AVFT), which reflects the spatial span of the future trajectory points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, the velocity variations indicate trajectory stationarity, which may have a significant influence on prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' For instance, a sudden start of a parked vehicle may be difficult for the model to predict timely and accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Therefore, the acceleration value at each moment is calculated to obtain the following sub-features: 1) average acceleration of the historical trajectory (AAHT), which K 5 represents the speed mutation degree of the input trajectory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=" 2) average acceleration of the future trajectory (AAFT), which reflects the overall situation of a TA's future speed mutation;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 3) maximum acceleration of the future trajectory (MAFT), considering that a sudden speed change at any moment can lead to severe deformation of the overall trajectory, it is necessary to extract the fastest speed change in the future as a feature for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Similarly, changes in the TA moving direction denote a potentially influential factor of prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' For instance, a vehicle going straight may suddenly swerve or make a U-turn, thus posing a serious challenge to the prediction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Therefore, the heading change speed (HCS) is extracted for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In detail, the absolute value of the change speed of the heading angle at each moment is calculated and used as a basic feature, and then the analysis of the following parameters is performed: 1) average HCS of the historical trajectory (AHCSHT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 2) average HCS of the future trajectory (AHCSFT), which reflects the overall curvature or volatility of the future trajectory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 3) maximum HCS of the future trajectory (MHCSFT), which increases when there is a sudden large change in direction at any point in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 2) Features of Surrounding TPs Convoluted interactions with other agents increase the difficulty in trajectory prediction, and although many of the existing prediction methods can explicitly or implicitly model interactions, it has not been fully discussed whether the performance of these black-box models is sensitive to actual interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' To examine this situation, a set of hierarchical prediction scenario complexity metrics is proposed to analyze the effect of a TA’s interactions with surrounding agents on the prediction algorithm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' First, with a TA as a center, the prediction scenario complexity has a positive correlation with the number of its surrounding TPs, and a basic feature, the number of TPs within x meters from the TA (NTPx), is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, the distance between the TA and its surrounding TPs directly affects the prediction scenario complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Assuming that a set of TPs within x meters of a TA i is denoted by \uf028 \uf029 x N i , then for any \uf028 \uf029 x j N i \uf0ce , its distance from i is \uf028 \uf029 , j j i dist s d s \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The density of TPs within x meters around TA (DTPx) is given by, \uf028 \uf029 1 DTPx j x N i d j e \uf06c \uf02d \uf03d \uf03d \uf0e5 ， (8) where \uf06c is a scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Furthermore, the potential conflicts due to the movement of surrounding TPs are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 3, for a TP \uf028 \uf029 x j N i \uf0ce within x meters around a TA i , its current state is given by \uf028 \uf029 \uf028 \uf029 0 0 [ , ] j j s v ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' then, its position after t seconds is expressed as \uf028 \uf029 \uf028 \uf029 t t j j s v t \uf02b , and the degree of conflict from TPs within x meters around TA (DCTPx) is defined as follows, \uf028 \uf029\uf028 \uf029 \uf028 \uf029 T ( ) ( ) agg 1 DCTPx X t t j N i d j e \uf06c \uf061 \uf02d \uf03d \uf03d \uf0e5 ， (9) where \uf028 \uf029 ( ) ( ) ( ) , t t t j i j d dist s s \uf03d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' T is the time horizon used for evaluation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' ( )t \uf061 is the scaling factor for the distance at time t, and in this study, it is set to grow faster over time to reinforce \uf028 \uf029 \uf028 \uf029 location , velocity , j j jx jy x y v v \uf0ec\uf0ef\uf0ed \uf0ef\uf0ee \uf028 \uf029 \uf028 \uf029 location , velocity , i i ix iy x y v v \uf0ec\uf0ef\uf0ed \uf0ef\uf0ee Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Schematic diagram of the conflict degree calculation from TPs within x meters from a TA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' (a) Agent type (d) Agent location Stage-1 Stage-2 Stage-3 Stage-4 Stage-5 Stage-6 gap (c) Compliance with traffic rules Red-light running Yellow-light running Obeying the rules For going straight and turning left vehicles (b) Behavior pattern U-turn Going straight Turning left Turning right Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Illustration of the other extracted scenario features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' the focus on the short-term risk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' \uf028 \uf029 T agg represents the aggregation operation, which is used to synthesize the conflicts at T times in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The two basic modes used in this study include the mean value (DCTPx_mean) and the maximum value (DCTPx_max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 3) Other Scenario Features In addition to the above two categories of features, the impact of several other representative features on the prediction error and epistemic uncertainty is studied, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The considered features include: \uf06c TA type: The GRIP++ can simultaneously predict future trajectories of multiple types of TAs, such as vehicles, pedestrians, and cyclists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Each type of agent has its movement pattern, which may cause different prediction performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Referring to the research presented in [33], TAs can be divided into four types: small vehicles, vehicles, pedestrians, motorcyclists, and bicyclists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' \uf06c TA behavior pattern: This study mainly focused on three basic behavior patterns of vehicles at intersections: going straight, turning left, and turning right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, this study extracts certain corner cases, such as U-turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The prediction errors and epistemic uncertainty under different behavioral patterns are compared and analyzed in a unified manner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=" 6 \uf06c TA's compliance with traffic rules: Traffic rules partially constrain the behaviors of participants, but in real-world scenarios, some of TAs may violate the rules, thus affecting the trajectory prediction performance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' For instance, in a signalized intersection, the behaviors of TAs can be classified based on their compliance with the traffic signal into obeying the rules, yellow-light running, and red-light running;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' \uf06c TA location: The whole process of a vehicle passing through the intersection is divided according to the time sequence into six stages: stage 1: ex-entering an intersection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' stage 2: in the gap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' stage 3: in the first crosswalk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' stage 4: inside an intersection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' stage 5: in the last crosswalk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' stage 6: exiting an intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Scenario Features Analysis To analyze the relationship between the above-mentioned features systematically, the qualitative and quantitative analysis methods are adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The main methods include feature correlation analysis and feature importance analysis based on random forest regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 1) Feature Correlation Analysis Correlation analysis is to calculate the degree of correlation between two or more feature variables using correlation coefficients as quantitative indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Typical correlation coefficients include the Pearson correlation coefficient and Spearman rank correlation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The Pearson correlation coefficient requires evaluated variables to conform to the normal distribution, but this is a strong assumption that experimental results can hardly satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In contrast, the Spearman rank correlation coefficient does not have such strict requirements on data as the Pearson correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Namely, it requires only that observed values of the two variables are paired rank data or rank data transformed from continuous variable observation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The Spearman rank coefficient can be mainly used in the monotonic relationship evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Specifically, it is assumed that the original data ( ix , iy ) are arranged in ascending order, and \uf028 \uf029 i R x and \uf028 \uf029 i R y are defined as the ranking of ix and iy in their corresponding data, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' ( ) R x and ( ) R y denote the mean of the ranking of the two groups of data, and n is the number of data pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Then, the Spearman rank correlation coefficient can be defined as follows: \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 1 2 2 1 1 2 1 2 ( ) ( ) ( ) ( ) 6 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 1 n i i i n n i i i i n i i i R x R x R y R y R x R x R y R y R x R y n n \uf072 \uf03d \uf03d \uf03d \uf02d \uf02d \uf02d \uf03d \uf02d \uf0d7 \uf02d \uf02d \uf03d \uf02d \uf02d \uf0e5 \uf0e5 \uf0e5 \uf0e5 (10) 2) Feature Importance Analysis Based on Random Forest Regression Feature correlation analysis can be used to assess linear and ordinal consistent correlations but cannot identify other types of correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' To address this limitation, this work proposes a feature importance analysis method to evaluate the impacts of different scenario features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The decision tree is a non-parametric supervised learning algorithm that can be used in solving both classification and regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' It is a hierarchical tree structure mainly composed of three types of nodes, root, internal, and leaf nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Ensemble learning uses multiple models to obtain accurate prediction results, and bootstrapping is one of its typical application techniques that refers to the process of randomly sampling of a sub-dataset through a given number of iterations and variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Random forest regression combines ensemble learning with the decision tree framework, creating multiple decision trees from data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' then, multiple outputs are averaged to obtain the final result, often achieving excellent performance for regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Random forest regression can be used to evaluate feature importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Particularly, in this study, the extracted scenario features are regarded as independent variables, while the prediction error and epistemic uncertainty of the prediction model under the corresponding conditions are regarded as dependent variables, and a random forest regression model is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Then, the contribution of each feature to the trees in the random forest is analyzed, as well as its importance to the performance of trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The variable importance measure is denoted as VIM, and it is assumed that there are J features and I decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=" Then, VIM j denotes the average change in node split impurity of the jth feature in all decision trees, and it is calculated by: \uf028 \uf029 \uf028 \uf029 1 ' ' 1 1 VIM VIM = , VIM I i j i j J I i j j i \uf03d \uf03d \uf03d \uf0e5 \uf0e5\uf0e5 (11) where \uf028 \uf029 VIM i j represents the importance of the jth feature in the ith decision tree, and it can be obtained by calculating the difference of Gini indices of nodes before and after branching." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Prediction across Different Intersection Datasets The cross-dataset analysis aims to analyze differences in scenarios between multiple datasets and the corresponding prediction algorithm performance disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In this study, the scenario type is limited to the intersection, and multiple intersection datasets involving various countries are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' First, the scenario features are extracted to analyze distributional shifts between different intersection datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Next, comprehensive cross-validation experiments are performed to investigate the prediction algorithm performance in terms of distributional shifts fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Particularly, N intersection datasets are selected, and each of them is divided into training and test subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Then, for each of the training subsets, a trajectory prediction model is developed and trained using the training subset first and then evaluated on the corresponding test subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Finally, 2 N sets of results are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' During the analysis, the following factors are mainly studied: 1) Trajectory prediction performance degradation due to distributional shifts: Combined with the differences in statistical features between different datasets obtained earlier, we analyze the impact of changes in the traffic environment on the prediction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' \uf072 7 2) Effect of the deep ensemble on prediction robustness across different datasets: The improvement in prediction accuracy and sensitivity of the estimated epistemic uncertainty to distributional shifts are analyzed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 3) Availability and complexity of different intersection datasets for trajectory prediction: By synthesizing multiple groups of results, the model performance is evaluated using different intersection datasets as a training set and the prediction challenge with different intersection datasets as a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' EXPERIMENTAL SETUP A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Intersection Datasets Focusing on the urban intersection scenario, multiple trajectory datasets were used for evaluation and analysis, involving different periods, weather, countries and regions, and many TP types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 1) SinD [55]: The SinD dataset is a typical drone dataset collected from a signalized intersection in Tianjin, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' These data were recorded from a static bird’s eye view at a sampling frequency of 10 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' This dataset contains about 420 minutes of traffic recordings, including over 13,000 TPs with seven types, including cars, trucks, buses, pedestrians, tricycles, bikes, and motorcycles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 2) INTERACTION [29]: The INTERACTION dataset contains 12 subsets covering merging, roundabout, and intersection scenarios, of which five intersection subsets are used in this study: USA_Intersection_EP1 (EP1), USA_Intersection_EP2 (EP2), USA_Intersection_MA (MA), USA_Intersection_GL (GL) and TC_Intersection_VA (VA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' They contain about 493 minutes of recordings in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The first four relate to unsignalized intersections in the US, mainly involving trajectories of vehicles, pedestrians, and bicycles recorded by drones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The VA denotes a signalized intersection in Bulgaria, which involves trajectories of cars, buses, trucks, motorcycles, and bicycles recorded by traffic cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Prediction Error Metrics The prediction error is a preferred metric for quantifying the performance of prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Following [2, 8, 25], the proposed model was evaluated using two error metrics: 1) Average Displacement Error (ADE): This is the mean square error of all predicted points of a trajectory compared to the ground truth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 2) Final Displacement Error (FDE): This is the distance between the predicted final destination and the true final destination at .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Predictive Uncertainty Evaluation As stated previously, the APE and FPE reflect the epistemic uncertainty of a prediction model, which can indicate situations where prediction models are performing poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Therefore, referring to [56, 57], this study uses the error-retention curves to evaluate the ability of the extracted epistemic uncertainty to detect prediction errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The curves depict the error over a dataset as a model’s predictions are replaced by ground-truth labels in order of decreasing uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The abscissa value of a point represents the proportion of the retained true error (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=', retention fraction), while the ordinate value represents the comprehensive error under this proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Similarly, the optimal and random curves are obtained by replacing the predictions in order of decreasing error and random order, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The area under the retention curves (AUC) is an evaluation metric of both the robustness of prediction models and the quality of uncertainty estimation, and an efficient uncertainty estimation is considered to achieve a low AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Implementation Details For achieving fair evaluation and transferability, the datasets were standardized to use up to 3 s of the previous data and 3 s of the future data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The trajectory data were resampled to 2 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The single GRIP++ model in the ensemble models was implemented in PyTorch, and its implementation details, including input preprocessing, graph convolution, and trajectory prediction model, mainly refer to the settings in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In the implementation of MC dropout and deep ensemble, the value of k was set to five, which was the result of a trade-off between uncertainty estimation quality and computational cost [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, during the training process of the MC dropout- model, a regularization term was added to the loss to improve its ability of uncertainty estimation [39], where the regularization coefficient was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='0001, and the dropout rate was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='5 In addition, the original dataset was further processed to obtain the labels for the subsequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The locations of TAs were classified by combining their raw coordinate data with the map in the Lanelet2 format [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Evaluation of Trajectory Prediction and Epistemic Uncertainty Estimation The training and test performances of the proposed model obtained by following the train-test process in the same intersection dataset are presented in TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Although the MC dropout can be used to estimate epistemic uncertainty, it increases the prediction error, which might be due to the modifications in the original loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In contrast, the deep ensemble-based method improves trajectory prediction accuracy while estimating epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' As shown in TABLE I, the error obtained by the deep ensemble-based method was lower than that of the single models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Thus, by integrating the results of multiple models, deep ensemble could effectively avoid the prediction performance degradation caused by the deviation of a single model, which is conducive to improving the prediction algorithm robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The evaluation results of the estimated epistemic uncertainty are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Compared with the MC dropout, the deep ensemble had obvious advantages in improving the model prediction accuracy and uncertainty estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Therefore, in the subsequent analysis, the epistemic uncertainty estimation framework based on the deep ensemble was adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 5, both uncertainty quantification metrics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=', APE and FPE) could accurately reflect the prediction model error (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=', ADE or FDE), and they showed a high degree of consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' By comparing the left and right sides of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 5, it can be concluded that although the prediction error on the test sets was slightly increased compared to that on the training set, there was a small difference in the epistemic uncertainty ft 1 TABLE I TRAJECTORY PREDICTION ERROR COMPARISON1 Dataset ADE1/FDE1 ADE1/FDE1 ADE1/FDE1 ADE1/FDE1 ADE1/FDE1 ADEdropout/FDEdropout ADE/FDE SinD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='405/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='404/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='402/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='403/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='861 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='627/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='709/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='591/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='386 (a) ADE for training set (b) FDE for training set (c) APE for test set (d) FPE for test set Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The ADE/FDE-retention curves (top) and retention scores curves (bottom) on the training and test sets of the SinD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=" The optimal curve (solid green line) was obtained by replacing the model's predictions with the ground-truth labels in order of decreasing error." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Similarly, the random curve (blue dotted line) was obtained by replacing the model’s predictions with the ground- truth labels in random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The red and yellow solid lines correspond to the results captured in order of decreasing APE and decreasing FPE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The retention scores corresponding to each retention fraction can be calculated by: (errorrandom – erroruncertainty)/(errorrandom – erroroptimal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' estimation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The results indicate that the deep ensemble-based method had good generalization ability In the second row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 5, a consistent trend where the retention scores first increase and then decrease with the retention fraction can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Scenario Features Analysis Results 1) Feature Correlation Comparison In general, it has been considered that scenario complexity increases with the values of the extracted TA’s kinematic features and the features of the TA’s surrounding TPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Therefore, the correlation between the model performance and these two types of features was calculated to analyze the influence of scenario complexity on trajectory prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Based on the model trained on the SinD training set, the feature correlation analysis experiment was performed on the SinD test set, where the prediction performance was 1 ADEk/FDEk represents the prediction error of model k among the ensemble models, ADEdropout/FDEdropout is the prediction error of the mc dropout-based method, and ADE/FDE denotes the prediction error of the deep ensemble-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' represented by prediction error and epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' For the features of the TA’s surrounding TPs, four groups of distances of x = 10, 20, 30, 50 were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The analysis results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 6, and based on them, the following conclusions can be drawn: \uf06c The distribution trends of environmental feature correlations corresponding to the two types of prediction errors (ADE and FDE) were highly consistent, as well as the distribution trends of environmental feature correlations corresponding to the two types of epistemic uncertainty (APE and FPE) estimates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' \uf06c The comparison of a TA’s kinematic features with the features of its surrounding TPs shows that the former had a strong positive correlation with the error and epistemic uncertainty of the prediction model, while the latter had weak correlations with the error and epistemic uncertainty (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='2 < \uf072 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Comparison of the correlation between prediction model performance and scenario features \uf06c The comparison of the correlation between different kinematic features and the prediction error shows that: \uf06e The sorting order in terms of the correlation was: acceleration-related features > velocity-related features > heading change speed-related features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' This order indicates that the mutation of a TA’s speed had a relatively large impact on the prediction error, while the change in the TA’s movement direction had a relatively low impact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' \uf06e The sorting order in terms of the correlation was: features related to future trajectories > features related to historical trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' This shows that the proposed trajectory prediction model had low adaptability to the speed and position mutations occurring at some certain points in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' \uf06c The correlation between different kinematic sub-features of a TA and the predictive uncertainty showed that the epistemic uncertainty was highly sensitive to the velocity and acceleration features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' namely, when a TA was driving at high speed or had a speed mutation, the model tended to show lower confidence in the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 2) Feature Importance Comparison As mentioned above, the feature correlation analysis only shows whether the relationship between two variables conforms to the order consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Therefore, the feature importance analysis experiment was performed based on the random forest regression to explore whether there were other types of correlation between the above scenario features and the prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The datasets and training settings used for the prediction algorithm were the same as in the feature correlation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The grid-based search was employed to obtain optimal random forest regression model, then the feature importance analysis was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 7, the distribution trend of feature importance was basically consistent with the feature correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' For instance, the features obtained from the surrounding TPs had little effect on the error and uncertainty of the prediction model uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In contrast, the kinematic features of a TA had a stronger influence, where velocity- and acceleration- related features had higher importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In random forest regression for the ADE, the AAFT had the highest importance, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Comparison of feature importance based on the random forest regression while in the random forest regression for the APE, the CV had the strongest influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 3) Other Scenario Features Analysis In addition to the above-mentioned features, the impacts of several other environmental features on the prediction algorithm performance were explored, including the type, behavior patterns, compliance with traffic rules, and location of a TA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Without loss of generality, the ADE was adopted as an error metric, and the APE was used for epistemic uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The prediction model was trained on the SinD training set and analyzed on the SinD test set The results indicated obvious differences in the prediction error distribution between different TAs, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Although the movement of pedestrians had high degrees of freedom and randomness, their speed and acceleration were generally low, so the corresponding prediction error was small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Meanwhile, the epistemic uncertainty distribution for different types of TAs showed high similarity with that of the prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The results of the trajectory prediction error and epistemic uncertainty of a vehicle under different behavior patterns are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 9, where it can be seen that the trajectory prediction performance tended to exhibit larger errors when the TA was turning left and right compared to the going-straight behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The error in the right-turn scenario was relatively (a) ρ for ADE (b) ρ for FDE (a) ρ for APE (b) ρ for FPE (a) feature importance for ADE (b) feature importance for APE 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The results of the trajectory prediction error and epistemic uncertainty of different TA types;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' sv: small vehicle, bv: large vehicle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' bi: motorcyclist or bicyclist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' pe: pedestrian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The results of the trajectory prediction error and epistemic uncertainty under different behavioral patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The results of the trajectory prediction error and epistemic uncertainty under different traffic rule compliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The results of the trajectory prediction error and epistemic uncertainty at different locations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 1: ex-entering the intersection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 2: in the gap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 3: in the first crosswalk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 4: inside the intersection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 5: in the last crosswalk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' and 6: exiting the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' large, and the reasons may be as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' First, the right-turn trajectory had a large curvature, and second, the vehicle was less affected by traffic lights and other TPs when turning right, compared to turning left and going straight, resulting in a higher driving speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, although the U-Turn represents a typical corner case, the results showed that the overall error in this pattern was small, which may be related to the generally low speed during the U-turning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Furthermore, the epistemic uncertainty distributions under different behavioral patterns were relatively consistent with the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The relationship between the vehicles’ compliance with traffic lights and the trajectory prediction performance is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 10, where it can be seen that the prediction error was larger when the vehicle ran red or yellow light than when there was no violation of traffic lights, and simultaneously the proposed model could output higher epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The impact of a TA’s location on the trajectory prediction performance is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' When the vehicle was in the gap area or first crosswalk before entering the intersection, there were many possible strategic options, referring to whether to enter the intersection and how to pass through the intersection, which increased the prediction complexity, and further increased the prediction error and epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' When the vehicle was inside the intersection, the prediction model exhibited considerable error and uncertainty due to a large number of interactions with other TPs and higher freedom of movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In contrast, the predominant behavior of the vehicles before entering and after exiting the intersection was to follow the lane, so these stages had lower prediction error and uncertainty than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Prediction evaluation across Different Intersection Datasets Different intersection datasets were collected at different times and locations, and the corresponding environmental conditions might be relatively different, resulting in shifts in data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The distributions of velocity, acceleration, heading, and HCS of objects in six intersection datasets are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 12, where it can be seen that there were obvious differences in the trajectory features between the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' For instance, the velocity and acceleration of a portion of trajectories in the SinD dataset were concentrated around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' One of the main reasons was the stopping of vehicles and pedestrians while waiting for the green light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Furthermore, the velocity in the SinD dataset exhibited a distinct multimodal distribution, which could be related to the multiple movement patterns caused by various TPs in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In contrast, the velocity and acceleration in the GL, MA, and VA datasets tended to be higher, reflecting more aggressive motions in these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, distributional shifts could also be observed by comparing the distribution of heading and its speed of change in different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The aforementioned differences in data distribution between datasets may bring application challenges of the trajectory prediction algorithms in real-world environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Therefore, experiments were performed on multiple intersection datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Specifically, based on the six intersection datasets mentioned above, a set of deep ensemble-based prediction models were trained on each dataset and evaluated on all six datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 13, distributional shifts in the real-traffic 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Distribution comparison of different intersection datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' (a) ADE1 (b) ADE (c) APE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The cross-dataset error and uncertainty matrix: (a) ADE1 is the error obtained by evaluating one of the ensemble models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' (b) ADE is the prediction error of the deep ensemble-based model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' (c) APE also relates to the deep ensemble-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The ith row and jth column of each matrix represent the results obtained by evaluating the model on the test subset at intersection j after training it on the training subset at intersection i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Comprehensive performances of the prediction algorithms on different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' (a) Comparison of prediction errors of all trained models on different test subsets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' (b) results of evaluating models trained on different training subsets on all test subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' environments had a strong impact on trajectory prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Even for the same type of scenario, for the model trained on one intersection dataset, it was difficult to generalize well to other intersection datasets directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' For instance, on the test subset of the SinD dataset, the model that achieved the best accuracy (ADE1/ADE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='405/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='389) was trained on the training subset of the SinD dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' in contrast, the prediction error (ADE1/ADE) of the model trained on the other intersection training subsets increased by 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='9%/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='61% - 265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='0%/221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Comparing the single prediction model error (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 13(a)) with the error of the deep ensemble-based model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 13(b)), it could be concluded that the deep ensemble-based approach performed systematically better than a single model, achieving significant improvements in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' As presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 13(c), the extracted epistemic uncertainty could indicate distributional shifts between different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Particularly, the proposed model could output higher epistemic uncertainty when the error increased due to changes in the test scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 14, the comprehensive performances of the trajectory prediction algorithm on specific single training or test set are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 14(a), the models trained on the training subsets of the SinD and GL datasets had better generalization ability than the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' This could be because of a larger amount of data and more diverse motion patterns of these two datasets compared to the other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Conversely, the overall error of the model based on the training subset of VA was the highest, and the main reasons were as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' First, this dataset contained less data than the other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Second, this dataset was constructed by data collected by roadside equipment, which might introduce more noise than drone-based data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The comprehensive performances of the prediction models on different intersection test subsets are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 14(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The GL and MA with higher velocity and acceleration features showed greater prediction difficulty, and the overall trajectory prediction error on the SinD test subset was the smallest among all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Qualitative Results The prediction results of the proposed framework in several scenarios on different intersection datasets are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 15, where each row corresponds to one intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 15, isolg 0 2 or J2 50 52 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content="0 0'T0 0'T2 AM 0'52 EbJ Ebo 0E." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='0 AV 2E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='0 1 2UD 926b2HC 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content="0 0'52 0'20 0'>2 J'00 J'52 J'32 5'00 0 AM 3 Eb Ebo AV 2!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='UD J926J6bnoit19l96 0 e 8 JO 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content="0 0'52 0'20 AM J'S2 EbJ Ebo J'20 AV 2IUD J'>2 926b-5 0 A 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='0 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content="0 0'4 D6U aer AM EbJ 8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content="0 Ebo AV J'O 2!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='UD J926J6b1 SinD VA EP0 EP1 MA GL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Qualitative results of trajectory prediction and uncertainty estimation on different datasets (corresponding to different rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The gray thick solid line represents the historical trajectory, and the black thin solid line represents the true future trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Different colors are used to denote predictions for different types of TAs: blue - small vehicle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' magenta - large vehicle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' yellow – pedestrian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' green - motorcycle or cyclist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, the focused object in each subgraph is highlighted in red, and the current prediction error and uncertainty estimation results of the object are presented above the subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' nipnonipno1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='.1 the first column shows cases where the prediction error was small and epistemic uncertainty was low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' These cases generally appeared when the TA exhibited obvious future intentions and moved relatively smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The second and third columns show scenarios with large trajectory prediction errors, which were generally accompanied by higher estimates of epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' This situation may occur when the TA was about to enter the intersection where it was difficult to determine its future behavior pattern, or its future motion showed a large pattern or speed change compared to the historical trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' CONCLUSION In this paper, a trajectory prediction framework that integrates the epistemic uncertainty estimation function is proposed, and the effects of the traffic environment and its changes on the prediction algorithm performance are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' A few typical scenario features are considered, and their influences on the prediction performance are examined by the feature correlation and importance analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Further, the distributional shifts between different intersection datasets and the resulting performance degradation of the prediction model are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=" Based on the obtained results, the following conclusions are drawn: (1) The extracted epistemic uncertainty is valuable for representing the model's confidence in its current predictions accurately, which has the potential to be used to identify unknown scenarios where the model may be underpowered." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Compared with the MC dropout-based method, the deep ensemble-based method performs significantly better in estimating epistemic uncertainty and improving the trajectory prediction robustness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' (2) Regarding the influence of different scenario features on the trajectory prediction performance, the feature correlation and importance analyses show similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Namely, there is a positive correlation between the kinematics of a TA and the prediction model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Higher velocity, acceleration, and speed of the heading change generally pose a greater challenge to the trajectory prediction process, while a prediction model tends to exhibit higher epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' However, one interesting finding is that the features of surrounding TPs, which reflect the complexity of interactions in the scenario, show little impact on the proposed prediction model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' In addition, the error and uncertainty of the prediction model vary with other abstract features, such as TA’s type, behavior pattern, compliance with traffic rules, and location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' The conducted analyses are helpful in locating the limitations in the prediction algorithms, thus providing guidance for the improvement of autonomous driving functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' (3) For the intersection scenario, different datasets show distributional shifts due to differences in local driving habits, road structures, and national cultures, thus posing great challenges to the prediction algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Fortunately, the proposed framework based on the deep ensemble is beneficial to improving the trajectory prediction robustness, and the extracted epistemic uncertainty can respond to the reduced confidence of the proposed model in a new environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' This improves the self-awareness ability of autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Although the used basic prediction algorithm has strong representativeness, it is still difficult to avoid the specificity of analysis conclusions completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' However, the proposed method is promising to analyze other prediction algorithms, subsequent work could consider combining more types of algorithms for systematic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Moreover, this work focuses on the extraction and analysis of epistemic uncertainty, but the existing works in the field of multimodal trajectory forecasting could be considered in the follow-up research to distinguish the epistemic uncertainty from the aleatoric uncertainty better and explore their practical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Furthermore, the role of uncertainty estimation of a trajectory prediction model in an autonomous driving decision-making mechanism needs to be studied further to improve the robustness against the risks of insufficient functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' ACKNOWLEDGMENT This work was supported in part by the National Science Foundation of China Project (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' 52072215 and 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='1109/ITSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='8569929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Wenbo Shao received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' degree in vehicle engineering from Tsinghua University, Beijing, China, in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' He is currently working toward the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' degree in Mechanical Engineering at Tsinghua University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' He is a member of Tsinghua Intelligent Vehicle Design And Safety Research Institute (IVDAS) and supervised by Professor Jun Li and Associate Research Professor Hong Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' His research interests include safety of the intended functionality of autonomous driving, trajectory prediction, decision-making, uncertainty theory and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Yanchao Xu received B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='E degree in vehicle engineering from Hainan University in China in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' He is currently pursuing the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' degree in Mechanical Engineering at Beijing Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' He is also one of the visiting students at IVDAS since 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' His research interest includes prediction, trajectory data mining, scenario parameterization for autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Jun Li received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' degree in vehicle engineering from Jilin University, Changchun, Jilin, China, in 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' He is currently an academician of the Chinese Academy of Engineering, a Professor at school of Vehicle and Mobility with Tsinghua University, president of the Society of Automotive Engineers of China, director of the expert committee of China Industry Innovation Alliance for the Intelligent and Connected Vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' His research interests include internal combustion engine, electric drive systems, electric vehicles, intelligent vehicles and connected vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Chen Lv (Senior Member, IEEE) received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' degree from the Department of Automotive Engineering, Tsinghua University, China, in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' From 2014 to 2015, he was a Joint Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Researcher at the EECS Department, University of California at Berkeley, Berkeley, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' From 2016 to 2018, he worked as a Research Fellow at the Advanced Vehicle Engineering Center, Cranfield University, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' He is currently an Assistant Professor with the School of Mechanical and Aerospace Engineering, and the Cluster Director of future mobility solutions at ERI@N, Nanyang Technological University, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' His research interests include advanced vehicles and human–machine systems, where he has contributed over 100 articles and obtained 12 granted patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Weida Wang received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' degree from Beihang University, Beijing, China, in 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' He is currently a Professor with the School of Mechanical Engineering, Beijing Institute of Technology, Beijing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' He is also the Director of the Research Institute of Special Vehicle, Beijing Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' His current research interests include electric vehicle, automated vehicle motion planning and control, and electromechanical transmission control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Hong Wang is Research Associate Professor at Tsinghua University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' She received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' degree in Beijing Institute of Technology, Beijing, China, in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' From the year 2015 to 2019, she was working as a Research Associate of Mechanical and Mechatronics Engineering with the University of Waterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Her research focuses on the safety of the on-board AI algorithm, the safe decision-making for intelligent vehicles, and the test and evaluation of SOTIF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' She becomes the IEEE member since the year 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' She has published over 60 papers on top international journals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} +page_content=' Her domestic and foreign academic part- time includes the associate editor for IEEE Transactions on Vehicular Technology and Intelligent Vehicles Symposium, Guest Editor of Special Issues on Intelligent Safety of Automotive Innovation, Young Communication Expert of Engineering, lead Guest Editor of Special Issues on Intelligent Safety of IEEE Intelligent Transportation Systems Magazine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE3T4oBgHgl3EQfQgnL/content/2301.04414v1.pdf'} diff --git a/5dE4T4oBgHgl3EQfbwyA/content/tmp_files/2301.05077v1.pdf.txt b/5dE4T4oBgHgl3EQfbwyA/content/tmp_files/2301.05077v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..444a68ee3aac0a42529a94009ad8155c786de5e6 --- /dev/null +++ b/5dE4T4oBgHgl3EQfbwyA/content/tmp_files/2301.05077v1.pdf.txt @@ -0,0 +1,2127 @@ +Incorporating time-dependent demand patterns in the +optimal location of capacitated charging stations +Carlo Filippi(1) +Gianfranco Guastaroba(1) +Lorenzo Peirano(1) +M. Grazia Speranza(1) +(1) University of Brescia, Department of Economics and Management, Brescia, Italy +{carlo.filippi, gianfranco.guastaroba, lorenzo.peirano, grazia.speranza}@unibs.it +January 13, 2023 +Abstract +A massive use of electric vehicles is nowadays considered to be a key element of a sustainable +transportation policy and the availability of charging stations is a crucial issue for their extensive +use. Charging stations in an urban area have to be deployed in such a way that they can satisfy a +demand that may dramatically vary in space and time. In this paper we present an optimization +model for the location of charging stations that takes into account the main specific features +of the problem, in particular the different charging technologies, and their associated service +time, and the fact that the demand depends on space and time. To measure the importance +of incorporating the time dependence in an optimization model, we also present a simpler +model that extends a classical location model and does not include the temporal dimension. A +worst-case analysis and extensive computational experiments show that ignoring the temporal +dimension of the problem may lead to a substantial amount of unsatisfied demand. +Keywords: Facility location, Charging stations, Electric vehicles, Demand patterns, Time-dependent +optimization. +1 +Introduction +Sustainable transportation is one of the major challenges that modern countries are facing. Several +sources indicate that the transportation sector generates the largest share of GreenHouse Gas +(GHG) emissions. According to the United States Environmental Protection Agency1, in 2020 the +transportation sector produced 27% of the total GHG emissions in the US, mostly generated from +burning fossil fuels by cars, trucks, ships, trains, and planes. Domestic statistics issued by the UK +government2 confirm that the transportation sector generated 27% of the total GHG emission. The +majority (91%) came from road transport vehicles, where the biggest contributors were cars and +taxis. Furthermore, data provided by the European Environment Agency3 highlight that in the +EU more than 22% of the GHG emissions came from the transportation sector. +1https://www.epa.gov/ghgemissions/sources-greenhouse-gas-emissions +2https://www.gov.uk/government/statistics/transport-and-environment-statistics-autumn-2021/tran +sport-and-environment-statistics-autumn-2021 +3https://www.eea.europa.eu/data-and-maps/data/data-viewers/eea-greenhouse-gas-projections-data- +viewer +arXiv:2301.05077v1 [math.OC] 12 Jan 2023 + +Despite technical advances have made available a range of options for sustainable mobility, there +are still important obstacles that must be overcome for their mass adoption. Among such options, +Electric Vehicles (EVs) are considered one of the major directions to reduce the environmental +impact of people mobility and make urban areas more sustainable. +In the 2021 edition of the +Global EV Outlook 20214, the International Energy Agency pointed out that at the end of 2020 +the global EVs stock hit 10 millions units, with 3 millions newly registered EVs. Europe was the +fastest growing market, with a sales share equal to 10% and some leading countries, such as Norway, +which registered a record high sales share of 75%. This trend was accelerated by many countries +of the European Union through substantial financial incentives. However, the decision of potential +EV buyers is still strongly affected by two major issues. On one hand, the purchase cost of an EV +is still higher than that of a traditional internal combustion engine vehicle. On the other hand, +the limited travel range of an EV and the long charging time are well-known to generate anxiety +in the potential buyers (e.g., Pevec et al., 2020). In fact, the willingness of drivers to purchase an +EV strongly depends on the availability of charging stations nearby their points of interests (e.g., +home and work). As the number of charging stations is growing, thanks to public and private +investments, the location problem of such stations has attracted much attention (see Section 2). +There are a number of factors that make the location of charging stations substantially different +from other, more classical, location problems, in particular the choice of the charger to install (e.g., +slow, quick, fast), and the characteristics of the charging demand. +The type of charger is a key factor to be taken into account, as it impacts the charging time. As of +the end of 2021, there exist three main types of charger (see Moloughney, 2021). Level 1 chargers, +also referred to as slow chargers, use common 120-volt outlets, and can take up to 40 hours to +raise the level of a standard battery EV (with a 60 kWh sized battery) from 10% to 80% of the +capacity. These chargers are most suitable for private usage. Level 2 chargers, sometimes called +quick chargers, can charge up to 10 times faster than a level 1 charger, and are the most commonly +used types for daily EV charging (see Moloughney, 2021). Given the same battery characteristics +mentioned above, the charging time is about 4.5 hours. The level 3 or fast chargers can reduce the +charging time to 40 minutes or even less. For a comprehensive study regarding the state of the art +on charging stations, the interested reader can refer to Pareek et al. (2020). The type of charger +demanded by EVs is affected by the urban layout. For example, slow chargers will be demanded +in residential areas so that EVs can be recharged over the night at low cost (an interesting study +of the factors influencing the charging demand is provided in Wolbertus et al., 2018). +In the classical location models a customer is characterized by the distance from any potential +location and by a single quantity - a measure of the demand. +The models do not consider a +temporal dimension of the problem which basically corresponds to assuming that the demand is +uniformly distributed over the time period of interest of the location decision. On the contrary, the +charging demand of EVs fluctuates over time, with peaks of demand in periods of time where the +traffic volume is high. Neglecting the demand dynamics may lead to solutions where the charging +capacity deployed is not sufficient to satisfy the demand during the peak times. +In this paper, we study the problem of determining an optimal deployment of charging stations for +EVs within an urban environment. Different types of chargers have to be located in pre-defined +potential locations, modeled as nodes of a network. The urban area is partitioned in sections. A +customer is associated with each section of the urban area. Its demand in a certain time interval +is the number of EVs in that section that need to be recharged. The customer is located in the +4https://www.iea.org/reports/global-ev-outlook-2021 +2 + +center of gravity of the section and is modeled as a node of the network. The urban area is also +partitioned in zones (e.g., commercial, industrial, or residential) which have different needs in terms +of minimum number of each type of charger deployed in the zone. +We have to determine, for each type of charger and each potential location, the number of chargers +to be deployed. Two criteria have a key role in this location problem: the cost of installing the +chargers and the distance the customers have to travel to be recharged. +We present, over a discretized time horizon, an optimization model that introduces a temporal +dimension which, to the best of our knowledge, has never been introduced in the literature on +location problems and captures the dynamics of the charging demand. Assuming that a charger can +take more than one period to fully recharge an EV, the proposed multi-period formulation includes +constraints to keep track of the usage of chargers across consecutive time periods and to ensure that +no other vehicles are assigned to any occupied charger. This novel approach guarantees a correct +sizing of the solution, in terms of number of stations opened and number of chargers installed, and +ensures that the demand is completely satisfied in all time periods. In order to assess the value of +introducing the temporal dimension in the location problem, which makes the optimization model +more complex, we present a single-period optimization model that captures the same specificities of +the problem but ignores the temporal aspect. In both models, the objective is the minimization of a +convex combination of two terms: the total cost of deploying the charging stations and installing the +chargers, and the average distance traveled by the customers to reach the assigned charging station. +The two optimization models turn out to be Mixed Integer Linear Programming (MILP) problems. +We compare the two models through a theoretical and a computational analysis. We show, through +worst-case analysis, that a solution to the single-period model may fail to satisfy a large portion of +the charging demand. Extensive computational experiments are run on different classes of randomly +generated instances. The results confirm the importance of explicitly considering the dependence +on time of the demand. +In fact, the single-period model is based on the common assumption +that the charging demand is uniformly distributed across the planning horizon. In an application +context such as the one at hand, where the demand fluctuates significantly during the day and +across different zones of the same urban area, the single-period model produces solutions that are +not capable of serving a large portion of the charging demand, especially in those time periods +where the demand is prominently concentrated. The computational experiments also include a +parametric analysis of the relative weight assigned to the objective function components. +Structure of the paper. +The remainder of the paper is organized as follows. +In Section 2, +the literature most closely related to our research is reviewed and the contribution of this paper +is highlighted. +In Section 3, after the presentation of the single-period extension of a classical +location model, we provide the multi-period mathematical formulation. In Section 4, we analyze +the worst-case performance of the single-period model in terms of portion of unsatisfied charging +demand. Section 5 reports extensive computational experiments conducted on instances gener- +ated to resemble demand dynamics frequently observed in different zones of a city. Finally, some +concluding remarks are outlined in Section 6. +2 +Literature review +The problem of determining an optimal location and size of charging stations for EVs has recently +attracted an increasing academic attention. Recent overviews of the main modeling and algorithmic +approaches employed in this research area are available in Deb et al. (2018), Zhang et al. (2019), and +3 + +Kchaou-Boujelben (2021). For a general introduction on location problems the interested reader +can refer to Laporte et al. (2019). In the following, we focus on the papers that are most closely +related to our research, and refer the interested reader to the above-mentioned surveys and the +references cited therein. +A first broad classification of the literature is based on the type of network considered (cf. Deb et al., +2018). When only the distribution network is considered, the optimal location of charging stations +must consider the potential adverse effects on the power grid, as an inappropriate placement of +charging stations can be a threat to the power system security and reliability. On the other hand, +when only the transportation network is taken into account, the main issue is to determine an +optimal location of charging stations over a road network. This paper lies in the latter category. +Within this category, the related literature can be further classified into two main streams of +models called flow-based and node-based demand models (e.g., see Kchaou-Boujelben, 2021). In +the literature, the majority of the research efforts are devoted to the flow-based demand models, +whereas the number of papers adopting a node-based approach is still relatively limited. To the +best of our knowledge, Anjos et al. (2020) are the only authors that integrated, within the same +optimization model, both a node-based and a flow-based approach. The flow-based demand models +are best suited for modeling long-haul (e.g., inter-urban) journeys where accounting for the limited +driving range of EVs is important (cf. Anjos et al., 2020). Contributions to this line of research +can be found, for example, in Kuby and Lim (2005), MirHassani and Ebrazi (2013), Yıldız et al. +(2016), and Hosseini et al. (2017). The present paper adopts a node-based demand model. +In the class of node-based demand models, drivers demanding to charge their EVs are associated +with one/few fixed locations, which represent, for instance, their residence, workplace or specific +service facilities (such as commercial activities). This approach is best suited for urban settings. +In fact, in such case EVs do not move much from the location where they need to be charged and +their limited driving range can be neglected (cf. Anjos et al., 2020). The most common modeling +approaches applied in the literature are based on the extension of classic discrete location models +(e.g., location-allocation as in Zhu et al. (2016), set covering as in Huang et al. (2016), and maximum +coverage problems as in Dong et al. (2019)) to incorporate technical constraints specific to EVs. +Characteristics of the charging demand (such as the population size, the penetration rate of EVs, +the type of zone, and the time of the day) are known to have a crucial impact on the optimal +location of charging stations. To position the present paper within the literature, we classify the +mathematical formulations into single-period and multi-period. In single-period optimization mod- +els all the decision variables are time independent. Although the spatial-temporal distribution of +the charging demands is described by different authors (e.g., see Yi et al., 2020, and the references +cited therein), only few authors have proposed multi-period optimization models where the alloca- +tion of the demand to the charging stations is time-dependent. The related stream of literature can +be classified according to the length of the planning horizon considered. A long planning horizon is +considered by some authors. The basic rationale of these models is that locating charging stations is +a long-term strategic decision. As a consequence, during these long periods of time the technology +available, as well as the charging demand, may change significantly. Along this line of research, +we mention the paper by Anjos et al. (2020) where it is assumed that the locating decisions taken +in a period have an impact on the charging demand in the subsequent periods. In fact, potential +EV buyers are influenced by the availability of charging opportunities. Some papers have proposed +multi-period optimization models that consider a short horizon, usually a day, divided in time +periods, usually hours. Our research belongs to this category of papers. +4 + +To the best of our knowledge, Cavadas et al. (2015) are the first authors to recognize the importance +of incorporating into an optimization model the dynamics of the charging demand across the day. +The aim of the proposed multi-period model is the maximization of the total demand served, +subject to a constraint on the budget available. The authors consider only one type of charger (i.e., +a slow type) and the sizing of the charging stations is not part of the optimization. In the model we +present in this paper, we address these shortcomings by considering multiple types of chargers and +optimizing the quantities installed in each opened station. Rajabi-Ghahnavieh and Sadeghi-Barzani +(2017) estimate the charging demand of EVs in different zones of a city and at different hours. The +authors consider the deployment of an unlimited number of fast chargers only and propose a non- +linear optimization model that includes three cost components: the total opening cost, the total +cost for the drivers to reach the assigned charging stations, and the cost of connecting the charging +stations to the electric grid substations. The variability of the demand across the day is taken +into consideration when determining the number of chargers to install. Nevertheless, the variables +assigning EVs to stations are not time-dependent, and, hence, drivers demanding to charge their +EVs at different hours are all assigned to the same station. In our paper, we allow the demand +arising from the same location during the day to be assigned to different stations, depending on +the evolution of the overall demand and the available cherging resources. Moreover, we consider +different types of chargers. Both short-term and long-term decisions are considered in Quddus et al. +(2019). The main long-term decisions are related to the year, the location, and the type of charging +stations to open. The short-term decisions are mainly related to the amount of power (provided +by different sources, such as electric grid and renewable sources) to satisfy the hourly charging +demand at a given location. Compared to our research, the drivers are, indirectly, pre-assigned to +a charging station and, hence, the assignment is not part of the optimization model. The authors +cast the problem as a two-stage stochastic programming model. Li and Jenn (2022) present an +optimization model based on the concept of charging opportunities, which is measured through +the time an individual stays at a given location within a day. The authors separate the charging +opportunities into home and non-home (i.e., public) categories, and allow the same individual to +charge the EV multiple times at different locations. The proposed optimization model determines +the number of home and non-home chargers to install, as well as the times and locations for each +individual to charge the EV. The model aims at minimizing the sum of the annual electricity cost +for charging the EVs and the total cost of locating the home and non-home chargers. The number +of chargers that can be installed in each location (called region by the authors) is unlimited. +Finally, we mention the growing body of literature that addresses the problem of determining an +optimal location of charging stations for EVs in car-sharing systems (e.g., cf. Brandst¨atter et al., +2017, 2020; Bekli et al., 2021). Although such problem has some characteristics in common with +ours, it includes some operational characteristics that make it considerably different, for example +the decisions about the number of EVs to acquire, the relocation of the EVs among stations, and +the assumption that charging occurs only between two consecutive trips. +Contributions of the paper. The contributions of this paper to the literature can be summarized +as follows. +✓ We present a node-based multi-period optimization model for the location of charging stations +that captures the dependence on time of the charging demand; +✓ the multi-period model takes into account several characteristics of the real problem: multiple +types of chargers (each with its own charging speed and installation cost), the capacitated +nature of the charging stations (in terms of maximum number of chargers that can be in- +5 + +stalled), a minimum number of chargers to be installed in different zones (e.g., commercial, +residential, industrial); +✓ the multi-period model is compared to a single-period model through a worst-case analysis; +✓ extensive computational experiments are presented that show, in particular, the importance +of incorporating the dependence on time of the charging demand. +3 +Problem definition and mathematical formulations +In this section, we first provide a general description of the location problem along with the notation +that is common to the two optimization models that will follow. Then, the single-period MILP +model is presented, together with the notation that is specific for the model, followed by the multi- +period formulation. +We consider the problem of determining, in an urban area, an optimal location of charging stations +for EVs, along with the type and number of chargers to deploy in each station. A maximum number +of chargers, of each type and in total, can be deployed in each station. The location for any station +can be selected from a pre-defined set of potential locations. We introduce a complete bipartite +network G = (I ∪ J , A), where I = {1, 2, . . . , I} is the set of demand nodes and J = {1, 2, . . . , J} +is the set of potential locations for the stations. Let cij be the travel distance from demand node i +to station j. +A fixed opening cost Fj is associated with each station j. The opening cost does not include the +cost of the chargers. We denote as K = {1, 2, . . . , K} the set of types of chargers considered, and +as fjk the cost of installing one charger of type k ∈ K in location j ∈ J . Let ujk be the maximum +number of chargers of type k that can be installed in station j. Similarly, uj denotes the maximum +number of chargers that can be installed in total in station j. The latter two parameters define, +implicitly, the maximum charging capacity of station j. +Each node i is the center of gravity of a section of the urban area where the demand of the section +is measured as the number of EVs that need to be recharged. We will introduce later, for each of +the two optimization models, the planning horizon and the notation for the demand of a customer. +For the sake of brevity, hereafter we refer to each potential location j simply as station j. The +demand must be entirely satisfied by the chargers that will be deployed. +To take into account that different parts of the urban area have different needs in terms of type of +charger desired, the urban area is partitioned in zones (e.g., commercial, residential, industrial). We +denote by L = {1, 2, . . . , L} the set of zones. We assume that, based on some preliminary analysis, +in each zone ℓ ∈ L a minimum percentage ρℓk of chargers of type k must be deployed. Each station +j ∈ J belongs to a zone as well as each customer i ∈ I. Thus, the zones imply a partition of both +the stations and the demand points. This partition does not restrict the allocation of demand to +stations, i.e., a demand point located in a zone can be assigned to a station located in a different +zone. +Two criteria have a key role in this location problem: the cost of opening the stations and installing +the chargers and the distance the customers have to travel to be recharged. The objective function +we consider, to be minimized, is a convex combination of these two criteria. The optimization +problem is aimed at determining, for each type of charger and each station, the number of chargers +to be deployed in such a way that the objective function is minimized. +6 + +Both MILP models include the following decision variables. Let zj ∈ {0, 1}, with j ∈ J , be a +binary variable that takes value 1 if station j is opened, and 0 otherwise. Let yjk ∈ Z+, with j ∈ J +and k ∈ K, be an integer variable that represents the number of chargers of type k installed in +station j. +3.1 +A single-period location model +This section presents a single-period model for the location of the charging stations. The MILP +formulation, denoted as SP-CFL, is an extension of a classical CFL model. Hereafter, we introduce +the notation needed for the formulation, in addition to the one introduced above. +We consider a single planning period of length H and denote as di the total demand in i ∈ I, that +is, the total number of EVs demanding to be recharged in i during H. Let pk denote the average +number of EVs fully recharged by one charger of type k during time period H. For the sake of +simplicity, we assume that pk does not depend on the type of EV. +The SP-CFL model also makes use of the following decision variables. Let xijk ∈ [0, 1], with i ∈ I, +j ∈ J , and k ∈ K, be the fraction of the demand of node i assigned to a charger of type k in station +j. Then, the SP-CFL model can be stated as the following MILP: +[SP-CFL] +min +λ · +� +� +1 +� +i∈I +di +� +i∈I +di +� +j∈J +cij +� +k∈K +xijk +� +� + (1 − λ) · +� +�� +j∈J +Fjzj + +� +j∈J +� +k∈K +fjkyjk +� +� +(1) +s.t. +yjk ≤ ujkzj +j ∈ J , k ∈ K +(2) +� +k∈K +yjk ≤ ujzj +j ∈ J +(3) +� +j∈J +� +k∈K +xijk = 1 +i ∈ I +(4) +� +i∈I +dixijk ≤ pkyjk +j ∈ J , k ∈ K +(5) +xijk ≤ yjk +i ∈ I, j ∈ J , k ∈ K +(6) +� +j∈Aℓ +yjk ≥ ρℓk +� +j∈Aℓ +� +k∈K +yjk +k ∈ K, ℓ ∈ L +(7) +zj ∈ {0, 1} +j ∈ J ; +yjk ∈ Z+ +j ∈ J , k ∈ K; +xijk ∈ [0, 1] +i ∈ I, j ∈ J , k ∈ K. +(8) +The objective function in (1) comprises two terms. The first one represents the average distance +traveled by the EVs to reach the assigned station. The second term is the total cost of opening the +7 + +stations and installing the chargers. The two terms represent criteria of a substantially different +nature: the first measures the quality of the service provided by the deployed stations and chargers +to the drivers, whereas the second the cost of the service. The two criteria are weighted by the +trade-off parameter λ ∈ [0, 1], which is used to balance their importance. +Constraints (2) and (3) limit the number of chargers that can be installed in station j. The former +set bounds the number of chargers of type k to be lower than or equal to ujk, whereas the second +set of constraints bounds the total number of chargers to be lower than or equal to uj. Both sets of +constraints (2) and (3) impose that no charger can be installed if station j is not open (i.e., zj = 0). +Constraints (4) ensure that the demand of each node i ∈ I is entirely satisfied. Constraints (5) +guarantee that the number of EVs assigned to the chargers of type k deployed in station j is not +greater than the charging capacity available (i.e., pkyjk). They also impose that no EV can be +assigned to a type k of chargers in station j if no charger of that type is available (i.e., yjk = 0). +Inequalities (6), which are redundant in this formulation, are well-known to yield a tighter Linear +Programming (LP) relaxation than the equivalent formulation without them (e.g., see Filippi et al., +2021). Constraints (7) guarantee that the number of chargers of type k installed in zone ℓ is at least +equal to the minimum percentage ρℓk. Finally, constraints (8) define the domain of the decision +variables. +3.2 +A multi-period location model +This section presents the MILP formulation for the multi-period model, henceforth denoted as the +MP-CFL model, for the problem defined at the beginning of this section. +The planning period H of the single-period model is here partitioned into a number T of time +periods. For example, if H is a day, we may partition the day in hours. Let T = {1, 2, . . . , T} +denote the set of time periods. We denote as Rk the number of consecutive time periods needed to +completely recharge a car using a charger of type k. Note that, similar to pk for the SP-CFL model, +Rk does not depend on the type of EV but only on the type of charger. Furthermore, parameters +pk and Rk are strictly related, as the latter is determined by dividing the length of the time horizon +by pk, i.e. Rk = T +pk . +The demand of each node i ∈ I is no longer identified by a single value (di in the SP-CFL model) +but by a time-dependent profile. Let dt +i denote the demand of node i ∈ I at the beginning of time +period t ∈ T . A more detailed discussion about the demand profiles can be found in Section 5.1.1. +We assume that the demand of a time period t must be served in that time period, i.e., it cannot +be postponed to a later time. We say that a node is served by a charger of type k at time t if a +charger is available at time t to start the charging which will occupy the charger for a total of Rk +time periods. The capacity installed in each station must be sufficient to serve the charging demand +assigned to that station in a time period and the demand assigned to the station in a previous time +period that has not yet completed the charging. Finally, let xt +ijk ∈ [0, 1], with i ∈ I, j ∈ J , k ∈ K, +and t ∈ T , be the fraction of the charging demand of node i to be served at time t that is assigned +to a charger of type k in station j. +The MP-CFL model is formulated as follows: +[MP-CFL] +8 + +min +λ · +� +� +1 +� +t∈T +� +i∈I +dt +i +� +t∈T +� +i∈I +dt +i +� +j∈J +cij +� +k∈K +xt +ijk +� +� + (1 − λ) · +� +�� +j∈J +Fjzj + +� +j∈J +� +k∈K +fjkyjk +� +� +(9) +s.t. (2), (3), and (7) +� +j∈J +� +k∈K +xt +ijk = 1 +i ∈ I, t ∈ T +(10) +xt +ijk ≤ yjk +i ∈ I, j ∈ J , k ∈ K, t ∈ T +(11) +� +i∈I +t−1 +� +τ=0 +dt−τ +i +xt−τ +ijk ≤ yjk +j ∈ J , k ∈ K, t ∈ T : t < Rk +(12) +� +i∈I +Rk−1 +� +τ=0 +dt−τ +i +xt−τ +ijk ≤ yjk +j ∈ J , k ∈ K, t ∈ T : t ≥ Rk +(13) +zj ∈ {0, 1} +j ∈ J ; +yjk ∈ Z+ +j ∈ J , k ∈ K; +xt +ijk ∈ [0, 1] +i ∈ I, j ∈ J , k ∈ K, t ∈ T . +(14) +The objective function in (9) is the multi-period extension of function (1). For each node i ∈ I, +constraints (10) ensure that the charging demand arising in each time period t is fully satisfied. +Akin to the objective function, also inequalities (11) are the multi-period extension of constraints +(6). +Constraints (12) and (13) guarantee that the number of EVs that are charging in time period t at +a charger of type k in station j is smaller than or equal to the number of available chargers of that +type (i.e., yjk). Note that the second sum in (12) and (13) is used to keep track of the EVs that +started to recharge in a previous time period but have not completed the charging in t. Constraints +(12) are defined for the first time periods in the planning horizon (such that t < Rk), whereas +(13) are defined for the remaining time periods. Finally, constraints (14) define the domain of the +decision variables. +4 +Worst-case analysis +In this section, we analyze the worst-case performance of the SP-CFL model in terms of the demand +that cannot be satisfied if the optimal solution produced is implemented in a context where the +demand fluctuates over time. In fact, in this case if an optimal solution to the SP-CFL model is +implemented, there is no guarantee that all the charging demand is satisfied. As the SP-CFL model +implicitly assumes that the charging demand is uniformly distributed across the planning horizon, +when the demand fluctuates over time, there may be peak time periods where the chargers installed +are not sufficient. +9 + +Theorem 1 When an optimal solution of the SP-CFL model is implemented, the fraction of the +demand that does not find an available charger to be served may be up to 1 − 1 +T , where T is the +number of time periods of the planning horizon. This bound is tight. +Proof To prove the theorem, we build the following instance. +Recalling constraint (4) and summing up all constraints (5), the following chain of inequalities +holds: +� +i∈I +di +(4) += +� +i∈I +di +� +j∈J +� +k∈K +xijk = +� +j∈J +� +k∈K +� +i∈I +dixijk +(5) +≤ +� +j∈J +� +k∈K +pkyjk +(15) +for any feasible solution to the SP-CFL model. +Consider an instance where the travel distances are all negligible compared to the fixed opening and +installing cost. In this situation, the SP-CFL model would open the minimum number of charging +stations and install the minimum number of chargers that are strictly necessary to satisfy the total +demand. As a consequence, the value of the right-hand side of the rightmost inequality in (15) +would be as small as possible. +Additionally, suppose there is a single type of charger (i.e., K = 1) and that the total demand +� +i∈I di is a multiple of p1. Recall that the latter parameter represents the number of EVs fully +recharged by one charger during the planning horizon. Note that it can be determined by dividing +the number of time periods T by the number of consecutive time periods needed to completely +recharge an EV (i.e., R1). Hence, at optimality, the inequality in (15) can be reformulated as +follows: +� +i∈I +di = +� +j∈J +p1yj1 = T +R1 +� +j∈J +yj1. +(16) +Thus, the total number of chargers deployed is � +j∈J yj1 = +R1 +� +i∈I di +T +, which, assuming that R1 = 1, +becomes � +j∈J yj1 = +� +i∈I di +T +. +Consider an extreme situation where the whole demand � +i∈I di arises in one time period, say ˆt, +whereas it is zero in the remaining periods. The demand that can be satisfied in such time period +is equal to the number of chargers installed (i.e., � +j∈J yj1). Given the assumptions above, this +value is also equal to +� +i∈I di +T +. Thus, the amount of demand that does not find an available charger +is equal to: +� +i∈I +di − +� +i∈I +di +T +. +The statement follows. +Figure 1 illustrates the construction for the special case where � +i∈I di = T, which implies that +� +j∈J yj1 = 1. The whole demand, equal to T, arises in time period ˆt (green bar), whereas it is zero +in the remaining periods. The SP-CFL model assumes that such demand is uniformly distributed +across the planning horizon (pink bars), and hence it opens one station equipped with one charger. +As a consequence, the charging demand that is not satisfied is T − 1 or, in percentage, T−1 +T . +10 + +Figure 1: An instance where the demand arises in time period ˆt (green bar). The pink bars show +a uniform distribution of the demand across the planning horizon, as implicitly assumed by the +SP-CFL model. +5 +Experimental Analysis +This section is devoted to the presentation and discussion of the computational experiments. They +were conducted on a Workstation HP Intel(R)-Xeon(R) at 3.5GHz with 64 GB RAM (Win 10 Pro, +64 bits). The processor is equipped with 6 physical cores, and all threads were used while solving +each instance. The MILP models were implemented in Java, compiled within Apache NetBeans +12.3, and solved by means of CPLEX 20.1. Each instance was solved with a CPU time limit of +3,600 seconds. All other CPLEX parameters were set at their default values. +The section is organized as follows. +First, we present the testing environment we used in our +experiments, then we compare the optimal solutions for two illustrative examples generated ac- +cording to two different urban structure models, and finally we provide detailed computational +results comparing the solutions produced by the single-period and the multi-period models. +5.1 +Testing environment +The generation of the charging demand and potential station locations follows the procedure de- +scribed in Section 5.1.1. All the remaining parameters defining the testing environment are detailed +in Section 5.1.2. +5.1.1 +Spatial and temporal charging demand generation +As far as the urban structure is concerned, we considered two classic models, the concentric zone +model and the sector model. The concentric zone model was proposed in 1925 by sociologist Ernest +Burgess on the base of his human ecology theory, and was initially applied to the city of Chicago (cf. +Burgess, 2008). It is, perhaps, the first theoretical model used to explain urban social structures. +The model depicts urban land usage as concentric rings: the business district is located in the +center, whereas the remainder of the city is expanded in rings, each corresponding to a different +land usage (such as industrial or residential). The sector model was proposed in 1939 by land +economist Homer Hoyt (see Hoyt, 1939). It is a modification of the Burgess’ model where the +city zones devoted to a specific land usage (e.g., business, residential, and productive) develop +in sectors expanding from the original city center. Though the actual structure of modern cities +can hardly be captured by models as simple as Burgess’ and Hoyt’s, they are the basis of more +11 + +T +: +1 +0 +t+1 +t-1 +