We study networks of communicating learning agents that cooperate to solve a common nonstochastic bandit problem. Agents use an underlying communication network to get messages about actions selected by other agents, and drop messages that took more than $d$ hops to arrive, where $d$ is a delay parameter. We introduce \textsc{Exp3-Coop}, a cooperative version of the {\sc Exp3} algorithm and prove that with $K$ actions and $N$ agents the average per-agent regret after $T$ rounds is at most of order $\sqrt{\bigl(d+1 + \tfrac{K}{N}α_{\le d}\bigr)(T\ln K)}$, where $α_{\le d}$ is the independence number of the $d$-th power of the connected communication graph $G$. We then show that for any connected graph, for $d=\sqrt{K}$ the regret bound is $K^{1/4}\sqrt{T}$, strictly better than the minimax regret $\sqrt{KT}$ for noncooperating agents. More informed choices of $d$ lead to bounds which are arbitrarily close to the full information minimax regret $\sqrt{T\ln K}$ when $G$ is dense. When $G$ has sparse components, we show that a variant of \textsc{Exp3-Coop}, allowing agents to choose their parameters according to their centrality in $G$, strictly improves the regret. Finally, as a by-product of our analysis, we provide the first characterization of the minimax regret for bandit learning with delay.
We study nonstochastic bandits and experts in a delayed setting where delays depend on both time and arms. While the setting in which delays only depend on time has been extensively studied, the arm-dependent delay setting better captures real-world applications at the cost of introducing new technical challenges. In the full information (experts) setting, we design an algorithm with a first-order regret bound that reveals an interesting trade-off between delays and losses. We prove a similar first-order regret bound also for the bandit setting, when the learner is allowed to observe how many …
Università degli Studi di Milano（米兰大学）；Paul Sabatier University (UT3)（保罗·萨巴蒂埃大学（UT3））；Toulouse School of Economics (TSE)（图卢兹经济学院（TSE））；Istituto Italiano di Tecnologia (IIT)（意大利技术研究院（IIT））；Google Research（谷歌研究院）；Tel Aviv University（特拉维夫大学）
We investigate a nonstochastic bandit setting in which the loss of an action is not immediately charged to the player, but rather spread over the subsequent rounds in an adversarial way. The instantaneous loss observed by the player at the end of each round is then a sum of many loss components of previously played actions. This setting encompasses as a special case the easier task of bandits with delayed feedback, a well-studied framework where the player observes the delayed losses individually. Our first contribution is a general reduction transforming a standard bandit algorithm into one t…
Princeton University（普林斯顿大学）；Università degli Studi di Milano（米兰大学）
Multi-armed bandit problems are the most basic examples of sequential decision problems with an exploration-exploitation trade-off. This is the balance between staying with the option that gave highest payoffs in the past and exploring new options that might give higher payoffs in the future. Although the study of bandit problems dates back to the Thirties, exploration-exploitation trade-offs arise in several modern applications, such as ad placement, website optimization, and packet routing. Mathematically, a multi-armed bandit is defined by the payoff process associated with each option. In …
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
We study agents communicating over an underlying network by exchanging messages, in order to optimize their individual regret in a common nonstochastic multi-armed bandit problem. We derive regret minimization algorithms that guarantee for each agent $v$ an individual expected regret of $\widetilde{O}\left(\sqrt{\left(1+\frac{K}{\left|\mathcal{N}\left(v\right)\right|}\right)T}\right)$, where $T$ is the number of time steps, $K$ is the number of actions and $\mathcal{N}\left(v\right)$ is the set of neighbors of agent $v$ in the communication graph. We present algorithms both for the case that t…
We study the problem of nonstochastic bandits with expert advice, extending the setting from finitely many experts to any countably infinite set: A learner aims to maximize the total reward by taking actions sequentially based on bandit feedback while benchmarking against a set of experts. We propose a variant of Exp4.P that, for finitely many experts, enables inference of correct expert rankings while preserving the order of the regret upper bound. We then incorporate the variant into a meta-algorithm that works on infinitely many experts. We prove a high-probability upper bound of $\tilde{\m…
Tel Aviv University（特拉维夫大学）；Institute for Advanced Study（高等研究院）；Microsoft Research（微软研究院）；Università degli Studi di Milano（米兰大学）；University of Insubria（因苏布里亚大学）；Technion（以色列理工学院）；Weizmann Institute of Science（魏茨曼科学研究所）
We present and study a partial-information model of online learning, where a decision maker repeatedly chooses from a finite set of actions, and observes some subset of the associated losses. This naturally models several situations where the losses of different actions are related, and knowing the loss of one action provides information on the loss of other actions. Moreover, it generalizes and interpolates between the well studied full-information setting (where all losses are revealed) and the bandit setting (where only the loss of the action chosen by the player is revealed). We provide se…
University of Copenhagen（哥本哈根大学）；Università degli Studi di Milano（米兰大学）
We investigate multiarmed bandits with delayed feedback, where the delays need neither be identical nor bounded. We first prove that "delayed" Exp3 achieves the $O(\sqrt{(KT + D)\ln K} )$ regret bound conjectured by Cesa-Bianchi et al. [2019] in the case of variable, but bounded delays. Here, $K$ is the number of actions and $D$ is the total delay over $T$ rounds. We then introduce a new algorithm that lifts the requirement of bounded delays by using a wrapper that skips rounds with excessively large delays. The new algorithm maintains the same regret bound, but similar to its predecessor requ…
Università degli Studi di Milano（米兰大学）；Weizmann Institute of Science（魏茨曼科学研究所）
We study how the regret guarantees of nonstochastic multi-armed bandits can be improved, if the effective range of the losses in each round is small (e.g. the maximal difference between two losses in a given round). Despite a recent impossibility result, we show how this can be made possible under certain mild additional assumptions, such as availability of rough estimates of the losses, or advance knowledge of the loss of a single, possibly unspecified arm. Along the way, we develop a novel technique which might be of independent interest, to convert any multi-armed bandit algorithm with regr…
This paper tackles a multi-agent bandit setting where $M$ agents cooperate together to solve the same instance of a $K$-armed stochastic bandit problem. The agents are \textit{heterogeneous}: each agent has limited access to a local subset of arms and the agents are asynchronous with different gaps between decision-making rounds. The goal for each agent is to find its optimal local arm, and agents can cooperate by sharing their observations with others. While cooperation between agents improves the performance of learning, it comes with an additional complexity of communication between agents.…
ETH Zurich（苏黎世联邦理工学院）；Massachusetts Institute of Technology（麻省理工学院）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
We consider decentralized stochastic multi-armed bandit problem with multiple players in the case of different communication probabilities between players. Each player makes a decision of pulling an arm without cooperation while aiming to maximize his or her reward but informs his or her neighbors in the end of every turn about the arm he or she pulled and the reward he or she got. Neighbors of players are determined according to an Erdos-Renyi graph with which is reproduced in the beginning of every turn. We consider i.i.d. rewards generated by a Bernoulli distribution and assume that players…
Technion - Israel Institute of Technology（以色列理工学院）
We consider the problem of multiple users targeting the arms of a single multi-armed stochastic bandit. The motivation for this problem comes from cognitive radio networks, where selfish users need to coexist without any side communication between them, implicit cooperation or common control. Even the number of users may be unknown and can vary as users join or leave the network. We propose an algorithm that combines an $ε$-greedy learning rule with a collision avoidance mechanism. We analyze its regret with respect to the system-wide optimum and show that sub-linear regret can be obtained in …
Microsoft Research（微软研究院）；University of California, Berkeley（加州大学伯克利分校）
We study the problem of information sharing and cooperation in Multi-Player Multi-Armed bandits. We propose the first algorithm that achieves logarithmic regret for this problem when the collision reward is unknown. Our results are based on two innovations. First, we show that a simple modification to a successive elimination strategy can be used to allow the players to estimate their suboptimality gaps, up to constant factors, in the absence of collisions. Second, we leverage the first result to design a communication protocol that successfully uses the small reward of collisions to coordinat…
Instance-Dependent Regret Bounds for Nonstochastic Linear Partial Monitoring
EPFL（苏黎世联邦理工学院）；Università degli Studi di Milano（米兰大学）；Politecnico di Milano（米兰理工大学）；ETH Zürich（苏黎世联邦理工学院）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Inria（法国国家信息与自动化技术研究院）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔INP）；LJK（实验室）
Alibaba Group（阿里巴巴集团）；University College London（伦敦大学学院）；Shanghai JiaoTong University（上海交通大学）
Real-time advertising allows advertisers to bid for each impression for a visiting user. To optimize specific goals such as maximizing revenue and return on investment (ROI) led by ad placements, advertisers not only need to estimate the relevance between the ads and user's interests, but most importantly require a strategic response with respect to other advertisers bidding in the market. In this paper, we formulate bidding optimization with multi-agent reinforcement learning. To deal with a large number of advertisers, we propose a clustering method and assign each cluster with a strategic b…
Nanyang Technological University（南洋理工大学）；NetEase Fuxi AI Lab（网易伏羲人工智能实验室）；City University of Hong Kong（香港城市大学）
Recent advances in multi-agent reinforcement learning (MARL) allow agents to coordinate their behaviors in complex environments. However, common MARL algorithms still suffer from scalability and sparse reward issues. One promising approach to resolving them is automatic curriculum learning (ACL). ACL involves a student (curriculum learner) training on tasks of increasing difficulty controlled by a teacher (curriculum generator). Despite its success, ACL's applicability is limited by (1) the lack of a general student framework for dealing with the varying number of agents across tasks and the s…
