We study a collaborative multi-agent stochastic linear bandit setting, where $N$ agents that form a network communicate locally to minimize their overall regret. In this setting, each agent has its own linear bandit problem (its own reward parameter) and the goal is to select the best global action w.r.t. the average of their reward parameters. At each round, each agent proposes an action, and one action is randomly selected and played as the network action. All the agents observe the corresponding rewards of the played actions and use an accelerated consensus procedure to compute an estimate of the average of the rewards obtained by all the agents. We propose a distributed upper confidence bound (UCB) algorithm and prove a high probability bound on its $T$-round regret in which we include a linear growth of regret associated with each communication round. Our regret bound is of order $\mathcal{O}\Big(\sqrt{\frac{T}{N \log(1/|λ_2|)}}\cdot (\log T)^2\Big)$, where $λ_2$ is the second largest (in absolute value) eigenvalue of the communication matrix.
Collaborative bandit learning, i.e., bandit algorithms that utilize collaborative filtering techniques to improve sample efficiency in online interactive recommendation, has attracted much research attention as it enjoys the best of both worlds. However, all existing collaborative bandit learning solutions impose a stationary assumption about the environment, i.e., both user preferences and the dependency among users are assumed static over time. Unfortunately, this assumption hardly holds in practice due to users' ever-changing interests and dependence relations, which inevitably costs a reco…
Multi-armed bandits (MAB) provide a principled online learning approach to attain the balance between exploration and exploitation. Due to the superior performance and low feedback learning without the learning to act in multiple situations, Multi-armed Bandits drawing widespread attention in applications ranging such as recommender systems. Likewise, within the recommender system, collaborative filtering (CF) is arguably the earliest and most influential method in the recommender system. Crucially, new users and an ever-changing pool of recommended items are the challenges that recommender sy…
CoCoB: Adaptive Collaborative Combinatorial Bandits for Online Recommendation
School of Computer Science and Technology, Donghua University（东华大学计算机科学与技术学院）；Shanghai Institute of Artificial Intelligence for Education, East China Normal University（华东师范大学上海人工智能教育研究院）
Institute for Interdisciplinary Information Sciences（交叉信息研究院）；Tsinghua University（清华大学）；Microsoft Research（微软研究院）；The University of Tokyo（东京大学）；RIKEN AIP（理化学研究所人工智能项目中心）；CENTAI Institute（CENTAI研究所）
In this paper, we formulate a Collaborative Pure Exploration in Kernel Bandit problem (CoPE-KB), which provides a novel model for multi-agent multi-task decision making under limited communication and general reward functions, and is applicable to many online learning tasks, e.g., recommendation systems and network scheduling. We consider two settings of CoPE-KB, i.e., Fixed-Confidence (FC) and Fixed-Budget (FB), and design two optimal algorithms CoopKernelFC (for FC) and CoopKernelFB (for FB). Our algorithms are equipped with innovative and efficient kernelized estimators to simultaneously ac…
University of Insubria（因苏布里亚大学）；Telefónica Research（西班牙电信研究中心）
Classical collaborative filtering, and content-based filtering methods try to learn a static recommendation model given training data. These approaches are far from ideal in highly dynamic recommendation domains such as news recommendation and computational advertisement, where the set of items and users is very fluid. In this work, we investigate an adaptive clustering technique for content recommendation based on exploration-exploitation strategies in contextual multi-armed bandit settings. Our algorithm takes into account the collaborative effects that arise due to the interaction of the us…
Indiana University（印第安纳大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Best arm identification (or, pure exploration) in multi-armed bandits is a fundamental problem in machine learning. In this paper we study the distributed version of this problem where we have multiple agents, and they want to learn the best arm collaboratively. We want to quantify the power of collaboration under limited interaction (or, communication steps), as interaction is expensive in many settings. We measure the running time of a distributed algorithm as the speedup over the best centralized algorithm where there is only one agent. We give almost tight round-speedup tradeoffs for this …
Université Paris Cité（巴黎西岱大学）；Inserm（法国国家健康与医学研究院）；MediaTek Research（联发科研究院）；Université de Lille（里尔大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；Centrale Lille（里尔中央理工学院）
This paper introduces a general multi-agent bandit model in which each agent is facing a finite set of arms and may communicate with other agents through a central controller in order to identify, in pure exploration, or play, in regret minimization, its optimal arm. The twist is that the optimal arm for each agent is the arm with largest expected mixed reward, where the mixed reward of an arm is a weighted sum of the rewards of this arm for all agents. This makes communication between agents often necessary. This general setting allows to recover and extend several recent models for collabora…
Montanuniversitaet Leoben（莱奥本矿业大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We present an algorithm that achieves almost optimal pseudo-regret bounds against adversarial and stochastic bandits. Against adversarial bandits the pseudo-regret is $O(K\sqrt{n \log n})$ and against stochastic bandits the pseudo-regret is $O(\sum_i (\log n)/Δ_i)$. We also show that no algorithm with $O(\log n)$ pseudo-regret against stochastic bandits can achieve $\tilde{O}(\sqrt{n})$ expected regret against adaptive adversarial bandits. This complements previous results of Bubeck and Slivkins (2012) that show $\tilde{O}(\sqrt{n})$ expected adversarial regret with $O((\log n)^2)$ stochastic …
We consider a collaborative online learning paradigm, wherein a group of agents connected through a social network are engaged in playing a stochastic multi-armed bandit game. Each time an agent takes an action, the corresponding reward is instantaneously observed by the agent, as well as its neighbours in the social network. We perform a regret analysis of various policies in this collaborative learning setting. A key finding of this paper is that natural extensions of widely-studied single agent learning policies to the network setting need not perform well in terms of regret. In particular,…
University of Insubria（因苏布里亚大学）；University of Cambridge（剑桥大学）；IIT Kanpur（坎普尔印度理工学院）；Telefonica Research（西班牙电信研究院）；Amazon Berlin（亚马逊柏林）
We investigate a novel cluster-of-bandit algorithm CAB for collaborative recommendation tasks that implements the underlying feedback sharing mechanism by estimating the neighborhood of users in a context-dependent manner. CAB makes sharp departures from the state of the art by incorporating collaborative effects into inference as well as learning processes in a manner that seamlessly interleaving explore-exploit tradeoffs and collaborative steps. We prove regret bounds under various assumptions on the data, which exhibit a crisp dependence on the expected number of clusters over the users, a …
We derive improved regret bounds for the Tsallis-INF algorithm of Zimmert and Seldin (2021). We show that in adversarial regimes with a $(Δ,C,T)$ self-bounding constraint the algorithm achieves $\mathcal{O}\left(\left(\sum_{i\neq i^*} \frac{1}{Δ_i}\right)\log_+\left(\frac{(K-1)T}{\left(\sum_{i\neq i^*} \frac{1}{Δ_i}\right)^2}\right)+\sqrt{C\left(\sum_{i\neq i^*}\frac{1}{Δ_i}\right)\log_+\left(\frac{(K-1)T}{C\sum_{i\neq i^*}\frac{1}{Δ_i}}\right)}\right)$ regret bound, where $T$ is the time horizon, $K$ is the number of arms, $Δ_i$ are the suboptimality gaps, $i^*$ is the best arm, $C$ is the co…
Media Lab and Institute for Data, Systems and Society, Massachusetts Institute of Technology（麻省理工学院媒体实验室与数据、系统与社会研究所）
We study the heavy-tailed stochastic bandit problem in the cooperative multi-agent setting, where a group of agents interact with a common bandit problem, while communicating on a network with delays. Existing algorithms for the stochastic bandit in this setting utilize confidence intervals arising from an averaging-based communication protocol known as~\textit{running consensus}, that does not lend itself to robust estimation for heavy-tailed settings. We propose \textsc{MP-UCB}, a decentralized multi-agent algorithm for the cooperative stochastic bandit that incorporates robust estimation wi…
Microsoft Research（微软研究院）；Stanford University（斯坦福大学）
We consider the non-stochastic version of the (cooperative) multi-player multi-armed bandit problem. The model assumes no communication at all between the players, and furthermore when two (or more) players select the same action this results in a maximal loss. We prove the first $\sqrt{T}$-type regret guarantee for this problem, under the feedback model where collisions are announced to the colliding players. Such a bound was not known even for the simpler stochastic version. We also prove the first sublinear guarantee for the feedback model where collision information is not available, namel…
Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Peking University（北京大学）
Multi-arm bandit (MAB) and stochastic linear bandit (SLB) are important models in reinforcement learning, and it is well-known that classical algorithms for bandits with time horizon $T$ suffer $Ω(\sqrt{T})$ regret. In this paper, we study MAB and SLB with quantum reward oracles and propose quantum algorithms for both models with $O(\mbox{poly}(\log T))$ regrets, exponentially improving the dependence in terms of $T$. To the best of our knowledge, this is the first provable quantum speedup for regrets of bandit problems and in general exploitation in reinforcement learning. Compared to previou…
We study the stochastic multi-armed bandits problem in the presence of adversarial corruption. We present a new algorithm for this problem whose regret is nearly optimal, substantially improving upon previous work. Our algorithm is agnostic to the level of adversarial contamination and can tolerate a significant amount of corruption with virtually no degradation in performance.
We provide a simple method to combine stochastic bandit algorithms. Our approach is based on a "meta-UCB" procedure that treats each of $N$ individual bandit algorithms as arms in a higher-level $N$-armed bandit problem that we solve with a variant of the classic UCB algorithm. Our final regret depends only on the regret of the base algorithm with the best regret in hindsight. This approach provides an easy and intuitive alternative strategy to the CORRAL algorithm for adversarial bandits, without requiring the stability conditions imposed by CORRAL on the base algorithms. Our results match lo…
Small steps no more: Global convergence of stochastic gradient bandits for arbitrary learning rates
Google DeepMind（谷歌DeepMind）；Google Research（谷歌研究院）；Georgia Institute of Technology（佐治亚理工学院）；University of Alberta（阿尔伯塔大学）；Simon Fraser University（西蒙菲莎大学）；Indian Institute of Science（印度科学学院）
Institute of Science and Technology Austria（奥地利科学技术研究所）
We consider the problem of minimizing the regret in stochastic multi-armed bandit, when the measure of goodness of an arm is not the mean return, but some general function of the mean and the variance.We characterize the conditions under which learning is possible and present examples for which no natural algorithm can achieve sublinear regret.
We present a new bandit algorithm, SAO (Stochastic and Adversarial Optimal), whose regret is, essentially, optimal both for adversarial rewards and for stochastic rewards. Specifically, SAO combines the square-root worst-case regret of Exp3 (Auer et al., SIAM J. on Computing 2002) and the (poly)logarithmic regret of UCB1 (Auer et al., Machine Learning 2002) for stochastic rewards. Adversarial rewards and stochastic rewards are the two main settings in the literature on (non-Bayesian) multi-armed bandits. Prior work on multi-armed bandits treats them separately, and does not attempt to jointly …
Microsoft Research（微软研究院）；UBC（不列颠哥伦比亚大学）；Stanford University（斯坦福大学）
We consider the cooperative multi-player version of the stochastic multi-armed bandit problem. We study the regime where the players cannot communicate but have access to shared randomness. In prior work by the first two authors, a strategy for this regime was constructed for two players and three arms, with regret $\tilde{O}(\sqrt{T})$, and with no collisions at all between the players (with very high probability). In this paper we show that these properties (near-optimal regret and no collisions at all) are achievable for any number of players and arms. At a high level, the previous strategy…
We study the problem of corralling stochastic bandit algorithms, that is combining multiple bandit algorithms designed for a stochastic environment, with the goal of devising a corralling algorithm that performs almost as well as the best base algorithm. We give two general algorithms for this setting, which we show benefit from favorable regret guarantees. We show that the regret of the corralling algorithms is no worse than that of the best algorithm containing the arm with the highest reward, and depends on the gap between the highest reward and other rewards.
Florida International University（佛罗里达国际大学）；IBM T.J. Watson Research Center（IBM托马斯·J·沃森研究中心）；St. John’s University（圣约翰大学）
Online interactive recommender systems strive to promptly suggest to consumers appropriate items (e.g., movies, news articles) according to the current context including both the consumer and item content information. However, such context information is often unavailable in practice for the recommendation, where only the users' interaction data on items can be utilized. Moreover, the lack of interaction records, especially for new users and items, worsens the performance of recommendation further. To address these issues, collaborative filtering (CF), one of the recommendation techniques rely…
Princeton University（普林斯顿大学）；University of Virginia（弗吉尼亚大学）
We study adversarial attacks on linear stochastic bandits: by manipulating the rewards, an adversary aims to control the behaviour of the bandit algorithm. Perhaps surprisingly, we first show that some attack goals can never be achieved. This is in sharp contrast to context-free stochastic bandits, and is intrinsically due to the correlation among arms in linear stochastic bandits. Motivated by this finding, this paper studies the attackability of a $k$-armed linear bandit environment. We first provide a complete necessity and sufficiency characterization of attackability based on the geometry…
The stochastic multi-armed bandit has provided a framework for studying decision-making in unknown environments. We propose a variant of the stochastic multi-armed bandit where the rewards are sampled from a stochastic linear dynamical system. The proposed strategy for this stochastic multi-armed bandit variant is to learn a model of the dynamical system while choosing the optimal action based on the learned model. Motivated by mathematical finance areas such as Intertemporal Capital Asset Pricing Model proposed by Merton and Stochastic Portfolio Theory proposed by Fernholz that both model ass…
University of Copenhagen（哥本哈根大学）；University of Southern California（南加州大学）
We develop the first general semi-bandit algorithm that simultaneously achieves $\mathcal{O}(\log T)$ regret for stochastic environments and $\mathcal{O}(\sqrt{T})$ regret for adversarial environments without knowledge of the regime or the number of rounds $T$. The leading problem-dependent constants of our bounds are not only optimal in some worst-case sense studied previously, but also optimal for two concrete instances of semi-bandit problems. Our algorithm and analysis extend the recent work of (Zimmert & Seldin, 2019) for the special case of multi-armed bandit, but importantly requires a …
Contextual bandit algorithms have become widely used for recommendation in online systems (e.g. marketplaces, music streaming, news), where they now wield substantial influence on which items get exposed to the users. This raises questions of fairness to the items -- and to the sellers, artists, and writers that benefit from this exposure. We argue that the conventional bandit formulation can lead to an undesirable and unfair winner-takes-all allocation of exposure. To remedy this problem, we propose a new bandit objective that guarantees merit-based fairness of exposure to the items while opt…
University of California, Davis（加州大学戴维斯分校）；University of California, Los Angeles（加州大学洛杉矶分校）；Amazon（亚马逊）
Stochastic linear contextual bandit algorithms have substantial applications in practice, such as recommender systems, online advertising, clinical trials, etc. Recent works show that optimal bandit algorithms are vulnerable to adversarial attacks and can fail completely in the presence of attacks. Existing robust bandit algorithms only work for the non-contextual setting under the attack of rewards and cannot improve the robustness in the general and popular contextual bandit environment. In addition, none of the existing methods can defend against attacked context. In this work, we provide t…
We introduce a simple and efficient algorithm for unconstrained zeroth-order stochastic convex bandits and prove its regret is at most $(1 + r/d)[d^{1.5} \sqrt{n} + d^3] polylog(n, d, r)$ where $n$ is the horizon, $d$ the dimension and $r$ is the radius of a known ball containing the minimiser of the loss.
University of Torontoland（多伦多兰大学）；University of Edenborrow（伊登伯勒大学）
We study the stochastic Multi-Armed Bandit (MAB) problem with random delays in the feedback received by the algorithm. We consider two settings: the reward-dependent delay setting, where realized delays may depend on the stochastic rewards, and the reward-independent delay setting. Our main contribution is algorithms that achieve near-optimal regret in each of the settings, with an additional additive dependence on the quantiles of the delay distribution. Our results do not make any assumptions on the delay distributions: in particular, we do not assume they come from any parametric family of …
UC San Diego（加州大学圣地亚哥分校）；West Virginia University（西弗吉尼亚大学）
We propose a novel modification of the standard upper confidence bound (UCB) method for the stochastic multi-armed bandit (MAB) problem which tunes the confidence bound of a given bandit based on its distance to others. Our UCB distance tuning (UCB-DT) formulation enables improved performance as measured by expected regret by preventing the MAB algorithm from focusing on non-optimal bandits which is a well-known deficiency of standard UCB. "Distance tuning" of the standard UCB is done using a proposed distance measure, which we call bandit distance, that is parameterizable and which therefore …
In this paper, we analyze the continuous armed bandit problems for nonconvex cost functions under certain smoothness and sublevel set assumptions. We first derive an upper bound on the expected cumulative regret of a simple bin splitting method. We then propose an adaptive bin splitting method, which can significantly improve the performance. Furthermore, a minimax lower bound is derived, which shows that our new adaptive method achieves locally minimax optimal expected cumulative regret.
National Chiao Tung University（国立交通大学）；Texas A&M University（德克萨斯农工大学）
Modifying the reward-biased maximum likelihood method originally proposed in the adaptive control literature, we propose novel learning algorithms to handle the explore-exploit trade-off in linear bandits problems as well as generalized linear bandits problems. We develop novel index policies that we prove achieve order-optimality, and show that they achieve empirical performance competitive with the state-of-the-art benchmark methods in extensive experiments. The new policies achieve this with low computation time per pull for linear bandits, and thereby resulting in both favorable regret as …
University of Copenhagen（哥本哈根大学）；ICREA（加泰罗尼亚高等研究院）；Pompeu Fabra University（庞培法布拉大学）
We present a new strategy for gap estimation in randomized algorithms for multiarmed bandits and combine it with the EXP3++ algorithm of Seldin and Slivkins (2014). In the stochastic regime the strategy reduces dependence of regret on a time horizon from $(\ln t)^3$ to $(\ln t)^2$ and eliminates an additive factor of order $Δe^{1/Δ^2}$, where $Δ$ is the minimal gap of a problem instance. In the adversarial regime regret guarantee remains unchanged.
Federated learning has emerged as an innovative paradigm of collaborative machine learning. Unlike conventional machine learning, a global model is collaboratively learned while data remains distributed over a tremendous number of client devices, thus not compromising user privacy. However, several challenges still remain despite its glowing popularity; above all, the global aggregation in federated learning involves the challenge of biased model averaging and lack of prior knowledge in client sampling, which, in turn, leads to high generalization error and slow convergence rate, respectively.…
University of California, Los Angeles（加州大学洛杉矶分校）；Purdue University（普渡大学）
We propose a new bootstrap-based online algorithm for stochastic linear bandit problems. The key idea is to adopt residual bootstrap exploration, in which the agent estimates the next step reward by re-sampling the residuals of mean reward estimate. Our algorithm, residual bootstrap exploration for stochastic linear bandit (\texttt{LinReBoot}), estimates the linear reward from its re-sampling distribution and pulls the arm with the highest reward estimate. In particular, we contribute a theoretical framework to demystify residual bootstrap-based exploration mechanisms in stochastic linear band…
We analyze the $K$-armed bandit problem where the reward for each arm is a noisy realization based on an observed context under mild nonparametric assumptions. We attain tight results for top-arm identification and a sublinear regret of $\widetilde{O}\Big(T^{\frac{1+D}{2+D}}\Big)$, where $D$ is the context dimension, for a modified UCB algorithm that is simple to implement ($k$NN-UCB). We then give global intrinsic dimension dependent and ambient dimension independent regret bounds. We also discuss recovering topological structures within the context space based on expected bandit performance …
The Impact of Batch Learning in Stochastic Linear Bandits
Eindhoven University of Technology（埃因霍温理工大学）；Jheronimus Academy of Data Science（赫罗尼姆斯数据科学学院）；Tilburg University（蒂尔堡大学）；University of Jyväskylä（于韦斯屈莱大学）
