We solve the COLT 2013 open problem of \citet{SCB} on minimizing regret in the setting of advice-efficient multiarmed bandits with expert advice. We give an algorithm for the setting of K arms and N experts out of which we are allowed to query and use only M experts' advices in each round, which has a regret bound of \tilde{O}\bigP{\sqrt{\frac{\min\{K, M\} N}{M} T}} after T rounds. We also prove that any algorithm for this problem must have expected regret at least \tildeΩ\bigP{\sqrt{\frac{\min\{K, M\} N}{M}T}}, thus showing that our upper bound is nearly tight.
We study the non-stationary stochastic multiarmed bandit (MAB) problem and propose two generic algorithms, namely, the limited memory deterministic sequencing of exploration and exploitation (LM-DSEE) and the Sliding-Window Upper Confidence Bound# (SW-UCB#). We rigorously analyze these algorithms in abruptly-changing and slowly-varying environments and characterize their performance. We show that the expected cumulative regret for these algorithms under either of the environments is upper bounded by sublinear functions of time, i.e., the time average of the regret asymptotically converges to z…
We study the multi-player stochastic multiarmed bandit (MAB) problem in an abruptly changing environment. We consider a collision model in which a player receives reward at an arm if it is the only player to select the arm. We design two novel algorithms, namely, Round-Robin Sliding-Window Upper Confidence Bound\# (RR-SW-UCB\#), and the Sliding-Window Distributed Learning with Prioritization (SW-DLP). We rigorously analyze these algorithms and show that the expected cumulative group regret for these algorithms is upper bounded by sublinear functions of time, i.e., the time average of the regre…
We study the nonstationary stochastic Multi-Armed Bandit (MAB) problem in which the distribution of rewards associated with each arm are assumed to be time-varying and the total variation in the expected rewards is subject to a variation budget. The regret of a policy is defined by the difference in the expected cumulative rewards obtained using the policy and using an oracle that selects the arm with the maximum mean reward at each time. We characterize the performance of the proposed policies in terms of the worst-case regret, which is the supremum of the regret over the set of reward distri…
Princeton University（普林斯顿大学）；University of Pennsylvania（宾夕法尼亚大学）
We consider the correlated multiarmed bandit (MAB) problem in which the rewards associated with each arm are modeled by a multivariate Gaussian random variable, and we investigate the influence of the assumptions in the Bayesian prior on the performance of the upper credible limit (UCL) algorithm and a new correlated UCL algorithm. We rigorously characterize the influence of accuracy, confidence, and correlation scale in the prior on the decision-making performance of the algorithms. Our results show how priors and correlation structure can be leveraged to improve performance.
We consider a multiobjective multiarmed bandit problem with lexicographically ordered objectives. In this problem, the goal of the learner is to select arms that are lexicographic optimal as much as possible without knowing the arm reward distributions beforehand. We capture this goal by defining a multidimensional form of regret that measures the loss of the learner due to not selecting lexicographic optimal arms, and then, consider two settings where the learner has prior information on the expected arm rewards. In the first setting, the learner only knows for each objective the lexicographi…
Max Planck Institute（马克斯·普朗克研究所）；Université Laval（拉瓦尔大学）；University College London（伦敦大学学院）；University of Leoben（莱奥本大学）
We present two alternative ways to apply PAC-Bayesian analysis to sequences of dependent random variables. The first is based on a new lemma that enables to bound expectations of convex functions of certain dependent random variables by expectations of the same functions of independent Bernoulli random variables. This lemma provides an alternative tool to Hoeffding-Azuma inequality to bound concentration of martingale values. Our second approach is based on integration of Hoeffding-Azuma inequality with PAC-Bayesian analysis. We also introduce a way to apply PAC-Bayesian analysis in situation …
University of Pennsylvania（宾夕法尼亚大学）；Duke University（杜克大学）；Google（谷歌）
In this paper we initiate the study of optimization of bandit type problems in scenarios where the feedback of a play is not immediately known. This arises naturally in allocation problems which have been studied extensively in the literature, albeit in the absence of delays in the feedback. We study this problem in the Bayesian setting. In presence of delays, no solution with provable guarantees is known to exist with sub-exponential running time. We show that bandit problems with delayed feedback that arise in allocation settings can be forced to have significant structure, with a slight los…
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…
In this paper we study the online learning problem involving rested and restless multiarmed bandits with multiple plays. The system consists of a single player/user and a set of K finite-state discrete-time Markov chains (arms) with unknown state spaces and statistics. At each time step the player can play M arms. The objective of the user is to decide for each step which M of the K arms to play over a sequence of trials so as to maximize its long term reward. The restless multiarmed bandit is particularly relevant to the application of opportunistic spectrum access (OSA), where a (secondary) …
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.
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Chair for Information Technology, University of Leoben（莱奥本大学信息技术教席）；Université Laval（拉瓦尔大学）；University College London（伦敦大学学院）
We develop a new tool for data-dependent analysis of the exploration-exploitation trade-off in learning under limited feedback. Our tool is based on two main ingredients. The first ingredient is a new concentration inequality that makes it possible to control the concentration of weighted averages of multiple (possibly uncountably many) simultaneously evolving and interdependent martingales. The second ingredient is an application of this inequality to the exploration-exploitation trade-off via importance weighted sampling. We apply the new tool to the stochastic multiarmed bandit problem, how…
University of Southern California（南加州大学）；University of California, Berkeley（加州大学伯克利分校）
We consider the problem of learning in single-player and multiplayer multiarmed bandit models. Bandit problems are classes of online learning problems that capture exploration versus exploitation tradeoffs. In a multiarmed bandit model, players can pick among many arms, and each play of an arm generates an i.i.d. reward from an unknown distribution. The objective is to design a policy that maximizes the expected reward over a time horizon for a single player setting and the sum of expected rewards for the multiplayer setting. In the multiplayer setting, arms may give different rewards to diffe…
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研究所）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立理工学院）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；IBM Research（IBM研究院）；University of Pennsylvania（宾夕法尼亚大学）
We address the problem of \emph{`Internal Regret'} in \emph{Sleeping Bandits} in the fully adversarial setup, as well as draw connections between different existing notions of sleeping regrets in the multiarmed bandits (MAB) literature and consequently analyze the implications: Our first contribution is to propose the new notion of \emph{Internal Regret} for sleeping MAB. We then proposed an algorithm that yields sublinear regret in that measure, even for a completely adversarial sequence of losses and availabilities. We further show that a low sleeping internal regret always implies a low ext…
In many platforms, user arrivals exhibit a self-reinforcing behavior: future user arrivals are likely to have preferences similar to users who were satisfied in the past. In other words, arrivals exhibit positive externalities. We study multiarmed bandit (MAB) problems with positive externalities. We show that the self-reinforcing preferences may lead standard benchmark algorithms such as UCB to exhibit linear regret. We develop a new algorithm, Balanced Exploration (BE), which explores arms carefully to avoid suboptimal convergence of arrivals before sufficient evidence is gathered. We also i…
We provide the first algorithm for online bandit linear optimization whose regret after T rounds is of order sqrt{Td ln N} on any finite class X of N actions in d dimensions, and of order d*sqrt{T} (up to log factors) when X is infinite. These bounds are not improvable in general. The basic idea utilizes tools from convex geometry to construct what is essentially an optimal exploration basis. We also present an application to a model of linear bandits with expert advice. Interestingly, these results show that bandit linear optimization with expert advice in d dimensions is no more difficult (i…
Online Resource Allocation: Bandits feedback and Advice on Time-varying Demands
Institute of Operations Research and Analytics, National University of Singapore（新加坡国立大学运筹学与分析研究院）；Department of Industrial Systems Engineering and Management, National University of Singapore（新加坡国立大学工业系统工程与管理系）
University of Edinburgh（爱丁堡大学）；University College London（伦敦大学学院）；Nara Institute of Science and Technology（奈良先端科学技术大学院大学）
Active learning aims to reduce annotation cost by predicting which samples are useful for a human teacher to label. However it has become clear there is no best active learning algorithm. Inspired by various philosophies about what constitutes a good criteria, different algorithms perform well on different datasets. This has motivated research into ensembles of active learners that learn what constitutes a good criteria in a given scenario, typically via multi-armed bandit algorithms. Though algorithm ensembles can lead to better results, they overlook the fact that not only does algorithm eff…
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…
Bilkent University（比尔肯特大学）；Massachusetts Institute of Technology（麻省理工学院）
We investigate the adversarial bandit problem with multiple plays under semi-bandit feedback. We introduce a highly efficient algorithm that asymptotically achieves the performance of the best switching $m$-arm strategy with minimax optimal regret bounds. To construct our algorithm, we introduce a new expert advice algorithm for the multiple-play setting. By using our expert advice algorithm, we additionally improve the best-known high-probability bound for the multi-play setting by $O(\sqrt{m})$. Our results are guaranteed to hold in an individual sequence manner since we have no statistical …
In this paper, we study a variant of the framework of online learning using expert advice with limited/bandit feedback. We consider each expert as a learning entity, seeking to more accurately reflecting certain real-world applications. In our setting, the feedback at any time $t$ is limited in a sense that it is only available to the expert $i^t$ that has been selected by the central algorithm (forecaster), \emph{i.e.}, only the expert $i^t$ receives feedback from the environment and gets to learn at time $t$. We consider a generic black-box approach whereby the forecaster does not control or…
We present a data-driven algorithm that advertisers can use to automate their digital ad-campaigns at online publishers. The algorithm enables the advertiser to search across available target audiences and ad-media to find the best possible combination for its campaign via online experimentation. The problem of finding the best audience-ad combination is complicated by a number of distinctive challenges, including (a) a need for active exploration to resolve prior uncertainty and to speed the search for profitable combinations, (b) many combinations to choose from, giving rise to high-dimensio…
University of Copenhagen（哥本哈根大学）；Università degli Studi di Milano（米兰大学）
We propose an algorithm for stochastic and adversarial multiarmed bandits with switching costs, where the algorithm pays a price $λ$ every time it switches the arm being played. Our algorithm is based on adaptation of the Tsallis-INF algorithm of Zimmert and Seldin (2021) and requires no prior knowledge of the regime or time horizon. In the oblivious adversarial setting it achieves the minimax optimal regret bound of $O\big((λK)^{1/3}T^{2/3} + \sqrt{KT}\big)$, where $T$ is the time horizon and $K$ is the number of arms. In the stochastically constrained adversarial regime, which includes the s…
Recent growing adoption of experimentation in practice has led to a surge of attention to multiarmed bandits as a technique to reduce the opportunity cost of online experiments. In this setting, a decision-maker sequentially chooses among a set of given actions, observes their noisy rewards, and aims to maximize her cumulative expected reward (or minimize regret) over a horizon of length $T$. In this paper, we introduce a general analysis framework and a family of algorithms for the stochastic linear bandit problem that includes well-known algorithms such as the optimism-in-the-face-of-uncerta…
Princeton University（普林斯顿大学）；Università degli Studi di Milano（米兰大学）；Microsoft Research New England（微软新英格兰研究院）；Wharton School, University of Pennsylvania（宾夕法尼亚大学沃顿商学院）
We address the online linear optimization problem with bandit feedback. Our contribution is twofold. First, we provide an algorithm (based on exponential weights) with a regret of order $\sqrt{d n \log N}$ for any finite action set with $N$ actions, under the assumption that the instantaneous loss is bounded by 1. This shaves off an extraneous $\sqrt{d}$ factor compared to previous works, and gives a regret bound of order $d \sqrt{n \log n}$ for any compact set of actions. Without further assumptions on the action set, this last bound is minimax optimal up to a logarithmic factor. Interestingl…
A multi-access wireless network with N transmitting nodes, each equipped with an energy harvesting (EH) device and a rechargeable battery of finite capacity, is studied. At each time slot (TS) a node is operative with a certain probability, which may depend on the availability of data, or the state of its channel. The energy arrival process at each node is modelled as an independent two-state Markov process, such that, at each TS, a node either harvests one unit of energy, or none. At each TS a subset of the nodes is scheduled by the access point (AP). The scheduling policy that maximises the …
University of Copenhagen（哥本哈根大学）；Google Research（谷歌研究院）
We present a modified tuning of the algorithm of Zimmert and Seldin [2020] for adversarial multiarmed bandits with delayed feedback, which in addition to the minimax optimal adversarial regret guarantee shown by Zimmert and Seldin simultaneously achieves a near-optimal regret guarantee in the stochastic setting with fixed delays. Specifically, the adversarial regret guarantee is $\mathcal{O}(\sqrt{TK} + \sqrt{dT\log K})$, where $T$ is the time horizon, $K$ is the number of arms, and $d$ is the fixed delay, whereas the stochastic regret guarantee is $\mathcal{O}\left(\sum_{i \neq i^*}(\frac{1}{…
Tel Aviv University（特拉维夫大学）；Microsoft Research（微软研究院）；Università degli Studi di Milano（米兰大学）；Technion—Israel Institute of Technology（以色列理工学院）
We study a general class of online learning problems where the feedback is specified by a graph. This class includes online prediction with expert advice and the multi-armed bandit problem, but also several learning problems where the online player does not necessarily observe his own loss. We analyze how the structure of the feedback graph controls the inherent difficulty of the induced $T$-round learning problem. Specifically, we show that any feedback graph belongs to one of three classes: strongly observable graphs, weakly observable graphs, and unobservable graphs. We prove that the first…
In this work we consider adversarial contextual bandits with risk constraints. At each round, nature prepares a context, a cost for each arm, and additionally a risk for each arm. The learner leverages the context to pull an arm and then receives the corresponding cost and risk associated with the pulled arm. In addition to minimizing the cumulative cost, the learner also needs to satisfy long-term risk constraints -- the average of the cumulative risk from all pulled arms should not be larger than a pre-defined threshold. To address this problem, we first study the full information setting wh…
Mathematics of statistical sequential decision-making: concentration, risk-awareness and modelling in stochastic bandits, with applications to bariatric surgery
Université de Lille（里尔大学）；Université de Technologies de Troyes（特鲁瓦技术大学）；ENSAI Rennes（雷恩国立统计与信息分析学校）；Inria Montpellier（法国国家信息与自动化研究所蒙彼利埃中心）；CWI Amsterdam（荷兰国家数学与计算机科学研究中心）；Université de Bordeaux（波尔多大学）；Inria Lille（法国国家信息与自动化研究所里尔中心）；CHU Lille（里尔大学附属医院）；Centre de Recherche en Informatique, Signal et Automatique de Lille（里尔计算机、信号与自动化研究中心）
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…
Algorithm Selection as a Bandit Problem with Unbounded Losses
IDSIA / USI-SUPSI Istituto Dalle Molle di studi sull’intelligenza artificiale（IDSIA / USI-SUPSI 达莱莫莱人工智能研究所）；Università della Svizzera italiana (USI)（瑞士意大利语区大学）；University of Lugano（卢加诺大学）；TU Munich（慕尼黑工业大学）
Università degli Studi di Milano（米兰大学）；Microsoft Research（微软研究院）；Weizmann Institute（魏茨曼科学研究所）
We study the power of different types of adaptive (nonoblivious) adversaries in the setting of prediction with expert advice, under both full-information and bandit feedback. We measure the player's performance using a new notion of regret, also known as policy regret, which better captures the adversary's adaptiveness to the player's behavior. In a setting where losses are allowed to drift, we characterize ---in a nearly complete manner--- the power of adaptive adversaries with bounded memories and switching costs. In particular, we show that with switching costs, the attainable rate with ban…
This paper considers two fundamental sequential decision-making problems: the problem of prediction with expert advice and the multi-armed bandit problem. We focus on stochastic regimes in which an adversary may corrupt losses, and we investigate what level of robustness can be achieved against adversarial corruptions. The main contribution of this paper is to show that optimal robustness can be expressed by a square-root dependency on the amount of corruption. More precisely, we show that two classes of algorithms, anytime Hedge with decreasing learning rate and algorithms with second-order r…
Università degli Studi di Milano（米兰大学）；Politecnico di Milano（米兰理工大学）
We investigate the problem of bandits with expert advice when the experts are fixed and known distributions over the actions. Improving on previous analyses, we show that the regret in this setting is controlled by information-theoretic quantities that measure the similarity between experts. In some natural special cases, this allows us to obtain the first regret bound for EXP4 that can get arbitrarily close to zero if the experts are similar enough. While for a different algorithm, we provide another bound that describes the similarity between the experts in terms of the KL-divergence, and we…
Improved Regret Bounds for Bandits with Expert Advice
Università degli Studi di Milano（米兰大学）；Politecnico di Milano（米兰理工大学）；Politecnico di Torino（都灵理工大学）；Istituto Italiano di Tecnologia（意大利技术研究院）；TU Delft（代尔夫特理工大学）
Peer Learning: Learning Complex Policies in Groups from Scratch via Action Recommendations
Hessian Center for Artificial Intelligence (hessian.AI)（黑森人工智能中心）；German Research Center for AI (DFKI)（德国人工智能研究中心）；Centre for Cognitive Science（认知科学中心）
Chennai Mathematical Institute（金奈数学研究所）；Indian Institute of Science（印度科学学院）
We study how to adapt to smoothly-varying ('easy') environments in well-known online learning problems where acquiring information is expensive. For the problem of label efficient prediction, which is a budgeted version of prediction with expert advice, we present an online algorithm whose regret depends optimally on the number of labels allowed and $Q^*$ (the quadratic variation of the losses of the best action in hindsight), along with a parameter-free counterpart whose regret depends optimally on $Q$ (the quadratic variation of the losses of all the actions). These quantities can be signifi…
We study the problem of expert advice under partial bandit feedback setting and create a sequential minimax optimal algorithm. Our algorithm works with a more general partial monitoring setting, where, in contrast to the classical bandit feedback, the losses can be revealed in an adversarial manner. Our algorithm adopts a universal prediction perspective, whose performance is analyzed with regret against a general expert selection sequence. The regret we study is against a general competition class that covers many settings (such as the switching or contextual experts settings) and the expert …
University of Utah（犹他大学）；Google Research（谷歌研究院）；Duke University（杜克大学）
We consider the classic online learning and stochastic multi-armed bandit (MAB) problems, when at each step, the online policy can probe and find out which of a small number ($k$) of choices has better reward (or loss) before making its choice. In this model, we derive algorithms whose regret bounds have exponentially better dependence on the time horizon compared to the classic regret bounds. In particular, we show that probing with $k=2$ suffices to achieve time-independent regret bounds for online linear and convex optimization. The same number of probes improve the regret bound of stochast…
