In $\mathcal{X}$-armed bandit problem an agent sequentially interacts with environment which yields a reward based on the vector input the agent provides. The agent's goal is to maximise the sum of these rewards across some number of time steps. The problem and its variations have been a subject of numerous studies, suggesting sub-linear and some times optimal strategies. The given paper introduces a novel variation of the problem. We consider an environment, which can abruptly change its behaviour an unknown number of times. To that end we propose a novel strategy and prove it attains sub-linear cumulative regret. Moreover, in case of highly smooth relation between an action and the corresponding reward, the method is nearly optimal. The theoretical result are supported by experimental study.
Princeton University（普林斯顿大学）；Université Paris Diderot（巴黎狄德罗大学）
We study the stochastic multi-armed bandit problem when one knows the value $μ^{(\star)}$ of an optimal arm, as a well as a positive lower bound on the smallest positive gap $Δ$. We propose a new randomized policy that attains a regret {\em uniformly bounded over time} in this setting. We also prove several lower bounds, which show in particular that bounded regret is not possible if one only knows $Δ$, and bounded regret of order $1/Δ$ is not possible if one only knows $μ^{(\star)}$
I analyse the frequentist regret of the famous Gittins index strategy for multi-armed bandits with Gaussian noise and a finite horizon. Remarkably it turns out that this approach leads to finite-time regret guarantees comparable to those available for the popular UCB algorithm. Along the way I derive finite-time bounds on the Gittins index that are asymptotically exact and may be of independent interest. I also discuss some computational issues and present experimental results suggesting that a particular version of the Gittins index strategy is a modest improvement on existing algorithms with…
The multi-armed bandit is a concise model for the problem of iterated decision-making under uncertainty. In each round, a gambler must pull one of $K$ arms of a slot machine, without any foreknowledge of their payouts, except that they are uniformly bounded. A standard objective is to minimize the gambler's regret, defined as the gambler's total payout minus the largest payout which would have been achieved by any fixed arm, in hindsight. Note that the gambler is only told the payout for the arm actually chosen, not for the unchosen arms. Almost all previous work on this problem assumed the pa…
Columbia University（哥伦比亚大学）；Stanford University（斯坦福大学）
In a multi-armed bandit (MAB) problem a gambler needs to choose at each round of play one of K arms, each characterized by an unknown reward distribution. Reward realizations are only observed when an arm is selected, and the gambler's objective is to maximize his cumulative expected earnings over some given horizon of play T. To do this, the gambler needs to acquire information about arms (exploration) while simultaneously optimizing immediate rewards (exploitation); the price paid due to this trade off is often referred to as the regret, and the main question is how small can this price be a…
INRIA Lille – Nord Europe（法国国家信息与自动化研究所里尔-北欧洲分所）；Ecole Normale Supérieure（巴黎高等师范学院）；CNRS（法国国家科学研究中心）；HEC Paris（巴黎高等商学院）
We consider the framework of stochastic multi-armed bandit problems and study the possibilities and limitations of forecasters that perform an on-line exploration of the arms. These forecasters are assessed in terms of their simple regret, a regret notion that captures the fact that exploration is only constrained by the number of available rounds (not necessarily known in advance), in contrast to the case when the cumulative regret is considered and when exploitation needs to be performed at the same time. We believe that this performance criterion is suited to situations when the cost of pul…
I present the first algorithm for stochastic finite-armed bandits that simultaneously enjoys order-optimal problem-dependent regret and worst-case regret. Besides the theoretical results, the new algorithm is simple, efficient and empirically superb. The approach is based on UCB, but with a carefully chosen confidence parameter that optimally balances the risk of failing confidence intervals against the cost of excessive optimism.
We consider a novel multi-armed bandit framework where the rewards obtained by pulling the arms are functions of a common latent random variable. The correlation between arms due to the common random source can be used to design a generalized upper-confidence-bound (UCB) algorithm that identifies certain arms as $non-competitive$, and avoids exploring them. As a result, we reduce a $K$-armed bandit problem to a $C+1$-armed problem, where $C+1$ includes the best arm and $C$ $competitive$ arms. Our regret analysis shows that the competitive arms need to be pulled $\mathcal{O}(\log T)$ times, whi…
University of Alberta（阿尔伯塔大学）；INRIA（法国国家信息与自动化研究所）
We study a new type of K-armed bandit problem where the expected return of one arm may depend on the returns of other arms. We present a new algorithm for this general class of problems and show that under certain circumstances it is possible to achieve finite expected cumulative regret. We also give problem-dependent lower bounds on the cumulative regret showing that at least in special cases the new algorithm is nearly optimal.
In this paper, we consider the distributed stochastic multi-armed bandit problem, where a global arm set can be accessed by multiple players independently. The players are allowed to exchange their history of observations with each other at specific points in time. We study the relationship between regret and communication. When the time horizon is known, we propose the Over-Exploration strategy, which only requires one-round communication and whose regret does not scale with the number of players. When the time horizon is unknown, we measure the frequency of communication through a new notion…
We discuss a multiple-play multi-armed bandit (MAB) problem in which several arms are selected at each round. Recently, Thompson sampling (TS), a randomized algorithm with a Bayesian spirit, has attracted much attention for its empirically excellent performance, and it is revealed to have an optimal regret bound in the standard single-play MAB problem. In this paper, we propose the multiple-play Thompson sampling (MP-TS) algorithm, an extension of TS to the multiple-play MAB problem, and discuss its regret analysis. We prove that MP-TS for binary rewards has the optimal regret upper bound that…
University of Cambridge（剑桥大学）；INRIA Lille - Nord Europe（法国国家信息与自动化研究所里尔-北欧中心）
We consider a stochastic bandit problem with infinitely many arms. In this setting, the learner has no chance of trying all the arms even once and has to dedicate its limited number of samples only to a certain number of arms. All previous algorithms for this setting were designed for minimizing the cumulative regret of the learner. In this paper, we propose an algorithm aiming at minimizing the simple regret. As in the cumulative regret setting of infinitely many armed bandits, the rate of the simple regret will depend on a parameter $β$ characterizing the distribution of the near-optimal arm…
Microsoft Research（微软研究院）；University of British Columbia（不列颠哥伦比亚大学）
We consider two agents playing simultaneously the same stochastic three-armed bandit problem. The two agents are cooperating but they cannot communicate. We propose a strategy with no collisions at all between the players (with very high probability), and with near-optimal regret $O(\sqrt{T \log(T)})$. We also argue that the extra logarithmic term $\sqrt{\log(T)}$ should be necessary by proving a lower bound for a full information variant of the problem.
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…
Technion – Institute of Technology（以色列理工学院）；Nvidia Research（英伟达研究院）
We consider the Multi-Armed Bandit (MAB) problem, where an agent sequentially chooses actions and observes rewards for the actions it took. While the majority of algorithms try to minimize the regret, i.e., the cumulative difference between the reward of the best action and the agent's action, this criterion might lead to undesirable results. For example, in large problems, or when the interaction with the environment is brief, finding an optimal arm is infeasible, and regret-minimizing algorithms tend to over-explore. To overcome this issue, algorithms for such settings should instead focus o…
Sri Lanka Technological Campus（斯里兰卡科技园区）；University of Peradeniya（佩拉德尼亚大学）
This paper considers a multi-armed bandit (MAB) problem in which multiple mobile agents receive rewards by sampling from a collection of spatially dispersed stochastic processes, called bandits. The goal is to formulate a decentralized policy for each agent, in order to maximize the total cumulative reward over all agents, subject to option availability and inter-agent communication constraints. The problem formulation is motivated by applications in which a team of autonomous mobile robots cooperates to accomplish an exploration and exploitation task in an uncertain environment. Bandit locati…
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 …
In this paper, we study the multi-armed bandit problem in the batched setting where the employed policy must split data into a small number of batches. While the minimax regret for the two-armed stochastic bandits has been completely characterized in \cite{perchet2016batched}, the effect of the number of arms on the regret for the multi-armed case is still open. Moreover, the question whether adaptively chosen batch sizes will help to reduce the regret also remains underexplored. In this paper, we propose the BaSE (batched successive elimination) policy to achieve the rate-optimal regrets (wit…
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 consider the problem of finitely parameterized multi-armed bandits where the model of the underlying stochastic environment can be characterized based on a common unknown parameter. The true parameter is unknown to the learning agent. However, the set of possible parameters, which is finite, is known a priori. We propose an algorithm that is simple and easy to implement, which we call Finitely Parameterized Upper Confidence Bound (FP-UCB) algorithm, which uses the information about the underlying parameter set for faster learning. In particular, we show that the FP-UCB algorithm achieves a …
National University of Singapore（新加坡国立大学）；California Institute of Technology（加州理工学院）
We study the regret of Thompson sampling (TS) algorithms for exponential family bandits, where the reward distribution is from a one-dimensional exponential family, which covers many common reward distributions including Bernoulli, Gaussian, Gamma, Exponential, etc. We propose a Thompson sampling algorithm, termed ExpTS, which uses a novel sampling distribution to avoid the under-estimation of the optimal arm. We provide a tight regret analysis for ExpTS, which simultaneously yields both the finite-time regret bound as well as the asymptotic regret bound. In particular, for a $K$-armed bandit …
We extend the classic multi-armed bandit (MAB) model to the setting of noncompliance, where the arm pull is a mere instrument and the treatment applied may differ from it, which gives rise to the instrument-armed bandit (IAB) problem. The IAB setting is relevant whenever the experimental units are human since free will, ethics, and the law may prohibit unrestricted or forced application of treatment. In particular, the setting is relevant in bandit models of dynamic clinical trials and other controlled trials on human interventions. Nonetheless, the setting has not been fully investigate in th…
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.
In this paper, we study the combinatorial multi-armed bandit problem (CMAB) with probabilistically triggered arms (PTAs). Under the assumption that the arm triggering probabilities (ATPs) are positive for all arms, we prove that a class of upper confidence bound (UCB) policies, named Combinatorial UCB with exploration rate $κ$ (CUCB-$κ$), and Combinatorial Thompson Sampling (CTS), which estimates the expected states of the arms via Thompson sampling, achieve bounded regret. In addition, we prove that CUCB-$0$ and CTS incur $O(\sqrt{T})$ gap-independent regret. These results improve the results…
In this paper, we propose a new multi-armed bandit problem called the Gambler's Ruin Bandit Problem (GRBP). In the GRBP, the learner proceeds in a sequence of rounds, where each round is a Markov Decision Process (MDP) with two actions (arms): a continuation action that moves the learner randomly over the state space around the current state; and a terminal action that moves the learner directly into one of the two terminal states (goal and dead-end state). The current round ends when a terminal state is reached, and the learner incurs a positive reward only when the goal state is reached. The…
Recent work on follow the perturbed leader (FTPL) algorithms for the adversarial multi-armed bandit problem has highlighted the role of the hazard rate of the distribution generating the perturbations. Assuming that the hazard rate is bounded, it is possible to provide regret analyses for a variety of FTPL algorithms for the multi-armed bandit problem. This paper pushes the inquiry into regret bounds for FTPL algorithms beyond the bounded hazard rate condition. There are good reasons to do so: natural distributions such as the uniform and Gaussian violate the condition. We give regret bounds f…
In this paper, we propose a constant word (RAM model) algorithm for regret minimisation for both finite and infinite Stochastic Multi-Armed Bandit (MAB) instances. Most of the existing regret minimisation algorithms need to remember the statistics of all the arms they encounter. This may become a problem for the cases where the number of available words of memory is limited. Designing an efficient regret minimisation algorithm that uses a constant number of words has long been interesting to the community. Some early attempts consider the number of arms to be infinite, and require the reward d…
We consider a stochastic multi-armed bandit setting and study the problem of constrained regret minimization over a given time horizon. Each arm is associated with an unknown, possibly multi-dimensional distribution, and the merit of an arm is determined by several, possibly conflicting attributes. The aim is to optimize a 'primary' attribute subject to user-provided constraints on other 'secondary' attributes. We assume that the attributes can be estimated using samples from the arms' distributions, and that the estimators enjoy suitable concentration properties. We propose an algorithm calle…
We study an important variant of the stochastic multi-armed bandit (MAB) problem, which takes penalization into consideration. Instead of directly maximizing cumulative expected reward, we need to balance between the total reward and fairness level. In this paper, we present some new insights in MAB and formulate the problem in the penalization framework, where rigorous penalized regret can be well defined and more sophisticated regret analysis is possible. Under such a framework, we propose a hard-threshold UCB-like algorithm, which enjoys many merits including asymptotic fairness, nearly opt…
We study an interesting variant of the stochastic multi-armed bandit problem, called the Fair-SMAB problem, where each arm is required to be pulled for at least a given fraction of the total available rounds. We investigate the interplay between learning and fairness in terms of a pre-specified vector denoting the fractions of guaranteed pulls. We define a fairness-aware regret, called r-Regret, that takes into account the above fairness constraints and naturally extends the conventional notion of regret. Our primary contribution is characterizing a class of Fair-SMAB algorithms by two paramet…
Northwestern University（西北大学）；University of Florida（佛罗里达大学）；New York University（纽约大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Linear contextual bandit is an important class of sequential decision making problems with a wide range of applications to recommender systems, online advertising, healthcare, and many other machine learning related tasks. While there is a lot of prior research, tight regret bounds of linear contextual bandit with infinite action sets remain open. In this paper, we address this open problem by considering the linear contextual bandit with (changing) infinite action sets. We prove a regret upper bound on the order of $O(\sqrt{d^2T\log T})\times \text{poly}(\log\log T)$ where $d$ is the domain d…
Boston University（波士顿大学）；Carnegie Mellon University（卡内基梅隆大学）
We investigate a natural but surprisingly unstudied approach to the multi-armed bandit problem under safety risk constraints. Each arm is associated with an unknown law on safety risks and rewards, and the learner's goal is to maximise reward whilst not playing unsafe arms, as determined by a given threshold on the mean risk. We formulate a pseudo-regret for this setting that enforces this safety constraint in a per-round way by softly penalising any violation, regardless of the gain in reward due to the same. This has practical relevance to scenarios such as clinical trials, where one must ma…
We study an interesting variant of the stochastic multi-armed bandit problem, called the Fair-SMAB problem, where each arm is required to be pulled for at least a given fraction of the total available rounds. We investigate the interplay between learning and fairness in terms of a pre-specified vector denoting the fractions of guaranteed pulls. We define a fairness-aware regret, called $r$-Regret, that takes into account the above fairness constraints and naturally extends the conventional notion of regret. Our primary contribution is characterizing a class of Fair-SMAB algorithms by two param…
Recently a multi-agent variant of the classical multi-armed bandit was proposed to tackle fairness issues in online learning. Inspired by a long line of work in social choice and economics, the goal is to optimize the Nash social welfare instead of the total utility. Unfortunately previous algorithms either are not efficient or achieve sub-optimal regret in terms of the number of rounds $T$. We propose a new efficient algorithm with lower regret than even previous inefficient ones. For $N$ agents, $K$ arms, and $T$ rounds, our approach has a regret bound of $\tilde{O}(\sqrt{NKT} + NK)$. This i…
I introduce and analyse an anytime version of the Optimally Confident UCB (OCUCB) algorithm designed for minimising the cumulative regret in finite-armed stochastic bandits with subgaussian noise. The new algorithm is simple, intuitive (in hindsight) and comes with the strongest finite-time regret guarantees for a horizon-free algorithm so far. I also show a finite-time lower bound that nearly matches the upper bound.
We consider the combinatorial multi-armed bandit (CMAB) problem, where the reward function is nonlinear. In this setting, the agent chooses a batch of arms on each round and receives feedback from each arm of the batch. The reward that the agent aims to maximize is a function of the selected arms and their expectations. In many applications, the reward function is highly nonlinear, and the performance of existing algorithms relies on a global Lipschitz constant to encapsulate the function's nonlinearity. This may lead to loose regret bounds, since by itself, a large gradient does not necessari…
We consider a variant of the multi-armed bandit model, which we call multi-armed bandit problem with known trend, where the gambler knows the shape of the reward function of each arm but not its distribution. This new problem is motivated by different online problems like active learning, music and interface recommendation applications, where when an arm is sampled by the model the received reward change according to a known trend. By adapting the standard multi-armed bandit algorithm UCB1 to take advantage of this setting, we propose the new algorithm named A-UCB that assumes a stochastic mod…
CentraleSupélec（中央理工-高等电力学院）；CNRS（法国国家科学研究中心）；Université de Lille（里尔大学）；Inria（法国国家信息与自动化研究所）
An online reinforcement learning algorithm is anytime if it does not need to know in advance the horizon T of the experiment. A well-known technique to obtain an anytime algorithm from any non-anytime algorithm is the "Doubling Trick". In the context of adversarial or stochastic multi-armed bandits, the performance of an algorithm is measured by its regret, and we study two families of sequences of growing horizons (geometric and exponential) to generalize previously known results that certain doubling tricks can be used to conserve certain regret bounds. In a broad setting, we prove that a ge…
Université Paris Diderot（巴黎第七大学）；Princeton University（普林斯顿大学）
We consider a multi-armed bandit problem in a setting where each arm produces a noisy reward realization which depends on an observable random covariate. As opposed to the traditional static multi-armed bandit problem, this setting allows for dynamically changing rewards that better describe applications where side information is available. We adopt a nonparametric model where the expected rewards are smooth functions of the covariate and where the hardness of the problem is captured by a margin parameter. To maximize the expected cumulative reward, we introduce a policy called Adaptively Binn…
In this paper, we consider stochastic multi-armed bandits (MABs) with heavy-tailed rewards, whose $p$-th moment is bounded by a constant $ν_{p}$ for $1<p\leq2$. First, we propose a novel robust estimator which does not require $ν_{p}$ as prior information, while other existing robust estimators demand prior knowledge about $ν_{p}$. We show that an error probability of the proposed estimator decays exponentially fast. Using this estimator, we propose a perturbation-based exploration strategy and develop a generalized regret analysis scheme that provides upper and lower regret bounds by revealin…
LINE Plus Corporation（LINE Plus公司）；Pohang University of Science and Technology（浦项科技大学）
We consider a multi-armed bandit problem in which a set of arms is registered by each agent, and the agent receives reward when its arm is selected. An agent might strategically submit more arms with replications, which can bring more reward by abusing the bandit algorithm's exploration-exploitation balance. Our analysis reveals that a standard algorithm indeed fails at preventing replication and suffers from linear regret in time $T$. We aim to design a bandit algorithm which demotivates replications and also achieves a small cumulative regret. We devise Hierarchical UCB (H-UCB) of replicatio…
Indian Institute of Science（印度科学学院）；Indian Institute of Technology, Kharagpur（印度理工学院克勒格布尔分校）
We extend the notion of regret with a welfarist perspective. Focussing on the classic multi-armed bandit (MAB) framework, the current work quantifies the performance of bandit algorithms by applying a fundamental welfare function, namely the Nash social welfare (NSW) function. This corresponds to equating algorithm's performance to the geometric mean of its expected rewards and leads us to the study of Nash regret, defined as the difference between the -- a priori unknown -- optimal mean (among the arms) and the algorithm's performance. Since NSW is known to satisfy fairness axioms, our approa…
Centre Borelli, ENS Paris-Saclay（巴黎萨克雷高等师范学院博雷利中心）；Cdiscount（Cdiscount公司）；CREST, ENSAE Paris（巴黎经济与统计高等学校统计与经济研究中心）；Criteo AI Lab（Criteo人工智能实验室）
Continuously learning and leveraging the knowledge accumulated from prior tasks in order to improve future performance is a long standing machine learning problem. In this paper, we study the problem in the multi-armed bandit framework with the objective to minimize the total regret incurred over a series of tasks. While most bandit algorithms are designed to have a low worst-case regret, we examine here the average regret over bandit instances drawn from some prior distribution which may change over time. We specifically focus on confidence interval tuning of UCB algorithms. We propose a band…
We consider a stochastic multi-armed bandit (MAB) problem motivated by ``large'' action spaces, and endowed with a population of arms containing exactly $K$ arm-types, each characterized by a distinct mean reward. The decision maker is oblivious to the statistical properties of reward distributions as well as the population-level distribution of different arm-types, and is precluded also from observing the type of an arm after play. We study the classical problem of minimizing the expected cumulative regret over a horizon of play $n$, and propose algorithms that achieve a rate-optimal finite-t…
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…
