In this paper, we consider the problem of multi-armed bandits with a large, possibly infinite number of correlated arms. We assume that the arms have Bernoulli distributed rewards, independent across time, where the probabilities of success are parametrized by known attribute vectors for each arm, as well as an unknown preference vector, each of dimension $n$. For this model, we seek an algorithm with a total regret that is sub-linear in time and independent of the number of arms. We present such an algorithm, which we call the Two-Phase Algorithm, and analyze its performance. We show upper bounds on the total regret which applies uniformly in time, for both the finite and infinite arm cases. The asymptotics of the finite arm bound show that for any $f \in ω(\log(T))$, the total regret can be made to be $O(n \cdot f(T))$. In the infinite arm case, the total regret is $O(\sqrt{n^3 T})$.
Technion – Institute of Technology（以色列理工学院）；Meta；Nvidia Research（英伟达研究院）
We consider a stochastic multi-armed bandit setting where reward must be actively queried for it to be observed. We provide tight lower and upper problem-dependent guarantees on both the regret and the number of queries. Interestingly, we prove that there is a fundamental difference between problems with a unique and multiple optimal arms, unlike in the standard multi-armed bandit problem. We also present a new, simple, UCB-style sampling concept, and show that it naturally adapts to the number of optimal arms and achieves tight regret and querying bounds.
We study the stochastic multi-armed bandit problem with non-equivalent multiple plays where, at each step, an agent chooses not only a set of arms, but also their order, which influences reward distribution. In several problem formulations with different assumptions, we provide lower bounds for regret with standard asymptotics $O(\log{t})$ but novel coefficients and provide optimal algorithms, thus proving that these bounds cannot be improved.
We consider online learning in partial-monitoring games against an oblivious adversary. We show that when the number of actions available to the learner is two and the game is nontrivial then it is reducible to a bandit-like game and thus the minimax regret is $Θ(\sqrt{T})$.
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…
We study Thompson Sampling algorithms for stochastic multi-armed bandits in the batched setting, in which we want to minimize the regret over a sequence of arm pulls using a small number of policy changes (or, batches). We propose two algorithms and demonstrate their effectiveness by experiments on both synthetic and real datasets. We also analyze the proposed algorithms from the theoretical aspect and obtain almost tight regret-batches tradeoffs for the two-arm case.
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.
The multi-armed bandit problems have been studied mainly under the measure of expected total reward accrued over a horizon of length $T$. In this paper, we address the issue of risk in multi-armed bandit problems and develop parallel results under the measure of mean-variance, a commonly adopted risk measure in economics and mathematical finance. We show that the model-specific regret and the model-independent regret in terms of the mean-variance of the reward process are lower bounded by $Ω(\log T)$ and $Ω(T^{2/3})$, respectively. We then show that variations of the UCB policy and the DSEE po…
University of Amsterdam（阿姆斯特丹大学）；INRIA Lille - Nord Europe（法国国家信息与自动化研究所里尔-北欧中心）
This paper proposes a new method for the K-armed dueling bandit problem, a variation on the regular K-armed bandit problem that offers only relative feedback about pairs of arms. Our approach extends the Upper Confidence Bound algorithm to the relative setting by using estimates of the pairwise probabilities to select a promising arm and applying Upper Confidence Bound with the winner as a benchmark. We prove a finite-time regret bound of order O(log t). In addition, our empirical results using real data from an information retrieval application show that it greatly outperforms the state of th…
In this paper, we consider a new Multi-Armed Bandit (MAB) problem where arms are nodes in an unknown and possibly changing graph, and the agent (i) initiates random walks over the graph by pulling arms, (ii) observes the random walk trajectories, and (iii) receives rewards equal to the lengths of the walks. We provide a comprehensive understanding of this problem by studying both the stochastic and the adversarial setting. We show that this problem is not easier than a standard MAB in an information theoretical sense, although additional information is available through random walk trajectorie…
Stochastic multi-armed bandits solve the Exploration-Exploitation dilemma and ultimately maximize the expected reward. Nonetheless, in many practical problems, maximizing the expected reward is not the most desirable objective. In this paper, we introduce a novel setting based on the principle of risk-aversion where the objective is to compete against the arm with the best risk-return trade-off. This setting proves to be intrinsically more difficult than the standard multi-arm bandit setting due in part to an exploration risk which introduces a regret associated to the variability of an algori…
We consider a dynamic pricing problem under unknown demand models. In this problem a seller offers prices to a stream of customers and observes either success or failure in each sale attempt. The underlying demand model is unknown to the seller and can take one of N possible forms. In this paper, we show that this problem can be formulated as a multi-armed bandit with dependent arms. We propose a dynamic pricing policy based on the likelihood ratio test. We show that the proposed policy achieves complete learning, i.e., it offers a bounded regret where regret is defined as the revenue loss wit…
This paper tackles the risk averse multi-armed bandits problem when incurred losses are non-stationary. The conditional value-at-risk (CVaR) is used as the objective function. Two estimation methods are proposed for this objective function in the presence of non-stationary losses, one relying on a weighted empirical distribution of losses and another on the dual representation of the CVaR. Such estimates can then be embedded into classic arm selection methods such as epsilon-greedy policies. Simulation experiments assess the performance of the arm selection algorithms based on the two novel es…
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）；LRI（计算机研究实验室）；Université Paris Sud（巴黎第十一大学）
Motivated by applications in energy management, this paper presents the Multi-Armed Risk-Aware Bandit (MARAB) algorithm. With the goal of limiting the exploration of risky arms, MARAB takes as arm quality its conditional value at risk. When the user-supplied risk level goes to 0, the arm quality tends toward the essential infimum of the arm distribution density, and MARAB tends toward the MIN multi-armed bandit algorithm, aimed at the arm with maximal minimal value. As a first contribution, this paper presents a theoretical analysis of the MIN algorithm under mild assumptions, establishing its…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We study identifying user clusters in contextual multi-armed bandits (MAB). Contextual MAB is an effective tool for many real applications, such as content recommendation and online advertisement. In practice, user dependency plays an essential role in the user's actions, and thus the rewards. Clustering similar users can improve the quality of reward estimation, which in turn leads to more effective content recommendation and targeted advertising. Different from traditional clustering settings, we cluster users based on the unknown bandit parameters, which will be estimated incrementally. In …
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 …
MARBLE: Multi-Armed Restless Bandits in Latent Markovian Environment
Department of Computer and Systems Sciences (DSV), Stockholm University（斯德哥尔摩大学计算机与系统科学系）；Inria Sophia Antipolis（法国国家信息与自动化研究所索菲亚·安蒂波利斯研究中心）；Smartprofile（Smartprofile公司）
We study exploration in Multi-Armed Bandits in a setting where $k$ players collaborate in order to identify an $ε$-optimal arm. Our motivation comes from recent employment of bandit algorithms in computationally intensive, large-scale applications. Our results demonstrate a non-trivial tradeoff between the number of arm pulls required by each of the players, and the amount of communication between them. In particular, our main result shows that by allowing the $k$ players to communicate only once, they are able to learn $\sqrt{k}$ times faster than a single player. That is, distributing learni…
Technion（以色列理工学院）；Microsoft Research New England（微软新英格兰研究院）
We consider a multi-armed bandit problem where the decision maker can explore and exploit different arms at every round. The exploited arm adds to the decision maker's cumulative reward (without necessarily observing the reward) while the explored arm reveals its value. We devise algorithms for this setup and show that the dependence on the number of arms, k, can be much better than the standard square root of k dependence, depending on the behavior of the arms' reward sequences. For the important case of piecewise stationary stochastic bandits, we show a significant improvement over existing …
We consider the stochastic bandit problem with a continuous set of arms, with the expected reward function over the arms assumed to be fixed but unknown. We provide two new Gaussian process-based algorithms for continuous bandit optimization-Improved GP-UCB (IGP-UCB) and GP-Thomson sampling (GP-TS), and derive corresponding regret bounds. Specifically, the bounds hold when the expected reward function belongs to the reproducing kernel Hilbert space (RKHS) that naturally corresponds to a Gaussian process kernel used as input by the algorithms. Along the way, we derive a new self-normalized conc…
We study the stream-based online active learning in a contextual multi-armed bandit framework. In this framework, the reward depends on both the arm and the context. In a stream-based active learning setting, obtaining the ground truth of the reward is costly, and the conventional contextual multi-armed bandit algorithm fails to achieve a sublinear regret due to this cost. Hence, the algorithm needs to determine whether or not to request the ground truth of the reward at current time slot. In our framework, we consider a stream-based active learning setting in which a query request for the gro…
This work explores adaptations of successful multi-armed bandits policies to the online contextual bandits scenario with binary rewards using binary classification algorithms such as logistic regression as black-box oracles. Some of these adaptations are achieved through bootstrapping or approximate bootstrapping, while others rely on other forms of randomness, resulting in more scalable approaches than previous works, and the ability to work with any type of classification algorithm. In particular, the Adaptive-Greedy algorithm shows a lot of promise, in many cases achieving better performanc…
Gittins indices provide an optimal solution to the classical multi-armed bandit problem. An obstacle to their use has been the common perception that their computation is very difficult. This paper demonstrates an accessible general methodology for the calculating Gittins indices for the multi-armed bandit with a detailed study on the cases of Bernoulli and Gaussian rewards. With accompanying easy-to-use open source software, this work removes computation as a barrier to using Gittins indices in these commonly found settings.
We consider the stochastic linear (multi-armed) contextual bandit problem with the possibility of hidden simple multi-armed bandit structure in which the rewards are independent of the contextual information. Algorithms that are designed solely for one of the regimes are known to be sub-optimal for the alternate regime. We design a single computationally efficient algorithm that simultaneously obtains problem-dependent optimal regret rates in the simple multi-armed bandit regime and minimax optimal regret rates in the linear contextual bandit regime, without knowing a priori which of the two m…
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…
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 study the problem of identifying the top $m$ arms in a multi-armed bandit game. Our proposed solution relies on a new algorithm based on successive rejects of the seemingly bad arms, and successive accepts of the good ones. This algorithmic contribution allows to tackle other multiple identifications settings that were previously out of reach. In particular we show that this idea of successive accepts and rejects applies to the multi-bandit best arm identification problem.
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.
The target of $\mathcal{X}$-armed bandit problem is to find the global maximum of an unknown stochastic function $f$, given a finite budget of $n$ evaluations. Recently, $\mathcal{X}$-armed bandits have been widely used in many situations. Many of these applications need to deal with large-scale data sets. To deal with these large-scale data sets, we study a distributed setting of $\mathcal{X}$-armed bandits, where $m$ players collaborate to find the maximum of the unknown function. We develop a novel anytime distributed $\mathcal{X}$-armed bandit algorithm. Compared with prior work on $\mathc…
In this paper we propose a multi-armed bandit inspired, pool based active learning algorithm for the problem of binary classification. By carefully constructing an analogy between active learning and multi-armed bandits, we utilize ideas such as lower confidence bounds, and self-concordant regularization from the multi-armed bandit literature to design our proposed algorithm. Our algorithm is a sequential algorithm, which in each round assigns a sampling distribution on the pool, samples one point from this distribution, and queries the oracle for the label of this sampled point. The design of…
IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）
In recent years, multi-armed bandit (MAB) framework has attracted a lot of attention in various applications, from recommender systems and information retrieval to healthcare and finance, due to its stellar performance combined with certain attractive properties, such as learning from less feedback. The multi-armed bandit field is currently flourishing, as novel problem settings and algorithms motivated by various practical applications are being introduced, building on top of the classical bandit problem. This article aims to provide a comprehensive review of top recent developments in multip…
Indiana University（印第安纳大学）；University of Alberta（阿尔伯塔大学）
Stochastic linear bandits are a natural and simple generalisation of finite-armed bandits with numerous practical applications. Current approaches focus on generalising existing techniques for finite-armed bandits, notably the optimism principle and Thompson sampling. While prior work has mostly been in the worst-case setting, we analyse the asymptotic instance-dependent regret and show matching upper and lower bounds on what is achievable. Surprisingly, our results show that no algorithm based on optimism or Thompson sampling will ever achieve the optimal rate, and indeed, can be arbitrarily …
This paper is devoted to the study of the max K-armed bandit problem, which consists in sequentially allocating resources in order to detect extreme values. Our contribution is twofold. We first significantly refine the analysis of the ExtremeHunter algorithm carried out in Carpentier and Valko (2014), and next propose an alternative approach, showing that, remarkably, Extreme Bandits can be reduced to a classical version of the bandit problem to a certain extent. Beyond the formal analysis, these two approaches are compared through numerical experiments.
Contextual multi-armed bandits are classical models in reinforcement learning for sequential decision-making associated with individual information. A widely-used policy for bandits is Thompson Sampling, where samples from a data-driven probabilistic belief about unknown parameters are used to select the control actions. For this computationally fast algorithm, performance analyses are available under full context-observations. However, little is known for problems that contexts are not fully observed. We propose a Thompson Sampling algorithm for partially observable contextual multi-armed ban…
The multi-armed bandit problem is a classical decision-making problem where an agent has to learn an optimal action balancing exploration and exploitation. Properly managing this trade-off requires a correct assessment of uncertainty; in multi-armed bandits, as in other machine learning applications, it is important to distinguish between stochasticity that is inherent to the system (aleatoric uncertainty) and stochasticity that derives from the limited knowledge of the agent (epistemic uncertainty). In this paper we consider the formalism of subjective logic, a concise and expressive framewor…
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…
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…
Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）
In this report, we survey Bayesian Optimization methods focussed on the Multi-Armed Bandit Problem. We take the help of the paper "Portfolio Allocation for Bayesian Optimization". We report a small literature survey on the acquisition functions and the types of portfolio strategies used in papers discussing Bayesian Optimization. We also replicate the experiments and report our findings and compare them to the results in the paper. Code link: https://colab.research.google.com/drive/1GZ14klEDoe3dcBeZKo5l8qqrKf_GmBDn?usp=sharing#scrollTo=XgIBau3O45_V.
Rotman School of Management, University of Toronto（多伦多大学罗特曼管理学院）
In many online learning or multi-armed bandit problems, the taken actions or pulled arms are ordinal and required to be monotone over time. Examples include dynamic pricing, in which the firms use markup pricing policies to please early adopters and deter strategic waiting, and clinical trials, in which the dose allocation usually follows the dose escalation principle to prevent dose limiting toxicities. We consider the continuum-armed bandit problem when the arm sequence is required to be monotone. We show that when the unknown objective function is Lipschitz continuous, the regret is $O(T)$.…
Contextual multi-armed bandit problems arise frequently in important industrial applications. Existing solutions model the context either linearly, which enables uncertainty driven (principled) exploration, or non-linearly, by using epsilon-greedy exploration policies. Here we present a deep learning framework for contextual multi-armed bandits that is both non-linear and enables principled exploration at the same time. We tackle the exploration vs. exploitation trade-off through Thompson sampling by exploiting the connection between inference time dropout and sampling from the posterior over …
Bridging Adversarial and Nonstationary Multi-armed Bandit
Rotman School of Management, University of Toronto（多伦多大学罗特曼管理学院）；The Hong Kong University of Science and Technology（香港科技大学）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）
Motivated by runtime verification of QoS requirements in self-adaptive and self-organizing systems that are able to reconfigure their structure and behavior in response to runtime data, we propose a QoS-aware variant of Thompson sampling for multi-armed bandits. It is applicable in settings where QoS satisfaction of an arm has to be ensured with high confidence efficiently, rather than finding the optimal arm while minimizing regret. Preliminary experimental results encourage further research in the field of QoS-aware decision making.
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 …
Bosch Center for Artificial Intelligence（博世人工智能中心）；University of Southern Denmark（南丹麦大学）；Technische Universität Darmstadt（达姆施塔特工业大学）
We present a PAC-Bayesian analysis of lifelong learning. In the lifelong learning problem, a sequence of learning tasks is observed one-at-a-time, and the goal is to transfer information acquired from previous tasks to new learning tasks. We consider the case when each learning task is a multi-armed bandit problem. We derive lower bounds on the expected average reward that would be obtained if a given multi-armed bandit algorithm was run in a new task with a particular prior and for a set number of steps. We propose lifelong learning algorithms that use our new bounds as learning objectives. O…
Adaptive and sequential experiment design is a well-studied area in numerous domains. We survey and synthesize the work of the online statistical learning paradigm referred to as multi-armed bandits integrating the existing research as a resource for a certain class of online experiments. We first explore the traditional stochastic model of a multi-armed bandit, then explore a taxonomic scheme of complications to that model, for each complication relating it to a specific requirement or consideration of the experiment design context. Finally, at the end of the paper, we present a table of know…
This work explores the idea of a causal contextual multi-armed bandit approach to automated marketing, where we estimate and optimize the causal (incremental) effects. Focusing on causal effect leads to better return on investment (ROI) by targeting only the persuadable customers who wouldn't have taken the action organically. Our approach draws on strengths of causal inference, uplift modeling, and multi-armed bandits. It optimizes on causal treatment effects rather than pure outcome, and incorporates counterfactual generation within data collection. Following uplift modeling results, we opti…
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…
Based on differential privacy (DP) framework, we introduce and unify privacy definitions for the multi-armed bandit algorithms. We represent the framework with a unified graphical model and use it to connect privacy definitions. We derive and contrast lower bounds on the regret of bandit algorithms satisfying these definitions. We leverage a unified proving technique to achieve all the lower bounds. We show that for all of them, the learner's regret is increased by a multiplicative factor dependent on the privacy level $ε$. We observe that the dependency is weaker when we do not require local …
The multi-armed bandit problem is a popular model for studying exploration/exploitation trade-off in sequential decision problems. Many algorithms are now available for this well-studied problem. One of the earliest algorithms, given by W. R. Thompson, dates back to 1933. This algorithm, referred to as Thompson Sampling, is a natural Bayesian algorithm. The basic idea is to choose an arm to play according to its probability of being the best arm. Thompson Sampling algorithm has experimentally been shown to be close to optimal. In addition, it is efficient to implement and exhibits several desi…
