We consider a multi-armed bandit problem motivated by situations where only the extreme values, as opposed to expected values in the classical bandit setting, are of interest. We propose distribution free algorithms using robust statistics and characterize the statistical properties. We show that the provided algorithms achieve vanishing extremal regret under weaker conditions than existing algorithms. Performance of the algorithms is demonstrated for the finite-sample setting using numerical experiments. The results show superior performance of the proposed algorithms compared to the well known algorithms.
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.
University of Pennsylvania（宾夕法尼亚大学）；Massachusetts Institute of Technology（麻省理工学院）
We present a new type of acquisition functions for online decision making in multi-armed and contextual bandit problems with extreme payoffs. Specifically, we model the payoff function as a Gaussian process and formulate a novel type of upper confidence bound (UCB) acquisition function that guides exploration towards the bandits that are deemed most relevant according to the variability of the observed rewards. This is achieved by computing a tractable likelihood ratio that quantifies the importance of the output relative to the inputs and essentially acts as an \textit{attention mechanism} th…
University of California, Berkeley（加州大学伯克利分校）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
We study the problem of batch learning from bandit feedback in the setting of extremely large action spaces. Learning from extreme bandit feedback is ubiquitous in recommendation systems, in which billions of decisions are made over sets consisting of millions of choices in a single day, yielding massive observational data. In these large-scale real-world applications, supervised learning frameworks such as eXtreme Multi-label Classification (XMC) are widely used despite the fact that they incur significant biases due to the mismatch between bandit feedback and supervised labels. Such biases c…
An automatic machine learning (AutoML) task is to select the best algorithm and its hyper-parameters simultaneously. Previously, the hyper-parameters of all algorithms are joint as a single search space, which is not only huge but also redundant, because many dimensions of hyper-parameters are irrelevant with the selected algorithms. In this paper, we propose a cascaded approach for algorithm selection and hyper-parameter optimization. While a search procedure is employed at the level of hyper-parameter optimization, a bandit strategy runs at the level of algorithm selection to allocate the bu…
We study regret minimization in a stochastic multi-armed bandit setting and establish a fundamental trade-off between the regret suffered under an algorithm, and its statistical robustness. Considering broad classes of underlying arms' distributions, we show that bandit learning algorithms with logarithmic regret are always inconsistent and that consistent learning algorithms always suffer a super-logarithmic regret. This result highlights the inevitable statistical fragility of all `logarithmic regret' bandit algorithms available in the literature---for instance, if a UCB algorithm designed f…
We investigate the optimality of perturbation based algorithms in the stochastic and adversarial multi-armed bandit problems. For the stochastic case, we provide a unified regret analysis for both sub-Weibull and bounded perturbations when rewards are sub-Gaussian. Our bounds are instance optimal for sub-Weibull perturbations with parameter 2 that also have a matching lower tail bound, and all bounded support perturbations where there is sufficient probability mass at the extremes of the support. For the adversarial setting, we prove rigorous barriers against two natural solution approaches us…
In this paper, we contribute to the Extreme Bandit problem, a variant of Multi-Armed Bandits in which the learner seeks to collect the largest possible reward. We first study the concentration of the maximum of i.i.d random variables under mild assumptions on the tail of the rewards distributions. This analysis motivates the introduction of Quantile of Maxima (QoMax). The properties of QoMax are sufficient to build an Explore-Then-Commit (ETC) strategy, QoMax-ETC, achieving strong asymptotic guarantees despite its simplicity. We then propose and analyze a more adaptive, anytime algorithm, QoMa…
Inria Lille（里尔法国国家信息与自动化研究所）；ENS Paris-Saclay（巴黎萨克雷高等师范学院）；ENSAE（法国国家统计与经济管理学院）；Criteo AI Lab（Criteo人工智能实验室）；DeepMind Paris（巴黎DeepMind）
We investigate stochastic combinatorial multi-armed bandit with semi-bandit feedback (CMAB). In CMAB, the question of the existence of an efficient policy with an optimal asymptotic regret (up to a factor poly-logarithmic with the action size) is still open for many families of distributions, including mutually independent outcomes, and more generally the multivariate sub-Gaussian family. We propose to answer the above question for these two families by analyzing variants of the Combinatorial Thompson Sampling policy (CTS). For mutually independent outcomes in $[0,1]$, we propose a tight analy…
Google Research（谷歌研究院）；Amazon（亚马逊）；MIT（麻省理工学院）；Stanford University（斯坦福大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
Motivated by modern applications, such as online advertisement and recommender systems, we study the top-$k$ extreme contextual bandits problem, where the total number of arms can be enormous, and the learner is allowed to select $k$ arms and observe all or some of the rewards for the chosen arms. We first propose an algorithm for the non-extreme realizable setting, utilizing the Inverse Gap Weighting strategy for selecting multiple arms. We show that our algorithm has a regret guarantee of $O(k\sqrt{(A-k+1)T \log (|\mathcal{F}|T)})$, where $A$ is the total number of arms and $\mathcal{F}$ is …
Tsinghua University（清华大学）；University of California, Davis（加利福尼亚大学戴维斯分校）；Shanghai Jiao Tong University（上海交通大学）；Ohio State University（俄亥俄州立大学）
The demand for seamless Internet access under extreme user mobility, such as on high-speed trains and vehicles, has become a norm rather than an exception. However, the 4G/5G mobile network is not always reliable to meet this demand, with non-negligible failures during the handover between base stations. A fundamental challenge of reliability is to balance the exploration of more measurements for satisfactory handover, and exploitation for timely handover (before the fast-moving user leaves the serving base station's radio coverage). This paper formulates this trade-off in extreme mobility as …
Uppsala University（乌普萨拉大学）；Department of Information Technology（信息技术系）；Division of Scientific Computing（科学计算部）；SE-751 05, Uppsala（乌普萨拉，SE-751 05）；Sweden（瑞典）
Approximate Bayesian computation is an established and popular method for likelihood-free inference with applications in many disciplines. The effectiveness of the method depends critically on the availability of well performing summary statistics. Summary statistic selection relies heavily on domain knowledge and carefully engineered features, and can be a laborious time consuming process. Since the method is sensitive to data dimensionality, the process of selecting summary statistics must balance the need to include informative statistics and the dimensionality of the feature vector. This p…
We present a novel distribution-free approach, the data-driven threshold machine (DTM), for a fundamental problem at the core of many learning tasks: choose a threshold for a given pre-specified level that bounds the tail probability of the maximum of a (possibly dependent but stationary) random sequence. We do not assume data distribution, but rather relying on the asymptotic distribution of extremal values, and reduce the problem to estimate three parameters of the extreme value distributions and the extremal index. We specially take care of data dependence via estimating extremal index sinc…
Massachusetts Institute of Technology（麻省理工学院）；Microsoft Research（微软研究院）；University of Southern California（南加州大学）
In statistical learning, algorithms for model selection allow the learner to adapt to the complexity of the best hypothesis class in a sequence. We ask whether similar guarantees are possible for contextual bandit learning.
Traditional multi-armed bandit (MAB) formulations usually make certain assumptions about the underlying arms' distributions, such as bounds on the support or their tail behaviour. Moreover, such parametric information is usually 'baked' into the algorithms. In this paper, we show that specialized algorithms that exploit such parametric information are prone to inconsistent learning performance when the parameter is misspecified. Our key contributions are twofold: (i) We establish fundamental performance limits of statistically robust MAB algorithms under the fixed-budget pure exploration setti…
Model misspecification is a major consideration in applications of statistical methods and machine learning. However, it is often neglected in contextual bandits. This paper studies a common form of misspecification, an inter-arm heterogeneity that is not captured by context. To address this issue, we assume that the heterogeneity arises due to arm-specific random variables, which can be learned. We call this setting a robust contextual bandit. The arm-specific variables explain the unknown inter-arm heterogeneity, and we incorporate them in the robust contextual estimator of the mean reward a…
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 …
Heidelberg University（海德堡大学）；Amazon Development Center（亚马逊开发中心）
The goal of counterfactual learning for statistical machine translation (SMT) is to optimize a target SMT system from logged data that consist of user feedback to translations that were predicted by another, historic SMT system. A challenge arises by the fact that risk-averse commercial SMT systems deterministically log the most probable translation. The lack of sufficient exploration of the SMT output space seemingly contradicts the theoretical requirements for counterfactual learning. We show that counterfactual learning from deterministic bandit logs is possible nevertheless by smoothing ou…
Contextual bandits often provide simple and effective personalization in decision making problems, making them popular tools to deliver personalized interventions in mobile health as well as other health applications. However, when bandits are deployed in the context of a scientific study -- e.g. a clinical trial to test if a mobile health intervention is effective -- the aim is not only to personalize for an individual, but also to determine, with sufficient statistical power, whether or not the system's intervention is effective. It is essential to assess the effectiveness of the interventio…
We study a nonparametric contextual bandit problem where the expected reward functions belong to a Hölder class with smoothness parameter $β$. We show how this interpolates between two extremes that were previously studied in isolation: non-differentiable bandits ($β\leq1$), where rate-optimal regret is achieved by running separate non-contextual bandits in different context regions, and parametric-response bandits (satisfying $β=\infty$), where rate-optimal regret can be achieved with minimal or no exploration due to infinite extrapolatability. We develop a novel algorithm that carefully adju…
Statistical Inference for Misspecified Contextual Bandits
Department of Statistics, University of Wisconsin–Madison（威斯康星大学麦迪逊分校统计系）；School of Data Science and Society, University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校数据科学与社会学院）
Online decision-making problem requires us to make a sequence of decisions based on incremental information. Common solutions often need to learn a reward model of different actions given the contextual information and then maximize the long-term reward. It is meaningful to know if the posited model is reasonable and how the model performs in the asymptotic sense. We study this problem under the setup of the contextual bandit framework with a linear reward model. The $\varepsilon$-greedy policy is adopted to address the classic exploration-and-exploitation dilemma. Using the martingale central…
An adversarial bandit problem with memory constraints is studied where only the statistics of a subset of arms can be stored. A hierarchical learning policy that requires only a sublinear order of memory space in terms of the number of arms is developed. Its sublinear regret orders with respect to the time horizon are established for both weak regret and shifting regret. This work appears to be the first on memory-constrained bandit problems under the adversarial setting.
Contextual bandits with linear payoffs, which are also known as linear bandits, provide a powerful alternative for solving practical problems of sequential decisions, e.g., online advertisements. In the era of big data, contextual data usually tend to be high-dimensional, which leads to new challenges for traditional linear bandits mostly designed for the setting of low-dimensional contextual data. Due to the curse of dimensionality, there are two challenges in most of the current bandit algorithms: the first is high time-complexity; and the second is extreme large upper regret bounds with hig…
Carnegie Mellon University（卡内基梅隆大学）；Google Research（谷歌研究院）；DeepMind；Duke University（杜克大学）；Norwegian University of Science and Technology（挪威科技大学）
We propose a novel framework for structured bandits, which we call an influence diagram bandit. Our framework captures complex statistical dependencies between actions, latent variables, and observations; and thus unifies and extends many existing models, such as combinatorial semi-bandits, cascading bandits, and low-rank bandits. We develop novel online learning algorithms that learn to act efficiently in our models. The key idea is to track a structured posterior distribution of model parameters, either exactly or approximately. To act, we sample model parameters from their posterior and the…
Multi-Armed-Bandit frameworks have often been used by researchers to assess educational interventions, however, recent work has shown that it is more beneficial for a student to provide qualitative feedback through preference elicitation between different alternatives, making a dueling bandits framework more appropriate. In this paper, we explore the statistical quality of data under this framework by comparing traditional uniform sampling to a dueling bandit algorithm and find that dueling bandit algorithms perform well at cumulative regret minimisation, but lead to inflated Type-I error rate…
Google Research（谷歌研究院）；DeepMind（深度思维）；University of Alberta（阿尔伯塔大学）
Most bandit policies are designed to either minimize regret in any problem instance, making very few assumptions about the underlying environment, or in a Bayesian sense, assuming a prior distribution over environment parameters. The former are often too conservative in practical settings, while the latter require assumptions that are hard to verify in practice. We study bandit problems that fall between these two extremes, where the learning agent has access to sampled bandit instances from an unknown prior distribution $\mathcal{P}$ and aims to achieve high reward on average over the bandit …
Mortal bandits have proven to be extremely useful for providing news article recommendations, running automated online advertising campaigns, and for other applications where the set of available options changes over time. Previous work on this problem showed how to regulate exploration of new arms when they have recently appeared, but they do not adapt when the arms are about to disappear. Since in most applications we can determine either exactly or approximately when arms will disappear, we can leverage this information to improve performance: we should not be exploring arms that are about …
We consider combinatorial semi-bandits over a set of arms ${\cal X} \subset \{0,1\}^d$ where rewards are uncorrelated across items. For this problem, the algorithm ESCB yields the smallest known regret bound $R(T) = {\cal O}\Big( {d (\ln m)^2 (\ln T) \over Δ_{\min} }\Big)$, but it has computational complexity ${\cal O}(|{\cal X}|)$ which is typically exponential in $d$, and cannot be used in large dimensions. We propose the first algorithm which is both computationally and statistically efficient for this problem with regret $R(T) = {\cal O} \Big({d (\ln m)^2 (\ln T)\over Δ_{\min} }\Big)$ and …
Criteo AI Lab（Criteo 人工智能实验室）；University of Arizona（亚利桑那大学）
Logistic Bandits have recently undergone careful scrutiny by virtue of their combined theoretical and practical relevance. This research effort delivered statistically efficient algorithms, improving the regret of previous strategies by exponentially large factors. Such algorithms are however strikingly costly as they require $Ω(t)$ operations at each round. On the other hand, a different line of research focused on computational efficiency ($\mathcal{O}(1)$ per-round cost), but at the cost of letting go of the aforementioned exponential improvements. Obtaining the best of both world is unfort…
Generalized Linear Bandits: Almost Optimal Regret with One-Pass Update
RIKEN AIP（日本理化学研究所高级研究所）；National Key Laboratory for Novel Software Technology（国家新型软件技术实验室）；School of Artificial Intelligence（人工智能学院）；The University of Tokyo（东京大学）
We present an approach to structured prediction from bandit feedback, called Bandit Structured Prediction, where only the value of a task loss function at a single predicted point, instead of a correct structure, is observed in learning. We present an application to discriminative reranking in Statistical Machine Translation (SMT) where the learning algorithm only has access to a 1-BLEU loss evaluation of a predicted translation instead of obtaining a gold standard reference translation. In our experiment bandit feedback is obtained by evaluating BLEU on reference translations without revealin…
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 …
Indian Institute of Science（印度科学研究所）；Tel Aviv University（特拉维夫大学）；Google Tel Aviv（谷歌特拉维夫）
We introduce the problem of regret minimization in Adversarial Dueling Bandits. As in classic Dueling Bandits, the learner has to repeatedly choose a pair of items and observe only a relative binary `win-loss' feedback for this pair, but here this feedback is generated from an arbitrary preference matrix, possibly chosen adversarially. Our main result is an algorithm whose $T$-round regret compared to the \emph{Borda-winner} from a set of $K$ items is $\tilde{O}(K^{1/3}T^{2/3})$, as well as a matching $Ω(K^{1/3}T^{2/3})$ lower bound. We also prove a similar high probability regret bound. We fu…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Microsoft Research（微软研究院）
Designing efficient general-purpose contextual bandit algorithms that work with large -- or even continuous -- action spaces would facilitate application to important scenarios such as information retrieval, recommendation systems, and continuous control. While obtaining standard regret guarantees can be hopeless, alternative regret notions have been proposed to tackle the large action setting. We propose a smooth regret notion for contextual bandits, which dominates previously proposed alternatives. We design a statistically and computationally efficient algorithm -- for the proposed smooth r…
New York University（纽约大学）；Microsoft Research（微软研究院）；Inria（法国国家信息与自动化研究所）
Contextual bandit algorithms are essential for solving many real-world interactive machine learning problems. Despite multiple recent successes on statistically and computationally efficient methods, the practical behavior of these algorithms is still poorly understood. We leverage the availability of large numbers of supervised learning datasets to empirically evaluate contextual bandit algorithms, focusing on practical methods that learn by relying on optimization oracles from supervised learning. We find that a recent method (Foster et al., 2018) using optimism under uncertainty works the b…
Mode estimation is a classical problem in statistics with a wide range of applications in machine learning. Despite this, there is little understanding in its robustness properties under possibly adversarial data contamination. In this paper, we give precise robustness guarantees as well as privacy guarantees under simple randomization. We then introduce a theory for multi-armed bandits where the values are the modes of the reward distributions instead of the mean. We prove regret guarantees for the problems of top arm identification, top m-arms identification, contextual modal bandits, and in…
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 paper studies regret minimization in a multi-armed bandit. It is well known that side information, such as the prior distribution of arm means in Thompson sampling, can improve the statistical efficiency of the bandit algorithm. While the prior is a blessing when correctly specified, it is a curse when misspecified. To address this issue, we introduce the assumption of a random-effect model to bandits. In this model, the mean arm rewards are drawn independently from an unknown distribution, which we estimate. We derive a random-effect estimator of the arm means, analyze its uncertainty, a…
We consider the general (stochastic) contextual bandit problem under the realizability assumption, i.e., the expected reward, as a function of contexts and actions, belongs to a general function class $\mathcal{F}$. We design a fast and simple algorithm that achieves the statistically optimal regret with only ${O}(\log T)$ calls to an offline regression oracle across all $T$ rounds. The number of oracle calls can be further reduced to $O(\log\log T)$ if $T$ is known in advance. Our results provide the first universal and optimal reduction from contextual bandits to offline regression, solving …
What is the most statistically efficient way to do off-policy evaluation and optimization with batch data from bandit feedback? For log data generated by contextual bandit algorithms, we consider offline estimators for the expected reward from a counterfactual policy. Our estimators are shown to have lowest variance in a wide class of estimators, achieving variance reduction relative to standard estimators. We then apply our estimators to improve advertisement design by a major advertisement company. Consistent with the theoretical result, our estimators allow us to improve on the existing ban…
Contextual bandit algorithms are extremely popular and widely used in recommendation systems to provide online personalised recommendations. A recurrent assumption is the stationarity of the reward function, which is rather unrealistic in most of the real-world applications. In the music recommendation scenario for instance, people's music taste can abruptly change during certain events, such as Halloween or Christmas, and revert to the previous music taste soon after. We would therefore need an algorithm which can promptly react to these changes. Moreover, we would like to leverage already ob…
Microsoft Research（微软研究院）；Columbia University（哥伦比亚大学）；Yahoo! Labs（雅虎研究院）；Princeton University（普林斯顿大学）
We present a new algorithm for the contextual bandit learning problem, where the learner repeatedly takes one of $K$ actions in response to the observed context, and observes the reward only for that chosen action. Our method assumes access to an oracle for solving fully supervised cost-sensitive classification problems and achieves the statistically optimal regret guarantee with only $\tilde{O}(\sqrt{KT/\log N})$ oracle calls across all $T$ rounds, where $N$ is the number of policies in the policy class we compete against. By doing so, we obtain the most practical contextual bandit learning a…
University of Cambridge（剑桥大学）；ENS Paris–Saclay（巴黎萨克雷高等师范学校）；Criteo Research, Paris（Criteo 研究院，巴黎）
We consider the problem of bandit optimization, inspired by stochastic optimization and online learning problems with bandit feedback. In this problem, the objective is to minimize a global loss function of all the actions, not necessarily a cumulative loss. This framework allows us to study a very general class of problems, with applications in statistics, machine learning, and other fields. To solve this problem, we analyze the Upper-Confidence Frank-Wolfe algorithm, inspired by techniques for bandits and convex optimization. We give theoretical guarantees for the performance of this algorit…
Buzz, Choose, Forget: A Meta-Bandit Framework for Bee-Like Decision Making
University of Toulouse, IRIT（图卢兹大学，IRIT）；University of Toulouse, CBI（图卢兹大学，CBI）；Regalia Team, INRIA University of Toulouse, France（Regalia团队，法国国家信息与自动化研究所图卢兹大学）
