Typical contextual bandit algorithms assume that the rewards at each round lie in some fixed range $[0, R]$, and their regret scales polynomially with this reward range $R$. However, many practical scenarios naturally involve heavy-tailed rewards or rewards where the worst-case range can be substantially larger than the variance. In this paper, we develop an algorithmic approach building on Catoni's estimator from robust statistics, and apply it to contextual bandits with general function approximation. When the variance of the reward at each round is known, we use a variance-weighted regression approach and establish a regret bound that depends only on the cumulative reward variance and logarithmically on the reward range $R$ as well as the number of rounds $T$. For the unknown-variance case, we further propose a careful peeling-based algorithm and remove the need for cumbersome variance estimation. With additional dependence on the fourth moment, our algorithm also enjoys a variance-based bound with logarithmic reward-range dependence. Moreover, we demonstrate the optimality of the leading-order term in our regret bound through a matching lower bound.
Catoni proposed a robust M-estimator and gave the deviation inequality for one fixed test function. The present paper is devoted to the uniform concentration inequality for a family of test functions. As an application, we consider empirical risk minimization for heavy-tailed losses.
Princeton University（普林斯顿大学）；Università degli Studi di Milano（米兰大学）；ICREA（加泰罗尼亚研究与高级研究院）；Universitat Pompeu Fabra（庞培法布拉大学）
The stochastic multi-armed bandit problem is well understood when the reward distributions are sub-Gaussian. In this paper we examine the bandit problem under the weaker assumption that the distributions have moments of order 1+ε, for some $ε\in (0,1]$. Surprisingly, moments of order 2 (i.e., finite variance) are sufficient to obtain regret bounds of the same order as under sub-Gaussian reward distributions. In order to achieve such regret, we define sampling strategies based on refined estimators of the mean such as the truncated empirical mean, Catoni's M-estimator, and the median-of-means e…
We study the stochastic Multi-Armed Bandit (MAB) problem under worst-case regret and heavy-tailed reward distribution. We modify the minimax policy MOSS for the sub-Gaussian reward distribution by using saturated empirical mean to design a new algorithm called Robust MOSS. We show that if the moment of order $1+ε$ for the reward distribution exists, then the refined strategy has a worst-case regret matching the lower bound while maintaining a distribution-dependent logarithm regret.
The purpose of this paper is to discuss empirical risk minimization when the losses are not necessarily bounded and may have a distribution with heavy tails. In such situations, usual empirical averages may fail to provide reliable estimates and empirical risk minimization may provide large excess risk. However, some robust mean estimators proposed in the literature may be used to replace empirical means. In this paper, we investigate empirical risk minimization based on a robust estimate proposed by Catoni. We develop performance bounds based on chaining arguments tailored to Catoni's mean es…
We revisit the classic regret-minimization problem in the stochastic multi-armed bandit setting when the arm-distributions are allowed to be heavy-tailed. Regret minimization has been well studied in simpler settings of either bounded support reward distributions or distributions that belong to a single parameter exponential family. We work under the much weaker assumption that the moments of order $(1+ε)$ are uniformly bounded by a known constant B, for some given $ε> 0$. We propose an optimal algorithm that matches the lower bound exactly in the first-order term. We also give a finite-time b…
In this paper, we study the problem of stochastic linear bandits with finite action sets. Most of existing work assume the payoffs are bounded or sub-Gaussian, which may be violated in some scenarios such as financial markets. To settle this issue, we analyze the linear bandits with heavy-tailed payoffs, where the payoffs admit finite $1+ε$ moments for some $ε\in(0,1]$. Through median of means and dynamic truncation, we propose two novel algorithms which enjoy a sublinear regret bound of $\widetilde{O}(d^{\frac{1}{2}}T^{\frac{1}{1+ε}})$, where $d$ is the dimension of contextual information and…
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…
In linear stochastic bandits, it is commonly assumed that payoffs are with sub-Gaussian noises. In this paper, under a weaker assumption on noises, we study the problem of \underline{lin}ear stochastic {\underline b}andits with h{\underline e}avy-{\underline t}ailed payoffs (LinBET), where the distributions have finite moments of order $1+ε$, for some $ε\in (0,1]$. We rigorously analyze the regret lower bound of LinBET as $Ω(T^{\frac{1}{1+ε}})$, implying that finite moments of order 2 (i.e., finite variances) yield the bound of $Ω(\sqrt{T})$, with $T$ being the total number of rounds to play b…
Satisficing Regret Minimization in Bandits: Constant Rate and Light-Tailed Distribution
School of Operations Research and Information Engineering, Cornell University（运营研究与信息工程学院，康奈尔大学）；SC Johnson College of Business, Cornell University（SC约翰逊商学院，康奈尔大学）
Peking University（北京大学）；Pazhou Lab（琶洲实验室）；University of California, Los Angles（加利福尼亚大学洛杉矶分校）
Despite a large amount of effort in dealing with heavy-tailed error in machine learning, little is known when moments of the error can become non-existential: the random noise $η$ satisfies Pr$\left[|η| > |y|\right] \le 1/|y|^α$ for some $α> 0$. We make the first attempt to actively handle such super heavy-tailed noise in bandit learning problems: We propose a novel robust statistical estimator, mean of medians, which estimates a random variable by computing the empirical mean of a sequence of empirical medians. We then present a generic reductionist algorithmic framework for solving bandit le…
KAUST（阿卜杜拉国王科技大学）；The Pennsylvania State University（宾夕法尼亚州立大学）；Purdue University（普渡大学）
In this paper, we study multi-armed bandits (MAB) and stochastic linear bandits (SLB) with heavy-tailed rewards and quantum reward oracle. Unlike the previous work on quantum bandits that assumes bounded/sub-Gaussian distributions for rewards, here we investigate the quantum bandits problem under a weaker assumption that the distributions of rewards only have bounded $(1+v)$-th moment for some $v\in (0,1]$. In order to achieve regret improvements for heavy-tailed bandits, we first propose a new quantum mean estimator for heavy-tailed distributions, which is based on the Quantum Monte Carlo Mea…
Implicitly normalized forecaster with clipping for linear and non-linear heavy-tailed multi-armed bandits
MSU Institute for Artificial Intelligence（莫斯科国立大学人工智能研究所）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Institute for Information Transmission Problems（信息传输问题研究所）；V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences（俄罗斯科学院特拉佩兹尼科夫控制科学研究所）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）；Skoltech（斯科尔科沃科学技术学院）；ISP RAS Research Center for Trusted Artificial Intelligence（俄罗斯科学院系统编程研究所可信人工智能研究中心）
Thompson Sampling provides an efficient technique to introduce prior knowledge in the multi-armed bandit problem, along with providing remarkable empirical performance. In this paper, we revisit the Thompson Sampling algorithm under rewards drawn from symmetric $α$-stable distributions, which are a class of heavy-tailed probability distributions utilized in finance and economics, in problems such as modeling stock prices and human behavior. We present an efficient framework for posterior inference, which leads to two algorithms for Thompson Sampling in this setting. We prove finite-time regret…
In this paper, we generalize the concept of heavy-tailed multi-armed bandits to adversarial environments, and develop robust best-of-both-worlds algorithms for heavy-tailed multi-armed bandits (MAB), where losses have $α$-th ($1<α\le 2$) moments bounded by $σ^α$, while the variances may not exist. Specifically, we design an algorithm \texttt{HTINF}, when the heavy-tail parameters $α$ and $σ$ are known to the agent, \texttt{HTINF} simultaneously achieves the optimal regret for both stochastic and adversarial environments, without knowing the actual environment type a-priori. When $α,σ$ are unkn…
University of Southern California（南加州大学）；Microsoft Research（微软研究院）
We initiate the study of learning in contextual bandits with the help of loss predictors. The main question we address is whether one can improve over the minimax regret $\mathcal{O}(\sqrt{T})$ for learning over $T$ rounds, when the total error of the predictor $\mathcal{E} \leq T$ is relatively small. We provide a complete answer to this question, including upper and lower bounds for various settings: adversarial versus stochastic environments, known versus unknown $\mathcal{E}$, and single versus multiple predictors. We show several surprising results, such as 1) the optimal regret is $\math…
Media Lab and Institute for Data, Systems and Society, Massachusetts Institute of Technology（麻省理工学院媒体实验室与数据、系统与社会研究所）
We study the heavy-tailed stochastic bandit problem in the cooperative multi-agent setting, where a group of agents interact with a common bandit problem, while communicating on a network with delays. Existing algorithms for the stochastic bandit in this setting utilize confidence intervals arising from an averaging-based communication protocol known as~\textit{running consensus}, that does not lend itself to robust estimation for heavy-tailed settings. We propose \textsc{MP-UCB}, a decentralized multi-agent algorithm for the cooperative stochastic bandit that incorporates robust estimation wi…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Wayne State University（韦恩州立大学）；Shandong University（山东大学）
We study private and robust multi-armed bandits (MABs), where the agent receives Huber's contaminated heavy-tailed rewards and meanwhile needs to ensure differential privacy. We first present its minimax lower bound, characterizing the information-theoretic limit of regret with respect to privacy budget, contamination level and heavy-tailedness. Then, we propose a meta-algorithm that builds on a private and robust mean estimation sub-routine \texttt{PRM} that essentially relies on reward truncation and the Laplace mechanism only. For two different heavy-tailed settings, we give specific scheme…
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…
Efficient Algorithms for Generalized Linear Bandits with Heavy-tailed Rewards
City University of Hong Kong（香港城市大学）；The City University of Hong Kong Shenzhen Research Institute（香港城市大学深圳研究院）；University of Waterloo（滑铁卢大学）；Zhejiang University（浙江大学）；JD AI Research（京东人工智能研究院）；Nanjing University（南京大学）；Peng Cheng Laboratory（鹏城实验室）
Motivated by problems in search and detection we present a solution to a Combinatorial Multi-Armed Bandit (CMAB) problem with both heavy-tailed reward distributions and a new class of feedback, filtered semibandit feedback. In a CMAB problem an agent pulls a combination of arms from a set $\{1,...,k\}$ in each round, generating random outcomes from probability distributions associated with these arms and receiving an overall reward. Under semibandit feedback it is assumed that the random outcomes generated are all observed. Filtered semibandit feedback allows the outcomes that are observed to …
Indian Institute of Technology Madras（印度理工学院马德拉斯分校）；ABInBev（百威英博）
Conditional Value-at-Risk (CVaR) is a widely used risk metric in applications such as finance. We derive concentration bounds for CVaR estimates, considering separately the cases of light-tailed and heavy-tailed distributions. In the light-tailed case, we use a classical CVaR estimator based on the empirical distribution constructed from the samples. For heavy-tailed random variables, we assume a mild `bounded moment' condition, and derive a concentration bound for a truncation-based estimator. Notably, our concentration bounds enjoy an exponential decay in the sample size, for heavy-tailed as…
Shandong University（山东大学）；KAUST（阿卜杜拉国王科技大学）；Nanjing University（南京大学）；East China Normal University（华东师范大学）
In this paper we investigate the problem of stochastic multi-armed bandits (MAB) in the (local) differential privacy (DP/LDP) model. Unlike previous results that assume bounded/sub-Gaussian reward distributions, we focus on the setting where each arm's reward distribution only has $(1+v)$-th moment with some $v\in (0, 1]$. In the first part, we study the problem in the central $ε$-DP model. We first provide a near-optimal result by developing a private and robust Upper Confidence Bound (UCB) algorithm. Then, we improve the result via a private and robust version of the Successive Elimination (…
Classical multi-armed bandit problems use the expected value of an arm as a metric to evaluate its goodness. However, the expected value is a risk-neutral metric. In many applications like finance, one is interested in balancing the expected return of an arm (or portfolio) with the risk associated with that return. In this paper, we consider the problem of selecting the arm that optimizes a linear combination of the expected reward and the associated Conditional Value at Risk (CVaR) in a fixed budget best-arm identification framework. We allow the reward distributions to be unbounded or even h…
Otto-von-Guericke University of Magdeburg（马格德堡奥托·冯·格里克大学）；GoogleDeepMind（谷歌深度思维）
Significant work has been recently dedicated to the stochastic delayed bandit setting because of its relevance in applications. The applicability of existing algorithms is however restricted by the fact that strong assumptions are often made on the delay distributions, such as full observability, restrictive shape constraints, or uniformity over arms. In this work, we weaken them significantly and only assume that there is a bound on the tail of the delay. In particular, we cover the important case where the delay distributions vary across arms, and the case where the delays are heavy-tailed. …
A confidence sequence (CS) is an anytime-valid sequential inference primitive which produces an adapted sequence of sets for a predictable parameter sequence with a time-uniform coverage guarantee. This work constructs a non-parametric non-asymptotic lower CS for the running average conditional expectation whose slack converges to zero given non-negative right heavy-tailed observations with bounded mean. Specifically, when the variance is finite the approach dominates the empirical Bernstein supermartingale of Howard et. al.; with infinite variance, can adapt to a known or unknown $(1 + δ)$-th…
The Ohio State University（俄亥俄州立大学）；UIUC（伊利诺伊大学厄巴纳-香槟分校）
We consider a budget-constrained bandit problem where each arm pull incurs a random cost, and yields a random reward in return. The objective is to maximize the total expected reward under a budget constraint on the total cost. The model is general in the sense that it allows correlated and potentially heavy-tailed cost-reward pairs that can take on negative values as required by many applications. We show that if moments of order $(2+γ)$ for some $γ> 0$ exist for all cost-reward pairs, $O(\log B)$ regret is achievable for a budget $B>0$. In order to achieve tight regret bounds, we propose alg…
In this paper we initiate a study of non parametric contextual bandits under shape constraints on the mean reward function. Specifically, we study a setting where the context is one dimensional, and the mean reward function is isotonic with respect to this context. We propose a policy for this problem and show that it attains minimax rate optimal regret. Moreover, we show that the same policy enjoys automatic adaptation; that is, for subclasses of the parameter space where the true mean reward functions are also piecewise constant with $k$ pieces, this policy remains minimax rate optimal simul…
Thompson Sampling has been demonstrated in many complex bandit models, however the theoretical guarantees available for the parametric multi-armed bandit are still limited to the Bernoulli case. Here we extend them by proving asymptotic optimality of the algorithm using the Jeffreys prior for 1-dimensional exponential family bandits. Our proof builds on previous work, but also makes extensive use of closed forms for Kullback-Leibler divergence and Fisher information (and thus Jeffreys prior) available in an exponential family. This allow us to give a finite time exponential concentration inequ…
Université de Lille（里尔大学）；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）；Centrale Lille（里尔中央理工学院）
We study the corrupted bandit problem, i.e. a stochastic multi-armed bandit problem with $k$ unknown reward distributions, which are heavy-tailed and corrupted by a history-independent adversary or Nature. To be specific, the reward obtained by playing an arm comes from corresponding heavy-tailed reward distribution with probability $1-\varepsilon \in (0.5,1]$ and an arbitrary corruption distribution of unbounded support with probability $\varepsilon \in [0,0.5)$. First, we provide $\textit{a problem-dependent lower bound on the regret}$ of any corrupted bandit algorithm. The lower bounds indi…
School of Mathematics, Nanjing University（南京大学数学系）；School of Artificial Intelligence, Nanjing University, National Key Laboratory for Novel Software Technology（南京大学人工智能学院，计算机软件新技术国家重点实验室）；School of Mathematics and Physics, Xi’an Jiaotong-Liverpool University（西交利物浦大学数理学院）
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…
Extended UCB Policies for Multi-armed Bandit Problems
School of Mathematics and Physics（数学与物理学院）；Xi’an Jiaotong-Liverpool University（西安交通大学利物浦大学）；School of Mathematics（数学学院）；Nanjing University（南京大学）；National Key Laboratory for Novel Software Technology（新型软件技术国家实验室）；School of Artificial Intelligence（人工智能学院）
The University of Tokyo（东京大学）；RIKEN AIP（理化学研究所人工智能研究中心）；Kyoto University（京都大学）
In the stochastic multi-armed bandit problem, a randomized probability matching policy called Thompson sampling (TS) has shown excellent performance in various reward models. In addition to the empirical performance, TS has been shown to achieve asymptotic problem-dependent lower bounds in several models. However, its optimality has been mainly addressed under light-tailed or one-parameter models that belong to exponential families. In this paper, we consider the optimality of TS for the Pareto model that has a heavy tail and is parameterized by two unknown parameters. Specifically, we discuss…
Pompeu Fabra University（庞培法布拉大学）；ICREA（加泰罗尼亚研究与高等研究院）；Barcelona Graduate School of Economics（巴塞罗那经济研究生院）；The Australian National University（澳大利亚国立大学）；Sorbonne University（索邦大学）
We survey some of the recent advances in mean estimation and regression function estimation. In particular, we describe sub-Gaussian mean estimators for possibly heavy-tailed data both in the univariate and multivariate settings. We focus on estimators based on median-of-means techniques but other methods such as the trimmed mean and Catoni's estimator are also reviewed. We give detailed proofs for the cornerstone results. We dedicate a section on statistical learning problems--in particular, regression function estimation--in the presence of possibly heavy-tailed data.
Consider the problem of finding a population or a probability distribution amongst many with the largest mean when these means are unknown but population samples can be simulated or otherwise generated. Typically, by selecting largest sample mean population, it can be shown that false selection probability decays at an exponential rate. Lately, researchers have sought algorithms that guarantee that this probability is restricted to a small $δ$ in order $\log(1/δ)$ computational time by estimating the associated large deviations rate function via simulation. We show that such guarantees are mis…
Peking University（北京大学）；University of Toronto（多伦多大学）
This paper presents two algorithms, AdaOFUL and VARA, for online sequential decision-making in the presence of heavy-tailed rewards with only finite variances. For linear stochastic bandits, we address the issue of heavy-tailed rewards by modifying the adaptive Huber regression and proposing AdaOFUL. AdaOFUL achieves a state-of-the-art regret bound of $\widetilde{O}\big(d\big(\sum_{t=1}^T ν_{t}^2\big)^{1/2}+d\big)$ as if the rewards were uniformly bounded, where $ν_{t}^2$ is the observed conditional variance of the reward at round $t$, $d$ is the feature dimension, and $\widetilde{O}(\cdot)$ h…
