A standard assumption in contextual multi-arm bandit is that the true context is perfectly known before arm selection. Nonetheless, in many practical applications (e.g., cloud resource management), prior to arm selection, the context information can only be acquired by prediction subject to errors or adversarial modification. In this paper, we study a contextual bandit setting in which only imperfect context is available for arm selection while the true context is revealed at the end of each round. We propose two robust arm selection algorithms: MaxMinUCB (Maximize Minimum UCB) which maximizes the worst-case reward, and MinWD (Minimize Worst-case Degradation) which minimizes the worst-case regret. Importantly, we analyze the robustness of MaxMinUCB and MinWD by deriving both regret and reward bounds compared to an oracle that knows the true context. Our results show that as time goes on, MaxMinUCB and MinWD both perform as asymptotically well as their optimal counterparts that know the reward function. Finally, we apply MaxMinUCB and MinWD to online edge datacenter selection, and run synthetic simulations to validate our theoretical analysis.
Contextual bandits are canonical models for sequential decision-making under uncertainty in environments with time-varying components. In this setting, the expected reward of each bandit arm consists of the inner product of an unknown parameter with the context vector of that arm. The classical bandit settings heavily rely on assuming that the contexts are fully observed, while study of the richer model of imperfectly observed contextual bandits is immature. This work considers Greedy reinforcement learning policies that take actions as if the current estimates of the parameter and of the unob…
Contextual bandits are widely-used in the study of learning-based control policies for finite action spaces. While the problem is well-studied for bandits with perfectly observed context vectors, little is known about the case of imperfectly observed contexts. For this setting, existing approaches are inapplicable and new conceptual and technical frameworks are required. We present an implementable posterior sampling algorithm for bandits with imperfect context observations and study its performance for learning optimal decisions. The provided numerical results relate the performance of the al…
Princeton University（普林斯顿大学）；Massachusetts Institute of Technology（麻省理工学院）
The cooperative bandit problem is increasingly becoming relevant due to its applications in large-scale decision-making. However, most research for this problem focuses exclusively on the setting with perfect communication, whereas in most real-world distributed settings, communication is often over stochastic networks, with arbitrary corruptions and delays. In this paper, we study cooperative bandit learning under three typical real-world communication scenarios, namely, (a) message-passing over stochastic time-varying networks, (b) instantaneous reward-sharing over a network with random dela…
We consider the actor-critic contextual bandit for the mobile health (mHealth) intervention. State-of-the-art decision-making algorithms generally ignore the outliers in the dataset. In this paper, we propose a novel robust contextual bandit method for the mHealth. It can achieve the conflicting goal of reducing the influence of outliers while seeking for a similar solution compared with the state-of-the-art contextual bandit methods on the datasets without outliers. Such performance relies on two technologies: (1) the capped-$\ell_{2}$ norm; (2) a reliable method to set the thresholding hyper…
Cascading Bandits Robust to Adversarial Corruptions
Hong Kong University of Science and Technology（香港科技大学）；East China Normal University（华东师范大学）；The Chinese University of Hong Kong（香港中文大学）；Shanghai Jiao Tong University（上海交通大学）
Microsoft Research（微软研究院）；University of Maryland（马里兰大学）；Yale University（耶鲁大学）
We investigate the feasibility of learning from a mix of both fully-labeled supervised data and contextual bandit data. We specifically consider settings in which the underlying learning signal may be different between these two data sources. Theoretically, we state and prove no-regret algorithms for learning that is robust to misaligned cost distributions between the two sources. Empirically, we evaluate some of these algorithms on a large selection of datasets, showing that our approach is both feasible and helpful in practice.
University of California, Davis（加州大学戴维斯分校）；University of California, Los Angeles（加州大学洛杉矶分校）；Amazon（亚马逊）
Stochastic linear contextual bandit algorithms have substantial applications in practice, such as recommender systems, online advertising, clinical trials, etc. Recent works show that optimal bandit algorithms are vulnerable to adversarial attacks and can fail completely in the presence of attacks. Existing robust bandit algorithms only work for the non-contextual setting under the attack of rewards and cannot improve the robustness in the general and popular contextual bandit environment. In addition, none of the existing methods can defend against attacked context. In this work, we provide t…
University California San Diego（加州大学圣地亚哥分校）；Shanghai Jiao Tong University（上海交通大学）；Nanjing University of Posts and Telecommunications（南京邮电大学）
We study the problem of stochastic bandits with adversarial corruptions in the cooperative multi-agent setting, where $V$ agents interact with a common $K$-armed bandit problem, and each pair of agents can communicate with each other to expedite the learning process. In the problem, the rewards are independently sampled from distributions across all agents and rounds, but they may be corrupted by an adversary. Our goal is to minimize both the overall regret and communication cost across all agents. We first show that an additive term of corruption is unavoidable for any algorithm in this probl…
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 kno…
We introduce robustness in \textit{restless multi-armed bandits} (RMABs), a popular model for constrained resource allocation among independent stochastic processes (arms). Nearly all RMAB techniques assume stochastic dynamics are precisely known. However, in many real-world settings, dynamics are estimated with significant \emph{uncertainty}, e.g., via historical data, which can lead to bad outcomes if ignored. To address this, we develop an algorithm to compute minimax regret -- robust policies for RMABs. Our approach uses a double oracle framework (oracles for \textit{agent} and \textit{nat…
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…
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…
Motivated by applications of bandit algorithms in education, we consider a stochastic multi-armed bandit problem with $\varepsilon$-contaminated rewards. We allow an adversary to give arbitrary unbounded contaminated rewards with full knowledge of the past and future. We impose the constraint that for each time $t$ the proportion of contaminated rewards for any action is less than or equal to $\varepsilon$. We derive concentration inequalities for two robust mean estimators for sub-Gaussian distributions in the $\varepsilon$-contamination context. We define the $\varepsilon$-contaminated stoch…
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…
Contextual multi-armed bandit algorithms are widely used in sequential decision tasks such as news article recommendation systems, web page ad placement algorithms, and mobile health. Most of the existing algorithms have regret proportional to a polynomial function of the context dimension, $d$. In many applications however, it is often the case that contexts are high-dimensional with only a sparse subset of size $s_0 (\ll d)$ being correlated with the reward. We consider the stochastic linear contextual bandit problem and propose a novel algorithm, namely the Doubly-Robust Lasso Bandit algori…
Pure Exploration in Asynchronous Federated Bandits
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Yale University（耶鲁大学）；Oregon State University（俄勒冈州立大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Indian Institute of Science（印度科学理工学院）；Inria Saclay – Île de France（法国国家信息与自动化研究所萨克雷-法兰西岛研究中心）
We study the task of maximizing rewards from recommending items (actions) to users sequentially interacting with a recommender system. Users are modeled as latent mixtures of C many representative user classes, where each class specifies a mean reward profile across actions. Both the user features (mixture distribution over classes) and the item features (mean reward vector per class) are unknown a priori. The user identity is the only contextual information available to the learner while interacting. This induces a low-rank structure on the matrix of expected rewards r a,b from recommending i…
This paper extends the Distributionally Robust Optimization (DRO) approach for offline contextual bandits. Specifically, we leverage this framework to introduce a convex reformulation of the Counterfactual Risk Minimization principle. Besides relying on convex programs, our approach is compatible with stochastic optimization, and can therefore be readily adapted tothe large data regime. Our approach relies on the construction of asymptotic confidence intervals for offline contextual bandits through the DRO framework. By leveraging known asymptotic results of robust estimators, we also show how…
We consider an adversarial variant of the classic $K$-armed linear contextual bandit problem where the sequence of loss functions associated with each arm are allowed to change without restriction over time. Under the assumption that the $d$-dimensional contexts are generated i.i.d.~at random from a known distributions, we develop computationally efficient algorithms based on the classic Exp3 algorithm. Our first algorithm, RealLinExp3, is shown to achieve a regret guarantee of $\widetilde{O}(\sqrt{KdT})$ over $T$ rounds, which matches the best available bound for this problem. Our second algo…
The stochastic multi-arm bandit problem has been extensively studied under standard assumptions on the arm's distribution (e.g bounded with known support, exponential family, etc). These assumptions are suitable for many real-world problems but sometimes they require knowledge (on tails for instance) that may not be precisely accessible to the practitioner, raising the question of the robustness of bandit algorithms to model misspecification. In this paper we study a generic Dirichlet Sampling (DS) algorithm, based on pairwise comparisons of empirical indices computed with re-sampling of the a…
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 (…
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…
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…
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…
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…
University of Tübingen（蒂宾根大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Applications of machine learning inform human decision makers in a broad range of tasks. The resulting problem is usually formulated in terms of a single decision maker. We argue that it should rather be described as a two-player learning problem where one player is the machine and the other the human. While both players try to optimize the final decision, the setup is often characterized by (1) the presence of private information and (2) opacity, that is imperfect understanding between the decision makers. We prove that both properties can complicate decision making considerably. A lower boun…
University of Oxford（牛津大学）；University of California San Diego（加利福尼亚大学圣迭戈分校）；University of Chicago（芝加哥大学）；University of Warwick（华威大学）
Motivated by cognitive radios, stochastic Multi-Player Multi-Armed Bandits has been extensively studied in recent years. In this setting, each player pulls an arm, and receives a reward corresponding to the arm if there is no collision, namely the arm was selected by one single player. Otherwise, the player receives no reward if collision occurs. In this paper, we consider the presence of malicious players (or attackers) who obstruct the cooperative players (or defenders) from maximizing their rewards, by deliberately colliding with them. We provide the first decentralized and robust algorithm…
This paper considers the actor-critic contextual bandit for the mobile health (mHealth) intervention. The state-of-the-art decision-making methods in mHealth generally assume that the noise in the dynamic system follows the Gaussian distribution. Those methods use the least-square-based algorithm to estimate the expected reward, which is prone to the existence of outliers. To deal with the issue of outliers, we propose a novel robust actor-critic contextual bandit method for the mHealth intervention. In the critic updating, the capped-$\ell_{2}$ norm is used to measure the approximation error,…
Imperfect-Information Extensive-Form Games (IIEFGs) is a prevalent model for real-world games involving imperfect information and sequential plays. The Extensive-Form Correlated Equilibrium (EFCE) has been proposed as a natural solution concept for multi-player general-sum IIEFGs. However, existing algorithms for finding an EFCE require full feedback from the game, and it remains open how to efficiently learn the EFCE in the more challenging bandit feedback setting where the game can only be learned by observations from repeated playing. This paper presents the first sample-efficient algorithm…
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…
In this work we revisit two classic high-dimensional online learning problems, namely linear regression and contextual bandits, from the perspective of adversarial robustness. Existing works in algorithmic robust statistics make strong distributional assumptions that ensure that the input data is evenly spread out or comes from a nice generative model. Is it possible to achieve strong robustness guarantees even without distributional assumptions altogether, where the sequence of tasks we are asked to solve is adaptively and adversarially chosen? We answer this question in the affirmative for b…
Université Paris-Saclay（巴黎萨克雷大学）；ENS Paris-Saclay（巴黎萨克雷高等师范学院）；CNRS（法国国家科学研究中心）；Centre Borelli（博雷利中心）；CREST（经济与统计研究中心）；ENSAE Paris（巴黎国立统计与经济管理学院）；Criteo AI Lab（Criteo人工智能实验室）
Motivated by cognitive radios, stochastic multi-player multi-armed bandits gained a lot of interest recently. In this class of problems, several players simultaneously pull arms and encounter a collision - with 0 reward - if some of them pull the same arm at the same time. While the cooperative case where players maximize the collective reward (obediently following some fixed protocol) has been mostly considered, robustness to malicious players is a crucial and challenging concern. Existing approaches consider only the case of adversarial jammers whose objective is to blindly minimize the coll…
Columbia University（哥伦比亚大学）；Seoul National University（首尔大学）；Shepherd23 Inc.（Shepherd23 公司）
We propose a novel contextual bandit algorithm for generalized linear rewards with an $\tilde{O}(\sqrt{κ^{-1} ϕT})$ regret over $T$ rounds where $ϕ$ is the minimum eigenvalue of the covariance of contexts and $κ$ is a lower bound of the variance of rewards. In several practical cases where $ϕ=O(d)$, our result is the first regret bound for generalized linear model (GLM) bandits with the order $\sqrt{d}$ without relying on the approach of Auer [2002]. We achieve this bound using a novel estimator called double doubly-robust (DDR) estimator, a subclass of doubly-robust (DR) estimator but with a …
Computationally efficient contextual bandits are often based on estimating a predictive model of rewards given contexts and arms using past data. However, when the reward model is not well-specified, the bandit algorithm may incur unexpected regret, so recent work has focused on algorithms that are robust to misspecification. We propose a simple family of contextual bandit algorithms that adapt to misspecification error by reverting to a good safe policy when there is evidence that misspecification is causing a regret increase. Our algorithm requires only an offline regression oracle to ensure…
