We study online meta-learning with bandit feedback, with the goal of improving performance across multiple tasks if they are similar according to some natural similarity measure. As the first to target the adversarial online-within-online partial-information setting, we design meta-algorithms that combine outer learners to simultaneously tune the initialization and other hyperparameters of an inner learner for two important cases: multi-armed bandits (MAB) and bandit linear optimization (BLO). For MAB, the meta-learners initialize and set hyperparameters of the Tsallis-entropy generalization of Exp3, with the task-averaged regret improving if the entropy of the optima-in-hindsight is small. For BLO, we learn to initialize and tune online mirror descent (OMD) with self-concordant barrier regularizers, showing that task-averaged regret varies directly with an action space-dependent measure they induce. Our guarantees rely on proving that unregularized follow-the-leader combined with two levels of low-dimensional hyperparameter tuning is enough to learn a sequence of affine functions of non-Lipschitz and sometimes non-convex Bregman divergences bounding the regret of OMD.
We study adversarial attacks that manipulate the reward signals to control the actions chosen by a stochastic multi-armed bandit algorithm. We propose the first attack against two popular bandit algorithms: $ε$-greedy and UCB, \emph{without} knowledge of the mean rewards. The attacker is able to spend only logarithmic effort, multiplied by a problem-specific parameter that becomes smaller as the bandit problem gets easier to attack. The result means the attacker can easily hijack the behavior of the bandit algorithm to promote or obstruct certain actions, say, a particular medical treatment. A…
We study the stochastic multi-armed bandits problem in the presence of adversarial corruption. We present a new algorithm for this problem whose regret is nearly optimal, substantially improving upon previous work. Our algorithm is agnostic to the level of adversarial contamination and can tolerate a significant amount of corruption with virtually no degradation in performance.
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）
We study online learning with bandit feedback across multiple tasks, with the goal of improving average performance across tasks if they are similar according to some natural task-similarity measure. As the first to target the adversarial setting, we design a unified meta-algorithm that yields setting-specific guarantees for two important cases: multi-armed bandits (MAB) and bandit linear optimization (BLO). For MAB, the meta-algorithm tunes the initialization, step-size, and entropy parameter of the Tsallis-entropy generalization of the well-known Exp3 method, with the task-averaged regret pr…
University of Copenhagen（哥本哈根大学）；University of Southern California（南加州大学）
We develop the first general semi-bandit algorithm that simultaneously achieves $\mathcal{O}(\log T)$ regret for stochastic environments and $\mathcal{O}(\sqrt{T})$ regret for adversarial environments without knowledge of the regime or the number of rounds $T$. The leading problem-dependent constants of our bounds are not only optimal in some worst-case sense studied previously, but also optimal for two concrete instances of semi-bandit problems. Our algorithm and analysis extend the recent work of (Zimmert & Seldin, 2019) for the special case of multi-armed bandit, but importantly requires a …
Microsoft Research NYC（微软研究院纽约分院）；Cornell University（康奈尔大学）；Google Research NYC（谷歌研究院纽约分院）
We study "adversarial scaling", a multi-armed bandit model where rewards have a stochastic and an adversarial component. Our model captures display advertising where the "click-through-rate" can be decomposed to a (fixed across time) arm-quality component and a non-stochastic user-relevance component (fixed across arms). Despite the relative stochasticity of our model, we demonstrate two settings where most bandit algorithms suffer. On the positive side, we show that two algorithms, one from the action elimination and one from the mirror descent family are adaptive enough to be robust to adver…
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…
We derive an algorithm that achieves the optimal (within constants) pseudo-regret in both adversarial and stochastic multi-armed bandits without prior knowledge of the regime and time horizon. The algorithm is based on online mirror descent (OMD) with Tsallis entropy regularization with power $α=1/2$ and reduced-variance loss estimators. More generally, we define an adversarial regime with a self-bounding constraint, which includes stochastic regime, stochastically constrained adversarial regime (Wei and Luo), and stochastic regime with adversarial corruptions (Lykouris et al.) as special case…
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 work we consider adversarial contextual bandits with risk constraints. At each round, nature prepares a context, a cost for each arm, and additionally a risk for each arm. The learner leverages the context to pull an arm and then receives the corresponding cost and risk associated with the pulled arm. In addition to minimizing the cumulative cost, the learner also needs to satisfy long-term risk constraints -- the average of the cumulative risk from all pulled arms should not be larger than a pre-defined threshold. To address this problem, we first study the full information setting wh…
We study the adversarial multi-armed bandit problem and create a completely online algorithmic framework that is invariant under arbitrary translations and scales of the arm losses. We study the expected performance of our algorithm against a generic competition class, which makes it applicable for a wide variety of problem scenarios. Our algorithm works from a universal prediction perspective and the performance measure used is the expected regret against arbitrary arm selection sequences, which is the difference between our losses and a competing loss sequence. The competition class can be d…
Contextual bandit algorithms are applied in a wide range of domains, from advertising to recommender systems, from clinical trials to education. In many of these domains, malicious agents may have incentives to attack the bandit algorithm to induce it to perform a desired behavior. For instance, an unscrupulous ad publisher may try to increase their own revenue at the expense of the advertisers; a seller may want to increase the exposure of their products, or thwart a competitor's advertising campaign. In this paper, we study several attack scenarios and show that a malicious agent can force a…
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…
Technion - Israel Institute of Technology（以色列理工学院）
We study meta-learning for adversarial multi-armed bandits. We consider the online-within-online setup, in which a player (learner) encounters a sequence of multi-armed bandit episodes. The player's performance is measured as regret against the best arm in each episode, according to the losses generated by an adversary. The difficulty of the problem depends on the empirical distribution of the per-episode best arm chosen by the adversary. We present an algorithm that can leverage the non-uniformity in this empirical distribution, and derive problem-dependent regret bounds. This solution compri…
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…
The RKHS bandit problem (also called kernelized multi-armed bandit problem) is an online optimization problem of non-linear functions with noisy feedback. Although the problem has been extensively studied, there are unsatisfactory results for some problems compared to the well-studied linear bandit case. Specifically, there is no general algorithm for the adversarial RKHS bandit problem. In addition, high computational complexity of existing algorithms hinders practical application. We address these issues by considering a novel amalgamation of approximation theory and the misspecified linear …
Best-of-both-worlds algorithms for online learning which achieve near-optimal regret in both the adversarial and the stochastic regimes have received growing attention recently. Existing techniques often require careful adaptation to every new problem setup, including specialised potentials and careful tuning of algorithm parameters. Yet, in domains such as linear bandits, it is still unknown if there exists an algorithm that can simultaneously obtain $O(\log(T))$ regret in the stochastic regime and $\tilde{O}(\sqrt{T})$ regret in the adversarial regime. In this work, we resolve this question …
Recent work on follow the perturbed leader (FTPL) algorithms for the adversarial multi-armed bandit problem has highlighted the role of the hazard rate of the distribution generating the perturbations. Assuming that the hazard rate is bounded, it is possible to provide regret analyses for a variety of FTPL algorithms for the multi-armed bandit problem. This paper pushes the inquiry into regret bounds for FTPL algorithms beyond the bounded hazard rate condition. There are good reasons to do so: natural distributions such as the uniform and Gaussian violate the condition. We give regret bounds f…
Tsinghua University（清华大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We study the linear contextual bandit problem in the presence of adversarial corruption, where the interaction between the player and a possibly infinite decision set is contaminated by an adversary that can corrupt the reward up to a corruption level $C$ measured by the sum of the largest alteration on rewards in each round. We present a variance-aware algorithm that is adaptive to the level of adversarial contamination $C$. The key algorithmic design includes (1) a multi-level partition scheme of the observed data, (2) a cascade of confidence sets that are adaptive to the level of the corrup…
Microsoft Research（微软研究院）；Facebook（脸书公司）；University of Maryland（马里兰大学）
We consider Bandits with Knapsacks (henceforth, BwK), a general model for multi-armed bandits under supply/budget constraints. In particular, a bandit algorithm needs to solve a well-known knapsack problem: find an optimal packing of items into a limited-size knapsack. The BwK problem is a common generalization of numerous motivating examples, which range from dynamic pricing to repeated auctions to dynamic ad allocation to network routing and scheduling. While the prior work on BwK focused on the stochastic version, we pioneer the other extreme in which the outcomes can be chosen adversariall…
Adversarial Attacks on Combinatorial Multi-Armed Bandits
Oregon State University（俄勒冈州立大学）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Pennsylvania State University（宾夕法尼亚州立大学）；Princeton University（普林斯顿大学）
University of Southern California（南加州大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We study high-probability regret bounds for adversarial $K$-armed bandits with time-varying feedback graphs over $T$ rounds. For general strongly observable graphs, we develop an algorithm that achieves the optimal regret $\widetilde{\mathcal{O}}((\sum_{t=1}^Tα_t)^{1/2}+\max_{t\in[T]}α_t)$ with high probability, where $α_t$ is the independence number of the feedback graph at round $t$. Compared to the best existing result [Neu, 2015] which only considers graphs with self-loops for all nodes, our result not only holds more generally, but importantly also removes any $\text{poly}(K)$ dependence …
We develop a novel and generic algorithm for the adversarial multi-armed bandit problem (or more generally the combinatorial semi-bandit problem). When instantiated differently, our algorithm achieves various new data-dependent regret bounds improving previous work. Examples include: 1) a regret bound depending on the variance of only the best arm; 2) a regret bound depending on the first-order path-length of only the best arm; 3) a regret bound depending on the sum of first-order path-lengths of all arms as well as an important negative term, which together lead to faster convergence rates fo…
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…
We propose a new algorithm for adversarial multi-armed bandits with unrestricted delays. The algorithm is based on a novel hybrid regularizer applied in the Follow the Regularized Leader (FTRL) framework. It achieves $\mathcal{O}(\sqrt{kn}+\sqrt{D\log(k)})$ regret guarantee, where $k$ is the number of arms, $n$ is the number of rounds, and $D$ is the total delay. The result matches the lower bound within constants and requires no prior knowledge of $n$ or $D$. Additionally, we propose a refined tuning of the algorithm, which achieves $\mathcal{O}(\sqrt{kn}+\min_{S}|S|+\sqrt{D_{\bar S}\log(k)})…
We consider the Scale-Free Adversarial Multi Armed Bandits(MAB) problem. At the beginning of the game, the player only knows the number of arms $n$. It does not know the scale and magnitude of the losses chosen by the adversary or the number of rounds $T$. In each round, it sees bandit feedback about the loss vectors $l_1,\dots, l_T \in \mathbb{R}^n$. The goal is to bound its regret as a function of $n$ and norms of $l_1,\dots, l_T$. We design a bandit Follow The Regularized Leader (FTRL) algorithm, that uses an adaptive learning rate and give two different regret bounds, based on the explorat…
NEC Corporation（NEC公司）；RIKEN AIP（理化学研究所人工智能研究中心）；Kyoto University（京都大学）
This paper considers the multi-armed bandit (MAB) problem and provides a new best-of-both-worlds (BOBW) algorithm that works nearly optimally in both stochastic and adversarial settings. In stochastic settings, some existing BOBW algorithms achieve tight gap-dependent regret bounds of $O(\sum_{i: Δ_i>0} \frac{\log T}{Δ_i})$ for suboptimality gap $Δ_i$ of arm $i$ and time horizon $T$. As Audibert et al. [2007] have shown, however, that the performance can be improved in stochastic environments with low-variance arms. In fact, they have provided a stochastic MAB algorithm with gap-variance-depen…
Wichita State University（威奇托州立大学）；Department of Electrical Engineering and Computer Science（电气工程与计算机科学系（不单独作为机构提取，此处主要机构为大学））
This paper addresses the problem of unknown delays in adversarial multi-armed bandit (MAB) with multiple play. Existing work on similar game setting focused on only the case where the learner selects an arm in each round. However, there are lots of applications in robotics where a learner needs to select more than one arm per round. It is therefore worthwhile to investigate the effect of delay when multiple arms are chosen. The multiple arms chosen per round in this setting are such that they experience the same amount of delay. There can be an aggregation of feedback losses from different com…
Research on the multi-armed bandit problem has studied the trade-off of exploration and exploitation in depth. However, there are numerous applications where the cardinal absolute-valued feedback model (e.g. ratings from one to five) is not suitable. This has motivated the formulation of the duelling bandits problem, where the learner picks a pair of actions and observes a noisy binary feedback, indicating a relative preference between the two. There exist a multitude of different settings and interpretations of the problem for two reasons. First, due to the absence of a total order of actions…
Microsoft Research, NYC（微软研究院，纽约市）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；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）；Grenoble INP（格勒诺布尔综合理工学院）
We study the problem of $K$-armed dueling bandit for both stochastic and adversarial environments, where the goal of the learner is to aggregate information through relative preferences of pair of decisions points queried in an online sequential manner. We first propose a novel reduction from any (general) dueling bandits to multi-armed bandits and despite the simplicity, it allows us to improve many existing results in dueling bandits. In particular, \emph{we give the first best-of-both world result for the dueling bandits regret minimization problem} -- a unified framework that is guaranteed…
We propose Banker-OMD, a novel framework generalizing the classical Online Mirror Descent (OMD) technique in online learning algorithm design. Banker-OMD allows algorithms to robustly handle delayed feedback, and offers a general methodology for achieving $\tilde{O}(\sqrt{T} + \sqrt{D})$-style regret bounds in various delayed-feedback online learning tasks, where $T$ is the time horizon length and $D$ is the total feedback delay. We demonstrate the power of Banker-OMD with applications to three important bandit scenarios with delayed feedback, including delayed adversarial Multi-armed bandits …
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Google Research（谷歌研究院）；The Hong Kong University of Science and Technology（香港科技大学）
We study the linear contextual bandit problem in the presence of adversarial corruption, where the reward at each round is corrupted by an adversary, and the corruption level (i.e., the sum of corruption magnitudes over the horizon) is $C\geq 0$. The best-known algorithms in this setting are limited in that they either are computationally inefficient or require a strong assumption on the corruption, or their regret is at least $C$ times worse than the regret without corruption. In this paper, to overcome these limitations, we propose a new algorithm based on the principle of optimism in the fa…
First- and Second-Order Bounds for Adversarial Linear Contextual Bandits
Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）；University of Amsterdam（阿姆斯特丹大学）；Universitat Pompeu Fabra（庞培法布拉大学）；Massachusetts Institute of Technology（麻省理工学院）
This paper investigates stochastic and adversarial combinatorial multi-armed bandit problems. In the stochastic setting under semi-bandit feedback, we derive a problem-specific regret lower bound, and discuss its scaling with the dimension of the decision space. We propose ESCB, an algorithm that efficiently exploits the structure of the problem and provide a finite-time analysis of its regret. ESCB has better performance guarantees than existing algorithms, and significantly outperforms these algorithms in practice. In the adversarial setting under bandit feedback, we propose \textsc{CombEXP}…
We consider a linear stochastic bandit problem involving $M$ agents that can collaborate via a central server to minimize regret. A fraction $α$ of these agents are adversarial and can act arbitrarily, leading to the following tension: while collaboration can potentially reduce regret, it can also disrupt the process of learning due to adversaries. In this work, we provide a fundamental understanding of this tension by designing new algorithms that balance the exploration-exploitation trade-off via carefully constructed robust confidence intervals. We also complement our algorithms with tight …
We study the notoriously difficult no-sensing adversarial multi-player multi-armed bandits (MP-MAB) problem from a new perspective. Instead of focusing on the hardness of multiple players, we introduce a new dimension of hardness, called attackability. All adversaries can be categorized based on the attackability and we introduce Adversary-Adaptive Collision-Communication (A2C2), a family of algorithms with forced-collision communication among players. Both attackability-aware and unaware settings are studied, and information-theoretic tools of the Z-channel model and error-correction coding a…
We consider the Scale-Free Adversarial Multi-Armed Bandit (MAB) problem with unrestricted feedback delays. In contrast to the standard assumption that all losses are $[0,1]$-bounded, in our setting, losses can fall in a general bounded interval $[-L, L]$, unknown to the agent beforehand. Furthermore, the feedback of each arm pull can experience arbitrary delays. We propose a novel approach named Scale-Free Delayed INF (SFD-INF) for this novel setting, which combines a recent "convex combination trick" together with a novel doubling and skipping technique. We then present two instances of SFD-I…
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…
University of Southern California（南加州大学）；VinUniversity（Vin大学）；VinAI Research（VinAI研究院）；DeepMind（深度思维）；Google Research（谷歌研究院）
We study a sequential decision problem where the learner faces a sequence of $K$-armed bandit tasks. The task boundaries might be known (the bandit meta-learning setting), or unknown (the non-stationary bandit setting). For a given integer $M\le K$, the learner aims to compete with the best subset of arms of size $M$. We design an algorithm based on a reduction to bandit submodular maximization, and show that, for $T$ rounds comprised of $N$ tasks, in the regime of large number of tasks and small number of optimal arms $M$, its regret in both settings is smaller than the simple baseline of $\t…
University of Maryland（马里兰大学）；Microsoft Research（微软研究院）
We describe MELEE, a meta-learning algorithm for learning a good exploration policy in the interactive contextual bandit setting. Here, an algorithm must take actions based on contexts, and learn based only on a reward signal from the action taken, thereby generating an exploration/exploitation trade-off. MELEE addresses this trade-off by learning a good exploration strategy for offline tasks based on synthetic data, on which it can simulate the contextual bandit setting. Based on these simulations, MELEE uses an imitation learning strategy to learn a good exploration policy that can then be a…
Tsinghua University（清华大学）；Georgia Institute of Technology（佐治亚理工学院）
We study the multi-armed bandit (MAB) problem with composite and anonymous feedback. In this model, the reward of pulling an arm spreads over a period of time (we call this period as reward interval) and the player receives partial rewards of the action, convoluted with rewards from pulling other arms, successively. Existing results on this model require prior knowledge about the reward interval size as an input to their algorithms. In this paper, we propose adaptive algorithms for both the stochastic and the adversarial cases, without requiring any prior information about the reward interval.…
This paper investigates the problem of best arm identification in $\textit{contaminated}$ stochastic multi-arm bandits. In this setting, the rewards obtained from any arm are replaced by samples from an adversarial model with probability $\varepsilon$. A fixed confidence (infinite-horizon) setting is considered, where the goal of the learner is to identify the arm with the largest mean. Owing to the adversarial contamination of the rewards, each arm's mean is only partially identifiable. This paper proposes two algorithms, a gap-based algorithm and one based on the successive elimination, for …
University of Milan（米兰大学）；Istituto Italiano di Tecnologia（意大利技术研究院）；University College London（伦敦大学学院）；Facebook AI Research（脸书人工智能研究院）
We investigate meta-learning procedures in the setting of stochastic linear bandits tasks. The goal is to select a learning algorithm which works well on average over a class of bandits tasks, that are sampled from a task-distribution. Inspired by recent work on learning-to-learn linear regression, we consider a class of bandit algorithms that implement a regularized version of the well-known OFUL algorithm, where the regularization is a square euclidean distance to a bias vector. We first study the benefit of the biased OFUL algorithm in terms of regret minimization. We then propose two strat…
