We investigate the \emph{linear contextual bandit problem} with independent and identically distributed (i.i.d.) contexts. In this problem, we aim to develop a \emph{Best-of-Both-Worlds} (BoBW) algorithm with regret upper bounds in both stochastic and adversarial regimes. We develop an algorithm based on \emph{Follow-The-Regularized-Leader} (FTRL) with Tsallis entropy, referred to as the $α$-\emph{Linear-Contextual (LC)-Tsallis-INF}. We show that its regret is at most $O(\log(T))$ in the stochastic regime under the assumption that the suboptimality gap is uniformly bounded from below, and at most $O(\sqrt{T})$ in the adversarial regime. Furthermore, our regret analysis is extended to more general regimes characterized by the \emph{margin condition} with a parameter $β\in (1, \infty]$, which imposes a milder assumption on the suboptimality gap. We show that the proposed algorithm achieves $O\left(\log(T)^{\frac{1+β}{2+β}}T^{\frac{1}{2+β}}\right)$ regret under the margin condition.
We derive improved regret bounds for the Tsallis-INF algorithm of Zimmert and Seldin (2021). We show that in adversarial regimes with a $(Δ,C,T)$ self-bounding constraint the algorithm achieves $\mathcal{O}\left(\left(\sum_{i\neq i^*} \frac{1}{Δ_i}\right)\log_+\left(\frac{(K-1)T}{\left(\sum_{i\neq i^*} \frac{1}{Δ_i}\right)^2}\right)+\sqrt{C\left(\sum_{i\neq i^*}\frac{1}{Δ_i}\right)\log_+\left(\frac{(K-1)T}{C\sum_{i\neq i^*}\frac{1}{Δ_i}}\right)}\right)$ regret bound, where $T$ is the time horizon, $K$ is the number of arms, $Δ_i$ are the suboptimality gaps, $i^*$ is the best arm, $C$ is the co…
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 define a novel family of algorithms for the adversarial multi-armed bandit problem, and provide a simple analysis technique based on convex smoothing. We prove two main results. First, we show that regularization via the \emph{Tsallis entropy}, which includes EXP3 as a special case, achieves the $Θ(\sqrt{TN})$ minimax regret. Second, we show that a wide class of perturbation methods achieve a near-optimal regret as low as $O(\sqrt{TN \log N})$ if the perturbation distribution has a bounded hazard rate. For example, the Gumbel, Weibull, Frechet, Pareto, and Gamma distributions all satisfy th…
Does 1/2-Tsallis-INF Also Work Well for Best-Arm Identification?
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Center for Applied Statistics and School of Statistics, Renmin University of China（中国人民大学应用统计中心与统计学院）
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 Copenhagen（哥本哈根大学）；Università degli Studi di Milano（米兰大学）
We propose an algorithm for stochastic and adversarial multiarmed bandits with switching costs, where the algorithm pays a price $λ$ every time it switches the arm being played. Our algorithm is based on adaptation of the Tsallis-INF algorithm of Zimmert and Seldin (2021) and requires no prior knowledge of the regime or time horizon. In the oblivious adversarial setting it achieves the minimax optimal regret bound of $O\big((λK)^{1/3}T^{2/3} + \sqrt{KT}\big)$, where $T$ is the time horizon and $K$ is the number of arms. In the stochastically constrained adversarial regime, which includes the s…
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…
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…
Università degli Studi di Milano（米兰大学）；Paul Sabatier University (UT3)（保罗·萨巴蒂埃大学（UT3））；Toulouse School of Economics (TSE)（图卢兹经济学院（TSE））；Istituto Italiano di Tecnologia (IIT)（意大利技术研究院（IIT））；Google Research（谷歌研究院）；Tel Aviv University（特拉维夫大学）
We investigate a nonstochastic bandit setting in which the loss of an action is not immediately charged to the player, but rather spread over the subsequent rounds in an adversarial way. The instantaneous loss observed by the player at the end of each round is then a sum of many loss components of previously played actions. This setting encompasses as a special case the easier task of bandits with delayed feedback, a well-studied framework where the player observes the delayed losses individually. Our first contribution is a general reduction transforming a standard bandit algorithm into one t…
Online learning to rank (OLTR) interactively learns to choose lists of items from a large collection based on certain click models that describe users' click behaviors. Most recent works for this problem focus on the stochastic environment where the item attractiveness is assumed to be invariant during the learning process. In many real-world scenarios, however, the environment could be dynamic or even arbitrarily changing. This work studies the OLTR problem in both stochastic and adversarial environments under the position-based model (PBM). We propose a method based on the follow-the-regular…
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…
Yale University（耶鲁大学）；University of Pennsylvania（宾夕法尼亚大学）
In this paper, we propose three online algorithms for submodular maximisation. The first one, Mono-Frank-Wolfe, reduces the number of per-function gradient evaluations from $T^{1/2}$ [Chen2018Online] and $T^{3/2}$ [chen2018projection] to 1, and achieves a $(1-1/e)$-regret bound of $O(T^{4/5})$. The second one, Bandit-Frank-Wolfe, is the first bandit algorithm for continuous DR-submodular maximization, which achieves a $(1-1/e)$-regret bound of $O(T^{8/9})$. Finally, we extend Bandit-Frank-Wolfe to a bandit algorithm for discrete submodular maximization, Responsive-Frank-Wolfe, which attains a …
Australian National University（澳大利亚国立大学）；Data61/NICTA（Data61/国家信息与通信技术研究院）；University of Alberta（阿尔伯塔大学）
We study the problem of using causal models to improve the rate at which good interventions can be learned online in a stochastic environment. Our formalism combines multi-arm bandits and causal inference to model a novel type of bandit feedback that is not exploited by existing approaches. We propose a new algorithm that exploits the causal feedback and prove a bound on its simple regret that is strictly better (in all quantities) than algorithms that do not use the additional causal information.
As bandit algorithms are increasingly utilized in scientific studies and industrial applications, there is an associated increasing need for reliable inference methods based on the resulting adaptively-collected data. In this work, we develop methods for inference on data collected in batches using a bandit algorithm. We first prove that the ordinary least squares estimator (OLS), which is asymptotically normal on independently sampled data, is not asymptotically normal on data collected using standard bandit algorithms when there is no unique optimal arm. This asymptotic non-normality result …
University of Michigan（密歇根大学）；Department of Computer Science（计算机科学系）；Department of Statistics（统计系）
We present online boosting algorithms for multiclass classification with bandit feedback, where the learner only receives feedback about the correctness of its prediction. We propose an unbiased estimate of the loss using a randomized prediction, allowing the model to update its weak learners with limited information. Using the unbiased estimate, we extend two full information boosting algorithms (Jung et al., 2017) to the bandit setting. We prove that the asymptotic error bounds of the bandit algorithms exactly match their full information counterparts. The cost of restricted feedback is refl…
We consider the stochastic linear (multi-armed) contextual bandit problem with the possibility of hidden simple multi-armed bandit structure in which the rewards are independent of the contextual information. Algorithms that are designed solely for one of the regimes are known to be sub-optimal for the alternate regime. We design a single computationally efficient algorithm that simultaneously obtains problem-dependent optimal regret rates in the simple multi-armed bandit regime and minimax optimal regret rates in the linear contextual bandit regime, without knowing a priori which of the two m…
Approximate information maximization for bandit games
Institut Pasteur（巴斯德研究所）；Université Paris Cité（巴黎西岱大学）；Epimethee（埃皮米修斯团队）；INRIA Paris（法国国家信息与自动化研究所巴黎分部）；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）；Université Paris Saclay（巴黎萨克雷大学）；LMO（奥赛数学实验室）
Northwestern University（西北大学）；University of Florida（佛罗里达大学）；New York University（纽约大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Linear contextual bandit is an important class of sequential decision making problems with a wide range of applications to recommender systems, online advertising, healthcare, and many other machine learning related tasks. While there is a lot of prior research, tight regret bounds of linear contextual bandit with infinite action sets remain open. In this paper, we address this open problem by considering the linear contextual bandit with (changing) infinite action sets. We prove a regret upper bound on the order of $O(\sqrt{d^2T\log T})\times \text{poly}(\log\log T)$ where $d$ is the domain d…
School of Information Science and Technology University of Science and Technology of China（中国科学技术大学信息科学技术学院）
We consider a variant of the classic multi-armed bandit problem where the expected reward of each arm is a function of an unknown parameter. The arms are divided into different groups, each of which has a common parameter. Therefore, when the player selects an arm at each time slot, information of other arms in the same group is also revealed. This regional bandit model naturally bridges the non-informative bandit setting where the player can only learn the chosen arm, and the global bandit model where sampling one arms reveals information of all arms. We propose an efficient algorithm, UCB-g,…
Assuming distributions are Gaussian often facilitates computations that are otherwise intractable. We study the performance of an agent that attains a bounded information ratio with respect to a bandit environment with a Gaussian prior distribution and a Gaussian likelihood function when applied instead to a Bernoulli bandit. Relative to an information-theoretic bound on the Bayesian regret the agent would incur when interacting with the Gaussian bandit, we bound the increase in regret when the agent interacts with the Bernoulli bandit. If the Gaussian prior distribution and likelihood functio…
Last-Iterate Analyses of FTRL with the 1/2-Tsallis Entropy in Stochastic Bandits
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Center for Applied Statistics and School of Statistics, Renmin University of China（中国人民大学统计学院）
Australian National University（澳大利亚国立大学）；Victoria University Wellington（惠灵顿维多利亚大学）
Motivated by clinical trials, we study bandits with observable non-compliance. At each step, the learner chooses an arm, after, instead of observing only the reward, it also observes the action that took place. We show that such noncompliance can be helpful or hurtful to the learner in general. Unfortunately, naively incorporating compliance information into bandit algorithms loses guarantees on sublinear regret. We present hybrid algorithms that maintain regret bounds up to a multiplicative factor and can incorporate compliance information. Simulations based on real data from the Internationa…
We consider a novel variant of the contextual bandit problem (i.e., the multi-armed bandit with side-information, or context, available to a decision-maker) where the context used at each decision may be corrupted ("useless context"). This new problem is motivated by certain on-line settings including clinical trial and ad recommendation applications. In order to address the corrupted-context setting,we propose to combine the standard contextual bandit approach with a classical multi-armed bandit mechanism. Unlike standard contextual bandit methods, we are able to learn from all iteration, eve…
NEC Corporation（日本电气公司）；Riken AIP（理化学研究所先进智能项目中心）；Tokyo Metropolitan University（东京都立大学）；Keio University（庆应义塾大学）；National Institute of Informatics（国立信息学研究所）
Bandit is a framework for designing sequential experiments. In each experiment, a learner selects an arm $A \in \mathcal{A}$ and obtains an observation corresponding to $A$. Theoretically, the tight regret lower-bound for the general bandit is polynomial with respect to the number of arms $|\mathcal{A}|$. This makes bandit incapable of handling an exponentially large number of arms, hence the bandit problem with side-information is often considered to overcome this lower bound. Recently, a bandit framework over a causal graph was introduced, where the structure of the causal graph is available…
In this paper we propose a flexible and efficient framework for handling multi-armed bandits, combining sequential Monte Carlo algorithms with hierarchical Bayesian modeling techniques. The framework naturally encompasses restless bandits, contextual bandits, and other bandit variants under a single inferential model. Despite the model's generality, we propose efficient Monte Carlo algorithms to make inference scalable, based on recent developments in sequential Monte Carlo methods. Through two simulation studies, the framework is shown to outperform other empirical methods, while also natural…
We study the neural-linear bandit model for solving sequential decision-making problems with high dimensional side information. Neural-linear bandits leverage the representation power of deep neural networks and combine it with efficient exploration mechanisms, designed for linear contextual bandits, on top of the last hidden layer. Since the representation is being optimized during learning, information regarding exploration with "old" features is lost. Here, we propose the first limited memory neural-linear bandit that is resilient to this phenomenon, which we term catastrophic forgetting. W…
Recommender systems should adapt to user interests as the latter evolve. A prevalent cause for the evolution of user interests is the influence of their social circle. In general, when the interests are not known, online algorithms that explore the recommendation space while also exploiting observed preferences are preferable. We present online recommendation algorithms rooted in the linear multi-armed bandit literature. Our bandit algorithms are tailored precisely to recommendation scenarios where user interests evolve under social influence. In particular, we show that our adaptations of the…
We study the attainable regret for online linear optimization problems with bandit feedback, where unlike the full-information setting, the player can only observe its own loss rather than the full loss vector. We show that the price of bandit information in this setting can be as large as $d$, disproving the well-known conjecture that the regret for bandit linear optimization is at most $\sqrt{d}$ times the full-information regret. Surprisingly, this is shown using "trivial" modifications of standard domains, which have no effect in the full-information setting. This and other results we pres…
In this paper, we propose an information-theoretic exploration strategy for stochastic, discrete multi-armed bandits that achieves optimal regret. Our strategy is based on the value of information criterion. This criterion measures the trade-off between policy information and obtainable rewards. High amounts of policy information are associated with exploration-dominant searches of the space and yield high rewards. Low amounts of policy information favor the exploitation of existing knowledge. Information, in this criterion, is quantified by a parameter that can be varied during search. We dem…
Technische Universität Dresden（德累斯顿工业大学）；Centre for Tactile Internet with Human-in-the-Loop (CeTI), Technische Universität Dresden（德累斯顿工业大学触觉互联网与人机闭环研究中心）；Max Planck UCL Centre for Computational Psychiatry and Ageing Research, University College London（伦敦大学学院马克斯·普朗克计算精神病学与衰老研究中心）；Secondmind；University College London（伦敦大学学院）
A key feature of sequential decision making under uncertainty is a need to balance between exploiting--choosing the best action according to the current knowledge, and exploring--obtaining information about values of other actions. The multi-armed bandit problem, a classical task that captures this trade-off, served as a vehicle in machine learning for developing bandit algorithms that proved to be useful in numerous industrial applications. The active inference framework, an approach to sequential decision making recently developed in neuroscience for understanding human and animal behaviour,…
The information ratio offers an approach to assessing the efficacy with which an agent balances between exploration and exploitation. Originally, this was defined to be the ratio between squared expected regret and the mutual information between the environment and action-observation pair, which represents a measure of information gain. Recent work has inspired consideration of alternative information measures, particularly for use in analysis of bandit learning algorithms to arrive at tighter regret bounds. We investigate whether quantification of information via such alternatives can improve…
We study fairness in linear bandit problems. Starting from the notion of meritocratic fairness introduced in Joseph et al. [2016], we carry out a more refined analysis of a more general problem, achieving better performance guarantees with fewer modelling assumptions on the number and structure of available choices as well as the number selected. We also analyze the previously-unstudied question of fairness in infinite linear bandit problems, obtaining instance-dependent regret upper bounds as well as lower bounds demonstrating that this instance-dependence is necessary. The result is a framew…
Queen Mary University（玛丽女王大学）；Boğaziçi University（博阿齐奇大学）；Google Research（谷歌研究院）
In this paper we present a new algorithm for online (sequential) inference in Bayesian neural networks, and show its suitability for tackling contextual bandit problems. The key idea is to combine the extended Kalman filter (which locally linearizes the likelihood function at each time step) with a (learned or random) low-dimensional affine subspace for the parameters; the use of a subspace enables us to scale our algorithm to models with $\sim 1M$ parameters. While most other neural bandit methods need to store the entire past dataset in order to avoid the problem of "catastrophic forgetting"…
Stochastic sparse linear bandits offer a practical model for high-dimensional online decision-making problems and have a rich information-regret structure. In this work we explore the use of information-directed sampling (IDS), which naturally balances the information-regret trade-off. We develop a class of information-theoretic Bayesian regret bounds that nearly match existing lower bounds on a variety of problem instances, demonstrating the adaptivity of IDS. To efficiently implement sparse IDS, we propose an empirical Bayesian approach for sparse posterior sampling using a spike-and-slab Ga…
We present algorithms for reducing the Dueling Bandits problem to the conventional (stochastic) Multi-Armed Bandits problem. The Dueling Bandits problem is an online model of learning with ordinal feedback of the form "A is preferred to B" (as opposed to cardinal feedback like "A has value 2.5"), giving it wide applicability in learning from implicit user feedback and revealed and stated preferences. In contrast to existing algorithms for the Dueling Bandits problem, our reductions -- named $\Doubler$, $\MultiSbm$ and $\DoubleSbm$ -- provide a generic schema for translating the extensive body …
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…
Multi-armed bandits (MAB) provide a principled online learning approach to attain the balance between exploration and exploitation. Due to the superior performance and low feedback learning without the learning to act in multiple situations, Multi-armed Bandits drawing widespread attention in applications ranging such as recommender systems. Likewise, within the recommender system, collaborative filtering (CF) is arguably the earliest and most influential method in the recommender system. Crucially, new users and an ever-changing pool of recommended items are the challenges that recommender sy…
How to explore efficiently is a central problem in multi-armed bandits. In this paper, we introduce the metadata-based multi-task bandit problem, where the agent needs to solve a large number of related multi-armed bandit tasks and can leverage some task-specific features (i.e., metadata) to share knowledge across tasks. As a general framework, we propose to capture task relations through the lens of Bayesian hierarchical models, upon which a Thompson sampling algorithm is designed to efficiently learn task relations, share information, and minimize the cumulative regrets. Two concrete example…
Bayesian strategies for contextual bandits have proved promising in single-state reinforcement learning tasks by modeling uncertainty using context information from the environment. In this paper, we propose Greedy Bandits with Sampled Context (GB-SC), a method for contextual multi-armed bandits to develop the prior from the context information using Thompson Sampling, and arm selection using an epsilon-greedy policy. The framework GB-SC allows for evaluation of context-reward dependency, as well as providing robustness for partially observable context vectors by leveraging the prior developed…
Restless multi-armed bandits with partially observable states has applications in communication systems, age of information and recommendation systems. In this paper, we study multi-state partially observable restless bandit models. We consider three different models based on information observable to decision maker -- 1) no information is observable from actions of a bandit 2) perfect information from bandit is observable only for one action on bandit, there is a fixed restart state, i.e., transition occurs from all other states to that state 3) perfect state information is available to decis…
Leiden University（莱顿大学）；Boston University（波士顿大学）；University of Southern California（南加州大学）
We study bandit convex optimization methods that adapt to the norm of the comparator, a topic that has only been studied before for its full-information counterpart. Specifically, we develop convex bandit algorithms with regret bounds that are small whenever the norm of the comparator is small. We first use techniques from the full-information setting to develop comparator-adaptive algorithms for linear bandits. Then, we extend the ideas to convex bandits with Lipschitz or smooth loss functions, using a new single-point gradient estimator and carefully designed surrogate losses.
