We study exploration in stochastic multi-armed bandits when we have access to a divisible resource that can be allocated in varying amounts to arm pulls. We focus in particular on the allocation of distributed computing resources, where we may obtain results faster by allocating more resources per pull, but might have reduced throughput due to nonlinear scaling. For example, in simulation-based scientific studies, an expensive simulation can be sped up by running it on multiple cores. This speed-up however, is partly offset by the communication among cores, which results in lower throughput than if fewer cores were allocated per trial to run more trials in parallel. In this paper, we explore these trade-offs in two settings. First, in a fixed confidence setting, we need to find the best arm with a given target success probability as quickly as possible. We propose an algorithm which trades off between information accumulation and throughput and show that the time taken can be upper bounded by the solution of a dynamic program whose inputs are the gaps between the sub-optimal and optimal arms. We also prove a matching hardness result. Second, we present an algorithm for a fixed deadline setting, where we are given a time deadline and need to maximize the probability of finding the best arm. We corroborate our theoretical insights with simulation experiments that show that the algorithms consistently match or outperform baseline algorithms on a variety of problem instances.
Microsoft Research（微软研究院）；Stanford University（斯坦福大学）
We consider the non-stochastic version of the (cooperative) multi-player multi-armed bandit problem. The model assumes no communication at all between the players, and furthermore when two (or more) players select the same action this results in a maximal loss. We prove the first $\sqrt{T}$-type regret guarantee for this problem, under the feedback model where collisions are announced to the colliding players. Such a bound was not known even for the simpler stochastic version. We also prove the first sublinear guarantee for the feedback model where collision information is not available, namel…
Institute of Science and Technology Austria（奥地利科学技术研究所）
We consider the problem of minimizing the regret in stochastic multi-armed bandit, when the measure of goodness of an arm is not the mean return, but some general function of the mean and the variance.We characterize the conditions under which learning is possible and present examples for which no natural algorithm can achieve sublinear regret.
We study the stream-based online active learning in a contextual multi-armed bandit framework. In this framework, the reward depends on both the arm and the context. In a stream-based active learning setting, obtaining the ground truth of the reward is costly, and the conventional contextual multi-armed bandit algorithm fails to achieve a sublinear regret due to this cost. Hence, the algorithm needs to determine whether or not to request the ground truth of the reward at current time slot. In our framework, we consider a stream-based active learning setting in which a query request for the gro…
We consider the stochastic bandit problem in the sublinear space setting, where one cannot record the win-loss record for all $K$ arms. We give an algorithm using $O(1)$ words of space with regret \[ \sum_{i=1}^{K}\frac{1}{Δ_i}\log \frac{Δ_i}Δ\log T \] where $Δ_i$ is the gap between the best arm and arm $i$ and $Δ$ is the gap between the best and the second-best arms. If the rewards are bounded away from $0$ and $1$, this is within an $O(\log 1/Δ)$ factor of the optimum regret possible without space constraints.
LINE Plus Corporation（LINE Plus公司）；Pohang University of Science and Technology（浦项科技大学）
We consider a multi-armed bandit problem in which a set of arms is registered by each agent, and the agent receives reward when its arm is selected. An agent might strategically submit more arms with replications, which can bring more reward by abusing the bandit algorithm's exploration-exploitation balance. Our analysis reveals that a standard algorithm indeed fails at preventing replication and suffers from linear regret in time $T$. We aim to design a bandit algorithm which demotivates replications and also achieves a small cumulative regret. We devise Hierarchical UCB (H-UCB) of replicatio…
We extend the classic multi-armed bandit (MAB) model to the setting of noncompliance, where the arm pull is a mere instrument and the treatment applied may differ from it, which gives rise to the instrument-armed bandit (IAB) problem. The IAB setting is relevant whenever the experimental units are human since free will, ethics, and the law may prohibit unrestricted or forced application of treatment. In particular, the setting is relevant in bandit models of dynamic clinical trials and other controlled trials on human interventions. Nonetheless, the setting has not been fully investigate in th…
This paper studies the problem of stochastic continuum-armed bandit with constraints (SCBwC), where we optimize a black-box reward function $f(x)$ subject to a black-box constraint function $g(x)\leq 0$ over a continuous space $\mathcal X$. We model reward and constraint functions via Gaussian processes (GPs) and propose a Rectified Pessimistic-Optimistic Learning framework (RPOL), a penalty-based method incorporating optimistic and pessimistic GP bandit learning for reward and constraint functions, respectively. We consider the metric of cumulative constraint violation $\sum_{t=1}^T(g(x_t))^{…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）
We study the effect of persistence of engagement on learning in a stochastic multi-armed bandit setting. In advertising and recommendation systems, repetition effect includes a wear-in period, where the user's propensity to reward the platform via a click or purchase depends on how frequently they see the recommendation in the recent past. It also includes a counteracting wear-out period, where the user's propensity to respond positively is dampened if the recommendation was shown too many times recently. Priming effect can be naturally modelled as a temporal constraint on the strategy space, …
We analyze the $K$-armed bandit problem where the reward for each arm is a noisy realization based on an observed context under mild nonparametric assumptions. We attain tight results for top-arm identification and a sublinear regret of $\widetilde{O}\Big(T^{\frac{1+D}{2+D}}\Big)$, where $D$ is the context dimension, for a modified UCB algorithm that is simple to implement ($k$NN-UCB). We then give global intrinsic dimension dependent and ambient dimension independent regret bounds. We also discuss recovering topological structures within the context space based on expected bandit performance …
Wayne State University（韦恩州立大学）；Virginia Tech（弗吉尼亚理工大学）
We study a stochastic bandit problem with a general unknown reward function and a general unknown constraint function. Both functions can be non-linear (even non-convex) and are assumed to lie in a reproducing kernel Hilbert space (RKHS) with a bounded norm. This kernelized bandit setup strictly generalizes standard multi-armed bandits and linear bandits. In contrast to safety-type hard constraints studied in prior works, we consider soft constraints that may be violated in any round as long as the cumulative violations are small, which is motivated by various practical applications. Our ultim…
The multi-armed bandits' framework is the most common platform to study strategies for sequential decision-making problems. Recently, the notion of fairness has attracted a lot of attention in the machine learning community. One can impose the fairness condition that at any given point of time, even during the learning phase, a poorly performing candidate should not be preferred over a better candidate. This fairness constraint is known to be one of the most stringent and has been studied in the stochastic multi-armed bandits' framework in a stationary setting for which regret bounds have been…
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…
The Best Arm Evades: Near-optimal Multi-pass Streaming Lower Bounds for Pure Exploration in Multi-armed Bandits
Cheriton School of Computer Science, University of Waterloo（滑铁卢大学切尔顿计算机科学学院）；Rutgers University（罗格斯大学）；Rice University（莱斯大学）；Texas A&M University（德克萨斯农工大学）；Google Research（谷歌研究院）
Centre Borelli, ENS Paris-Saclay（巴黎萨克雷高等师范学院博雷利中心）；Cdiscount（Cdiscount公司）；CREST, ENSAE Paris（巴黎国立统计与经济管理学院经济与统计研究中心）；Criteo AI Lab（Criteo人工智能实验室）
The Greedy algorithm is the simplest heuristic in sequential decision problem that carelessly takes the locally optimal choice at each round, disregarding any advantages of exploring and/or information gathering. Theoretically, it is known to sometimes have poor performances, for instance even a linear regret (with respect to the time horizon) in the standard multi-armed bandit problem. On the other hand, this heuristic performs reasonably well in practice and it even has sublinear, and even near-optimal, regret bounds in some very specific linear contextual and Bayesian bandit models. We buil…
University of Washington（华盛顿大学）；Harvard University（哈佛大学）
We study a multi-armed bandit problem in a dynamic environment where arm rewards evolve in a correlated fashion according to a Markov chain. Different than much of the work on related problems, in our formulation a learning algorithm does not have access to either a priori information or observations of the state of the Markov chain and only observes smoothed reward feedback following time intervals we refer to as epochs. We demonstrate that existing methods such as UCB and $\varepsilon$-greedy can suffer linear regret in such an environment. Employing mixing-time bounds on Markov chains, we d…
The stochastic multi-armed bandit setting has been recently studied in the non-stationary regime, where the mean payoff of each action is a non-decreasing function of the number of rounds passed since it was last played. This model captures natural behavioral aspects of the users which crucially determine the performance of recommendation platforms, ad placement systems, and more. Even assuming prior knowledge of the mean payoff functions, computing an optimal planning in the above model is NP-hard, while the state-of-the-art is a $1/4$-approximation algorithm for the case where at most one ar…
Pompeu Fabra University（庞培法布拉大学）；ICREA（加泰罗尼亚高级研究学院）；Barcelona Graduate School of Economics（巴塞罗那经济学研究生院）；Imperial College London（伦敦帝国理工学院）；Cisco Systems, Inc.（思科系统公司）
The fidelity bandits problem is a variant of the $K$-armed bandit problem in which the reward of each arm is augmented by a fidelity reward that provides the player with an additional payoff depending on how 'loyal' the player has been to that arm in the past. We propose two models for fidelity. In the loyalty-points model the amount of extra reward depends on the number of times the arm has previously been played. In the subscription model the additional reward depends on the current number of consecutive draws of the arm. We consider both stochastic and adversarial problems. Since single-arm…
In this paper, we propose a new multi-objective contextual multi-armed bandit (MAB) problem with two objectives, where one of the objectives dominates the other objective. Unlike single-objective MAB problems in which the learner obtains a random scalar reward for each arm it selects, in the proposed problem, the learner obtains a random reward vector, where each component of the reward vector corresponds to one of the objectives and the distribution of the reward depends on the context that is provided to the learner at the beginning of each round. We call this problem contextual multi-armed …
IIT Bombay（印度理工学院孟买分校）；Boston University（波士顿大学）；University of Alberta（阿尔伯塔大学）
In many security and healthcare systems a sequence of features/sensors/tests are used for detection and diagnosis. Each test outputs a prediction of the latent state, and carries with it inherent costs. Our objective is to {\it learn} strategies for selecting tests to optimize accuracy \& costs. Unfortunately it is often impossible to acquire in-situ ground truth annotations and we are left with the problem of unsupervised sensor selection (USS). We pose USS as a version of stochastic partial monitoring problem with an {\it unusual} reward structure (even noisy annotations are unavailable). Un…
We consider systems that require timely monitoring of sources over a communication network, where the cost of delayed information is unknown, time-varying and possibly adversarial. For the single source monitoring problem, we design algorithms that achieve sublinear regret compared to the best fixed policy in hindsight. For the multiple source scheduling problem, we design a new online learning algorithm called Follow-the-Perturbed-Whittle-Leader and show that it has low regret compared to the best fixed scheduling policy in hindsight, while remaining computationally feasible. The algorithm an…
We introduce a multi-armed bandit model where the reward is a sum of multiple random variables, and each action only alters the distributions of some of them. After each action, the agent observes the realizations of all the variables. This model is motivated by marketing campaigns and recommender systems, where the variables represent outcomes on individual customers, such as clicks. We propose UCB-style algorithms that estimate the uplifts of the actions over a baseline. We study multiple variants of the problem, including when the baseline and affected variables are unknown, and prove subli…
We consider a variant of the stochastic multi-armed bandit problem, where multiple players simultaneously choose from the same set of arms and may collide, receiving no reward. This setting has been motivated by problems arising in cognitive radio networks, and is especially challenging under the realistic assumption that communication between players is limited. We provide a communication-free algorithm (Musical Chairs) which attains constant regret with high probability, as well as a sublinear-regret, communication-free algorithm (Dynamic Musical Chairs) for the more difficult setting of pla…
The Chinese University of Hong Kong（香港中文大学）；Shanghai Jiao Tong University（上海交通大学）；Adobe Research（奥多比研究院）
Contextual multi-armed bandit (MAB) is an important sequential decision-making problem in recommendation systems. A line of works, called the clustering of bandits (CLUB), utilize the collaborative effect over users and dramatically improve the recommendation quality. Owing to the increasing application scale and public concerns about privacy, there is a growing demand to keep user data decentralized and push bandit learning to the local server side. Existing CLUB algorithms, however, are designed under the centralized setting where data are available at a central server. We focus on studying …
ENS Paris-Saclay（巴黎萨克雷高等师范学校）；Univ. Lille（里尔大学）；Inria（法国国家信息与自动化研究所）；ENSAE Paris（巴黎经济与统计学院）；McGill University（麦吉尔大学）；CNRS（法国国家科学研究中心）；Centre Borelli（博雷利中心）；CRIStAL（里尔信号、自动化、计算机与图像处理研究中心）；Criteo AI Lab（卡塔戈人工智能实验室）
We study a multiplayer stochastic multi-armed bandit problem in which players cannot communicate, and if two or more players pull the same arm, a collision occurs and the involved players receive zero reward. We consider the challenging heterogeneous setting, in which different arms may have different means for different players, and propose a new and efficient algorithm that combines the idea of leveraging forced collisions for implicit communication and that of performing matching eliminations. We present a finite-time analysis of our algorithm, giving the first sublinear minimax regret boun…
Standard Multi-Armed Bandit (MAB) problems assume that the arms are independent. However, in many application scenarios, the information obtained by playing an arm provides information about the remainder of the arms. Hence, in such applications, this informativeness can and should be exploited to enable faster convergence to the optimal solution. In this paper, we introduce and formalize the Global MAB (GMAB), in which arms are globally informative through a global parameter, i.e., choosing an arm reveals information about all the arms. We propose a greedy policy for the GMAB which always sel…
Indian Institute of Science（印度科学学院）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）
We study the effect of impairment on stochastic multi-armed bandits and develop new ways to mitigate it. Impairment effect is the phenomena where an agent only accrues reward for an action if they have played it at least a few times in the recent past. It is practically motivated by repetition and recency effects in domains such as advertising (here consumer behavior may require repeat actions by advertisers) and vocational training (here actions are complex skills that can only be mastered with repetition to get a payoff). Impairment can be naturally modelled as a temporal constraint on the s…
We extend the adversarial/non-stochastic multi-play multi-armed bandit (MPMAB) to the case where the number of arms to play is variable. The work is motivated by the fact that the resources allocated to scan different critical locations in an interconnected transportation system change dynamically over time and depending on the environment. By modeling the malicious hacker and the intrusion monitoring system as the attacker and the defender, respectively, we formulate the problem for the two players as a sequential pursuit-evasion game. We derive the condition under which a Nash equilibrium of…
University of California, Berkeley（加州大学伯克利分校）；Indian Institute of Science（印度科学院）
We consider the problem of online learning in misspecified linear stochastic multi-armed bandit problems. Regret guarantees for state-of-the-art linear bandit algorithms such as Optimism in the Face of Uncertainty Linear bandit (OFUL) hold under the assumption that the arms expected rewards are perfectly linear in their features. It is, however, of interest to investigate the impact of potential misspecification in linear bandit models, where the expected rewards are perturbed away from the linear subspace determined by the arms features. Although OFUL has recently been shown to be robust to r…
