A novel reinforcement learning algorithm is introduced for multiarmed restless bandits with average reward, using the paradigms of Q-learning and Whittle index. Specifically, we leverage the structure of the Whittle index policy to reduce the search space of Q-learning, resulting in major computational gains. Rigorous convergence analysis is provided, supported by numerical experiments. The numerical experiments show excellent empirical performance of the proposed scheme.
Whittle index policy is a powerful tool to obtain asymptotically optimal solutions for the notoriously intractable problem of restless bandits. However, finding the Whittle indices remains a difficult problem for many practical restless bandits with convoluted transition kernels. This paper proposes NeurWIN, a neural Whittle index network that seeks to learn the Whittle indices for any restless bandits by leveraging mathematical properties of the Whittle indices. We show that a neural network that produces the Whittle index is also one that produces the optimal control for a set of Markov deci…
University of Oxford（牛津大学）；Google Research（谷歌研究院）；Harvard University（哈佛大学）
We study the problem of planning restless multi-armed bandits (RMABs) with multiple actions. This is a popular model for multi-agent systems with applications like multi-channel communication, monitoring and machine maintenance tasks, and healthcare. Whittle index policies, which are based on Lagrangian relaxations, are widely used in these settings due to their simplicity and near-optimality under certain conditions. In this work, we first show that Whittle index policies can fail in simple and practically relevant RMAB settings, even when the RMABs are indexable. We discuss why the optimalit…
Restless bandits are an important class of problems with applications in recommender systems, active learning, revenue management and other areas. We consider infinite-horizon discounted restless bandits with many arms where a fixed proportion of arms may be pulled in each period and where arms share a finite state space. Although an average-case-optimal policy can be computed via stochastic dynamic programming, the computation required grows exponentially with the number of arms $N$. Thus, it is important to find scalable policies that can be computed efficiently for large $N$ and that are ne…
University of California, Los Angeles（加州大学洛杉矶分校）；University of Michigan（密歇根大学）
In this paper we consider the problem of learning the optimal policy for uncontrolled restless bandit problems. In an uncontrolled restless bandit problem, there is a finite set of arms, each of which when pulled yields a positive reward. There is a player who sequentially selects one of the arms at each time step. The goal of the player is to maximize its undiscounted reward over a time horizon T. The reward process of each arm is a finite state Markov chain, whose transition probabilities are unknown by the player. State transitions of each arm is independent of the selection of the player. …
In this paper we study the online learning problem involving rested and restless multiarmed bandits with multiple plays. The system consists of a single player/user and a set of K finite-state discrete-time Markov chains (arms) with unknown state spaces and statistics. At each time step the player can play M arms. The objective of the user is to decide for each step which M of the K arms to play over a sequence of trials so as to maximize its long term reward. The restless multiarmed bandit is particularly relevant to the application of opportunistic spectrum access (OSA), where a (secondary) …
University of Michigan（密歇根大学）；Criteo AI Lab（科韬人工智能实验室）
Restless bandit problems assume time-varying reward distributions of the arms, which adds flexibility to the model but makes the analysis more challenging. We study learning algorithms over the unknown reward distributions and prove a sub-linear, $O(\sqrt{T}\log T)$, regret bound for a variant of Thompson sampling. Our analysis applies in the infinite time horizon setting, resolving the open question raised by Jung and Tewari (2019) whose analysis is limited to the episodic case. We adopt their policy mapping framework, which allows our algorithm to be efficient and simultaneously keeps the re…
Tsinghua University（清华大学）；The Chinese University of Hong Kong（香港中文大学）
We study the online restless bandit problem, where the state of each arm evolves according to a Markov chain, and the reward of pulling an arm depends on both the pulled arm and the current state of the corresponding Markov chain. In this paper, we propose Restless-UCB, a learning policy that follows the explore-then-commit framework. In Restless-UCB, we present a novel method to construct offline instances, which only requires $O(N)$ time-complexity ($N$ is the number of arms) and is exponentially better than the complexity of existing learning policy. We also prove that Restless-UCB achieves…
SUNY-Binghamton University（纽约州立大学宾汉姆顿分校）；Indian Institute of Science（印度科学学院）
We study a finite-horizon restless multi-armed bandit problem with multiple actions, dubbed R(MA)^2B. The state of each arm evolves according to a controlled Markov decision process (MDP), and the reward of pulling an arm depends on both the current state of the corresponding MDP and the action taken. The goal is to sequentially choose actions for arms so as to maximize the expected value of the cumulative rewards collected. Since finding the optimal policy is typically intractable, we propose a computationally appealing index policy which we call Occupancy-Measured-Reward Index Policy. Our po…
MARBLE: Multi-Armed Restless Bandits in Latent Markovian Environment
Department of Computer and Systems Sciences (DSV), Stockholm University（斯德哥尔摩大学计算机与系统科学系）；Inria Sophia Antipolis（法国国家信息与自动化研究所索菲亚·安蒂波利斯研究中心）；Smartprofile（Smartprofile公司）
University of Potsdam（波茨坦大学）；Otto von Güricke Universität Magdeburg（奥托·冯·格里克马格德堡大学）
We study the stochastic multi-armed bandit problem in the case when the arm samples are dependent over time and generated from so-called weak $\cC$-mixing processes. We establish a $\cC-$Mix Improved UCB agorithm and provide both problem-dependent and independent regret analysis in two different scenarios. In the first, so-called fast-mixing scenario, we show that pseudo-regret enjoys the same upper bound (up to a factor) as for i.i.d. observations; whereas in the second, slow mixing scenario, we discover a surprising effect, that the regret upper bound is similar to the independent case, with…
Restless multi-armed bandits are often used to model budget-constrained resource allocation tasks where receipt of the resource is associated with an increased probability of a favorable state transition. Prior work assumes that individual arms only benefit if they receive the resource directly. However, many allocation tasks occur within communities and can be characterized by positive externalities that allow arms to derive partial benefit when their neighbor(s) receive the resource. We thus introduce networked restless bandits, a novel multi-armed bandit setting in which arms are both restl…
We obtain the conditions for the emergence of the swarm intelligence effect in an interactive game of restless multi-armed bandit (rMAB). A player competes with multiple agents. Each bandit has a payoff that changes with a probability $p_{c}$ per round. The agents and player choose one of three options: (1) Exploit (a good bandit), (2) Innovate (asocial learning for a good bandit among $n_{I}$ randomly chosen bandits), and (3) Observe (social learning for a good bandit). Each agent has two parameters $(c,p_{obs})$ to specify the decision: (i) $c$, the threshold value for Exploit, and (ii) $p_{…
We consider the restless multi-armed bandit (RMAB) problem with unknown dynamics in which a player chooses M out of N arms to play at each time. The reward state of each arm transits according to an unknown Markovian rule when it is played and evolves according to an arbitrary unknown random process when it is passive. The performance of an arm selection policy is measured by regret, defined as the reward loss with respect to the case where the player knows which M arms are the most rewarding and always plays the M best arms. We construct a policy with an interleaving exploration and exploitat…
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…
We consider finite-horizon restless bandits with multiple pulls per period, which play an important role in recommender systems, active learning, revenue management, and many other areas. While an optimal policy can be computed, in principle, using dynamic programming, the computation required scales exponentially in the number of arms $N$. Thus, there is substantial value in understanding the performance of index policies and other policies that can be computed efficiently for large $N$. We study the growth of the optimality gap, i.e., the loss in expected performance compared to an optimal p…
Constrained Restless Bandits for Dynamic Scheduling in Cyber-Physical Systems
Department of Electrical Engineering, Indian Institute of Technology Bombay（印度理工学院班加罗尔电子工程系）；Polytechnique Montreal（蒙特利尔理工学院）；IIIT Allahabad（阿哈迈德纳巴德印度理工学院）；University of Ottawa（渥太华大学）；IIT Bombay（印度理工学院班加罗尔）
Harvard University（哈佛大学）；University of Southern California（南加利福尼亚大学）；Washington University in St. Louis（圣路易斯华盛顿大学）
Motivated by a broad class of mobile intervention problems, we propose and study restless multi-armed bandits (RMABs) with network effects. In our model, arms are partially recharging and connected through a graph, so that pulling one arm also improves the state of neighboring arms, significantly extending the previously studied setting of fully recharging bandits with no network effects. In mobile interventions, network effects may arise due to regular population movements (such as commuting between home and work). We show that network effects in RMABs induce strong reward coupling that is no…
Restless Multi-Armed Bandits (RMAB) is an apt model to represent decision-making problems in public health interventions (e.g., tuberculosis, maternal, and child care), anti-poaching planning, sensor monitoring, personalized recommendations and many more. Existing research in RMAB has contributed mechanisms and theoretical results to a wide variety of settings, where the focus is on maximizing expected value. In this paper, we are interested in ensuring that RMAB decision making is also fair to different arms while maximizing expected value. In the context of public health settings, this would…
Lagrangian Relaxation for Multi-Action Partially Observable Restless Bandits: Heuristic Policies and Indexability
Department of Electrical Engineering, Indian Institute of Technology Madras（印度理工学院马德拉斯分校电子工程系）；Department of Electrical Engineering and Computer Science, University of Ottawa（渥太华大学电子工程与计算机科学系）
We consider multi-dimensional Markov decision processes and formulate a long term discounted reward optimization problem. Two simulation based algorithms---Monte Carlo rollout policy and parallel rollout policy are studied, and various properties for these policies are discussed. We next consider a restless multi-armed bandit (RMAB) with multi-dimensional state space and multi-actions bandit model. A standard RMAB consists of two actions for each arms whereas in multi-actions RMAB, there are more that two actions for each arms. A popular approach for RMAB is Whittle index based heuristic polic…
The multi-armed restless bandit problem is studied in the case where the pay-off distributions are stationary $φ$-mixing. This version of the problem provides a more realistic model for most real-world applications, but cannot be optimally solved in practice, since it is known to be PSPACE-hard. The objective of this paper is to characterize a sub-class of the problem where {\em good} approximate solutions can be found using tractable approaches. Specifically, it is shown that under some conditions on the $φ$-mixing coefficients, a modified version of UCB can prove effective. The main challeng…
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…
In this paper we present a model for the hidden Markovian bandit problem with linear rewards. As opposed to current work on Markovian bandits, we do not assume that the state is known to the decision maker before making the decision. Furthermore, we assume structural side information where the decision maker knows in advance that there are two types of hidden states; one is common to all arms and evolves according to a Markovian distribution, and the other is unique to each arm and is distributed according to an i.i.d. process that is unique to each arm. We present an algorithm and regret anal…
We consider an opportunistic spectrum access (OSA) problem where the time-varying condition of each channel (e.g., as a result of random fading or certain primary users' activities) is modeled as an arbitrary finite-state Markov chain. At each instance of time, a (secondary) user probes a channel and collects a certain reward as a function of the state of the channel (e.g., good channel condition results in higher data rate for the user). Each channel has potentially different state space and statistics, both unknown to the user, who tries to learn which one is the best as it goes and maximize…
We consider a class of restless multi-armed bandit (RMAB) problems with unknown arm dynamics. At each time, a player chooses an arm out of N arms to play, referred to as an active arm, and receives a random reward from a finite set of reward states. The reward state of the active arm transits according to an unknown Markovian dynamics. The reward state of passive arms (which are not chosen to play at time t) evolves according to an arbitrary unknown random process. The objective is an arm-selection policy that minimizes the regret, defined as the reward loss with respect to a player that alway…
GINO-Q: Learning an Asymptotically Optimal Index Policy for Restless Multi-armed Bandits
Department of Electrical and Electronic Engineering, Imperial College London（帝国理工学院伦敦分校电子与电气工程系）；Department of Information Engineering, The Chinese University of Hong Kong（香港中文大学信息工程系）
Optimal Control of Fluid Restless Multi-armed Bandits: A Machine Learning Approach
Sloan School of Management, Massachusetts Institute of Technology（麻省理工学院斯隆管理学院）；Operations Research Center, Massachusetts Institute of Technology（麻省理工学院运筹学中心）；Departamento de Estadı́stica, Universidad Carlos III de Madrid（马德里卡斯蒂利亚-拉曼查大学统计系）
