Reinforcement learning is an attractive approach to learn good resource allocation and scheduling policies based on data when the system model is unknown. However, the cumulative regret of most RL algorithms scales as $\tilde O(\mathsf{S} \sqrt{\mathsf{A} T})$, where $\mathsf{S}$ is the size of the state space, $\mathsf{A}$ is the size of the action space, $T$ is the horizon, and the $\tilde{O}(\cdot)$ notation hides logarithmic terms. Due to the linear dependence on the size of the state space, these regret bounds are prohibitively large for resource allocation and scheduling problems. In this paper, we present a model-based RL algorithm for such problems which has scalable regret. In particular, we consider a restless bandit model, and propose a Thompson-sampling based learning algorithm which is tuned to the underlying structure of the model. We present two characterizations of the regret of the proposed algorithm with respect to the Whittle index policy. First, we show that for a restless bandit with $n$ arms and at most $m$ activations at each time, the regret scales either as $\tilde{O}(mn\sqrt{T})$ or $\tilde{O}(n^2 \sqrt{T})$ depending on the reward model. Second, under an additional technical assumption, we show that the regret scales as $\tilde{O}(n^{1.5} \sqrt{T})$ or $\tilde{O}(\max\{m\sqrt{n}, n\} \sqrt{T})$. We present numerical examples to illustrate the salient features of the algorithm.
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…
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）
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…
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…
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）
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…
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…
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…
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）
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 propose Streaming Bandits, a Restless Multi Armed Bandit (RMAB) framework in which heterogeneous arms may arrive and leave the system after staying on for a finite lifetime. Streaming Bandits naturally capture the health intervention planning problem, where health workers must manage the health outcomes of a patient cohort while new patients join and existing patients leave the cohort each day. Our contributions are as follows: (1) We derive conditions under which our problem satisfies indexability, a precondition that guarantees the existence and asymptotic optimality of the Whittle Index …
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…
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…
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…
Ben-Gurion University of the Negev（内盖夫本-古里安大学）；Princeton University（普林斯顿大学）
We consider an extension to the restless multi-armed bandit (RMAB) problem with unknown arm dynamics, where an unknown exogenous global Markov process governs the rewards distribution of each arm. Under each global state, the rewards process of each arm evolves according to an unknown Markovian rule, which is non-identical among different arms. At each time, a player chooses an arm out of $N$ arms to play, and receives a random reward from a finite set of reward states. The arms are restless, that is, their local state evolves regardless of the player's actions. Motivated by recent studies on …
Ben-Gurion University of the Negev（内盖夫本-古里安大学）；Princeton University（普林斯顿大学）
We consider an extension to the restless multi-armed bandit (RMAB) problem with unknown arm dynamics, where an unknown exogenous global Markov process governs the rewards distribution of each arm. Under each global state, the rewards process of each arm evolves according to an unknown Markovian rule, which is non-identical among different arms. At each time, a player chooses an arm out of N arms to play, and receives a random reward from a finite set of reward states. The arms are restless, that is, their local state evolves regardless of the player's actions. Motivated by recent studies on re…
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…
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…
A sensing policy for the restless multi-armed bandit problem with stationary but unknown reward distributions is proposed. The work is presented in the context of cognitive radios in which the bandit problem arises when deciding which parts of the spectrum to sense and exploit. It is shown that the proposed policy attains asymptotically logarithmic weak regret rate when the rewards are bounded independent and identically distributed or finite state Markovian. Simulation results verifying uniformly logarithmic weak regret are also presented. The proposed policy is a centrally coordinated index …
Binghamton University, State University of New York（纽约州立大学宾汉姆顿分校）；Indian Institute of Science（印度科学学院）
We consider the problem of content caching at the wireless edge to serve a set of end users via unreliable wireless channels so as to minimize the average latency experienced by end users due to the constrained wireless edge cache capacity. We formulate this problem as a Markov decision process, or more specifically a restless multi-armed bandit problem, which is provably hard to solve. We begin by investigating a discounted counterpart, and prove that it admits an optimal policy of the threshold-type. We then show that this result also holds for average latency problem. Using this structural …
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…
In the classic Bayesian restless multi-armed bandit (RMAB) problem, there are $N$ arms, with rewards on all arms evolving at each time as Markov chains with known parameters. A player seeks to activate $K \geq 1$ arms at each time in order to maximize the expected total reward obtained over multiple plays. RMAB is a challenging problem that is known to be PSPACE-hard in general. We consider in this work the even harder non-Bayesian RMAB, in which the parameters of the Markov chain are assumed to be unknown \emph{a priori}. We develop an original approach to this problem that is applicable when…
The Non-Bayesian Restless Multi-Armed Bandit: A Case of Near-Logarithmic Strict Regret
Department of Aeronautics；Astronautics Massachusetts Institute of Technology, Cambridge, MA 02139, USA Ming Hsieh Department of Electrical Engineering University of Southern California, Los Angeles, CA 90089, USA Department of Electrical；Computer Engineering University of California, Davis, CA 95616, USA Email
In many public health settings, it is important for patients to adhere to health programs, such as taking medications and periodic health checks. Unfortunately, beneficiaries may gradually disengage from such programs, which is detrimental to their health. A concrete example of gradual disengagement has been observed by an organization that carries out a free automated call-based program for spreading preventive care information among pregnant women. Many women stop picking up calls after being enrolled for a few months. To avoid such disengagements, it is important to provide timely intervent…
We consider the problem of learning the optimal threshold policy for control problems. Threshold policies make control decisions by evaluating whether an element of the system state exceeds a certain threshold, whose value is determined by other elements of the system state. By leveraging the monotone property of threshold policies, we prove that their policy gradients have a surprisingly simple expression. We use this simple expression to build an off-policy actor-critic algorithm for learning the optimal threshold policy. Simulation results show that our policy significantly outperforms othe…
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…
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. …
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…
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…
Adaptive Client Selection via Q-Learning-based Whittle Index in Wireless Federated Learning
Guangdong Prov./Zhuhai Key Lab of IRADS, Beijing Normal-Hong Kong Baptist University, Guangdong, P. R. China（广东/珠海IRADS实验室，北京师范大学-香港 Baptist大学，广东，中华人民共和国）；College of Engineering, Carnegie Mellon University, Pittsburgh, PA, USA（工程学院，卡内基梅隆大学，匹兹堡，PA，美国）；Department of Statistics, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, U.K.（统计学系，牛津大学，牛津OX1 3LB，英国）
Multi-action restless multi-armed bandits (RMABs) are a powerful framework for constrained resource allocation in which $N$ independent processes are managed. However, previous work only study the offline setting where problem dynamics are known. We address this restrictive assumption, designing the first algorithms for learning good policies for Multi-action RMABs online using combinations of Lagrangian relaxation and Q-learning. Our first approach, MAIQL, extends a method for Q-learning the Whittle index in binary-action RMABs to the multi-action setting. We derive a generalized update rule …
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 consider the problem of service placement at the network edge, in which a decision maker has to choose between $N$ services to host at the edge to satisfy the demands of customers. Our goal is to design adaptive algorithms to minimize the average service delivery latency for customers. We pose the problem as a Markov decision process (MDP) in which the system state is given by describing, for each service, the number of customers that are currently waiting at the edge to obtain the service. However, solving this $N$-services MDP is computationally expensive due to the curse of dimensionalit…
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…
