A Hidden Markov Restless Multi-armed Bandit Model for Playout Recommendation Systems

We consider a restless multi-armed bandit (RMAB) in which there are two types of arms, say A and B. Each arm can be in one of two states, say $0$ or $1.$ Playing a type A arm brings it to state $0$ with probability one and not playing it induces state transitions with arm-dependent probabilities. Whereas playing a type B arm leads it to state $1$ with probability $1$ and not playing it gets state that dependent on transition probabilities of arm. Further, play of an arm generates a unit reward with a probability that depends on the state of the arm. The belief about the state of the arm can be calculated using a Bayesian update after every play. This RMAB has been designed for use in recommendation systems where the user's preferences depend on the history of recommendations. This RMAB can also be used in applications like creating of playlists or placement of advertisements. In this paper we formulate the long term reward maximization problem as infinite horizon discounted reward and average reward problem. We analyse the RMAB by first studying discounted reward scenario. We show that it is Whittle-indexable and then obtain a closed form expression for the Whittle index for each arm calculated from the belief about its state and the parameters that describe the arm. We next analyse the average reward problem using vanishing discounted approach and derive the closed form expression for Whittle index. For a RMAB to be useful in practice, we need to be able to learn the parameters of the arms. We present an algorithm derived from Thompson sampling scheme, that learns the parameters of the arms and also illustrate its performance numerically

Constrained Restless Bandits for Dynamic Scheduling in Cyber-Physical Systems

This paper studies a class of constrained restless multi-armed bandits (CRMAB). The constraints are in the form of time varying set of actions (set of available arms). This variation can be either stochastic or semi-deterministic. Given a set of arms, a fixed number of them can be chosen to be played in each decision interval. The play of each arm yields a state dependent reward. The current states of arms are partially observable through binary feedback signals from arms that are played. The current availability of arms is fully observable. The objective is to maximize long term cumulative reward. The uncertainty about future availability of arms along with partial state information makes this objective challenging. Applications for CRMAB abound in the domain of cyber-physical systems. First, this optimization problem is analyzed using Whittle's index policy. To this end, a constrained restless single-armed bandit is studied. It is shown to admit a threshold-type optimal policy and is also indexable. An algorithm to compute Whittle's index is presented. An alternate solution method with lower complexity is also presented in the form of an online rollout policy. Further, upper bounds on the value function are derived in order to estimate the degree of sub-optimality of various solutions. The simulation study compares the performance of Whittle's index, online rollout, myopic and modified Whittle's index policies.Comment: 14 pages, 2 figure

Sequential Decision Making with Limited Observation Capability: Application to Wireless Networks

This work studies a generalized class of restless multi-armed bandits with hidden states and allow cumulative feedback, as opposed to the conventional instantaneous feedback. We call them lazy restless bandits (LRB) as the events of decision-making are sparser than events of state transition. Hence, feedback after each decision event is the cumulative effect of the following state transition events. The states of arms are hidden from the decision-maker and rewards for actions are state dependent. The decision-maker needs to choose one arm in each decision interval, such that long term cumulative reward is maximized. As the states are hidden, the decision-maker maintains and updates its belief about them. It is shown that LRBs admit an optimal policy which has threshold structure in belief space. The Whittle-index policy for solving LRB problem is analyzed; indexability of LRBs is shown. Further, closed-form index expressions are provided for two sets of special cases; for more general cases, an algorithm for index computation is provided. An extensive simulation study is presented; Whittle-index, modified Whittle-index and myopic policies are compared. Lagrangian relaxation of the problem provides an upper bound on the optimal value function; it is used to assess the degree of sub-optimality various policies

Simulation Based Algorithms for Markov Decision Processes and Multi-Action Restless Bandits

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 policy. Indexability is an important requirement to use index based policy. Based on this, an RMAB is classified into indexable or non-indexable bandits. Our interest is in the study of Monte-Carlo rollout policy for both indexable and non-indexable restless bandits. We first analyze a standard indexable RMAB (two-action model) and discuss an index based policy approach. We present approximate index computation algorithm using Monte-Carlo rollout policy. This algorithm's convergence is shown using two-timescale stochastic approximation scheme. Later, we analyze multi-actions indexable RMAB, and discuss the index based policy approach. We also study non-indexable RMAB for both standard and multi-actions bandits using Monte-Carlo rollout policy.Comment: 3 Figure