Relaxed Indexability and Index Policy for Partially Observable Restless Bandits

Abstract: This paper addresses an important class of restless multi-armed bandit (RMAB) problems that finds broad application in operations research, stochastic optimization, and reinforcement learning. There are $N$ independent Markov processes that may be operated, observed and offer rewards. Due to the resource constraint, we can only choose a subset of $M~(M<N)$ processes to operate and accrue reward determined by the states of selected processes. We formulate the problem as a partially observable RMAB with an infinite state space and design an algorithm that achieves a near-optimal performance with low complexity. Our algorithm is based on a generalization of Whittle's original idea of indexability. Referred to as the relaxed indexability, the extended definition leads to the efficient online verifications and computations of the approximate Whittle index under the proposed algorithmic framework.