The Non-Bayesian Restless Multi-Armed Bandit: A Case of Near-Logarithmic Strict Regret

TL;DR

This paper introduces a novel approach for non-Bayesian restless multi-armed bandit problems with unknown parameters, achieving near-logarithmic regret and matching the performance of an optimal model-aware policy in certain structured settings.

Contribution

It proposes a meta-policy framework that learns the optimal policy by treating candidate policies as arms in a multi-armed bandit, and demonstrates near-logarithmic regret in non-Bayesian RMABs.

Findings

Achieves near-logarithmic regret in non-Bayesian RMABs.

Develops a novel sensing policy for spectrum access with unknown channels.

Introduces a generalized Chernoff-Hoeffding bound for analysis.

Abstract

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 corresponding Bayesian problem has the structure that, depending on the known parameter values, the optimal solution is one of a prescribed finite set of policies. In such settings, we propose to learn the optimal policy for the non-Bayesian RMAB by employing a suitable meta-policy which treats each policy from this finite set as an arm in a different non-Bayesian multi-armed bandit problem for which a single-arm selection policy is optimal. We demonstrate this approach by developing a novel sensing policy for opportunistic spectrum access over unknown dynamic channels. We prove that our policy achieves near-logarithmic regret (the difference in expected reward compared to a model-aware genie), which leads to the same average reward that can be achieved by the optimal policy under a known model. This is the first such result in the literature for a non-Bayesian RMAB. For our proof, we also develop a novel generalization of the Chernoff-Hoeffding bound.