Achieving Exponential Asymptotic Optimality in Average-Reward Restless Bandits without Global Attractor Assumption

TL;DR

This paper introduces a novel two-set policy for infinite-horizon average-reward restless bandits that achieves exponential asymptotic optimality without requiring a global attractor assumption, outperforming prior methods.

Contribution

The paper presents the first policy to attain exponential asymptotic optimality under mild assumptions, avoiding the strong global attractor condition required by previous approaches.

Findings

Two-set policy achieves $O( ext{exp}(-C N))$ optimality gap.

Policy outperforms previous methods in simulations.

Exponential asymptotic optimality is linked to local stability, not global attractors.

Abstract

We consider the infinite-horizon average-reward restless bandit problem. We propose a novel \emph{two-set policy} that maintains two dynamic subsets of arms: one subset of arms has a nearly optimal state distribution and takes actions according to an Optimal Local Control routine; the other subset of arms is driven towards the optimal state distribution and gradually merged into the first subset. We show that our two-set policy is asymptotically optimal with an $O(\exp(-C N))$ optimality gap for an $N$-armed problem, under the mild assumptions of aperiodic-unichain, non-degeneracy, and local stability. Our policy is the first to achieve \emph{exponential asymptotic optimality} under the above set of easy-to-verify assumptions, whereas prior work either requires a strong \emph{global attractor} assumption or only achieves an $O(1/\sqrt{N})$ optimality gap. We further discuss obstacles in weakening the assumptions by demonstrating examples where exponential asymptotic optimality is not achievable when any of the three assumptions is violated. Notably, we prove a lower bound for a large class of locally unstable restless bandits, showing that local stability is particularly fundamental for exponential asymptotic optimality. Finally, we use simulations to demonstrate that the two-set policy outperforms previous policies on certain RB problems and performs competitively overall.