Restless Bandits with Average Reward: Breaking the Uniform Global Attractor Assumption

We study the infinite-horizon restless bandit problem with the average reward criterion, under both discrete-time and continuous-time settings. A fundamental question is how to design computationally efficient policies that achieve a diminishing optimality gap as the number of arms, $N$, grows large. Existing results on asymptotical optimality all rely on the uniform global attractor property (UGAP), a complex and challenging-to-verify assumption. In this paper, we propose a general, simulation-based framework that converts any single-armed policy into a policy for the original $N$-armed problem. This is accomplished by simulating the single-armed policy on each arm and carefully steering the real state towards the simulated state. Our framework can be instantiated to produce a policy with an $O(1/\sqrt{N})$ optimality gap. In the discrete-time setting, our result holds under a simpler synchronization assumption, which covers some problem instances that do not satisfy UGAP. More notably, in the continuous-time setting, our result does not require any additional assumptions beyond the standard unichain condition. In both settings, we establish the first asymptotic optimality result that does not require UGAP.