Achieving an $O(1\mathrm{/}N)$ Optimality Gap in Average-Reward Weakly-Coupled MDPs

We study average-reward weakly-coupled Markov decision processes (WCMDPs), where a WCMDP consists of $N$ smaller MDPs, called arms, that share multiple per-step budget constraints. We consider the setting where the arms have identical model parameters, multiple actions, and state- and action-dependent costs. For restless bandits (RBs), a well-studied special case of WCMDPs, prior work has developed policies that achieve an $O(1\mathrm{/}\sqrt{N})$ optimality gap under general conditions, and has further identified conditions under which policies can achieve a better-than-$1\mathrm{/}\sqrt{N}$ optimality gap. However, for general WCMDPs, no prior result achieves an optimality gap better than $1\mathrm{/}\sqrt{N}$. In this paper, we identify conditions analogous to those for RBs under which a better-than-$1\mathrm{/}\sqrt{N}$ optimality gap is achievable, and design a policy that attains an $O(1\mathrm{/}N)$ optimality gap. Notably, unlike prior approaches based on generalizing priority orderings, our policy is not priority-based but rather is designed to induce locally linear mean-field dynamics.