Weakly-Coupled Multi-Action Restless Bandits -- Exponential Convergence in Probability

Referee: [§2.3] §2.3 (Definition of admissible policies): The abstract asserts exponential convergence 'for any policy in a rather general class' without non-degenerate assumptions, yet standard concentration arguments for mean-field limits require either uniform minorization of the transition kernel or Lipschitz continuity of the policy map with respect to the empirical measure. The manuscript must state the precise conditions (e.g., uniform bounds away from 0/1 on action probabilities or Lipschitz constants independent of N) that define the class; otherwise the claimed generality is not supported and the 'no non-degenerate assumptions' statement becomes circular.

Authors: We agree that the definition of the admissible policy class in §2.3 requires more explicit emphasis to avoid ambiguity. The class consists of policies whose action-probability mappings are Lipschitz continuous in the empirical measure, with the Lipschitz constant independent of N. This regularity condition is imposed on the policy (not the transition kernels), which is what permits the exponential convergence result without any non-degeneracy or minorization assumptions on the kernels. We will revise both the abstract and §2.3 to state this condition explicitly and to clarify that the absence of non-degenerate assumptions refers specifically to the kernels. revision: yes

Referee: [Theorem 4.1] Theorem 4.1 (Exponential convergence in probability): The proof sketch relies on direct analysis of the stochastic process and its fluid limit, but the exponential tail bound is load-bearing for all subsequent optimality claims. The derivation should explicitly identify the martingale or concentration inequality used and verify that it holds uniformly over the admissible policy class without invoking minorization; if the bound depends on a hidden Lipschitz constant, the result reduces to the degenerate case already covered in the literature.

Authors: The proof of Theorem 4.1 (given in full in the appendix) proceeds by representing the deviation process as a martingale-difference sequence whose increments are controlled by the Lipschitz continuity of the policy. We then apply Freedman's inequality to obtain the exponential tail bounds. Because the Lipschitz constant is uniform in N by definition of the admissible class, the concentration holds uniformly over the class and does not rely on any minorization of the transition kernels. We will expand the main-text proof sketch to name Freedman's inequality explicitly and to highlight the role of the policy Lipschitz condition. revision: yes

Referee: [§5] §5 (Proposed policy and optimality): The if-and-only-if statement linking fluid optimality to asymptotic optimality inherits the same restriction on the policy class. The manuscript should provide a counter-example policy outside the class for which the exponential rate fails, to demonstrate that the class is not artificially narrow.

Authors: We acknowledge the value of an explicit counter-example. While constructing a fully worked, setting-specific policy outside the Lipschitz class that demonstrably loses the exponential rate would require substantial additional technical development, we will add a remark in the revised §5 that explains the necessity of the Lipschitz condition by reference to standard counter-examples in the mean-field literature (where non-Lipschitz policies yield only sub-exponential or slower convergence). This discussion will clarify that the class is natural and not artificially restrictive. revision: partial