A Unified Framework for Locality in Scalable MARL

Scalable methods for networked multi-agent reinforcement learning let each agent plan using only a small neighborhood of the agent graph. This works only when the system is value-local, meaning a perturbation at one agent affects the long-run value at another agent weakly when the two are far apart. In the average-reward setting, the standard way to certify locality is the Dobrushin row-sum bound on a single matrix Cπ that captures how each agent's next state depends on each other agent's current state. To make this matrix easy to work with, prior work bounds it by a supremum over joint actions. The resulting bound is independent of the policy, but it is loose whenever the policy never picks the worst-case action. We split Cπ into pieces that separately track environment sensitivity and policy sensitivity, Cπ preceq Emathrm s+Emathrm aPi(π), where Emathrm s measures how the next state moves with the current state, Emathrm a measures how it moves with the current action, and Pi(π) measures how reactive the policy is to changes in state. The spectral radius of Hπ := Emathrm s+Emathrm aPi(π) then controls the decay of the average-reward Poisson solution, and the spectral certificate ρ(Hπ)<1 is strictly weaker than the row-sum condition |Hπ|_∞<1 on the same matrix and applies in regimes where policy-independent action-supremum bounds used in prior Dobrushin-style work cannot. For temperature-τ softmax policies we get Pi(π)≤ L/(2τ), so the softmax temperature directly controls locality. We use this decay result to give a deterministic oracle guarantee for a block-coordinate KL-proximal policy-improvement template whose truncation bias decays exponentially in the message-passing radius kappa.