Infinite Horizon Average Cost Optimality Criteria for Mean-Field Control

TL;DR

This paper investigates mean-field control problems under infinite horizon average cost criteria, establishing existence of solutions and optimal policies for both finite and infinite populations, and analyzing their convergence properties.

Contribution

It proves the existence of solutions to the average cost optimality equation and optimal policies for mean-field control in both finite and infinite populations, and links their solutions.

Findings

Existence of solutions to the ACOE under certain conditions.

Optimal policies exist for finite and infinite population problems.

Finite population optimal value converges to infinite population optimal value as population grows.

Abstract

We study mean-field control problems in discrete-time under the infinite horizon average cost optimality criteria. We focus on both the finite population and the infinite population setups. We show the existence of a solution to the average cost optimality equation (ACOE) and the existence of optimal stationary Markov policies for finite population problems under (i) a minorization condition that provides geometric ergodicity on the collective state process of the agents, and (ii) under standard Lipschitz continuity assumptions on the stage-wise cost and transition function of the agents when the Lipschitz constant of the transition function satisfies a certain bound. For the infinite population problem, we establish the existence of a solution to the ACOE, and the existence of optimal policies under the continuity assumptions on the cost and the transition functions. Finally, we relate the finite population and infinite population control problems: (i) we prove that the optimal value of the finite population problem converges to the optimal value of the infinite population problem as the number of agents grows to infinity; (ii) we show that the accumulation points of the finite population optimal solution corresponds to an optimal solution for the infinite population problem, and finally (iii), we show that one can use the solution of the infinite population problem for the finite population problem symmetrically across the agents to achieve near optimal performance when the population is sufficiently large.