Can Mean Field Control (MFC) Approximate Cooperative Multi Agent Reinforcement Learning (MARL) with Non-Uniform Interaction?

TL;DR

This paper extends mean-field control methods to non-uniform multi-agent reinforcement learning by modeling arbitrary interactions, providing approximation guarantees and a practical algorithm with theoretical error bounds and sample complexity.

Contribution

It introduces a framework for non-uniform agent interactions in MARL using doubly stochastic matrices, and develops an NPG algorithm with proven approximation error and sample complexity.

Findings

Approximation error of order $1/\sqrt{N}$ for non-uniform interactions.

Development of an NPG algorithm with $\mathcal{O}(\epsilon^{-3})$ sample complexity.

Theoretical bounds for mean-field approximation in non-exchangeable agent settings.

Abstract

Mean-Field Control (MFC) is a powerful tool to solve Multi-Agent Reinforcement Learning (MARL) problems. Recent studies have shown that MFC can well-approximate MARL when the population size is large and the agents are exchangeable. Unfortunately, the presumption of exchangeability implies that all agents uniformly interact with one another which is not true in many practical scenarios. In this article, we relax the assumption of exchangeability and model the interaction between agents via an arbitrary doubly stochastic matrix. As a result, in our framework, the mean-field `seen' by different agents are different. We prove that, if the reward of each agent is an affine function of the mean-field seen by that agent, then one can approximate such a non-uniform MARL problem via its associated MFC problem within an error of $e=\mathcal{O}(\frac{1}{\sqrt{N}}[\sqrt{|\mathcal{X}|} + \sqrt{|\mathcal{U}|}])$ where $N$ is the population size and $|\mathcal{X}|$, $|\mathcal{U}|$ are the sizes of state and action spaces respectively. Finally, we develop a Natural Policy Gradient (NPG) algorithm that can provide a solution to the non-uniform MARL with an error $\mathcal{O}(\max\{e,\epsilon\})$ and a sample complexity of $\mathcal{O}(\epsilon^{-3})$ for any $\epsilon >0$.