Robustness and sample complexity of model-based MARL for general-sum Markov games

TL;DR

This paper analyzes the sample complexity and robustness of model-based multi-agent reinforcement learning in general-sum Markov games, providing bounds on the number of samples needed for approximate equilibria and showing equilibrium stability under model perturbations.

Contribution

It offers the first sample complexity bounds for model-based MARL in general-sum Markov games and studies equilibrium robustness to model approximations.

Findings

Hoeffding-based sample bound: ( (1-\u03b3)^{-4} bb bb bb bb)

Bernstein-based sample bound: ( (1-b3)^{-1} bb bb)

Markov perfect equilibrium is stable under model perturbations.

Abstract

Multi-agent reinforcement learning (MARL) is often modeled using the framework of Markov games (also called stochastic games or dynamic games). Most of the existing literature on MARL concentrates on zero-sum Markov games but is not applicable to general-sum Markov games. It is known that the best-response dynamics in general-sum Markov games are not a contraction. Therefore, different equilibria in general-sum Markov games can have different values. Moreover, the Q-function is not sufficient to completely characterize the equilibrium. Given these challenges, model based learning is an attractive approach for MARL in general-sum Markov games. In this paper, we investigate the fundamental question of \emph{sample complexity} for model-based MARL algorithms in general-sum Markov games. We show two results. We first use Hoeffding inequality based bounds to show that $\tilde{\mathcal{O}}( (1-\gamma)^{-4} \alpha^{-2})$ samples per state-action pair are sufficient to obtain a $\alpha$-approximate Markov perfect equilibrium with high probability, where $\gamma$ is the discount factor, and the $\tilde{\mathcal{O}}(\cdot)$ notation hides logarithmic terms. We then use Bernstein inequality based bounds to show that $\tilde{\mathcal{O}}( (1-\gamma)^{-1} \alpha^{-2} )$ samples are sufficient. To obtain these results, we study the robustness of Markov perfect equilibrium to model approximations. We show that the Markov perfect equilibrium of an approximate (or perturbed) game is always an approximate Markov perfect equilibrium of the original game and provide explicit bounds on the approximation error. We illustrate the results via a numerical example.