Abstract:Mean-field games (MFG) have become significant tools for solving large-scale multi-agent reinforcement learning problems under symmetry. However, the assumption of exact symmetry limits the applicability of MFGs, as real-world scenarios often feature inherent heterogeneity. Furthermore, most works on MFG assume access to a known MFG model, which might not be readily available for real-world finite-agent games. In this work, we broaden the applicability of MFGs by providing a methodology to extend any finite-player, possibly asymmetric, game to an "induced MFG". First, we prove that $N$-player dynamic games can be symmetrized and smoothly extended to the infinite-player continuum via explicit Kirszbraun extensions. Next, we propose the notion of $\alpha,\beta$-symmetric games, a new class of dynamic population games that incorporate approximate permutation invariance. For $\alpha,\beta$-symmetric games, we establish explicit approximation bounds, demonstrating that a Nash policy of the induced MFG is an approximate Nash of the $N$-player dynamic game. We show that TD learning converges up to a small bias using trajectories of the $N$-player game with finite-sample guarantees, permitting symmetrized learning without building an explicit MFG model. Finally, for certain games satisfying monotonicity, we prove a sample complexity of $\widetilde{\mathcal{O}}(\varepsilon^{-6})$ for the $N$-agent game to learn an $\varepsilon$-Nash up to symmetrization bias. Our theory is supported by evaluations on MARL benchmarks with thousands of agents.
Abstract:In this paper, we study the statistical efficiency of Reinforcement Learning in Mean-Field Control (MFC) and Mean-Field Game (MFG) with general function approximation. We introduce a new concept called Mean-Field Model-Based Eluder Dimension (MBED), which subsumes a rich family of Mean-Field RL problems. Additionally, we propose algorithms based on Optimistic Maximal Likelihood Estimation, which can return an $\epsilon$-optimal policy for MFC or an $\epsilon$-Nash Equilibrium policy for MFG, with sample complexity polynomial w.r.t. relevant parameters and independent of the number of states, actions and the number of agents. Notably, our results only require a mild assumption of Lipschitz continuity on transition dynamics and avoid strong structural assumptions in previous work. Finally, in the tabular setting, given the access to a generative model, we establish an exponential lower bound for MFC setting, while providing a novel sample-efficient model elimination algorithm to approximate equilibrium in MFG setting. Our results reveal a fundamental separation between RL for single-agent, MFC, and MFG from the sample efficiency perspective.
Abstract:Mean-field games have been used as a theoretical tool to obtain an approximate Nash equilibrium for symmetric and anonymous $N$-player games in literature. However, limiting applicability, existing theoretical results assume variations of a "population generative model", which allows arbitrary modifications of the population distribution by the learning algorithm. Instead, we show that $N$ agents running policy mirror ascent converge to the Nash equilibrium of the regularized game within $\tilde{\mathcal{O}}(\varepsilon^{-2})$ samples from a single sample trajectory without a population generative model, up to a standard $\mathcal{O}(\frac{1}{\sqrt{N}})$ error due to the mean field. Taking a divergent approach from literature, instead of working with the best-response map we first show that a policy mirror ascent map can be used to construct a contractive operator having the Nash equilibrium as its fixed point. Next, we prove that conditional TD-learning in $N$-agent games can learn value functions within $\tilde{\mathcal{O}}(\varepsilon^{-2})$ time steps. These results allow proving sample complexity guarantees in the oracle-free setting by only relying on a sample path from the $N$ agent simulator. Furthermore, we demonstrate that our methodology allows for independent learning by $N$ agents with finite sample guarantees.