Last Iterate Convergence in Monotone Mean Field Games

Mean Field Game (MFG) is a framework for modeling and approximating the behavior of large numbers of agents. Computing equilibria in MFG has been of interest in multi-agent reinforcement learning. The theoretical guarantee that the last updated policy converges to an equilibrium has been limited. We propose the use of a simple, proximal-point (PP) type method to compute equilibria for MFGs. We then provide the first last-iterate convergence (LIC) guarantee under the Lasry--Lions-type monotonicity condition. We also propose an approximation of the update rule of PP ($\mathtt{APP}$) based on the observation that it is equivalent to solving the regularized MFG, which can be solved by mirror descent. We further establish that the regularized mirror descent achieves LIC at an exponential rate. Our numerical experiment demonstrates that $\mathtt{APP}$ efficiently computes the equilibrium.