Learning Regularized Monotone Graphon Mean-Field Games

TL;DR

This paper proves the existence of Nash Equilibria in regularized Graphon Mean-Field Games under weaker conditions and introduces an efficient discrete-time learning algorithm with convergence guarantees, supported by empirical results.

Contribution

It establishes NE existence in regularized GMFGs under weaker assumptions and develops a novel discrete-time learning algorithm with convergence analysis.

Findings

Proved NE existence under weaker conditions.

Designed a discrete-time learning algorithm with proven convergence.

Empirically validated the efficiency of the proposed algorithm.

Abstract

This paper studies two fundamental problems in regularized Graphon Mean-Field Games (GMFGs). First, we establish the existence of a Nash Equilibrium (NE) of any $\lambda$-regularized GMFG (for $\lambda\geq 0$). This result relies on weaker conditions than those in previous works for analyzing both unregularized GMFGs ($\lambda=0$) and $\lambda$-regularized MFGs, which are special cases of GMFGs. Second, we propose provably efficient algorithms to learn the NE in weakly monotone GMFGs, motivated by Lasry and Lions [2007]. Previous literature either only analyzed continuous-time algorithms or required extra conditions to analyze discrete-time algorithms. In contrast, we design a discrete-time algorithm and derive its convergence rate solely under weakly monotone conditions. Furthermore, we develop and analyze the action-value function estimation procedure during the online learning process, which is absent from algorithms for monotone GMFGs. This serves as a sub-module in our optimization algorithm. The efficiency of the designed algorithm is corroborated by empirical evaluations.