Learning Regularized Monotone Graphon Mean-Field Games

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 previous works 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.