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.

翻译：本文研究了正则化图平均场博弈中的两个基本问题。首先，我们建立了任意$\lambda$-正则化GMFG（对于$\lambda\geq 0$）纳什均衡的存在性。这一结果依赖于比先前工作更弱的条件，这些先前工作分别分析了非正则化GMFG（$\lambda=0$）和$\lambda$-正则化MFG（作为GMFG的特例）。其次，受Lasry和Lions [2007]的启发，我们提出了在弱单调GMFG中学习纳什均衡的可证明高效算法。现有文献要么仅分析连续时间算法，要么需要额外条件来分析离散时间算法。相比之下，我们设计了一种离散时间算法，并仅在弱单调条件下推导出其收敛速率。此外，我们在在线学习过程中开发并分析了动作-价值函数估计过程，这在单调GMFG算法中尚属空白。该过程作为我们优化算法中的子模块。通过实证评估验证了所设计算法的效率。