学习正则化单调图平均场博弈
Learning Regularized Monotone Graphon Mean-Field Games
AI总结：
本文研究正则化图平均场博弈，证明了任意λ-正则化GMFG纳什均衡的存在性（条件更弱），并提出了弱单调GMFG下仅依赖弱单调条件的离散时间学习算法，附带动作-价值函数估计子模块，实证验证了算法效率。
摘要：
本文研究了正则化图平均场博弈（GMFGs）中的两个基本问题。首先，我们建立了任意 $\lambda$-正则化 GMFG（对于 $\lambda\geq 0$）的纳什均衡（NE）的存在性。这一结果所依赖的条件比先前工作中分析非正则化 GMFG（$\lambda=0$）和 $\lambda$-正则化 MFG（它们是 GMFG 的特例）所需的条件更弱。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的算法。先前的文献要么只分析了连续时间算法，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 的算法中是缺失的。这作为我们优化算法中的一个子模块。所设计算法的效率通过实证评估得到了证实。
英文摘要：
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 $λ$-regularized GMFG (for $λ\geq 0$). This result relies on weaker conditions than those in previous works for analyzing both unregularized GMFGs ($λ=0$) and $λ$-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.
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