提出新算法，无需严格单调性即可收敛到均衡。
Last Iterate Convergence in Monotone Mean Field Games
- 用带KL正则的近端点更新，每步等价于求解一个正则化博弈均衡。
- 证明该方法具有指数级最后迭代收敛速度。
- 适合研究多智能体系统收敛性或需精确轨迹的应用者。
在拉斯里-利翁框架下，均值场博弈（MFG）刻画无限数量智能体间的交互。现有算法要么要求严格单调性，要么仅保证平均迭代序列收敛（如连续时间虚构博弈）。本文填补这一空白：首先，证明带KL正则的近端点（PP）更新的最后迭代策略，在非严格单调条件下收敛至MFG均衡；其次，发现每次PP更新等价于求解一个KL正则化后的MFG均衡。进一步证明，该均衡可通过镜面下降（MD）以指数速率求解。基于此，提出近似近端点（APP）算法，通过少量MD步骤近似实现PP更新。标准基准上的数值实验表明，APP算法能可靠收敛至未正则化的均值场均衡，且无需时间平均。
原文摘要 · Abstract (English)
In the Lasry--Lions framework, Mean-Field Games (MFGs) model interactions among an infinite number of agents. However, existing algorithms either require strict monotonicity or only guarantee the convergence of averaged iterates, as in Fictitious Play in continuous time. We address this gap with the following theoretical result. First, we prove that the last-iterated policy of a proximal-point (PP) update with KL regularization converges to an equilibrium of MFG under non-strict monotonicity. Second, we see that each PP update is equivalent to finding the equilibria of a KL-regularized MFG. We then prove that this equilibrium can be found using Mirror Descent (MD) with an exponential last-iterate convergence rate. Building on these insights, we propose the Approximate Proximal-Point ($\mathtt{APP}$) algorithm, which approximately implements the PP update via a small number of MD steps. Numerical experiments on standard benchmarks confirm that the $\mathtt{APP}$ algorithm reliably converges to the unregularized mean-field equilibrium without time-averaging.
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