证明了均值场优化在混合纳什均衡附近局部指数稳定
Local exponential stability of mean-field Langevin descent-ascent and associated particle system
- 在概率测度空间中分析耦合优化动力学，基于Wasserstein空间的局部收敛性
- 初始点足够接近均衡时，均值场系统以量化速率指数收敛，粒子系统稳定时间指数于N
- 揭示了混合纳什均衡的鲁棒吸引域，适合研究博弈优化与粒子系统的学者
我们研究了用于熵正则化双人零和博弈的均值场Langevin下降-上升（MFL-DA）动力学及其对应的相互作用粒子系统。针对一般非凸-非凹收益函数，Wang和Chizat（COLT 2024）提出了原始单时间尺度MFL-DA是否收敛至混合纳什均衡及收敛速率的问题。本文在Wasserstein空间中给出局部肯定答案：若初始数据足够接近混合纳什均衡，则均值场动力学以可量化的速率指数收敛。进一步表明，有限-$N$粒子系统在时间指数于$N$的范围内继承此稳定性，且具有$N$-无关的指数收敛率，仅受有限粒子误差地板影响。结合Mourrat和Pillaud-Vivien近期对MFL-DA的反例（表明全局收敛不成立），本定理完整给出了Wang-Chizat问题的正向局部结果：混合纳什均衡具有鲁棒吸引域，对均值场流及其有限粒子近似均稳定。
原文摘要 · Abstract (English)
We study the mean-field Langevin descent-ascent (MFL-DA), a coupled optimization dynamics on the space of probability measures for entropically regularized two-player zero-sum games, together with its associated interacting particle system. For general nonconvex-nonconcave payoffs, Wang and Chizat (COLT 2024) asked whether the original single-timescale MFL-DA converges to the mixed Nash equilibrium and, if so, at what rate. We prove a local affirmative answer in Wasserstein space: if the initial datum is sufficiently close to the mixed Nash equilibrium, then the mean-field dynamics converges to it exponentially fast at a quantitative rate. We further show that the finite-$N$ particle system inherits this stability up to times exponential in $N$, with an $N$-independent exponential rate modulo a finite-particle error floor. Combined with the recent counterexample of Mourrat and Pillaud-Vivien for MFL-DA, which shows that global convergence cannot hold in general, our theorem completes the positive local counterpart of the Wang-Chizat question: the mixed Nash equilibrium has a robust basin of attraction, stable under both the mean-field flow and its finite-particle approximation.
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