小增益纳什：可微博弈中纳什均衡的认证收缩方法

Small-Gain Nash: Certified Contraction to Nash Equilibria in Differentiable Games

December 7, 2025

作者: Vedansh Sharma

cs.AI

摘要

基于梯度的学习在博弈中的经典收敛性保证要求伪梯度在欧几里得几何中满足（强）单调性条件（如Rosen于1965年所示），但该条件即使在具有强玩家间耦合的简单博弈中也常不成立。我们提出小增益纳什条件（SGN），这是一种在定制块加权几何中的块小增益条件。SGN将局部曲率和玩家间Lipschitz耦合边界转化为可处理的收缩证明：它构建了一个加权块度量，使得伪梯度在这些边界成立的任意区域均呈现强单调性——即使其在欧几里得意义下非单调。连续流在此设计的几何中呈指数收缩，且投影欧拉法与RK4离散化在由SGN裕度和局部Lipschitz常数导出的显式步长范围内收敛。我们的分析揭示了一个经认证的“时间尺度带”，这是一种非渐近的、基于度量的证明，其作用类似于TTUR机制：SGN并非通过渐近不等步长强制实现时间尺度分离，而是识别出有限范围的相对度量权重，使得单一步长动力学可证明具有收缩性。我们在二次博弈上验证该框架——当欧几里得单调性分析无法预测收敛时，SGN仍能成功认证收敛性，并将该构造扩展至马尔可夫博弈中熵正则化策略梯度的镜像/费希尔几何。最终形成离线认证流程：在紧致区域估计曲率、耦合及Lipschitz参数，优化块权重以扩大SGN裕度，并返回一个包含度量、收缩率及安全步长的结构化可计算收敛证明，适用于非单调博弈。

English

Classical convergence guarantees for gradient-based learning in games require the pseudo-gradient to be (strongly) monotone in Euclidean geometry as shown by rosen(1965), a condition that often fails even in simple games with strong cross-player couplings. We introduce Small-Gain Nash (SGN), a block small-gain condition in a custom block-weighted geometry. SGN converts local curvature and cross-player Lipschitz coupling bounds into a tractable certificate of contraction. It constructs a weighted block metric in which the pseudo-gradient becomes strongly monotone on any region where these bounds hold, even when it is non-monotone in the Euclidean sense. The continuous flow is exponentially contracting in this designed geometry, and projected Euler and RK4 discretizations converge under explicit step-size bounds derived from the SGN margin and a local Lipschitz constant. Our analysis reveals a certified ``timescale band'', a non-asymptotic, metric-based certificate that plays a TTUR-like role: rather than forcing asymptotic timescale separation via vanishing, unequal step sizes, SGN identifies a finite band of relative metric weights for which a single-step-size dynamics is provably contractive. We validate the framework on quadratic games where Euclidean monotonicity analysis fails to predict convergence, but SGN successfully certifies it, and extend the construction to mirror/Fisher geometries for entropy-regularized policy gradient in Markov games. The result is an offline certification pipeline that estimates curvature, coupling, and Lipschitz parameters on compact regions, optimizes block weights to enlarge the SGN margin, and returns a structural, computable convergence certificate consisting of a metric, contraction rate, and safe step-sizes for non-monotone games.