We study the convergence to local Nash equilibria of gradient methods for two-player zero-sum differentiable games. It is well-known that such dynamics converge locally when $S \succ 0$ and may diverge when $S=0$, where $S\succeq 0$ is the symmetric part of the Jacobian at equilibrium that accounts for the "potential" component of the game. We show that these dynamics also converge as soon as $S$ is nonzero (partial curvature) and the eigenvectors of the antisymmetric part $A$ are in general position with respect to the kernel of $S$. We then study the convergence rates when $S \ll A$ and prove that they typically depend on the average of the eigenvalues of $S$, instead of the minimum as an analogy with minimization problems would suggest. To illustrate our results, we consider the problem of computing mixed Nash equilibria of continuous games. We show that, thanks to partial curvature, conic particle methods -- which optimize over both weights and supports of the mixed strategies -- generically converge faster than fixed-support methods. For min-max games, it is thus beneficial to add degrees of freedom "with curvature": this can be interpreted as yet another benefit of over-parameterization.

翻译：我们研究了梯度方法在两人零和可微博弈中收敛到局部纳什均衡的问题。众所周知，当$S \succ 0$时，此类动力学过程局部收敛，而当$S=0$时可能发散，其中$S\succeq 0$是均衡点处雅可比矩阵的对称部分，对应博弈的“势”分量。我们证明，只要$S$非零（即存在局部曲率），且反对称部分$A$的特征向量相对于$S$的核处于一般位置，这些动力学过程也会收敛。随后，我们研究了$S \ll A$情况下的收敛速率，并证明该速率通常取决于$S$特征值的平均值，而非如最小化问题中类比所暗示的最小值。为阐明结果，我们考虑了连续博弈中混合纳什均衡的计算问题。研究表明，得益于局部曲率，圆锥粒子方法——同时对混合策略的权重和支撑集进行优化——通常比固定支撑方法收敛更快。对于极小极大博弈，增加“具有曲率”的自由度是有益的：这可以理解为过参数化的另一优势。