Minimax problems are notoriously challenging to optimize. However, we demonstrate that the two-timescale extragradient can be a viable solution. By utilizing dynamical systems theory, we show that it converges to points that satisfy the second-order necessary condition of local minimax points, under a mild condition. This work surpasses all previous results as we eliminate a crucial assumption that the Hessian, with respect to the maximization variable, is nondegenerate.

翻译：极小极大问题以优化难度大而著称。然而，我们证明了双时间尺度外梯度法是一种可行的解决方案。利用动力系统理论，我们展示了在温和条件下，该方法收敛到满足局部极小极大点二阶必要条件的点。这项工作超越了所有先前的结果，因为我们消除了一个关键假设——即关于最大化变量的海森矩阵是非退化的。