Motivated by applications in Optimization, Game Theory, and the training of Generative Adversarial Networks, the convergence properties of first order methods in min-max problems have received extensive study. It has been recognized that they may cycle, and there is no good understanding of their limit points when they do not. When they converge, do they converge to local min-max solutions? We characterize the limit points of two basic first order methods, namely Gradient Descent/Ascent (GDA) and Optimistic Gradient Descent Ascent (OGDA). We show that both dynamics avoid unstable critical points for almost all initializations. Moreover, for small step sizes and under mild assumptions, the set of \{OGDA\}-stable critical points is a superset of \{GDA\}-stable critical points, which is a superset of local min-max solutions (strict in some cases). The connecting thread is that the behavior of these dynamics can be studied from a dynamical systems perspective.

研究第一阶段方法在极小极大问题中的收敛属性，证明了基本的GD和OGD方法可以避免不稳定的临界点，并在初始状态下几乎所有的点都是OGDA稳定的临界点，而OGDA稳定的临界点集合是包含GDA稳定的临界点的超集，这些动态的行为可以从动态系统的角度进行研究。

极大极小优化问题中 (乐观) 梯度下降的极限点