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。