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Motivated by applications in Optimization, Game Theory, and the training ofGenerative Adversarial Networks, the convergence properties of first ordermethods in min-max problems have received extensive study. It has beenrecognized that they may cycle, and there is no good understanding of theirlimit points when they do not. When they converge, do they converge to localmin-max solutions? We characterize the limit points of two basic first ordermethods, namely Gradient Descent/Ascent (GDA) and Optimistic Gradient DescentAscent (OGDA). We show that both dynamics avoid unstable critical points foralmost all initializations. Moreover, for small step sizes and under mildassumptions, 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 thesedynamics can be studied from a dynamical systems perspective.

The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization