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We study a wide class of non-convex non-concave min-max games thatgeneralizes over standard bilinear zero-sum games. In this class, playerscontrol the inputs of a smooth function whose output is being applied to abilinear zero-sum game. This class of games is motivated by the indirect natureof the competition in Generative Adversarial Networks, where players controlthe parameters of a neural network while the actual competition happens betweenthe distributions that the generator and discriminator capture. We establishtheoretically, that depending on the specific instance of the problemgradient-descent-ascent dynamics can exhibit a variety of behaviorsantithetical to convergence to the game theoretically meaningful min-maxsolution. Specifically, different forms of recurrent behavior (includingperiodicity and Poincar\'e recurrence) are possible as well as convergence tospurious (non-min-max) equilibria for a positive measure of initial conditions.At the technical level, our analysis combines tools from optimization theory,game theory and dynamical systems.

Poincar\'e Recurrence, Cycles and Spurious Equilibria in Gradient-Descent-Ascent for Non-Convex Non-Concave Zero-Sum Games