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

TL;DR

This paper analyzes the complex dynamics of gradient-based algorithms in non-convex non-concave min-max games, revealing recurrence, cycles, and convergence to spurious equilibria, which challenge traditional convergence expectations.

Contribution

It introduces a theoretical framework showing diverse dynamic behaviors, including recurrence and non-min-max equilibria, in gradient descent-ascent for complex non-convex non-concave games.

Findings

Gradient descent-ascent can exhibit recurrence and cycles.

Spurious equilibria are common in these game dynamics.

Convergence to true min-max solutions is not guaranteed.

Abstract

We study a wide class of non-convex non-concave min-max games that generalizes over standard bilinear zero-sum games. In this class, players control the inputs of a smooth function whose output is being applied to a bilinear zero-sum game. This class of games is motivated by the indirect nature of the competition in Generative Adversarial Networks, where players control the parameters of a neural network while the actual competition happens between the distributions that the generator and discriminator capture. We establish theoretically, that depending on the specific instance of the problem gradient-descent-ascent dynamics can exhibit a variety of behaviors antithetical to convergence to the game theoretically meaningful min-max solution. Specifically, different forms of recurrent behavior (including periodicity and Poincar\'e recurrence) are possible as well as convergence to spurious (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.