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.

本文研究了一类非凸非凹的极小极大博弈，应用于生成对抗网络中。作者从优化理论、博弈理论和动态系统的角度展开分析，证明了针对特定的问题实例，梯度下降升力动力学可能表现出多种不收敛至极小极大解的行为，包括周期性和波恩卡雷复发。