GANs May Have No Nash Equilibria

TL;DR

This paper investigates the theoretical existence of Nash equilibria in GANs, showing they may not exist, and introduces the concept of proximal equilibrium as a more suitable solution concept, along with a new training method.

Contribution

The paper proves that GAN zero-sum games may lack Nash equilibria and introduces proximal equilibrium as an alternative, along with a novel proximal training approach.

Findings

GAN zero-sum games may have no local Nash equilibria

Optimal Wasserstein GANs provide proximal equilibria

Proximal training can find equilibrium solutions in practice

Abstract

Generative adversarial networks (GANs) represent a zero-sum game between two machine players, a generator and a discriminator, designed to learn the distribution of data. While GANs have achieved state-of-the-art performance in several benchmark learning tasks, GAN minimax optimization still poses great theoretical and empirical challenges. GANs trained using first-order optimization methods commonly fail to converge to a stable solution where the players cannot improve their objective, i.e., the Nash equilibrium of the underlying game. Such issues raise the question of the existence of Nash equilibrium solutions in the GAN zero-sum game. In this work, we show through several theoretical and numerical results that indeed GAN zero-sum games may not have any local Nash equilibria. To characterize an equilibrium notion applicable to GANs, we consider the equilibrium of a new zero-sum game with an objective function given by a proximal operator applied to the original objective, a solution we call the proximal equilibrium. Unlike the Nash equilibrium, the proximal equilibrium captures the sequential nature of GANs, in which the generator moves first followed by the discriminator. We prove that the optimal generative model in Wasserstein GAN problems provides a proximal equilibrium. Inspired by these results, we propose a new approach, which we call proximal training, for solving GAN problems. We discuss several numerical experiments demonstrating the existence of proximal equilibrium solutions in GAN minimax problems.