On the Nash equilibrium of moment-matching GANs for stationary Gaussian processes

TL;DR

This paper investigates the conditions under which Nash equilibria exist in moment-matching GANs for stationary Gaussian processes, highlighting the impact of discriminator design and symmetry properties on equilibrium existence and stability.

Contribution

It provides a theoretical analysis of Nash equilibrium existence in moment-matching GANs, emphasizing the role of discriminator moment choices and symmetry considerations.

Findings

Discriminator choice affects Nash equilibrium existence.

Symmetry properties influence equilibrium uniqueness.

Gradient methods' stability depends on equilibrium characteristics.

Abstract

Generative Adversarial Networks (GANs) learn an implicit generative model from data samples through a two-player game. In this paper, we study the existence of Nash equilibrium of the game which is consistent as the number of data samples grows to infinity. In a realizable setting where the goal is to estimate the ground-truth generator of a stationary Gaussian process, we show that the existence of consistent Nash equilibrium depends crucially on the choice of the discriminator family. The discriminator defined from second-order statistical moments can result in non-existence of Nash equilibrium, existence of consistent non-Nash equilibrium, or existence and uniqueness of consistent Nash equilibrium, depending on whether symmetry properties of the generator family are respected. We further study empirically the local stability and global convergence of gradient descent-ascent methods towards consistent equilibrium.