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

Résumé

This talk is about the existence of meaningful Nash equilibrium in an idealized mathematical setting of Generative Adversarial Networks (GANs). The first part introduces GANs from the perspective of likelihood-free density estimation in statistics, and gives an overview of the associated 2-player game and some results on the non-existence of Nash equilibrium in practice. To introduce the idealized setting, the stationary Gaussian process and its consistent estimator will be reviewed using the classical Fourier analysis. Existing algorithms used in GANs training will be briefly reviewed. The second part introduces the notion of consistent of Nash equilibrium. We study the existence and uniqueness of consistent Nash or non-Nash equilibrium, by varying the discriminator family in the game of GANs. Three discriminator families are considered based on a real-valued linear transform, complex-valued linear transform, or convolutional transform. We shall present proofs and numerical results to show the rich structure of Nash equilibrium in these cases.