Dynamics of Fourier Modes in Torus Generative Adversarial Networks

TL;DR

This paper introduces a Fourier series-based method to analyze GAN training dynamics, providing insights into convergence, stability, and the nature of Nash equilibria, which explains the slow and unstable training observed.

Contribution

The paper proposes a novel Fourier analysis approach to study GAN convergence and stability, linking the dynamics to spiral attractors and perturbations of periodic orbits.

Findings

Convergent GAN orbits are small perturbations of periodic orbits.

Nash equilibria in GANs act as spiral attractors.

The method accurately models the training flow of a 2-parametric GAN.

Abstract

Generative Adversarial Networks (GANs) are powerful Machine Learning models capable of generating fully synthetic samples of a desired phenomenon with a high resolution. Despite their success, the training process of a GAN is highly unstable and typically it is necessary to implement several accessory heuristics to the networks to reach an acceptable convergence of the model. In this paper, we introduce a novel method to analyze the convergence and stability in the training of Generative Adversarial Networks. For this purpose, we propose to decompose the objective function of the adversary min-max game defining a periodic GAN into its Fourier series. By studying the dynamics of the truncated Fourier series for the continuous Alternating Gradient Descend algorithm, we are able to approximate the real flow and to identify the main features of the convergence of the GAN. This approach is confirmed empirically by studying the training flow in a $2$-parametric GAN aiming to generate an unknown exponential distribution. As byproduct, we show that convergent orbits in GANs are small perturbations of periodic orbits so the Nash equillibria are spiral attractors. This theoretically justifies the slow and unstable training observed in GANs.