Instability and Local Minima in GAN Training with Kernel Discriminators

TL;DR

This paper investigates the training dynamics of GANs with kernel discriminators, revealing conditions for convergence and failure modes like mode collapse and divergence through a simplified model.

Contribution

It introduces the Isolated Points Model to analyze GAN training with kernel discriminators, providing insights into convergence and failure modes.

Findings

Conditions for convergence to good minima identified

Explanation of mode collapse and divergence phenomena

Numerical simulations validate theoretical predictions

Abstract

Generative Adversarial Networks (GANs) are a widely-used tool for generative modeling of complex data. Despite their empirical success, the training of GANs is not fully understood due to the min-max optimization of the generator and discriminator. This paper analyzes these joint dynamics when the true samples, as well as the generated samples, are discrete, finite sets, and the discriminator is kernel-based. A simple yet expressive framework for analyzing training called the $\textit{Isolated Points Model}$ is introduced. In the proposed model, the distance between true samples greatly exceeds the kernel width, so each generated point is influenced by at most one true point. Our model enables precise characterization of the conditions for convergence, both to good and bad minima. In particular, the analysis explains two common failure modes: (i) an approximate mode collapse and (ii) divergence. Numerical simulations are provided that predictably replicate these behaviors.