Convergence dynamics of Generative Adversarial Networks: the dual metric flows

TL;DR

This paper investigates the convergence behavior of GAN training dynamics in the small learning rate limit, introducing dual metric flows to analyze probability evolution and address mode collapse.

Contribution

It introduces the concept of dual metric flows for GANs, providing a theoretical framework for understanding their convergence and mode collapse mitigation.

Findings

GAN dynamics tend to a limit as learning rate approaches zero

Dual flows are formalized as evolution equations in metric spaces

The theory offers insights into mode collapse mitigation

Abstract

Fitting neural networks often resorts to stochastic (or similar) gradient descent which is a noise-tolerant (and efficient) resolution of a gradient descent dynamics. It outputs a sequence of networks parameters, which sequence evolves during the training steps. The gradient descent is the limit, when the learning rate is small and the batch size is infinite, of this set of increasingly optimal network parameters obtained during training. In this contribution, we investigate instead the convergence in the Generative Adversarial Networks used in machine learning. We study the limit of small learning rate, and show that, similar to single network training, the GAN learning dynamics tend, for vanishing learning rate to some limit dynamics. This leads us to consider evolution equations in metric spaces (which is the natural framework for evolving probability laws) that we call dual flows. We give formal definitions of solutions and prove the convergence. The theory is then applied to specific instances of GANs and we discuss how this insight helps understand and mitigate the mode collapse. Keywords: GAN; metric flow; generative network