Despite the growing prominence of generative adversarial networks (GANs), optimization in GANs is still a poorly understood topic. In this paper, we analyze the ``gradient descent'' form of GAN optimization, i.e., the natural setting where we simultaneously take small gradient steps in both generator and discriminator parameters. We show that even though GAN optimization does \emph{not} correspond to a convex-concave game (even for simple parameterizations), under proper conditions, equilibrium points of this optimization procedure are still \emph{locally asymptotically stable} for the traditional GAN formulation. On the other hand, we show that the recently proposed Wasserstein GAN can have non-convergent limit cycles near equilibrium. Motivated by this stability analysis, we propose an additional regularization term for gradient descent GAN updates, which \emph{is} able to guarantee local stability for both the WGAN and the traditional GAN, and also shows practical promise in speeding up convergence and addressing mode collapse.

The authors present a dynamical system based analysis of simultaneous gradient descent updates for GANs, by considering the limit dynamical system that corresponds to the discrete updates. They show that under a series of assumptions, an equilibrium point of the dynamical system is locally asymptotically stable, implying convergence to the equilibrium if the system is initialized in a close neighborhood of it. Then they show how some types of GANs fail to satisfy some of their conditions and propose a fix to the gradient updates that re-instate local stability. They give experimental evidence that the local-stability inspired fix yields improvements in practice on MNIST digit generation and simple multi-modal distributions. However, I do think that these drawbacks are remedied by the fact that their modification, based on local asymptotic theory, did give noticeable improvements.

Despite the growing prominence of generative adversarial networks (GANs), optimization in GANs is still a poorly understood topic. In this paper, we analyze the gradient descent'' form of GAN optimization, i.e., the natural setting where we simultaneously take small gradient steps in both generator and discriminator parameters. We show that even though GAN optimization does \emph{not} correspond to a convex-concave game (even for simple parameterizations), under proper conditions, equilibrium points of this optimization procedure are still \emph{locally asymptotically stable} for the traditional GAN formulation. On the other hand, we show that the recently proposed Wasserstein GAN can have non-convergent limit cycles near equilibrium. Motivated by this stability analysis, we propose an additional regularization term for gradient descent GAN updates, which \emph{is} able to guarantee local stability for both the WGAN and the traditional GAN, and also shows practical promise in speeding up convergence and addressing mode collapse.