Gradient descent GAN optimization is locally stable (1706.04156v3)

Abstract: 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 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.

Gradient Descent GAN Optimization is Locally Stable

The research undertaken by Vaishnavh Nagarajan and J. Zico Kolter from Carnegie-Mellon University explores the local stability of gradient descent optimization in the context of Generative Adversarial Networks (GANs). The focus of the paper lies in mathematically evaluating the commonly employed training methodologies of GANs, specifically via gradient descent, and determining the conditions under which stability can be assured.

GANs, comprising a generator and a discriminator, are known for their complex dynamics and potential instability during training, governed by a minimax optimization problem. The paper rigorously analyzes this dynamic using a theoretical framework to ascertain aspects of stability that have major implications for practical training regimes.

Main Contributions

The paper provides several notable contributions to the understanding of GAN optimization:

1. Theoretical Analysis: The authors provide a definitive mathematical exploration of the dynamics induced by gradient descent optimization in GANs. This analysis is underpinned by the introduction of robust theoretical theorems which delineate under what circumstances stability can be maintained.
2. Minimax Optimization Stability: Through the derivation of key propositions and corollaries, the paper presents conditions related to local stability within the minimax framework. This includes insights into the behavior of the loss landscape and the interplay between the generator and discriminator models.
3. Implications for Practice: Although theoretical in nature, the results have direct implications for how GANs should be trained in practice. Understanding these stability conditions could lead to more efficient training algorithms, which reduce issues related to non-convergence and mode collapse prevalent in GAN training.

Implications and Speculations

From a theoretical standpoint, this work deepens the community's understanding of the intricate dynamics within GAN training. This, in turn, places emphasis on the importance of cautious design and adjustment of training parameters and strategies according to the specified stability conditions. Practically, the research suggests pathways to developing more stable GAN training algorithms, reducing empirical trial-and-error approaches and favoring informed, theoretically-backed choices.

Future Directions

Emerging from this paper are several open-ended questions which drive future research. Chief among these is the exploration of whether the established theoretical conditions apply across different GAN architectures and experimental configurations. There's an opportunity to extend these local stability findings to global stability concerns or broader areas of adversarial learning.

Additionally, further empirical evaluation could be conducted to validate these theoretical findings across various datasets and real-world applications, confirming their robustness and translating them into concrete methodological advancements.

In summary, Nagarajan and Kolter's paper provides significant insights into controlling the famously unstable training process of GANs, offering both theoretical depth and practical guidance to improve the development of generative models and advancing the overall field of deep learning.