[논문 리뷰] Gradient descent GAN optimization is locally stable

Vaishnavh Nagarajan, J. Zico Kolter|arXiv (Cornell University)|2017. 06. 13.

Model Reduction and Neural Networks인용 수 76

한 줄 요약

본 논문은 GAN에서 생성기와 판별기가 동시에 업데이트될 때의 그래디언트 업데이트를 분석하고, 특정 조건에서 국소적 지수 안정성을 보이며, 전통적 GAN과 WGAN 모두의 안정성을 보장하기 위한 그래디언트 기반 정규화를 제안한다.

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.

연구 동기 및 목표

• Motivate and analyze the stability of gradient-descent GAN optimization where both G and D update simultaneously.
• Show that equilibrium points are locally exponentially stable under suitable conditions.
• Demonstrate that WGANs can exhibit non-convergent limit cycles in gradient descent.
• Propose a regularization term on the discriminator gradient that ensures local stability for both GAN and WGAN setups.
• Relate the proposed regularization to unrolled GANs and practical training improvements.

제안 방법

• Formulate a generalized GAN objective with a concave function f and analyze the gradient-descent updates as a dynamical system.
• Apply the ODE method and linearization to study local stability around equilibrium points.
• Define and utilize matrices K_DD and K_DG to characterize the Jacobian at equilibrium.
• Prove local exponential stability under a set of assumptions (I–IV) and discuss cases with multiple equilibria.
• Show that Wasserstein GANs may have non-convergent limit cycles in this gradient-descent setting (no global convergence).
• Introduce a gradient-penalty regularization for the generator updates that preserves equilibrium and enforces local stability (double backprop).

실험 결과

연구 질문

• RQ1 Under what conditions are gradient-descent GAN updates locally exponentially stable around equilibrium points?
• RQ2 Do traditional GANs and WGANs both admit local stability under gradient-descent dynamics?
• RQ3 Can a regularization term guarantee local stability for both GAN and WGAN formulations?
• RQ4 How does the presence of limit cycles differ between GANs and WGANs in gradient-descent updates?
• RQ5 What is the relationship between the proposed gradient penalty and existing stabilization methods like unrolled GANs?

주요 결과

• GAN gradient-descent dynamics can be locally exponentially stable around good equilibria under suitable curvature conditions.
• WGANs can exhibit non-convergent limit cycles in gradient-descent updates near equilibrium.
• A gradient-based regularization term on the discriminator gradient guarantees local exponential stability for both GANs and WGANs.
• The regularizer is simple to implement and can speed up convergence and help prevent mode collapse.
• The analysis connects to and differentiates from unrolled GANs and improved WGAN training approaches.