Global Convergence to the Equilibrium of GANs using Variational Inequalities

Abstract

In optimization, the negative gradient of a function denotes the direction of steepest descent. Furthermore, traveling in any direction orthogonal to the gradient maintains the value of the function. In this work, we show that these orthogonal directions that are ignored by gradient descent can be critical in equilibrium problems. Equilibrium problems have drawn heightened attention in machine learning due to the emergence of the Generative Adversarial Network (GAN). We use the framework of Variational Inequalities to analyze popular training algorithms for a fundamental GAN variant: the Wasserstein Linear-Quadratic GAN. We show that the steepest descent direction causes divergence from the equilibrium, and convergence to the equilibrium is achieved through following a particular orthogonal direction. We call this successful technique Crossing-the-Curl, named for its mathematical derivation as well as its intuition: identify the game's axis of rotation and move "across" space in the direction towards smaller "curling".

References in corpus (9)

• Progressive Growing of GANs for Improved Quality, Stability, and Variation
• Energy-based Generative Adversarial Network
• Which Training Methods for GANs do actually Converge?
• A Variational Inequality Perspective on Generative Adversarial Networks
• The Mechanics of n-Player Differentiable Games
• Generative Adversarial Nets from a Density Ratio Estimation Perspective
• Understanding GANs: the LQG Setting
• Online Monotone Optimization
• Online Monotone Games

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• Finding Mixed Nash Equilibria of Generative Adversarial Networks
• Implicit competitive regularization in GANs
• Training Generative Adversarial Networks by Solving Ordinary Differential Equations
• Stable Opponent Shaping in Differentiable Games
• Smooth markets: A basic mechanism for organizing gradient-based learners
• Latent-Optimized Adversarial Neural Transfer for Sarcasm Detection
• Convergence and Sample Complexity of SGD in GANs
• Competitive Mirror Descent
• Competitive Policy Optimization
• Causal Inference in Network Economics
• Bridging Explicit and Implicit Deep Generative Models via Neural Stein Estimators
• Exploiting the Hidden Tasks of GANs: Making Implicit Subproblems Explicit
• LRS-DAG: Low Resource Supervised Domain Adaptation with Generalization Across Domains