A Convex Duality Framework for GANs

Abstract

Generative adversarial network (GAN) is a minimax game between a generator mimicking the true model and a discriminator distinguishing the samples produced by the generator from the real training samples. Given an unconstrained discriminator able to approximate any function, this game reduces to finding the generative model minimizing a divergence measure, e.g. the Jensen-Shannon (JS) divergence, to the data distribution. However, in practice the discriminator is constrained to be in a smaller class $\mathcal{F}$ such as neural nets. Then, a natural question is how the divergence minimization interpretation changes as we constrain $\mathcal{F}$. In this work, we address this question by developing a convex duality framework for analyzing GANs. For a convex set $\mathcal{F}$, this duality framework interprets the original GAN formulation as finding the generative model with minimum JS-divergence to the distributions penalized to match the moments of the data distribution, with the moments specified by the discriminators in $\mathcal{F}$. We show that this interpretation more generally holds for f-GAN and Wasserstein GAN. As a byproduct, we apply the duality framework to a hybrid of f-divergence and Wasserstein distance. Unlike the f-divergence, we prove that the proposed hybrid divergence changes continuously with the generative model, which suggests regularizing the discriminator's Lipschitz constant in f-GAN and vanilla GAN. We numerically evaluate the power of the suggested regularization schemes for improving GAN's training performance.

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• Reciprocal Adversarial Learning via Characteristic Functions
• Hidden Convexity of Wasserstein GANs: Interpretable Generative Models with Closed-Form Solutions
• A New Primal-Dual Algorithm for a Class of Nonlinear Compositional Convex Optimization Problems
• Generative Adversarial Networks and Adversarial Autoencoders: Tutorial and Survey
• A Limited-Capacity Minimax Theorem for Non-Convex Games or: How I Learned to Stop Worrying about Mixed-Nash and Love Neural Nets
• Parallel Deep Neural Networks Have Zero Duality Gap
• Wasserstein GAN Can Perform PCA
• Scalable Personalised Item Ranking through Parametric Density Estimation
• Bridging the Gap Between $f$-GANs and Wasserstein GANs
• A Wasserstein Minimax Framework for Mixed Linear Regression