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 score, e.g. the Jensen-Shannon (JS) divergence, to the data distribution. However, in practice the discriminator is constrained to be in a smaller class F such as convolutional neural nets. Then, a natural question is how the divergence minimization interpretation will change as we constrain F. In this work, we address this question by developing a convex duality framework for analyzing GAN minimax problems. For a convex set F, this duality framework interprets the original vanilla GAN problem as finding the generative model with the minimum JS-divergence to the distributions penalized to match the moments of the data distribution, with the moments specified by the discriminators in F. We show that this interpretation more generally holds for f-GAN and Wasserstein GAN. We further apply the convex duality framework to explain why regularizing the discriminator’s Lipschitz constant, e.g. via spectral normalization or gradient penalty, can greatly improve the training performance in a general f-GAN problem including the vanilla GAN formulation. We prove that Lipschitz regularization can be interpreted as convolving the original divergence score with the first-order Wasserstein distance, which results in a continuously-behaving target divergence measure. We numerically explore the power of Lipschitz regularization for improving the continuity behavior and training performance in GAN problems.