  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.
