  While Generative Adversarial Networks (GANs) have demonstrated promising
performance on multiple vision tasks, their learning dynamics are not yet well
understood, both in theory and in practice. To address this issue, we study GAN
dynamics in a simple yet rich parametric model that exhibits several of the
common problematic convergence behaviors such as vanishing gradients, mode
collapse, and diverging or oscillatory behavior. In spite of the non-convex
nature of our model, we are able to perform a rigorous theoretical analysis of
its convergence behavior. Our analysis reveals an interesting dichotomy: a GAN
with an optimal discriminator provably converges, while first order
approximations of the discriminator steps lead to unstable GAN dynamics and
mode collapse. Our result suggests that using first order discriminator steps
(the de-facto standard in most existing GAN setups) might be one of the factors
that makes GAN training challenging in practice.
