A convenient infinite dimensional framework for generative adversarial learning

In recent years, generative adversarial networks (GANs) have demonstrated impressive experimental results while there are only a few works that foster statistical learning theory for GANs. In this work, we propose an infinite dimensional theoretical framework for generative ad-versarial learning. We assume that the probability density functions of the underlying measure are uniformly bounded, k-times alpha-Holder differentiable (Ck,alpha) and uniformly bounded away from zero. Under these assumptions, we show that the Rosenblatt transformation induces an optimal generator, which is realizable in the hypothesis space of Ck,alpha-generators. With a con-sistent definition of the hypothesis space of discriminators, we further show that the Jensen-Shannon divergence between the distribution induced by the generator from the adversarial learning procedure and the data gener-ating distribution converges to zero. Under certain regularity assumptions on the density of the data generating process, we also provide rates of convergence based on chaining and concentration.