This paper studies the rates of convergence for learning distributions implicitly with the adversarial framework and Generative Adversarial Networks (GANs), which subsume Wasserstein, Sobolev, MMD GAN, and Generalized/Simulated Method of Moments (GMM/SMM) as special cases. We study a wide range of parametric and nonparametric target distributions under a host of objective evaluation metrics. We investigate how to obtain valid statistical guarantees for GANs through the lens of regularization.

Modern statistical inference tasks often require iterative optimization methods to compute the solution. Convergence analysis from an optimization viewpoint only informs us how well the solution is approximated numerically but overlooks the sampling nature of the data. We introduce the moment-adjusted stochastic gradient descents, a new stochastic optimization method for statistical inference.