An error analysis of generative adversarial networks for learning distributions

This paper studies how well generative adversarial networks (GANs) learn probability distributions from finite samples. Our main results estimate the convergence rates of GANs under a collection of integral probability metrics defined through H\"older classes, including the Wasserstein distance as a special case. We also show that GANs are able to adaptively learn data distributions with low-dimensional structure or have H\"older densities, when the network architectures are chosen properly. In particular, for distributions concentrate around a low-dimensional set, it is proved that the learning rates of GANs do not depend on the high ambient dimension, but on the lower intrinsic dimension. Our analysis is based on a new oracle inequality decomposing the estimation error into generator and discriminator approximation error and statistical error, which may be of independent interest.