An Error Analysis of Generative Adversarial Networks for Learning Distributions

Jian Huang, Yuling Jiao, Zhen Li, Shiao Liu, Yang Wang, Yunfei Yang.

Year: 2022, Volume: 23, Issue: 116, Pages: 1−43


This paper studies how well generative adversarial networks (GANs) learn probability distributions from finite samples. Our main results establish the convergence rates of GANs under a collection of integral probability metrics defined through H\"{o}lder classes, including the Wasserstein distance as a special case. We also show that GANs are able to adaptively learn data distributions with low-dimensional structures or have H\"{o}lder densities, when the network architectures are chosen properly. In particular, for distributions concentrated around a low-dimensional set, we show 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 the generator and discriminator approximation error and the statistical error, which may be of independent interest.