How Well Generative Adversarial Networks Learn Distributions

32 Pages Posted: 19 Oct 2020

See all articles by Tengyuan Liang

Tengyuan Liang

University of Chicago - Booth School of Business

Date Written: October 16, 2020


This paper studies the rates of convergence for learning distributions implicitly with the adversarial framework and Generative Adversarial Networks (GAN), 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 a good statistical guarantee for GANs through the lens of regularization. On the nonparametric end, we derive the optimal minimax rates for distribution estimation under the adversarial framework. On the parametric end, we establish a theory for general neural network classes (including deep leaky ReLU networks), that characterizes the interplay on the choice of generator and discriminator pair. We discover and isolate a new notion of regularization, called the generator-discriminator-pair regularization, that sheds light on the advantage of GANs compared to classical parametric and nonparametric approaches for explicit distribution estimation. We develop novel oracle inequalities as the main technical tools for analyzing GANs, which is of independent interest.

Keywords: Generative Adversarial Networks, Implicit Distribution Estimation, Simulated Method of Moments, Oracle Inequality, Neural Network Learning, Minimax Problem, Pair Regularization

Suggested Citation

Liang, Tengyuan, How Well Generative Adversarial Networks Learn Distributions (October 16, 2020). University of Chicago, Becker Friedman Institute for Economics Working Paper No. 2020-154, Available at SSRN: or

Tengyuan Liang (Contact Author)

University of Chicago - Booth School of Business ( email )

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