Rates of convergence for density estimation with generative adversarial networks

Nikita Puchkin, Sergey Samsonov, Denis Belomestny, Eric Moulines, Alexey Naumov

Abstract: In this work we undertake a thorough study of the non-asymptotic properties of the vanilla generative adversarial networks (GANs). We prove an oracle inequality for the Jensen-Shannon (JS) divergence between the underlying density $p$ and the GAN estimate with a significantly better statistical error term compared to the previously known results. The advantage of our bound becomes clear in application to nonparametric density estimation. We show that the JS-divergence between the GAN estimate and $p$ decays as fast as $(\log{n}/n)^{2\beta/(2\beta + d)}$, where $n$ is the sample size and $\beta$ determines the smoothness of $p$. This rate of convergence coincides (up to logarithmic factors) with minimax optimal for the considered class of densities.