Rates of convergence for density estimation with generative adversarial networks

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)²β/(2β+d⁾, where n is the sample size and β determines the smoothness of p^∗. This rate of convergence coincides (up to logarithmic factors) with minimax optimal for the considered class of densities.