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 a sharp oracle inequality for the Jensen-Shannon (JS) divergence between the underlying density $\mathsf{p}^*$ and the GAN estimate. We also study the rates of convergence in the context of nonparametric density estimation. In particular, we show that the JS-divergence between the GAN estimate and $\mathsf{p}^*$ decays as fast as $(\log{n}/n)^{2\beta/(2\beta+d)}$ where $n$ is the sample size and $\beta$ determines the smoothness of $\mathsf{p}^*$. To the best of our knowledge, this is the first result in the literature on density estimation using vanilla GANs with JS convergence rates faster than $n^{-1/2}$ in the regime $\beta > d/2$. Moreover, we show that the obtained rate is minimax optimal (up to logarithmic factors) for the considered class of densities.