Rates of convergence for density estimation with GANs

In this work we undertake a thorough study of the non-asymptotic properties of the vanilla generative adversarial networks (GANs). We derive theoretical guarantees for the density estimation with GANs under a proper choice of the deep neural networks classes representing generators and discriminators. In particular, we prove that the resulting estimate converges to the true density $\mathsf{p}^*$ in terms of Jensen-Shannon (JS) divergence at the rate $(\log{n}/n)^{2\beta/(2\beta+d)}$ 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.