  We undertake a precise study of the non-asymptotic properties of vanilla
generative adversarial networks (GANs) and derive theoretical guarantees in the
problem of estimating an unknown $d$-dimensional density $p^*$ under a proper
choice of the class of generators and discriminators. We prove that the
resulting density estimate converges to $p^*$ 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 $p^*.$ This is the first result
in the literature on density estimation using vanilla GANs with JS rates faster
than $n^{-1/2}$ in the regime $\beta>d/2.$
