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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) 2β/(2β+d) where n is the sample size and β 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 β > d/2.

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