Rates of convergence for density estimation with generative adversarial networks

TL;DR

It is shown 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 $\mathsf{p}^*$.

Abstract

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 $\mathsf{p}^*$ 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 $\mathsf{p}^*$ decays as fast as $(\log{n}/n)^{2β/(2β+ d)}$, where $n$ is the sample size and $β$ determines the smoothness of $\mathsf{p}^*$. This rate of convergence coincides (up to logarithmic factors) with minimax optimal for the considered class of densities.

Theorems & Definitions (23)

• Theorem 1
• Remark 1
• Lemma 1
• Remark 2
• Lemma 2
• Lemma 3
• Lemma 4
• Theorem 2
• Remark 3
• Theorem 3