We provide non asymptotic rates of convergence of the Wasserstein Generative Adversarial networks (WGAN) estimator. We build neural networks classes representing the generators and discriminators which yield a GAN that achieves the minimax optimal rate for estimating a certain probability measure $\mu$ with support in $\mathbb{R}^p$. The probability $\mu$ is considered to be the push forward of the Lebesgue measure on the $d$-dimensional torus $\mathbb{T}^d$ by a map $g^\star:\mathbb{T}^d\rightarrow \mathbb{R}^p$ of smoothness $\beta+1$. Measuring the error with the $\gamma$-H\"older Integral Probability Metric (IPM), we obtain up to logarithmic factors, the minimax optimal rate $O(n^{-\frac{\beta+\gamma}{2\beta +d}}\vee n^{-\frac{1}{2}})$ where $n$ is the sample size, $\beta$ determines the smoothness of the target measure $\mu$, $\gamma$ is the smoothness of the IPM ($\gamma=1$ is the Wasserstein case) and $d\leq p$ is the intrinsic dimension of $\mu$. In the process, we derive a sharp interpolation inequality between H\"older IPMs. This novel result of theory of functions spaces generalizes classical interpolation inequalities to the case where the measures involved have densities on different manifolds.

翻译：我们提供了Wasserstein生成对抗网络（WGAN）估计器的非渐近收敛速率。我们构建了表示生成器和判别器的神经网络类，使得该GAN在估计支集位于$\mathbb{R}^p$中的某个概率测度$\mu$时达到最小最大最优速率。该概率测度$\mu$被视为通过光滑度为$\beta+1$的映射$g^\star:\mathbb{T}^d\rightarrow \mathbb{R}^p$将$d$维环面$\mathbb{T}^d$上的勒贝格测度前推得到。使用$\gamma$-Hölder积分概率度量（IPM）衡量误差时，我们得到了忽略对数因子后的最小最大最优速率$O(n^{-\frac{\beta+\gamma}{2\beta +d}}\vee n^{-\frac{1}{2}})$，其中$n$为样本量，$\beta$决定目标测度$\mu$的光滑性，$\gamma$为IPM的光滑性（$\gamma=1$对应Wasserstein情形），而$d\leq p$为$\mu$的本质维度。在此过程中，我们推导了Hölder IPM之间的一个尖锐插值不等式。这一函数空间理论的新结果将经典插值不等式推广至所涉测度在不同流形上具有密度函数的情形。