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：我们给出了Wasserstein生成对抗网络（WGAN）估计器的非渐近收敛速率。通过构建表示生成器和判别器的神经网络类，我们得到了一个能够以最小极大最优速率估计具有$\mathbb{R}^p$中支撑集的特定概率测度$\mu$的GAN。该概率测度$\mu$被视为$d$维环面$\mathbb{T}^d$上勒贝格测度经由光滑性为$\beta+1$的映射$g^\star:\mathbb{T}^d\rightarrow \mathbb{R}^p$推出的前推测度。采用$\gamma$-H\"older积分概率度量（IPM）衡量误差，我们得到（忽略对数因子）最小极大最优速率$O(n^{-\frac{\beta+\gamma}{2\beta +d}}\vee n^{-\frac{1}{2}})$，其中$n$为样本量，$\beta$决定目标测度$\mu$的光滑性，$\gamma$为IPM的光滑度（$\gamma=1$对应Wasserstein情形），$d\leq p$是$\mu$的内在维度。在此过程中，我们推导了H\"older IPM之间的尖锐插值不等式。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及测度在不同流形上具有密度函数的情形。