The empirical success of Generative Adversarial Networks (GANs) caused an increasing interest in theoretical research. The statistical literature is mainly focused on Wasserstein GANs and generalizations thereof, which especially allow for good dimension reduction properties. Statistical results for Vanilla GANs, the original optimization problem, are still rather limited and require assumptions such as smooth activation functions and equal dimensions of the latent space and the ambient space. To bridge this gap, we draw a connection from Vanilla GANs to the Wasserstein distance. By doing so, existing results for Wasserstein GANs can be extended to Vanilla GANs. In particular, we obtain an oracle inequality for Vanilla GANs in Wasserstein distance. The assumptions of this oracle inequality are designed to be satisfied by network architectures commonly used in practice, such as feedforward ReLU networks. By providing a quantitative result for the approximation of a Lipschitz function by a feedforward ReLU network with bounded H\"older norm, we conclude a rate of convergence for Vanilla GANs as well as Wasserstein GANs as estimators of the unknown probability distribution.

翻译：生成对抗网络（GANs）在经验上的成功引发了理论研究的日益关注。统计文献主要集中于Wasserstein GAN及其推广形式，这些方法尤其具有良好的降维特性。而针对原始优化问题的原始GAN，其统计结果仍然较为有限，且需要诸如平滑激活函数、潜空间与背景空间维度相等之类的假设。为了弥合这一差距，我们将原始GAN与Wasserstein距离联系起来。通过这一方法，现有的Wasserstein GAN结果可以推广至原始GAN。特别地，我们得到了原始GAN在Wasserstein距离下的预言不等式。该预言不等式的假设条件被设计为能够满足实践中常用的网络架构，例如前馈ReLU网络。通过给出具有有界Hölder范数的前馈ReLU网络逼近Lipschitz函数的定量结果，我们推导了原始GAN以及Wasserstein GAN作为未知概率分布估计量的收敛速度。