This paper studies how well generative adversarial networks (GANs) learn probability distributions from finite samples. Our main results estimate the convergence rates of GANs under a collection of integral probability metrics defined through H\"older classes, including the Wasserstein distance as a special case. We also show that GANs are able to adaptively learn data distributions with low-dimensional structure or have H\"older densities, when the network architectures are chosen properly. In particular, for distributions concentrate around a low-dimensional set, it is proved that the learning rates of GANs do not depend on the high ambient dimension, but on the lower intrinsic dimension. Our analysis is based on a new oracle inequality decomposing the estimation error into generator and discriminator approximation error and statistical error, which may be of independent interest.

本文研究了生成对抗网络（GAN）如何从有限样本中学习概率分布，得到了GAN在一组H"older类定义的积分概率度量下的收敛速度和Wasserstein距离特殊情况下的学习率，并证明了当网络结构适当选择时，GAN能够自适应地学习低维结构或具有H"older密度的数据分布。特别是对于集中在低维集合周围的分布，我们展示了GAN的学习速率不取决于高环境维度，而取决于较低的内在维度。我们的分析基于一种新的神谕不等式，将估计误差分解为生成器和鉴别器逼近误差和统计误差，这可能是具有独立研究价值的。

生成对抗网络学习分布的误差分析