This paper studies how well generative adversarial networks (GANs) learn
probability distributions from finite samples. Our main results establish 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 structures or have H\"older densities, when
the network architectures are chosen properly. In particular, for distributions
concentrated around a low-dimensional set, we show 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 the generator and discriminator approximation error and
the 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。