We study in this paper the rate of convergence for learning densities under the Generative Adversarial Networks (GANs) framework, borrowing insights from nonparametric statistics. We introduce an improved GAN estimator that achieves a faster rate, through leveraging the level of smoothness in the target density and the evaluation metric, which in theory remedies the mode collapse problem reported in the literature. A minimax lower bound is constructed to show that when the dimension is large, the exponent in the rate for the new GAN estimator is near optimal. One can view our results as answering in a quantitative way how well GAN learns a wide range of densities with different smoothness properties, under a hierarchy of evaluation metrics. As a byproduct, we also obtain improved bounds for GAN with deeper ReLU discriminator network.

本文研究了属于生成对抗网络框架下学习密度函数的收敛速度，通过借鉴非参数统计学中的思路引入了一种改进型生成对抗网络估算器，同时利用目标密度的光顺程度和度量结果，实现更快速的学习速率，并理论上解决了文献中报道的模式崩溃问题。作者还构建了最小极大下界来证明，在维数较大的情况下，新生成对抗网络估算器的速率指数是接近最优的。因此我们的结果可以以量化的方式回答生成对抗网络在多种光滑程度具有不同密度下的学习效果，并从中获得了更好的泛化界限。

生成对抗网络能否学习密度：一种非参数方法