Abstract
We study in this paper the rate of convergence for learning densities under
the Generative Adversarial Networks (GAN) framework, borrowing insights from
nonparametric statistics. We introduce an improved GAN estimator that achieves
a faster rate, through simultaneously 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 generalization bounds for GAN with
deeper ReLU discriminator network.
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