How Well Can Generative Adversarial Networks Learn Densities: A Nonparametric View
This work addresses the mode collapse problem in GANs for density estimation, providing theoretical insights for researchers in machine learning and statistics, though it is incremental as it builds on existing GAN frameworks.
The paper tackles the problem of quantifying how well Generative Adversarial Networks (GANs) can learn densities by analyzing convergence rates from a nonparametric statistical perspective, introducing an improved GAN estimator that achieves a faster rate and is shown to be near-optimal in high dimensions.
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.