Distribution Approximation and Statistical Estimation Guarantees of Generative Adversarial Networks
This provides foundational theoretical support for GANs in unsupervised learning, addressing a key gap for researchers and practitioners in machine learning.
The paper tackles the lack of theoretical guarantees for Generative Adversarial Networks (GANs) by proving they can consistently estimate data distributions with densities in Hölder spaces under strong metrics like Wasserstein-1 distance, and show fast convergence free of the curse of dimensionality for low-dimensional data structures.
Generative Adversarial Networks (GANs) have achieved a great success in unsupervised learning. Despite its remarkable empirical performance, there are limited theoretical studies on the statistical properties of GANs. This paper provides approximation and statistical guarantees of GANs for the estimation of data distributions that have densities in a Hölder space. Our main result shows that, if the generator and discriminator network architectures are properly chosen, GANs are consistent estimators of data distributions under strong discrepancy metrics, such as the Wasserstein-1 distance. Furthermore, when the data distribution exhibits low-dimensional structures, we show that GANs are capable of capturing the unknown low-dimensional structures in data and enjoy a fast statistical convergence, which is free of curse of the ambient dimensionality. Our analysis for low-dimensional data builds upon a universal approximation theory of neural networks with Lipschitz continuity guarantees, which may be of independent interest.