Nonparametric Density Estimation & Convergence Rates for GANs under Besov IPM Losses
This work addresses the theoretical understanding of convergence rates in density estimation and GANs for researchers in statistics and machine learning, offering incremental improvements by generalizing and unifying existing results.
The paper tackles the problem of nonparametric density estimation under Besov IPM losses, providing lower and upper bounds that identify minimax optimal convergence rates and showing that linear estimators often fail to achieve these rates. It applies these results to GANs, demonstrating that GANs can outperform the best linear estimator in statistical error.
We study the problem of estimating a nonparametric probability density under a large family of losses called Besov IPMs, which include, for example, $\mathcal{L}^p$ distances, total variation distance, and generalizations of both Wasserstein and Kolmogorov-Smirnov distances. For a wide variety of settings, we provide both lower and upper bounds, identifying precisely how the choice of loss function and assumptions on the data interact to determine the minimax optimal convergence rate. We also show that linear distribution estimates, such as the empirical distribution or kernel density estimator, often fail to converge at the optimal rate. Our bounds generalize, unify, or improve several recent and classical results. Moreover, IPMs can be used to formalize a statistical model of generative adversarial networks (GANs). Thus, we show how our results imply bounds on the statistical error of a GAN, showing, for example, that GANs can strictly outperform the best linear estimator.