Robust Density Estimation under Besov IPM Losses

We study minimax convergence rates of nonparametric density estimation under the Huber contamination model, in which a ``contaminated'' proportion of the data comes from an unknown outlier distribution. We provide the first results for this problem under a large family of losses, called Besov integral probability metrics (IPMs), that include L^p, Wasserstein, Kolmogorov-Smirnov, Cramer-von Mises, and other commonly used metrics. Under a range of smoothness assumptions on the population and outlier distributions, we show that a re-scaled thresholding wavelet estimator converges at the minimax optimal rate under a wide variety of losses and also exhibits optimal dependence on the contamination proportion. We also provide a purely data-dependent extension of the estimator that adapts to both an unknown contamination proportion and the unknown smoothness of the true density. Finally, based on connections shown recently between density estimation under IPM losses and generative adversarial networks (GANs), we show that certain GAN architectures are robustly minimax optimal.