The majority of work in robust statistics has focused on providing guarantees under the Huber -contamination model [Huber, 1965].

The majority of work in robust statistics has focused on providing guarantees under the Huber -contamination model [Huber, 1965].

We study the problem of estimating a nonparametric probability distribution under a family of losses called Besov IPMs. This family is quite large, including, for example, L^p distances, total variation distance, and generalizations of both Wasserstein (earthmover's) 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 distribution interact to determine the mini-max optimal convergence rate. We also show that, in many cases, linear distribution estimates, such as the empirical distribution or kernel density estimator, cannot converge at the optimal rate. These bounds generalize, unify, or improve on 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, in many cases, GANs can strictly outperform the best linear estimator.

We consider density estimation for Besov spaces when the estimator is restricted to use only a limited number of bits about each sample. We provide a noninteractive adaptive estimator which exploits the sparsity of wavelet bases, along with a simulate-and-infer technique from parametric estimation under communication constraints. We show that our estimator is nearly rate-optimal by deriving minmax lower bounds that hold even when interactive protocols are allowed. Interestingly, while our wavelet-based estimator is almost rate-optimal for Sobolev spaces as well, it is unclear whether the standard Fourier basis, which arise naturally for those spaces, can be used to achieve the same performance.

The construction and theoretical analysis of the most popular universally consistent nonparametric density estimators hinge on one functional property: smoothness. In this paper we investigate the theoretical implications of incorporating a multi-view latent variable model, a type of low-rank model, into nonparametric density estimation. To do this we perform extensive analysis on histogram-style estimators that integrate a multi-view model. Our analysis culminates in showing that there exists a universally consistent histogram-style estimator that converges to any multi-view model with a finite number of Lipschitz continuous components at a rate of $\widetilde{O}(1/\sqrt[3]{n})$ in $L^1$ error.

We study minimax convergence rates of nonparametric density estimation under a large class of loss functions called ``adversarial losses'', which, besides classical L^p losses, includes maximum mean discrepancy (MMD), Wasserstein distance, and total variation distance. These losses are closely related to the losses encoded by discriminator networks in generative adversarial networks (GANs). In a general framework, we study how the choice of loss and the assumed smoothness of the underlying density together determine the minimax rate. We also discuss implications for training GANs based on deep ReLU networks, and more general connections to learning implicit generative models in a minimax statistical sense.

Such concerns are shared by politics and industry, leading to the adoption of France's "Loi pour une République numérique (Law for the Digital Republic)" in October 2016 (Algan et al., 2016), EU's General Data Protection Regulation in May 2018, and the California Consumer Privacy Act (Wang et al., 2022), all of which regulate data protection, collection, and processing.

As shown in several recent papers [Liu et al., 2017, Liang, 2018, Singh et al., 2018, Uppal Namely, the estimators used in past work rely on the unrealistic assumption that the practitioner knows the Besov space in which the true density lies. The rest of this paper is organized as follows.

As shown in several recent papers [Liu et al., 2017, Liang, 2018, Singh et al., 2018, Uppal Namely, the estimators used in past work rely on the unrealistic assumption that the practitioner knows the Besov space in which the true density lies. The rest of this paper is organized as follows.