On the Sample Complexity of Privately Learning Unbounded High-Dimensional Gaussians

TL;DR

This paper establishes near-optimal sample complexity bounds for differentially private learning of high-dimensional Gaussian distributions without parameter restrictions, advancing understanding of private distribution learning.

Contribution

It provides the first finite sample upper bounds for general Gaussians under privacy constraints, introducing new analytic tools for global cover construction.

Findings

Bounds are near-optimal when covariance is identity.

Bounds are conjectured to be near-optimal in the general case.

Techniques may extend to other distribution classes without finite covers.

Abstract

We provide sample complexity upper bounds for agnostically learning multivariate Gaussians under the constraint of approximate differential privacy. These are the first finite sample upper bounds for general Gaussians which do not impose restrictions on the parameters of the distribution. Our bounds are near-optimal in the case when the covariance is known to be the identity, and conjectured to be near-optimal in the general case. From a technical standpoint, we provide analytic tools for arguing the existence of global "locally small" covers from local covers of the space. These are exploited using modifications of recent techniques for differentially private hypothesis selection. Our techniques may prove useful for privately learning other distribution classes which do not possess a finite cover.