On the Sample Complexity of Privately Learning Gaussians and their Mixtures

Abstract

Multivariate Gaussians: We provide sample complexity upper bounds for semi-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 for differentially private hypothesis selection. Mixtures of Gaussians: We consider the problem of learning mixtures of Gaussians under the constraint of approximate differential privacy. We provide the first sample complexity upper bounds for privately learning mixtures of unbounded axis-aligned (or even unbounded univariate) Gaussians. To prove our results, we design a new technique for privately learning mixture distributions. A class of distributions F is said to be list-decodable if there is an algorithm that, given "heavily corrupted" samples from a distribution f in F, outputs a list of distributions, H, such that one of the distributions in H approximates f. We show that if F is privately list-decodable then we can privately learn mixtures of distributions in F. Finally, we show axis-aligned Gaussian distributions are privately list-decodable, thereby proving mixtures of such distributions are privately learnable.