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We provide sample complexity upper bounds for agnostically learningmultivariate Gaussians under the constraint of approximate differentialprivacy. These are the first finite sample upper bounds for general Gaussianswhich do not impose restrictions on the parameters of the distribution. Ourbounds are near-optimal in the case when the covariance is known to be theidentity, and conjectured to be near-optimal in the general case. From atechnical standpoint, we provide analytic tools for arguing the existence ofglobal "locally small" covers from local covers of the space. These areexploited using modifications of recent techniques for differentially privatehypothesis selection. Our techniques may prove useful for privately learningother distribution classes which do not possess a finite cover.

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