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

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.

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Cited by in corpus (6)

• Efficient Mean Estimation with Pure Differential Privacy via a Sum-of-Squares Exponential Mechanism
• Robust and Differentially Private Mean Estimation
• Covariance-Aware Private Mean Estimation Without Private Covariance Estimation
• Differential privacy and robust statistics in high dimensions
• A Private and Computationally-Efficient Estimator for Unbounded Gaussians
• Private and polynomial time algorithms for learning Gaussians and beyond