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# A Private and Computationally-Efficient Estimator for Unbounded Gaussians

*Proceedings of Thirty Fifth Conference on Learning Theory*, PMLR 178:544-572, 2022.

#### Abstract

We give the first polynomial-time, polynomial-sample, differentially private estimator for the mean and covariance of an arbitrary Gaussian distribution $N(\mu,\Sigma)$ in $\R^d$. All previous estimators are either nonconstructive, with unbounded running time, or require the user to specify a priori bounds on the parameters $\mu$ and $\Sigma$. The primary new technical tool in our algorithm is a new differentially private preconditioner that takes samples from an arbitrary Gaussian $N(0,\Sigma)$ and returns a matrix $A$ such that $A \Sigma A^T$ has constant condition number