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We give the first polynomial-time, polynomial-sample, differentially privateestimator for the mean and covariance of an arbitrary Gaussian distribution$\mathcal{N}(\mu,\Sigma)$ in $\mathbb{R}^d$. All previous estimators are eithernonconstructive, with unbounded running time, or require the user to specify apriori bounds on the parameters $\mu$ and $\Sigma$. The primary new technicaltool in our algorithm is a new differentially private preconditioner that takessamples from an arbitrary Gaussian $\mathcal{N}(0,\Sigma)$ and returns a matrix$A$ such that $A \Sigma A^T$ has constant condition number.

A Private and Computationally-Efficient Estimator for Unbounded Gaussians