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We investigate unbiased high-dimensional mean estimators in differentialprivacy. We consider differentially private mechanisms whose expected outputequals the mean of the input dataset, for every dataset drawn from a fixedbounded $d$-dimensional domain $K$. A classical approach to private meanestimation is to compute the true mean and add unbiased, but possiblycorrelated, Gaussian noise to it. In the first part of this paper, we study theoptimal error achievable by a Gaussian noise mechanism for a given domain $K$when the error is measured in the $\ell_p$ norm for some $p \ge 2$. We givealgorithms that compute the optimal covariance for the Gaussian noise for agiven $K$ under suitable assumptions, and prove a number of nice geometricproperties of the optimal error. These results generalize the theory offactorization mechanisms from domains $K$ that are symmetric and finite (or,equivalently, symmetric polytopes) to arbitrary bounded domains.  In the second part of the paper we show that Gaussian noise mechanismsachieve nearly optimal error among all private unbiased mean estimationmechanisms in a very strong sense. In particular, for every input dataset, anunbiased mean estimator satisfying concentrated differential privacy introducesapproximately at least as much error as the best Gaussian noise mechanism. Weextend this result to local differential privacy, and to approximatedifferential privacy, but for the latter the error lower bound holds either fora dataset or for a neighboring dataset, and this relaxation is necessary.

General Gaussian Noise Mechanisms and Their Optimality for Unbiased Mean Estimation