Covariance-Aware Private Mean Estimation Without Private Covariance Estimation

present two sample-efficient differentially private mean estimators for ddimensional (sub)Gaussian distributions with unknown covariance. Informally, given n greater than or similar to d/alpha(2) samples from such a distribution with mean mu and covariance mu(similar to), our estimators output broken vertical bar broken vertical bar mu such that k vertical bar vertical bar(similar to)mu - mu vertical bar vertical bar Sigma <= alpha, where vertical bar vertical bar (.) vertical bar vertical bar Sigma is the Mahalanobis distance. All previous estimators with the same guarantee either require strong a priori bounds on the covariance matrix or require Omega(d(3/2)) samples. Each of our estimators is based on a simple, general approach to designing differentially private mechanisms, but with novel technical steps to make the estimator private and sample-efficient. Our first estimator samples a point with approximately maximum Tukey depth using the exponential mechanism, but restricted to the set of points of large Tukey depth. Proving that this mechanism is private requires a novel analysis. Our second estimator perturbs the empirical mean of the data set with noise calibrated to the empirical covariance, without releasing the covariance itself. Its sample complexity guarantees hold more generally for subgaussian distributions, albeit with a slightly worse dependence on the privacy parameter. For both estimators, careful preprocessing of the data is required to satisfy differential privacy.