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Covariance-Aware Private Mean Estimation Without Private Covariance Estimation
March 27, 2024, 4:43 a.m. | Gavin Brown, Marco Gaboardi, Adam Smith, Jonathan Ullman, Lydia Zakynthinou
cs.LG updates on arXiv.org arxiv.org
Abstract: We present two sample-efficient differentially private mean estimators for $d$-dimensional (sub)Gaussian distributions with unknown covariance. Informally, given $n \gtrsim d/\alpha^2$ samples from such a distribution with mean $\mu$ and covariance $\Sigma$, our estimators output $\tilde\mu$ such that $\| \tilde\mu - \mu \|_{\Sigma} \leq \alpha$, where $\| \cdot \|_{\Sigma}$ 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.
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