A Fast Algorithm for Adaptive Private Mean Estimation

• URL: http://arxiv.org/abs/2301.07078v1
• Date: Tue, 17 Jan 2023 18:44:41 GMT
• Title: A Fast Algorithm for Adaptive Private Mean Estimation
• Authors: John Duchi and Saminul Haque and Rohith Kuditipudi
• Abstract summary: We design an $(varepsilon, delta)$-differentially private algorithm that is adaptive to $Sigma$. The estimator achieves optimal rates of convergence with respect to the induced Mahalanobis norm $||cdot||_Sigma$.
• Score: 5.090363690988394
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We design an $(\varepsilon, \delta)$-differentially private algorithm to estimate the mean of a $d$-variate distribution, with unknown covariance $\Sigma$, that is adaptive to $\Sigma$. To within polylogarithmic factors, the estimator achieves optimal rates of convergence with respect to the induced Mahalanobis norm $||\cdot||_\Sigma$, takes time $\tilde{O}(n d^2)$ to compute, has near linear sample complexity for sub-Gaussian distributions, allows $\Sigma$ to be degenerate or low rank, and adaptively extends beyond sub-Gaussianity. Prior to this work, other methods required exponential computation time or the superlinear scaling $n = \Omega(d^{3/2})$ to achieve non-trivial error with respect to the norm $||\cdot||_\Sigma$.

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