A Fast Algorithm for Adaptive Private Mean Estimation

01/17/2023

∙

by   John Duchi, et al.

∙

0

∙

share

We design an (ε, δ)-differentially private algorithm to estimate the mean of a d-variate distribution, with unknown covariance Σ, that is adaptive to Σ. To within polylogarithmic factors, the estimator achieves optimal rates of convergence with respect to the induced Mahalanobis norm ||·||_Σ, takes time Õ(n d^2) to compute, has near linear sample complexity for sub-Gaussian distributions, allows Σ 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 = Ω(d^3/2) to achieve non-trivial error with respect to the norm ||·||_Σ.