Computational efficiency of pure DP high-dimensional mean estimation with bounded moments

Develop computationally efficient person-level ε-differentially private algorithms for multivariate mean estimation of distributions over ℝ^d with bounded k-th moments that achieve the near-optimal sample complexity guarantees established for the computationally inefficient pure DP estimator (Theorem 1.3). Specifically, design efficient procedures that, given n users each holding m i.i.d. samples, output an estimate of the mean with ℓ2 error at most α while matching the sample complexity up to polylogarithmic factors in d, m, k, α, and ε.

Background

The paper studies mean estimation under person-level differential privacy when each user contributes m samples, focusing on distributions in ℝ^d with bounded k-th moments. It establishes sample complexity upper and lower bounds and presents algorithms for both approximate and pure differential privacy. The pure DP multivariate estimator (Theorem 1.3) achieves near-optimal sample complexity but is computationally inefficient, relying on reductions to many univariate mean testing problems as in Kamath, Singhal, and Ullman (2020).

The authors note that existing approaches yielding pure DP efficiency often use semidefinite programming and sum-of-squares techniques, which scale poorly as the order of moments increases. Because their methods generally require higher-order moment information, translating the theoretical sample complexity guarantees into computationally efficient algorithms remains unresolved and is posed explicitly as an open question.