Tight and Robust Private Mean Estimation with Few Users

TL;DR

This paper introduces a nearly optimal, robust private mean estimation method that minimizes user count, independent of data dimension, and applies broadly to distribution learning and optimization tasks.

Contribution

It presents a new private mean estimation mechanism that is nearly optimal, robust to user corruptions, and independent of data dimension, solving a key open problem.

Findings

Achieves near-optimal user-to-sample trade-off for private mean estimation.

Mechanism is robust against up to 49% user corruptions.

Applicable to private distribution learning and stochastic optimization.

Abstract

In this work, we study high-dimensional mean estimation under user-level differential privacy, and design an $(\varepsilon,\delta)$-differentially private mechanism using as few users as possible. In particular, we provide a nearly optimal trade-off between the number of users and the number of samples per user required for private mean estimation, even when the number of users is as low as $O(\frac{1}{\varepsilon}\log\frac{1}{\delta})$. Interestingly, this bound on the number of \emph{users} is independent of the dimension (though the number of \emph{samples per user} is allowed to depend polynomially on the dimension), unlike the previous work that requires the number of users to depend polynomially on the dimension. This resolves a problem first proposed by Amin et al. Moreover, our mechanism is robust against corruptions in up to $49\%$ of the users. Finally, our results also apply to optimal algorithms for privately learning discrete distributions with few users, answering a question of Liu et al., and a broader range of problems such as stochastic convex optimization and a variant of stochastic gradient descent via a reduction to differentially private mean estimation.