Private Mean Estimation with Person-Level Differential Privacy

TL;DR

This paper investigates the fundamental limits and algorithms for privately estimating the mean of a distribution when each individual has multiple data points, providing tight bounds and efficient methods under different privacy regimes.

Contribution

It establishes tight sample complexity bounds for person-level differential privacy in mean estimation with multiple samples per person, and introduces new algorithms and tail bounds for this setting.

Findings

Optimal sample complexity bounds for person-level DP mean estimation.

Efficient algorithms under approximate DP with nearly matching lower bounds.

New tail bounds for sums of vector-valued bounded-moments random variables.

Abstract

We study person-level differentially private (DP) mean estimation in the case where each person holds multiple samples. DP here requires the usual notion of distributional stability when $\textit{all}$ of a person's datapoints can be modified. Informally, if $n$ people each have $m$ samples from an unknown $d$-dimensional distribution with bounded $k$-th moments, we show that \[n = \tilde \Theta\left(\frac{d}{\alpha^2 m} + \frac{d}{\alpha m^{1/2} \varepsilon} + \frac{d}{\alpha^{k/(k-1)} m \varepsilon} + \frac{d}{\varepsilon}\right)\] people are necessary and sufficient to estimate the mean up to distance $\alpha$ in $\ell_2$-norm under $\varepsilon$-differential privacy (and its common relaxations). In the multivariate setting, we give computationally efficient algorithms under approximate-DP and computationally inefficient algorithms under pure DP, and our nearly matching lower bounds hold for the most permissive case of approximate DP. Our computationally efficient estimators are based on the standard clip-and-noise framework, but the analysis for our setting requires both new algorithmic techniques and new analyses. In particular, our new bounds on the tails of sums of independent, vector-valued, bounded-moments random variables may be of interest.