Optimal Survey Design for Private Mean Estimation

TL;DR

This paper develops an optimal stratified sampling scheme for private mean estimation under differential privacy, minimizing variance while respecting privacy constraints across different mechanisms.

Contribution

It introduces the first privacy-aware stratified sampling method that optimally allocates subsampling sizes to minimize variance under DP mechanisms.

Findings

Proposes an efficient algorithm for optimal subsampling sizes.

Establishes strong convexity of the variance objective.

Provides structural insights into the optimal survey design.

Abstract

This work identifies the first privacy-aware stratified sampling scheme that minimizes the variance for general private mean estimation under the Laplace, Discrete Laplace (DLap) and Truncated-Uniform-Laplace (TuLap) mechanisms within the framework of differential privacy (DP). We view stratified sampling as a subsampling operation, which amplifies the privacy guarantee; however, to have the same final privacy guarantee for each group, different nominal privacy budgets need to be used depending on the subsampling rate. Ignoring the effect of DP, traditional stratified sampling strategies risk significant variance inflation. We phrase our optimal survey design as an optimization problem, where we determine the optimal subsampling sizes for each group with the goal of minimizing the variance of the resulting estimator. We establish strong convexity of the variance objective, propose an efficient algorithm to identify the integer-optimal design, and offer insights on the structure of the optimal design.