# Optimal Survey Design for Private Mean Estimation

arXiv: https://arxiv.org/abs/2501.18121 · HTML: https://mlsanity.com/paper/2501.18121/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 50 references that are in the map.
- First posted on arXiv 30 Jan 2025.

## Builds on (most cited first, 19 of 19)

- [Concentrated Differential Privacy: Simplifications, Extensions, and Lower Bounds](https://mlsanity.com/paper/1605.02065/) (2016)
- [Distributed Differential Privacy via Shuffling](https://mlsanity.com/paper/1808.01394/) (2018)
- [Differentially Private Chi-Squared Hypothesis Testing: Goodness of Fit and Independence Testing](https://mlsanity.com/paper/1602.03090/) (2016)
- [Differentially Private Bayesian Linear Regression](https://mlsanity.com/paper/1910.13153/) (2019)
- [Parametric Bootstrap for Differentially Private Confidence Intervals](https://mlsanity.com/paper/2006.07749/) (2020)
- [The Impact of the U.S. Census Disclosure Avoidance System on Redistricting and Voting Rights Analysis](https://mlsanity.com/paper/2105.14197/) (2021)
- [Differentially Private Bayesian Inference for Exponential Families](https://mlsanity.com/paper/1809.02188/) (2018)
- [Differentially Private Bayesian Inference for Generalized Linear Models](https://mlsanity.com/paper/2011.00467/) (2020)
- [Local Private Hypothesis Testing: Chi-Square Tests](https://mlsanity.com/paper/1709.07155/) (2017)
- [Differentially Private Bootstrap: New Privacy Analysis and Inference Strategies](https://mlsanity.com/paper/2210.06140/) (2022)
- [Locally Private Bayesian Inference for Count Models](https://mlsanity.com/paper/1803.08471/) (2018)
- [Fine-grained Poisoning Attack to Local Differential Privacy Protocols for Mean and Variance Estimation](https://mlsanity.com/paper/2205.11782/) (2022)
- [Canonical Noise Distributions and Private Hypothesis Tests](https://mlsanity.com/paper/2108.04303/) (2021)
- [Data Augmentation MCMC for Bayesian Inference from Privatized Data](https://mlsanity.com/paper/2206.00710/) (2022)
- [Simulation-based, Finite-sample Inference for Privatized Data](https://mlsanity.com/paper/2303.05328/) (2023)
- [Random Sampling Plus Fake Data: Multidimensional Frequency Estimates With Local Differential Privacy](https://mlsanity.com/paper/2109.07269/) (2021)
- [Differentially Private Confidence Intervals for Proportions under Stratified Random Sampling](https://mlsanity.com/paper/2301.08324/) (2023)
- [Controlling Privacy Loss in Sampling Schemes: an Analysis of Stratified and Cluster Sampling](https://mlsanity.com/paper/2007.12674/) (2020)
- [Differential Privacy By Sampling](https://mlsanity.com/paper/1708.01884/) (2017)

## Abstract (arXiv)

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.