# Locally Private Gaussian Estimation

arXiv: https://arxiv.org/abs/1811.08382 · HTML: https://mlsanity.com/paper/1811.08382/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Nov 2018.

## Built on by (most cited first, 7 of 7)

- [Covariance-Aware Private Mean Estimation Without Private Covariance Estimation](https://mlsanity.com/paper/2106.13329/) (2021)
- [DP-PCA: Statistically Optimal and Differentially Private PCA](https://mlsanity.com/paper/2205.13709/) (2022)
- [Privately Learning Mixtures of Axis-Aligned Gaussians](https://mlsanity.com/paper/2106.02162/) (2021)
- [Nonparametric extensions of randomized response for private confidence sets](https://mlsanity.com/paper/2202.08728/) (2022)
- [Sample-Optimal Locally Private Hypothesis Selection and the Provable Benefits of Interactivity](https://mlsanity.com/paper/2312.05645/) (2023)
- [Locally Differentially Private Bayesian Inference](https://mlsanity.com/paper/2110.14426/) (2021)
- [Bayesian Frequency Estimation Under Local Differential Privacy With an Adaptive Randomized Response Mechanism](https://mlsanity.com/paper/2405.07020/) (2024)

## Abstract (arXiv)

We study a basic private estimation problem: each of n users draws a single i.i.d. sample from an unknown Gaussian distribution, and the goal is to estimate the mean of this Gaussian distribution while satisfying local differential privacy for each user. Informally, local differential privacy requires that each data point is individually and independently privatized before it is passed to a learning algorithm. Locally private Gaussian estimation is therefore difficult because the data domain is unbounded: users may draw arbitrarily different inputs, but local differential privacy nonetheless mandates that different users have (worst-case) similar privatized output distributions. We provide both adaptive two-round solutions and nonadaptive one-round solutions for locally private Gaussian estimation. We then partially match these upper bounds with an information-theoretic lower bound. This lower bound shows that our accuracy guarantees are tight up to logarithmic factors for all sequentially interactive (varepsilon,δ)-locally private protocols.