Locally Private Mean Estimation: Z-test and Tight Confidence Intervals

Abstract

This work provides tight upper- and lower-bounds for the problem of mean estimation under $\epsilon$-differential privacy in the local model, when the input is composed of $n$ i.i.d. drawn samples from a normal distribution with variance $\sigma$. Our algorithms result in a $(1-\beta )$-confidence interval for the underlying distribution's mean $\mu$ of length $\tilde{O}\left(\frac{\sigma \sqrt{\log (\frac{1}{\beta })}}{\epsilon \sqrt{n}}\right)$. In addition, our algorithms leverage binary search using local differential privacy for quantile estimation, a result which may be of separate interest. Moreover, we prove a matching lower-bound (up to poly-log factors), showing that any one-shot (each individual is presented with a single query) local differentially private algorithm must return an interval of length $\Omega \left(\frac{\sigma \sqrt{\log (1/\beta )}}{\epsilon \sqrt{n}}\right)$.

Cited by in corpus (6)

• A Comprehensive Survey on Local Differential Privacy Toward Data Statistics and Analysis
• Locally Private Gaussian Estimation
• Parametric Bootstrap for Differentially Private Confidence Intervals
• Instance-optimal Mean Estimation Under Differential Privacy
• Private Prediction Sets
• Differentially Private Confidence Intervals for Proportions under Stratified Random Sampling