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arXiv · 1810.08054

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

Marco Gaboardi·Ryan Rogers·Or Sheffet
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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 $\Omega\left( \frac{\sigma\sqrt{\log(1/\beta)}}{\epsilon\sqrt{n}}\right)$.

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Marco Gaboardi, Ryan Rogers, Or Sheffet. 2018-10-18. Locally Private Mean Estimation: Z-test and Tight Confidence Intervals. https://arxiv.org/abs/1810.08054

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