Locally Private Mean Estimation: $Z$test and Tight Confidence Intervals
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Proceedings of Machine Learning Research, PMLR 89:25452554, 2019.
Abstract
This work provides tight upper and lowerbounds for the problem of mean estimation under differential privacy in the localmodel, when the input is composed of $n$ i.i.d. drawn samples from a Gaussian. Our algorithms result in a $(1\beta)$confidence interval for the underlying distribution’s mean of length $O(\sigma *sqrt(log(n/beta)log(1/\beta))/(\epsilon*sqrt(n))$. In addition, our algorithms leverage on binary search using local differential privacy for quantile estimation, a result which may be of separate interest. Moreover, our algorithms have a matching lowerbound, where we prove that any oneshot (each individual is presented with a single query) local differentially private algorithm must return an interval of length $\Omega(\sigma*sqrt(\log(1/\beta))/(\epsilon*sqrt(n)))$.
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