Locally Private Mean Estimation: $Z$-test and Tight Confidence Intervals
Marco Gaboardi, Ryan Rogers, Or Sheffet
Abstract
This work provides tight upper- and lower-bounds for the problem of mean estimation under differential privacy in the local-model, 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 lower-bound, where we prove that any one-shot (each individual is presented with a single query) local differentially private algorithm must return an interval of length $\Omega(\sigma*sqrt(\log(1/\beta))/(\epsilon*sqrt(n)))$.
BibTeX
@InProceedings{pmlr-v89-gaboardi19a,
title = {Locally Private Mean Estimation: $Z$-test and Tight Confidence Intervals},
author = {Gaboardi, Marco and Rogers, Ryan and Sheffet, Or},
booktitle = {Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics},
pages = {2545--2554},
year = {2019},
editor = {Chaudhuri, Kamalika and Sugiyama, Masashi},
volume = {89},
series = {Proceedings of Machine Learning Research},
month = {16--18 Apr},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v89/gaboardi19a/gaboardi19a.pdf},
url = {https://proceedings.mlr.press/v89/gaboardi19a.html},
abstract = {This work provides tight upper- and lower-bounds for the problem of mean estimation under differential privacy in the local-model, 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 lower-bound, where we prove that any one-shot (each individual is presented with a single query) local differentially private algorithm must return an interval of length $\Omega(\sigma*sqrt(\log(1/\beta))/(\epsilon*sqrt(n)))$.}
}