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

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)) ).