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We tackle the problem of estimating a location parameter with differentialprivacy guarantees and sub-Gaussian deviations. Recent work in statistics hasfocused on the study of estimators that achieve sub-Gaussian type deviationseven for heavy tailed data. We revisit some of these estimators through thelens of differential privacy and show that a naive application of the Laplacemechanism can lead to sub-optimal results. We design two private algorithms forestimating the median that lead to estimators with sub-Gaussian type errors.Unlike most existing differentially private median estimators, both algorithmsare well defined for unbounded random variables that are not even required tohave finite moments. We then turn to the problem of sub-Gaussian meanestimation and show that under heavy tails natural differentially privatealternatives lead to strictly worse deviations than their non-privatesub-Gaussian counterparts. This is in sharp contrast with recent results thatshow that from an asymptotic perspective the cost of differential privacy isnegligible.

Differentially private sub-Gaussian location estimators