Differentially private depth functions and their associated medians

In this paper, we investigate the differentially private estimation of data depth functions and their associated medians. We introduce several methods for privatizing depth values at a fixed point, and show that for some depth functions, when the depth is computed at an out of sample point, privacy can be gained for free when $n\rightarrow \infty$. We also present a method for privately estimating the vector of sample point depth values. Additionally, we introduce estimation methods for depth-based medians for both depth functions with low global sensitivity and depth functions with only highly probable, low local sensitivity. We provide a general result (Lemma 1) which can be used to prove consistency of an estimator produced by the exponential mechanism, provided the limiting cost function is sufficiently smooth at a unique minimizer. We also introduce a general algorithm to privately estimate a minimizer of a cost function which has, with high probability, low local sensitivity. This algorithm combines the propose-test-release algorithm with the exponential mechanism. An application of this algorithm to generate consistent estimates of the projection depth-based median is presented. Thus, for these private depth-based medians, we show that it is possible for privacy to be obtained for free when $n\rightarrow \infty$.