  Statistical tools which satisfy rigorous privacy guarantees are necessary for
modern data analysis. It is well-known that robustness against contamination is
linked to differential privacy. Despite this fact, using multivariate medians
for differentially private and robust multivariate location estimation has not
been systematically studied. We develop novel finite-sample performance
guarantees for differentially private multivariate depth-based medians, which
are essentially sharp. Our results cover commonly used depth functions, such as
the halfspace (or Tukey) depth, spatial depth, and the integrated dual depth.
We show that under Cauchy marginals, the cost of heavy-tailed location
estimation outweighs the cost of privacy. We demonstrate our results
numerically using a Gaussian contamination model in dimensions up to d = 100,
and compare them to a state-of-the-art private mean estimation algorithm. As a
by-product of our investigation, we prove concentration inequalities for the
output of the exponential mechanism about the maximizer of the population
objective function. This bound applies to objective functions that satisfy a
mild regularity condition.
