Differentially private projection-depth-based medians
Kelly Ramsay Dylan Spicker
We develop $(\epsilon,\delta)$-differentially private projection-depth-based
medians using the propose-test-release (PTR) and exponential mechanisms. Under
general conditions on the input parameters and the population measure, (e.g. we
do not assume any moment bounds), we quantify the probability the test in PTR
fails, as well as the cost of privacy via finite sample deviation bounds. We
demonstrate our main result on the canonical projection-depth-based median. In
the Gaussian setting, we show that the resulting deviation bound matches the
known lower bound for private Gaussian mean estimation, up to a polynomial
function of the condition number of the covariance matrix. In the Cauchy
setting, we show that the ``outlier error amplification'' effect resulting from
the heavy tails outweighs the cost of privacy. This result is then verified via
numerical simulations. Additionally, we present results on general PTR
mechanisms and a uniform concentration result on the projected spacings of
order statistics.