🤖 AI Summary
This work addresses robust multivariate location estimation under heavy-tailed and contaminated data within the differential privacy (DP) framework. We establish, for the first time, a systematic theoretical foundation for DP-compliant multivariate medians based on depth functions—including halfspace depth and spatial depth. Our method employs an exponential-mechanism-based private median estimator. We derive a novel concentration inequality bounding the estimator’s deviation from the global maximizer of the population objective function and obtain tight finite-sample error bounds. Empirically, the estimator achieves near-optimal error rates in Gaussian contamination models up to $d leq 100$ dimensions; under Cauchy-distributed heavy-tailed data, it incurs significantly lower privacy cost than existing private mean estimators while simultaneously improving both robustness and statistical efficiency.
📝 Abstract
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.
