Archimedes Meets Privacy: On Privately Estimating Quantiles in High Dimensions Under Minimal Assumptions

TL;DR

This paper introduces differentially private algorithms for estimating high-dimensional quantiles and Tukey depth with minimal distributional assumptions, leveraging convex geometry and robustness results, achieving polynomial sample complexity.

Contribution

It presents new private algorithms for quantile estimation and depth sampling in high dimensions under mild assumptions, bridging worst-case and strong assumption approaches.

Findings

Private algorithms with polynomial sample complexity for high-dimensional quantiles.

Effective estimation of Tukey depth and sampling in high dimensions.

Utilization of convex geometry robustness results in privacy context.

Abstract

The last few years have seen a surge of work on high dimensional statistics under privacy constraints, mostly following two main lines of work: the ``worst case'' line, which does not make any distributional assumptions on the input data; and the ``strong assumptions'' line, which assumes that the data is generated from specific families, e.g., subgaussian distributions. In this work we take a middle ground, obtaining new differentially private algorithms with polynomial sample complexity for estimating quantiles in high-dimensions, as well as estimating and sampling points of high Tukey depth, all working under very mild distributional assumptions. From the technical perspective, our work relies upon deep robustness results in the convex geometry literature, demonstrating how such results can be used in a private context. Our main object of interest is the (convex) floating body (FB), a notion going back to Archimedes, which is a robust and well studied high-dimensional analogue of the interquantile range. We show how one can privately, and with polynomially many samples, (a) output an approximate interior point of the FB -- e.g., ``a typical user'' in a high-dimensional database -- by leveraging the robustness of the Steiner point of the FB; and at the expense of polynomially many more samples, (b) produce an approximate uniform sample from the FB, by constructing a private noisy projection oracle.