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Archimedes Meets Privacy: On Privately Estimating Quantiles in High Dimensions Under Minimal Assumptions
Omri Ben-Eliezer · Dan Mikulincer · Ilias Zadik

Thu Dec 01 09:00 AM -- 11:00 AM (PST) @ Hall J #329

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 fundamental 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 of a distribution. 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.

Author Information

Omri Ben-Eliezer (Massachusetts Institute of Technology)
Dan Mikulincer (MIT)
Ilias Zadik (MIT)

I am a CDS Moore-Sloan (postdoctoral) fellow at the Center for Data Science of NYU and a member of it's Math and Data (MaD) group. I received my PhD on September 2019 from MIT , where I was advised by David Gamarnik. My research lies broadly in the interface of high dimensional statistics, the theory of machine learning and applied probability.

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