How to Find a Point in the Convex Hull Privately
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We study the question of how to compute a point in the convex hull of an input set S of n points in ℝ^d in a differentially private manner. This question, which is trivial non-privately, turns out to be quite deep when imposing differential privacy. In particular, it is known that the input points must reside on a fixed finite subset G⊆ℝ^d, and furthermore, the size of S must grow with the size of G. Previous works focused on understanding how n needs to grow with |G|, and showed that n=O(d^2.5·8^log^*|G|) suffices (so n does not have to grow significantly with |G|). However, the available constructions exhibit running time at least |G|^d^2, where typically |G|=X^d for some (large) discretization parameter X, so the running time is in fact Ω(X^d^3). In this paper we give a differentially private algorithm that runs in O(n^d) time, assuming that n=Ω(d^4log X). To get this result we study and exploit some structural properties of the Tukey levels (the regions D_≥ k consisting of points whose Tukey depth is at least k, for k=0,1,...). In particular, we derive lower bounds on their volumes for point sets S in general position, and develop a rather subtle mechanism for handling point sets S in degenerate position (where the deep Tukey regions have zero volume). A naive approach to the construction of the Tukey regions requires n^O(d^2) time. To reduce the cost to O(n^d), we use an approximation scheme for estimating the volumes of the Tukey regions (within their affine spans in case of degeneracy), and for sampling a point from such a region, a scheme that is based on the volume estimation framework of Lovász and Vempala (FOCS 2003) and of Cousins and Vempala (STOC 2015). Making this framework differentially private raises a set of technical challenges that we address.
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