How to Find a Point in the Convex Hull Privately

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 without privacy requirements, 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 [Amos Beimel et al., 2010; Amos Beimel et al., 2019; Amos Beimel et al., 2013; Mark Bun et al., 2018; Mark Bun et al., 2015; Haim Kaplan et al., 2019] 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|^d², where typically |G|=X^d for some (large) discretization parameter X, so the running time is in fact Ω(X^d³). In this paper we give a differentially private algorithm that runs in O(n^d) time, assuming that n=Ω(d⁴ log 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(d²) 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 [László Lovász and Santosh S. Vempala, 2006] and of Cousins and Vempala [Ben Cousins and Santosh S. Vempala, 2018]. Making this framework differentially private raises a set of technical challenges that we address.