Private coresets

A coreset of a point set P is a small weighted set of points that captures some geometric properties of P. Coresets have found use in a vast host of geometric settings. We forge a link between coresets, and differentially private sanitizations that can answer any number of queries without compromising privacy. We define the notion of private coresets, which are simultaneously both coresets and differentially private, and show how they may be constructed. We first show that the existence of a small coreset with low generalized sensitivity (i.e., replacing a single point in the original point set slightly affects the quality of the coreset) implies (in an inefficient manner) the existence of a private coreset for the same queries. This greatly extends the works of Blum, Ligett, and Roth [STOC 2008] and McSherry and Talwar [FOCS 2007]. We also give an efficient algorithm to compute private coresets for k-median and k-mean queries in R ^d, immediately implying efficient differentially private sanitizations for such queries. Following McSherry and Talwar, this construction also gives efficient coalition proof (approximately dominant strategy) mechanisms for location problems. Unlike coresets which only have a multiplicative approximation factor, we prove that private coresets must exhibit additive error. We present a new technique for showing lower bounds on this error.