Locally Private k-Means Clustering with Constant Multiplicative Approximation and Near-Optimal Additive Error

Given a data set of size n in d ′ -dimensional Euclidean space, the k-means problem asks for a set of k points (called centers) so that the sum of the ℓ 2 2 -distances between points of a given data set of size n and the set of k centers is minimized. Recent work on this problem in the locally private setting achieves constant multiplicative approximation with additive error Õ(n ) and proves a lower bound of Ω( √ n) on the additive error for any solution with a constant number of rounds. In this work we bridge the gap between the exponents of n in the upper and lower bounds on the additive error with two new algorithms. Given any α > 0, our first algorithm achieves a multiplicative approximation guarantee which is at most a (1 + α) factor greater than that of any non-private k-means clustering algorithm with k Õ(1/α 2 ) √ d ′ n poly log n additive error. Given any c > √ 2, our second algorithm achieves O(k 1+ Õ(1/(2c 2 -1)) √ d ′ n poly log n) additive error with constant multiplicative approximation. Both algorithms go beyond the Ω(n 1/2+a ) factor that occurs in the additive error for arbitrarily small parameters a in previous work, and the second algorithm in particular shows for the first time that it is possible to solve the locally private k-means problem in a constant number of rounds with constant factor multiplicative approximation and polynomial dependence on k in the additive error arbitrarily close to linear.