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 $\ell_2^2$-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 $\tilde{ O} (n^{ 1/2 + a} \cdot k \cdot \max \{ \sqrt{ d}, \sqrt{ k} \})$ and proves a lower bound of $\Omega(\sqrt{ n})$ 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 $\alpha>0$, our first algorithm achieves a multiplicative approximation guarantee which is at most a $(1+\alpha)$ factor greater than that of any non-private $k$-means clustering algorithm with $k^{ \tilde{ O}(1/\alpha^2)} \sqrt{ d' n} \mbox{ poly}\log n$ additive error. Given any $c>\sqrt{ 2}$, our second algorithm achieves $O(k^{ 1 + \tilde{ O}(1/(2c^2-1))} \sqrt{ d' n} \mbox{ poly} \log n)$ additive error with constant multiplicative approximation. Both algorithms go beyond the $\Omega(n^{ 1/2 + a})$ 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.
Categories: cs.DS cs.CR cs.LG