Locally Private k-Means Clustering

We design a new algorithm for the Euclidean $k$-means problem that operates in the local model of differential privacy. Unlike in the non-private literature, differentially private algorithms for the $k$-means incur both additive and multiplicative errors. Our algorithm significantly reduces the additive error while keeping the multiplicative error the same as in previous state-of-the-art results. Specifically, on a database of size $n$, our algorithm guarantees $O(1)$ multiplicative error and $\approx n^{1/2+a}$ additive error for an arbitrarily small constant $a$, whereas all previous algorithms in the local model on had additive error $\approx n^{2/3+a}$. We give a simple lower bound showing that additive error of $\approx\sqrt{n}$ is necessary for $k$-means algorithms in the local model (at least for algorithms with a constant number of interaction rounds, which is the setting we consider in this paper).