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 objective 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 ≈ n 1/2+a additive error for an arbitrarily small constant a > 0. All previous algorithms in the local model had additive error ≈ n 2/3+a . Our techniques extend to k-median clustering.

We show that the additive error we obtain is almost optimal in terms of its dependency on the database size n. Specifically, we give a simple lower bound showing that every locally-private algorithm for the k-means objective must have additive error at least ≈ √ n.