# Locally Private k-Means Clustering

SODA (2020)

## Abstract

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 $n^{1/2+a}$ additive error for an arbitrarily small constant a>0. All previous algorithms in the local model had additive error $n^{2/3+a}$.

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