Differentially private $k$-means clustering via exponential mechanism and max cover

We introduce a new $(\epsilon_p, \delta_p)$-differentially private algorithm for the $k$-means clustering problem. Given a dataset in Euclidean space, the $k$-means clustering problem requires one to find $k$ points in that space such that the sum of squares of Euclidean distances between each data point and its closest respective point among the $k$ returned is minimised. Although there exist privacy-preserving methods with good theoretical guarantees to solve this problem [Balcan et al., 2017; Kaplan and Stemmer, 2018], in practice it is seen that it is the additive error which dictates the practical performance of these methods. By reducing the problem to a sequence of instances of maximum coverage on a grid, we are able to derive a new method that achieves lower additive error then previous works. For input datasets with cardinality $n$ and diameter $\Delta$, our algorithm has an $O(\Delta^2 (k \log^2 n \log(1/\delta_p)/\epsilon_p + k\sqrt{d \log(1/\delta_p)}/\epsilon_p))$ additive error whilst maintaining constant multiplicative error. We conclude with some experiments and find an improvement over previously implemented work for this problem.