Computational Hardness of Private Coreset

We study the problem of differentially private (DP) computation of coreset for the $k$-means objective. For a given input set of points, a coreset is another set of points such that the $k$-means objective for any candidate solution is preserved up to a multiplicative $(1 \pm \alpha)$ factor (and some additive factor). We prove the first computational lower bounds for this problem. Specifically, assuming the existence of one-way functions, we show that no polynomial-time $(\epsilon, 1/n^{\omega(1)})$-DP algorithm can compute a coreset for $k$-means in the $\ell_\infty$-metric for some constant $\alpha > 0$ (and some constant additive factor), even for $k=3$. For $k$-means in the Euclidean metric, we show a similar result but only for $\alpha = \Theta\left(1/d^2\right)$, where $d$ is the dimension.