Computational Hardness of Private Coreset
Abstract
In this work, we prove computational lower bounds against differentially private (DP) coreset. Specifically, assuming the existence of one-way functions, we show that no polynomial-time $(\epsilon, 1/n^{\omega(1)})$-DP algorithm can compute $(\alpha, \beta)$-coreset for $k$-means in the $\ell_\infty$ metric for some constant $\alpha > 1$. For the Euclidean metric, we show a similar result but only for $\alpha = 1 + \Theta\left(\frac{1}{d^2}\right)$ where $d$ is the dimension.
