Computational Hardness of Private Coreset

TL;DR

This paper establishes the first computational lower bounds for differentially private coreset construction for k-means, showing that such coresets are computationally hard to compute under standard cryptographic assumptions.

Contribution

It proves that no polynomial-time differentially private algorithm can compute approximate coresets for k-means in certain metrics, assuming one-way functions exist, highlighting fundamental computational limitations.

Findings

No polynomial-time DP algorithms for k-means coreset in $ ext{ell}_ ext{infty}$-metric.

Hardness results for Euclidean k-means coresets with dimension-dependent approximation.

Assumes existence of one-way functions for the hardness proofs.

Abstract

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.