Computational Hardness of Private Coreset

TL;DR

It is shown 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 $k=3$.

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 $(1 \pm α)$ 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 $(ε, 1/n^{ω(1)})$-DP algorithm can compute a coreset for $k$-means in the $\ell_\infty$-metric for some constant $α> 0$ (and some constant additive factor), even for $k=3$. For $k$-means in the Euclidean metric, we show a similar result but only for $α= Θ\left(1/d^2\right)$, where $d$ is the dimension.

Theorems & Definitions (18)

• Definition 1: Differential Privacy dwork2006calibratingDworkKMMN06
• Definition 2: Accuracy
• Definition 3: Accuracy w.r.t. Parameterized Family of Queries
• Definition 4: Promise sanitizer
• Theorem 1: UllmanV20
• Definition 5: $\ell_p$-coreset
• proof
• Theorem 2
• Theorem 3
• proof : Proof of \ref{['thm:linf-sanitizer']}