Computational Hardness of Private Coreset

2026-02-19

AI Summary

This work investigates the computational feasibility of efficiently constructing coresets for the $k$-means problem under differential privacy constraints. Under standard cryptographic assumptions—such as the existence of one-way functions—it establishes, for the first time, that even for $k=3$, no polynomial-time algorithm can produce a private coreset with a constant approximation factor in the $\ell_\infty$ metric. Furthermore, in Euclidean space, the paper provides dimension-dependent lower bounds on the achievable approximation quality. These results reveal a fundamental tension between privacy preservation and efficient approximation, thereby establishing strong computational complexity lower bounds for the construction of differentially private coresets.

Technology Category

• Machine Learning: Privacy
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• differentially private
• coreset
• k-means
• computational hardness
• one-way functions

Innovation

• differentially private
• coreset
• computational hardness
• k-means
• one-way functions