公平 $k$-均值的一个低于4的近似算法

中文摘要 AI 辅助

聚类中的公平性引起了持续的研究兴趣，其动机是在机器学习应用中确保受保护群体的公平代表性。我们研究欧几里得空间中的公平 $k$-均值聚类问题，其中每个聚类中每个受保护群体的比例必须位于指定的下界和上界之间。这些约束使得确定聚类中心和点分配变得具有挑战性。我们提出了一种近似算法，该算法将线性规划松弛与输入的几何变换相结合，以构造候选中心集。给定一个用于加权 $k$-均值的 $\rho$-近似算法和任意 $\epsilon>0$，我们的算法返回一个分数解，其成本至多为最优整数公平成本的 $1+(3-1/\Gamma)\rho+O(\epsilon)$ 倍，其中 $\Gamma\approx6.357$ 是标准欧几里得 $k$-均值线性规划积分间隙的上界。以PTAS作为子程序，近似比变为 $3.8427+O(\epsilon)$，将之前的 $5+O(\epsilon)$ 因子改进到4以下。该解精确满足所有公平性约束，并且可以舍入为整数分配，具有有界的公平性加性违反且成本不增加。相同的近似保证扩展到 $k$-稀疏Wasserstein重心问题。

英文摘要

Fairness in clustering has attracted sustained research interest, motivated by the need to ensure equitable representation of protected groups in machine learning applications. We study fair $k$-means clustering in Euclidean space, where the proportion of each protected group in every cluster must lie within specified lower and upper bounds. These constraints make it challenging to determine both cluster centers and point assignments. We propose an approximation algorithm that combines a linear programming relaxation with geometric transformations of the input to construct candidate center sets. Given a $ρ$-approximate algorithm for weighted $k$-means and any $ε>0$, our algorithm returns a fractional solution whose cost is at most $1+(3-1/Γ)ρ+O(ε)$ times the optimal integral fair cost, where $Γ\approx6.357$ is an upper bound on the integrality gap of the standard Euclidean $k$-means LP. With a PTAS as the subroutine, the approximation ratio becomes $3.8427+O(ε)$, improving the previous factor of $5+O(ε)$ to below $4$. The solution satisfies all fairness constraints exactly and can be rounded to an integral assignment with a bounded additive violation of fairness and no increase in cost. The same approximation guarantee extends to the $k$-sparse Wasserstein barycenter problem.

发表机构

• University of Science and Technology of China（中国科学技术大学）
• School of Informatics, University of Edinburgh（爱丁堡大学信息学院）

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