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arXiv · 2506.06990

Modified K-means Algorithm with Local Optimality Guarantees

Mingyi Li·Michael R. Metel·Akiko Takeda
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Abstract

The K-means algorithm is one of the most widely studied clustering algorithms in machine learning. While extensive research has focused on its ability to achieve a globally optimal solution, there still lacks a rigorous analysis of its local optimality guarantees. In this paper, we first present conditions under which the K-means algorithm converges to a locally optimal solution. Based on this, we propose simple modifications to the K-means algorithm which ensure local optimality in both the continuous and discrete sense, with the same computational complexity as the original K-means algorithm. As the dissimilarity measure, we consider a general Bregman divergence, which is an extension of the squared Euclidean distance often used in the K-means algorithm. Numerical experiments confirm that the K-means algorithm does not always find a locally optimal solution in practice, while our proposed methods provide improved locally optimal solutions with reduced clustering loss. Our code is available at https://github.com/lmingyi/LO-K-means.

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Mingyi Li, Michael R. Metel, Akiko Takeda. 2025-06-11. Modified K-means Algorithm with Local Optimality Guarantees. https://arxiv.org/abs/2506.06990

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