Randomizing the Number of Centers in k-means++

2026-07-28

AI Summary

This work addresses the limitations of k-means++ under fixed cluster count $k$, where its worst-case expected approximation ratio is $\Theta(\log k)$, offering weak performance guarantees. The authors introduce a budget-smoothed setting in which $k$ is drawn uniformly at random from $\{K, \dots, 2K - 1\}$. Within this randomized framework, they establish—for the first time—that k-means++ achieves an $O(1)$ approximation ratio with constant probability. Through rigorous theoretical analysis grounded in probabilistic methods, the paper derives tight approximation bounds under this smoothed model, thereby overcoming the classical logarithmic barrier and demonstrating substantially improved performance compared to the worst-case behavior under a fixed $k$.

Abstract

The $k$-means++ algorithm is a standard and widely used seeding method for $k$-means clustering, but for a fixed number $k$ of centers its worst-case expected approximation ratio is $Θ(\log k)$. We consider the same algorithm when an adversary first fixes the dataset and some $K$; the number of centers $k$ is then chosen uniformly from $\{K,\ldots,2K-1\}$. We prove that $k$-means++ is an $O(1)$-approximation with constant probability in this budget-smoothed setup.

Problem

• k-means++
• approximation ratio
• randomized centers
• clustering
• budget-smoothed

Innovation

• k-means++
• randomized seeding
• approximation ratio
• budget-smoothed analysis
• clustering