An interpretable Good--Turing restart criterion for k-means++

2026-07-09

AI Summary

This work addresses the limitation of k-means++ with a fixed number of restarts, which often leads to either excessive computation on easy datasets or insufficient optimization on difficult ones. To overcome this, the authors propose the Good-Turing Restart Criterion (GTRC), a novel adaptive stopping mechanism that, for the first time, integrates Good-Turing frequency estimation into k-means++ restart decisions. GTRC combines an unconditional error bound with an upper confidence bound to yield a data-driven and interpretable criterion for terminating restarts. Experimental results across 36 datasets demonstrate that GTRC achieves clustering quality comparable to that obtained with the best fixed restart count while substantially reducing computational overhead on simpler datasets and automatically allocating more restarts to challenging ones.

Abstract

The k-means++ algorithm is commonly restarted multiple times to avoid poor local optima, yet the number of restarts is almost always chosen arbitrarily and applied uniformly regardless of data set difficulty. This undermines any comparison relying on such a choice and wastes computation on easy data sets while potentially under-serving hard ones. We introduce GTRC, a restart criterion combining a Good-Turing estimate, a proven unconditional bound, and a confidence-based bound on the probability that a further restart would improve on the current result, stopping once this probability falls below a user-specified tolerance $\varepsilon$. Across 36 data sets, GTRC reached clustering quality competitive with well-chosen fixed restart counts, while the number of restarts used varied considerably and appropriately with data set difficulty, governed by an interpretable, data-dependent signal rather than a fixed rule. GTRC offers a principled and reportable alternative to fixing the number of $k$-means++ restarts in advance. Software:https://github.com/RCdeAmorim/Good-Turing-Restart-Criterion.

Problem

• k-means++
• restart criterion
• clustering
• Good-Turing
• local optima

Innovation

• Good-Turing estimate
• k-means++ restart criterion
• adaptive stopping
• clustering quality
• data-dependent restart