Noisy k-means++ Revisited

TL;DR

This paper proves that the $k$-means++ algorithm maintains an $O( ext{log }k)$ approximation ratio even when small adversarial noise is introduced in the sampling process, closing a gap in previous research.

Contribution

It establishes a tight $O( ext{log }k)$ approximation guarantee for noisy $k$-means++ algorithms, improving upon the previous weaker bounds.

Findings

The $k$-means++ algorithm retains its $O( ext{log }k)$ approximation under noisy conditions.

Previous weaker bounds of $O( ext{log}^2 k)$ are improved to tight bounds.

The analysis confirms robustness of $k$-means++ against small adversarial perturbations.

Abstract

The $k$-means++ algorithm by Arthur and Vassilvitskii [SODA 2007] is a classical and time-tested algorithm for the $k$-means problem. While being very practical, the algorithm also has good theoretical guarantees: its solution is $O(\log k)$-approximate, in expectation. In a recent work, Bhattacharya, Eube, Roglin, and Schmidt [ESA 2020] considered the following question: does the algorithm retain its guarantees if we allow for a slight adversarial noise in the sampling probability distributions used by the algorithm? This is motivated e.g. by the fact that computations with real numbers in $k$-means++ implementations are inexact. Surprisingly, the analysis under this scenario gets substantially more difficult and the authors were able to prove only a weaker approximation guarantee of $O(\log^2 k)$. In this paper, we close the gap by providing a tight, $O(\log k)$-approximate guarantee for the $k$-means++ algorithm with noise.