Can Evolutionary Clustering Have Theoretical Guarantees?

TL;DR

This paper provides the first theoretical guarantees for the approximation performance of a simple evolutionary algorithm, GSEMO, across multiple clustering formulations including fairness considerations, filling a significant gap in the understanding of evolutionary clustering methods.

Contribution

It proves that GSEMO can theoretically guarantee approximation performance for various clustering problems, including fairness-aware clustering, which was previously only supported by empirical results.

Findings

GSEMO achieves theoretical approximation guarantees for $k$-tMM, $k$-center, $k$-median, and $k$-means clustering.

GSEMO's performance guarantees extend to fairness-aware clustering under individual fairness constraints.

The work bridges the gap between empirical success and theoretical understanding of evolutionary clustering algorithms.

Abstract

Clustering is a fundamental problem in many areas, which aims to partition a given data set into groups based on some distance measure, such that the data points in the same group are similar while that in different groups are dissimilar. Due to its importance and NP-hardness, a lot of methods have been proposed, among which evolutionary algorithms are a class of popular ones. Evolutionary clustering has found many successful applications, but all the results are empirical, lacking theoretical support. This paper fills this gap by proving that the approximation performance of the GSEMO (a simple multi-objective evolutionary algorithm) for solving four formulations of clustering, i.e., $k$-tMM, $k$-center, discrete $k$-median and $k$-means, can be theoretically guaranteed. Furthermore, we consider clustering under fairness, which tries to avoid algorithmic bias, and has recently been an important research topic in machine learning. We prove that for discrete $k$-median clustering under individual fairness, the approximation performance of the GSEMO can be theoretically guaranteed with respect to both the objective function and the fairness constraint.