Near-Optimal Explainable $k$-Means for All Dimensions

TL;DR

This paper presents an efficient algorithm for explainable $k$-means clustering in all dimensions, achieving near-optimal cost bounds and significantly improving over previous methods, especially in two dimensions.

Contribution

The authors develop a new algorithm that guarantees near-optimal explainable $k$-means clustering costs across all dimensions, improving theoretical bounds and practical performance.

Findings

Achieves $k^{1 - 2/d} ext{poly}(d ext{log}k)$ approximation ratio.

Improves the previous bound to $k^{1 - 2/d} ext{polylog}(k)$, which is near-optimal.

For $d=2$, obtains an $O( ext{log}k ext{log} ext{log}k)$ bound, exponentially better than previous work.

Abstract

Many clustering algorithms are guided by certain cost functions such as the widely-used $k$-means cost. These algorithms divide data points into clusters with often complicated boundaries, creating difficulties in explaining the clustering decision. In a recent work, Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020) introduced explainable clustering, where the cluster boundaries are axis-parallel hyperplanes and the clustering is obtained by applying a decision tree to the data. The central question here is: how much does the explainability constraint increase the value of the cost function? Given $d$-dimensional data points, we show an efficient algorithm that finds an explainable clustering whose $k$-means cost is at most $k^{1 - 2/d}\,\mathrm{poly}(d\log k)$ times the minimum cost achievable by a clustering without the explainability constraint, assuming $k,d\ge 2$. Taking the minimum of this bound and the $k\,\mathrm{polylog} (k)$ bound in independent work by Makarychev-Shan (ICML 2021), Gamlath-Jia-Polak-Svensson (2021), or Esfandiari-Mirrokni-Narayanan (2021), we get an improved bound of $k^{1 - 2/d}\,\mathrm{polylog}(k)$, which we show is optimal for every choice of $k,d\ge 2$ up to a poly-logarithmic factor in $k$. For $d = 2$ in particular, we show an $O(\log k\log\log k)$ bound, improving near-exponentially over the previous best bound of $O(k\log k)$ by Laber and Murtinho (ICML 2021).