Almost Tight Approximation Algorithms for Explainable Clustering

TL;DR

This paper develops nearly optimal approximation algorithms for explainable clustering, specifically for $k$-medians and $k$-means, improving previous bounds and establishing tight lower bounds, especially in low-dimensional spaces.

Contribution

It introduces nearly tight approximation algorithms for explainable $k$-medians and $k$-means, improving bounds and providing bounds that are close to optimal.

Findings

An $O( ext{log} k ext{ log log} k)$-approximation for explainable $k$-medians.

An $O(d ext{ log}^2 d)$-approximation for low-dimensional explainable $k$-medians.

An $O(k ext{ log} k)$-approximation for explainable $k$-means.

Abstract

Recently, due to an increasing interest for transparency in artificial intelligence, several methods of explainable machine learning have been developed with the simultaneous goal of accuracy and interpretability by humans. In this paper, we study a recent framework of explainable clustering first suggested by Dasgupta et al.~\cite{dasgupta2020explainable}. Specifically, we focus on the $k$-means and $k$-medians problems and provide nearly tight upper and lower bounds. First, we provide an $O(\log k \log \log k)$-approximation algorithm for explainable $k$-medians, improving on the best known algorithm of $O(k)$~\cite{dasgupta2020explainable} and nearly matching the known $\Omega(\log k)$ lower bound~\cite{dasgupta2020explainable}. In addition, in low-dimensional spaces $d \ll \log k$, we show that our algorithm also provides an $O(d \log^2 d)$-approximate solution for explainable $k$-medians. This improves over the best known bound of $O(d \log k)$ for low dimensions~\cite{laber2021explainable}, and is a constant for constant dimensional spaces. To complement this, we show a nearly matching $\Omega(d)$ lower bound. Next, we study the $k$-means problem in this context and provide an $O(k \log k)$-approximation algorithm for explainable $k$-means, improving over the $O(k^2)$ bound of Dasgupta et al. and the $O(d k \log k)$ bound of \cite{laber2021explainable}. To complement this we provide an almost tight $\Omega(k)$ lower bound, improving over the $\Omega(\log k)$ lower bound of Dasgupta et al. Given an approximate solution to the classic $k$-means and $k$-medians, our algorithm for $k$-medians runs in time $O(kd \log^2 k )$ and our algorithm for $k$-means runs in time $ O(k^2 d)$.