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On Optimal Coreset Construction for Euclidean $(k,z)$-Clustering

Lingxiao Huang, Jian Li, Xuan Wu

TL;DR

A new coreset lower bound Ω(k ε−z−2) for Euclidean (k,z)-clustering when ε ≥ Ω(k−1/(z+2)).

Abstract

Constructing small-sized coresets for various clustering problems in different metric spaces has attracted significant attention for the past decade. A central problem in the coreset literature is to understand what is the best possible coreset size for $(k,z)$-clustering in Euclidean space. While there has been significant progress in the problem, there is still a gap between the state-of-the-art upper and lower bounds. For instance, the best known upper bound for $k$-means ($z=2$) is $\min \{O(k^{3/2} \varepsilon^{-2}),O(k \varepsilon^{-4})\}$ [1,2], while the best known lower bound is $Ω(k\varepsilon^{-2})$ [1]. In this paper, we make significant progress on both upper and lower bounds. For a large range of parameters (i.e., $\varepsilon, k$), we have a complete understanding of the optimal coreset size. In particular, we obtain the following results: (1) We present a new coreset lower bound $Ω(k \varepsilon^{-z-2})$ for Euclidean $(k,z)$-clustering when $\varepsilon \geq Ω(k^{-1/(z+2)})$. In view of the prior upper bound $\tilde{O}_z(k \varepsilon^{-z-2})$ [1], the bound is optimal. The new lower bound also implies improved lower bounds for $(k,z)$-clustering in doubling metrics. (2) For the upper bound, we provide efficient coreset construction algorithms for $(k,z)$-clustering with improved or optimal coreset sizes in several metric spaces. In particular, we provide an $\tilde{O}_z(k^{\frac{2z+2}{z+2}} \varepsilon^{-2})$-sized coreset, with a unfied analysis, for $(k,z)$-clustering for all $z\geq 1$ in Euclidean space. [1] Cohen-Addad, Larsen, Saulpic, Schwiegelshohn. STOC'22. [2] Cohen-Addad, Larsen, Saulpic, Schwiegelshohn, Sheikh-Omar, NeurIPS'22.

On Optimal Coreset Construction for Euclidean $(k,z)$-Clustering

TL;DR

A new coreset lower bound Ω(k ε−z−2) for Euclidean (k,z)-clustering when ε ≥ Ω(k−1/(z+2)).

Abstract

Constructing small-sized coresets for various clustering problems in different metric spaces has attracted significant attention for the past decade. A central problem in the coreset literature is to understand what is the best possible coreset size for -clustering in Euclidean space. While there has been significant progress in the problem, there is still a gap between the state-of-the-art upper and lower bounds. For instance, the best known upper bound for -means () is [1,2], while the best known lower bound is [1]. In this paper, we make significant progress on both upper and lower bounds. For a large range of parameters (i.e., ), we have a complete understanding of the optimal coreset size. In particular, we obtain the following results: (1) We present a new coreset lower bound for Euclidean -clustering when . In view of the prior upper bound [1], the bound is optimal. The new lower bound also implies improved lower bounds for -clustering in doubling metrics. (2) For the upper bound, we provide efficient coreset construction algorithms for -clustering with improved or optimal coreset sizes in several metric spaces. In particular, we provide an -sized coreset, with a unfied analysis, for -clustering for all in Euclidean space. [1] Cohen-Addad, Larsen, Saulpic, Schwiegelshohn. STOC'22. [2] Cohen-Addad, Larsen, Saulpic, Schwiegelshohn, Sheikh-Omar, NeurIPS'22.
Paper Structure (51 sections, 23 theorems, 132 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 51 sections, 23 theorems, 132 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. $(k, z)$-Clustering.
  3. Coresets.
  4. Our Contributions
  5. Technical Overview
  6. Lower bounds
  7. Prior lower bound instances cohenaddad2022towards.
  8. A failed attempt.
  9. Our lower bound instance.
  10. Upper Bounds
  11. A new chaining argument.
  12. Bounding the covering number.
  13. Other Related Work
  14. Coresets for variants of clustering.
  15. Coresets for other problems.
  16. ...and 36 more sections

Key Result

Theorem 1.2

For any integer $k\geq 1$, constant $z\geq 1$, and error parameter $\varepsilon \geq \Omega(k^{-1/(z+2)})$, there exists an instance $P$ in Euclidean metric $\mathbb{R}^d$ (for $d = O(k \varepsilon^z)$) such that any $\varepsilon$-coreset of $P$ is of size at least $\Omega(k \varepsilon^{-z-2})$. Mo

Figures (3)

  • Figure 1: Comparison of prior coreset bounds and the new ones for Euclidean $(k, z)$-Clustering. Note that this figure is based on the case that $z>2$ in which $k^{-1/z}> k^{-0.5}$.
  • Figure 2: An example of Definition \ref{['def:group']}
  • Figure 3: An example of Definition \ref{['def:main_partition']}

Theorems & Definitions (54)

  • Definition 1.1: Coreset langberg2010universalfeldman2011unified
  • Theorem 1.2: Coreset lower bound for Euclidean $(k, z)$-Clustering
  • Corollary 1.3: Coreset lower bound for $(k, z)$-Clustering in doubling metrics
  • Theorem 1.4: Improved upper bound for Euclidean $(k, z)$-Clustering; see also Theorem \ref{['thm:Euclidean']}
  • proof
  • Claim 2.1: Weights of $S_i$
  • proof
  • Claim 2.2: Properties of $C$
  • proof
  • proof
  • ...and 44 more