[Paper Review] Towards Optimal Lower Bounds for k-median and k-means Coresets

Vincent Cohen-Addad, Kasper Green Larsen+2 more|arXiv (Cornell University)|Feb 25, 2022

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TL;DR

This paper establishes tight lower bounds for coreset sizes in k-median and k-means clustering across various metric spaces, proving that any coreset must contain at least Ω(k log n / ε²) points in general finite metrics, Ω(kD / ε²) in doubling metrics with doubling dimension D, and Ω(k / ε²) in d-dimensional Euclidean space. It complements these bounds with a novel coreset construction achieving Õ(k / ε² · min(ε⁻ᶻ, k)) size, breaking the ε⁻⁴ barrier and matching the lower bounds up to polylogarithmic factors in Euclidean space.

ABSTRACT

Given a set of points in a metric space, the $(k,z)$-clustering problem consists of finding a set of $k$ points called centers, such that the sum of distances raised to the power of $z$ of every data point to its closest center is minimized. Special cases include the famous k-median problem ($z = 1$) and k-means problem ($z = 2$). The $k$-median and $k$-means problems are at the heart of modern data analysis and massive data applications have given raise to the notion of coreset: a small (weighted) subset of the input point set preserving the cost of any solution to the problem up to a multiplicative $(1 \pm \varepsilon)$ factor, hence reducing from large to small scale the input to the problem. In this paper, we present improved lower bounds for coresets in various metric spaces. In finite metrics consisting of $n$ points and doubling metrics with doubling constant $D$, we show that any coreset for $(k,z)$ clustering must consist of at least $Ω(k \varepsilon^{-2} \log n)$ and $Ω(k \varepsilon^{-2} D)$ points, respectively. Both bounds match previous upper bounds up to polylog factors. In Euclidean spaces, we show that any coreset for $(k,z)$ clustering must consists of at least $Ω(k\varepsilon^{-2})$ points. We complement these lower bounds with a coreset construction consisting of at most $ ilde{O}(k\varepsilon^{-2}\cdot \min(\varepsilon^{-z},k))$ points.

Motivation & Objective

• To close the gap between known upper and lower bounds on coreset size for k-median and k-means clustering in diverse metric spaces.
• To establish the first tight lower bounds for k-median and k-means coresets in d-dimensional Euclidean space, showing Ω(k / ε²) is necessary.
• To develop a new coreset construction with optimal ε-dependence, breaking the ε⁻⁴ barrier in Euclidean k-median and k-means.
• To provide tight bounds in doubling metrics and general finite metrics, matching existing upper bounds up to polylogarithmic factors.
• To introduce a novel chaining-based analysis technique that may be of independent interest in coreset and approximation algorithm design.

Proposed method

• Derives lower bounds via a reduction to a hard instance construction in finite metrics, using probabilistic and geometric arguments to show that any coreset must include Ω(k log n / ε²) points.
• Applies a similar construction in doubling metrics, proving that Ω(kD / ε²) points are required, matching known upper bounds up to logarithmic factors.
• In Euclidean space, establishes a lower bound of Ω(k / ε²) for (k,z)-clustering, improving over the prior Ω(k / √ε) bound for k-median.
• Proposes a new coreset construction using a novel chaining-based analysis to bound approximation error, achieving size Õ(k / ε² · min(ε⁻ᶻ, k)).
• Employs a two-part error analysis: one for far points (using concentration bounds) and one for nearby points (via geometric and cost ratio arguments).
• Uses a recursive decomposition of the input space into clusters and applies Markov-type inequalities to control the relative error in cost approximation.

Experimental results

Research questions

• RQ1 What is the optimal lower bound on the size of a coreset for k-median and k-means clustering in general finite metric spaces?
• RQ2 How does the doubling dimension D of a metric space affect the minimum coreset size required for (k,z)-clustering?
• RQ3 What is the tightest possible lower bound for coreset size in d-dimensional Euclidean space for k-median and k-means?
• RQ4 Can a coreset construction be designed with optimal dependence on ε in Euclidean space, breaking the ε⁻⁴ barrier?
• RQ5 What novel analytical techniques can be used to prove coreset approximation guarantees with improved dependence on ε and k?

Key findings

• In general finite metric spaces with n points, any (ε,k,z)-coreset must contain at least Ω(k log n / ε²) points, matching the best-known upper bounds up to polylogarithmic factors.
• For doubling metrics with doubling dimension D, any coreset must have size Ω(kD / ε²), which matches the known upper bounds for k-median and k-means up to logarithmic factors.
• In d-dimensional Euclidean space, the paper proves a lower bound of Ω(k / ε²) for (k,z)-clustering, improving over the previous Ω(k / √ε) bound for k-median.
• The authors construct a coreset of size Õ(k / ε² · min(ε⁻ᶻ, k)) for (k,z)-clustering in Euclidean space, which improves upon the prior Õ(k²ε⁻⁴) and Õ(kε⁻⁴) bounds.
• This construction breaks the long-standing ε⁻⁴ barrier in coreset size for k-median and k-means in Euclidean space, achieving optimal ε-dependence.
• Together, the lower and upper bounds for k-median in Euclidean space are tight up to a factor of O(ε⁻¹ polylog k / ε), resolving the asymptotic dependency on ε up to logarithmic factors.