Distributed Center-Based Clustering: A Unified Framework

We develop a family of distributed center-based clustering algorithms that work over connected networks of users. In the proposed scenario, users contain a local dataset and communicate only with their immediate neighbours, with the aim of finding a clustering of the full, joint data. The proposed f...

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Published in:IEEE transactions on signal processing Vol. 73; pp. 903 - 918
Main Authors: Armacki, Aleksandar, Bajovic, Dragana, Jakovetic, Dusan, Kar, Soummya
Format: Journal Article
Language:English
Published: New York IEEE 2025
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Clustering
Clustering algorithms
consensus
Convergence
Costs
Distributed databases
Europe
first-order methods
fixed points
Logistics
networks
peer-to-peer
Peer-to-peer computing
Privacy
Servers
Signal processing algorithms
ISSN:1053-587X, 1941-0476
Online Access:Get full text
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Abstract We develop a family of distributed center-based clustering algorithms that work over connected networks of users. In the proposed scenario, users contain a local dataset and communicate only with their immediate neighbours, with the aim of finding a clustering of the full, joint data. The proposed family, termed Distributed Gradient Clustering (DGC-<inline-formula><tex-math notation="LaTeX">\mathcal{F}_{\rho}</tex-math></inline-formula>), is parametrized by <inline-formula><tex-math notation="LaTeX">\rho\geq 1</tex-math></inline-formula>, controlling the proximity of users' center estimates, with <inline-formula><tex-math notation="LaTeX">\mathcal{F}</tex-math></inline-formula> determining the clustering loss. Our framework allows for a broad class of smooth convex loss functions, including popular clustering losses like <inline-formula><tex-math notation="LaTeX">K</tex-math></inline-formula>-means and Huber loss. Specialized to <inline-formula><tex-math notation="LaTeX">K</tex-math></inline-formula>-means and Huber loss, DGC-<inline-formula><tex-math notation="LaTeX">\mathcal{F}_{\rho}</tex-math></inline-formula> gives rise to novel distributed clustering algorithms DGC-KM<inline-formula><tex-math notation="LaTeX">{}_{\rho}</tex-math></inline-formula> and DGC-HL<inline-formula><tex-math notation="LaTeX">{}_{\rho}</tex-math></inline-formula>, while novel clustering losses based on the logistic and fair loss lead to DGC-LL<inline-formula><tex-math notation="LaTeX">{}_{\rho}</tex-math></inline-formula> and DGC-FL<inline-formula><tex-math notation="LaTeX">{}_{\rho}</tex-math></inline-formula>. We provide a unified analysis and establish several strong results, under mild assumptions. First, the sequence of centers generated by the methods converges to a well-defined notion of fixed point, under any center initialization and value of <inline-formula><tex-math notation="LaTeX">\rho</tex-math></inline-formula>. Second, as <inline-formula><tex-math notation="LaTeX">\rho</tex-math></inline-formula> increases, the family of fixed points produced by DGC-<inline-formula><tex-math notation="LaTeX">\mathcal{F}_{\rho}</tex-math></inline-formula> converges to a notion of consensus fixed points. We show that consensus fixed points of DGC-<inline-formula><tex-math notation="LaTeX">\mathcal{F}_{\rho}</tex-math></inline-formula> are equivalent to fixed points of gradient clustering over the full data, guaranteeing a clustering of the full data is produced. For the special case of Bregman losses, we show that our fixed points converge to the set of Lloyd points. Numerical experiments on real data confirm our theoretical findings and demonstrate strong performance of the methods.
AbstractList We develop a family of distributed center-based clustering algorithms that work over connected networks of users. In the proposed scenario, users contain a local dataset and communicate only with their immediate neighbours, with the aim of finding a clustering of the full, joint data. The proposed family, termed Distributed Gradient Clustering (DGC-[Formula Omitted]), is parametrized by [Formula Omitted], controlling the proximity of users’ center estimates, with [Formula Omitted] determining the clustering loss. Our framework allows for a broad class of smooth convex loss functions, including popular clustering losses like [Formula Omitted]-means and Huber loss. Specialized to [Formula Omitted]-means and Huber loss, DGC-[Formula Omitted] gives rise to novel distributed clustering algorithms DGC-KM[Formula Omitted] and DGC-HL[Formula Omitted], while novel clustering losses based on the logistic and fair loss lead to DGC-LL[Formula Omitted] and DGC-FL[Formula Omitted]. We provide a unified analysis and establish several strong results, under mild assumptions. First, the sequence of centers generated by the methods converges to a well-defined notion of fixed point, under any center initialization and value of [Formula Omitted]. Second, as [Formula Omitted] increases, the family of fixed points produced by DGC-[Formula Omitted] converges to a notion of consensus fixed points. We show that consensus fixed points of DGC-[Formula Omitted] are equivalent to fixed points of gradient clustering over the full data, guaranteeing a clustering of the full data is produced. For the special case of Bregman losses, we show that our fixed points converge to the set of Lloyd points. Numerical experiments on real data confirm our theoretical findings and demonstrate strong performance of the methods.
We develop a family of distributed center-based clustering algorithms that work over connected networks of users. In the proposed scenario, users contain a local dataset and communicate only with their immediate neighbours, with the aim of finding a clustering of the full, joint data. The proposed family, termed Distributed Gradient Clustering (DGC-<inline-formula><tex-math notation="LaTeX">\mathcal{F}_{\rho}</tex-math></inline-formula>), is parametrized by <inline-formula><tex-math notation="LaTeX">\rho\geq 1</tex-math></inline-formula>, controlling the proximity of users' center estimates, with <inline-formula><tex-math notation="LaTeX">\mathcal{F}</tex-math></inline-formula> determining the clustering loss. Our framework allows for a broad class of smooth convex loss functions, including popular clustering losses like <inline-formula><tex-math notation="LaTeX">K</tex-math></inline-formula>-means and Huber loss. Specialized to <inline-formula><tex-math notation="LaTeX">K</tex-math></inline-formula>-means and Huber loss, DGC-<inline-formula><tex-math notation="LaTeX">\mathcal{F}_{\rho}</tex-math></inline-formula> gives rise to novel distributed clustering algorithms DGC-KM<inline-formula><tex-math notation="LaTeX">{}_{\rho}</tex-math></inline-formula> and DGC-HL<inline-formula><tex-math notation="LaTeX">{}_{\rho}</tex-math></inline-formula>, while novel clustering losses based on the logistic and fair loss lead to DGC-LL<inline-formula><tex-math notation="LaTeX">{}_{\rho}</tex-math></inline-formula> and DGC-FL<inline-formula><tex-math notation="LaTeX">{}_{\rho}</tex-math></inline-formula>. We provide a unified analysis and establish several strong results, under mild assumptions. First, the sequence of centers generated by the methods converges to a well-defined notion of fixed point, under any center initialization and value of <inline-formula><tex-math notation="LaTeX">\rho</tex-math></inline-formula>. Second, as <inline-formula><tex-math notation="LaTeX">\rho</tex-math></inline-formula> increases, the family of fixed points produced by DGC-<inline-formula><tex-math notation="LaTeX">\mathcal{F}_{\rho}</tex-math></inline-formula> converges to a notion of consensus fixed points. We show that consensus fixed points of DGC-<inline-formula><tex-math notation="LaTeX">\mathcal{F}_{\rho}</tex-math></inline-formula> are equivalent to fixed points of gradient clustering over the full data, guaranteeing a clustering of the full data is produced. For the special case of Bregman losses, we show that our fixed points converge to the set of Lloyd points. Numerical experiments on real data confirm our theoretical findings and demonstrate strong performance of the methods.
Author Kar, Soummya
Bajovic, Dragana
Jakovetic, Dusan
Armacki, Aleksandar
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Snippet We develop a family of distributed center-based clustering algorithms that work over connected networks of users. In the proposed scenario, users contain a...
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SubjectTerms Algorithms
Clustering
Clustering algorithms
consensus
Convergence
Costs
Distributed databases
Europe
first-order methods
fixed points
Logistics
networks
peer-to-peer
Peer-to-peer computing
Privacy
Servers
Signal processing algorithms
Title Distributed Center-Based Clustering: A Unified Framework
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