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Uniform Deviation Bounds for k-Means Clustering
Olivier Bachem · Mario Lucic · Hamed Hassani · Andreas Krause

Mon Aug 07 08:30 PM -- 08:48 PM (PDT) @ C4.8
in Learning theory 2 »

Uniform deviation bounds limit the difference between a model's expected loss and its loss on an empirical sample uniformly for all models in a learning problem. In this paper, we provide a novel framework to obtain uniform deviation bounds for loss functions which are unbounded. As a result, we obtain competitive uniform deviation bounds for k-Means clustering under weak assumptions on the underlying distribution. If the fourth moment is bounded, we prove a rate of O(m^(-1/2)) compared to the previously known O(m^(-1/4)) rate. Furthermore, we show that the rate also depends on the kurtosis - the normalized fourth moment which measures the "tailedness" of a distribution. We also provide improved rates under progressively stronger assumptions, namely, bounded higher moments, subgaussianity and bounded support of the underlying distribution.

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Author Information

Olivier Bachem (ETH Zurich)
Mario Lucic (ETH Zurich)
Hamed Hassani (ETH Zurich)

I am an assistant professor in the Department of Electrical and Systems Engineering (as of July 2017). I hold a secondary appointment in the Department of Computer and Information Systems. I am also a faculty affiliate of the Warren Center for Network and Data Sciences. Before joining Penn, I was a research fellow at the Simons Institute, UC Berkeley (program: Foundations of Machine Learning). Prior to that, I was a post-doctoral scholar and lecturer in the Institute for Machine Learning at ETH Zürich. I received my Ph.D. degree in Computer and Communication Sciences from EPFL.

Andreas Krause (ETH Zurich)
Andreas Krause

Andreas Krause is a Professor of Computer Science at ETH Zurich, where he leads the Learning & Adaptive Systems Group. He also serves as Academic Co-Director of the Swiss Data Science Center and Chair of the ETH AI Center, and co-founded the ETH spin-off LatticeFlow. Before that he was an Assistant Professor of Computer Science at Caltech. He received his Ph.D. in Computer Science from Carnegie Mellon University (2008) and his Diplom in Computer Science and Mathematics from the Technical University of Munich, Germany (2004). He is a Max Planck Fellow at the Max Planck Institute for Intelligent Systems, an ELLIS Fellow, a Microsoft Research Faculty Fellow and a Kavli Frontiers Fellow of the US National Academy of Sciences. He received the Rössler Prize, ERC Starting Investigator and ERC Consolidator grants, the German Pattern Recognition Award, an NSF CAREER award as well as the ETH Golden Owl teaching award. His research has received awards at several premier conferences and journals, including the ACM SIGKDD Test of Time award 2019 and the ICML Test of Time award 2020. Andreas Krause served as Program Co-Chair for ICML 2018, and currently serves as General Chair for ICML 2023 and as Action Editor for the Journal of Machine Learning Research.

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