Sharper Generalization Bounds for Clustering

Shaojie Li, Yong Liu
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:6392-6402, 2021.

Abstract

Existing generalization analysis of clustering mainly focuses on specific instantiations, such as (kernel) $k$-means, and a unified framework for studying clustering performance is still lacking. Besides, the existing excess clustering risk bounds are mostly of order $\mathcal{O}(K/\sqrt{n})$ provided that the underlying distribution has bounded support, where $n$ is the sample size and $K$ is the cluster numbers, or of order $\mathcal{O}(K^2/n)$ under strong assumptions on the underlying distribution, where these assumptions are hard to be verified in general. In this paper, we propose a unified clustering learning framework and investigate its excess risk bounds, obtaining state-of-the-art upper bounds under mild assumptions. Specifically, we derive sharper bounds of order $\mathcal{O}(K^2/n)$ under mild assumptions on the covering number of the hypothesis spaces, where these assumptions are easy to be verified. Moreover, for the hard clustering scheme, such as (kernel) $k$-means, if just assume the hypothesis functions to be bounded, we improve the upper bounds from the order $\mathcal{O}(K/\sqrt{n})$ to $\mathcal{O}(\sqrt{K}/\sqrt{n})$. Furthermore, state-of-the-art bounds of faster order $\mathcal{O}(K/n)$ are obtained with the covering number assumptions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-li21k, title = {Sharper Generalization Bounds for Clustering}, author = {Li, Shaojie and Liu, Yong}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {6392--6402}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/li21k/li21k.pdf}, url = {https://proceedings.mlr.press/v139/li21k.html}, abstract = {Existing generalization analysis of clustering mainly focuses on specific instantiations, such as (kernel) $k$-means, and a unified framework for studying clustering performance is still lacking. Besides, the existing excess clustering risk bounds are mostly of order $\mathcal{O}(K/\sqrt{n})$ provided that the underlying distribution has bounded support, where $n$ is the sample size and $K$ is the cluster numbers, or of order $\mathcal{O}(K^2/n)$ under strong assumptions on the underlying distribution, where these assumptions are hard to be verified in general. In this paper, we propose a unified clustering learning framework and investigate its excess risk bounds, obtaining state-of-the-art upper bounds under mild assumptions. Specifically, we derive sharper bounds of order $\mathcal{O}(K^2/n)$ under mild assumptions on the covering number of the hypothesis spaces, where these assumptions are easy to be verified. Moreover, for the hard clustering scheme, such as (kernel) $k$-means, if just assume the hypothesis functions to be bounded, we improve the upper bounds from the order $\mathcal{O}(K/\sqrt{n})$ to $\mathcal{O}(\sqrt{K}/\sqrt{n})$. Furthermore, state-of-the-art bounds of faster order $\mathcal{O}(K/n)$ are obtained with the covering number assumptions.} }
Endnote
%0 Conference Paper %T Sharper Generalization Bounds for Clustering %A Shaojie Li %A Yong Liu %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-li21k %I PMLR %P 6392--6402 %U https://proceedings.mlr.press/v139/li21k.html %V 139 %X Existing generalization analysis of clustering mainly focuses on specific instantiations, such as (kernel) $k$-means, and a unified framework for studying clustering performance is still lacking. Besides, the existing excess clustering risk bounds are mostly of order $\mathcal{O}(K/\sqrt{n})$ provided that the underlying distribution has bounded support, where $n$ is the sample size and $K$ is the cluster numbers, or of order $\mathcal{O}(K^2/n)$ under strong assumptions on the underlying distribution, where these assumptions are hard to be verified in general. In this paper, we propose a unified clustering learning framework and investigate its excess risk bounds, obtaining state-of-the-art upper bounds under mild assumptions. Specifically, we derive sharper bounds of order $\mathcal{O}(K^2/n)$ under mild assumptions on the covering number of the hypothesis spaces, where these assumptions are easy to be verified. Moreover, for the hard clustering scheme, such as (kernel) $k$-means, if just assume the hypothesis functions to be bounded, we improve the upper bounds from the order $\mathcal{O}(K/\sqrt{n})$ to $\mathcal{O}(\sqrt{K}/\sqrt{n})$. Furthermore, state-of-the-art bounds of faster order $\mathcal{O}(K/n)$ are obtained with the covering number assumptions.
APA
Li, S. & Liu, Y.. (2021). Sharper Generalization Bounds for Clustering. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:6392-6402 Available from https://proceedings.mlr.press/v139/li21k.html.

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