Spatial properties of Bayesian unsupervised trees

Linxi Liu, Li Ma
Proceedings of Thirty Seventh Conference on Learning Theory, PMLR 247:3556-3581, 2024.

Abstract

Tree-based methods are popular nonparametric tools for capturing spatial heterogeneity and making predictions in multivariate problems. In unsupervised learning, trees and their ensembles have also been applied to a wide range of statistical inference tasks, such as multi-resolution sketching of distributional variations, localization of high-density regions, and design of efficient data compression schemes. In this paper, we study the spatial adaptation property of Bayesian tree-based methods in the unsupervised setting, with a focus on the density estimation problem. We characterize spatial heterogeneity of the underlying density function by using anisotropic Besov spaces, region-wise anisotropic Besov spaces, and two novel function classes as their extensions. For two types of commonly used prior distributions on trees under the context of unsupervised learning—the optional P{ó}lya tree (Wong and Ma, 2010) and the Dirichlet prior (Lu et al., 2013)—we calculate posterior concentration rates when the density function exhibits different types of heterogeneity. In specific, we show that the posterior concentration rate for trees is near minimax over the anisotropic Besov space. The rate is adaptive in the sense that to achieve such a rate we do not need any prior knowledge of the parameters of the Besov space.

Cite this Paper

BibTeX
@InProceedings{pmlr-v247-liu24b, title = {Spatial properties of {B}ayesian unsupervised trees}, author = {Liu, Linxi and Ma, Li}, booktitle = {Proceedings of Thirty Seventh Conference on Learning Theory}, pages = {3556--3581}, year = {2024}, editor = {Agrawal, Shipra and Roth, Aaron}, volume = {247}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--03 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v247/liu24b/liu24b.pdf}, url = {https://proceedings.mlr.press/v247/liu24b.html}, abstract = {Tree-based methods are popular nonparametric tools for capturing spatial heterogeneity and making predictions in multivariate problems. In unsupervised learning, trees and their ensembles have also been applied to a wide range of statistical inference tasks, such as multi-resolution sketching of distributional variations, localization of high-density regions, and design of efficient data compression schemes. In this paper, we study the spatial adaptation property of Bayesian tree-based methods in the unsupervised setting, with a focus on the density estimation problem. We characterize spatial heterogeneity of the underlying density function by using anisotropic Besov spaces, region-wise anisotropic Besov spaces, and two novel function classes as their extensions. For two types of commonly used prior distributions on trees under the context of unsupervised learning—the optional P{ó}lya tree (Wong and Ma, 2010) and the Dirichlet prior (Lu et al., 2013)—we calculate posterior concentration rates when the density function exhibits different types of heterogeneity. In specific, we show that the posterior concentration rate for trees is near minimax over the anisotropic Besov space. The rate is adaptive in the sense that to achieve such a rate we do not need any prior knowledge of the parameters of the Besov space.} }
Endnote
%0 Conference Paper %T Spatial properties of Bayesian unsupervised trees %A Linxi Liu %A Li Ma %B Proceedings of Thirty Seventh Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2024 %E Shipra Agrawal %E Aaron Roth %F pmlr-v247-liu24b %I PMLR %P 3556--3581 %U https://proceedings.mlr.press/v247/liu24b.html %V 247 %X Tree-based methods are popular nonparametric tools for capturing spatial heterogeneity and making predictions in multivariate problems. In unsupervised learning, trees and their ensembles have also been applied to a wide range of statistical inference tasks, such as multi-resolution sketching of distributional variations, localization of high-density regions, and design of efficient data compression schemes. In this paper, we study the spatial adaptation property of Bayesian tree-based methods in the unsupervised setting, with a focus on the density estimation problem. We characterize spatial heterogeneity of the underlying density function by using anisotropic Besov spaces, region-wise anisotropic Besov spaces, and two novel function classes as their extensions. For two types of commonly used prior distributions on trees under the context of unsupervised learning—the optional P{ó}lya tree (Wong and Ma, 2010) and the Dirichlet prior (Lu et al., 2013)—we calculate posterior concentration rates when the density function exhibits different types of heterogeneity. In specific, we show that the posterior concentration rate for trees is near minimax over the anisotropic Besov space. The rate is adaptive in the sense that to achieve such a rate we do not need any prior knowledge of the parameters of the Besov space.
APA
Liu, L. & Ma, L.. (2024). Spatial properties of Bayesian unsupervised trees. Proceedings of Thirty Seventh Conference on Learning Theory, in Proceedings of Machine Learning Research 247:3556-3581 Available from https://proceedings.mlr.press/v247/liu24b.html.

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