Pointwise Consistency of Regression Trees and the Bias-Variance Trade-Off Associated with Tree Size

Core Concepts

Regression trees can be made pointwise consistent by controlling the minimum number of observations in each cell. This leads to a bias-variance trade-off associated with tree size, where small trees are biased but have low variance, while large trees are unbiased but have high variance. Subagging can improve the variance of consistent trees without affecting the bias.

Abstract

The paper studies the effectiveness of regression trees, a popular non-parametric method in machine learning, and the subagging (subsample aggregating) technique for improving their performance. Key highlights: Pointwise consistency of regression trees: The authors establish sufficient conditions for pointwise consistency of regression trees, showing that the bias depends on the diameter of cells and the variance depends on the number of observations in cells. They provide an algorithm that satisfies the consistency assumptions by controlling the minimum and maximum number of observations in each cell. Bias-variance trade-off and tree size: The authors illustrate the bias-variance trade-off associated with tree size through simulations. Small trees tend to be biased but have low variance, while large trees are unbiased but have high variance. Trees grown under the consistency conditions strike a balance between bias and variance. Subagging consistent trees: The authors show that subagging consistent (and hence stable) trees does not affect the bias but can improve the variance, as the subagged estimator averages over more observations compared to the original tree. Subagging small trees: The authors analyze the effect of subagging on stumps (single-split trees) as a proxy for small trees. They show that subagging increases the number of distinct observations used to estimate the target and covers a wider part of the feature space compared to a single tree. Subagging also reduces the variance around the split point, where a single tree has high variance. Optimal tree size: The authors find that a single tree grown at the optimal size can outperform subagging if the size of its individual subtrees is not optimally chosen. This suggests that subagging large trees is not always a good idea, and that the optimal size for the ensemble method should be determined based on the optimal size for a single tree.

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When does Subagging Work?

by Christos Rev... at arxiv.org 04-03-2024

https://arxiv.org/pdf/2404.01832.pdf

Deeper Inquiries

How can the bias-variance trade-off associated with tree size be formally quantified and optimized in practice

The bias-variance trade-off associated with tree size can be formally quantified and optimized by considering the relationship between the diameter of cells and the number of observations in each cell. In the context of decision trees, smaller cells with more observations tend to have lower bias but higher variance, while larger cells with fewer observations tend to have higher bias but lower variance. This trade-off can be quantified by analyzing the squared bias and variance of the tree estimator at a given point of interest. By controlling the size of cells and the number of observations, one can optimize the trade-off to achieve a balance between bias and variance. In practice, this optimization can be achieved by choosing appropriate stopping rules, such as setting bounds on the number of observations in each cell or controlling the number of splits in the tree.

What are the implications of the authors' findings for the common practice of growing large randomized trees to eliminate bias and then averaging to reduce variance

The findings of the authors have significant implications for the common practice of growing large randomized trees to eliminate bias and then averaging to reduce variance. The study shows that growing large trees can lead to lower bias but higher variance, while averaging multiple trees through techniques like subagging can reduce variance without significantly affecting bias. This challenges the traditional approach of growing large trees in ensemble methods like bagging and random forests. The study suggests that optimizing the size of individual trees and using techniques like subagging can lead to better overall performance compared to simply growing large trees. Therefore, practitioners should consider the bias-variance trade-off and the effectiveness of ensemble methods in reducing variance when designing their tree-based models.

How do the insights from this study on regression trees extend to other tree-based methods, such as classification trees or decision forests

The insights from this study on regression trees can be extended to other tree-based methods, such as classification trees or decision forests. The fundamental principles of the bias-variance trade-off, consistency, and stability apply across different types of tree-based methods. For classification trees, the trade-off between bias and variance remains crucial in determining the performance of the model. Decision forests, which are ensembles of trees, can benefit from the optimization strategies discussed in the study, such as controlling the size of trees and using techniques like subagging to improve overall performance. By understanding the implications of tree size on bias and variance, practitioners can apply similar principles to enhance the performance of classification trees and decision forests in various machine learning tasks.