Consistency of Honest Decision Trees and Random Forests

2026-01-21

AI Summary

This study investigates the weak, almost sure, and uniform convergence of honest decision trees and random forests in regression tasks. By constructing a unified analytical framework based on honest tree structures, subsampling ensembles, and a two-stage bootstrap procedure—combined with classical nonparametric statistical convergence techniques—the authors establish an asymptotic approximation theory for the estimator’s convergence to the true regression function under compact covariate domains and mild regularity conditions. Employing an elementary and self-contained proof strategy, this work clarifies the intrinsic connection between data-adaptive partitioning and kernel methods, thereby not only simplifying existing theoretical results but also extending their applicability.

Technology Category

• Machine Learning: Ensemble Methods
• Reasoning under Uncertainty: Decision/Utility Theory
• Search and Optimization: Non-convex Optimization

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We study various types of consistency of honest decision trees and random forests in the regression setting. In contrast to related literature, our proofs are elementary and follow the classical arguments used for smoothing methods. Under mild regularity conditions on the regression function and data distribution, we establish weak and almost sure convergence of honest trees and honest forest averages to the true regression function, and moreover we obtain uniform convergence over compact covariate domains. The framework naturally accommodates ensemble variants based on subsampling and also a two-stage bootstrap sampling scheme. Our treatment synthesizes and simplifies existing analyses, in particular recovering several results as special cases. The elementary nature of the arguments clarifies the close relationship between data-adaptive partitioning and kernel-type methods, providing an accessible approach to understanding the asymptotic behavior of tree-based methods.

Problem

• consistency
• honest decision trees
• random forests
• regression
• asymptotic behavior

Innovation

• honest decision trees
• random forests
• consistency
• uniform convergence
• asymptotic analysis