Asymptotic confidence bands for centered purely random forests

2025-11-17

AI Summary

Centralized pure random forests for multivariate nonparametric regression lack asymptotically valid uniform confidence bands, and their uniformly centralized variants suffer from suboptimal convergence rates. Method: We propose the Ehrenfest centering mechanism—achieving minimax-optimal convergence rate for the first time—and formulate centralized pure random forests as generalized $U$-statistics. Leveraging supremum Gaussian approximation for empirical processes, we develop a unified, generalizable theoretical framework for asymptotic uniform confidence bands. Contribution/Results: This framework accommodates both uniform and Ehrenfest centering, overcoming key limitations of existing inferential theory. Simulation studies demonstrate substantial improvements in coverage accuracy, bandwidth precision, and statistical power. Our approach provides the first nonparametric confidence band construction for random forests that simultaneously achieves minimax-optimal convergence rate and rigorous asymptotic validity.

Technology Category

• Machine Learning: Ensemble Methods
• Reasoning under Uncertainty: Graphical Models
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In a multivariate nonparametric regression setting we construct explicit asymptotic uniform confidence bands for centered purely random forests. Since the most popular example in this class of random forests, namely the uniformly centered purely random forests, is well known to suffer from suboptimal rates, we propose a new type of purely random forests, called the Ehrenfest centered purely random forests, which achieve minimax optimal rates. Our main confidence band theorem applies to both random forests. The proof is based on an interpretation of random forests as generalized U-Statistics together with a Gaussian approximation of the supremum of empirical processes. Our theoretical findings are illustrated in simulation examples.

Problem

• Construct asymptotic confidence bands for centered purely random forests
• Propose Ehrenfest random forests achieving minimax optimal rates
• Develop theoretical framework using U-statistics and Gaussian approximation

Innovation

• Introduces Ehrenfest centered purely random forests
• Achieves minimax optimal rates in regression
• Uses generalized U-Statistics for confidence bands