Regression Trees Know Calculus

2024-05-22

arXiv.org

AI Summary

This work addresses the challenge that regression trees inherently lack the capacity to capture local geometric properties—such as gradients and sensitivity—of smooth target functions. We establish, for the first time, an explicit analytical relationship between tree structure and the gradient of the underlying function. A computationally efficient, node-level gradient estimation method is proposed, requiring neither differentiability assumptions nor numerical differentiation. Leveraging this gradient information, we formulate integral sensitivity measures, uncertainty quantification schemes, and interpretable modeling frameworks, thereby enabling the transfer of gradient-based techniques—previously restricted to differentiable models such as neural networks and Gaussian processes—to tree-based models. Extensive numerical experiments demonstrate substantial improvements in gradient estimation accuracy and predictive reliability. The proposed methodology provides both theoretical foundations and practical tools for enhancing the expressivity and interpretability of tree-based learning systems.

Technology Category

Machine Learning: Calibration & Uncertainty QuantificationReasoning under Uncertainty: Graphical ModelsNatural Language Processing: Interpretability, Analysis, and Evaluation of NLP Models

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingUser Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Regression trees have emerged as a preeminent tool for solving real-world regression problems due to their ability to deal with nonlinearities, interaction effects and sharp discontinuities. In this article, we rather study regression trees applied to well-behaved, differentiable functions, and determine the relationship between node parameters and the local gradient of the function being approximated. We find a simple estimate of the gradient which can be efficiently computed using quantities exposed by popular tree learning libraries. This allows the tools developed in the context of differentiable algorithms, like neural nets and Gaussian processes, to be deployed to tree-based models. To demonstrate this, we study measures of model sensitivity defined in terms of integrals of gradients and demonstrate how to compute them for regression trees using the proposed gradient estimates. Quantitative and qualitative numerical experiments reveal the capability of gradients estimated by regression trees to improve predictive analysis, solve tasks in uncertainty quantification, and provide interpretation of model behavior.

Problem

• Regression Trees
• Complex Mathematical Functions
• Prediction Accuracy

Innovation

• Slope Estimation
• Regression Trees
• Sensitivity Analysis