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.

翻译：回归树因其处理非线性、交互效应及尖锐间断性的能力，已成为解决现实世界回归问题的卓越工具。本文则研究回归树应用于性质良好、可微函数时的情况，并探究节点参数与被逼近函数局部梯度之间的关系。我们提出一种简单的梯度估计方法，可利用主流树学习库公开的计算量高效实现。这使得在可微算法（如神经网络和高斯过程）背景下开发的工具能够应用于基于树的模型。为验证此方法，我们研究了基于梯度积分定义的模型敏感度度量，并演示如何利用所提出的梯度估计为回归树计算这些度量。定量与定性的数值实验表明，回归树估计的梯度能够改进预测分析、解决不确定性量化任务，并为模型行为提供解释。