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.

翻译：本文研究回归设定下诚实决策树与随机森林的多种一致性。与相关文献不同，我们的证明采用初等方法，遵循平滑方法中的经典论证。在回归函数与数据分布的温和正则性条件下，我们建立了诚实树及诚实森林平均值弱收敛与几乎必然收敛于真实回归函数，并进一步获得了紧致协变量域上的一致收敛性。该框架自然地适用于基于子采样的集成变体以及两阶段自助采样方案。我们的分析综合并简化了现有研究，特别将若干结果恢复为特例。论证的初等性揭示了数据自适应划分与核方法之间的紧密联系，为理解基于树的方法的渐近行为提供了可入门的途径。