Convergence Rates of Oblique Regression Trees for Flexible Function Libraries
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灵活函数库下斜向回归树的收敛率
- 研究问题
- 按协变量线性组合分割的斜向回归树，能否对丰富回归函数库获得理论误差保障，并解释其预测精度与可解释性的关系？
- 方法与思路
- 作者分析递归最小化平方误差的斜向树，利用自适应分割与序贯贪心凸优化的联系，建立预言机不等式并研究斜向森林。
- 材料陈述的贡献
- 摘要称斜向树能适应由脊函数线性组合及其极限构成的函数库，在适当条件下达到与针对同一函数库的神经网络相近的精度。
- 阅读关注点
- 应核对函数库与误差度量、树复杂度和样本量的关系、近似最优分割的计算要求，以及与神经网络比较所需的共同条件。
- 阅读边界
- 这是 arXiv 预印本，同行评审或期刊接收未经核验。摘要未列出具体速率及比较条件，不能据此认定斜向树普遍优于其他预测方法。
原文摘要
We develop a theoretical framework for the analysis of oblique decision trees, where the splits at each decision node occur at linear combinations of the covariates (as opposed to conventional tree constructions that force axis-aligned splits involving only a single covariate). While this methodology has garnered significant attention from the computer science and optimization communities since the mid-80s, the advantages they offer over their axis-aligned counterparts remain only empirically justified, and explanations for their success are largely based on heuristics. Filling this long-standing gap between theory and practice, we show that oblique regression trees (constructed by recursively minimizing squared error) satisfy a type of oracle inequality and can adapt to a rich library of regression models consisting of linear combinations of ridge functions and their limit points. This provides a quantitative baseline to compare and contrast decision trees with other less interpretable methods, such as projection pursuit regression and neural networks, which target similar model forms. Contrary to popular belief, one need not always trade-off interpretability with accuracy. Specifically, we show that, under suitable conditions, oblique decision trees achieve similar predictive accuracy as neural networks for the same library of regression models. To address the combinatorial complexity of finding the optimal splitting hyperplane at each decision node, our proposed theoretical framework can accommodate many existing computational tools in the literature. Our results rely on (arguably surprising) connections between recursive adaptive partitioning and sequential greedy approximation algorithms for convex optimization problems (e.g., orthogonal greedy algorithms), which may be of independent theoretical interest. Using our theory and methods, we also study oblique random forests.