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.

翻译：我们为倾斜决策树分析建立了一个理论框架，其中每个决策节点的分裂发生在协变量的线性组合上（不同于仅涉及单一协变量的传统轴对齐分裂树结构）。尽管自80年代中期以来，这种方法已引起计算机科学和优化领域的广泛关注，但其相较于轴对齐方法所展现的优势仍然仅得到经验验证，且对其成功的解释主要基于启发式方法。为弥补这一长期存在的理论与实践的鸿沟，我们证明了（通过递归最小化平方误差构建的）倾斜回归树满足一类预言不等式，并能适应由岭函数线性组合及其极限点组成的丰富回归模型库。这为决策树与其他针对类似模型形式的可解释性较弱的方法（如投影追踪回归和神经网络）的定量比较提供了基准。与传统观点相反，我们不必总在可解释性与准确性之间权衡取舍。具体而言，在适当条件下，倾斜决策树在相同回归模型库上能达到与神经网络相当的预测精度。为解决每个决策节点寻找最优分裂超平面的组合复杂度问题，我们提出的理论框架可兼容现有文献中多种计算工具。我们的结果依赖于递归自适应划分与凸优化问题（如正交贪婪算法）的序贯贪婪逼近算法之间（颇具启发性的）联系，这本身可能具有独立的理论价值。基于我们的理论和方法，我们还对倾斜随机森林进行了研究。