Convergence Rates of Oblique Regression Trees for Flexible Function Libraries

TL;DR

This paper provides a theoretical analysis of oblique regression trees, demonstrating their ability to adapt to complex models and achieve accuracy comparable to neural networks, thus bridging the gap between empirical success and theoretical understanding.

Contribution

It introduces a theoretical framework for oblique decision trees, showing they satisfy oracle inequalities and can adapt to rich model libraries, with implications for interpretability and accuracy.

Findings

Oblique trees can adapt to linear combinations of ridge functions.

They achieve similar accuracy to neural networks under certain conditions.

The framework accommodates existing computational tools for hyperplane optimization.

Abstract

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.