Consistency of The Oblique Decision Tree and Its Random Forest

The classification and regression tree (CART) and Random Forest (RF) are arguably the most popular pair of statistical learning methods. However, their statistical consistency can only be proved under very restrictive assumption on the underlying regression function. As an extension of the standard CART, Breiman (1984) suggested using linear combinations of predictors as splitting variables. The method became known as the oblique decision tree (ODT) and has received lots of attention. ODT tends to perform better than CART and requires fewer partitions. In this paper, we further show that ODT is consistent for very general regression functions as long as they are continuous. We also prove the consistency of ODT-based random forests (ODRF) that uses either fixed-size or random-size subset of features in the features bagging, the latter of which is also guaranteed to be consistent for general regression functions, but the former is consistent only for functions with specific structures. After refining the existing software package according to the established theory, our numerical experiments also show that ODRF has a noticeable overall improvement over RF and other decision forests.