Arman Zharmagambetov, Miguel Carreira-Perpinan
Regression forests (ensembles of regression trees) have long been recognized as the leading off-the-shelf method for regression, where the task is to predict a continuous scalar or vector output. The main approaches are based on bagging, where individual trees are trained independently on bootstrap samples of the data; or on boosting, where individual trees are trained sequentially on the whole data but with adaptively weighted instances. However, both approaches rely on a greedy top-down procedure such as CART to learn an axis-aligned tree, where each decision node tests for a single feature. We instead use the recently proposed Tree Alternating Optimization (TAO) algorithm. This is able to learn an oblique tree, where each decision node tests for a linear combination of features, and which has much higher accuracy than axis-aligned trees. We show that using TAO with the bagging approach produces much better forests than random forests, Adaboost or gradient boosting in every dataset we have tried across a wide range of input and output dimensionality and sample size. The resulting forest has significantly lower test regression error while using shallower trees with fewer parameters and lower inference time overall. This result has an immense practical impact and advocates for the power of optimization in ensemble learning.