Minimax Rates for High-Dimensional Random Tessellation Forests

TL;DR

This paper demonstrates that a broad class of random forests with oblique splits, including STIT forests and Poisson hyperplane tessellations, achieve minimax optimal rates in high-dimensional settings, extending previous axis-aligned results.

Contribution

It proves minimax optimal convergence rates for random forests with general split directions, including oblique splits, in arbitrary dimensions, using stochastic geometry techniques.

Findings

Oblique split forests achieve minimax optimality in high dimensions.

STIT forests and Poisson hyperplane tessellations are included in the optimal class.

First results showing minimax rates for oblique split random forests.

Abstract

Random forests are a popular class of algorithms used for regression and classification. The algorithm introduced by Breiman in 2001 and many of its variants are ensembles of randomized decision trees built from axis-aligned partitions of the feature space. One such variant, called Mondrian forests, was proposed to handle the online setting and is the first class of random forests for which minimax rates were obtained in arbitrary dimension. However, the restriction to axis-aligned splits fails to capture dependencies between features, and random forests that use oblique splits have shown improved empirical performance for many tasks. In this work, we show that a large class of random forests with general split directions also achieve minimax optimal convergence rates in arbitrary dimension. This class includes STIT forests, a generalization of Mondrian forests to arbitrary split directions, as well as random forests derived from Poisson hyperplane tessellations. These are the first results showing that random forest variants with oblique splits can obtain minimax optimality in arbitrary dimension. Our proof technique relies on the novel application of the theory of stationary random tessellations in stochastic geometry to statistical learning theory.