Analysis of purely random forests bias

TL;DR

This paper analyzes the bias of purely random forest models in regression, showing that infinite forests outperform single trees in bias reduction and establishing the minimum number of trees needed for optimal risk rates.

Contribution

It provides the first theoretical analysis of the bias in purely random forests, revealing how the number of trees influences approximation error and linking it to kernel estimators.

Findings

Infinite forests have faster bias decay than single trees.

A minimum number of trees can achieve the same risk rate as an infinite forest.

Bias of purely random forests relates to certain kernel estimators.

Abstract

Random forests are a very effective and commonly used statistical method, but their full theoretical analysis is still an open problem. As a first step, simplified models such as purely random forests have been introduced, in order to shed light on the good performance of random forests. In this paper, we study the approximation error (the bias) of some purely random forest models in a regression framework, focusing in particular on the influence of the number of trees in the forest. Under some regularity assumptions on the regression function, we show that the bias of an infinite forest decreases at a faster rate (with respect to the size of each tree) than a single tree. As a consequence, infinite forests attain a strictly better risk rate (with respect to the sample size) than single trees. Furthermore, our results allow to derive a minimum number of trees sufficient to reach the same rate as an infinite forest. As a by-product of our analysis, we also show a link between the bias of purely random forests and the bias of some kernel estimators.