Analysis of purely random forests bias
This provides theoretical insights into random forests, a widely used method, but is incremental as it focuses on simplified models rather than practical implementations.
The paper analyzes the approximation error (bias) of purely random forest models in regression, showing that infinite forests achieve a faster bias reduction rate than single trees, leading to strictly better risk rates, and derives a minimum number of trees needed to match this performance.
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.