Analysis of purely random forests bias

Sylvain Arlot (1, 2) , Robin Genuer (3, 4)

1 DI-ENS - Département d'informatique - ENS Paris
2 SIERRA - Statistical Machine Learning and Parsimony
3 ISPED - Institut de Santé Publique, d'Epidémiologie et de Développement
4 SISTM - Statistics In System biology and Translational Medicine

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.

Keywords: Random forests; Purely random forests; Non-parametric regression; approximation error

Domains: Statistics [math.ST]; Statistics Theory [stat.TH]; Methodology [stat.ME]; Machine Learning [cs.LG]

Sylvain Arlot, Robin Genuer. Analysis of purely random forests bias. [Research Report] -. 2014. ⟨hal-01023596⟩