Smoothing Effects of Bagging: Von Mises Expansions of Bagged Statistical Functionals

TL;DR

This paper extends bagging to statistical functionals, showing that the resample size acts as a smoothing parameter and that bagging induces smoothness even for rough or unstable functionals, with a von Mises expansion of finite length.

Contribution

It generalizes the concept of bagging to statistical functionals and establishes the smoothing effect via von Mises expansions related to the Efron-Stein ANOVA.

Findings

Bagging induces smoothness in statistical functionals.

The von Mises expansion of bagged functionals is finite and related to resample size.

Smaller resample size M results in more smoothing.

Abstract

Bagging is a device intended for reducing the prediction error of learning algorithms. In its simplest form, bagging draws bootstrap samples from the training sample, applies the learning algorithm to each bootstrap sample, and then averages the resulting prediction rules. We extend the definition of bagging from statistics to statistical functionals and study the von Mises expansion of bagged statistical functionals. We show that the expansion is related to the Efron-Stein ANOVA expansion of the raw (unbagged) functional. The basic observation is that a bagged functional is always smooth in the sense that the von Mises expansion exists and is finite of length 1 + resample size $M$. This holds even if the raw functional is rough or unstable. The resample size $M$ acts as a smoothing parameter, where a smaller $M$ means more smoothing.