Smoothing Effects of Bagging: Von Mises Expansions of Bagged Statistical Functionals
This provides theoretical insights into bagging for statisticians and machine learning practitioners, but it is incremental as it extends existing bagging concepts to functionals.
The authors tackled the problem of understanding the smoothing effects of bagging on statistical functionals, showing that bagged functionals are always smooth with a von Mises expansion of finite length 1 + M, even if the raw functional is rough or unstable.
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.