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

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.

Kernel Additive Principal Components

Additive principal components (APCs for short) are a nonlinear generalization of linear principal components. We focus on smallest APCs to describe additive nonlinear constraints that are approximately satisfied by the data. Thus APCs fit data with implicit equations that treat the variables symmetrically, as opposed to regression analyses which fit data with explicit equations that treat the data asymmetrically by singling out a response variable. We propose a regularized data-analytic procedure for APC estimation using kernel methods. In contrast to existing approaches to APCs that are based on regularization through subspace restriction, kernel methods achieve regularization through shrinkage and therefore grant distinctive flexibility in APC estimation by allowing the use of infinite-dimensional functions spaces for searching APC transformation while retaining computational feasibility. To connect population APCs and kernelized finite-sample APCs, we study kernelized population APCs and their associated eigenproblems, which eventually lead to the establishment of consistency of the estimated APCs. Lastly, we discuss an iterative algorithm for computing kernelized finite-sample APCs.