Statistical Learnability of Generalized Additive Models based on Total Variation Regularization

Abstract

A generalized additive model (GAM, Hastie and Tibshirani (1987)) is a nonparametric model by the sum of univariate functions with respect to each explanatory variable, i.e., $f(\mathbf{x})=\sum f_j(x_j)$, where $x_j\in \mathbb{R}$ is $j$-th component of a sample $\mathbf{x}\in {\mathbb{R}}^p$. In this paper, we introduce the total variation (TV) of a function as a measure of the complexity of functions in $L_{\mathrm{c}}^1(\mathbb{R})$-space. Our analysis shows that a GAM based on TV-regularization exhibits a Rademacher complexity of $O(\sqrt{\frac{\log p}{m}})$, which is tight in terms of both $m$ and $p$ in the agnostic case of the classification problem. In result, we obtain generalization error bounds for finite samples according to work by Bartlett and Mandelson (2002).