Consistency and robustness of kernel-based regression in convex risk minimization

TL;DR

This paper analyzes the statistical properties, consistency, and robustness of kernel-based regression methods inspired by convex risk minimization, providing theoretical guarantees and practical guidelines for their application.

Contribution

It establishes $L$-risk consistency and robustness properties for a broad class of kernel regression methods, linking loss functions, kernel choices, and influence functions.

Findings

Kernel methods are $L$-risk consistent, ensuring they can learn effectively.

Robustness properties are characterized, allowing for bounded influence functions.

Bounds for bias and sensitivity curves are derived, connecting KBR with classical estimators.

Abstract

We investigate statistical properties for a broad class of modern kernel-based regression (KBR) methods. These kernel methods were developed during the last decade and are inspired by convex risk minimization in infinite-dimensional Hilbert spaces. One leading example is support vector regression. We first describe the relationship between the loss function $L$ of the KBR method and the tail of the response variable. We then establish the $L$-risk consistency for KBR which gives the mathematical justification for the statement that these methods are able to ``learn''. Then we consider robustness properties of such kernel methods. In particular, our results allow us to choose the loss function and the kernel to obtain computationally tractable and consistent KBR methods that have bounded influence functions. Furthermore, bounds for the bias and for the sensitivity curve, which is a finite sample version of the influence function, are developed, and the relationship between KBR and classical $M$ estimators is discussed.