On the Connection between $L_p$ and Risk Consistency and its Implications on Regularized Kernel Methods

TL;DR

This paper explores the relationship between risk consistency and $L_p$-consistency across various loss functions, revealing that shifted loss functions do not significantly alter the assumptions needed, with implications for kernel methods like SVMs.

Contribution

It establishes a broader connection between risk and $L_p$-consistency for diverse loss functions and examines the effects of shifted loss functions on these assumptions.

Findings

Risk consistency implies $L_p$-consistency for a wide class of loss functions.

Shifted loss functions do not substantially reduce the assumptions on the probability measure.

Results have implications for regularized kernel methods such as support vector machines.

Abstract

As a predictor's quality is often assessed by means of its risk, it is natural to regard risk consistency as a desirable property of learning methods, and many such methods have indeed been shown to be risk consistent. The first aim of this paper is to establish the close connection between risk consistency and $L_p$-consistency for a considerably wider class of loss functions than has been done before. The attempt to transfer this connection to shifted loss functions surprisingly reveals that this shift does not reduce the assumptions needed on the underlying probability measure to the same extent as it does for many other results. The results are applied to regularized kernel methods such as support vector machines.