Error Scaling Laws for Kernel Classification under Source and Capacity Conditions

TL;DR

This paper derives error decay rates for kernel classifiers under source and capacity conditions, providing accurate predictions of learning curves for real datasets and contrasting SVM and ridge classification.

Contribution

It introduces explicit error scaling laws for kernel classification under source and capacity conditions, validated on real data and two standard classifiers.

Findings

Derived decay rates for misclassification error as a function of source and capacity coefficients.

Rates accurately describe learning curves for real datasets.

Contrasted SVM and ridge classification, showing differences in error decay.

Abstract

We consider the problem of kernel classification. While worst-case bounds on the decay rate of the prediction error with the number of samples are known for some classifiers, they often fail to accurately describe the learning curves of real data sets. In this work, we consider the important class of data sets satisfying the standard source and capacity conditions, comprising a number of real data sets as we show numerically. Under the Gaussian design, we derive the decay rates for the misclassification (prediction) error as a function of the source and capacity coefficients. We do so for two standard kernel classification settings, namely margin-maximizing Support Vector Machines (SVM) and ridge classification, and contrast the two methods. We find that our rates tightly describe the learning curves for this class of data sets, and are also observed on real data. Our results can also be seen as an explicit prediction of the exponents of a scaling law for kernel classification that is accurate on some real datasets.