A Comprehensive Analysis on the Learning Curve in Kernel Ridge Regression

TL;DR

This paper provides a detailed analysis of the learning curves in kernel ridge regression, examining kernel properties, validating the Gaussian Equivalent Property, and deriving improved bounds for generalization performance across various settings.

Contribution

It introduces a comprehensive analysis of KRR learning curves, validates the Gaussian Equivalent Property, and derives sharper bounds applicable to diverse scenarios.

Findings

Validation of the Gaussian Equivalent Property for KRR

Improved bounds on generalization error in various regimes

Insights into the impact of kernel spectral properties

Abstract

This paper conducts a comprehensive study of the learning curves of kernel ridge regression (KRR) under minimal assumptions. Our contributions are three-fold: 1) we analyze the role of key properties of the kernel, such as its spectral eigen-decay, the characteristics of the eigenfunctions, and the smoothness of the kernel; 2) we demonstrate the validity of the Gaussian Equivalent Property (GEP), which states that the generalization performance of KRR remains the same when the whitened features are replaced by standard Gaussian vectors, thereby shedding light on the success of previous analyzes under the Gaussian Design Assumption; 3) we derive novel bounds that improve over existing bounds across a broad range of setting such as (in)dependent feature vectors and various combinations of eigen-decay rates in the over/underparameterized regimes.