A Theoretical Analysis of the Test Error of Finite-Rank Kernel Ridge Regression

TL;DR

This paper provides sharp, non-asymptotic bounds for the test error of finite-rank kernel ridge regression, improving upon previous bounds and applicable across all regularization parameters.

Contribution

It derives tighter, non-asymptotic bounds for finite-rank KRR test error that are valid for any regularization, filling a gap in theoretical understanding.

Findings

Bounds are tighter than previous results.

Bounds are valid for any regularization parameter.

Results are applicable to transfer learning scenarios.

Abstract

Existing statistical learning guarantees for general kernel regressors often yield loose bounds when used with finite-rank kernels. Yet, finite-rank kernels naturally appear in several machine learning problems, e.g.\ when fine-tuning a pre-trained deep neural network's last layer to adapt it to a novel task when performing transfer learning. We address this gap for finite-rank kernel ridge regression (KRR) by deriving sharp non-asymptotic upper and lower bounds for the KRR test error of any finite-rank KRR. Our bounds are tighter than previously derived bounds on finite-rank KRR, and unlike comparable results, they also remain valid for any regularization parameters.