Learning Curves and Benign Overfitting of Spectral Algorithms in Large Dimensions

TL;DR

This paper characterizes the learning curve and benign overfitting of spectral algorithms in high dimensions, revealing three regimes and demonstrating benign overfitting occurs under certain conditions.

Contribution

It provides a comprehensive asymptotic analysis of spectral kernel methods across all regularization regimes in large dimensions, including the under-regularized and interpolation regimes.

Findings

Learning curve has three regimes: over-regularized, under-regularized, and interpolation.

Benign overfitting occurs in under-regularized and interpolation regimes for positive source smoothness.

Kernel learning curve can be approximated by a sequence model in the regularized regime.

Abstract

Existing large-dimensional theory for spectral algorithms resolves either the optimally tuned point or the interpolation limit, but leaves the under-regularized regime unexplored. We study the learning curve and benign overfitting of spectral algorithms in the large-dimensional setting where the sample size and dimension are of comparable order, i.e., $n \asymp d^{\gamma}$ for some $\gamma>0$. We first consider inner-product kernels on the sphere $\mathbb{S}^{d-1}$ and establish a sharp asymptotic characterization of the excess risk across the full regularization path under various source conditions $s \geq 0$, where $s$ measures the relative smoothness of the regression function. Our results reveal that the learning curve is not simply U-shaped but instead consists of three distinct regimes: over-regularized, under-regularized, and interpolation regimes. This characterization allows us to fully capture the benign overfitting phenomenon, demonstrating that benign overfitting arises consistently across both the under-regularized and interpolation regimes whenever $s$ is positive but no larger than a critical threshold. We further show that, in the sufficiently regularized regime, the kernel learning curve is recovered by an associated sequence model. Finally, we extend the learning-curve analysis to large-dimensional KRR for a class of kernels on general domains in $\mathbb{R}^d$ whose low-degree eigenspaces satisfy spectral-scaling and hyper-contractivity conditions.