Optimal Learning Rates for Regularized Least-Squares with a Fourier Capacity Condition

TL;DR

This paper establishes minimax adaptive learning rates for Tikhonov-regularized problems in Hilbert spaces, using a novel Fourier isocapacitary condition that relates kernel capacities and spectral properties without traditional assumptions.

Contribution

It introduces a new Fourier isocapacitary condition to analyze spectral properties and derive adaptive rates for regularized learning without standard kernel eigendecay assumptions.

Findings

Derived minimax adaptive rates under broad conditions.

Introduced a Fourier isocapacitary condition linking kernel capacities and spectral inference.

Achieved analysis without requiring the true function to be in the hypothesis class.

Abstract

We derive minimax adaptive rates for a new, broad class of Tikhonov-regularized learning problems in Hilbert scales under general source conditions. Our analysis does not require the regression function to be contained in the hypothesis class, and most notably does not employ the conventional \textit{a priori} assumptions on kernel eigendecay. Using the theory of interpolation, we demonstrate that the spectrum of the Mercer operator can be inferred in the presence of ``tight'' $L^{\infty}(\mathcal{X})$ embeddings of suitable Hilbert scales. Our analysis utilizes a new Fourier isocapacitary condition, which captures the interplay of the kernel Dirichlet capacities and small ball probabilities via the optimal Hilbert scale function.