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

Autor:	Talwai, Prem, Simchi-Levi, David
Rok vydání:	2022
Předmět:	Mathematics - Statistics Theory Mathematics - Functional Analysis Statistics - Machine Learning
Druh dokumentu:	Working Paper
Popis:	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. Comment: The first version of this paper contained an error -- refer to current version
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2204.07856 Zobrazit plný text záznamu View this record from Arxiv