Optimal rates for the regularized learning algorithms under general source condition

TL;DR

This paper analyzes the convergence rates of regularized learning algorithms under general source conditions, focusing on polynomial eigenvalue decay and operator monotone index functions, to establish optimal error bounds.

Contribution

It introduces a unified framework for analyzing convergence rates of various regularization schemes under general source conditions with polynomial eigenvalue decay.

Findings

Derived upper convergence rates for Tikhonov regularization.

Established minimax optimal error bounds for learning algorithms.

Extended analysis to general regularization schemes using operator monotone index functions.

Abstract

We consider the learning algorithms under general source condition with the polynomial decay of the eigenvalues of the integral operator in vector-valued function setting. We discuss the upper convergence rates of Tikhonov regularizer under general source condition corresponding to increasing monotone index function. The convergence issues are studied for general regularization schemes by using the concept of operator monotone index functions in minimax setting. Further we also address the minimum possible error for any learning algorithm.