Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm

TL;DR

This paper extends learning rate analysis for least-squares regression to stronger Sobolev norms, allowing estimation of functions and derivatives without altering the algorithm, and proves their asymptotic optimality.

Contribution

It introduces finite sample bounds for Sobolev norms in least-squares regression, combining integral operator techniques with embedding properties for the first time.

Findings

Finite sample bounds for Sobolev norms are established.

The approach allows estimation of derivatives without changing the algorithm.

Results are asymptotically optimal in many cases.

Abstract

Learning rates for least-squares regression are typically expressed in terms of $L_2$-norms. In this paper we extend these rates to norms stronger than the $L_2$-norm without requiring the regression function to be contained in the hypothesis space. In the special case of Sobolev reproducing kernel Hilbert spaces used as hypotheses spaces, these stronger norms coincide with fractional Sobolev norms between the used Sobolev space and $L_2$. As a consequence, not only the target function but also some of its derivatives can be estimated without changing the algorithm. From a technical point of view, we combine the well-known integral operator techniques with an embedding property, which so far has only been used in combination with empirical process arguments. This combination results in new finite sample bounds with respect to the stronger norms. From these finite sample bounds our rates easily follow. Finally, we prove the asymptotic optimality of our results in many cases.