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.
%0 Journal Article
%1 FiSt2020
%A Fischer, Simon
%A Steinwart, Ingo
%D 2020
%E Rosasco, Lorenzo
%I JMLR
%J J. Mach. Learn. Res.
%K Interpolation_Norms Learning_Rates Least-Squares_Regression Regularized_Kernel_Methods Statistical_Learning_Theory Uniform_Convergence myown
%N 205
%P 1--38
%T Sobolev norm learning rates for regularized least-squares algorithms
%U http://jmlr.org/papers/v21/19-734.html
%V 21
%X 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.
@article{FiSt2020,
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.},
added-at = {2020-10-26T14:51:01.000+0100},
author = {Fischer, Simon and Steinwart, Ingo},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2b574398996f20641d9d51fbb803a1688/simonfischer},
editor = {Rosasco, Lorenzo},
interhash = {014f5853ed11000c9f6b2b0992bd5428},
intrahash = {b574398996f20641d9d51fbb803a1688},
issn = {1533-7928},
journal = {J. Mach. Learn. Res.},
keywords = {Interpolation_Norms Learning_Rates Least-Squares_Regression Regularized_Kernel_Methods Statistical_Learning_Theory Uniform_Convergence myown},
month = oct,
number = 205,
pages = {1--38},
publisher = {JMLR},
timestamp = {2021-07-07T12:18:41.000+0200},
title = {Sobolev norm learning rates for regularized least-squares algorithms},
url = {http://jmlr.org/papers/v21/19-734.html},
volume = 21,
year = 2020
}