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Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm

Simon Fischer, Ingo Steinwart

Journal of Machine Learning Research (JMLR), 21, pp. 1–38, 2020.


Learning rates for least-squares regression are typically expressed in terms of L2-norms. In this paper we extend these rates to norms stronger than the L2-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 L2. 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{fischer20_jmlr, title = {Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm}, author = {Fischer, Simon and Steinwart, Ingo}, year = {2020}, journal = {Journal of Machine Learning Research (JMLR)}, volume = {21}, pages = {1--38}, url = {https://www.jmlr.org/papers/volume21/19-734/19-734.pdf} }