mlsanityData to 7 Oct 2026

Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm

arXiv 1702.07254 ↗First posted 23 Feb 2017stat.MLcited by 113

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Abstract (arXiv)

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.

Key facts. Cited by 113 papers in the map (data through 7 Oct 2026). First posted on arXiv 23 Feb 2017.

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Citing papers per year: 2021: 16; 2022: 11; 2023: 26; 2024: 28; 2025: 14; 2026: 18.

113 papers.

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  10. 2023Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithmcited by 22
  11. 2021The Three Stages of Learning Dynamics in High-Dimensional Kernel Methodscited by 21
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  14. 2024Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithmscited by 12
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