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Zbl 1165.68388
Wu, Qiang; Zhou, Ding-Xuan
Learning with sample dependent hypothesis spaces. (English)
[J] Comput. Math. Appl. 56, No. 11, 2896-2907 (2008). ISSN 0898-1221

Summary: Many learning algorithms use hypothesis spaces which are trained from samples, but little theoretical work has been devoted to the study of these algorithms. In this paper we show that mathematical analysis for these algorithms is essentially different from that for algorithms with hypothesis spaces independent of the sample or depending only on the sample size. The difficulty lies in the lack of a proper characterization of approximation error. To overcome this difficulty, we propose an idea of using a larger function class (not necessarily linear space) containing the union of all possible hypothesis spaces (varying with the sample) to measure the approximation ability of the algorithm. We show how this idea provides error analysis for two particular classes of learning algorithms in kernel methods: learning the kernel via regularization and coefficient based regularization. We demonstrate the power of this approach by its wide applicability.

MSC 2000:: *68Q32 Computational learning theory
62J02 General nonlinear regression
68T05 Learning and adaptive systems

Keywords: learning theory; regularization scheme; error analysis; sample dependent hypothesis spaces; approximation error

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