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On regularization algorithms in learning theory. (English) Zbl 1109.68088
Summary: In this paper we discuss a relation between Learning Theory and Regularization of linear ill-posed inverse problems. It is well known that Tikhonov regularization can be profitably used in the context of supervised learning, where it usually goes under the name of regularized least-squares algorithm. Moreover, the gradient descent algorithm was studied recently, which is an analog of Landweber regularization scheme. In this paper we show that a notion of regularization defined according to what is usually done for ill-posed inverse problems allows to derive learning algorithms which are consistent and provide a fast convergence rate. It turns out that for priors expressed in term of variable Hilbert scales in reproducing kernel Hilbert spaces our results for Tikhonov regularization match those of {\it S. Smale} and {\it D. Zhou} [Learning theory estimates via integral operators and their approximations, submitted for publication, retrievable at $\langle$http://www.tti-c.org/smale.html$\rangle$ (2005)] and improve the results for Landweber iterations obtained by {\it Y. Yao, L. Rosasco} and {\it A. Caponnetto} [On early stopping in gradient descent learning, Constructive Approximation (2005), submitted for publication]. The remarkable fact is that our analysis shows that the same properties are shared by a large class of learning algorithms which are essentially all the linear regularization schemes. The concept of operator monotone functions turns out to be an important tool for the analysis.

##### MSC:
 68T05 Learning and adaptive systems
Full Text:
##### References:
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