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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 S. Smale and D. Zhou [Learning theory estimates via integral operators and their approximations, submitted for publication, retrievable at $〈$http://www.tti-c.org/smale.html$〉$ (2005)] and improve the results for Landweber iterations obtained by Y. Yao, L. Rosasco and 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.