Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 640758, 16 pages
Research Article

Estimates of the Approximation Error Using Rademacher Complexity: Learning Vector-Valued Functions

Giorgio Gnecco1,2 and Marcello Sanguineti2

1Department of Mathematics (DIMA), University of Genova, Via Dodecaneso 35, 16146 Genova, Italy
2Department of Communications, Computer, and System Sciences (DIST), University of Genova, Via Opera Pia 13, 16145 Genova, Italy

Received 15 January 2008; Revised 13 August 2008; Accepted 20 October 2008

Academic Editor: Ulrich Abel

Copyright © 2008 Giorgio Gnecco and Marcello Sanguineti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


For certain families of multivariable vector-valued functions to be approximated, the accuracy of approximation schemes made up of linear combinations of computational units containing adjustable parameters is investigated. Upper bounds on the approximation error are derived that depend on the Rademacher complexities of the families. The estimates exploit possible relationships among the components of the multivariable vector-valued functions. All such components are approximated simultaneously in such a way to use, for a desired approximation accuracy, less computational units than those required by componentwise approximation. An application to N-stage optimization problems is discussed.