Complexity of pattern classes and Lipschitz property
Complexity of pattern classes and Lipschitz property
Rademacher and Gaussian complexities are successfully used in learning theory for measuring the capacity of the class of functions to be learned. One of the most important properties for these complexities is their Lipschitz property: a composition of a class of functions with a fixed Lipschitz function may increase its complexity by at most twice the Lipschitz constant. The proof of this property is non-trivial (in contrast to the other properties) and it is believed that the proof in the Gaussian case is conceptually more difficult then the one for the Rademacher case. In this paper we give a detailed prove of the Lipschitz property for the Rademacher case and generalize the same idea to an arbitrary complexity (including the Gaussian). We also discuss a related topic about the Rademacher complexity of a class consisting of all the Lipschitz functions with a given Lipschitz constant. We show that the complexity is surprisingly low in the one-dimensional case. The question for higher dimensions remains open.
181-193
Springer Berlin, Heidelberg
Ambroladze, Amiran
86f53c6f-a258-4f34-b7e1-4af90e4d4324
Shawe-Taylor, John
b1931d97-fdd0-4bc1-89bc-ec01648e928b
October 2004
Ambroladze, Amiran
86f53c6f-a258-4f34-b7e1-4af90e4d4324
Shawe-Taylor, John
b1931d97-fdd0-4bc1-89bc-ec01648e928b
Ambroladze, Amiran and Shawe-Taylor, John
(2004)
Complexity of pattern classes and Lipschitz property.
David, Shai Ben, Case, John and Maruoka, Akira
(eds.)
In Algorithmic Learning Theory: 15th International Conference, ALT 2004, Padova, Italy, October 2-5, 2004. Proceedings.
vol. 3244,
Springer Berlin, Heidelberg.
.
Record type:
Conference or Workshop Item
(Paper)
Abstract
Rademacher and Gaussian complexities are successfully used in learning theory for measuring the capacity of the class of functions to be learned. One of the most important properties for these complexities is their Lipschitz property: a composition of a class of functions with a fixed Lipschitz function may increase its complexity by at most twice the Lipschitz constant. The proof of this property is non-trivial (in contrast to the other properties) and it is believed that the proof in the Gaussian case is conceptually more difficult then the one for the Rademacher case. In this paper we give a detailed prove of the Lipschitz property for the Rademacher case and generalize the same idea to an arbitrary complexity (including the Gaussian). We also discuss a related topic about the Rademacher complexity of a class consisting of all the Lipschitz functions with a given Lipschitz constant. We show that the complexity is surprisingly low in the one-dimensional case. The question for higher dimensions remains open.
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Published date: October 2004
Organisations:
Electronics & Computer Science
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Local EPrints ID: 259921
URI: http://eprints.soton.ac.uk/id/eprint/259921
PURE UUID: a6c28a03-595f-410b-86bd-1261a327ee04
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Date deposited: 09 Sep 2004
Last modified: 15 Mar 2024 21:32
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Contributors
Author:
Amiran Ambroladze
Author:
John Shawe-Taylor
Editor:
Shai Ben David
Editor:
John Case
Editor:
Akira Maruoka
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