Complexity of Pattern Classes and Lipschitz Property
Amiran Ambroladze, Emilio Parrado-Hernandez and John Shawe-Taylor
Theoretical Computer Science 2005.

Abstract

Rademacher and Gaussian complexities are successfully used in learning theory for measuring the capacity of the class of functions to be learnt. 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 dificult then the one for the Rademacher case. In this paper we give a detailed prove of the Lipschitz property for the general case (with the only assumption which makes the complexity notion meaningful) including the Rademacher and Gaussian cases. 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 onedimensional case.

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