PASCAL - Pattern Analysis, Statistical Modelling and Computational Learning

Entropy conditions for Lr-convergence of empirical processes
Andrea Caponnetto, Ernesto De Vito and Massimiliano Pontil
(2007) Other. UCL.

Abstract

The Law of Large Numbers (LLN) over classes of functions is a classical topic of Empirical Processes Theory. The properties characterizing classes of functions on which the LLN holds uniformly (i.e. Glivenko-Cantelli classes) have been widely studied in the literature. An elegant sufficient condition for such a property is finiteness of the Koltchinskii-Pollard entropy integral, and other conditions have been formulated in terms of suitable combinatorial complexities (e.g. the Vapnik-Chervonenkis dimension). In this paper, we endow the class of functions F with a probability measure and consider the LLN relative to the associated Lr metric. This framework extends the case of uniform convergence over F, which is recovered when r goes to infinity. The main result is a Lr-LLN in terms of a suitable uniform entropy integral which generalizes the Koltchinskii-Pollard entropy integral.

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EPrint Type:Monograph (Other)
Project Keyword:Project Keyword UNSPECIFIED
Subjects:Learning/Statistics & Optimisation
ID Code:3781
Deposited By:Massimiliano Pontil
Deposited On:22 February 2008