Minimax nonparametric classification - Part I: Rates of convergence

Research output: Contribution to journalArticle

59 Scopus citations

Abstract

This paper studies minimax aspects of nonparametric classification. We first study minimax estimation of the conditional probability of a class label, given the feature variable. This function, say f, is assumed to be in a general nonparametric class. We show the minimax rate of convergence under square L2 loss is determined by the massiveness of the class as measured by metric entropy. The second part of the paper studies minimax classification. The loss of interest is the difference between the probability of misclassification of a classifier and that of the Bayes decision. As is well known, an upper bound on risk for estimating f gives an upper bound on the risk for classification, but the rate is known to be suboptimal for the class of monotone functions. This suggests that one does not have to estimate f well in order to classify well. However, we show that the two problems are in fact of the same difficulty in terms of rates of convergence under a sufficient condition, which is satisfied by many function classes including Besov (Sobolev), Lipschitz, and bounded variation. This is somewhat surprising in view of a result of Devroye, Gyorfi, and Lugosi (1996).

Original languageEnglish (US)
Pages (from-to)2271-2284
Number of pages14
JournalIEEE Transactions on Information Theory
Volume45
Issue number7
DOIs
StatePublished - 1999

Fingerprint Dive into the research topics of 'Minimax nonparametric classification - Part I: Rates of convergence'. Together they form a unique fingerprint.

  • Cite this