Minimax Nonparametric Classi cation|Part I: Rates of Convergence

|This paper studies minimax aspects of nonparametric classi cation. We rst 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 classi cation. The loss of interest is the di erence between the probability of misclassi cation of a classi er 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 classi cation, 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 di culty in terms of rates of convergence under a su cient condition, which is satis ed by many function classes including Besov (Sobolev), Lipschitz, and bounded variation. This is somewhat surprising in view of a result of Devroye, Gyor , and Lugosi (1996). Index Terms|Conditional probability estimation, mean error probability regret, metric entropy, minimax rates of convergence, nonparametric classi cation, neural network classes, sparse approximation.