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Minimax nonparametric classification .I. Rates of convergence

机译：Minimax非参数分类收敛速度

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摘要

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 L/sub 2/ 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, Gorfi, and Lugosi (see A Probabilistic Theory of Pattern Recognition, New York: Springer-Verlag, 1996).
机译：本文研究非参数分类的minimax方面。给定特征变量，我们首先研究类别标签条件概率的极小极大估计。假设此函数（例如f）处于一般的非参数类中。我们显示平方L / sub 2 /损失下的最小最大收敛速率由度量熵衡量的类的质量决定。本文的第二部分研究最小极大分类。利息损失是分类器分类错误的概率与贝叶斯决策分类错误的概率之间的差。众所周知，估计f的风险上限给出了分类风险的上限，但是对于单调函数的类别，该比率次优。这表明人们不必为了很好地进行分类就可以很好地估计f。但是，我们证明，在充分条件下，就收敛速度而言，这两个问题实际上都存在相同的困难，这由许多函数类（包括Besov（Sobolev），Lipschitz和有界变异）满足。鉴于Devroye，Gorfi和Lugosi的研究结果，这有点令人惊讶（请参阅模式识别的概率论，纽约：Springer-Verlag，1996年）。

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