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Square root penalty: Adaption to the margin in classification and in edge estimation. (English) Zbl 1080.62047

From the paper: Consider obervations \((X_1,Y_1),\dots,(X_n,Y_n)\), where \(Y_i\) is a bounded response random variable and \(X_i\in{\mathcal X}\) is the corresponding instance. We regard \(\{(X_i,Y_i)\}^n_{i=1}\) as i.i.d. copies of a population version \((X,Y)\). The goal is to predict the response \(Y\) given the value of the instance \(X\). We consider two statistical problems: binary classification and boundary estimation in binary images (edge estimation). In the classification setup \(Y_i\in \{0,1\}\) is a label (e.g., {ill, healthy}, {white, black}, etc.), while in edge estimation \(Y_i\) can be either a label or a general bounded random variable. Most of the paper will be concerned with the model of binary classification. The results for edge estimation are quite analogous and they will be stated as corollaries.
We consider the problem of adaptation to the margin in binary classification. We suggest a penalized empirical risk minimization classifier that adaptively attains, up to a logarithmic factor, fast optimal rates of convergence for the excess risk, that is, rates that can be faster than \(n^{-1/2}\), where \(n\) is the sample size. We show that our method also gives adaptive estimators for the problem of edge estimation.

MSC:

62H30 Classification and discrimination; cluster analysis (statistical aspects)
62G07 Density estimation
62G08 Nonparametric regression and quantile regression
68T10 Pattern recognition, speech recognition
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