## AbstractWe consider the binary classification problem. Given an i.i.d. sample drawn from the distribution of an X x {0,1}-valued random pair, we propose to estimate the so-called Bayes classifier by minimizing the sum of the empirical classification error and a penalty term based on Efron's or i.i.d. weighted bootstrap samples of the data. We obtain exponential inequalities for such bootstrap type penalties, which allow us to derive non-asymptotic properties for the corresponding estimators. In particular, we prove that these estimators achieve the global minimax risk over sets of functions built from Vapnik-Chervonenkis classes. The obtained results generalize Koltchinskii's and Bartlett, Boucheron, Lugosi's ones for Rademacher penalties that can thus be seen as special examples of bootstrap type penalties.
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