Square root penalty: Adaptation to the margin in classification and in edge estimation

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.