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

We consider the problem of adaptation to the margin in binary classi-fication. We suggest a penalized empirical risk minimization classifier that adaptively attains, up to a logarithmic factor, fast optimal rates of conver-gence 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. 1. Introduction. Consider observations (X1, Y1),..., (Xn,Yn