Simultaneous Adaptation to the Margin and to Complexity in Classification

We consider the problem of adaptation to the margin and to complexity in binary classification. We suggest an exponential weighting aggregation scheme. We use this aggregation procedure to construct classifiers which adapt automatically to margin and complexity. Two main examples are worked out in which adaptivity is achieved in frameworks proposed by Steinwart and Scovel [Learning Theory. Lecture Notes in Comput. Sci. 3559 (2005) 279-294. Springer, Berlin; Ann. Statist. 35 (2007) 575-607] and Tsybakov [Ann. Statist. 32 (2004) 135-166]. Adaptive schemes, like ERM or penalized ERM, usually involve a minimization step. This is not the case for our procedure.