  It has been recently shown that, under the margin (or low noise) assumption,
there exist classifiers attaining fast rates of convergence of the excess Bayes
risk, i.e., the rates faster than $n^{-1/2}$. The works on this subject
suggested the following two conjectures: (i) the best achievable fast rate is
of the order $n^{-1}$, and (ii) the plug-in classifiers generally converge
slower than the classifiers based on empirical risk minimization. We show that
both conjectures are not correct. In particular, we construct plug-in
classifiers that can achieve not only the fast, but also the {\it super-fast}
rates, i.e., the rates faster than $n^{-1}$. We establish minimax lower bounds
showing that the obtained rates cannot be improved.
