  We consider the problem of optimality, in a minimax sense, and adaptivity to
the margin and to regularity in binary classification. We prove an oracle
inequality, under the margin assumption (low noise condition), satisfied by an
aggregation procedure which uses exponential weights. This oracle inequality
has an optimal residual: $(\log M/n)^{\kappa/(2\kappa-1)}$ where $\kappa$ is
the margin parameter, $M$ the number of classifiers to aggregate and $n$ the
number of observations. We use this inequality first to construct minimax
classifiers under margin and regularity assumptions and second to aggregate
them to obtain a classifier which is adaptive both to the margin and
regularity. Moreover, by aggregating plug-in classifiers (only $\log n$), we
provide an easily implementable classifier adaptive both to the margin and to
regularity.
