Aggregation procedures: optimality and fast rates.

Summary In this thesis we deal with aggregation procedures under the margin assumption. We prove that the margin assumption improves the rate of aggregation. Another contribution of this thesis is to show that some empirical risk minimization procedures are suboptimal when the loss function is convex, even under the margin assumption. Contrarily to some aggregation procedures with exponential weights, these model selection methods cannot benefit from the large margin. Then, we apply aggregation methods to construct adaptive estimators in several different problems. The final contribution of this thesis is to purpose a new approach to the control of the bias term in classification by introducing some spaces of sparse prediction rules. Minimax rates of convergence have been obtained for these classes of functions and, by using an aggregation method, we provide an adaptive version of these estimators.