Generalized mirror averaging and D-convex aggregation

We study the problem of aggregation of estimators. Given a collection of M different estimators, we construct a new estimator, called aggregate, which is nearly as good as the best linear combination over an l ₁-ball of ℝ ^M of the initial estimators. The aggregate is obtained by a particular version of the mirror averaging algorithm. We show that our aggregation procedure statisfies sharp oracle inequalities under general assumptions. Then we apply these results to a new aggregation problem: D-convex aggregation. Finally we implement our procedure in a Gaussian regression model with random design and we prove its optimality in a minimax sense up to a logarithmic factor.