Learning by mirror averaging

Given a collection of $M$ different estimators or classifiers, we study the problem of model selection type aggregation, i.e., we construct a new estimator or classifier, called aggregate, which is nearly as good as the best among them with respect to a given risk criterion. We define our aggregate by a simple recursive procedure which solves an auxiliary stochastic linear programming problem related to the original non-linear one and constitutes a special case of the mirror averaging algorithm. We show that the aggregate satisfies sharp oracle inequalities under some general assumptions. The results allow one to construct in an easy way sharp adaptive nonparametric estimators for several problems including regression, classification and density estimation