Recursive Aggregation of Estimators by Mirror Descent Algorithm with Averaging

We consider a recursive algorithm to construct an aggregated estimator from a finite number of base decision rules in the classification problem. The estimator approximately minimizes a convex risk functional under the $\ell_1$-constraint. It is defined by a stochastic version of the mirror descent algorithm which performs descent of the gradient type in the dual space with an additional averaging. The main result of the paper is an upper bound for the expected accuracy of the proposed estimator. This bound is of the order $C\sqrt{(\log M)/t}$ with an explicit and small constant factor $C$, where $M$ is the dimension of the problem and $t$ stands for the sample size. A similar bound is proved for a more general setting, which covers, in particular, the regression model with squared loss.