Hyper-Sparse Optimal Aggregation

Given a finite set F of functions and a learning sample, the aim of an aggregation procedure is to have a risk as close as possible to risk of the best function in F . Up to now, optimal aggregation procedures are convex combinations of every elements of F . In this paper, we prove that optimal aggregation procedures combining only two functions in F exist. Such algorithms are of particular interest when F contains many irrelevant functions that should not appear in the aggregation procedure. Since selectors are suboptimal aggregation procedures, this proves that two is the minimal number of elements of F required for the construction of an optimal aggregation procedure in every situations. Then, we perform a numerical study for the problem of selection of the regularization parameters of the Lasso and the Elastic-net estimators. We compare on simulated examples our aggregation algorithms to aggregation with exponential weights, to Mallow’s Cp and to cross-validation selection procedures.