Aggregating estimates by convex optimization

Abstract

We discuss the approach to estimate aggregation and adaptive estimation based upon (nearly optimal) testing of convex hypotheses. We show that in the situation where the observations stem from simple observation schemes (Juditsky and Nemirovski, 2020) and where the set of unknown signals is a finite union of convex and compact sets, the proposed approach leads to aggregation and adaptation routines with nearly optimal performance. As an illustration, we consider application of the proposed estimates to the problem of recovery of unknown signal known to belong to a union of ellitopes (Juditsky and Nemirovski, 2018 and 2020) in Gaussian observation scheme. The proposed approach can be implemented efficiently when the number of sets in the union is “not very large.” We conclude the paper with a small simulation study illustrating practical performance of the proposed procedures in the problem of signal estimation in the single-index model.