Aggregating estimates by convex optimization

TL;DR

This paper introduces a convex optimization-based method for aggregation and adaptive estimation of signals from convex hypotheses, achieving near-optimal performance in certain observation schemes and practical efficiency.

Contribution

It proposes a novel convex optimization approach for aggregation and adaptation, applicable to unions of convex sets, with demonstrated effectiveness and practical implementation considerations.

Findings

Near-optimal aggregation routines for convex hypotheses

Efficient implementation for a limited number of sets

Successful application to signal recovery in Gaussian models

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 {\em simple observation schemes} and where 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 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.