Aggregation of Affine Estimators

TL;DR

This paper investigates aggregation methods for affine estimators in fixed design regression, establishing high-probability oracle inequalities and demonstrating the effectiveness of a generalized $Q$-aggregation scheme.

Contribution

It introduces high-probability oracle inequalities for exponential weighting and $Q$-aggregation, advancing theoretical understanding of estimator aggregation.

Findings

High-probability oracle inequalities for EW aggregation.

Sharp oracle inequalities for $Q$-aggregation.

Universal aggregation achieves all known bounds.

Abstract

We consider the problem of aggregating a general collection of affine estimators for fixed design regression. Relevant examples include some commonly used statistical estimators such as least squares, ridge and robust least squares estimators. Dalalyan and Salmon (2012) have established that, for this problem, exponentially weighted (EW) model selection aggregation leads to sharp oracle inequalities in expectation, but similar bounds in deviation were not previously known. While results indicate that the same aggregation scheme may not satisfy sharp oracle inequalities with high probability, we prove that a weaker notion of oracle inequality for EW that holds with high probability. Moreover, using a generalization of the newly introduced $Q$-aggregation scheme we also prove sharp oracle inequalities that hold with high probability. Finally, we apply our results to universal aggregation and show that our proposed estimator leads simultaneously to all the best known bounds for aggregation, including $\ell_q$-aggregation, $q \in (0,1)$, with high probability.