Optimal learning with $Q$-aggregation

TL;DR

This paper develops an optimal model selection aggregation method using $Q$-aggregation for supervised learning with convex and Lipschitz loss functions, extending previous Gaussian regression results.

Contribution

It generalizes $Q$-aggregation to broader supervised learning settings with convex, Lipschitz loss, achieving optimal oracle inequalities.

Findings

Estimator satisfies optimal oracle inequalities in expectation.

Estimator performs well with high probability.

Method extends previous Gaussian regression results.

Abstract

We consider a general supervised learning problem with strongly convex and Lipschitz loss and study the problem of model selection aggregation. In particular, given a finite dictionary functions (learners) together with the prior, we generalize the results obtained by Dai, Rigollet and Zhang [Ann. Statist. 40 (2012) 1878-1905] for Gaussian regression with squared loss and fixed design to this learning setup. Specifically, we prove that the $Q$-aggregation procedure outputs an estimator that satisfies optimal oracle inequalities both in expectation and with high probability. Our proof techniques somewhat depart from traditional proofs by making most of the standard arguments on the Laplace transform of the empirical process to be controlled.