Optimal bounds for aggregation of affine estimators

TL;DR

This paper establishes that the optimal aggregation rate for affine estimators remains logarithmic in the number of estimators, even when estimators depend on the data, using a penalized convex hull approach.

Contribution

It proves that the minimax aggregation rate for affine estimators is unaffected by data dependence and introduces a penalized procedure to achieve this optimal rate.

Findings

Minimax rate for affine estimators is of order log(M).

Dependence on data does not increase the aggregation rate.

A penalized convex hull method attains the minimax rate.

Abstract

We study the problem of aggregation of estimators when the estimators are not independent of the data used for aggregation and no sample splitting is allowed. If the estimators are deterministic vectors, it is well known that the minimax rate of aggregation is of order $\log(M)$, where $M$ is the number of estimators to aggregate. It is proved that for affine estimators, the minimax rate of aggregation is unchanged: it is possible to handle the linear dependence between the affine estimators and the data used for aggregation at no extra cost. The minimax rate is not impacted either by the variance of the affine estimators, or any other measure of their statistical complexity. The minimax rate is attained with a penalized procedure over the convex hull of the estimators, for a penalty that is inspired from the $Q$-aggregation procedure. The results follow from the interplay between the penalty, strong convexity and concentration.