Optimal bounds for aggregation of affine estimators

Understand

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.