Honest and adaptive confidence sets in Lp

TL;DR

This paper develops honest and adaptive confidence sets in Lp-loss for non-parametric Gaussian regression, identifying regimes where adaptation is possible and establishing lower bounds on the size of necessary exclusions, with a focus on the impact of p.

Contribution

It characterizes regimes for adaptive confidence sets in Lp-loss, especially for p >= 2, and provides lower bounds on the size of regions that must be excluded, revealing a transition as p varies.

Findings

Two main regimes for p >= 2: unrestricted adaptation and restricted adaptation.

Lower bounds show the minimal size of excluded regions cannot be significantly smaller.

A continuous transition in behavior is observed from p=2 to p=infinity.

Abstract

We consider the problem of constructing honest and adaptive confidence sets in Lp-loss (with p>=1 and p < infinity) over sets of Sobolev-type classes, in the setting of non-parametric Gaussian regression. The objective is to adapt the diameter of the confidence sets with respect to the smoothness degree of the underlying function, while ensuring that the true function lies in the confidence interval with high probability.When p >=2, we identify two main regimes, (i) one where adaptation is possible without any restrictions on the model, and (ii) one where critical regions have to be removed. We also prove by a matching lower bound that the size of the regions that we remove can not be chosen significantly smaller. These regimes are shown to depend in a qualitative way on the index p, and a continuous transition from p = 2 to p = infinity is exhibited.