ARK: Robust Knockoffs Inference with Coupling

TL;DR

This paper studies the robustness of the model-X knockoffs framework when feature distributions are misspecified, introducing the approximate knockoffs (ARK) procedure and proving its asymptotic control of FDR and $k$-FWER through coupling techniques.

Contribution

It introduces the ARK procedure that uses estimated feature distributions and proves its asymptotic error rate control via coupling with the ideal model-X knockoffs.

Findings

ARK achieves asymptotic FDR control under misspecification.

Coupling technique links approximate and ideal knockoffs procedures.

Three constructions of coupled knockoffs verify robustness.

Abstract

We investigate the robustness of the model-X knockoffs framework with respect to the misspecified or estimated feature distribution. We achieve such a goal by theoretically studying the feature selection performance of a practically implemented knockoffs algorithm, which we name as the approximate knockoffs (ARK) procedure, under the measures of the false discovery rate (FDR) and $k$-familywise error rate ($k$-FWER). The approximate knockoffs procedure differs from the model-X knockoffs procedure only in that the former uses the misspecified or estimated feature distribution. A key technique in our theoretical analyses is to couple the approximate knockoffs procedure with the model-X knockoffs procedure so that random variables in these two procedures can be close in realizations. We prove that if such coupled model-X knockoffs procedure exists, the approximate knockoffs procedure can achieve the asymptotic FDR or $k$-FWER control at the target level. We showcase three specific constructions of such coupled model-X knockoff variables, verifying their existence and justifying the robustness of the model-X knockoffs framework. Additionally, we formally connect our concept of knockoff variable coupling to a type of Wasserstein distance.