  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.
