The Benjamini-Hochberg (BH) procedure is widely used to control the false detection rate (FDR) in multiple testing.

It imposes lower and upper bound constraints on the number of facilities opened for each label, ensuring fair representation of all demographic groups by the selected facilities.

We still need to demonstrate that the properties in P AC-Bayes analysis hold for both the margin operator and the robust margin operator. Then we complete the proof of Lemma 6.1. The proof of Lemma 7.1 and 7.2 is similar. We provide the proof of Lemma 7.2 below. Lemma 7.1 follows the proof of Lemma 7.2 by replacing the robust margin operator by the margin Since the above bound holds for any x in the domain X, we can get the following a.s.: R The second inequality is the tail bound above.

Then, giventhenumberofroundsT, forany 2 (0,1), >1, 2 (0,1), >0, suppose m e (poly(T, k, L, )), 1 = 2 = ( pTm), andµ satisfiestheconditionsin Theorem 4.1 andmax