e 2E all

The language of causal inference provides further intuition for the structure imposed on Problem 3.1 Recall that in Assumption 4.1 imposes that Assumption 4.2, we assume that Then given Proposition B.1, it follows that | P Observe that by Proposition B.1, we have that P However, as we will show in Remark B.4, when strong duality holds for This useful result, which follows from a simple one-line proof in 5.6.2 of [ The idea here is to apply Lemma B.3 for the constant function defined by B.3 Relationship to constrained PAC learning In contrast, the optimization problem in Problem 4.6 contains a family In this appendix, we provide the proofs that were omitted in the main text. Under Assumptions 4.1 and 4.2, Problem 3.1 is equivalent to minimize The main idea in this proof is the following. Finally, we undo our expansion to arrive at at the statement of the proposition. Then, recall that by Assumption 4.2, we have that Now observe that under Assumption 4.1, we have that Under Assumptions 4.1 and 4.2, if we restrict the feasible set to the set of Before proving Proposition 5.2, we formally state the assumptions we require on We make the following assumptions: 1. Given these assumptions, we restate Proposition 5.2 below: Proposition 5.2.