Appendix

This appendix contains the following sections: Proofs: Section A contains proofs of theoretical results presented in the main paper. We estimate the compute time to be on the order of 100 hours. Based on the definition of Y ( π( x)), we can write it as follows, using the fact that Y is binary. First, we will prove the following lemma: Lemma 1. Let U, V be binary random variables, then Cov(U, V | X) = E[(U E[ U | X ])(V E[V | X ]) | X ] = E[(U E[U | X ])V | X ] E [(U E[ U | X ])E[ V | X ] | X ] = E[(U E[ U | X ])V | X ]. (10) Here, Eq. (10) follows since for any bounded f ( X), E [(U E[ U | X ]) f ( X) | X ] = f ( X) E[ U E[ U | X ] | X ] = 0.