Appendix A Proofs

This result is well-known [29] but we include a proof here for completeness. First, let's define the conditional risk of h at x, denoted by R Next, suppose q ( x) null= c and h( x) null= h( x). From the proof of Lemma 4.1, we know that The proof of Lemma 4.5 depends on another lemma, which will also be useful in the unknown hypothesis class setting. This is the first claim of the lemma. Thus, ( null) must be false!