d800149d2f947ad4d64f34668f8b20f6-Supplemental.pdf

Appendix of "Does enforcing fairness mitigate biases caused by subpopulation shift?" Assume 1. there are only two groups and the set of risk profiles R R Next we provide a proof of Theorem 4.3 under the additional assumption that The risk set R is convex. The unconstrained risk minimizer on unbiased data is algorithmically fair; i.e. FRM problem (4.3) is convex, so (1.2) implies R A sufficient condition for (4.5) is null P In this section, we state and prove a more general verion of Theorem 4.3 that permits continuous discriminative attributes. Let A be a complemented subspace in B. Then A Finally, we review some relevant background on infinite dimensional optimization.