Contents of Appendix

Bayes-consistency only holds for the full family of measurable functions, which of course is distinct from the more restricted hypothesis set used by a learning algorithm. Therefore, a hypothesis setdependent notion of H-consistency has been proposed by Long and Servedio (2013) in the realizable setting, used by Zhang and Agarwal (2020) for linear models, and generalized by Kuznetsov et al. (2014) to the structured prediction case. Long and Servedio (2013) showed that there exists a case where a Bayes-consistent loss is not H-consistent while inconsistent losses can be H-consistent. Zhang and Agarwal (2020) further investigated the phenomenon in (Long and Servedio, 2013) and showed that the situation of losses that are not H-consistent with linear models can be remedied by carefully choosing a larger piecewise linear hypothesis set. Kuznetsov et al. (2014) proved positive results for the H-consistency of several multi-class ensemble algorithms, as an extension of H-consistency results in (Long and Servedio, 2013). Recently, the notions of H-calibration and H-consistency have been used by Bao et al. (2020); Awasthi et al. (2021a) in the study of adversarial binary classification losses, as defined in (Goodfellow et al., 2014; Madry et al., 2017; Tsipras et al., 2018; Carlini and Wagner, 2017; Awasthi et al., 2023).