Realizable $H$-Consistent and Bayes-Consistent Loss Functions for Learning to Defer

We present a comprehensive study of surrogate loss functions for learning to defer. We introduce a broad family of surrogate losses, parameterized by a non-increasing function $\Psi$, and establish their realizable $H$-consistency under mild conditions. For cost functions based on classification error, we further show that these losses admit $H$-consistency bounds when the hypothesis set is symmetric and complete, a property satisfied by common neural network and linear function hypothesis sets. Our results also resolve an open question raised in previous work [Mozannar et al., 2023] by proving the realizable $H$-consistency and Bayes-consistency of a specific surrogate loss. Furthermore, we identify choices of $\Psi$ that lead to $H$-consistent surrogate losses for *any general cost function*, thus achieving Bayes-consistency, realizable $H$-consistency, and $H$-consistency bounds *simultaneously*. We also investigate the relationship between $H$-consistency bounds and realizable $H$-consistency in learning to defer, highlighting key differences from standard classification. Finally, we empirically evaluate our proposed surrogate losses and compare them with existing baselines.