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.

我们对学习延迟的替代损失函数进行了全面的研究，引入了一类由非递增函数Ψ参数化的广泛替代损失，并在温和条件下建立了其实现的H一致性。对于基于分类误差的成本函数，我们进一步展示了在假设集对称且完备（常见的神经网络和线性函数假设集满足此属性）时，这些损失具有H一致性界限。我们的结果还解决了先前工作(Mozannar et al., 2023)中提出的一个开放问题，通过证明了一个特定替代损失的实现H一致性和贝叶斯一致性。此外，我们还确定了Ψ的选择，以实现任何通用成本函数的H一致性替代损失，从而同时实现了贝叶斯一致性、实现的H一致性和H一致性界限。我们还研究了学习延迟中H一致性界限与实现的H一致性之间的关系，突出了与标准分类的关键差异。最后，我们对我们提出的替代损失进行了经验评估，并将其与现有基准进行了比较。

可实现的H-一致和贝叶斯一致的损失函数用于推迟学习