This paper aims to provide a better understanding of a symmetric loss. First,
we emphasize that using a symmetric loss is advantageous in the balanced error
rate (BER) minimization and area under the receiver operating characteristic
curve (AUC) maximization from corrupted labels. Second, we prove general
theoretical properties of symmetric losses, including a
classification-calibration condition, excess risk bound, conditional risk
minimizer, and AUC-consistency condition. Third, since all nonnegative
symmetric losses are non-convex, we propose a convex barrier hinge loss that
benefits significantly from the symmetric condition, although it is not
symmetric everywhere. Finally, we conduct experiments to validate the relevance
of the symmetric condition.

本文旨在提供关于对称损失函数的更好理解。论文首先强调在平衡错误率（BER）最小化和受损标签下的接收者操作特性曲线（AUC）最大化中使用对称损失函数的优势。其次，我们证明对称损失函数的一般性质，包括分类校准条件、风险超出界限、条件风险最小化以及 AUC 一致性条件。第三，由于所有非负对称损失函数都是非凸的，我们提出了一个凸性障碍铰链损失函数，它从对称条件中受益很大，尽管它不是在所有地方都对称。最后，我们进行实验以验证对称条件的相关性。