  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.
