We study losses for binary classification and class probability estimation
and extend the understanding of them from margin losses to general composite
losses which are the composition of a proper loss with a link function. We
characterise when margin losses can be proper composite losses, explicitly show
how to determine a symmetric loss in full from half of one of its partial
losses, introduce an intrinsic parametrisation of composite binary losses and
give a complete characterisation of the relationship between proper losses and
``classification calibrated'' losses. We also consider the question of the
``best'' surrogate binary loss. We introduce a precise notion of ``best'' and
show there exist situations where two convex surrogate losses are
incommensurable. We provide a complete explicit characterisation of the
convexity of composite binary losses in terms of the link function and the
weight function associated with the proper loss which make up the composite
loss. This characterisation suggests new ways of ``surrogate tuning''. Finally,
in an appendix we present some new algorithm-independent results on the
relationship between properness, convexity and robustness to misclassification
noise for binary losses and show that all convex proper losses are non-robust
to misclassification noise.

本文研究了二元分类和类概率估计中的损失，将对它们的理解从边缘损失扩展到了一般的组合损失，文中表征了何时边缘损失可以成为适当的组合损失，明确地展示了如何从部分损失的一半中完整地确定对称损失，并介绍了组合二元损失的内在参数化的概念，并给出了适当损失和 “分类校准” 损失之间关系的完整的特征。本文还探讨了 “最佳” 替代二元损失的问题，并介绍了精确的 “最佳” 的概念，以及存在两种凸代理损失不可比较的情况。最后，文中在附录中提供了一些新的算法无关的关于二元损失适当性、凸性和对于误分类噪声的鲁棒性之间关系的结果，并且证明了所有凸适当损失都对误分类噪声不鲁棒。