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Abstract: We study losses for binary classification and class probability estimationand extend the understanding of them from margin losses to general compositelosses which are the composition of a proper loss with a link function. Wecharacterise when margin losses can be proper composite losses, explicitly showhow to determine a symmetric loss in full from half of one of its partiallosses, introduce an intrinsic parametrisation of composite binary losses andgive 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- andshow there exist situations where two convex surrogate losses areincommensurable. We provide a complete explicit characterisation of theconvexity of composite binary losses in terms of the link function and theweight function associated with the proper loss which make up the compositeloss. This characterisation suggests new ways of ``surrogate tuning-. Finally,in an appendix we present some new algorithm-independent results on therelationship between properness, convexity and robustness to misclassificationnoise for binary losses and show that all convex proper losses are non-robustto misclassification noise.

Author: Mark D. Reid, Robert C. Williamson


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