Composite Multiclass Losses

We consider loss functions for multiclass prediction problems. We show when a multiclass loss can be expressed as a proper composite loss'', which is the composition of a proper loss and a link function. We extend existing results for binary losses to multiclass losses. We determine the stationarity condition, Bregman representation, order-sensitivity, existence and uniqueness of the composite representation for multiclass losses. We also show that the integral representation for binary proper losses can not be extended to multiclass losses. We subsume existing results onclassification calibration'' by relating it to properness. We draw conclusions concerning the design of multiclass losses.