  A supervised learning framework has been proposed for the situation where
each training data is provided with a complementary label that represents a
class to which the pattern does not belong. In the existing literature,
complementary-label learning has been studied independently from ordinary-label
learning, which assumes that each training data is provided with a label
representing the class to which the pattern belongs. However, providing a
complementary label should be treated as equivalent to providing the rest of
all the labels as the candidates of the one true class. In this paper, we focus
on the fact that the loss functions for one-versus-all and pairwise
classification corresponding to ordinary-label learning and complementary-label
learning satisfy certain additivity and duality, and provide a framework which
directly bridge those existing supervised learning frameworks. Further, we
derive classification risk and error bound for any loss functions which satisfy
additivity and duality.
