# Coherence Functions for Multicategory Margin-based Classification Methods

[edit]

Zhihua Zhang,
Michael Jordan,
Wu-Jun Li,
Dit-Yan Yeung
;

Proceedings of the Twelth International Conference on Artificial Intelligence and Statistics, PMLR 5:647-654, 2009.

#### Abstract

Margin-based classification methods are typically devised based on a majorization-minimization procedure, which approximately solves an otherwise intractable minimization problem defined with the 0-l loss. However, extension of such methods from the binary classification setting to the more general multicategory setting turns out to be non-trivial. In this paper, our focus is to devise margin-based classification methods that can be seamlessly applied to both settings, with the binary setting simply as a special case. In particular, we propose a new majorization loss function that we call the coherence function, and then devise a new multicategory margin-based boosting algorithm based on the coherence function. Analogous to deterministic annealing, the coherence function is characterized by a temperature factor. It is closely related to the multinomial log-likelihood function and its limit at zero temperature corresponds to a multicategory hinge loss function.

#### Related Material