Coherence Functions with Applications in Large-Margin Classification Methods

TL;DR

This paper introduces coherence functions as smooth, convex surrogates for hinge loss in large-margin classifiers, enabling direct probability estimation and bridging hinge and logistic functions, with efficient algorithms for training.

Contribution

It proposes a novel family of coherence functions derived from the maximum-entropy principle, connecting hinge and logistic functions, and develops algorithms for their efficient training.

Findings

Coherence functions smoothly approximate hinge loss.

C-learning enables probability estimation in large-margin classifiers.

Efficient coordinate descent algorithms are developed for training.

Abstract

Support vector machines (SVMs) naturally embody sparseness due to their use of hinge loss functions. However, SVMs can not directly estimate conditional class probabilities. In this paper we propose and study a family of coherence functions, which are convex and differentiable, as surrogates of the hinge function. The coherence function is derived by using the maximum-entropy principle and is characterized by a temperature parameter. It bridges the hinge function and the logit function in logistic regression. The limit of the coherence function at zero temperature corresponds to the hinge function, and the limit of the minimizer of its expected error is the minimizer of the expected error of the hinge loss. We refer to the use of the coherence function in large-margin classification as C-learning, and we present efficient coordinate descent algorithms for the training of regularized ${\cal C}$-learning models.