Coherence Functions with Applications in Large-Margin Classification Methods
This work addresses a limitation in large-margin classification for practitioners needing probability estimates, offering an incremental improvement over existing methods.
The paper tackles the inability of support vector machines (SVMs) to estimate conditional class probabilities by proposing coherence functions as convex, differentiable surrogates for the hinge loss, bridging it with logistic regression. It introduces C-learning with efficient coordinate descent algorithms for training regularized models, showing that the minimizer of the coherence function's expected error converges to that of the hinge loss.
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.