A Unified View on Multi-class Support Vector Classification

A unified view on multi-class support vector machines (SVMs) is presented, covering most prominent variants including the one-vs-all approach and the algorithms proposed by Weston & Watkins, Crammer & Singer, Lee, Lin, & Wahba, and Liu & Yuan. The unification leads to a template for the quadratic training problems and new multi-class SVM formulations. Within our framework, we provide a comparative analysis of the various notions of multi-class margin and margin-based loss. In particular, we demonstrate limitations of the loss function considered, for instance, in the Crammer & Singer machine. We analyze Fisher consistency of multi-class loss functions and universal consistency of the various machines. On the one hand, we give examples of SVMs that are, in a particular hyperparameter regime, universally consistent without being based on a Fisher consistent loss. These include the canonical extension of SVMs to multiple classes as proposed by Weston & Watkins and Vapnik as well as the one-vs-all approach. On the other hand, it is demonstrated that machines based on Fisher consistent loss functions can fail to identify proper decision boundaries in low-dimensional feature spaces. We compared the performance of nine different multi-class SVMs in a thorough empirical study. Our results suggest to use the Weston & Watkins SVM, which can be trained comparatively fast and gives good accuracies on benchmark functions. If training time is a major concern, the one-vs-all approach is the method of choice.