Multicategory large-margin unified machines

Hard and soft classifiers are two important groups of techniques for classification problems. Logistic regression and Support Vector Machines are typical examples of soft and hard classifiers respectively. The essential difference between these two groups is whether one needs to estimate the class conditional probability for the classification task or not. In particular, soft classifiers predict the label based on the obtained class conditional probabilities, while hard classifiers bypass the estimation of probabilities and focus on the decision boundary. In practice, for the goal of accurate classification, it is unclear which one to use in a given situation. To tackle this problem, the Large-margin Unified Machine (LUM) was recently proposed as a unified family to embrace both groups. The LUM family enables one to study the behavior change from soft to hard binary classifiers. For multicategory cases, however, the concept of soft and hard classification becomes less clear. In that case, class probability estimation becomes more involved as it requires estimation of a probability vector. In this paper, we propose a new Multicategory LUM (MLUM) framework to investigate the behavior of soft versus hard classification under multicategory settings. Our theoretical and numerical results help to shed some light on the nature of multicategory classification and its transition behavior from soft to hard classifiers. The numerical results suggest that the proposed tuned MLUM yields very competitive performance.