A Conjugate Property between Loss Functions and Uncertainty Sets in Classification Problems

In binary classification problems, mainly two approaches have been proposed; one is loss function approach and the other is uncertainty set approach. The loss function approach is applied to major learning algorithms such as support vector machine (SVM) and boosting methods. The loss function represents the penalty of the decision function on the training samples. In the learning algorithm, the empirical mean of the loss function is minimized to obtain the classifier. Against a backdrop of the development of mathematical programming, nowadays learning algorithms based on loss functions are widely applied to real-world data analysis. In addition, statistical properties of such learning algorithms are well-understood based on a lots of theoretical works. On the other hand, the learning method using the socalled uncertainty set is used in hard-margin SVM, mini-max probability machine (MPM) and maximum margin MPM. In the learning algorithm, firstly, the uncertainty set is defined for each binary label based on the training samples. Then, the best separating hyperplane between the two uncertainty sets is employed as the decision function. This is regarded as an extension of the maximum-margin approach. The uncertainty set approach has been studied as an application of robust optimization in the field of mathematical programming. The statistical properties of learning algorithms with uncertainty sets have not been intensively studied. In this paper, we consider the relation between the above two approaches. We point out that the uncertainty set is described by using the level set of the conjugate of the loss function. Based on such relation, we study statistical properties of learning algorithms using uncertainty sets.