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In binary classification problems, mainly two approaches have been proposed;one is loss function approach and the other is uncertainty set approach. Theloss function approach is applied to major learning algorithms such as supportvector machine (SVM) and boosting methods. The loss function represents thepenalty of the decision function on the training samples. In the learningalgorithm, the empirical mean of the loss function is minimized to obtain theclassifier. Against a backdrop of the development of mathematical programming,nowadays learning algorithms based on loss functions are widely applied toreal-world data analysis. In addition, statistical properties of such learningalgorithms are well-understood based on a lots of theoretical works. On theother hand, the learning method using the so-called uncertainty set is used inhard-margin SVM, mini-max probability machine (MPM) and maximum margin MPM. Inthe learning algorithm, firstly, the uncertainty set is defined for each binarylabel based on the training samples. Then, the best separating hyperplanebetween the two uncertainty sets is employed as the decision function. This isregarded as an extension of the maximum-margin approach. The uncertainty setapproach has been studied as an application of robust optimization in the fieldof mathematical programming. The statistical properties of learning algorithmswith uncertainty sets have not been intensively studied. In this paper, weconsider the relation between the above two approaches. We point out that theuncertainty set is described by using the level set of the conjugate of theloss function. Based on such relation, we study statistical properties oflearning algorithms using uncertainty sets.

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