Learning multiclass rules with class-selective rejection and performance constraints

The task of classification occurs in a wide range of human activity. The problem concerns learning a decision rule that allows to assign a pattern to a decision option on the basis of observed attributes or features. Contexts in which a classification task is fundamental include, sorting letters on the basis of machine-read postcodes, the preliminary diagnosis of a patient’s disease or the fraud currency and documents detection. In the classical framework, decision options are given by the pre-defined classes and a decision rule is designed by optimizing a given loss function, for instance the misclassification rate. In some cases, the loss function should be more general. First, for some applications, like face identification or cancer diagnosis, one may favor withholding decision instead of taking a wrong decision. In such cases, the introduction of rejection options should be considered in order to ensure a higher reliability Ha (1997); Horiuchi (1998); Jrad, Grall-Maës & Beauseroy (2008); Jrad et al. (2009d). Basic rejection consists of assigning a pattern to all classes which means that no decision is taken. More advanced rejection methods enable to assign a pattern ambiguously to a subset of classes. In this class-selective rejection scheme, decision options are given by the pre-defined classes as well as by defined subsets of different combinations among these classes. In order to define a decision rule, a general loss function can be defined by costs that penalize differently the wrong decisions and the ambiguous ones. Some applications may require to control the performance of the decision rule or more specifically, the performance measured by indicators related to the decision rule. These latter could be formulated as the performance constraints. Hence, the decision problem should also take into account these constraints. A general formulation of this problem was proposed in GrallMaës & Beauseroy (2009). The decision problem is formulated as an optimization problem with constraints. It was shown that the optimal rule can be obtained by optimizing its Lagrangian dual function which consists of finding the saddle point of this Lagrangian function. This optimal theoretical rule is applicable when the probability distributions are known. However, in many applications, only amounts of training set is available. Therefore, one should infer a classifier from a more or less limited set of training examples. In the classical decision framework, referred as the classical framework, many historical strands of research can be identified: statistical, Support Vector Machines, Neural Network Bishop (2006); Guobin & Lu (2007); Hao & Lin (2007); Husband & Lin (2002); Vapnik (1998); Yang et al. (2007)... In the class-selective rejection scheme, fewer works have been done Ha (1997); Horiuchi (1998). 1