Empirical Risk Minimization for Variable Consistency Dominance-Based Rough Set Approach

Abstract

The paper concerns reasoning about partially inconsistent ordinal data. We reveal the relation between Variable Consistency Dominance-based Rough Set Approach (VC-DRSA) and the empirical risk minimization problem. VC-DRSA is an extension of DRSA that admits some degree of inconsistency with respect to dominance, which is controlled by thresholds on a consistency measure. To prove this relation, we first solve an optimization problem to find thresholds ensuring assignment of a maximum number of objects under disjoint and balanced setting of extended lower approximations of two complementary unions of ordered decision classes: “at least” class i, and “at most” class $i-1$, for a given $i \in \{2, \ldots , p\}$, where p is the total number of classes. For a given i, each object is supposed to be assigned to at most one of the two extended lower approximations. Moreover, the assignment is not influenced by unions’ cardinalities. Second, we prune the set of objects not assigned to any extended lower approximation. Then, from a definable set, for a given i, we derive a classification function, which indicates assignment of an object to one of the two unions of decision classes. We define empirical risk associated with the classification function as a hinge loss function. We prove that the classification function minimizing the empirical risk function corresponds to the extended lower approximation in VC-DRSA involving thresholds obtained from the above optimization problem, followed by the pruning.