부산대학교 도서관

In the standard classification problems, the objective is to categorize points into different classes. Multiple instance learning (MIL), instead, is aimed at classifying bags of points , each point being an instance . The main peculiarity of a MIL problem is that, in the learning phase, only the label of each bag is known whereas the labels of the instances are unknown. We discuss an instance-level learning approach for a binary MIL classification problem characterized by two classes of instances, positive and negative, respectively. In such a problem, a negative bag is constituted only by negative instances, while a bag is positive if it contains at least one positive instance. We start from a mixed integer nonlinear optimization model drawn from the literature and the main result we obtain is to prove that a Lagrangian relaxation approach, equipped with a dual ascent scheme, allows us to obtain an optimal solution of the original problem. The relaxed problem is tackled by means of a block coordinate descent (BCD) algorithm. We provide, finally, the results of our implementation on some benchmark data sets.