On multiple-instance learning of halfspaces

Dimitrios I. Diochnos, Robert H. Sloan, György Turán

Abstract: In multiple-instance learning the learner receives bags, i.e., sets of instances. A bag is labeled positive if it contains a positive example of the target. An Ω ( d log r ) lower bound is given for the VC-dimension of bags of size r for d-dimensional halfspaces and it is shown that the same lower bound holds for halfspaces over any large point set in general position. This lower bound improves an Ω ( log r ) lower bound of Sabato and Tishby, and it is sharp in order of magnitude. We also show that the hypothesis finding problem is NP-complete and formulate several open problems. Highlights ► We study the complexity of learning halfspaces, a basic learning problem, in the multi-instance model of learning. ► Give a lower bound for the VC-dimension (sharp in order of magnitude). ► The lower bound uses results on cyclic polytopes. ► Also consider both the complexity of hypothesis finding and active learning.