Multiple-Instance Learning of Real-Valued Geometric Patterns

Recently there has been significant research in multiple-instance learning, yet most of this work has only considered this model when there are Boolean labels. However, in many of the application areas for which the multiple-instance model fits, real-valued labels are more appropriate than Boolean labels. We define and study a real-valued multiple-instance model in which each multiple-instance example (bag) is given a real-valued label in [0, 1] that indicates the degree to which the bag satisfies the target concept. To provide additional structure to the learning problem, we associate a real-valued label with each point in the bag. These values are then combined using a real-valued aggregation operator to obtain the label for the bag. We then present on-line agnostic algorithms for learning real-valued multiple-instance geometric concepts defined by axis-aligned boxes in constant-dimensional space and describe several possible applications of these algorithms. We obtain our learning algorithms by reducing the problem to one in which the exponentiated gradient or gradient descent algorithm can be used. We also give a novel application of the virtual weights technique. In typical applications of the virtual weights technique, all of the concepts in a group have the same weight and prediction, allowing a single representative concept from each group to be tracked. However, in our application there are an exponential number of different weights and possible predictions. Hence, boxes in each group have different weights and predictions, making the computation of the contribution of a group significantly more involved. However, we are able to both keep the number of groups polynomial in the number of trials and efficiently compute the overall prediction.