On Learning with Label Proportions

We study a binary learning setting called Learning with Label Proportions (LLP), in which the training data is provided in groups, and for each group only the proportion of the positive instances are given -- The task is to learn a model to predict the labels of the individual instances. LLP has broad applications in political science, marketing, healthcare, and computer vision. Though there are several works about effective algorithms for LLP, one fundamental question is left unanswered: When and why the instances labels can be learned under LLP. To answer this question, we propose a two-step analysis. We first provide a VC-type bound on the generalization error of the bag proportions. We then show that under some conditions, a good bag proportion predictor guarantees a good instance label predictor. We discuss applications of the analysis, including learning with population proportions, and justification of LLP algorithms. We also show one application of LLP, predicting income, based on real census data.