On Learning from Label Proportions
  Learning from Label Proportions (LLP) is a learning setting, where the training data is provided in groups, or "bags", and only the proportion of each class in each bag is known. The task is to learn a model to predict the class labels of the individual instances. LLP has broad applications in political science, marketing, healthcare, and computer vision. This work answers the fundamental question, when and why LLP is possible, by introducing a general framework, Empirical Proportion Risk Minimization (EPRM). EPRM learns an instance label classifier to match the given label proportions on the training data. Our result is based on a two-step analysis. First, we provide a VC bound on the generalization error of the bag proportions. We show that the bag sample complexity is only mildly sensitive to the bag size. Second, we show that under some mild assumptions, good bag proportion prediction guarantees good instance label prediction. The results together provide a formal guarantee that the individual labels can indeed be learned in the LLP setting. We discuss applications of the analysis, including justification of LLP algorithms, learning with population proportions, and a paradigm for learning algorithms with privacy guarantees. We also demonstrate the feasibility of LLP based on a case study in real-world setting: predicting income based on census data. 
Felix X. Yu, Krzysztof Choromanski, Sanjiv Kumar, Tony Jebara, Shih-Fu   Chang
stat.ML cs.LG