  In the problem of learning with label proportions, which we call LLP
learning, the training data is unlabeled, and only the proportions of examples
receiving each label are given. The goal is to learn a hypothesis that predicts
the proportions of labels on the distribution underlying the sample. This model
of learning is applicable to a wide variety of settings, including predicting
the number of votes for candidates in political elections from polls.
  In this paper, we formally define this class and resolve foundational
questions regarding the computational complexity of LLP and characterize its
relationship to PAC learning. Among our results, we show, perhaps surprisingly,
that for finite VC classes what can be efficiently LLP learned is a strict
subset of what can be leaned efficiently in PAC, under standard complexity
assumptions. We also show that there exist classes of functions whose
learnability in LLP is independent of ZFC, the standard set theoretic axioms.
This implies that LLP learning cannot be easily characterized (like PAC by VC
dimension).
