This paper provides learning theoretic guarantees for a semi-supervised or transfer-type problem of learning from "label proportions" where several bags of data are provided but without labels for every point, instead having access to the label proportion within each bag. The problem is clearly challenging to analyze theoretically, so the authors have taken on a difficult task and done a decent job. Three reviewers were warmly receptive (a fourth has low confidence), but nobody strongly arguing for rejection nor acceptance in the discussion. Overall, it is a nice paper, and regardless of the way this paper finally goes, the authors would benefit from discussing some of the finer details of their assumptions and theorems.