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In the problem of learning a class ratio from unlabeled data, which we call CR learning, the training data is unlabeled, and only the ratios, or 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 CR learning and characterize its relationship to PAC learning. Among our results, we show, perhaps surprisingly, that for finite VC classes what can be efficiently CR learned is a strict subset of what can be learned efficiently in PAC, under standard complexity assumptions. We also show that there exist classes of functions whose CR learnability is independent of ZFC, the standard set theoretic axioms. This implies that CR learning cannot be easily characterized (like PAC by VC dimension). 

The study of influence maximization in social networks has largely ignored disparate effects these algorithms might have on the individuals contained in the social network. Individuals may place a high value on receiving information, e.g. job openings or advertisements for loans. While well-connected individuals at the center of the network are likely to receive the information that is being distributed through the network, poorly connected individuals are systematically less likely to receive the information, producing a gap in access to the information between individuals. In this work, we study how best to spread information in a social network while minimizing this access gap. We propose to use the maximin social welfare function as an objective function, where we maximize the minimum probability of receiving the information under an intervention. We prove that in this setting this welfare function constrains the access gap whereas maximizing the expected number of nodes reached does not. We also investigate the difficulties of using the maximin, and present hardness results and analysis for standard greedy strategies. Finally, we investigate practical ways of optimizing for the maximin, and give empirical evidence that a simple greedy-based strategy works well in practice.