  Crowdsourcing has become a popular method for collecting labeled training
data. However, in many practical scenarios traditional labeling can be
difficult for crowdworkers (for example, if the data is high-dimensional or
unintuitive, or the labels are continuous).
  In this work, we develop a novel model for crowdsourcing that can complement
standard practices by exploiting people's intuitions about groups and relations
between them. We employ a recent machine learning setting, called Ballpark
Learning, that can estimate individual labels given only coarse, aggregated
signal over groups of data points. To address the important case of continuous
labels, we extend the Ballpark setting (which focused on classification) to
regression problems. We formulate the problem as a convex optimization problem
and propose fast, simple methods with an innate robustness to outliers.
  We evaluate our methods on real-world datasets, demonstrating how useful
constraints about groups can be harnessed from a crowd of non-experts. Our
methods can rival supervised models trained on many true labels, and can obtain
considerably better results from the crowd than a standard label-collection
process (for a lower price). By collecting rough guesses on groups of instances
and using machine learning to infer the individual labels, our lightweight
framework is able to address core crowdsourcing challenges and train machine
learning models in a cost-effective way.
