  Curation of large fully supervised datasets has become one of the major
roadblocks for machine learning. Weak supervision provides an alternative to
supervised learning by training with cheap, noisy, and possibly correlated
labeling functions from varying sources. The key challenge in weakly supervised
learning is combining the different weak supervision signals while navigating
misleading correlations in their errors. In this paper, we propose a simple
data-free approach for combining weak supervision signals by defining a
constrained space for the possible labels of the weak signals and training with
a random labeling within this constrained space. Our method is efficient and
stable, converging after a few iterations of gradient descent. We prove
theoretical conditions under which the worst-case error of the randomized label
decreases with the rank of the linear constraints. We show experimentally that
our method outperforms other weak supervision methods on various text- and
image-classification tasks.
