We present an objective function for learning with unlabeled data that utilizes auxiliary expectation constraints. We optimize this objective function using a procedure that alternates between information and moment projections. Our method provides an alternate interpretation of the posterior regularization framework (Graca et al., 2008), maintains uncertainty during optimization unlike constraint-driven learning (Chang et al., 2007), and is more efficient than generalized expectation criteria (Mann & McCallum, 2008). Applications of this framework include minimally supervised learning, semisupervised learning, and learning with constraints that are more expressive than the underlying model. In experiments, we demonstrate comparable accuracy to generalized expectation criteria for minimally supervised learning, and use expressive structural constraints to guide semi-supervised learning, providing a 3%-6% improvement over stateof- the-art constraint-driven learning.