  As larger deep learning models are hard to interpret, there has been a recent
focus on generating explanations of these black-box models. In contrast, we may
have apriori explanations of how models should behave. In this paper, we
formalize this notion as learning from explanation constraints and provide a
learning theoretic framework to analyze how such explanations can improve the
learning of our models. One may naturally ask, "When would these explanations
be helpful?" Our first key contribution addresses this question via a class of
models that satisfies these explanation constraints in expectation over new
data. We provide a characterization of the benefits of these models (in terms
of the reduction of their Rademacher complexities) for a canonical class of
explanations given by gradient information in the settings of both linear
models and two layer neural networks. In addition, we provide an algorithmic
solution for our framework, via a variational approximation that achieves
better performance and satisfies these constraints more frequently, when
compared to simpler augmented Lagrangian methods to incorporate these
explanations. We demonstrate the benefits of our approach over a large array of
synthetic and real-world experiments.
