  This paper develops a novel methodology for using symbolic knowledge in deep
learning. From first principles, we derive a semantic loss function that
bridges between neural output vectors and logical constraints. This loss
function captures how close the neural network is to satisfying the constraints
on its output. An experimental evaluation shows that it effectively guides the
learner to achieve (near-)state-of-the-art results on semi-supervised
multi-class classification. Moreover, it significantly increases the ability of
the neural network to predict structured objects, such as rankings and paths.
These discrete concepts are tremendously difficult to learn, and benefit from a
tight integration of deep learning and symbolic reasoning methods.
