  Structured output prediction problems are ubiquitous in machine learning. The
prominent approach leverages neural networks as powerful feature extractors,
otherwise assuming the independence of the outputs. These outputs, however,
jointly encode an object, e.g. a path in a graph, and are therefore related
through the structure underlying the output space. We discuss the semantic
loss, which injects knowledge about such structure, defined symbolically, into
training by minimizing the network's violation of such dependencies, steering
the network towards predicting distributions satisfying the underlying
structure. At the same time, it is agnostic to the arrangement of the symbols,
and depends only on the semantics expressed thereby, while also enabling
efficient end-to-end training and inference. We also discuss key improvements
and applications of the semantic loss. One limitations of the semantic loss is
that it does not exploit the association of every data point with certain
features certifying its membership in a target class. We should therefore
prefer minimum-entropy distributions over valid structures, which we obtain by
additionally minimizing the neuro-symbolic entropy. We empirically demonstrate
the benefits of this more refined formulation. Moreover, the semantic loss is
designed to be modular and can be combined with both discriminative and
generative neural models. This is illustrated by integrating it into generative
adversarial networks, yielding constrained adversarial networks, a novel class
of deep generative models able to efficiently synthesize complex objects
obeying the structure of the underlying domain.
