  Injecting discrete logical constraints into neural network learning is one of
the main challenges in neuro-symbolic AI. We find that a
straight-through-estimator, a method introduced to train binary neural
networks, could effectively be applied to incorporate logical constraints into
neural network learning. More specifically, we design a systematic way to
represent discrete logical constraints as a loss function; minimizing this loss
using gradient descent via a straight-through-estimator updates the neural
network's weights in the direction that the binarized outputs satisfy the
logical constraints. The experimental results show that by leveraging GPUs and
batch training, this method scales significantly better than existing
neuro-symbolic methods that require heavy symbolic computation for computing
gradients. Also, we demonstrate that our method applies to different types of
neural networks, such as MLP, CNN, and GNN, making them learn with no or fewer
labeled data by learning directly from known constraints.
