We present DL2, a system for training and querying neural networks with logical constraints. DL2 is more expressive than prior work and can capture a richer class of constraints on inputs, outputs and internals of models. Using DL2, one can declaratively specify domain knowledge to be enforced during training or pose queries on the model with the goal of finding inputs that satisfy a given constraint. DL2 works by translating logical constraints into a differentiable loss with desirable mathematical properties, then minimized with standard gradient-based methods. Our evaluation demonstrates that DL2 can express interesting constraints beyond the reach of prior work, leading to improved prediction accuracy.
Summary
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This paper proposes DL2, a framework for turning queries over parameters and input, output pairs to neural networks into differentiable loss functions, and an associated declarative language for specifying these queries. The motivation for this work is twofold. The first is to allow for the specification of additional domain knowledge during training. For example, if a user expects that the predicted probabilities of some output classes should be correlated for all predictions, this constraint can be enforced during weight learning. Second, it allows users to search for specific inputs that satisfy specified conditions. In this way, DL2 can capture popular applications like searching for adversarial examples by querying for inputs close to a known input of class A but that the network predicts is class B with high confidence.
The paper provides a concise specification of the query language (a mixture of logical and numeric operators) and asserts a theorem that the given procedure for constructing the query loss produces a function such that anytime the function is 0, the constraints are satisfied. No proof is given, but I cannot see a counterexample. There is also a statement about the converse relationship, that when the loss is above some threshold it implies that the query is not satisfied. 
Experiments are conducted on supervised, semi-supervised, and unsupervised computer vision tasks. I particularly liked the experiment on semi-supervised learning with CIFAR-100. By replacing labeled examples with domain knowledge about the relationships among classes in CIFAR-100, the paper demonstrates a compelling use case for DL2.
The primary technical challenge is the non-convex optimization required to search for a solution to a query. Experiments show that the loss functions created by DL2 are often solved quickly and correctly, but not always
Strengths
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The framework is expressive enough that many interesting use cases are clear, from specifying background knowledge during training to model inspection. The experiments cover a range of these use cases, demonstrating that the constructed optimization objectives usually work as intended.
Weaknesses
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The statement in Theorem 1 regarding the converse case is unclear, because it says that the limit of \delta as \epsilon approaches zero is zero, but it is not explained what \epsilon is or how it changes. If \epsilon is the threshold that can often be used in the query, it is not obvious that every query contains exactly one \epsilon. If other cases exist, it is unclear how Theorem 1 applies.
It remains unknown how to handle the case when queries fail. AS the paper points out, if a query fails, it cannot be determined whether no solution exists or if the optimization simply failed to find a solution. Of course, this is a computationally hard in general.
Related Work
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There are a couple of points from related work that would be good to add to the paper.
First, the paper "Adversarial Sets for Regularising Neural Link Predictors" (Minervini et al., UAI17) is a prior paper that generates adversarial examples to handle restrictions on inputs which may not exist in the training set. The paper claims DL2 is the first to do this, but I believe this paper is an earlier example that does so, albeit for a particular problem. DL2 is certainly more general.
Second, the description of the limitations of rule distillation (Hu et al., ACL16), particularly in Appendix A is not fully accurate. The expressivity of PSL is greater than stated (see Bach et al., JMLR17 for a full description). In particular, the DL2 loss function for z = (1, 1) can be expressed exactly in PSL using what it calls arithmetic rules. It is not clear that this affects the findings of the semi-supervised learning experiment significantly, although I would appreciate a clarification of the authors. PSL by construction produces convex loss functions, and so the constraint that all outputs for a group of classes is either high OR low would probably not work well.