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 our semantic loss function 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.
SUMMARY 
The paper proposes a new form of regularization utilizing logical constraints. The semantic loss function is built on the exploitation of symbolic knowledge extracted from data and connecting the logical constraints to the outputs of a neural network. The use of Boolean logic as a constraint provides a secondary regularization term to prevent over-fitting and improve predictions. The benefit of using the function is found primarily with semi-supervised tasks where data is partially unlabelled. The logical constraints provided by the semantic loss function allow for improved classification of unlabeled data.
Output constraints for the semantic loss function are represented with one-hot encoding, prefer- ence rankings, and paths in a grid. These three different output constraints are designed to explore different learning purposes. The semantic function was tested on both semi-supervised classifica- tion tasks as well as structure learning. The paper primarily focuses on the one-hot encoding constraint as it is viewed as a capable technique for multi-class classification.
POSITIVES 
In terms of structure, the paper was written very well. Sufficient background information was con- veyed which helped in understanding the proposed semantic loss function. A thorough breakdown is also carried out on the semantic loss function itself by explaining its axioms which help explain how the outputs of a neural network match a given constraint.
As a scientific contribution, I would say results from the experiments were able to justify the proposal of the semantic loss function. The function was able to perform better than most other implementations for semi-supervised learning tasks, and the function was tested on multiple datasets. The paper also made use of testing the function against other notable machine learning approaches, and in most cases the function performed better, but this usually was confined to semi-supervised learning tasks. During supervised learning tasks the function did not perform markedly better than older implementations. Given that, the semantic loss function did prove to be a seemingly simple approach to improving semi-supervised classification tasks.
• The background section covers the knowledge required in understanding the semantic loss function. The paper also clearly explains the meaning for some of the notation used in the definitions.
• Experiments which clearly show the benefit of using the semantic loss function. Multiple experiment types were done as well which showed evidence of the broad applicability of the function.
• In depth description of the definitions, axioms, and propositions of the semantic loss function.
• A large number of experiments exploring the usefulness of the function for multiple learning tasks, and on multiple datasets.
NEGATIVES 
I was not clear if the logical constraints are to be instantiated before learning, i.e. they are defined by hand prior to being implemented in the neural network. This is a pretty important question and drastically changes the nature of the learning process. Beyond that complaint, the paper did not suffer from any critical issues. There were some issues with spelling, and the section titled ’Algorithm’ fails to clearly define a complete algorithm using the semantic loss function. It would have helped to have two algorithms. One defining the pipeline for the semantic loss function, and another showing the implementation of the function in a machine learning framework. The semantic loss function found success only in cases were the learning task was semi-supervised, and not in cases of total supervised learning. This is not a true negative, but an observation on the effectiveness of the function.
- A few typos in the paper.
- The axioms for the semantic loss function where defined but there seemed to be a lack of a clear algorithm provided showing the pipeline implementation of the semantic loss function.
- While the semantic loss function does improve learning performance in most cases, the im- provements are confined to semi-supervised learning tasks, and with the MNIST dataset another methodology, Ladder Nets, was able to outperform the semantic loss function.
RELATED WORK
The paper proposed that logic constraints applied to the output of neural networks have the capacity to improve semi-supervised classification tasks as well as finding the shortest path. In the introduction, the paper lists Zhiting Hu et al. paper titled Harnessing Deep Neural Networks with Logic Rules as an example of a similar approach. Hu et al. paper utilized logic constraints in conjunction with neural nets as well. A key difference was that Hu et al. applied their network architecture to supervised classification tasks. Since the performance of the current papers semantic loss function with supervised tasks did not improve upon other methods, it may benefit to utilize the research by Hu et al. as a means of direct comparison for supervised learning tasks, and possibly incorporate their methods with the semantic loss function in order to improve upon supervised learning tasks.
CONCLUSION
Given the success of the semantic loss function with semi-supervised tasks, I would accept this paper. The semantic loss was able to improve learning with respect to the tested datasets, and the paper clearly described the properties of the functions. The paper would benefit by including a more concrete algorithm describing the flow of data through a given neural net to the semantic loss function, as well as the process by which the semantic loss function constrains the data based on propositional logic, but in general this complaint is more nit picking. The semantic loss function and the experiments which tested the function showed clearly that there is a benefit to this research and there are areas for it to improve.