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1. Injecting Logical Constraints into Neural Networks via Straight-Through Estimators

   Yang, Zhun ; Lee, Joohyung ; Park, Chiyoun ( July 2022 , Proceedings of Machine Learning Research)

   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. 

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2. A Semantic Loss Function for Deep Learning with Symbolic Knowledge

   Xu, Jingyi ; Zhang, Zilu ; Friedman, Tal ; Liang, Yitao ; Van den Broeck, Guy ( July 2018 , Proceedings of the 35th International Conference on Machine Learning (ICML))

   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. 

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3. Pylon: A PyTorch Framework for Learning with Constraints

   https://doi.org/10.1609/aaai.v36i11.21711  

   Ahmed, Kareem ; Li, Tao ; Ton, Thy ; Guo, Quan ; Chang, Kai-Wei ; Kordjamshidi, Parisa ; Srikumar, Vivek ; Van den Broeck, Guy ; Singh, Sameer ( January 2022 , Proceedings of the NeurIPS 2021 Competitions and Demonstrations Track)

   Kiela, Douwe ; Ciccone, Marco ; Caputo, Barbara (Ed.)

   Deep learning excels at learning low-level task information from large amounts of data, but struggles with learning high-level domain knowledge, which can often be directly and succinctly expressed. In this work, we introduce Pylon, a neuro-symbolic training framework that builds on PyTorch to augment procedurally trained neural networks with declaratively specified knowledge. Pylon allows users to programmatically specify constraints as PyTorch functions, and compiles them into a differentiable loss, thus training predictive models that fit the data whilst satisfying the specified constraints. Pylon includes both exact as well as approximate compilers to efficiently compute the loss, employing fuzzy logic, sampling methods, and circuits, ensuring scalability even to complex models and constraints. A guiding principle in designing Pylon has been the ease with which any existing deep learning codebase can be extended to learn from constraints using only a few lines: a function expressing the constraint and a single line of code to compile it into a loss. We include case studies from natural language processing, computer vision, logical games, and knowledge graphs, that can be interactively trained, and highlights Pylon’s usage. 

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4. Prompt2DeModel: Declarative Neuro-Symbolic Modeling with Natural Language

   Faghihi, Hossein Rajaby ; Nafar, Aliakbar ; Uszok, Andrzej ; Karimian, Hamid ; Kordjamshidi, Parisa ( September 2024 , Springer-Verlag)

   This paper presents a conversational pipeline for crafting domain knowledge for complex neuro-symbolic models through natural language prompts. It leverages large language models to generate declarative programs in the DomiKnowS framework. The programs in this framework express concepts and their relationships as a graph in addition to logical constraints between them. The graph, later, can be connected to trainable neural models according to those specifications. Our proposed pipeline utilizes techniques like dynamic in-context demonstration retrieval, model refinement based on feedback from a symbolic parser, visualization, and user interaction to generate the tasks’ structure and formal knowledge representation. This approach empowers domain experts, even those not well-versed in ML/AI, to formally declare their knowledge to be incorporated in customized neural models in the DomiKnowS framework. 

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5. NeuPSL: Neural Probabilistic Soft Logic

   Pryor, Connor ; Dickens, Charles Andrew ; Augustine, Eriq ; Albalak ; Alon ; Wang ; William Yang ; Getoor, Lise ( June 2022 , ArXivorg)

   We present Neural Probabilistic Soft Logic (NeuPSL), a novel neuro-symbolic (NeSy) framework that unites state-of-the-art symbolic reasoning with the low-level perception of deep neural networks. To explicitly model the boundary between neural and symbolic representations, we introduce NeSy Energy-Based Models, a general family of energy-based models that combine neural and symbolic reasoning. Using this framework, we show how to seamlessly integrate neural and symbolic parameter learning and inference. We perform an extensive empirical evaluation and show that NeuPSL outperforms existing methods on joint inference and has significantly lower variance in almost all settings. 

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