Convex and Bilevel Optimization for Neuro-Symbolic Inference and Learning

Further, we propose a novel inference algorithm and establish theoretical properties for a state-of-the-art NeSy system that are crucial for learning. Our proposed learning framework builds upon NeSy energy-based models (NeSy-EBMs) (Pryor et al., 2023), a general class of NeSy systems that encompasses a variety of existing NeSy methods, including DeepProblog (Manhaeve et al., 2018; 2021), SATNet (Wang et al., 2019), logic tensor networks (Badreddine et al., 2022), and NeuPSL (Pryor et al., 2023). NeSy-EBMs use neural network outputs to parameterize an energy function and formulate an inference problem that may be non-smooth and constrained. Thus, predictions are not guaranteed to be a function of the inputs and parameters with an explicit form or to be differentiable, and traditional deep learning techniques are not directly applicable. We therefore equivalently formulate NeSy-EBM learning as a bilevel problem and, to support smooth first-order gradient-based optimization, propose a smoothing strategy that is novel to NeSy learning. Specifically, we replace the constrained NeSy energy function with its Moreau envelope. The augmented Lagrangian method for equality-constrained minimization is then applied with the new formulation.