Rule Induction and Approximate Symbolic Guidance in Neuro-Symbolic Learning

While machine learning techniques, particularly deep learning, have achieved remarkable success across diverse domains, they often lack interpretability, robustness, and the ability to incorporate prior knowledge. On the other hand, symbolic reasoning methods provide a structured and verifiable approach to problem-solving, but struggle with noisy data and the reliance on expert-crafted knowledge bases. Neuro-symbolic approaches seek to combine the strengths of neural and symbolic methods to overcome these limitations. This thesis explores two complementary axes of neuro-symbolic integration. First, we study how symbolic knowledge can be learned within differentiable neural architectures. To this end, we propose RL-Net, a neural framework for learning interpretable ordered rule lists that supports multi-class and multi-label classification while remaining fully differentiable. Second, we investigate how neural models can be trained under the guidance of symbolic knowledge. Here, the challenge we focus on lies in the high computational cost of evaluating complex logical rules. We explore approximation-based methods for incorporating such constraints into training, analyse their impact on learning, and identify limitations of existing approaches. Building on this, we introduce LUBAC, a framework that makes use of both lower- and upper-bounds of complex logical evaluations, compiled into arithmetic circuits. We also provide theoretical guarantees on the quality of gradient approximations obtained through these methods.