让神经网络与逻辑推理协同训练，实现端到端学习。
Neuro-symbolic Learning Yielding Logical Constraints
- 用凸差编程松弛逻辑约束，打通连续与离散的鸿沟。
- 通过基数约束和信任域方法，避免逻辑规则退化。
- 适合需要可解释性推理的AI系统研究者。
神经符号系统融合神经感知与逻辑推理能力，但其端到端学习仍是未解难题。本文提出一种自然框架，将神经网络训练、符号接地与逻辑约束合成整合为高效统一的端到端学习流程。该框架的优势源于训练与推理阶段神经与符号模块间的增强交互。技术上，为弥合连续神经网络与离散逻辑约束之间的差距，引入差-凸规划（difference-of-convex programming）对逻辑约束进行松弛，同时保持其精度。采用基数约束作为逻辑约束学习的语言，并结合信任区域方法防止逻辑约束在学习过程中退化。理论分析与实证评估均验证了该框架的有效性。
原文摘要 · Abstract (English)
Neuro-symbolic systems combine the abilities of neural perception and logical reasoning. However, end-to-end learning of neuro-symbolic systems is still an unsolved challenge. This paper proposes a natural framework that fuses neural network training, symbol grounding, and logical constraint synthesis into a coherent and efficient end-to-end learning process. The capability of this framework comes from the improved interactions between the neural and the symbolic parts of the system in both the training and inference stages. Technically, to bridge the gap between the continuous neural network and the discrete logical constraint, we introduce a difference-of-convex programming technique to relax the logical constraints while maintaining their precision. We also employ cardinality constraints as the language for logical constraint learning and incorporate a trust region method to avoid the degeneracy of logical constraint in learning. Both theoretical analyses and empirical evaluations substantiate the effectiveness of the proposed framework.
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