We address a key challenge for neuro-symbolic (NeSy) systems by leveraging convex and bilevel optimization techniques to develop a general gradient-based framework for end-to-end neural and symbolic parameter learning. The applicability of our framework is demonstrated with NeuPSL, a state-of-the-art NeSy architecture. To achieve this, we propose a smooth primal and dual formulation of NeuPSL inference and show learning gradients are functions of the optimal dual variables. Additionally, we develop a dual block coordinate descent algorithm for the new formulation that naturally exploits warm-starts. This leads to over 100x learning runtime improvements over the current best NeuPSL inference method. Finally, we provide extensive empirical evaluations across $8$ datasets covering a range of tasks and demonstrate our learning framework achieves up to a 16% point prediction performance improvement over alternative learning methods.

翻译：我们通过利用凸优化和双层优化技术，解决了神经符号（NeSy）系统的一个关键挑战，开发了一种通用的基于梯度的端到端神经与符号参数学习框架。通过最先进的NeSy架构NeuPSL，我们证明了该框架的适用性。为实现这一目标，我们提出了NeuPSL推理的光滑原始-对偶形式，并证明学习梯度是最优对偶变量的函数。此外，我们为该新形式开发了一种自然利用热启动的对偶块坐标下降算法。这使得学习运行时间相比当前最佳的NeuPSL推理方法提升了100倍以上。最后，我们在涵盖多种任务的8个数据集上进行了广泛的实证评估，结果表明我们的学习框架相比其他替代学习方法，预测性能最多提升了16个百分点。