Neurosymbolic (NeSy) models integrate neural networks and symbolic reasoning for robust and interpretable AI. State-of-the-art NeSy models require that the symbolic component is expressed in a differentiable way, often complicating the use of approximate inference. We propose EM-NeSy which casts probabilistic NeSy learning as an instance of the Expectation-Maximization (EM) algorithm. In the expectation step, we compute the posterior over the neurally predicted symbols conditioned on the label via probabilistic inference. In the maximization step, we update the neural parameters based on this posterior using gradient descent only through the neural component. This formulation unlocks the full potential of the EM algorithm for NeSy learning. It allows NeSy to extend naturally to approximate reasoning without any additional modifications or differentiability requirements of the symbolic component. Furthermore, it recovers the standard end-to-end gradient-based NeSy setting under exact inference. Our experimental results demonstrate the scalability and computational efficiency of EM-NeSy.

翻译：神经符号（NeSy）模型通过整合神经网络与符号推理，实现鲁棒且可解释的人工智能。当前最先进的NeSy模型要求符号组件以可微形式表达，这通常限制了近似推理的应用。我们提出EM-NeSy方法，将概率性NeSy学习建模为期望最大化（EM）算法的一个实例。在期望步骤中，我们通过概率推理计算以标签为条件、由神经网络预测的符号的后验分布；在最大化步骤中，我们基于该后验仅通过神经组件进行梯度下降，更新神经参数。该公式释放了EM算法在NeSy学习中的全部潜力，使NeSy能够自然地扩展到近似推理，无需对符号组件进行额外修改或施加可微性要求。此外，在精确推理条件下，EM-NeSy可恢复标准的端到端梯度基NeSy设置。实验结果表明了EM-NeSy的可扩展性与计算效率。