Neuro-symbolic learning (NSL) models complex symbolic rule patterns into latent variable distributions by neural networks, which reduces rule search space and generates unseen rules to improve downstream task performance. Centralized NSL learning involves directly acquiring data from downstream tasks, which is not feasible for federated learning (FL). To address this limitation, we shift the focus from such a one-to-one interactive neuro-symbolic paradigm to one-to-many Federated Neuro-Symbolic Learning framework (FedNSL) with latent variables as the FL communication medium. Built on the basis of our novel reformulation of the NSL theory, FedNSL is capable of identifying and addressing rule distribution heterogeneity through a simple and effective Kullback-Leibler (KL) divergence constraint on rule distribution applicable under the FL setting. It further theoretically adjusts variational expectation maximization (V-EM) to reduce the rule search space across domains. This is the first incorporation of distribution-coupled bilevel optimization into FL. Extensive experiments based on both synthetic and real-world data demonstrate significant advantages of FedNSL compared to five state-of-the-art methods. It outperforms the best baseline by 17% and 29% in terms of unbalanced average training accuracy and unseen average testing accuracy, respectively.

翻译：神经符号学习（NSL）通过神经网络将复杂的符号规则模式建模为隐变量分布，从而缩减规则搜索空间并生成未见规则以提升下游任务性能。集中式NSL学习需要直接从下游任务获取数据，这在联邦学习（FL）场景中难以实现。为突破此限制，我们将研究重心从这种一对一的交互式神经符号范式，转向以隐变量作为FL通信媒介的一对多联邦神经符号学习框架（FedNSL）。基于我们对NSL理论的全新重构，FedNSL能够通过对规则分布施加适用于FL设置的简洁有效的KL散度约束，识别并处理规则分布异质性问题。该框架进一步从理论上调整变分期望最大化（V-EM）算法，以缩减跨领域的规则搜索空间。这是首次将分布耦合的双层优化引入FL的研究。基于合成数据与真实数据的大量实验表明，FedNSL相较于五种前沿方法具有显著优势：在不平衡平均训练准确率与未见平均测试准确率两项指标上，分别超出最佳基线方法17%和29%。