  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.
