将代数模型计数的半环思想用于学习，实现跨逻辑与概率模型的统一梯度计算。
The Gradient of Algebraic Model Counting
- 用半环统一框架推广梯度与反向传播至逻辑-概率混合模型
- 利用半环的消去与排序性质，显著降低反向传播内存开销
- 适用于神经符号与统计关系学习，尤其适合高效优化复杂逻辑结构
代数模型计数通过半环统一了多种逻辑公式的推理任务。本文转向学习问题，特别是在结合逻辑、概率与神经表示的统计关系和神经符号人工智能中。我们证明，同样的半环视角同样适用于学习过程，从而将各类学习算法统一于一个通用框架中。该方法可将梯度与反向传播推广到不同的半环空间。此外，我们展示了如何利用半环的消去与排序特性来实现更高效的反向传播，进而获得若干面向概率逻辑模型的先进梯度优化方法变体。我们还分析为何在可满足性电路上的代数模型计数无法提升二阶优化效率。实验表明，所提出的代数反向传播相比现有方法展现出显著的速度提升。
原文摘要 · Abstract (English)
Algebraic model counting unifies many inference tasks on logic formulas by exploiting semirings. Rather than focusing on inference, we consider learning, especially in statistical-relational and neurosymbolic AI, which combine logical, probabilistic and neural representations. Concretely, we show that the very same semiring perspective of algebraic model counting also applies to learning. This allows us to unify various learning algorithms by generalizing gradients and backpropagation to different semirings. Furthermore, we show how cancellation and ordering properties of a semiring can be exploited for more memory-efficient backpropagation. This allows us to obtain some interesting variations of state-of-the-art gradient-based optimisation methods for probabilistic logical models. We also discuss why algebraic model counting on tractable circuits does not lead to more efficient second-order optimization. Empirically, our algebraic backpropagation exhibits considerable speed-ups as compared to existing approaches.
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