Symbolic knowledge can provide crucial inductive bias for training neural models, especially in low data regimes. A successful strategy for incorporating such knowledge involves relaxing logical statements into sub-differentiable losses for optimization. In this paper, we study the question of how best to relax logical expressions that represent labeled examples and knowledge about a problem; we focus on sub-differentiable t-norm relaxations of logic. We present theoretical and empirical criteria for characterizing which relaxation would perform best in various scenarios. In our theoretical study driven by the goal of preserving tautologies, the Lukasiewicz t-norm performs best. However, in our empirical analysis on the text chunking and digit recognition tasks, the product t-norm achieves best predictive performance. We analyze this apparent discrepancy, and conclude with a list of best practices for defining loss functions via logic.

本文研究如何最佳地放松逻辑表达式，以便将符号知识纳入到神经网络中，重点是利用亚微分的 t-范数放松逻辑表达式。通过理论和实证研究，我们为表征在不同情况下哪种松弛方式表现最佳提供了理论和实证标准，并针对在保留重言式方面，Lukasiewicz t-范数表现最佳的观点进行了理论研究，但在对文本分块和数字识别任务的实证分析中，产品 t-范数实现了最佳预测性能。最后，我们总结了利用逻辑定义损失函数的最佳实践。

评估神经网络逻辑松弛：一项全面研究