Numerous neuro-symbolic approaches have recently been proposed typically with
the goal of adding symbolic knowledge to the output layer of a neural network.
Ideally, such losses maximize the probability that the neural network's
predictions satisfy the underlying domain. Unfortunately, this type of
probabilistic inference is often computationally infeasible. Neuro-symbolic
approaches therefore commonly resort to fuzzy approximations of this
probabilistic objective, sacrificing sound probabilistic semantics, or to
sampling which is very seldom feasible. We approach the problem by first
assuming the constraint decomposes conditioned on the features learned by the
network. We iteratively strengthen our approximation, restoring the dependence
between the constraints most responsible for degrading the quality of the
approximation. This corresponds to computing the mutual information between
pairs of constraints conditioned on the network's learned features, and may be
construed as a measure of how well aligned the gradients of two distributions
are. We show how to compute this efficiently for tractable circuits. We test
our approach on three tasks: predicting a minimum-cost path in Warcraft,
predicting a minimum-cost perfect matching, and solving Sudoku puzzles,
observing that it improves upon the baselines while sidestepping
intractability.

该论文探讨如何利用神经符号方法求解具有约束条件的优化问题，为此提出一种计算约束条件之间的互信息的方法，并在三个任务中进行了测试，证明了该方法在提高性能的同时，避免了计算上的复杂性。

神经符号学习的语义强化

Semantic Strengthening of Neuro-Symbolic Learning

This paper describes a revision of the classic Lazy Probabilistic Roadmaps
algorithm (Lazy PRM), that results from pairing PRM and a novel Branch-and-Cut
(BC) algorithm. Cuts are dynamically generated constraints that are imposed on
minimum cost paths over the geometric graphs selected by PRM. Cuts eliminate
paths that cannot be mapped into smooth plans that satisfy suitably defined
kinematic constraints. We generate candidate smooth plans by fitting splines to
vertices in minimum-cost path. Plans are validated with a recently proposed
algorithm that maps them into finite traces, without need to choose a fixed
discretization step. Trace elements exactly describe when plans cross
constraint boundaries modulo arithmetic precision. We evaluate several planners
using our methods over the recently proposed BARN benchmark, and we report
evidence of the scalability of our approach.

本文采用新的 Branch-and-Cut 算法来修订传统的 Lazy Probabilistic Roadmaps 算法，通过在 PRM 选择的几何图上动态生成约束来阻止无法满足运动学约束的路径，评估了我们的方法在最近提出的 BARN 基准测试上的可伸缩性。