FourierCSP: 通过沃尔什-傅里叶展开实现可微约束满足问题求解
FourierCSP: Differentiable Constraint Satisfaction Problem Solving by Walsh-Fourier Expansion
AI总结：
FourierCSP通过沃尔什-傅里叶展开实现对通用约束满足问题的高效求解，扩展了连续局部搜索框架，提高了求解效率并推动神经符号整合的发展。
摘要：
约束满足问题（CSP）在数学、物理和理论计算机科学中具有基础性地位。连续局部搜索（CLS）求解器作为近期进展，在某些布尔满足性（SAT）问题上能取得高度竞争性的结果。受这些进展的启发，我们将CLS框架从布尔SAT扩展到具有有限域变量和表达性强的约束形式的通用CSP。我们提出了FourierCSP，一种连续优化框架，将沃尔什-傅里叶变换推广到CSP，允许将各种约束转换为紧凑的多项式，从而避免使用辅助变量和内存密集型编码。我们采用具有可证明收敛保证的投影子梯度和镜像下降算法，并进一步结合它们以加速基于梯度的优化。在基准套件上的实验证明，FourierCSP具有可扩展性和竞争力，显著扩大了可以被可微CLS技术高效解决的问题类别，并为端到端神经符号整合铺平了道路。
英文摘要：
The Constraint-satisfaction problem (CSP) is fundamental in mathematics, physics, and theoretical computer science. Continuous local search (CLS) solvers, as recent advancements, can achieve highly competitive results on certain classes of Boolean satisfiability (SAT) problems. Motivated by these advances, we extend the CLS framework from Boolean SAT to general CSP with finite-domain variables and expressive constraint formulations. We present FourierCSP, a continuous optimization framework that generalizes the Walsh-Fourier transform to CSP, allowing for transforming versatile constraints to compact multilinear polynomials, thereby avoiding the need for auxiliary variables and memory-intensive encodings. We employ projected subgradient and mirror descent algorithms with provable convergence guarantees, and further combine them to accelerate gradient-based optimization. Empirical results on benchmark suites demonstrate that FourierCSP is scalable and competitive, significantly broadening the class of problems that can be efficiently solved by differentiable CLS techniques and paving the way toward end-to-end neurosymbolic integration.
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