伊辛自旋玻璃稀有事件的计算复杂性与模拟
Computational complexity and simulation of rare events of Ising spin glasses
摘要：
我们讨论了随机二维伊辛自旋玻璃的计算复杂性，这代表了一类有趣的黑盒优化约束满足问题。考虑了两种极端情况：（1）+/- J自旋玻璃，和（2）高斯自旋玻璃。我们还研究了这两种极端情况之间的平滑过渡。所有研究的自旋玻璃系统的计算复杂性被发现是由极硬自旋玻璃样品的罕见事件。我们发现，所有研究的自旋玻璃系统的复杂性是密切相关的Frechet极值分布。在一个混合算法，结合了层次贝叶斯优化算法（hBOA）与一个确定性的位翻转爬山，由全局搜索（hBOA）和本地搜索执行的步骤数遵循弗雷歇分布。尽管如此，与纯粹基于局部搜索的方法不同，这些分布的参数证实了hBOA与局部搜索的良好可扩展性。我们进一步认为，优化算法的标准性能指标-例如直到收敛的平均评估次数-可能会产生误导。最后，我们的研究结果表明，高度多模态约束满足的问题，如伊辛自旋眼镜，重组为基础的搜索可以提供更好的质量比基于变异的搜索结果。
英文摘要：
We discuss the computational complexity of random 2D Ising spin glasses, which represent an interesting class of constraint satisfaction problems for black box optimization. Two extremal cases are considered: (1) the +/- J spin glass, and (2) the Gaussian spin glass. We also study a smooth transition between these two extremal cases. The computational complexity of all studied spin glass systems is found to be dominated by rare events of extremely hard spin glass samples. We show that complexity of all studied spin glass systems is closely related to Frechet extremal value distribution. In a hybrid algorithm that combines the hierarchical Bayesian optimization algorithm (hBOA) with a deterministic bit-flip hill climber, the number of steps performed by both the global searcher (hBOA) and the local searcher follow Frechet distributions. Nonetheless, unlike in methods based purely on local search, the parameters of these distributions confirm good scalability of hBOA with local search. We further argue that standard performance measures for optimization algorithms--such as the average number of evaluations until convergence--can be misleading. Finally, our results indicate that for highly multimodal constraint satisfaction problems, such as Ising spin glasses, recombination-based search can provide qualitatively better results than mutation-based search.
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