The Overlap Gap Property：随机结构优化的几何障碍
The Overlap Gap Property: a Geometric Barrier to Optimizing over Random Structures
摘要：
优化随机结构的问题出现在科学和工程的许多领域，从统计物理到机器学习和人工智能。对于许多这样的结构，通过快速算法找到最优解是未知的，并且通常被认为是不可能的。与此同时，这些问题的形式上的硬度，比如说复杂性理论的$NP$硬度是缺乏的。在这篇介绍性的文章中，描述了一种新的方法在随机结构中的算法棘手性，这是基于拓扑不连通性的一组成对的距离近最佳的解决方案，称为重叠间隙属性。这篇文章演示了这个属性a）如何出现在大多数已知表现出明显算法硬度的模型中b）与许多模型的硬度/易处理性相变相一致，重要的是c）允许在数学上严格排除大类算法作为潜在竞争者，特别是表现出输入稳定性（不敏感性）的算法。
英文摘要：
The problem of optimizing over random structures emerges in many areas of science and engineering, ranging from statistical physics to machine learning and artificial intelligence. For many such structures finding optimal solutions by means of fast algorithms is not known and often is believed not possible. At the same time the formal hardness of these problems in form of say complexity-theoretic $NP$-hardness is lacking. In this introductory article a new approach for algorithmic intractability in random structures is described, which is based on the topological disconnectivity property of the set of pair-wise distances of near optimal solutions, called the Overlap Gap Property. The article demonstrates how this property a) emerges in most models known to exhibit an apparent algorithmic hardness b) is consistent with the hardness/tractability phase transition for many models analyzed to the day, and importantly c) allows to mathematically rigorously rule out large classes of algorithms as potential contenders, in particular the algorithms exhibiting the input stability (insensitivity).
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