贝叶斯网络结构学习：多面体、刻面和复杂性
Bayesian Network Structure Learning with Integer Programming: Polytopes, Facets, and Complexity
摘要：
学习概率图模型结构的挑战性任务是现代AI研究中的一个重要问题。近年来，贝叶斯网络的结构学习算法取得了一些重大进展-可以说是最核心的一类图形模型-特别是在所谓的基于分数的设置中。在GOBNILP系统中实现了一种基于整数规划（IP）的最优贝叶斯网络结构学习（BNSL）的通用方法。尽管最近的算法的进步，目前的理解的基础方面的IP为基础的方法BNSL仍然有些缺乏。理解切割平面的基本方面和相关的分离问题（不仅从纯理论的角度来看很重要，而且因为它提供了进一步提高最先进方法的效率的希望，以准确地解决BNSL。在本文中，我们对这些目标作出了一些理论贡献：（i）我们研究了分离问题的计算复杂性，证明了该问题是NP-难的;（ii）我们形式化和分析了基于IP的BNSL方法的三个关键多面体之间的关系;（iii）我们从理论和实践的角度研究了三个多面体的方面，通过穷举计算，一个完整的枚举方面的低维家庭可变多面体;和，此外，（四）我们建立了紧密的连接的BNSL问题的无环子图问题。
英文摘要：
The challenging task of learning structures of probabilistic graphical models is an important problem within modern AI research. Recent years have witnessed several major algorithmic advances in structure learning for Bayesian networks---arguably the most central class of graphical models---especially in what is known as the score-based setting. A successful generic approach to optimal Bayesian network structure learning (BNSL), based on integer programming (IP), is implemented in the GOBNILP system. Despite the recent algorithmic advances, current understanding of foundational aspects underlying the IP based approach to BNSL is still somewhat lacking. Understanding fundamental aspects of cutting planes and the related separation problem( is important not only from a purely theoretical perspective, but also since it holds out the promise of further improving the efficiency of state-of-the-art approaches to solving BNSL exactly. In this paper, we make several theoretical contributions towards these goals: (i) we study the computational complexity of the separation problem, proving that the problem is NP-hard; (ii) we formalise and analyse the relationship between three key polytopes underlying the IP-based approach to BNSL; (iii) we study the facets of the three polytopes both from the theoretical and practical perspective, providing, via exhaustive computation, a complete enumeration of facets for low-dimensional family-variable polytopes; and, furthermore, (iv) we establish a tight connection of the BNSL problem to the acyclic subgraph problem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
