Simge Küçükyavuz, Ali Shojaie, Hasan Manzour, Linchuan Wei, Hao-Hsiang Wu
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引用次数: 0
摘要
贝叶斯网络(BN)以有向无环图(DAG)的形式表示一组随机变量(节点)之间的条件概率关系,在知识发现领域有着广泛的应用。我们研究的问题是从连续观测数据中学习 BN 的稀疏 DAG 结构。这个核心问题可以建模为一个混合整数程序,其目标函数由一个凸二次损失函数和一个正则化惩罚组成,并受到线性约束。众所周知,该数学程序的最优解在某些条件下具有理想的统计特性。然而,对于中等规模的问题,最先进的优化求解器无法在合理的计算时间内获得现有数学公式的公认最优解。为解决这一难题,我们从计算和统计两个角度着手。一方面,我们提出了一个具体的早期停止准则来终止分支与边界过程,从而获得混合整数程序的近似最优解,并建立了该近似解的一致性。另一方面,我们用二阶圆锥约束取代了表示连续和二进制指标变量之间关系的线性 "big- M "约束,从而改进了现有公式。我们的数值结果证明了所提方法的有效性。
Consistent Second-Order Conic Integer Programming for Learning Bayesian Networks.
Bayesian Networks (BNs) represent conditional probability relations among a set of random variables (nodes) in the form of a directed acyclic graph (DAG), and have found diverse applications in knowledge discovery. We study the problem of learning the sparse DAG structure of a BN from continuous observational data. The central problem can be modeled as a mixed-integer program with an objective function composed of a convex quadratic loss function and a regularization penalty subject to linear constraints. The optimal solution to this mathematical program is known to have desirable statistical properties under certain conditions. However, the state-of-the-art optimization solvers are not able to obtain provably optimal solutions to the existing mathematical formulations for medium-size problems within reasonable computational times. To address this difficulty, we tackle the problem from both computational and statistical perspectives. On the one hand, we propose a concrete early stopping criterion to terminate the branch-and-bound process in order to obtain a near-optimal solution to the mixed-integer program, and establish the consistency of this approximate solution. On the other hand, we improve the existing formulations by replacing the linear "big- " constraints that represent the relationship between the continuous and binary indicator variables with second-order conic constraints. Our numerical results demonstrate the effectiveness of the proposed approaches.
期刊介绍:
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