Consistent Second-Order Conic Integer Programming for Learning Bayesian Networks

• URL: http://arxiv.org/abs/2005.14346v3
• Date: Fri, 6 May 2022 19:51:52 GMT
• Title: Consistent Second-Order Conic Integer Programming for Learning Bayesian Networks
• Authors: Simge Kucukyavuz, Ali Shojaie, Hasan Manzour, Linchuan Wei, Hao-Hsiang Wu
• Abstract summary: We study the problem of learning the sparse DAG structure of a BN from continuous observational data. The optimal solution to this mathematical program is known to have desirable statistical properties under certain conditions. We propose a concrete early stopping criterion to terminate the branch-and-bound process in order to obtain a near-optimal solution.
• Score: 2.7473982588529653
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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-$M$" 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.

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