Consistent Second-Order Conic Integer Programming for Learning Bayesian Networks

TL;DR

This paper introduces a novel second-order conic formulation and an early stopping criterion for learning sparse Bayesian network structures, improving computational efficiency while maintaining statistical consistency.

Contribution

It proposes a second-order conic reformulation and an early stopping rule for mixed-integer programs in Bayesian network learning, enhancing solution quality and computational speed.

Findings

The new formulation outperforms linear constraints in efficiency.

The early stopping criterion yields near-optimal solutions.

The approach maintains statistical consistency.

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.