Inexact Column Generation for Bayesian Network Structure Learning via Difference-of-Submodular Optimization

TL;DR

This paper introduces an inexact column generation method using difference-of-submodular optimization for Bayesian network structure learning, improving solution quality and scalability over existing methods.

Contribution

It reformulates the pricing problem as a difference of submodular functions and applies the DCA for efficient inexact solutions, enhancing BNSL IP approaches.

Findings

Higher quality solutions for continuous Gaussian data with increased graph density

Comparable performance to benchmark methods on larger graphs

Efficiently solves the complex pricing problem in BNSL IP formulations

Abstract

In this paper, we consider a score-based Integer Programming (IP) approach for solving the Bayesian Network Structure Learning (BNSL) problem. State-of-the-art BNSL IP formulations suffer from the exponentially large number of variables and constraints. A standard approach in IP to address such challenges is to employ row and column generation techniques, which dynamically generate rows and columns, while the complex pricing problem remains a computational bottleneck for BNSL. For the general class of $\ell_0$-penalized likelihood scores, we show how the pricing problem can be reformulated as a difference of submodular optimization problem, and how the Difference of Convex Algorithm (DCA) can be applied as an inexact method to efficiently solve the pricing problems. Empirically, we show that, for continuous Gaussian data, our row and column generation approach yields solutions with higher quality than state-of-the-art score-based approaches, especially when the graph density increases, and achieves comparable performance against benchmark constraint-based and hybrid approaches, even when the graph size increases.