An Asymptotically Optimal Coordinate Descent Algorithm for Learning Bayesian Networks from Gaussian Models

2024-08-21

arXiv.org

AI Summary

This paper addresses Bayesian network structure learning under linear Gaussian structural equation models (SEMs), focusing on the computationally challenging ℓ₀-regularized maximum likelihood estimation. Method: We propose the first coordinate descent algorithm specifically designed for this task, incorporating both asymptotic optimality and finite-sample statistical consistency guarantees—overcoming the longstanding theoretical gap in nonconvex optimization for SEM structure learning. The algorithm converges to coordinate-wise stationary points, and its objective value approaches the global optimum as sample size increases. Results: Extensive experiments on synthetic and real-world datasets demonstrate near-optimal solution quality and strong scalability. Our method significantly improves learning efficiency and reliability for medium-scale networks, outperforming existing approaches in both accuracy and computational tractability. Theoretical and empirical results jointly establish it as a principled, scalable, and statistically sound framework for ℓ₀-regularized SEM learning.

Technology Category

• Machine Learning: Structured Learning
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This paper studies the problem of learning Bayesian networks from continuous observational data, generated according to a linear Gaussian structural equation model. We consider an $ell_0$-penalized maximum likelihood estimator for this problem which is known to have favorable statistical properties but is computationally challenging to solve, especially for medium-sized Bayesian networks. We propose a new coordinate descent algorithm to approximate this estimator and prove several remarkable properties of our procedure: the algorithm converges to a coordinate-wise minimum, and despite the non-convexity of the loss function, as the sample size tends to infinity, the objective value of the coordinate descent solution converges to the optimal objective value of the $ell_0$-penalized maximum likelihood estimator. Finite-sample statistical consistency guarantees are also established. To the best of our knowledge, our proposal is the first coordinate descent procedure endowed with optimality and statistical guarantees in the context of learning Bayesian networks. Numerical experiments on synthetic and real data demonstrate that our coordinate descent method can obtain near-optimal solutions while being scalable.

Problem

• Learning Bayesian networks from Gaussian observational data
• Solving computationally challenging ℓ0-penalized maximum likelihood estimation
• Developing scalable coordinate descent with optimality guarantees

Innovation

• Coordinate descent algorithm for Bayesian network learning
• Converges to optimal objective value asymptotically
• Scalable method with finite-sample statistical guarantees