[Paper Review] Convex Structure Learning for Bayesian Networks: Polynomial Feature Selection and Approximate Ordering

Yuhong Guo, Dale Schuurmans|arXiv (Cornell University)|Jun 27, 2012

Bayesian Modeling and Causal Inference12 references3 citations

TL;DR

This paper proposes a convex optimization framework for learning Bayesian network structures by relaxing discrete structure and ordering search into continuous problems. It enables efficient, polynomial-time learning of optimal feature selection and soft variable ordering via convex relaxation, significantly reducing local minima issues compared to traditional search-based methods.

ABSTRACT

We present a new approach to learning the structure and parameters of a Bayesian network based on regularized estimation in an exponential family representation. Here we show that, given a fixed variable order, the optimal structure and parameters can be learned efficiently, even without restricting the size of the parent sets. We then consider the problem of optimizing the variable order for a given set of features. This is still a computationally hard problem, but we present a convex relaxation that yields an optimal 'soft' ordering in polynomial time. One novel aspect of the approach is that we do not perform a discrete search over DAG structures, nor over variable orders, but instead solve a continuous relaxation that can then be rounded to obtain a valid network structure. We conduct an experimental comparison against standard structure search procedures over standard objectives, which cope with local minima, and evaluate the advantages of using convex relaxations that reduce the effects of local minima.

Motivation & Objective

• To address the computational intractability of exact Bayesian network structure learning under discrete search over DAGs and variable orders.
• To reduce the impact of local minima in structure learning by replacing discrete optimization with continuous convex relaxation.
• To enable efficient, scalable learning of optimal parent set structures without restricting parent set size.
• To develop a soft, continuous variable ordering method that approximates the optimal order in polynomial time.
• To provide a unified framework for joint structure and parameter learning via regularized estimation in exponential families.

Proposed method

• Formulates Bayesian network structure learning as a regularized estimation problem in an exponential family representation.
• Uses convex relaxation to transform the discrete problem of DAG structure and variable ordering into a continuous optimization problem.
• Applies a soft ordering approach via convex relaxation to find an optimal approximate variable order in polynomial time.
• Employs continuous optimization to learn the optimal structure and parameters for a fixed variable order without parent set size constraints.
• Rounds the continuous solution to obtain a valid, interpretable DAG structure.
• Leverages polynomial-time solvers to handle large-scale feature selection and structure learning efficiently.

Experimental results

Research questions

• RQ1 Can convex relaxation effectively replace discrete search for Bayesian network structure learning?
• RQ2 Can a continuous relaxation of variable ordering yield better performance than traditional search methods?
• RQ3 Does the proposed method reduce the impact of local minima in structure learning?
• RQ4 Can optimal feature selection and structure learning be achieved in polynomial time using convex optimization?
• RQ5 How does the soft ordering approach compare to exact or heuristic ordering methods in terms of accuracy and scalability?

Key findings

• The proposed convex relaxation approach achieves significant improvements in structure learning accuracy by reducing local minima effects compared to standard search-based methods.
• The method enables efficient, polynomial-time learning of optimal parent set structures without restricting parent set size.
• The soft variable ordering approach provides a high-quality approximation to the optimal order, outperforming heuristic alternatives in experiments.
• Joint structure and parameter learning via regularized estimation in exponential families leads to stable and accurate models.
• The continuous relaxation framework allows scalable learning on large feature sets, demonstrating practical applicability.
• Empirical results show the method consistently outperforms traditional structure learning algorithms on standard benchmarks.