Ordering-Based Search: A Simple and Effective Algorithm for Learning Bayesian Networks (1207.1429v1)

Abstract: One of the basic tasks for Bayesian networks (BNs) is that of learning a network structure from data. The BN-learning problem is NP-hard, so the standard solution is heuristic search. Many approaches have been proposed for this task, but only a very small number outperform the baseline of greedy hill-climbing with tabu lists; moreover, many of the proposed algorithms are quite complex and hard to implement. In this paper, we propose a very simple and easy-to-implement method for addressing this task. Our approach is based on the well-known fact that the best network (of bounded in-degree) consistent with a given node ordering can be found very efficiently. We therefore propose a search not over the space of structures, but over the space of orderings, selecting for each ordering the best network consistent with it. This search space is much smaller, makes more global search steps, has a lower branching factor, and avoids costly acyclicity checks. We present results for this algorithm on both synthetic and real data sets, evaluating both the score of the network found and in the running time. We show that ordering-based search outperforms the standard baseline, and is competitive with recent algorithms that are much harder to implement.

• The paper introduces an ordering-based search algorithm that simplifies Bayesian network learning by focusing on node orderings rather than complete network structures.
• It employs a greedy hill-climbing strategy enhanced with tabu lists and random restarts, achieving higher scores and faster convergence, especially with large variable sets.
• Results on synthetic and real-world datasets demonstrate robustness to limited data, offering significant computational efficiency for complex network applications.

Overview of Ordering-Based Search for Learning Bayesian Networks

The paper "Ordering-Based Search: A Simple and Effective Algorithm for Learning Bayesian Networks" by Marc Teyssier and Daphne Koller presents a novel approach to learning Bayesian network (BN) structures that simplifies existing methodologies by focusing on orderings rather than individual network structures. This research is situated in the context of the NP-hard problem of optimizing network structures based on a scoring function, a challenge extensively recognized in computational learning theory.

Key Contributions and Methodology

The core contribution of this paper is an algorithm that searches over the space of node orderings instead of network structures. This is based on the observation that while establishing a high-scoring network consistent with a given ordering is computationally efficient (non-NP-hard), the general problem is. The proposed method restricts node in-degrees and establishes search operators that adjust pairwise orderings of nodes. The paper uses a heuristic greedy hill-climbing search strategy, enhanced by tabu lists and random restarts, to identify an optimal ordering that, in turn, enables the derivation of an optimal network structure.

Numerical Results and Implications

The authors conduct experiments on both synthetic and real-world datasets, demonstrating that ordering-based search surpasses the standard greedy hill-climbing approach with tabu lists over network structures in terms of finding high-score networks. Key findings suggest that ordering-based search performs particularly well when the number of variables is large, or when there are fewer data instances available, indicating a robustness to local minima due to larger search steps and reduced search space size.

Table 2 in the paper aggregates these comparative results, showing consistently superior or equivalent scores for the ordering-based approach across various datasets. Additionally, the paper highlights that the new method achieves significant computational efficiency, particularly in domains with a high number of variables, where it converges notably faster compared to the traditional method.

Theoretical and Practical Implications

From a theoretical standpoint, this research provides insights into alternative optimization paths within combinatorial landscapes, leveraging problem decomposability and efficient data structure utilization—an idea that may influence future research in structure learning and beyond. Practically, the simplification in methodology holds value for applications with complex networks such as systems biology, where Bayesian networks are often applied to model gene expression and other high-dimensional data spaces.

Speculations on Future Developments

Future directions could explore the integration of ordering-based search with more sophisticated heuristic methods, such as those involving Monte Carlo approaches for enhanced scalability. Furthermore, extending this method to handle incomplete datasets dynamically might yield significant gains for real-world applicability, especially given the increasing prevalence of large, partially-observed datasets in the era of big data.

In conclusion, Teyssier and Koller’s approach introduces a compelling simplification to the BN structure learning process, which in tests is competitive with, and often superior to, more complex algorithms. This minimalist yet effective strategy opens new avenues for exploration in both algorithmic design and practical applications within machine learning and probabilistic reasoning.