Large-Sample Learning of Bayesian Networks is NP-Hard (1212.2468v1)

Abstract: In this paper, we provide new complexity results for algorithms that learn discrete-variable Bayesian networks from data. Our results apply whenever the learning algorithm uses a scoring criterion that favors the simplest model able to represent the generative distribution exactly. Our results therefore hold whenever the learning algorithm uses a consistent scoring criterion and is applied to a sufficiently large dataset. We show that identifying high-scoring structures is hard, even when we are given an independence oracle, an inference oracle, and/or an information oracle. Our negative results also apply to the learning of discrete-variable Bayesian networks in which each node has at most k parents, for all k > 3.

• The paper proves that large-sample Bayesian network structure learning is NP-hard by reducing it to the NP-complete Degree-Bounded Feedback Arc Set problem.
• It demonstrates that even with independence, inference, and information oracles, the combinatorial challenge of structure identification persists.
• The findings underscore the need for heuristic methods in practical applications and guide future research on efficient, domain-specific approximations.

Large-Sample Learning of Bayesian Networks is NP-Hard: An Essay

In the paper "Large-Sample Learning of Bayesian Networks is NP-Hard," Chickering, Meek, and Heckerman present a thorough complexity analysis of learning Bayesian networks from data. Their findings affirm that the learning task remains NP-hard even under the asymptotic assumption of large datasets and consistent scoring criteria.

Introduction and Background

Understanding Bayesian networks' structure-learning complexities has been a central topic within the Uncertainty in Artificial Intelligence (UAI) community. Prior research has already established that without restrictive assumptions, learning Bayesian networks is generally NP-hard. Motivated by these findings, Chickering et al. extend the complexity analysis to large-sample scenarios using consistent scoring criteria such as the Bayesian Information Criterion (BIC) and Minimum Description Length (MDL).

Main Findings and Results

The authors demonstrate that identifying high-scoring Bayesian network structures remains NP-hard, even when provided with substantial computational resources such as independence, inference, or information oracles. This result holds for networks where nodes can have at most $k$ parents, for any $k \geq 3$.

Complexity Analysis

The core argument hinges on the following reduction: large-sample learning is reduced to a known NP-complete problem, the Degree-Bounded Feedback Arc Set (DBFAS). The reduction process allows them to import the complexity from DBFAS directly into the learning problem. The primary decision problem LEARN is defined, which asks whether there exists a Directed Acyclic Graph (DAG) model that includes a given probability distribution and supports $\leq d$ parameters. By proving that LEARN is NP-hard, the authors successfully extend this hardness to the problem of learning network structures.

Oracular Extensions

Significantly, the paper provides proofs that this complexity result holds even when learning algorithms have access to sophisticated oracles:

• Independence Oracle: Determines conditional independence between variables in constant time.
• Constrained Inference Oracle: Computes joint probability for a constant-sized set of variables in constant time.
• Constrained Information Oracle: Computes mutual information between variables given a constant-sized conditioning set in constant time.

The authors prove that the learning problem remains NP-hard despite these oracular aids, highlighting that the inherent difficulty lies in the combinatorial nature of structure identification rather than inference per se.

Theoretical and Practical Implications

The implications of this research are profound both theoretically and practically. From a theoretical perspective, it establishes a solid foundation for understanding the limitations and challenges of Bayesian network learning. Practically, it suggests that in real-world scenarios, where assumptions such as DAG-perfection and known sort orders do not hold, one must resort to heuristic methods. Such heuristics may include the Greedy Equivalence Search (GES) algorithm, which works well under specific conditions but can falter in the general case.

Future Developments

While the NP-hardness result is discouraging, it opens the door to exploring new, more efficient heuristics and approximations for practical scenarios. Future research might focus on identifying and leveraging domain-specific properties or partial constraints that could lead to efficient learning algorithms under certain practical assumptions.

Conclusion

Chickering, Meek, and Heckerman's findings solidify the understanding that large-sample learning of Bayesian network structures is computationally intractable, aligning with the complexity observed in finite-sample scenarios. This work lays a critical groundwork for ongoing research in developing practical approaches for learning Bayesian networks, acknowledging the computational boundaries imposed by their NP-hard nature.