Learning Identifiable Gaussian Bayesian Networks in Polynomial Time and Sample Complexity

TL;DR

This paper introduces a polynomial-time algorithm for learning sparse Gaussian Bayesian networks with equal noise variance, capable of uniquely identifying the DAG structure in high-dimensional settings with fewer samples than previous methods.

Contribution

The paper presents a new polynomial-time algorithm that recovers the true DAG structure of Gaussian Bayesian networks under weaker conditions than existing methods, with optimal sample complexity.

Findings

Requires $O(k^4 \, \log p)$ samples for high-probability recovery

Outperforms existing methods in accuracy of DAG structure recovery

Operates efficiently in high-dimensional, sparse settings

Abstract

Learning the directed acyclic graph (DAG) structure of a Bayesian network from observational data is a notoriously difficult problem for which many hardness results are known. In this paper we propose a provably polynomial-time algorithm for learning sparse Gaussian Bayesian networks with equal noise variance --- a class of Bayesian networks for which the DAG structure can be uniquely identified from observational data --- under high-dimensional settings. We show that $O(k^4 \log p)$ number of samples suffices for our method to recover the true DAG structure with high probability, where $p$ is the number of variables and $k$ is the maximum Markov blanket size. We obtain our theoretical guarantees under a condition called Restricted Strong Adjacency Faithfulness, which is strictly weaker than strong faithfulness --- a condition that other methods based on conditional independence testing need for their success. The sample complexity of our method matches the information-theoretic limits in terms of the dependence on $p$. We show that our method out-performs existing state-of-the-art methods for learning Gaussian Bayesian networks in terms of recovering the true DAG structure while being comparable in speed to heuristic methods.