Learning discrete Bayesian networks in polynomial time and sample complexity
This provides a polynomial-time solution for structure learning in Bayesian networks, which is significant for fields like machine learning and statistics, though it is incremental as it builds on existing methods with specific assumptions.
The paper tackles the NP-hard problem of learning discrete Bayesian network structures by proposing a method that, under certain conditions, recovers the true structure with sample complexity growing logarithmically with the number of nodes and running in polynomial time.
In this paper, we study the problem of structure learning for Bayesian networks in which nodes take discrete values. The problem is NP-hard in general but we show that under certain conditions we can recover the true structure of a Bayesian network with sufficient number of samples. We develop a mathematical model which does not assume any specific conditional probability distributions for the nodes. We use a primal-dual witness construction to prove that, under some technical conditions on the interaction between node pairs, we can do exact recovery of the parents and children of a node by performing group l_12-regularized multivariate regression. Thus, we recover the true Bayesian network structure. If degree of a node is bounded then the sample complexity of our proposed approach grows logarithmically with respect to the number of nodes in the Bayesian network. Furthermore, our method runs in polynomial time.