# Learning discrete Bayesian networks in polynomial time and sample complexity

## TL;DR

This paper presents a polynomial-time, sample-efficient method for learning the structure of discrete Bayesian networks under certain conditions, using group l12-regularized regression without assuming specific conditional distributions.

## Contribution

It introduces a novel approach that guarantees exact structure recovery with polynomial time and logarithmic sample complexity for bounded-degree networks, under certain technical conditions.

## Key findings

- Exact structure recovery under technical conditions
- Polynomial time algorithm for Bayesian network learning
- Logarithmic sample complexity for bounded-degree networks

## Abstract

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.