This paper studies the problem of learning Bayesian networks from continuous observational data, generated according to a linear Gaussian structural equation model. We consider an $\ell_0$-penalized maximum likelihood estimator for this problem which is known to have favorable statistical properties but is computationally challenging to solve, especially for medium-sized Bayesian networks. We propose a new coordinate descent algorithm to approximate this estimator and prove several remarkable properties of our procedure: the algorithm converges to a coordinate-wise minimum, and despite the non-convexity of the loss function, as the sample size tends to infinity, the objective value of the coordinate descent solution converges to the optimal objective value of the $\ell_0$-penalized maximum likelihood estimator. Finite-sample optimality and statistical consistency guarantees are also established. To the best of our knowledge, our proposal is the first coordinate descent procedure endowed with optimality and statistical guarantees in the context of learning Bayesian networks. Numerical experiments on synthetic and real data demonstrate that our coordinate descent method can obtain near-optimal solutions while being scalable.

本研究解决了从连续观测数据中学习贝叶斯网络的问题，该数据根据线性高斯结构方程模型生成。我们提出了一种新的坐标下降算法，证明了该算法能够收敛到坐标最小值，并在样本量趋于无穷大时，目标值收敛于管理的最优目标值，首次在学习贝叶斯网络中提出了具有最优性和统计保证的坐标下降程序。

从高斯模型中学习贝叶斯网络的渐近最优坐标下降算法