We study the problem of learning directed acyclic graphs from continuous observational data, generated according to a linear Gaussian structural equation model. State-of-the-art structure learning methods for this setting have at least one of the following shortcomings: i) they cannot provide optimality guarantees and can suffer from learning sub-optimal models; ii) they rely on the stringent assumption that the noise is homoscedastic, and hence the underlying model is fully identifiable. We overcome these shortcomings and develop a computationally efficient mixed-integer programming framework for learning medium-sized problems that accounts for arbitrary heteroscedastic noise. We present an early stopping criterion under which we can terminate the branch-and-bound procedure to achieve an asymptotically optimal solution and establish the consistency of this approximate solution. In addition, we show via numerical experiments that our method outperforms three state-of-the-art algorithms and is robust to noise heteroscedasticity, whereas the performance of the competing methods deteriorates under strong violations of the identifiability assumption. The software implementation of our method is available as the Python package \emph{micodag}.

翻译：研究了从连续观测数据中学习有向无环图的问题，这些数据根据线性高斯结构方程模型生成。针对该场景，现有最先进的结构学习方法至少存在以下缺陷之一：i) 无法提供最优性保证，且可能学习到次优模型；ii) 依赖噪声同方差的严格假设，导致底层模型完全可识别。我们克服了这些缺陷，提出了一种计算高效的混合整数规划框架，用于处理中等规模问题，可适应任意异方差噪声。我们提出了一种早期停止准则，在此准则下可终止分支定界过程，从而获得渐近最优解，并建立了该近似解的一致性。此外，通过数值实验表明，我们的方法优于三种最先进算法，且对噪声异方差具有鲁棒性，而对比方法在强违反可识别性假设时性能显著下降。本方法的软件实现已作为Python包\emph{micodag}发布。