We deal with the combinatorial problem of learning directed acyclic graph (DAG) structure from observational data adhering to a linear structural equation model (SEM). Leveraging advances in differentiable, nonconvex characterizations of acyclicity, recent efforts have advocated a continuous constrained optimization paradigm to efficiently explore the space of DAGs. Most existing methods employ lasso-type score functions to guide this search, which (i) require expensive penalty parameter retuning when the $\textit{unknown}$ SEM noise variances change across problem instances; and (ii) implicitly rely on limiting homoscedasticity assumptions. In this work, we propose a new convex score function for sparsity-aware learning of linear DAGs, which incorporates concomitant estimation of scale and thus effectively decouples the sparsity parameter from the exogenous noise levels. Regularization via a smooth, nonconvex acyclicity penalty term yields CoLiDE ($\textbf{Co}$ncomitant $\textbf{Li}$near $\textbf{D}$AG $\textbf{E}$stimation), a regression-based criterion amenable to efficient gradient computation and closed-form estimation of noise variances in heteroscedastic scenarios. Our algorithm outperforms state-of-the-art methods without incurring added complexity, especially when the DAGs are larger and the noise level profile is heterogeneous. We also find CoLiDE exhibits enhanced stability manifested via reduced standard deviations in several domain-specific metrics, underscoring the robustness of our novel linear DAG estimator.

翻译：摘要：本文研究从符合线性结构方程模型（SEM）的观测数据中学习有向无环图（DAG）结构的组合优化问题。利用可微非凸无环性约束表征的最新进展，近期研究倡导采用连续约束优化范式来高效探索DAG空间。现有方法大多采用套索型评分函数来引导搜索，但存在以下问题：（i）当问题实例中$\textit{未知}$的SEM噪声方差发生变化时，需要重新调整代价高昂的惩罚参数；（ii）隐含地依赖于同方差性假设。本文提出一种新的凸评分函数用于稀疏感知的线性DAG学习，该函数融合了尺度的伴随估计，从而有效将稀疏参数与外生噪声水平解耦。通过光滑非凸无环性惩罚项的正则化，得到CoLiDE（$\textbf{Co}$ncomitant $\textbf{Li}$near $\textbf{D}$AG $\textbf{E}$stimation）方法——一种适用于高效梯度计算和异方差场景下噪声方差闭式估计的回归准则。我们的算法在不增加额外复杂度的前提下，优于现有最先进方法，尤其在DAG规模更大且噪声水平分布异质的情况下表现突出。此外，CoLiDE展现出增强的稳定性，在多个领域特定指标上标准差显著降低，这凸显了本新型线性DAG估计器的鲁棒性。