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.

我们提出一种新的凸得分函数，用于稀疏感知线性有向无环图（DAG）的学习，在异方差场景中通过平滑的非凸无环性惩罚项进行正则化，以实现峰值同时估计，并有效地将稀疏参数与外生噪声水平解耦，该算法在大型DAG和异质噪声水平时展现出更高的性能，且表现出增强的稳定性。

CoLiDE：Concomitant线性DAG估计