Optimal estimation of Gaussian DAG models
This provides foundational statistical guarantees for causal inference and graphical modeling, with incremental extensions to subgaussian errors.
The paper tackles the problem of learning Gaussian directed acyclic graph (DAG) models from observational data, establishing minimax optimal sample complexity of n ∝ q log(d/q) for two settings and showing that learning directed models is statistically no harder than undirected ones under equal variances.
We study the optimal sample complexity of learning a Gaussian directed acyclic graph (DAG) from observational data. Our main results establish the minimax optimal sample complexity for learning the structure of a linear Gaussian DAG model in two settings of interest: 1) Under equal variances without knowledge of the true ordering, and 2) For general linear models given knowledge of the ordering. In both cases the sample complexity is $n\asymp q\log(d/q)$, where $q$ is the maximum number of parents and $d$ is the number of nodes. We further make comparisons with the classical problem of learning (undirected) Gaussian graphical models, showing that under the equal variance assumption, these two problems share the same optimal sample complexity. In other words, at least for Gaussian models with equal error variances, learning a directed graphical model is statistically no more difficult than learning an undirected graphical model. Our results also extend to more general identification assumptions as well as subgaussian errors.