We study a family of regularized score-based estimators for learning the
structure of a directed acyclic graph (DAG) for a multivariate normal
distribution from high-dimensional data with $p\gg n$. Our main results
establish support recovery guarantees and deviation bounds for a family of
penalized least-squares estimators under concave regularization without
assuming prior knowledge of a variable ordering. These results apply to a
variety of practical situations that allow for arbitrary nondegenerate
covariance structures as well as many popular regularizers including the MCP,
SCAD, $\ell_{0}$ and $\ell_{1}$. The proof relies on interpreting a DAG as a
recursive linear structural equation model, which reduces the estimation
problem to a series of neighbourhood regressions. We provide a novel
statistical analysis of these neighbourhood problems, establishing uniform
control over the superexponential family of neighbourhoods associated with a
Gaussian distribution. We then apply these results to study the statistical
properties of score-based DAG estimators, learning causal DAGs, and inferring
conditional independence relations via graphical models. Our results
yield---for the first time---finite-sample guarantees for structure learning of
Gaussian DAGs in high-dimensions via score-based estimation.

研究了一种正则化的基于得分的评估器族，可以从高维数据中学习多元正态分布的有向无环图（DAG）的结构，通过损失函数惩罚最小二乘估计，支持多种正则化方法，并且首次提出了在高维条件下基于得分的有向无环图结构学习的有限样本保证。