We consider the problem of estimating a directed acyclic graph (DAG) for a multivariate normal distribution from high-dimensional data with $p\gg n$. Our main results establish nonasymptotic deviation bounds on the estimation error, sparsity bounds, and model selection consistency for a penalized least squares estimator under concave regularization. The proofs rely on interpreting the graphical model as a recursive linear structural equation model, which reduces the estimation problem to a series of tractable neighbourhood regressions and allows us to avoid making any assumptions regarding faithfulness. In doing so, we provide some novel techniques for handling general nonidentifiable and nonconvex problems. These techniques are used to guarantee uniform control over a superexponential number of neighbourhood regression problems by exploiting various notions of monotonicity among them. Our results apply to a wide variety of practical situations that allow for arbitrary nondegenerate covariance structures as well as many popular regularizers including the MCP, SCAD, $\ell_{0}$ and $\ell_{1}$.

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。

利用惩罚邻域回归学习有向无环图