Learning linear structural equation models in polynomial time and sample complexity

Abstract: The problem of learning structural equation models (SEMs) from data is a fundamental problem in causal inference. We develop a new algorithm --- which is computationally and statistically efficient and works in the high-dimensional regime --- for learning linear SEMs from purely observational data with arbitrary noise distribution. We consider three aspects of the problem: identifiability, computational efficiency, and statistical efficiency. We show that when data is generated from a linear SEM over $p$ nodes and maximum degree $d$, our algorithm recovers the directed acyclic graph (DAG) structure of the SEM under an identifiability condition that is more general than those considered in the literature, and without faithfulness assumptions. In the population setting, our algorithm recovers the DAG structure in $\mathcal{O}(p(d² + \log p))$ operations. In the finite sample setting, if the estimated precision matrix is sparse, our algorithm has a smoothed complexity of $\widetilde{\mathcal{O}}(p³ + pd<sup>7)$, while if the estimated precision matrix is dense, our algorithm has a smoothed complexity of $\widetilde{\mathcal{O}}(p<sup>5)$. For sub-Gaussian noise, we show that our algorithm has a sample complexity of $$\mathcal{O}(\frac{d^{8}{\varepsilon^2}} \log (\frac{p}{\sqrt{\delta}}))$$ to achieve $\varepsilon$ element-wise additive error with respect to the true autoregression matrix with probability at most $1 - \delta$, while for noise with bounded $(4m)$-th moment, with $m$ being a positive integer, our algorithm has a sample complexity of $$\mathcal{O}(\frac{d^{8}{\varepsilon^2}} (\frac{p<sup>2}{\delta})<sup>{1/m})$$.