Efficient Nonlinear DAG Learning Under Projection Framework

Naiyu Yin, Yue Yu, Tian Gao, Qiang Ji

Abstract: Directed Acyclic Graphs (DAGs) are foundational in machine learning, causal inference, and probabilistic modeling. Recovering the underlying DAG structure from observational data is crucial in these areas. The DAG learning can be approached as a constrained optimization problem with a continuous acyclicity constraint, often solved iteratively through sub-problem optimization. A recent breakthrough has shown that the set of DAGs can be represented as the weighted gradients of graph potential functions. Hence, one may search for a DAG in the equivalent space, whereby the acyclicity constraint is guaranteed to be satisfied. However, the original work, DAG-NoCurl, is limited to (generalized) linear structural equation models (SEMs) where explicit weighted adjacency matrices are defined. Herein, we theoretically derive a nonlinear projection formulation and propose an efficient two-step nonlinear DAG learning method, which we coined DAG-NCMLP. The proposed approach first obtains a non-acyclic graph and then projects it to the equivalent space of DAGs to obtain the acyclic graph. Experimental studies on benchmark datasets demonstrate that our proposed method provides similar accuracy, if not better, compared to state-of-the-art nonparametric DAG learning methods with hard-constrained optimization, while substantially reducing the computational time.