Linear Causal Discovery with Interventional Constraints
Zhigao Guo, Feng Dong
TL;DR
This work introduces interventional constraints as a new form of high-level causal knowledge that bounds total causal effects in linear SEMs, going beyond purely structural priors. It formalizes a constrained optimization framework, where the objective combines data fit and sparsity while interventional constraints enforce sign and magnitude on total effects via $T=(I-W)^{-1}-I$ and a DAG constraint $h(W)=0$. To solve the resulting non-convex problem, a two-stage approach (L-BFGS-B followed by SLSQP) yields a feasible weight matrix $W^*$ that respects both acyclicity and interventional constraints, with theoretical KKT convergence guarantees. Empirically, Lin-CDIC improves accuracy and interpretability on synthetic data and the Sachs dataset, outperforming path-constrained and unconstrained baselines in recovering causal structure and effects, and revealing new plausible interactions under realistic constraint settings. The work highlights practical gains in explainability and downstream inference, while also outlining scalability and extension challenges toward nonlinear, latent, and cyclic causal models.
Abstract
Incorporating causal knowledge and mechanisms is essential for refining causal models and improving downstream tasks such as designing new treatments. In this paper, we introduce a novel concept in causal discovery, termed interventional constraints, which differs fundamentally from interventional data. While interventional data require direct perturbations of variables, interventional constraints encode high-level causal knowledge in the form of inequality constraints on causal effects. For instance, in the Sachs dataset (Sachs et al.\ 2005), Akt has been shown to be activated by PIP3, meaning PIP3 exerts a positive causal effect on Akt. Existing causal discovery methods allow enforcing structural constraints (for example, requiring a causal path from PIP3 to Akt), but they may still produce incorrect causal conclusions such as learning that "PIP3 inhibits Akt". Interventional constraints bridge this gap by explicitly constraining the total causal effect between variable pairs, ensuring learned models respect known causal influences. To formalize interventional constraints, we propose a metric to quantify total causal effects for linear causal models and formulate the problem as a constrained optimization task, solved using a two-stage constrained optimization method. We evaluate our approach on real-world datasets and demonstrate that integrating interventional constraints not only improves model accuracy and ensures consistency with established findings, making models more explainable, but also facilitates the discovery of new causal relationships that would otherwise be costly to identify.