Differentiable Structure Learning for Cyclic Linear Gaussian Models with Latent Confounders

2026-09-29

AI Summary

This study addresses the challenging problem of linear Gaussian causal structure learning in the presence of cycles and unknown latent confounders. It proposes the concept of marginal quasi-equivalence, reformulating structure learning as a Gaussian negative log-likelihood minimization problem regularized by a complexity penalty. By parameterizing edges and variables via Bernoulli gates, the method constructs a continuously relaxed, differentiable objective function that enables end-to-end optimization. Furthermore, it establishes that the expected objective coincides with the global infimum of its discrete counterpart. Under the assumption of algebraic faithfulness, experimental results demonstrate that the proposed approach significantly outperforms existing baselines in terms of structure recovery error, effectively solving the causal discovery problem in complex scenarios involving cyclic dependencies and latent confounding.

Abstract

We study causal structure learning from observational data in linear Gaussian structural causal models in the presence of directed cycles and an unknown number of exogenous latent confounders, bounded by a given maximum. We derive the covariance of the observed variables and introduce marginal quasi-equivalence, which characterizes when different causal models share a full-dimensional subset of the observational distributions they can generate. We formulate structure learning as minimization of the Gaussian negative log-likelihood with a logarithmically scaled complexity penalty that counts directed edges and latent variables. For a fixed number of observed variables and a fixed upper bound on latent variables, we establish consistency of global score minimizers up to marginal quasi-equivalence under algebraic faithfulness, structural minimality, and model-overlap assumptions. We parameterize the inclusion of directed edges and candidate latent variables using Bernoulli gates, whose continuous probabilities are optimized jointly with the structural coefficients. Averaging the penalized negative log-likelihood over these gates yields an objective with a closed-form differentiable complexity penalty. We prove that this expected objective has the same global infimum as the corresponding discrete structure-learning objective. Experimental results show that our approach achieves lower recovery error than previous methods in several experimental settings.

Problem

• causal structure learning
• cyclic linear Gaussian models
• latent confounders
• observational data

Innovation

• Differentiable Structure Learning
• Cyclic Causal Models
• Latent Confounders
• Bernoulli Gates
• Marginal Quasi-equivalence