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DAG DECORation: Continuous Optimization for Structure Learning under Hidden Confounding

Samhita Pal, James O'quinn, Kaveh Aryan, Heather Pua, James P. Long, Amir Asiaee

TL;DR

This work addresses learning causal structure from observational data in linear Gaussian SEMs with latent confounding by proposing DECOR, a single differentiable estimator that jointly learns a DAG and a correlated noise model. DECOR relies on identifiability under bow-free structure with a uniform eigenvalue margin on the noise covariance, and optimizes a smooth, likelihood-aligned objective with acyclicity and bow constraints, coupled with a post-hoc bow reconciliation. The method alternates between a NOTEARS-style directed-edge update and a convex noise-update (covariance or precision route), enabling a practical, scalable approach that handles non-pervasive confounding. Empirical results on synthetic benchmarks show DECOR matches or outperforms strong baselines, with adaptive regularization providing robust performance as confounding density grows, highlighting the method’s potential for reliable structure learning in realistic settings where latent confounders are present but not pervasive.

Abstract

We study structure learning for linear Gaussian SEMs in the presence of latent confounding. Existing continuous methods excel when errors are independent, while deconfounding-first pipelines rely on pervasive factor structure or nonlinearity. We propose \textsc{DECOR}, a single likelihood-based and fully differentiable estimator that jointly learns a DAG and a correlated noise model. Our theory gives simple sufficient conditions for global parameter identifiability: if the mixed graph is bow free and the noise covariance has a uniform eigenvalue margin, then the map from $(\B,\OmegaMat)$ to the observational covariance is injective, so both the directed structure and the noise are uniquely determined. The estimator alternates a smooth-acyclic graph update with a convex noise update and can include a light bow complementarity penalty or a post hoc reconciliation step. On synthetic benchmarks that vary confounding density, graph density, latent rank, and dimension with $n<p$, \textsc{DECOR} matches or outperforms strong baselines and is especially robust when confounding is non-pervasive, while remaining competitive under pervasiveness.

DAG DECORation: Continuous Optimization for Structure Learning under Hidden Confounding

TL;DR

This work addresses learning causal structure from observational data in linear Gaussian SEMs with latent confounding by proposing DECOR, a single differentiable estimator that jointly learns a DAG and a correlated noise model. DECOR relies on identifiability under bow-free structure with a uniform eigenvalue margin on the noise covariance, and optimizes a smooth, likelihood-aligned objective with acyclicity and bow constraints, coupled with a post-hoc bow reconciliation. The method alternates between a NOTEARS-style directed-edge update and a convex noise-update (covariance or precision route), enabling a practical, scalable approach that handles non-pervasive confounding. Empirical results on synthetic benchmarks show DECOR matches or outperforms strong baselines, with adaptive regularization providing robust performance as confounding density grows, highlighting the method’s potential for reliable structure learning in realistic settings where latent confounders are present but not pervasive.

Abstract

We study structure learning for linear Gaussian SEMs in the presence of latent confounding. Existing continuous methods excel when errors are independent, while deconfounding-first pipelines rely on pervasive factor structure or nonlinearity. We propose \textsc{DECOR}, a single likelihood-based and fully differentiable estimator that jointly learns a DAG and a correlated noise model. Our theory gives simple sufficient conditions for global parameter identifiability: if the mixed graph is bow free and the noise covariance has a uniform eigenvalue margin, then the map from to the observational covariance is injective, so both the directed structure and the noise are uniquely determined. The estimator alternates a smooth-acyclic graph update with a convex noise update and can include a light bow complementarity penalty or a post hoc reconciliation step. On synthetic benchmarks that vary confounding density, graph density, latent rank, and dimension with , \textsc{DECOR} matches or outperforms strong baselines and is especially robust when confounding is non-pervasive, while remaining competitive under pervasiveness.

Paper Structure

This paper contains 29 sections, 3 theorems, 19 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Problem formulation
  4. Connections to existing objectives.
  5. Mixed-graph notation.
  6. Related Work
  7. DAG Discovery via Continuous Optimization
  8. Deconfounding in Causal Discovery
  9. Low-rank plus sparse precision decomposition.
  10. DeCAMfounder: deconfounding via additive-noise identifiability.
  11. Global identifiability related to continuous optimization
  12. Graphical intuition.
  13. Algebraic rank form.
  14. Injectivity.
  15. A simple sufficient characterization for global identifiability
  16. ...and 14 more sections

Key Result

Lemma 3.3

Under acyclicity, reorder variables in a topological order so that $\mathbf{T}$ is unit lower triangular. Then, for any node $i$, the submatrix $\mathbf{T}_{[i],\,P(i)}$ has full column rank, and the rows indexed by $P(i)$ contain an identity on the parent columns.

Figures (2)

  • Figure 1: Pervasive versus bow-free structures and their PAG summary. Latent-to-$X_i$ edges are labeled with $L_{1i}$.
  • Figure 2: Performance under varying confounding density ($p{=}20$, $q{=}5$, $n{=}200$, $B_{\text{density}}{=}0.1$). Each curve shows mean across 10 replicates; error bars indicate standard errors.

Theorems & Definitions (6)

  • Lemma 3.3: Effect block is full column rank
  • Lemma 3.4: Noise block retains a margin
  • Theorem 3.5: Deterministic identifiability under a bow and a margin
  • proof : Proof of Lemma \ref{['lem:effect']}
  • proof : Proof of Lemma \ref{['lem:noise']}
  • proof : Proof of Theorem \ref{['thm:ident']}