DAG Structure Learning with Node-Varying Heavy-Tailed Disturbances

Robin Francis; Sundeep Prabhakar Chepuri

cs.LGOct 9, 2026

DAG Structure Learning with Node-Varying Heavy-Tailed Disturbances

Authors: Robin Francis, Sundeep Prabhakar Chepuri

Organizations: Indian Institute of Science, Bangalore, India

Abstract

DAG structures can be learned from observations by incorporating acyclicity via smooth constraints, such as DAGMA's log-determinant function. Typically, observations and the underlying DAG are related through linear structural equation models, with Gaussianity assumption on the disturbances at each node. Extending DAG learning methods that account for scale and tail behavior of disturbances that vary across nodes remains a challenge. We propose a node-varying Student- $t$ DAGMA that jointly estimates the graph structure, node-specific disturbance scales, and the degrees-of-freedom parameter. A Gaussian scale-mixture representation yields an iteratively reweighted formulation that down-weights large standardized residuals. Building on existing results, we also establish identifiability of the graph and disturbance parameters. Synthetic experiments demonstrate improved structural recovery under heterogeneous heavy-tailed disturbances, and experiments on real finance data show that the proposed method yields lower test negative log-likelihood than the evaluated baselines.

Figures & tables

Panel of Figure 1 — Figure 1: Illustration of node-varying heavy-tailed disturbances and the benefit of node-varying Student- $t$ DAGMA. $(a)$ Different nodes exhibit different disturbance characteristics. $(b)$ Difference in NLL between fixed and adaptive models as the fraction of heavy-tailed nodes varies.

Figure 2: Recovered DAG structures obtained using the competing methods. TP, FP, and FN denote true positive, false positive, and false negative, respectively.

Method	Test NLL	Test NRMSE	Edge count
DAGMA	$32.84$	$0.469$	$14$
Huber-DAGMA	$31.61$	$\mathbf{0.464}$	$18$
CoLiDE-NV	$32.17$	$0.467$	$23$
Ours	$\mathbf{26.40}$	$0.465$	$15$

Table 1: Test-set performance and recovered edge counts on the 49 Industry Portfolios dataset. Bold values indicate the lowest NLL and NRMSE.

Causal Structure Learning

Causal structure learning infers directed relationships among variables from observational, interventional, or time-series data.

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May 19, 2026J/K move · Enter open · S save

Exploiting Non-Negativity in DAG Structure Learning

Samuel Rey, Madeline navarro, Santiago Segarra, Gonzalo Mateos

Dept. of Electrical and Computer Engineering, University of Rochester, Rochester, NY 14627, USA

Graph Structure Learning Structural Causal Models Causal Structure Learning Nonconvex Optimization Augmented Lagrangian Method Constrained Optimization

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This work addresses the problem of learning directed acyclic graphs (DAGs) from nodal observations adhering to a linear structural equation model. DAG structure learning is a central task in signal processing, machine learning, and causal inference, but it remains challenging because acyclicity is a global combinatorial property and the search space grows superexponentially in the number of nodes. Continuous acyclicity characterizations have led to important algorithmic advances by replacing the discrete DAG restriction with smooth equality constraints on the candidate adjacency matrix. However, existing formulations still involve difficult non-convex optimization landscapes and may suffer from degenerate first-order optimality conditions. Here, we restrict attention to DAGs with non-negative edge weights and exploit this additional structure to obtain a simpler acyclicity characterization devoid of Hadamard products. We propose a new non-negative constrained DAG estimator and develop a tractable, locally convergent algorithm based on the method of multipliers. We further analyze the benign optimization landscape induced by the non-negativity constraints. In the large-sample regime, we show that the true DAG is the unique global minimizer of the population augmented Lagrangian. Moreover, the landscape contains no spurious interior stationary points, and the true DAG is the only acyclic Karush--Kuhn--Tucker (KKT) point. Critically, this allows us to guarantee local convergence to the true DAG given infinite samples. Comprehensive numerical experiments show that the proposed structure identification method improves over state-of-the-art continuous DAG-learning baselines.

Abstract

Causal Structure Learning

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