Causal Discovery in Equal Variance Linear Gaussian DAGs via SURE-Tuned Ridge Regression

Sambit Mishra; Urbashi Mitra

cs.LGAug 17, 2026

Causal Discovery in Equal Variance Linear Gaussian DAGs via SURE-Tuned Ridge Regression

Authors: Sambit Mishra, Urbashi Mitra

Organizations: Department of Electrical and Computer Engineering University of Southern California

Abstract

Recovering the directed acyclic graph (DAG) of a structural equation model (SEM) from observational data is a central problem in causal discovery. The iterative gradient descent and per-problem hyperparameter tuning of continuous-optimization methods are poorly suited to two practically important regimes: the sample-limited regime, where the number of samples is comparable to or smaller than the number of nodes in the DAG, and the compute-limited regime. This work proposes SURE-Ridge, a non-iterative, closed-form estimator for equal variance linear Gaussian SEM. The method performs parallel node-wise regressions with regularization parameters chosen adaptively by Stein's unbiased risk estimate (SURE), and applies an adaptive thresholding procedure to extract a DAG from the resulting soft adjacency matrix. Numerical results show that SURE-Ridge achieves the lowest structural Hamming distance in the small-sample regime and the lowest run time across all sample sizes tested, compared with NOTEARS, DAGMA, and GBNSL baselines.

Figures & tables

Fig. 1: SHD vs. sample size

n

across two problem sizes (

d=20,50

, top to bottom). Lines and shaded bands show the mean and one standard deviation across 100 Monte Carlo trials per setting.

Fig. 2: Per-DAG run time on an Intel i9-13900F CPU at

d=50

, on a logarithmic scale.

Causal Discovery

Causal discovery infers cause-and-effect structures from observational and interventional data, distinguishing causal relationships from correlation.

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MARCEDES: Score-based causal discovery under non-Gaussianity with continuous optimization

Anamitra Chaudhuri, Anirban Bhattacharya, Yang Ni

Department of Statistics Florida State University Tallahassee, FL 32306 · Department of Statistics The University of Texas at Austin · Department of Statistics Texas A&M University College Station, TX 77843+1

Causal Discovery Structural Causal Models

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We consider the problem of learning the underlying causal directed acyclic graph (DAG) structure corresponding to a structural equation model (SEM) with non-Gaussian errors. Motivated by an intentionally misspecified non-Gaussian SEM with all Laplace errors, we first introduce the mean absolute residual risk, defined over the space of all real matrices, and show that, asymptotically, the risk of the true weighted causal DAG matrix is strictly smaller than that of any other matrix. Nevertheless, to enhance generality and account for high-dimensional and finite-sample settings, we further incorporate row-specific sparsity penalties along with a soft DAG constraint to derive a continuous score function over the space of real matrices. Accordingly, we propose a score-based DAG learning method, named MARCEDES, formulated as an unconstrained score minimization problem, which can be efficiently solved using gradient-based optimization techniques, thereby circumventing the challenges associated with constrained optimization. Furthermore, we develop a computational algorithm to handle the non-smoothness of the score objective and to enable optimal tuning of row-specific sparsity penalties under a generalized Bayes framework. Finally, we demonstrate the efficiency and improved performance of the proposed method over existing approaches through an extensive simulation study.

Abstract

Causal Discovery

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