Scalable Contrastive Causal Discovery under Unknown Soft Interventions

Mingxuan Zhang; Khushi Desai; Sopho Kevlishvili; Elham Azizi

stat.MLMar 3, 2026

Scalable Contrastive Causal Discovery under Unknown Soft Interventions

Authors: Mingxuan Zhang, Khushi Desai, Sopho Kevlishvili, Elham Azizi

Organizations: Department of Systems Biology, Columbia University · Irving Institute for Cancer Dynamics, Columbia University · Department of Computer Science, Columbia University · Department of Biomedical Engineering, Columbia University ·

Abstract

Observational causal discovery is only identifiable up to the Markov equivalence class. While interventions can reduce this ambiguity, in practice interventions are often soft with multiple unknown targets. In many realistic scenarios, only a single intervention regime is observed. We propose a scalable causal discovery model for paired observational and interventional settings with shared underlying causal structure and unknown soft interventions. The model aggregates subset-level PDAGs and applies contrastive cross-regime orientation rules to construct a globally consistent maximal PDAG under Meek closure, enabling generalization to both in-distribution and out-of-distribution settings. Theoretically, we prove that our model is sound with respect to a restricted $Ψ$ equivalence class induced solely by the information available in the subset-restricted setting. We further show that the model asymptotically recovers the corresponding identifiable PDAG and can orient additional edges compared to non-contrastive subset-restricted methods. Experiments on synthetic data demonstrate improved causal structure recovery, generalization to unseen graphs with held-out causal mechanisms, and scalability to larger graphs, with ablations supporting the theoretical results.

Figures & tables

Figure 1: F1 (harmonic mean of precision and recall) and SHD (structural hamming distance) for state-of-the-art methods that only perform in distribution graph prediction on 20 nodes 20 edges. 10 graphs were generated using the polynomial mechanism. The horizontal red dashed line indicates mean across 10 graphs on predictions from running classic causal discovery baseline.

Figure 2: F1 and SHD for state-of-the-art methods compared to SCONE on predictions over 10 graphs with 20 nodes 20 edges, 20 nodes 30 edges, and 20 graphs for 50 nodes 50 edges. SEA-FCI obtains an F1 score of 0.0 in the 20 node 20 edge example, and is represented by a circle on the bottom of the axis.

Model	SHD $\downarrow$	F1 $\uparrow$
Scone	$\textbf{126.7}\pm 25.0$	$\textbf{0.237}\pm 0.046$
Avici	$131.3\pm 18.0$	$0.100\pm 0.042$
Dcd-Fg	$1001.3\pm 537.9$	$0.052\pm 0.011$
Sea (FCI)	$3178.8\pm 425.2$	$0.022\pm 0.004$
Sea (GIES)	$1797.2\pm 296.6$	$0.022\pm 0.004$

Table 1: Comparison of causal discovery methods on graphs with

N=100

nodes and

E=100

edges. Results are averaged over 20 graphs (mean

\pm

std).

$N$	$E$	Model	SHD $\downarrow$	F1 $\uparrow$
		Scone	$\textbf{39.0}\pm 9.7$	$\textbf{0.469}\pm 0.143$
20	40	Scone-NB	$39.6\pm 9.0$	$0.453\pm 0.123$
		Scone	53.85 $\pm$ 11.60	0.424 $\pm$ 0.102
50	50	Scone-NB	56.60 $\pm$ 10.79	0.393 $\pm$ 0.092

Table 2: Comparison of SCONE with and without using orientation rules. Results are averaged over 20 graphs (mean

\pm

std). SCONE-NB is the model with bias head ablated and thus do not incur rules.

$N$	$E$	Model	SHD $\downarrow$	F1 $\uparrow$
		Scone	$\textbf{39.0}\pm 9.7$	$\textbf{0.469}\pm 0.143$
20	40	Scone-NC	$48.6\pm 9.3$	$0.341\pm 0.136$
		Scone	$\textbf{53.85}\pm 11.60$	$\textbf{0.424}\pm 0.102$
50	50	Scone-NC	$66.55\pm 9.1$	$0.322\pm 0.076$

Table 3: Comparison of SCONE with and without using contrastive features. Results for

N=20

are averaged over 10 graphs and

N=50

over 20 graphs (mean

\pm

std). Scone-NC is the model with contrastive features ablated.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: Precision, recall, F1 and SHD for state-of-the-art methods that only perform in-distribution graph prediction on 20 nodes 20 edges. 10 graphs were generated using the polynomial mechanism. The horizontal red dashed line indicates mean across 10 graphs on predictions from running Polynomial-BIC baseline only.

Model	SHD $\downarrow$	F1 $\uparrow$	Precision $\uparrow$	Recall $\uparrow$
Scone	$\textbf{14.6}\pm 4.4$	$0.655\pm 0.094$	$\textbf{0.694}\pm 0.098$	$0.624\pm 0.106$
Notears	$19.7\pm 7.9$	$\textbf{0.675}\pm 0.073$	$0.544\pm 0.081$	$\textbf{0.899}\pm 0.058$
Dcdi-Dsf	$63.9\pm 14.9$	$0.195\pm 0.048$	$0.396\pm 0.069$	$0.130\pm 0.036$
Dcd-Fg	$86.7\pm 25.1$	$0.157\pm 0.057$	$0.101\pm 0.035$	$0.377\pm 0.190$
Poly-BIC CD	22.1	0.524	0.511	0.546

Appendix

Table 4: Comparison of causal discovery methods on polynomial in-distribution graphs with

N=20

nodes and

E=20

edges. Results are averaged over 10 graphs (mean

\pm

std).

Figure 4: Precision, recall, F1 and SHD for state-of-the-art methods compared to SCONE on predictions over 10 graphs with 20 nodes 20 edges, 20 nodes 30 edges and 20 graphs for 50 nodes 50 edges. SEA-FCI obtains a precision, recall and F1 score of 0.0 in the 20 node 20 edge example, and is represented by a circle on the bottom of the axis.

$N$	$E$	Model	SHD $\downarrow$	F1 $\uparrow$	Precision $\uparrow$	Recall $\uparrow$
		Scone	$\textbf{16.3}\pm 3.7$	$\textbf{0.571}\pm 0.098$	$\textbf{0.610}\pm 0.110$	$\textbf{0.537}\pm 0.091$
20	20	Avici	$23.3\pm 6.2$	$0.319\pm 0.099$	$0.346\pm 0.098$	$0.310\pm 0.127$
		Notears	$18.0\pm 4.6$	$0.468\pm 0.105$	$0.434\pm 0.091$	$0.517\pm 0.139$
		Dcd-Fg	$98.8\pm 22.2$	$0.155\pm 0.044$	$0.096\pm 0.024$	$0.450\pm 0.183$
		Dcdi-Dsf	$70.3\pm 13.5$	$0.221\pm 0.082$	$0.512\pm 0.181$	$0.142\pm 0.057$
		Sea (FCI)	$21.9\pm 2.0$	$0.000\pm 0.000$	$0.000\pm 0.000$	$0.000\pm 0.000$

Appendix

Table 5: Comparison of causal discovery methods across graph sizes. Results are averaged over 10 graphs for

N=20

and 20 graphs for

N=50

(mean

\pm

std).

Model	SHD $\downarrow$	F1 $\uparrow$	Precision $\uparrow$	Recall $\uparrow$
Scone	$\textbf{126.7}\pm 25.0$	$\textbf{0.237}\pm 0.046$	$\textbf{0.345}\pm 0.074$	$0.186\pm 0.046$
Avici	$131.3\pm 18.0$	$0.100\pm 0.042$	$0.210\pm 0.082$	$0.068\pm 0.031$
Dcd-Fg	$1001.3\pm 537.9$	$0.052\pm 0.011$	$0.030\pm 0.008$	$0.230\pm 0.076$
Sea (FCI)	$3178.8\pm 425.2$	$0.022\pm 0.004$	$0.012\pm 0.002$	$\textbf{0.325}\pm 0.056$
Sea (GIES)	$1797.2\pm 296.6$	$0.022\pm 0.004$	$0.012\pm 0.002$	$0.185\pm 0.044$

Appendix

Table 6: Comparison of SHD, F1, precision and recall for causal discovery methods on graphs with

N=100

nodes and

E=100

edges. Results are averaged over 20 graphs (mean

\pm

std).

$N$	$E$	Model	SHD $\downarrow$	F1 $\uparrow$	Precision $\uparrow$	Recall $\uparrow$
		Scone	$\textbf{39.0}\pm 9.7$	$\textbf{0.469}\pm 0.143$	$\textbf{0.473}\pm 0.138$	$\textbf{0.466}\pm 0.153$
20	40	Scone-NB	$39.6\pm 9.0$	$0.453\pm 0.123$	$0.460\pm 0.126$	$0.449\pm 0.125$
		Scone	$\textbf{53.85}\pm 11.60$	$\textbf{0.424}\pm 0.102$	$\textbf{0.455}\pm 0.102$	$\textbf{0.398}\pm 0.103$
50	50	Scone-NB	$56.60\pm 10.79$	$0.393\pm 0.092$	$0.424\pm 0.094$	$0.368\pm 0.092$

Appendix

Table 7: Comparison of SCONE with and without orientation rules. Results for

N=20

are averaged over 10 graphs and

N=50

over 20 graphs (mean

\pm

std). Scone-NB ablates the contrastive bias heads.

$N$	$E$	Model	SHD $\downarrow$	F1 $\uparrow$	Precision $\uparrow$	Recall $\uparrow$
		Scone	$\textbf{39.0}\pm 9.7$	$\textbf{0.469}\pm 0.143$	$\textbf{0.473}\pm 0.138$	$\textbf{0.466}\pm 0.153$
20	40	Scone-NC	$48.6\pm 9.3$	$0.341\pm 0.136$	$0.350\pm 0.136$	$0.335\pm 0.139$
		Scone	$\textbf{53.85}\pm 11.60$	$\textbf{0.424}\pm 0.102$	$\textbf{0.455}\pm 0.102$	$\textbf{0.398}\pm 0.103$
50	50	Scone-NC	$66.55\pm 9.1$	$0.322\pm 0.076$	$0.350\pm 0.076$	$0.300\pm 0.077$

Appendix

Table 8: Comparison of SCONE with and without any contrastive features. Results for

N=20

are averaged over 10 graphs and

N=50

over 20 graphs (mean

\pm

std). Scone-NC ablates the contrastive features in the model.

Causal Discovery

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

See all 221 papers

PACER: Acyclic Causal Discovery from Large-Scale Interventional Data
May 14, 2026Ramon Viñas Torné, Sílvia Fàbregas Salazar, Soyon Park +4Causal Discovery Causal Structure Learning
Swiss Federal Technology Institute of Lausanne (EPFL), Switzerland · ETH Zurich, Zurich, Switzerland · Cornell University, USA
Decoupled Causal Discovery
Sep 20, 2026Zhengkang Guan, Fei Wu, Kun KuangCausal Discovery Causal Structure Learning
College of Computer Science and Technology Zhejiang University
Causal Discovery in Action: Learning Chain-Reaction Mechanisms from Interventions
Mar 23, 2026Panayiotis Panayiotou, Özgür ŞimşekCausal Discovery Causal Identifiability
Department of Computer Science, University of Bath, UK

May 14, 2026J/K move · Enter open · S save

PACER: Acyclic Causal Discovery from Large-Scale Interventional Data

Ramon Viñas Torné, Sílvia Fàbregas Salazar, Soyon Park, Ivo Alexander Ban+3

Swiss Federal Technology Institute of Lausanne (EPFL), Switzerland · ETH Zurich, Zurich, Switzerland · Cornell University, USA

Causal Discovery Causal Structure Learning

Open paper AskPDF arXiv

Inferring the structure of directed acyclic graphs (DAGs) from data is a central challenge in causal discovery, particularly in modern high-dimensional settings where large-scale interventional data are increasingly available. While interventional data can improve identifiability, existing methods remain limited by soft acyclicity constraints, leading to optimization over invalid cyclic graphs, numerical instability, and reduced scalability. We introduce PACER (Perturbation-driven Acyclic Causal Edge Recovery), a scalable framework for causal discovery that guarantees acyclicity by construction. PACER parameterizes a distribution over DAGs through a joint model of variable permutations and edge probabilities, enabling direct optimization over valid causal structures without surrogate penalties. The framework supports a unified likelihood-based treatment of observational and interventional data, flexible conditional density models, and the incorporation of structural prior knowledge. For linear-Gaussian mechanisms, we derive closed-form expressions for the expected interventional log-likelihood and its gradients, yielding substantial computational gains. Empirically, PACER matches or exceeds state-of-the-art methods on protein signaling and large-scale genetic perturbation benchmarks, while scaling efficiently to networks with thousands of variables and achieving up to two orders of magnitude speedups over penalty-based differentiable approaches. These results demonstrate that exact and scalable causal discovery from high-dimensional perturbation data is achievable through principled search space design.

Abstract

Causal Discovery

Explore similar work

PACER: Acyclic Causal Discovery from Large-Scale Interventional Data

Decoupled Causal Discovery

Causal Discovery in Action: Learning Chain-Reaction Mechanisms from Interventions