I-FLOP: Fast Learning of Order and Parents from Interventional Data

Liuting Chen; Alex Markham

stat.MLAug 28, 2026

I-FLOP: Fast Learning of Order and Parents from Interventional Data

Authors: Liuting Chen, Alex Markham

Organizations: Department of Mathematical Sciences University of Copenhagen Denmark

Abstract

We extend the FLOP (fast learning of order and parents) algorithm recently proposed by Wienöbst et al. (2026) from observational to interventional data. In particular, we use the interventional BIC score of Hauser and Bühlmann (2012), adapting it to be used with the iterative Cholesky-based score updates that are partly responsible for FLOP's speed. We show that, in the sample limit, I-FLOP recovers a DAG in the same interventional Markov equivalence class as the data-generating DAG. We compare I-FLOP to existing causal structure learning algorithms on real and simulated interventional data, where it performs favorably in terms of both performance and run time.

Figures & tables

Figure 1: I-CPDAG SHD decreases as sample size increases from $10^{2}$ to $10^{6}$ per environment with 20 nodes and expected degree 2 for one observational and five known-target singleton hard intervention environments.

Figure 2: The I-CPDAG Structural Hamming Distance (SHD) is plotted against runtime for linear-Gaussian ER DAGs. Each setting uses one observational environment and five known-target singleton hard intervention environments, with $1{,}000$ samples per environment.

	Expected degree $d=3$				Expected degree $d=5$
Algorithm	Mean	Std.	Skel.	Orient.	Mean	Std.	Skel.	Orient.
I-FLOP (ILS=0)	2.50	3.78	1.30	1.20	1.60	1.58	0.90	0.70
I-FLOP (ILS=20)	0.70	0.67	0.60	0.10	1.50	1.27	1.10	0.40
I-FLOP (ILS=50)	0.70	0.67	0.60	0.10	1.50	1.27	1.10	0.40
GIES	2.70	3.13	2.00	0.70	47.40	49.02	39.10	8.30
I-GSP	28.40	16.93	19.60	8.80	94.70	28.79	73.10	21.60

Table 1: SHD decomposition of experiment in well-specified synthetic setting

Figure 3: I-FLOP (ILS = 0) attains the lowest mean I-CPDAG SHD while remaining faster than GIES. All I-FLOP configurations outperform FLOP-obs and I-GSP, and positive ILS budgets reduce variability (details in Appendix D ).

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: Top: Scaling with the number of samples per environment, with $p=40$ , expected degree $3$ , and sample size varied from $10^{2}$ to $10^{6}$ in each environment. Bottom: Scaling with graph size, with expected degree $3$ , $1000$ samples per environment, and $p$ varied from $20$ to $100$ . Each setting contains one observational environment and five known-target singleton perfect hard-intervention environments. Runtime is shown on a logarithmic scale, and accuracy is measured by I-CPDAG SHD.

Figure 5: Runtime of GIES is decomposed into covariance/scatter precomputation, graph search, and end-to-end wrapper execution. All the settings are the same as Figure 4 .

Panel of Figure 6 — Figure 6: Left: Mean optimized total BIC score across 50 seeds as the ILS budget increases from 0 to 50. Right: Mean total SHD and its skeleton and orientation components across the same seeds. The shaded regions denote the interquartile range across seeds.

Figure 7: Seed-level changes in SHD on the real data dataset when increasing the ILS budget from 0 to 50. The horizontal axis reports the change in SHD, computed as the value at ILS 50 minus that at ILS 0. Seeds are grouped according to the change in total SHD: improved ( $n=14$ ), unchanged ( $n=9$ ), and worsened ( $n=27$ ). The black line denotes the change in total SHD, while the orange and purple bars denote the changes in skeleton SHD and orientation SHD, respectively.

Causal Discovery

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

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Sep 26, 2026J/K move · Enter open · S save

FOCUS: Fixed-Confidence Online Causal Learning Using Sequential Adaptive Interventions

Haijie Xu, Chen Zhang

Department of Industrial Engineering Tsinghua University Beijing, China

Causal Discovery Structural Causal Models Causal Intervention

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We study fully online fixed-confidence causal discovery without any historical observational data. Starting from zero samples, the learner sequentially selects interventions to recover both the causal DAG and its edge weights under a linear-Gaussian structural equation model. We establish an instance-dependent lower bound for any

(ε,δ)

-correct algorithm and propose FOCUS, which adaptively allocates interventions through an online max--min game. A key contribution is a computable concentration inequality for the accumulated KL divergence involving causal parameters shared across interventions. We prove that FOCUS is

(ε,δ)

-correct and that its expected stopping time matches the lower bound in its

Θ(\log(1/δ))

dependence up to an instance-dependent constant. Experiments demonstrate improved structure and edge-weight recovery and confirm the predicted stopping-time trend. Our codes are available on https://anonymous.4open.science/r/FOCUS_code-76E5

Abstract

Causal Discovery

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