I-FLOP: Fast Learning of Order and Parents from Interventional Data
I-FLOP is a fast causal structure learning algorithm from interventional data, leveraging target-filtered BIC scores with theoretical consistency guarantees.
Key Findings
Methodology
I-FLOP extends Wienöbst et al.'s FLOP by integrating Hauser and Bühlmann's (2012) interventional BIC score. It employs iterative Cholesky-based score updates for efficiency, constructing target-filtered local scores that pool environment-specific data where variables are not directly intervened. The algorithm uses order-based search with reinsertion moves and warm-start grow-shrink procedures, ensuring asymptotic global consistency. The process involves precomputing environment statistics, building local scores, optimizing parent sets, and applying local and global theoretical guarantees.
Key Results
- On simulated and real interventional datasets, I-FLOP outperforms GIES and I-GSP in structure accuracy and runtime. As sample size increases to 10^6, SHD approaches zero, demonstrating asymptotic recovery of the true I-MEC.
- In linear Gaussian models with expected degrees 3 and 5, I-FLOP reduces SHD by over 30% compared to GIES, with halved runtime in denser graphs, showing excellent scalability.
- On real data from the Causal Chamber Light Tunnel, I-FLOP achieves the highest accuracy and lowest SHD, validating practical effectiveness.
Significance
This work advances causal inference by effectively incorporating known intervention targets, overcoming computational bottlenecks of prior methods. It provides a scalable, theoretically guaranteed approach for large-scale, multi-environment data, enabling more accurate causal discovery in complex systems such as biological networks and engineered systems. The method bridges the gap between theory and application, offering a robust tool for real-world causal analysis.
Technical Contribution
Key innovations include the target-filtered pooled score integrating intervention information, Cholesky-based incremental score updates, and a grow-shrink parent selection strategy with global consistency proofs. The approach guarantees asymptotic recovery of the true I-MEC, combining efficiency with rigorous theoretical backing, and extends the scope of order-based causal discovery to multi-environment interventional data.
Novelty
This is the first method to embed known intervention targets directly into a FLOP-like order search framework, using target-filtered scores and Cholesky updates to ensure efficiency and consistency. It surpasses existing algorithms like GIES and I-GSP by providing a unified, scalable, and theoretically sound approach for multi-environment causal discovery, especially under known interventions.
Limitations
- The approach assumes known, hard intervention targets; uncertain or soft interventions are not supported, limiting applicability in some real-world scenarios.
- Model assumptions restrict to linear Gaussian SEMs, making it less suitable for nonlinear or non-Gaussian systems.
- Performance may degrade with limited sample sizes or poorly designed interventions, requiring further robustness enhancements.
Future Work
Future research will focus on extending the framework to handle uncertain, soft, or unknown interventions, as well as nonlinear and non-Gaussian models. Incorporating deep learning techniques for nonparametric causal discovery, and developing adaptive intervention strategies to optimize data collection, will further broaden its applicability and robustness.
AI Executive Summary
Causal structure learning is crucial for understanding complex systems across sciences and engineering. Traditional algorithms rely heavily on observational data, which often cannot distinguish causal directions due to Markov equivalence. Interventional data, where specific variables are manipulated, can resolve these ambiguities but pose computational challenges. Existing methods like GIES and I-GSP leverage intervention information but struggle with scalability and theoretical guarantees in large, high-dimensional settings.
Wienöbst et al. introduced FLOP, an efficient order-based algorithm that combines local score updates with greedy search, significantly speeding up causal discovery. Building on this, the current work introduces I-FLOP, a novel extension that incorporates known intervention targets via a target-filtered BIC score. This approach pools environment-specific data where variables are not directly intervened, ensuring consistency and efficiency.
The core technical innovation involves Cholesky-based incremental score updates and a grow-shrink parent set selection strategy, which guarantees asymptotic recovery of the true causal structure. Theoretical proofs confirm both local and global consistency under standard assumptions, including faithfulness and intervention conservativeness.
Empirical evaluations on synthetic and real datasets demonstrate that I-FLOP outperforms state-of-the-art algorithms in structure accuracy and runtime, especially in dense networks with large sample sizes. It achieves near-perfect recovery of the interventional Markov equivalence class in large samples, validating its practical utility.
This method represents a significant step forward in scalable, theoretically grounded causal discovery from multi-environment intervention data. Future directions include relaxing assumptions on intervention knowledge, extending to nonlinear models, and integrating adaptive experimental design, aiming to make causal inference more robust and widely applicable across scientific disciplines.
Deep Dive
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.