Title: Near-Optimal Experiment Design in Linear non-Gaussian Cyclic Models
Venue: NeurIPS 2025
Accepted: yes
Keywords: Causal Discovery

Adaptive Experiment Design

Linear Non-Gaussian SCMs

Cyclic Causal Models

Adaptive Submodularity

Greedy Optimization
Abstract: We study the problem of causal structure learning from a combination of observational and interventional data generated by a linear non-Gaussian structural equation model that might contain cycles. Recent results show that using mere observational data identifies the causal graph only up to a permutation-equivalence class. We obtain a combinatorial characterization of this class by showing that each equivalence class corresponds to a perfect matching in a bipartite graph. This bipartite representation allows us to analyze how interventions modify or constrain the matchings. Specifically, we show that each atomic intervention reveals one edge of the true matching and eliminates all incompatible causal graphs. Consequently, we formalize the optimal experiment design task as an adaptive stochastic optimization problem over the set of equivalence classes with a natural reward function that quantifies how many graphs are eliminated from the equivalence class by an intervention. We show that this reward function is adaptive submodular and provide a greedy policy with a provable near-optimal performance guarantee. A key technical challenge is to efficiently estimate the reward function without having to explicitly enumerate all the graphs in the equivalence class. We propose a sampling-based estimator using random matchings and analyze its bias and concentration behavior. Our simulation results show that performing a small number of interventions guided by our stochastic optimization framework recovers the true underlying causal structure.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors characterize graphically the equivalence class of graphs learnable from observational data as a bipartite graph where each node represents the ith row of the matrix (I-W) and an edge in the matching only exists if the jth entry of W is nonzero. Each valid permutation over rows defines a perfect matching. So distribution-equivalent graphs are row permutations consistent with matchings in the bipartite graph.\n\nSTRENGTHS AND WEAKNESSES:\nstrengths:\n\n- significance: the authors nicely set up the technical difficulty of learning with observational and interventional data with cycles. \n- quality: the paper contributes nice constructions and characterizations to this problem. The graphical characterization is nice, and the characterization of what is learned by intervening is nice. (I am not enough of an expert to know whether these characterizations are immediate or not). This reduces the combinatorial search problem quite a bit. The authors construct an adaptive search procedure with appealing properties, given access to a sampler over the space of perfect matchings, which is a relatively friendly object. The paper is a nice end-to-end procedure (given the output of ICA) that relates to fairly classical graphical ideas. \n\nweaknesses: \n\n- relevance: it's not so clear when causal discovery in cyclic graphs is relevant, what are the new potential real-world cases of causal discovery where you have both observational data and the capacity to run interventions that this can enable? the simulations are on random graphs and therefore not illuminating of the potential real-world relevance. \n\n\n- significance: \n\n-- The paper is a good characterization of the \"interventional\" part of causal discovery under observational+interventional data with cycles. It's not how the characterizations and approach of this paper are connected to the \"observational\" part of learning, ie from finite samples. The paper could be improved by exploring this front. \n-- The development of the adaptive algorithm enjoys a lot of relatively tractalbe structure. The objective is to find the set I that minimizes the size of the reduced equivalence class |Ω(I,ϕ)|. However, a lot of later characterizations seem to rely quite heavily on the uniform prior assumption. For example, I'm a little confused by this, \"Therefore, prior to an intervention, the probability that v is matched to row rzi is pi = ni/N.\", is this at the first step with the uniform prior, or at intermediate later timesteps as well? \n\nThe empirical analysis is a good proof of concept but not super thorough\n\n- clarity\n\nOverall good, but the paper could benefit from some explanations at certain places. I will put some questions here because they are more for expository purposes (and not really questions that I need answers to \"in order to improve my score\" as in the other box.)\n\nQuestions: \n- what is the SCC that is preserve in Fig 4b? \n- Why does intervening on variable xi, 218 identifies the row corresponding to xi. how should i interpret this / why is this different than interventions in the acyclic case? ]\n- What is the interpretation of (I-W) and its rows/permutations? \n- this is confusing: \nposterior distribution over realizations conditioned on consistency with all\n274 observations in ψ\n- \"Figures 2a and 2b show the average number of interventions under each strategy.\"\nTo do what? Recover the full graph? Exhaust the nodes? What feedback does the algorithm get to know that it's discovered the whole graph?\n\nQUESTIONS:\nQuestions: \n\nIn appendix E the algorithm seems to do less favorably on dense graphs with a max of 60 nodes. Why is that? It seems the graphical structure might have something to do with it since all the methods appear to be doing nearly the same (qualitatively the algorithm is performing like max degree). Since erdos renyi graphs are so well-studied, is there some graphical statistic that could be used to explain when the adaptive algorithm will in particular do better than a max degree heuristic?\n\nLIMITATIONS:\nn/a", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper tackles causal discovery in linear non-Gaussian cyclic models—systems where  cycles make structure learning tricky. Previous work shows that observational data alone can't fully identify the causal graph; instead, we're left with a set of permutation-equivalent candidates. Authors show that each atomic intervention cuts through the uncertainty by revealing one true edge, pruning incompatible graphs from the equivalence class. The authors frame experiment design as an adaptive stochastic optimization problem, proving their reward function is adaptive submodular and that a greedy policy achieves near-optimal efficiency. Simulations confirm that their method, aided by a sampling estimator, recovers the true causal structure with minimal interventions.\n\nSTRENGTHS AND WEAKNESSES:\nThis work establishes that linear non-Gaussian cyclic models are identifiable only up to an equivalence class, characterized combinatorially via perfect matchings in bipartite graphs. Furthermore, the authors prove that each intervention reveals one true edge, enabling adaptive experiment design with submodular guarantees, and develop an efficient sampling estimator to overcome the computational challenge of graph enumeration. They also provide experimental validation of the proposed approach, demonstrating near-optimal recovery of causal structure with minimal interventions.\n\nWeaknesses: The problem setup is restricted to linear models and assumes non-Gaussian noise (with at most one Gaussian component). Additionally, the noise components must be independent, which limits practical applicability. The paper also lacks comparison with Ehsan Mokhtarian et al. (2023), which, while focusing on general SCMs, remains a valid baseline for the linear case. Furthermore, the framework limits each experiment to single-variable interventions, potentially restricting applications where multi-node intervention data is available.\n\nQUESTIONS:\nQuestions/Comments for the Authors:\n\nComparison with Mokhtarian et al. (2023):\nCould you compare your proposed approach against Mokhtarian et al. (2023) for the linear case? Demonstrating superior performance for this specific class of models would help readers better appreciate the work’s value.\n\nAssumption 3 (Non-Gaussian Noise):\nWhy is the restriction to at most one Gaussian noise component (Line 125) necessary? Does the model fail to admit a unique solution if multiple noise components are Gaussian? Could you justify this theoretically or with an example or a reference?\n\nSingle-Node Intervention Limitation:\nWhat prevents extending your method to multi-node interventions when such data is available? Could a similar approach be adapted for this case?\n\nSampling-Based Estimator Concerns:\nWhile you characterize the statistical accuracy of the sampling-based estimator, how does its use affect the optimality of solutions obtained via the greedy policy? Specifically, could estimation errors in the reward function lead to suboptimal intervention sequences?\n\nExperimental Details:\nHow many samples (M) were used for the sampling-based estimation in your experiments? This parameter’s choice seems critical for balancing accuracy and computational cost.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper tackles causal structure learning in linear non-Gaussian models with cycles, where observational data alone only identifies graphs up to an equivalence class. The authors provide a novel combinatorial characterization showing that each equivalence class corresponds to a perfect matching in a bipartite graph, and prove that each intervention reveals one edge of the true matching while eliminating incompatible graphs. They formalize optimal experiment design as an adaptive stochastic optimization problem with a submodular reward function, enabling a greedy algorithm with near-optimal performance guarantees. To make this computationally tractable, they develop a sampling-based estimator that avoids explicitly enumerating all graphs in the equivalence class and provide theoretical analysis of its accuracy. Their experiments in synthetic linear non-Gaussian data demonstrate that, when provided with the oracle weight matrix up to row-permutation and scaling, this approach recovers true causal structures with few interventions, often performing close to the theoretical lower bound.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nClarity - This submission is very clearly written, and extremely well organized. Assumptions are clearly stated in assumption formatting in Section 3, all reocurring terminology is defined and explained, and the groundwork for upcoming sections. The major sections are sensibily split into mapping each observational equivalence class into a bipartite graph (4), characterizing how interventions pair down the equivalence class (5), developing an adaptive greedy interventional design that is provably near optimal (6), and suggesting a reasonable statistical estimator for the theoretical method in 6, with corresponding analysis of estimator error. In each section, the authors very seamlessly weave between definitions and mathematical results, and explaining the intuition behind each step.\n\nQuality - Although I did not check the proofs in the appendix closely, the intuition behind the major results (Theorem 4.4, Theorem 6.6, and Theorem 7.1) seems clear, and proof sketches throughout the work seem accurate. The authors claims seem generally reasonable, as they outline weaknesses (such as the computational intractability of the theoretical reward function in Section 6), and take steps to try and mitigate them (developing a statistical estimator via sampling in Section 7). The authors adress a wide range of parts of the discovery problem, providing a novel framing of equivalence classes as bipartite matchings, connecting interventional results to the matching, and finally prove near statistical optimality of their approach. Therefore, the claimed contributions in Section 1 are largely substantiated by the details of the work.\n\nOriginality - The key novelty in the work lies first in finding a bipartite matching representation of the equivalence class of potentially cyclic linear SEMs (Section 4), then showing how this bipartite matching can be combined with interventions to eliminate graphs from the class (Section 5). The most important idea is perhaps the move in Section 4.1 to reduce the typical $d$-dimensional graph to a lower level representation  (condensation graph) via strongly connected components, and then proving that condensation graphs are sufficient for acharacterizing all distribution-equivalent graphs. This analysis is quite neat, and potentially more broadly applicable beyond linear SEMs, as strongly connected components may be useful (but perhaps not sufficient) for characterizing other distributions.\n\nWeaknesses:\n\nSignificance\n\n- The main simplifying assumption int his work is linearity (and also causal sufficiency to some degree), which limit the scope of applicability. While this is fine for foundational results, and particularly as there are not many works addressing interventional experiment design in cyclic settings, authors should make clear where they leverage linearity heavily. Additional a discussion on what parts of this are potentially transferable to the nonlinear setting, and perhaps some speculation on what parts are more or less likely to generalize would also be valuable to other researchers in this area.\n- The other major drawback of the experimental approach here is that it seems to depends somewhat strongly on the accuracy of the original estimation of the equivalence class. After all, the way interventions are chosen depends on calculating the value of the  intervention subject to knowledge of the class. This motivates the need to characterize how their approach interacts with error in estimating the original observational class. This can be tackled theoretically in different ways; either you could show that consistency can still be achieved even if original test was wrong (optimality is probably no longer true though), or you could provide a way to loosen the current implementation to get consistency, perhaps by randomly sampling from graphs in nearby equivalence classes (just a thought). This could also be addressed empirically, by running tests in synthetic data to see how bad a problem misspecification of the original equivalence class is - this would help inform future work as to whether this is an subarea of this methodology that would benefit a lot from improvement.\n\nQUESTIONS:\nQ1. How does your interventional design approach perform when the initial observational equivalence class is incorrectly estimated? To address this, please consider doing the following:\n   - Provide theoretical analysis showing consistency can still be achieved even with incorrect initial equivalence class estimation (acknowledging optimality may be lost) \n   - Develop a modified implementation that accounts for uncertainty in the equivalence class (e.g., by sampling from nearby equivalence classes)\n   - Present empirical results on synthetic data quantifying how sensitive your approach is to misspecification\n\nQ2. Which components of your approach could potentially extend to nonlinear causal models, and where does linearity play a fundamental role? To address this, please consider doing the following:\n   - Clearly identify which theoretical results (bipartite matching characterization, strongly connected components analysis, intervention selection) rely essentially on linearity vs. those that might generalize\n   - Discuss whether the strongly connected components framework could be useful for characterizing equivalence classes in nonlinear settings\n   - Provide informed speculation about what modifications would be needed for nonlinear extensions\n\nIf the authors address my concerns in the Strengths/Weaknesses and Questions sections, I am very happy to increase my score.\n\nLIMITATIONS:\nLimitations, outside of the additional empirical experiments I requested above, are adequately addressed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work addresses interventional experiment design in cyclic causal models under linear non-Gaussian functional models. \nIt contributes a characterization of the (observational) Markov Equivalence class in this setting through a direct correspondence between graphs in the MEC and matchings \nin bipartite graphs. The authors give an analysis of how targeted interventions on the system constrain this class, and motivate a reward function for optimal experiment design \nfor which they establish near-optimality guarantees. A sampling-based estimator is also suggested and analyzed, and simulation experiments support both the exact and sampling-based strategies.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths** The problem setting is sufficiently novel compared to the existing work  (e.g. Mokhtarian, 2023)  making this work a valuable addition to the literature. \nIn particular, the theoretical insights are original, both regarding the graphical characterizations as well as the proposed adaptive greedy policy with guarantees and sampling-based estimator. \nTheoretical justifications of all of these aspects are given where I found no major issues. There are also no issues regarding clarity and presentation of the work. \n\n**Weaknesses**  Regarding the quality of the experiments, \nit would be interesting  to see a more comprehensive experimental comparison to adaptive methods assuming acyclicity, perhaps on both  cyclic and acyclic settings, to better judge the empirical benefits of considering this setting.\nExperiments are conducted assuming access to an optimal ICA strategy which, while  reasonable  to isolate the effect of the proposed strategy itself, leave it unclear whether the proposal allows to address real-world settings.\nThis affects the practical significance of this work, which may be somewhat limited by to the strong assumptions on the generating process, \nnamely the linear non-Gaussian model and perfect, single-node interventions without off-target effects  which may not always be realistic.  \n\nNevertheless, existing work addressing cyclic causal graphs is indeed limited and this work can provide an important basis for future work, in particular with the given characterization of the MEC which may be of independent interest.\n\nQUESTIONS:\nThe main questions are related to the concerns mentioned above, \n- How does the method perform against previous proposals for optimal experimental design? Even when these assume acyclicity, it would be nevertheless be interesting to see the empirical performance in a cyclic setting. Similarly, it would be interesting to see a comparison between the acyclic vs. cyclic setting to judge the effect of this. \n- Did you perform any experiments without access to an ideal ICA procedure to judge the effect of this, and if not do you expect the version without ICA-oracle to still outperform the baselines?\n\nLIMITATIONS:\nThere a no limitations besides what is noted above and no negative impacts.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-03"}