Title: $\psi$DAG: Projected Stochastic Approximation Iteration for Linear DAG Structure Learning
Venue: NeurIPS 2025
Accepted: no
Keywords: Structure learning

Continuous Optimization

Directed Acyclic Graphs
Abstract: Learning the structure of Directed Acyclic Graphs (DAGs) presents a significant challenge due to the vast combinatorial search space of possible graphs, which scales exponentially with the number of nodes. Recent advancements have redefined this problem as a continuous optimization task by incorporating differentiable acyclicity constraints. These methods commonly rely on algebraic characterizations of DAGs, such as matrix exponentials, to enable the use of gradient-based optimization techniques. Despite these innovations, existing methods often face optimization difficulties due to the highly non-convex nature of DAG constraints and the per-iteration computational complexity. In this work, we present a novel framework for learning DAGs, employing a Stochastic Approximation approach integrated with Stochastic Gradient Descent (SGD)-based optimization techniques. Our framework introduces new projection methods tailored to efficiently enforce DAG constraints, ensuring that the algorithm converges to a feasible local minimum. With its low iteration complexity, the proposed method is well-suited for handling large-scale problems with improved computational efficiency. We demonstrate the effectiveness and scalability of our framework through comprehensive experimental evaluations, which confirm its superior performance across various settings.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes $\\psi$DAG, a stochastic gradient descent (SGD) method for learning Directed Acyclic Graphs (DAGs) using differentiable acyclicity constraints. It reformulates DAG learning with linear Structural Equation Models (SEMs) in a stochastic manner (Section 3) and leverages vertex ordering to reduce the constraint space (Section 4). Experimental results demonstrate that $\\psi$DAG scales effectively to large graphs with up to 10,000 nodes.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n- S1 [Quality] The paper is clearly written and well-structured, making the overall approach easy to follow.\n- S2 [Clarity] The motivation and methodological components are well articulated. The claims and corresponding experimental results are also clearly presented.\n- S3 [Significance] The experiments thoroughly evaluate the proposed method using synthetic data, structure recovery tasks, scalability comparisons, and a real-world application.\n- S4 [Significance] DAG learning with differentiable acyclicity constraints is a highly active area of research, and scalability remains a major challenge. $\\psi$DAG addresses this issue via stochastic optimization, and its contribution is likely to have significant impact.\n- S5 [Originality] The integration of SGD with DAG learning via differentiable acyclicity constraints is novel and original.\n\n**Weaknesses**\n- W1 [Significance] The role and justification of vertex ordering require further clarification. Lines 172--193 suggest that using the true vertex ordering for the true DAG (solution) can improve the objective value, but in real-world scenarios, the true ordering is generally unavailable during optimization.\n- W2 [Significance] Extending $\\psi$DAG to nonlinear SEMs would further increase the method’s utility, as many recent approaches support both linear and nonlinear formulations.\n\nQUESTIONS:\nMy main concern is W1 described above. It would be pleasure to answer W1.\n\nIn addition, I have the following questions:\n\n- Q1. Does a vertex ordering always exist for &W_{k}^{(1/3)}&?\n- Q2. As I understand it, &W_{k}^{(1/3)}&, &W_{k}^{(2/3)}& and &W_{k+1}}& become dense matrices. Is this correct? If so, how is sparsity induced in &W_{K}&? I could not find a sparsity-inducing regularization term similar to the one used in NOTEARS.\n- Q3. Can the projection method described in Algorithm 3 also be applied to deterministic settings such as NOTEARS and DAGMA? I am particularly interested in a comparison of the computational time between regularization-based approaches (e.g., NOTEARS, DAGMA) and the projection-based method proposed in Algorithm 3 under a deterministic setting.\n- Q4. What is the role of weights L in Algorithm 3?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-24"}

comments: SUMMARY:
The paper proposes a new linear DAG structure learning algorithm based on stochastic approximation with novel projection methods to enforce the DAG constraints so the problem can be solved using SGD. The empirical results are impressive compared to existing continuous optimization approaches, both in terms of accuracy and runtime/scalability (the proposed method can be applied to larger graphs).

STRENGTHS AND WEAKNESSES:
Strengths:
1. The approach is novel and the methodological contribution appears to be significant (I am unable to check proofs).
2. The new optimization method appears to be more stable than existing continuous optimization approaches, leading to both better accuracy and runtime.
3. The approach enables DAG structure learning on larger graphs than other existing methods

Weaknesses:
1. The approach is limited to linear DAGs
2. Since the focus is on linear models, there may be some other baselines worth including that also focus on the linear case, e.g. high-dimensional PC, LiNGAM
3. There's several assumptions/approximations required at different steps in the process, which make it unclear what kind of correctness guarantee there may be

QUESTIONS:
1. Is there a correctness guarantee that can be stated formally?
2. Have the authors tried running their approach in the nonlinear setting? If so, how do these results compare?
3. Did the authors consider any additional experiments to specifically measure the stability of the optimization to other approaches?

LIMITATIONS:
The linear assumption should be stated more clearly in the paper.
TITLE: Official Review
CLARITY: 3
IMPACT: 4
ORIGINALITY: 4
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a gradient-based method to learn the structure of directed acyclic graphs (DAG). The authors formulate the DAG structure learning as a set of linear structural equation models and formulate DAG learning as stochastic approximation. The novelty of the method is to add a projection after SGD steps, and an estimated adjacency matrix is introduced as a mask to ensure computational efficiency and scalability. The authors evaluated the proposed method against multiple baselines and on various DAG learning tasks. The proposed method demonstrates consistent improvements in accuracy (structural hamming distance) and runtime or scalability compared to the baselines.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n- The key strengths are the scalability and accuracy of the proposed method compared to many baselines. \n- The principles and deduction of the solution seem to be reasonable and mathematically sound. \n\n\n## Weaknesses\n- There lacks intuitive explanation of the applicability of the linear SEM and the corresponding DAG. It seems that the linear SEM only applies to stochastic graphical models, but in many real world applications, conditional SEM is more common, in other words, some of the nodes have deterministic inputs or features rather than purely stochastic. The proposed method seems not to be able to cover these applications. \n- It is unclear how many observations are required for the proposed approach to estimate DAG. In many real-world applications, the observations or data samples for DAG are limited, and may be partial or unbalanced across nodes. How does it perform when the observations are limited? The SGD method seems to rely on a large number of observations X. \n- It is unclear if or how the proposed method works for non-i.i.d. noise. The paper seems to be evaluated on well behaved noise, such as Gaussian, exponential, or Gumbel distributions, and the noise is i.i.d. across nodes. This is questionable since non-i.i.d. Noise would be more common in such causal inference or DAG models, since different nodes typically represent different entities. \n- The presentation could be improved from the organization and consistency in notations. It is unclear from the first reading about the typical applications of the method, what are the required observations X and how do they relate to real world data, and what are the assumptions for the method. The notations in Algorithm 3 lines 1, 3, 4 are not clear (11^T, dots between W and L, and [:]).\n\nQUESTIONS:\n- What are the typical applications or applicable problems for the linear SEM / DAG, where random noise serves as external inputs on all the nodes?\n- What assumptions are required for the formulation and method? Do i.i.d. noise or known distribution required? Does it support discrete noise distribution?\n- Many systems represented by DAG has deterministic or semi-deterministic inputs, and/or the inputs for different nodes follow different distributions. Is the proposed method capable of working on these conditional SEM / DAG? \n- What are the requirements on the amount of samples for the proposed method to work? \n- What are the limitations of the proposed method?\n\nLIMITATIONS:\n- May not work well for problems with limited sample/observations. \n- May not work well for non-i.i.d. noise inputs or when some nodes have deterministic external inputs or the inputs follow discrete distributions.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors propose  $\\Psi$-DAG a novel structure learning method for linear Additive Noise Models. Differently from existing methods, $\\Psi$-DAG consists of a three-step algorithm that: (i.) finds a possibly cyclic preliminary solution, (ii.) refines it into an acyclic solution from which to extract the topological order, and (iii.) further refines it by keeping the topological order. Overall, the method outperforms on the selected benchmark competing structure learning methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- Empirically, the $\\Psi$ -DAG outperforms existing structure learning methods (PC, FGES, NOCURL, GOLEM, DAGMA) and does not incur high computational costs as the others, apart from DAGMA, which is still faster. However, I have concerns on the chosen benchmark (see questions).\n\nWeaknesses:\n- The idea of extracting a topological order from a preliminary solution before fitting the final solution after fixing the order is similar to the NOCURL approach. A detailed comparison between the two is not discussed in the paper.\n- The authors do not discuss whether simulated data has been normalized or not. This is important as non-normalized benchmarks are known to be flawed for the evaluation of structure learning algorithms (\"Beware of the simulated DAG!\", Reisach et al. 2021).\n- I might have missed it, but it is not clear to me how the \"stochastic reformulation\" build in Section 3 differs from the usual MSE problem.\n\nQUESTIONS:\n- Why are results of NOTEARS on the synthetic dataset missing in the main body? When reported in the Appendix (Figure 7), NOTEARS seems to perform better than $\\Psi$ -DAG.\n- How is the acyclic optimization for the second step of your algorithm performed? Which acyclicity constraint are you employing?\n- How does the extraction of the topological ordering differs from the procedure used by NOCURL? In their work, they also find a preliminary solution and then extract a topological order to then refit the final weighted adjacency matrix.\n- How would the method perform on normalized data?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-05"}