Title: Differentiable Structure Learning with Partial Orders
Venue: NeurIPS 2024
Accepted: yes
Keywords: Causal discovery

Continuous optimization

Differentiable Structure Learning

Partial Orders
Abstract: Differentiable structure learning is a novel line of causal discovery research that transforms the combinatorial optimization of structural models into a continuous optimization problem. However, the field has lacked feasible methods to integrate partial order constraints, a critical prior information typically used in real-world scenarios, into the differentiable structure learning framework. The main difficulty lies in adapting these constraints, typically suited for the space of total orderings, to the continuous optimization context of structure learning in the graph space. To bridge this gap, this paper formalizes a set of equivalent constraints that map partial orders onto graph spaces and introduces a plug-and-play module for their efficient application. This module preserves the equivalent effect of partial order constraints in the graph space, backed by theoretical validations of correctness and completeness. It significantly enhances the quality of recovered structures while maintaining good efficiency, which learns better structures using 90\% fewer samples than the data-based method on a real-world dataset. This result, together with a comprehensive evaluation on synthetic cases, demonstrates our method's ability to effectively improve differentiable structure learning with partial orders.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of learning DAGs from observational data with incorporating prior knowledge represented as partial order constraints by extending from existing continuous DAG learning methods such as NOTEARS and DAGMA. The paper first presents a method that shuts down the corresponding cells of the adjacency matrix of the DAG according to the permutations of the partial orders and then proposes an augmented acyclicity-based method with improved efficiency. The paper conducts experiments on simulated datasets and a real-world dataset. In addition, the paper also provides comprehensive theoretical motivations and analysis to the research problem and the proposed method.\n\nSTRENGTHS:\n1. The problem studied in the paper is novel and has significance in practice. To my knowledge, it is a new (sub-)problem of DAG discovery from observational data, which has not been studied before.\n\n2.  The proposed method is well motivated both conceptually and theoretically. The notations and definitions are rigorously defined.\n\n3. The experiments are comprehensive and support the claims of the paper.\n\nWEAKNESSES:\nNot all my comments below are weaknesses and some of them are questions.\n\n1. In Eq 8a, a (relatively) straightforward method is introduced, which is argued to be less efficient, motivating the development of the augmented acyclicity-based method. Is the straightforward method implementable? If so, it would be better to compare it with augmented acyclicity-based method in terms of the performance and efficiency.\n\n2. I find it hard to fully understand Eq 9c and 9d. What's the definition of $\\mathcal{A}(W, o)$ and why does it have the formulation in 9c? What's $W_{o,i,j}$? Does it mean $W$ is a three dimensional tensor?\n\n3. To learn $W$, one needs the algorithm to be differentiable in terms of $W$. For the method in Eq 8a, one might need to backpropagate gradients through permutations. If so, how can we do this without continuous relaxation? For the augmented acyclicity-based method, it seems that one needs to conduct a few algorithms (Algorithm 2, 3, 4). Are the operations in these algorithms differentiable in terms of $W$? More discussions are needed.\n\n4. The proposed method introduces a new hyperparameter $\\gamma$. How to determine $\\gamma$ in practice as there is no ground truth to do validation?\n\n5. In the experiments of real-word data, Linear NOTEARS is used as the baseline. How do we know the (non)linearity in the real data?\n\nQUESTIONS:\nPlease see my comments above.\n\nLIMITATIONS:\nThe authors adequately addressed the limitations", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper contributes interesting new theoretical results for the field of differentiable graph structure learning and showcases how to practically exploit those results for improved structure learning. Differentiable structure learning converts the combinatorial optimisation problem of finding the correct graph structure of a problem into an optimisation problem by parametrising an adjacency matrix according to the structural equation model (SEM). Existing methods focused on enforcing DAG structures on these parametrised matrices such that DAG graphs can be learned. This paper instead focuses on a new type of constraints, trying to enforce that the learned graphs adhere to a given set of orders on the underlying variables. It is first proven (Theorem 3) that adherence to a set of orders $\\mathcal{O}$ is equivalent to a constraint comprised of $|\\mathcal{O}^+|$ terms, which is deemed computationally infeasible. Then the paper continues to prove its main result, showing that adherence to the set of orders $\\mathcal{O}$ is also equivalent to a constraint over maximal paths in the transitive *reduction* $\\mathcal{O}^-$ of $\\mathcal{O}$. Finally, this theoretical result is implemented such that it can take any existing SEM-based structure learning algorithm and augment it with given orders. The increase in performance is shown on a number of synthetic and more realistic structure learning benchmarks when comparing with and without the exploitation of given orders.\n\nSTRENGTHS:\n1. The structure of the paper is very clear and hence reads well. As someone who is not deeply familiar with the details of differentiable structure learning, the incremental and logical build-up of the preliminaries is also appreciated. In general, the authors put significant effort into making the paper self-contained, improving readability further. \n\n2. The paper is very formal in its notation and methodology, without going overboard on mathematical notation or jargon. Again, this allows for someone not intimately familiar with the topic to more easily follow the proofs and reasoning of the paper.\n\n3. The idea of the paper is certainly interesting and novel, especially considering the reference that incorporating orders can significantly reduce the search space of possible structures and the hardness of structure learning [1]. The overall idea of trying to incorporate prior knowledge of any form also ties in well with the rising popularity of neurosymbolic methods [2, 3].\n\n4. The authors clearly also put in a lot of effort to ensure a reader can follow the theoretical arguments on a high level throughout the text without getting lost in mathematical intricacies during reading. The many intermediate comments and examples are very helpful to keep the story focused, which is never easy when the road to a general result requires multiple intermediate results.\n\n5. The empirical evidence is very convincing. I thank the authors for including aggregates and variability metrics over multiple runs, which is sadly not a given anymore.\n\n[1] Teyssier, M., & Koller, D. (2005, July). Ordering-based search: a simple and effective algorithm for learning Bayesian networks. In Proceedings of the Twenty-First Conference on Uncertainty in Artificial Intelligence (pp. 584-590).\n\n[2] Garcez, A. D. A., & Lamb, L. C. (2023). Neurosymbolic AI: The 3 rd wave. Artificial Intelligence Review, 56(11), 12387-12406.\n\n[3] Marra, G., Dumančić, S., Manhaeve, R., & De Raedt, L. (2024). From statistical relational to neurosymbolic artificial intelligence: A survey. Artificial Intelligence, 104062.\n\nWEAKNESSES:\nWhile I overall did enjoy reading the paper, I do have a number of questions and concerns:\n\n1. The paper does not clearly distinguish its own theoretical contributions from previously known results. While I appreciate all lemmata and theorems are proven either in the main body or in the appendix, I would like to see clear statements of which results were already known, as some seem to relate to the provided references. In particular, I suspect Theorems 1, 2, and 3 were already proven before, though I am unsure about 3. Same for lemma 1. If some or all of these results are indeed novel, then explicitly stating they are can only further increase the impact of the paper.\n\n2. Theorem 2 is a nice result showing how knowing a total ordering can indeed considerably reduce the complexity of a DAG structure learning task. However, it is not used in the rest of the paper. If Theorem 2 is not a new result, I would remove it to improve the flow of the paper and give more space for further clarifications.\n\n3. I am not sure if I can agree with Remarks 2 and 6, where it is stated that the result of Theorem 3 does not lead to a computationally feasible solution while that of Theorem 4 does. It seems that using Equation 8b introduces a number of terms that is quadratic in the number of variables, as it is bounded by the number of possible pairs over those variables. While I can see that using Equation 9b is certainly more efficient for a single sequential ordering, the picture is less clear when $\\mathcal{O}$ consists of many (sequential) orderings. In general, the trade-off between orderings in $\\mathcal{O}^+$ or maximal paths in $\\mathcal{O}^-$ does not seem clear, especially since no bounds on $|\\mathcal{P}(\\mathcal{O}^-)|$ are given.\n\n4. While I generally agree with the provided proofs of all statements, there are a couple of places where I do have concerns about the validity of the results (see precise questions below). Importantly, I am unsure whether the exact statement of the main Theorem 4 holds, leading me to maintain a lower score for now until my questions are clarified.\n\n5. The experimental section certainly considers enough datasets and data-generating configurations, but it does seem to miss one important comparison. I would have liked to see an experimental confirmation of the trade-off between using Equation 8b compared to Equation 9b, for both run time and general performance. Especially since the complexity of 8b is used as a motivation to discard it in favour of 9b. Moreover, some of the metrics could be better explained, even if only in the appendix. For example, I am not familiar with the details of the structural Hamming distance (SHD) or False Discovery Rate (FDR) and the notion of run time is also ambiguous.\n\nQUESTIONS:\n1. Can you clarify which theoretical statements are novel by specifying if Theorems 1, 2, and 3, and Lemma 1 were already known?\n\n2. Theoretical clarifications:\n   + I believe the proof of Lemma 2, specifically the necessity part, is not fully complete. The second case (line 543), where some edges of the cycle are contained in $o$, implicitly seems to assume that there are at least 2 disjoint paths in the cycle with edges in $o$, i.e. $|r_o| = l \\geq 2$ because it is written that $q_i  < r_{i + 1}$. If $l = 1$, then $r_2$ does not exist and this inequality does not make sense. Hence, the case where $l = 1$ seems missing, although I think that a shorter version of a similar argument could apply.\n   + I do not see how the argument in lines 567-569 follows from Lemma 3 together with Equation 16. Concretely, Lemma 3 **only** seems to say that simultaneous adherence to $o^+$ for all $o \\in \\mathcal{P}(\\mathcal{O}^-)$ is equivalent to adherence to $\\mathcal{O}^+$, **not** that adherence to $o$ for all $o \\in \\mathcal{P}(\\mathcal{O}^-)$ is equivalent to adherence to $\\mathcal{O}$. This confusion might be due to the ambiguity of notation as described in Remark 3, but it is important to validate Theorem 4. From Remark 3, it seems that you do not consider a sequential ordering, such as $o$, to be a partial ordering as it is not transitive (line 199). However, then $o$ and $o^+$ are not the same, implying that adherence to $o^+$ is stronger than adhering to $o$. Similarly, it is unclear whether $\\mathcal{O}^+ = \\mathcal{O}$. **This confusion can have strong implications on the impact of the paper and is my primary concern. If the reasoning in lines 567-569 is invalid, the proposed objective in Equation 9b only enforces a weaker version of ordering adherence than the one enforced by Equation 8b.**\n\n3. Experimental clarification: What is the precise meaning of \"run time\" here? Is it the time required to learn the provided solutions? If so, what is the stopping condition for the algorithm?\n\n4. Can you elaborate on the computational trade-off between using Equation 8b compared to Equation 9b? Do you believe there are cases where this trade-off is negligable or even in favour of Equation 8b?\n\n5. Example 1 is not clear to me as it discusses the effect of removing **step 1** from the overall algorithm. However, it seems to be that **step 2** does not make sense without  **step 1**. So to be precise, why does $h'$ degenerate into $h(\\mathcal{A}(W, \\mathcal{O}^-)$ when removing **step 1**?\n\nI want to stress that I am very open to significantly increase my overall score (and my scores of soundness and contribution) to a full accept if the authors can address my questions. In particular, the seeming uncertainty around the validity of Theorem 4 is my main reason for borderline rejection because of its potential impact. If Theorem 4 does end up being weaker than stated, I believe an empirical comparison between Equation 8b (which surely enforces strict partial order adherence) and Equation 9b is warranted.\n\nLIMITATIONS:\nThe limitations are very well described in the appendix. I would only have like for it to be (partly) present somewhere in the main body of the paper, especially how any optimisation-based paradigm can not fully guarantee constraint satisfaction.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper provides a solution for imposing partial ordering information into differentiable DAG learning, which is a very important problem. It also proposes an efficient implementation based on rigorous theoretical justification. With this prior information, even with fewer samples, better structural recovery can be achieved in experiments.\n\nSTRENGTHS:\n1. This paper imposes partial ordering into the differentiable DAG learning problem. It is a very interesting problem to the community.\n2. The paper is theoretically well-supported.\n3. This paper addresses computational issues when imposing partial ordering into differentiable DAG learning. The experiments indicate that better structural recovery can be achieved with the information of partial order.\n\nWEAKNESSES:\n1. I found there are some points in the paper that are hard to understand. A mild suggestion is to add examples to illustrate these concepts, which would make the paper more readable. For instance, toy examples to explain what $\\mathcal{O}^+$ and $\\mathcal{O}^-$ are. Especially in section 3.3, where there are many definitions, theorems, and lemmas, examples would help readers grasp the points and have a better understanding of the paper.\n2. Typo: Line 204 $\\mathcal{O}^-)$ should be corrected to $\\mathcal{O}^-$.\n\nQUESTIONS:\n1. Lines 164-166: \"Merely forbidding edges...not contained in $\\mathcal{O}$\" I'm not sure I understand this point. Could you provide some examples to illustrate this argument?\n2. In Eq (6), $(i,j)\\in \\mathcal{O}^+$ indicates $i\\prec j$, meaning there is no directed edge from $j$ to $i$. However, in Equation (8b), $(i,j)\\in \\mathcal{O}^+$ implies there is no directed edge from $i$ to $j$. Do I understand this correctly? If so, please make notation is consistent throughout the paper.\n3. What Equation (9c) means here? $W - W\\circ W_o$ can be regarded as remove edge in path $o$, then we add $\\tau W_o$ back?\n4. In the experiments, do you test $p$ is small, such as $p = 10$?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces an approach to integrate partial order constraints into differentiable structure learning for causal discovery. The key contributions are:\n* Formulating an equivalent constraint set of path prohibitions to implement partial order constraints in the graph space.\n* Proposing an efficient method to integrate partial orders by augmenting the acyclicity constraint.\n* Proving the theoretical correctness and completeness of the proposed augmented acyclicity constraint.\n* Demonstrating the effectiveness of the method through experiments on both synthetic and real-world datasets.\n\nThe authors show that their method can significantly improve the quality of recovered causal structures while maintaining computational efficiency, especially for long sequential orderings. They also demonstrate that using partial order constraints can reduce the required sample size for accurate causal discovery on real-world data.\n\nSTRENGTHS:\n* The paper addresses an important gap in differentiable structure learning by enabling the integration of partial order constraints.\n* The augmented acyclicity approach handles long sequential orderings more efficiently, addressing a key limitation of a naive implementation.\n* The experiments cover both synthetic and real-world datasets, demonstrating the method's effectiveness across various scenarios.\n* Results on the Sachs dataset show significant improvements in structure recovery with reduced sample sizes, highlighting the method's impact for real-world applications.\n* The proposed method is designed as a plug-and-play module that can be integrated with various differentiable structure learning algorithms.\n\nWEAKNESSES:\n* Several statements are known or follow easily from existing results. \n* As noted in the limitations section, the method's efficiency can degrade with complex partial order structures involving multiple chains.\n* The paper doesn't extensively discuss how sensitive the method is to incorrect or conflicting partial order priors, which is possible in applications.\n* While the paper compares to baselines without priors, it doesn't compare to other methods that might incorporate different types of prior knowledge.\n* The paper doesn't provide an in-depth analysis of how sensitive the results are to the choice of hyperparameters, particularly τ in the augmented acyclicity term.\n\nQUESTIONS:\nMy overall take of this paper is that it could be a nice contribution as it appears to effectively handle partial ordering constraints for differentiable structure learning methods. However, some rewriting might be deemed necessary.\n\n* The definition of the SEM is confusing, in eq.(1) you might want to define $z$ as well, and I also find eq.(2) confusing, the definition of W_kj is not a real number but then you are taking an inner product in the function for D_ij. First, why is it necessary to write the sample version? Second, it is clearer if you state what kind of models work for the results in the sequel.\n\n* Several results are known or are trivial. Theorem 1 is already proved in Wei et al. (2020); Lemma 1 and Theorem 2 are straightforward results; Theorem 3 follows also directly from prior results in Wei et al. (2020). My point here being that theorems in a paper are typically representative of novel results and that correspond to the main contributions. In this case, I failed to see this. Theorem 4 should perhaps be Theorem 1 and the only theorem in the paper.\n\nMinor things:\n* L21: reform -> reformed\n* L25: Ng et al., 2020, do not propose an aciclicity characterization.\n* You can also consider citing Deng et al. 2023 (\"Global Optimality in Bivariate Gradient-based DAG Learning\") as they formally show, albeit in the bivariate case, how the continuous framework can succeed in recovering the underlying DAG.\n\nLIMITATIONS:\nLimitations are addressed in Appendix F.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-18"}