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Consistency of Neural Causal Partial Identification

Recent progress in Neural Causal Models (NCMs) showcased how identification and partial identification of causal effects can be automatically carried out via training of neural generative models that respect the constraints encoded in a given causal graph [Xia et al. 2022, Balazadeh et al. 2022]. However, formal consistency of these methods has only been proven for the case of discrete variables or only for linear causal models. In this work, we prove the consistency of partial identification via NCMs in a general setting with both continuous and categorical variables. Further, our results highlight the impact of the design of the underlying neural network architecture in terms of depth and connectivity as well as the importance of applying Lipschitz regularization in the training phase. In particular, we provide a counterexample showing that without Lipschitz regularization this method may not be asymptotically consistent. Our results are enabled by new results on the approximability of Structural Causal Models (SCMs) via neural generative models, together with an analysis of the sample complexity of the resulting architectures and how that translates into an error in the constrained optimization problem that defines the partial identification bounds.

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Peer review

Accept (poster)NeurIPS 2024 · via OpenReview
Reviewer Mk7v5/10 · confidence 2/52024-07-12

Summary

This paper studies theoretically the partial identification capabilities of neural causal models (NCMs), that is, the extent to which this type of models can approximate the interval on which a certain causal query falls on, $\theta(\mathcal{M}) \in [\underline{F}, \overline{F}]$. To this end, the authors develop a lot of results showing that under ideal scenarios with oracle knowledge on the exogenous distributions, NCMs can approximate any SCM if the proper architecture and regularization is used. Interestingly, the authors show that regularization is key in these settings, as otherwise the observational likelihood can be modelled arbitrarily well, while the interventional distribution do not look alike. It is worth-noting that the theory developed consider general scenarios with confounders and mixed-type data.

Strengths

- **S1.** This work addresses an intellectually interesting question of whether NCMs can properly perform causal partial identification. - **S2.** The theory developed looks general, sound, and it makes sense on an intuitive level (disclaimer: I have _not_ looked into the proofs). - **S3.** The work properly studies mixed-type data, which I have rarely seen done. - **S4.** The fact that NCMs _need_ a form of regularization to properly approximate the SCMs is really interesting.

Weaknesses

- **W1.** My personal main concern is that I do not immediately see any practical implication of the theory developed here. What I mean is that, while I find it interesting, if I have understood it correctly it works under the assumption that we know the exogenous distribution of the ground-truth model, which we never know, and which we could be arbitrarily different from whatever we model (we can always change the exogenous distribution and push the transformation to the generator $f$). - **W2.** While trying to be general, I similarly feel that some of the assumptions can be quite restrictive in practice. E.g., global Lipschitz-continuity is quite a constraining assumption from my point of view. - **W3.** I know it is a dense theory paper, but the presentation leaves a lot to be desired. There is a lot of similar notation, non-standard definitions (e.g. SCMs), and a lack of intuition/examples that makes the results hard to parse. The paper is simply not accessible to a large percentage of the community. - **W4.** The experimental section is extremely short, and it would require of more settings, an ablation study of the different components of the theory, and a proper analysis of the results.

Questions

I have no specific questions, but I want to make clear that a dense paper like this would require many more hours for me to fully understand it and I may have missed some important points. I will reflect this in my confidence score.

Rating

5

Confidence

2

Soundness

3

Presentation

2

Contribution

3

Limitations

Limitations are **not** discussed. In the checklist, it is said to be discussed on the Conclusion section, which does not exist.

Reviewer VZrh6/10 · confidence 3/52024-07-12

Summary

This paper provides a novel perspective and solid contribution to the case where continuous and categorical variables both exist.

Strengths

The presentation and organization are concise and clear; the contribution is solid.

Weaknesses

The assumption is somehow strong (e.g., assumption 2).

Questions

How do we argue the importance of solving the problem of ``the approximation error and consistency of the optimization-based approach to partial identification via NCMs''?

Rating

6

Confidence

3

Soundness

3

Presentation

3

Contribution

2

Limitations

See above.

Reviewer VZrh2024-08-11

Thanks

It is a concise and high-quality response. Thank you! I raise my score to 6. Besides, in fact, I'm quite familiar with Partial Identification, one of my main research topics. I also share two nice new papers and recommend you cite them as follows. Of course, I acknowledge that this literature is somehow different from your settings, but they all make meaningful contributions to general PI. For instance, after my careful proofreading, Literature [1] 's main IFF theorem holds in discrete/continuous/mixed cases. Finally, my last question is: Why is the extension from the discrete case to the continuous case highly non-trivial? For instance, when we get the continuous variable, why not conduct naive discretization, and the corresponding result may not change significantly? If this holds true, it will not be interesting. So, I wonder about some more insightful reasons/counterexamples. Thank you! I will continuously consider raising my score. [1] tight partial identification of causal effects with marginal distribution of unmeasured confounders. (Zhang, ICML2024 spotlight) [2] Model-Agnostic Covariate-Assisted INference on Partially Identified Causal effects (Wen et al)

Authorsrebuttal2024-08-11

Thank you so much for your feedback and for sharing the recent literature with us. We sincerely appreciate your contributions to improving our work. Regarding the two papers you mentioned, they indeed focus on a specific type of causal graph, which differs from the general causal graph setting our algorithm addresses. However, we agree that these papers offer valuable insights for the partial identification literature. We will cite them in the camera-ready version of our paper. As for your question about discretization, we appreciate the opportunity to clarify this point. As we mention in the paper, for a general discrete SCM, the Partial Identification (PI) problem becomes a Polynomial Programming (PP) problem [1], which is NP-hard in general. While discretizing continuous data and solving the PP problem to obtain bounds is possible, the problem's size grows *exponentially* with the cardinality of the support. This implies that the finer the discretization of the continuous data, the larger and more computationally expensive the resulting PP problem becomes. In our experiments, we found that the PP problem quickly becomes intractable when we increase the support's cardinality of continuous variables. For instance, in a continuous IV setting, we compared our algorithm with Autobounds [1], the state-of-the-art algorithm for solving discrete PI problems, by discretizing all continuous variables such that their support's cardinality is 8 (details are in Section D.2). This produces a PP problem of size approximately $2^{14}$. However, even we solved such a large problem, the bound obtained by this approach was not as tight as the one obtained using the NCM approach. While finer discretization could potentially improve the bound, the resulting PP problem becomes so large that we were unable to obtain a solution after running the Autobounds algorithm for over a day. Therefore, we believe it is crucial to generalize the NCM approach to include continuous (and mixed) variables, as it proves to be more efficient and powerful compared to the discretization approach in these settings. Thank you again for your detailed review and valuable questions. We are glad to take any further questions. [1] Guilherme Duarte, Noam Finkelstein, Dean Knox, Jonathan Mummolo, and Ilya Shpitser. An automated approach to causal inference in discrete settings, September 2021

Reviewer 85217/10 · confidence 4/52024-07-12

Summary

The paper extends neural causal models (NCMs) to the continuous and mixed-type variables. NCMs is a neural-networks based tool to perform an automated point/partial identification and estimation of causal queries. The authors provided new theoretic results (1) on how to construct a canonical representation of an SCM, (2) on how to approximate it with sufficiently expressive Lipshitz neural networks, and (3) on additional assumptions to perform the consistent identification/estimation of an average treatment effect (ATE) from an arbitrary causal digram and an observational distribution. The paper provides experimental results for the partial identification of the ATE in the discrete and continuous outcomes settings.

Strengths

To the best of my knowledge, this paper is the first one to advance neural causal models (NCMs) to the mixed-type variables setting. NCMs are a universal tool for general automated partial identification and estimation of causal quantities (a very important problem in causal inference and treatment effect estimation). The theory provided in the paper is very general and extends over simple linear SCMs. The paper is clearly written (as far as a massive theoretic contribution allows) and well-structured. The authors rigorous proofs for all the theoretic statements.

Weaknesses

I didn’t find any major or minor weaknesses in the paper. Yet, I have several suggestions on how to improve the paper’s understandability for the general public of the NeurIPS conference: 1. I encourage the authors to provide more examples of how to construct canonical representations for some simple SCMs and then the corresponding neural architectures of the NCMs. 2. I would provide a more precise explanation of Figures 1 and 5, e.g., the number of layers in the yellow and blue blocks. Also, multiple notation elements could potentially be added to the Figures, e.g., the number of connected components in the latent space or the number of confounded components. 3. I also encourage the authors to provide the code of the proposed method. 4. Seems like the conclusion is missing in the main part of the text. I am willing to increase my score if the authors implement the above-mentioned suggestions.

Questions

I appreciate the provided continuous IV experiment. I wonder, what is the performance of the method in other confounded ATE settings where the ground-truth is available, e.g., a no-assumptions bound for the ATE with hidden confounding?

Rating

7

Confidence

4

Soundness

4

Presentation

3

Contribution

4

Limitations

The authors have fully stated the limitations of their theory.

Reviewer zYjd6/10 · confidence 2/52024-07-13

Summary

This paper develop consistency results for partial identification via neural causal model with both continuous and categorical variables. Their results shed light on the impact of the neural network architecture and Lipschitz regularization during training. The resulting method can be trained via gradient-based optimization algorithms, and is validated on synthetic data.

Strengths

- The paper is clearly written. The assumptions used are clearly stated. - The identification result is useful and a good complement to existing ones, by making them more general. - The identification result and algorithm are technically sound.

Weaknesses

- The empirical validation of the algorithm/result is not extensive, but I understand that the goal of the work is to establish identification results and a thorough empirical study might not be necessary. - The paper would benefit from giving some proof sketches (or a brief overview of the proof strategy) in the main paper. - Many of the assumptions and results are not well explained, making it not straightforward to understand the assumptions and implications of the results.

Questions

- A conclusion section does not seem to be provided. - It would enrich the paper to briefly discuss the connection with optimization-based approaches in causal learning (many of which also used similar techniques including augmented Lagrangian and gumbel techniques), although the task is not completely the same (see e.g. https://arxiv.org/abs/2007.01754, https://arxiv.org/abs/1910.08527, https://openreview.net/forum?id=HsSLdHuAmnY). - It would be helpful to elaborate more on the assumptions required and discuss some examples where the assumptions are satisfied/violated. - Other related work that could be worth mentioning: https://arxiv.org/abs/2105.12891

Rating

6

Confidence

2

Soundness

3

Presentation

2

Contribution

3

Limitations

The limitations have not been well discussed.

Reviewer zYjd2024-08-11

Thank you for the helpful replies. Most of my concerns have been addressed and I would like to maintain my rating.

Reviewer Mk7v2024-08-11

I thank the authors for their kind and detailed response, it should be clear by now that I am an outsider on this particular problem, and the initial barrier is a bit too tough to pass in the reviewing period. Clearly, other reviewers were much more positive than I initially was. With that said, I understand the practical implications of this work, and I applaud it. However, I still have some questions for which I would appreciate an answer (I apologize I don't have currently the time to go through the proofs in the appendix, yet I'll try to go through the main paper one more time). To my understanding, one key aspect of this work is splitting the estimation error in two terms, the error in the exogenous distribution and that of the causal mechanisms. From there, one can use the theory of NN approximation to estimate the error committed in each part, depending on depth and width. To achieve this result, it is necessary to assume a specific $\mathcal{M}^*$ in its canonical form, which would have a specific Lipschitz constant $L^*$ that the proofs should somehow use. What I don't understand is that, without assuming that one knows the exogenous distribution of $\mathcal{M}^*$, it is possible to find alternative canonical representations by applying arbitrary transformations $\phi$ to the canonical form, i.e., having $(\phi \circ P_U, \phi^{-1} \circ f)$ instead of $(P_U, f)$, with arbitrary Lipschitz constants for the target SCM. As far as I can see, the definition of the canonical form does allow these cases. How does your work rule them out? I guess some of the assumptions do it? Thanks in advance.

Authorsrebuttal2024-08-13

Thank you for your question. We want to clarify that the approximation theorems (Theorem 1 and Corollary 1) are tools we use to prove the consistency theorem (Theorem 4), which is the central result of our paper. These approximation theorems state that given an SCM (with known latent distribution and structure functions) that satisfies some assumptions, we can approximate it using NCMs. However, in the consistency theorem, *we do not assume that we know the latent distribution or the structure functions (nor does the consistency theorem claim that we recovered the true latent distribution; just that the true latent distribution is part of the “search space”)*. The consistency theorem proved the consistency of the bounds for the target estimand. As we mentioned before, the goal of partial identification is to determine bounds for a causal quantity across all SCMs that have the same observation distribution as the ground-truth model. Since it is difficult to search over all SCMs, we argue that it is sufficient—and more practical—to search over all NCMs. The approximation theorems are used to prove that the set of NCMs is ‘close’ to the set of SCMs so that the errors caused by this substitution in the partial identification problem can be controlled. Therefore, the short answer to your question is that we cannot rule out alternative causal models that result from transformations, as long as they satisfy all the assumptions (particularly Assumptions 3 and 4). Since we only observe a subset of the variables, multiple causal models could potentially lead to the same observational distribution. Without additional information about the true underlying model, it is often difficult to uniquely identify a single causal model from observations alone. This is a typical case in causal inference. Even when the target estimand is point identified (e.g. ATE with no unobserved confounding), and hence our bounds converge to a singleton, that does not mean that the latent distribution of the structural causal model nor the structural functions are identifiable, due to the ambiguity you mention. Still, in such unique identification cases, all such transformations yield the same target estimand. We hope this clarification helps you better understand our work, and we are happy to address any further questions you might have.

Reviewer Mk7v2024-08-14

Many thanks once again for the detailed response. I think I now get a bit better the intuition of the work. Since the discussion period is over, I am going to increase my score to borderline accept, as I do not feel comfortable giving more score to a work for which I don't have a full grasp on the results. This is mostly on me, coming from a different area, and the "mathematical heaviness" of the work, which would require many more hours of me to fully grasp. However, this should be not a problem, as my confidence score is already quite low.

Reviewer 85212024-08-12

Thank you for the clarifications. I still have an important concern regarding the no-assumptions bound experiments (5). For the outcome $Y \in [a, b]$, the no-assumption (Manski) bounds on the ATE always have the width of $b - a$ [1]. In the example, provided by the authors, $Y \in [-11, 11]$ and, thus, the ground-truth width is $b - a = 22$. Therefore, it seems that the method proposed by the authors yields invalid bounds with a width of $8.32$ (whereas the Autobounds package provides a more valid width of $23.53$). Could the authors explain the source of the invalidity? **References**: - [1] Manski, Charles F. "Nonparametric bounds on treatment effects." The American Economic Review 80.2 (1990): 319-323.

Authorsrebuttal2024-08-13

Thank you for your question. We would like to point out that the Manski bound [1] is derived specifically for binary treatment settings, whereas in our example, the treatment is continuous. Besides, the causal graph of [1] is different from the leaky mediation graph. For these reasons, it is not immediately clear whether the same bound can be applied directly in our context. A more comparable setting is the binary IV example in our paper, where both algorithms yield similar bounds. In the binary IV example, we use the algorithm from [2] to verify that the bounds we get are correct. Additionally, we impose Lipschitz constraints on the structure functions when solving the PI problem, which may result in tighter bounds. However, the analytic characterization of the optimal bounds under Lipschitz constraints has not been established and seems hard to establish in analytic form. We are happy to address any further questions you may have. [1] Manski, Charles F. "Nonparametric bounds on treatment effects." The American Economic Review 80.2 (1990): 319-323. [2] Balke, Alexander, and Judea Pearl. "Bounds on treatment effects from studies with imperfect compliance." Journal of the American statistical Association 92.439 (1997): 1171-1176.

Program Chairsdecision2024-09-25

Decision

Accept (poster)

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