Causal Bayesian Optimization with Unknown Causal Graphs

Thank you for the feedback and for the positive review. We address the questions as follows

• The red line is the performance of the CBO-U method. This was just to make it easier to compare our method with the other methods.
• In the CBO case, the model has exact knowledge of the causal structure and found the optimal interventions after very few iterations. This happens quite often in CBO, as the surrogate model tends to fit data generating process pretty well. This is a strong motivation for using this methodology. Furthermore, CBO-W also has a straight line since the graph is misspecified, it only selects uninformative interventions, meaning it selects intervention values that does not influence the target value, and the algorithm gets stuck in this minimum. The algorithm will have no way of getting out this local minimum. This further motivates why we incorporated uncertainty into the algorithm to ensure that this does not happen in our proposed solution. (The random baseline is the one that moves around a lot)

We are happy to answer any remaining questions you might have.

Thank you for the review and the feedback. We address the questions as follows:

• Question about novelty: We believe the novelty of the approach is two-fold. The first novel insights we made is that we do not need full knowledge of the causal graph for CBO. This makes the structure learning aspect of the problem easier, and allows our method to adapt to these general settings. This is where the novelty of Section 4.1 comes in as the results are derived specifically for this problem based on this insight. It incorporates uncertainty and it extends to nonlinear cases. Although there is a range of work with regards to Bayesian Causal Discovery for the full graph, these approaches do not fit into the framework as well as the results presented in this section. For example in those cases, each time an interventional sample is obtained, the full optimization process needs to be run again to determine the new probabilities of all the edges. This is wasteful especially for larger graphs. Our approach only requires simple Bayesian updates and focuses on relevant edges at each iteration of the algorithm. If we were to include methods, we believe a better approach would be to use these methods to determine the prior probabilities over all the edges, and then still use the update rules from Section 4.1 for the algorithm to determine the direct parents. This will also require us to adjust our assumptions based on the assumptions for these methods. This includes causal sufficiency.
• Question about Assumption 4.1: Assumption 4.1 is needed as distinguishing parents and children from observational data alone is not possible and the doubly robust methodology requires this assumption for the theoretical guarantees. This is nicely shown in [1]. However, if we have some prior beliefs about some of the edges and not sure whether it is a child or a parent, then the rest of the theory in Section 4.1 will still hold, and the interventional samples should be able to distinguish between parents and children if we intervene on the children.
• We addressed this as a general question to all reviewers. Although [2] gives some conditions for optimal intervention in the case of confounders, the bigger challenge lies in the causal discovery aspect of the algorithm. We will now need to learn a slightly larger part of the causal graph in the presence of unobserved confounders which is a very challenging problem in causal discovery.
• Question about Assumption 4.3: We addressed this as a general question to all reviewers. Although [2] gives some conditions for optimal intervention in the case of confounders, the bigger challenge lies in the causal discovery aspect of the algorithm. We will now need to learn a slightly larger part of the causal graph in the presence of unobserved confounders.
• Questions about other methods: To further illustrate why we did not include these types of algorithms we added another experiment. As an extreme case, we included a graph with 100 nodes in the experiments section. Our method is still able to perform optimization. The other methods will try to learn a full causal graph, where most of the edges are not relevant to this optimization problem.

[1] Francesco Quinzan, Ashkan Soleymani, Patrick Jaillet, Cristian R Rojas, and Stefan Bauer. Drcfs: Doubly robust causal feature selection. In International Conference on Machine Learning, pp. 28468–28491. PMLR, 2023. [2] Sanghack Lee and Elias Bareinboim. Structural causal bandits: Where to intervene? Advances in neural information processing systems, 31, 2018.

We hope this addresses your concern and we are happy to answer any further questions.

Thank you to the authors for their response. I acknowledge the contribution of this paper, which addresses the Causal Bayesian Optimization (CBO) problem under unknown graphs using uncertainty estimates over parent sets. The empirical results presented are appreciated.

I have also reviewed the authors’ responses and the comments of other reviewers. The authors noted that the main challenge lies in learning the Markov Blanket of Y in the presence of confounders. However, this observation is not entirely accurate, as when the reward is confounded with any of its ancestors, the candidate optimal intervention targets are specific subsets of the reward's ancestors. The paper titled "Structural Causal Bandits: Where to Intervene?" provides a graphical characterization of the potentially optimal intervention targets in such cases. Including a discussion on this point would enhance the paper.

Upon re-evaluating, I am lowering my score to reject, as I believe the paper lacks sufficient contributions to meet the standards required for acceptance. I also want to emphasize that the numerical score is only a guideline, and my detailed comments should be given greater consideration in the final decision.

Thank you for the review. We appreciate the feedback and we address the questions as follows.

• Question 1 about the bandit literature: Thank you for the suggestion, we have updated our related work with a short section about causal bandit literature.
• Question about Equation 2: $C$ (which could be $\emptyset$) refers to non-manipulative variables. We want to minimize the expectation with regards to the interventional distribution of the target variable. $C$ is included since $Y$ can be influenced by both the manipulative variables and the non-manipulative variables. This is essentially why the interventional distribution of Y is written as $P(Y | \varepsilon, C)$. Since $Y$ can be influenced by $C$ it is included in the expectation. $G$ is included in the expectation as we do not know the true graph and we need to learn it.
• Question about Assumption 4.3: We addressed this question as a general answer to all reviewers. If we relax this assumption, we can still use results from [1], but in this case a slightly larger part of the graph needs to be learned. This is the main challenge when relaxing this assumption as causal sufficiency is often an underlying assumption in causal discovery literature.

[1] Sanghack Lee and Elias Bareinboim. Structural causal bandits: Where to intervene? Advances in neural information processing systems, 31, 2018.

We hope this addresses your concerns and we would be happy to answer any further questions.

Thank you for your response. I have read the comments from all reviewers, particularly reviewer qxLe. I agree with reviewer qxLe regarding the contribution of the paper and believe it could be improved by relaxing Assumption 4.3. The findings in recent work [1] may be helpful in this regard. I also agree with reviewer qxLe that the main challenge of the problem in the presence of confounders is not learning Markov Blanket of $Y$.

[1] Muhammad Qasim Elahi, Mahsa Ghasemi and Murat Kocaoglu, Partial Structure Discovery is Sufficient for No-regret Learning in Causal Bandits, 2024, https://arxiv.org/abs/2411.04054.

Thank you for the review. We appreciate the positive feedback and the suggestions. We address the questions as follows:

Algorithm questions:

• Question about type 1 vs type 2 errors: Yes, you are correct the approach does require the groundtruth parents to be discovered during the initial phase. This is why we used the Bayesian approach with uncertainty estimation. It reduces the chances of not ‘discovering’ the correct parents during the initial phase. It also then updates as it gets new interventional samples. The posterior distribution will then become tighter as the number of samples increases. Currently, the CBO-W shows what will happen if the correct parents are not identified in the algorithm. This essentially shows how the algorithm will get stuck if the correct parents are not determined. This shows what will happen if we cannot determine the direct parents. We will work on incorporating a breakdown when this happens.
• Question about the Dream graphs: We are referring the Dream4 causal graph.
• Selecting the plausible parents: Algorithmically it would not be an issue to use an approach like this, especially if we have both interventional data and observational data available. The reason we did not go that route is that often in practice in this type of setup we would not have interventional samples available or very few interventional samples. Thus, using an approach that works well on observational data made more sense. Furthermore, the goal of CBO-U being to find the optimal intervention in as few interventions as possible. Thus we want to collect the interventional samples in a very targeted manner right from the start and then update the probability based on these samples.

Assumption questions:

• We made the discussion about the limitations clearer in that section. We believe the general framework in those cases will be similar to the one we proposed here. You would want to learn the posterior over the relevant part of the causal graph, and then select interventions in a targeted manner. In those cases, the structure learning aspect will become more challenging.
• The question about Assumption 4.3 we addressed as a general question to all reviewers. The causal discovery aspect of the algorithm will become more challenging, since we the parents will not necessarily be the optimal intervention set. In this case, we still do not need to learn the full causal graph and we would need to learn the Markov Blanket for the target variable in the presence of confounders. When we relax Assumption 4.3 there are similar conditions we can exploit as referred to in this paper by [1]. This could be a nice follow-up from this paper. The approach can then work in a similar way to CBO-U. Learning this type of structure is however a very challenging problem in causal discovery.
• There are other approaches for CBO with soft interventions [2, 3]. Both these methods require full knowledge of the graph, but do not have optimality conditions. [3] is a model-based approach and [2] shows cases when these contextual interventions are better than hard interventions. Thus, work needs to be done on determining optimal interventions in this case. One could consider a joint structure-learning and model-based approach.

[1] Sanghack Lee and Elias Bareinboim. Structural causal bandits: Where to intervene? Advances in neural information processing systems, 31, 2018.

[2] Limor Gultchin, Virginia Aglietti, Alexis Bellot, and Silvia Chiappa. Functional causal bayesian optimization. In Uncertainty in Artificial Intelligence, pp. 756–765. PMLR, 2023.

[3] Scott Sussex, Anastasiia Makarova, and Andreas Krause. Model-based causal bayesian optimization. International Conference on Learning Representations, 2023

We hope this answers your questions and we are happy to answer any further questions.