Representation Learning from Interventional Data

Comparison to ERM-Resampled: Please refer to Table T1 for the comparison between RepLIn and ERM-Resampled.

Use of resampling: We observed that resampling improves robustness in general. A comparison between our method without resampling and other methods can be found in Table T1. Additionally, resampling will help our method obtain better gradients from dependence-related losses as the estimate of the dependence improves with the number of samples.

Clarity for Section 3: The phrase from Section 3 that the reviewer found unclear was intended to argue about the danger of leaving spurious information in the representation. In a recent line of work, it was proposed to fine-tune the classifier to achieve robustness after observing that the learned representation contained both invariant and spurious features [R12-15]. However, in the absence of a fine-tuning dataset that exactly matches the test distribution, some spurious correlations may still slip through. We demonstrate this in a case where the support of the interventional distribution does not match that of the test distribution (Sections 3 and 4.3 in the main paper).

Terminology in the title: The larger goal of our work is to learn useful representations from interventional data. The term ``representation learning" is not limited to unsupervised/self-supervised learning. For example, representations learned from ImageNet classification have proved useful in several downstream applications.

Test dataset in Fig. 3: In Fig. 3, $A$ and $B$ are binary variables. During observations, $B\coloneqq A$ and during interventions, $B$ is set to values independent of $A$. The training data consists of both observational and interventional data, so that the model may learn the true invariant features. Since we are interested in evaluating the robustness of our models to interventional distributional shifts, our test set is built entirely from interventional data.

Multi-variate case: To demonstrate the advantage of our method in the multi-variate case, we construct a causal graph with five binary variables. The latents are generated according to the following SCM.

$$A = \text{Bern}(0.4) \\$$ $$B = \text{Bern}(0.6) \\$$ $$C = A \vee B \\$$ $$D = A \wedge C \\$$ $$E = \neg B \wedge C$$

where $\text{Bern}(p)$ indicates a Bernoulli distribution parameterized by $p$. The observed data $X$ is obtained as,

$$X\_A = \text{NN}\_{4,3}(A)$$ $$X\_B = \text{NN}\_{4,3}(B)$$ $$X\_C = \text{NN}\_{2,0.1}(C)$$ $$X\_D = \text{NN}\_{2,0.1}(D)$$ $$X\_E = \text{NN}\_{2,0.1}(E)$$ $$X = \text{concat}\left(X\_A, X\_B, X\_C, X\_D, X\_E\right)$$

where $\text{NN}_{l,\sigma}$ indicates a randomly initialized neural network with $l$ hidden layers, each with 100 units, and an added noise layer which adds a Gaussian noise with zero mean and $\sigma$ standard deviation. Each latent variable adds information about itself to the observed signal. Our intuition here is that, due to more hidden layers and larger added noise, the model may struggle to use the invariant information from $X$ and instead learn spurious features.

To collect interventional data, we intervene on $C$, $D$, and $E$ separately. Our evaluation metrics, therefore, will be the accuracy of the model in predicting $A$ and $B$ during these interventions. We compare ERM-Resampled and RepLIn-Resampled on their predictive accuracy during intervention on $C$, $D$, and $E$ in Tables T2, T3, and T4.

Common responses: The relevance and applications of our problem setting can be found in common response A. We discuss the difference between our goal and that in domain generalization in common response B.

Origin of dependence loss function: Our dependence loss is motivated by the observed correlation between the drop in accuracy and feature dependence during interventions. Since dependence between learned features is commonly measured and adjusted using kernel-based measures of dependence such as HSIC and KCC, we use the same to design our loss function.

Origin of self-dependence loss function: A caveat in learning independent features is that the presence of irrelevant/degenerate information in the representation can minimize the dependence loss while maintaining strong performance. To prevent this, we use our self-dependency loss which penalizes the representations for not fully aligned with its corresponding attribute of interest.

Common responses: The existing works on OOD generalization are designed for general distribution shifts. However, we are interested in a specific form of distribution shift induced by causal interventions. Please refer to common response B, where we describe how our goals and approach differ from standard invariant learning.

Our approach and d-separation: d-separation only tells us what the true dependence relations in the ideal causal model are. It does not translate to learning systems where no checks are employed during training to enforce true causal relations. One of our contributions is the observation that d-separation is not inherently followed by these models. Our method was driven by the observed correlation between the drop in accuracy and dependence between the features during interventions, as discussed in Section 2.1 of our paper. This observation is non-trivial and has not been previously discussed in the literature on learning from interventional data, although the question of whether learned models respect causal relationships has been asked [R21-23].

Knowledge of corruption labels: Robustness to label-dependent corruption is one of the several applications of our work. As we mentioned in common response A, we may often have this information. In practice, it is possible to infer the type of image corruption [R11].

Correction in related works: We apologize for the error in the description of Ahuja et al. (2022). Their paper indeed says it “does not require any structural assumptions about the dependency between the latents”. We have made this correction in the newly added section that contrasts our setting with that of identifiable causal representation learning.

Common Response B

Regarding the common response B, I'm afraid I have to disagree that datasets like PACS are not a result of intervention. Changes in the style of the image (photo, sketch, etc.) can be attributed to intervening on the latent variables that control to the style variations in the image. The goal of domain generalization methods is more difficult since they do not observe these latent variables and the causal graph for the data generation process, but the proposed approach in this work relies heavily on them. I do understand the difference between the proposed setup and the domain generalization setup, but the former does not seem challenging due to the reasons I mentioned before.

Comparison with related works from Causal Representation Learning

After reading the common response A, perhaps a case could be made that assuming the knowledge of intervened variables can be practical in some cases. But then a proper comparison needs to be made with other approaches from causal representation learning that can do so. For example, some works enforce support independence via Hausdorff distance [1, 2] between latent representations, which is a richer penalty than enforcing statistical independence as it can handle the case of correlation due to unobserved confounders as well. The proposed setup of authors can be been similar to the work of Ahuja et al. [2], the only difference being that the authors consider a supervised setup with a classification loss; while the setup of Ahuja et al. is unsupervised and they have a reconstruction objective. However, the access to interventional data is similar and the penalty of enforcing support distance in Ahuja et al. can be used by the authors in the proposed setup as well.

I do agree with the authors when they say: "The general objective in identifiable causal representation learning is to “learn both the true joint distribution over both observed and latent variables. The objective of this work is to provide a method to learn representations that are robust to interventional distribution shifts under the assumption of known interventional targets and their parents."

However, there are rich ideas in the causal representation learning about enforcing (statistical/support) independence, etc, which can serve as competitive baselines in the proposed setup.

I do appreciate the effort made by the authors but at this time the submission indeed requires more work and unfortunately I would not be able to increase my score. In the current form, the authors have technically sound results but I think the overall contribution is limited. I encourage authors to incorporate my feedback about causal rep learning baselines since I think it can make a strong case for downstream applications of the various causal rep learning works that have mostly dealt with identification and small-scale experiments for now.

References

[1] Roth, Karsten, Mark Ibrahim, Zeynep Akata, Pascal Vincent, and Diane Bouchacourt. "Disentanglement of correlated factors via hausdorff factorized support." arXiv preprint arXiv:2210.07347 (2022).

[2] Ahuja, Kartik, Divyat Mahajan, Yixin Wang, and Yoshua Bengio. "Interventional causal representation learning." In International conference on machine learning, pp. 372-407. PMLR, 2023.

Common responses: We justify the relevance and practicality of our setting in common response A. We apologize for the confusion due to the term "causal representation learning,” which is more commonly used in the identifiable learning community. Our response to your questions about the missing related works from identifiable causal representation learning can be found in common response C.

About assumptions: The primary objective of our work is to develop a method that can efficiently exploit the limited amounts of interventional data that may be available for training. To this end, we make a few assumptions. However, our assumptions are justified given that our work is the first to explicitly exploit interventions to achieve robustness to interventional distribution shift.

Imperfect/soft interventions: We developed our method under the assumption of perfect interventions. Constraining the dependence between intervened variables can be posed as a constrained optimization problem for which closed-form solutions may be obtained. For example, [R10] models the allowed sensitive information leakage as a constraint and obtains a closed-form solution. Nonetheless, we believe minimizing dependence between the interventional features can improve robustness against such distribution shifts.

Additional Results

Table T1: Comparison of our method against different baselines on the proposed Windmill dataset (Section 2.1 in the main paper)

References

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[R2] Azizi et al., "Synthetic Data from Diffusion Models Improves ImageNet Classification,” TMLR 2023

[R3] Gao et al., "Out-of-Domain Robustness via Targeted Augmentations,” ICML 2023

[R4] Sauer and Geiger, "Counterfactual Generative Networks,” ICLR 2021

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[R8] Shi et al., "Gradient Matching for Domain Generalization,” ICLR 2022

[R9] Nguyen et al., "Quality Not Quantity: On the Interaction between Dataset Design and Robustness of CLIP,” NeurIPS 2022

[R10] Sadeghi et al., "On the Global Optima of Kernelized Adversarial Representation Learning,” ICCV 2019

[R11] Li et al., "All-In-One Image Restoration for Unknown Corruption,” CVPR 2022

[R12] Menon et al., "Overparameterisation and Worst-Case Generalisation,” ICLR 2021

[R13] Kirichenko et al., "Last Layer Re-Training is Sufficient for Robustness to Spurious Correlations,” ICLR 2023

[R14] Rosenfeld et al., "Domain-Adjusted Regression or: ERM May Already Learn Features Sufficient for Out-of-Distribution Generalization,” NeurIPS DistShift 2022

[R15] Qiu et al., "Simple and Fast Group Robustness by Automatic Feature Reweighting,” ICML 2023

[R16] Li et al., "Deeper, Broader and Artier Domain Generalization,” ICCV 2017

[R17] Ilse et al., “Selecting Data Augmentation for Simulating Interventions”, ICML 2021

[R18] Mitrovic et al., “Representation Learning via Invariant Causal Mechanisms”, ICLR 2021

[R19] Lagemann et al., “Deep learning of causal structures in high dimensions under data limitations”, Nature Machine Intelligence 2023

[R20] Kemmern et al., “Large-Scale Genetic Perturbations Reveal Regulatory Networks and an Abundance of Gene-Specific Repressors”, Cell 2014

[R21] Jin et al., “Can Large Language Models Infer Causation from Correlation?”, ArXiv 2023

[R22] Wang and Boddeti, “Do Learned Representations Respect Causal Relationships?”, CVPR 2022

[R23] Kıcıman et al., “Causal Reasoning and Large Language Models: Opening a New Frontier for Causality”, ArXiv 2023