Paper: Representation Learning from Interventional Data (ICLR-2024)

Reviewer comment (Experiments):
It seems that RepLin algorithm assumes knowledge of which datapoint is interventional, whereas the ERM and other methods do not use this assumption. ERM-Resampled uses that knowledge, but not as directly as RepLin. However, experimental comparisons with ERM-Resampled is absent.

Author response:
**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.

Reviewer comment (Writing):
In Section 3, it is unclear what the author(s) are trying to discuss, especially the phrase “it is advisable to remove spurious information from the representations entirely”.

Author response:
**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).

Reviewer comment (Evaluation):
The title of the paper is about representation learning, but the paper evaluates the quality of the representation only via a single downstream task, making the problem more like supervised learning.

Author response:
**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.

Reviewer comment (Experiments):
What is the test dataset for the problem in Figure 3? If in Figure 3(b) the test is on the full support, then it is a significantly harder problem then Figure 3(a), making the motivation unclear.

Author response:
**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.

Reviewer comment (Experiments):
Why is there no comparison between RepLin and ERM-Resampled?

Author response:
**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.

Reviewer comment (Experiments):
Does the method work when there are more than two features that directly affect X? What about when the intervention is on two variables?

Author response:
**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.

Reviewer comment (Theory):
It is unclear exactly where the dependency and self-dependency losses come from. In particular, they seem to be only heuristically motivated, and it's unclear what minimizers of this loss should look like. Why should the "self-dependence loss" prevent the model from learning only irrelevant features, and how do we know that this is the right loss to do so? A rigorous mathematical treatment of the proposed loss would be very helpful here.

Author response:
**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.

Reviewer comment (Novelty):
My major concern with the work is that the technical contribution of the work is weak since the problem setup is very simple. The authors assume they know exactly the label that has been intervened and its non-descendants, which makes the proposed approach a simple application of the d-separation criteria in causal graphs. In contrast, the common setups in OOD generalization involving distribution shifts [1, 2] do not assume that we observe the intervened variables (and their causes) explicitly.

Author response:
**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].

Reviewer comment (Novelty):
The description of the prior work Ahuja et al. (2022) in the learning using interventional data section is incorrect. Their work is not limited to independent latent factors, rather they allowed for general causal relationships between the latents.

Author response:
**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.

Reviewer comment (Novelty):
The authors should have a detailed discussion on the array of 10+ works on causal representation learning from interventional data, e.g. [1], [2] (see also related works section in [2] which contains a wide list of such works). I'm surprised the authors seem unaware of this very closely related line of work.

Author response:
**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.

Reviewer comment (Novelty):
Related to the above point, while the authors compare to IRM in app. F and highlight the differences, I feel like the works on causal representation learning are more apt to contrast with, due to various modeling assumptions.

Author response:
**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.

Reviewer comment (Theory):
The authors consider perfect interventions. How robust are their methods to imperfect or soft interventions (with the usual meanings from the causal literature)?

Author response:
**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.