Thank you for your valuable feedback and questions. Please see our responses for the other comments.

However, it can be seen that these values are generally much lower for BiasBios than for CIFAR10. We would like to emphasize that the results from the CIFAR10 and BiasBios datasets are not directly comparable, as they are based on slightly different metrics. The metric for the CIFAR10 dataset is specific to our controlled study design, where we measure the degree to which the representations of two classes have collapsed with each other due to under-representation by comparing representations learned using balanced data to those learned using imbalanced data. In our experiments with the BiasBios dataset, we wish to demonstrate the potential harmfulness of collapse in a real data scenario where underrepresentation occurs naturally, which requires a slightly different measure of collapse. Specifically, we measure the ratio of the similarity of samples with the same gender and different occupations to the similarity of samples with different gender and the same occupation. This measure allows us to notice harmful cases of collapse, such as female surgeons being more similar to female dentists than to male surgeons.

Regarding ImageNet: Thank you for the suggestion. We started looking into such an experiment, but realized that the imagenet-1k dataset used for training publicly available CL models is indeed pretty balanced. Most classes have 1300 samples, with the least frequent class being black-and-tan_coonhound with 732 samples. Therefore, it is not suitable for studying the effect of underrepresentation. Artificially subsampling the data and training our own CL model as we did on CIFAR10 is unfortunately computationally too challenging for our computational resources.

Large difference between deer and horse collapse: Although we are not completely certain, a possible explanation is that the representations of deer are more dispersed compared to horse when undersampled, as seen in Figure 2 that the diagonal RH for deer ($1.126 \pm 0.012$) is significantly higher than horse ($1.068 \pm 0.009$). Since deer becomes more scattered, the representation harm metric between deer and horse is less compared to when horse is undersampled.

Training on 75% of the test set: The linear heads are trained with a randomly chosen 75% of the test dataset and evaluated with the remaining 25% to calculate the metrics in eq. (4.2).

Typos and figures Thank you for your suggestions. We have corrected the typos, are updating the figures, and will update the manuscript soon. Current heatmaps already use a divergent heatmap (1 is light green, above one are cold colors like blue, and below 1 are warm colors like red), but are not "centered". We will center the 1 to be white to improve readability.

Thank you for your insightful comments and questions. Please see our responses below.

Intuitive definition of allocative harms and representation harms: We added an intuitive definition and an example of allocative harms in the first paragraph of the Introduction. Allocation harm refers to the demographic disparities in resource allocations, and this harm is easily traced back to the rate of misclassification of the underlying resource allocation mechanism. In our eq. (4.2) we use this difference miclassification rate to quantify the allocation harm caused by under-representation.

We also extended the second paragraph of the Introduction with an example of representation harm to facilitate the intuition behind it. In the third paragraph and in Figure 1 we also present an example using the CIFAR10 dataset, where an undersampling of the automobile images causes their representations to cluster with those of trucks. In Section 2.1.2, we measure this clustering by calculating the ratio of the average cosine distances between the representations of two groups when one of them is under-represented to the average distance between the same two groups when they are class-balanced.

Different representation harm metrics in Sections 2.1.2 and 2.2.2: Although the exact quantification of representation harm metric may depend on specific instances of under-representation, the main idea behind the metrics are similar: they measure the collapse of the representations of an under-represented group with semantically similar groups. In our experiments with the CIFAR10 dataset in Section 2.1 we control the undersampling for each class and wish to demonstrate that the collapse grows/emerges when the underrepresentation occurs. In our experiments with the BiasBios dataset, we wish to demonstrate the potential harmfulness of collapse in a real data scenario where underrepresentation occurs naturally, which requires a slightly different measure of collapse. Specifically, we measure the ratio of the similarity of samples with the same gender and different occupations to the similarity of samples with different gender and the same occupation. This measure allows us to notice harmful cases of collapse, such as female surgeons being more similar to female dentists than to male surgeons.

Justification of the representation harm metric: We reiterate that the key idea behind measuring representation harm is to measure collapse between groups in the data, while relevant measures of collapse, the definitions of groups, and the kinds of groups between which collapse is most problematic may vary across domains and applications.

Examples used in Section 2 are not very useful: The collapse between automobile and truck representations (in the third paragraph of Section 1 and Section 2.1) is an example to illustrate the idea of representation harm. For a more realistic example, we refer to the Wall Street Journal article titled "Google Mistakenly Tags Black People as 'Gorillas,' Showing Limits of Algorithms" [3]. We have included this example in the second paragraph in the Introduction and in the first paragraph of Section 2.1.2. Additionally, we provide realistic examples in our experiments with the BiasBios dataset in Section 2.2.2, where the representations of under-represented female surgeons are closer to the representation of female dentists than they are to male surgeons. Similar harm is also observed between the attorney and paralegal professions. For a more detailed discussion, see our last paragraph in Section 2.2.2 named Results.

Why do sections 5 and 6 require a different data setup? We assume that the reviewer meant Sections 3 and 4. If so, then we point out that the synthetic experiments in Section 3 are used to motivate and interpret our theoretical analysis (Section 3.2) and to draw its connections with our empirical studies in Section 2. On the other hand, the causal mediation analysis in Section 4 shows that in CIFAR10 the downstream allocation harm in the classification task is partly caused by the representation harm.

Motivation for causal framework: As we explain in the second line of the first paragraph of Section 4 and in Section 4.1, causal mediation analysis [1] is the statistically principled way to dissect total allocation harm (TE) and tease out the part caused by representation harm through the natural indirect effect (NIE).

Thank you for your insightful comments and questions. Please see our responses below.

Enlarge texts in figures: Thank you for the suggestion. We have updated some of the figures in the main draft with a larger text and shall update our other figures (in the main and supplementary draft) shortly. If there are any figures remaining that you find difficult to read, let us know which ones and we will try to improve their readability.

Justification of the representation harm metric: We reiterate that the key idea behind measuring representation harm is to measure collapse between groups in the data, while relevant measures of collapse, the definitions of groups, and the kinds of groups between which collapse is most problematic may vary across domains and applications.

Different data setups in Sections 5 and 6: We assume that this was a typo, and you meant Sections 3 and 4. If so, then we point out that the synthetic experiments in Section 3 are used to motivate and interpret our theoretical analysis (Section 3.2) and to draw its connections with our empirical studies in Section 2. On the other hand, the causal mediation analysis in Section 4 shows that in CIFAR10 the downstream allocation harm in the classification task is partly caused by the representation harm.

Motivation for causal framework: As we explain in the second line of the first paragraph of Section 4 and in Section 4.1, causal mediation analysis [1] is the statistically principled way to dissect total allocation harm (TE) and tease out the part caused by representation harm through the natural indirect effect (NIE).

Absence of a comprehensive large-scale study: We believe that the experiments conducted with the two datasets, the synthetic dataset, and the theoretical analysis were sufficient to demonstrate that under-representation can lead to representation and allocation harm in contrastive learning settings. In addition to various CL methods, such as SimCLR, SimSiam, and SimCSE, we have also included the boosted contrastive learning method [2] (in Section 5) to further strengthen our experimental results. Although we acknowledge that additional experiments would be beneficial, the current paper already consists of nine pages of main text and 11 pages of supplementary material, not including references.

References

[1] Pearl, J. (2022). Direct and indirect effects. In Probabilistic and causal inference: the works of Judea Pearl (pp. 373-392).

[2] Zhou, Z., Yao, J., Wang, Y. F., Han, B., & Zhang, Y. (2022, June). Contrastive learning with boosted memorization. In International Conference on Machine Learning (pp. 27367-27377). PMLR.