Small Variance, Big Fairness: A Path to Harmless Fairness without Demographics

Dear Reviewer,

We appreciate your thoughtful review of our paper. Your feedback is instrumental in refining our work, and we appreciate the time and consideration you have given to our manuscript. Here are our responses to your comments and questions:

1. The position of Proposition 2

• Regarding the motivation of minimizing the variance of losses, we suggest readers also refer to the Idea​ paragraph in the Introduction section, where variance is also viewed as the second moment of loss distribution. We thought it is straightforward and useful to character accuracy disparities when demographics are unknown.
• Note that Proposition 2 is derived by using MAD metric, and it also expects that loss should be (approximately) independent of input, which is used to help understand the problem only. Thus, we left it to the appendix as a supplementary.

2. Concern about Proposition 1

Thank you for your detailed reading.

• In Appendix B, we presented that Eq. (10) formulated as a pairwise loss which is tighter than variance without blowing up one side by a factor of $n$. However, shown as in Table 3 on Page 13, the pairwise loss has not achieved better results because the global information (e.g., mean loss) is hard to be incorporated for a minibatch update in practical implementation.
• Although loose, experimental results in Table 3 demonstrated that variance minimizing is useful to characterize accuracy disparity without demographics. We understand tight bound is a pursuit in most cases, while it is not the only criterion in this scenario.

3. Experimental Results

We have conducted numerous detailed new experiments, please refer to the global comments for more information.

4. Relationship with Federated Learning

We understand that learning on data without demographics is closely related to the data privacy community. However, please note that our problem setting does not introduce any prior about demographics, which indicates the solution should adapt to any demographic distribution. Thus, there is no need to set distributed groups that are based on federated learning. If each local group is exactly a group divided by interested attributes, then the central model at least knows how many local groups there exist (when it executes gradient aggregation), which can be some side information to demographics and thus beyond our problem setting.

5. Training time of each method

Each method shares similar training steps. While explicit details of the training steps are not provided, a comprehensive account of the experimental configurations is presented in Appendix F, specifically within the section titled "Harmless Implementation of Baselines." In this section, we meticulously elucidate the selection of steps within a harmless fairness context. To expound further, each method is characterized by an empirical loss denoted as $\hat{\mu}$ in our methodology and as the learner loss (in comparison to adversarial loss) in the ARL approach. This empirical loss serves as the criterion for step selection, whereby we opt for the harmless step that manifests the loss value most proximate to that of a well-trained ERM model. This methodological choice ensures a judicious alignment with established benchmarks, fostering a fair evaluation of the experimental outcomes.

We hope these planned revisions and clarifications will address your concerns. Please let us know if you have any further questions. We are inclined to address them to improve the quality of this work.

Dear Reviewer,

We would like to express our gratitude for taking the time to thoroughly review our work. We have carefully considered each of your comments, and we are committed to addressing them to enhance the overall quality.

1. Experimental Results

We have conducted numerous detailed new experiments, please refer to the global comments for more information.

2. Code and Experimental Artifacts

Following your suggestion, we have included our code in the supplementary materials, which showcases a simple yet effective approach to achieving fairness. We will release the full code once this paper is accepted.

3. Comparison with Constrained Optimization Approaches

• We reminded that utilizing constrained optimization essentially requires a proper $\delta$ (an upper bound of the minimum of loss in Eq. 1&2) which is not accessible unless minimizing the average losses on the entire training set beforehand.
• We clarified that the targeted problem is treated as a bi-objective problem in this paper, with two objectives having different priorities. In this sense, the proposed algorithm does not have to handle $\delta$ separately, while “the constraint” is implicitly satisfied. While the suggested [1] and TFCO focus on solving constrained optimization problems. To avoid confusion, we will cite these works and offer a reminder in the main body of the paper.
• Using a fixed λ, which is simply set to 1 in Table 2, we linearly combined two objectives, which serve as an ablation study in our experiments. The results showed that the model performance may significantly drop, e.g., on COMPAS.
• We apologize for the description of "a fixed multiplier". Our intended meaning is that the advanced Lagrange method aims at the optimal multiplier while it is dynamic along with the optimization process in our method.

4. Details on the model structure

Following your advice, we have included the necessary description in the paper.

All the models, excluding FairRF, which operates within a distinct problem setting, conform to a shared neural network framework. Specifically, for binary classification tasks, the core neural network architecture consists of an embedding layer followed by two hidden layers, with 64 and 32 neurons, respectively. In the ARL model, an additional adversarial component is integrated, detailed in its respective paper, featuring one hidden layer with 32 neurons. For multi-classification tasks, the primary neural network transforms into resnet18, and the embedding layer transitions to a Conv2d-based frontend. Throughout these experiments, the Adagrad optimizer was employed. FairRF, utilizing its officially published code implementation, maintains the same backbone network with nuanced variations in specific details.

5. Details about Loss Function

Thank you once again for your detailed reading. Throughout the experiments, we implemented binary cross-entropy (BCE) for binary classification and categorical cross-entropy (CE) for multi-class classification. However, we would like to underscore that our method is general and is compatible with other forms of loss as well. We will incorporate this information into our main paper.

6. Grayed Out Rows in Table 2

Table 2 shows that applying a fixed $\lambda$ or using $\lambda_1$ only might result in uniform classifiers. This is because without adjustment of $\lambda_2$, the model is at risk of providing very uncertain predictions to training samples, that is, most of the data lies near the decision boundary. Fig. 3 observes the same phenomenon by monitoring the distance between samples to the decision boundary, which demonstrates the necessity of $\lambda_2$. Mathematically, $\lambda_2$ lifts up the importance of the average loss from the “loss perspective” even if the gradients of two objectives do not conflict with each other by much. Following your advice, we added the plot of $\lambda$ in Appendix G.

We hope these planned revisions and clarifications will address your concerns. Please let us know if you have any further questions. We are inclined to address them to improve the quality of this work.

[1]: Cotter, Andrew, Heinrich Jiang, and Karthik Sridharan. "Two-player games for efficient non-convex constrained optimization." Algorithmic Learning Theory. PMLR, 2019.

[2] Gong, Chengyue, and Xingchao Liu. "Bi-objective trade-off with dynamic barrier gradient descent." NeurIPS 2021 (2021).

Dear Reviewer,

We appreciate your thorough review of our paper and your insightful comments. Your feedback is invaluable in refining and strengthening our work. Here are our responses to your comments and questions:

1. Other Fairness metrics

• As we claimed in the Introduction, utility disparity across different groups is our interest, which also aligns with the recent studies based on Rawlsian fairness where predictive accuracy of the worst-off group is concerned and they do not include equal odds and statistical parity either.
• Note that equal odds and statistical parity measure fairness over predicted y, which requires the latent representation or final prediction to be somehow invariant to groups. On the one hand, these metrics are not compatible with the concept of model utility we advocated. On the other hand, to achieve such fairness we need to reformulate the objective because our original goal is to let loss value (determined by both predicted y and ground truth y) be approximately invariant to groups.

2. Experimental Results

We have conducted numerous detailed new experiments, please refer to the global comments for more information.

3. Discussion of related works

Thanks for suggesting the related works. We will incorporate relevant discussions and references in the related work section to provide a more comprehensive background for our work.

• We understand variance regularization on ERM aims at improving the model reliability but may not necessarily imply a flat minimum in the loss landscape [1]. A flat minimum like sharpness-aware minimization [4] expects that average loss should not change by much if a model is perturbed within a small region. However mathematically, variance minimization cannot ensure such a property.
• Based on the concept of Rashomon [3], research [2] also proposed a method that achieves fairness from good-utility models, which is quite similar to the branch of “Harmless Fairness” in Related Work. Our work differs from theirs by two key differences. (1) We further focused on “Harmless fairness without demographic” and thus paid much attention to how to describe fairness under such a restrictive scenario. (2) Research [2] solves a constrained optimization problem, which means we need a proper upper bound for the average loss. One may obtain it by running an ERM model beforehand and then setting it, which is not needed as we treat it as a bi-objective problem and the constraint is satisfied by adjusting the weight factor $\lambda$.

We hope these planned revisions and clarifications will address your concerns. Please let us know if you have any further questions. We are inclined to address them to improve the quality of this work.

[1] Hao Li et al. Visualizing the Loss Landscape of Neural Nets. 2018.

[2] Amanda Coston et al. Characterizing Fairness Over the Set of Good Models Under Selective Labels. 2021.

[3] Lucas Monteiro Paes. On the Inevitability of the Rashomon Effect. 2023.

[4] Sharpness-Aware Minimization for Efficiently Improving Generalization, ICLR 2021.

Dear Reviewer,

We appreciate your detailed and insightful feedback on our manuscript, and we would like to express our gratitude for taking the time to thoroughly review our work. We have carefully considered each of your points and suggestions, and we are committed to addressing them to enhance the quality and clarity of our contribution.

1. Discussion about related works

Thank you for suggesting the related studies. As we discussed in Section 5, we agreed that penalizing the variance of losses can be motivated by different applications, and we would add its advantage of robustness [1]. Particularly, in the submitted manuscript, we have pointed out the relation between variance regularization and instance-reweight learning inspired by the work [2], which is not a fairness study though. TERM [3] reformulates the standard ERM which improves the worst-performance fairness and thus would be included as a worst-case fairness method.

2. Additional experimental details

Your suggestion regarding additional experimental details is well-received. We apologize for the omission of the model structure and will include the necessary details in the paper.

• All the models, excluding FairRF, which operates within a distinct problem setting, conform to a shared neural network framework. Specifically, for binary classification tasks, the core neural network architecture consists of an embedding layer followed by two hidden layers, with 64 and 32 neurons, respectively. In the ARL model, an additional adversarial component is integrated, detailed in its respective paper, featuring one hidden layer with 32 neurons. For multi-classification tasks, the primary neural network transforms into resnet18, and the embedding layer transitions to a Conv2d-based frontend. Throughout these experiments, the Adagrad optimizer was employed. FairRF, utilizing its officially published code implementation, maintains the same backbone network with nuanced variations in specific details.
• To address the ambiguity in the explanation of the hyperparameter epsilon, we will provide a more detailed clarification. When the angle of the gradient reaches an orthogonal angle, Formula 4 can be simplified as $\lambda \ge \epsilon$, indicating that in the absence of conflicts in gradient updating, to what extent do we still want the primary objective to be updated? We designate this as a hyperparameter because we have observed variations in the quality of data across different datasets. For instance, on the COMPAS dataset, known for its numerous noisy samples [4], a larger epsilon is needed to control the model to maintain utility. In this case, the regularization of the second moment of the distribution is constrained, preventing the model from focusing more on outlier samples. For reproducibility, it is recommended to set the value of epsilon as 0, 0.5, 0.5, and 3 for Law School, UCI Adult, CelebA, and COMPAS, respectively.

3. The ambiguity of symbols

We apologize for the ambiguity of objectives. To be clearer, we used $\pi, \hat{\sigma}, \hat{\sigma}^2$ to denote the 3 objectives derived in B.1 and Table 3:

• $\pi=\sum_{i=1}^{N-1}|\ell_i - \ell_{i+1}|$
• $\hat{\sigma} = \frac{1}{\sqrt{N}}\sqrt{\sum_{i=1}^N(\ell_i - \hat{\mu})^2}$
• $\hat{\sigma}^2 = \frac{1}{N}\sum_{i=1}^N(\ell_i - \hat{\mu})^2$

4. Experimental Results

We have conducted numerous detailed new experiments, please refer to the global comments for more information.

5. Details about the filtered samples

Your interpretation is accurate. This subset of samples will not be utilized for training during the current iteration. Nevertheless, in the subsequent iteration, we will reevaluate the selection of samples

6. The derivation of $\lambda=0$

We appreciate your inquiry and apologize for the confusion we may have caused. Actually, the statement is not derived from Equation 6. Here, we aim to provide an intuitive explanation of the update strategy. If the primary objective is satisfactory, attention should shift to the secondary objective. In this case, the lambda should be relatively small, potentially becoming zero as it is non-negative. The design of $\lambda_2$, as outlined in Equation 6, aligns with this strategy. When the primary objective is satisfactory, the value of \mu tends to be close to the minimum loss of the selected samples, resulting in a value close to zero.

We hope these planned revisions and clarifications will address your concerns. Please let us know if you have any further questions. We are inclined to address them to improve the quality of this work.

Table 2: Ablation experiments on four benchmark datasets.

Maintaining superiority consistently under random partitioning.

Given that our focus is on fairness without demographics accessible during the training stage, our model excels not only in the specific subgroups split by the attributes of "race" and "gender" but also in other possible divisions. Thus, we conducted additional experiments involving over 100 iterations, randomly splitting the data into K groups, ranking the metrics in terms of accuracy and prediction error, and calculating the average rank. For each iteration, we rank the results from the top to the bottom as 1, 2, 3, 4. It's essential to note that FairRF is not included because this method relies on selecting related attributes and is not suitable for this setting. The following results indicate that our method consistently performs well regardless of the choice of K.

Table 3: Average rank of randomly splitting into 4, 10, 20 groups.