Cauchy-Schwarz Fairness Regularizer

We thank Reviewer yoVn for the time and efforts in review our paper. To address the reviewer's questions, we provide detailed responses below.

[Equation 1 hold in expectation over all X.] Thank you for your suggestion. In our current paper, we introduced $X$ prior to presenting the fairness metrics.

[Fairness Coefficient.] Thank you for pointing this out. To clarify, the coefficient $\lambda$ is introduced to represent fairness in general frameworks and baseline methods, while $\alpha$ and $\beta$ are specifically used as fairness coefficients in our proposed framework.

[Equation 6.]

Thank you for pointing this out. You are correct that Equation 6 should still involve ${\bf x}_i^p$ and ${\bf x}_i^p$. The simplification is introduced only after Equation 6 for clarity. Specifically, as stated in the paper:

In the following, we represent ${\bf x}_i$ with distribution $p$ and ${\bf x}_j$ with distribution $q$ as ${\bf x}_i^p$ and ${\bf x}_j^q$, respectively, for simplicity.

We have addressed this issue in our paper. Thank you for bringing this to our attention.

[Undefined ${\mu}_p$ and ${\mu}_q$.]

Thank you for the question. To improve clarity and logical flow, we have moved the definitions of $\mu_p$ and $\mu_q$ from the proof of Remark 1 to the section where they are first introduced in Remark 1. The empirical mean embeddings are defined as: $\mu\_p = \frac{1}{N\_1} \sum\_{i=1}^{N\_1} f({\bf x}\_{i}^p), \ \text{and} \ \mu\_q = \frac{1}{N\_2} \sum_{j=1}^{N\_2} f({\bf x}\_j^q)$.

[Equation 9.] Thanks. Updated as $L$.

[Equation 12: Same $\theta$.] We have added the following explanation after Equation 12 in the updated paper to clarify the comparison: It is important to emphasize that the divergences are compared under the same model parameter, $\theta$.

[Definitions of the baselines should be in the main body, not in the appendix.]

Thank you for your suggestion. We will include an overall introduction to the baselines in the main body of the paper, and put detailed descriptions about the baselines in the appendix.

[Section 3 overall] Thank you for the suggestion. We have reorganized section 3. And we appreciate your recognition that 'CS divergence really is a worthwhile object of study'.

[Section 3: Why does the divergence from CS being smaller than KL provide any particular advantage?]

Thank you for your question. Both the CS divergence and the KL divergence measure the dependency between the predictions and the sensitive attribute. A smaller divergence value indicates that the predictions are more independent of the sensitive attribute. This reduction in dependency directly improves fairness metrics, such as achieving smaller values for both $\Delta_{DP}$ and $\Delta_{EO}$, leading to a fairer model overall.

[Section 3: Gaussian Data Assumption.]

Thank you for your question. The boundary relationship between CS divergence and KL divergence can be extended beyond the Gaussian assumption. Additionally, the proposed CS divergence does not rely on any specific distributional assumptions. In addition, its effectiveness has been validated through the experimental results presented in the paper.

[Section 3: The explanation behind the mechanism of the CS regularizer is unclear.]

Thank you for your feedback. The motivation for proposing the CS regularizer is to minimize the dependency between the predictions $\hat{Y}$ and the sensitive attribute $S$, thereby promoting fairness. The CS divergence is chosen because it provides a tighter bound on the dependency, so it offers a stronger fairness guarantee compared to other measures.

Dear Reviewer yoVn,

The authors have provided detailed responses to your review. Could you please indicate if the authors have addressed (some of) your concerns, and let us know if you will keep or modify your assessment on this submission?

Thank you very much.

Area Chair

[Question 3: Combining Multiple Regularizers.]

Thank you for the insightful question. To address this, we conducted additional experiments where we combined KL divergence and CS divergence as regularizers. The experiments were performed on the Adult dataset, with gender as the sensitive attribute.

Table: Fairness performance on the Adult dataset with gender as the sensitive attribute. CS+KL indicates equal weights for the CS and KL regularizers, while CS+0.5KL denotes weights of $1$ and $0.5$ for the CS and KL regularizers, respectively.

Actually, combining multiple fairness objectives has several drawbacks, which is why most existing studies avoid using multiple regularizers. Instead, they often choose to add simple constraint terms. The key drawbacks of combining fairness regularizers with the CS regularizer are summarized as follows:

• The CS divergence is upper-bounded by the KL divergence. Therefore, adding KL as an additional fairness objective is theoretically redundant and will not provide further benefits.
• Adding KL or other fairness metrics increases computational complexity, making the optimization process more challenging.

This experiment further shows the significance of our contribution: proposing a suitable, tighter-bounded fairness regularizer.

Reference:

[1] Fair prediction with disparate impact: A study of bias in recidivism prediction instruments.

[2] Fairness constraints: Mechanisms for fair classification.

[3] Inherent Trade-Offs in the Fair Determination of Risk Scores.

We sincerely appreciate your constructive feedback and hope our response adequately addresses your concerns. If so, we kindly request your reconsideration and support in reassessing our paper.

Thanks,

Authors

Thank you Reviewer 4GYJ for your time and efforts in reviewing our paper, and recognizing our paper has comprehensive experimental results. To address Reviewer 4GYJ's concerns and questions, we provided detailed point-by-point responses below.

[Weakness 1: Justification in Section 3.2 is about general properties of CS divergence rather than directly connecting these to fairness goals.]

We thank the reviewer for pointing this out. We have updated Section 3.2 to focus more specifically on the fairness problem. Our main contribution is to derive propositions under the fairness setting, so we fully agree with the suggestion to align the writing more closely with fairness objectives. Thank you again for your suggestion, which has helped us to improve our paper!

[Weakness 2: Limited Fairness Notions.]

We thank the reviewer for the constructive feedback. We agree that demonstrating the generalizability of our framework to other fairness notions is crucial. To address this, we provide additional results comparing our framework with baselines under the following fairness notions: Predictive Parity (PPV) [1], p%-Rule (PRULE) [2], Balance for Positive Class (BFP) [3], and Balance for Negative Class (BFN) [3]. The dataset is Adult, using gender as the sensitive attribute. All other experimental settings are consistent with Table 1 in the paper.

Table: Fairness performance comparison on the Adult dataset, with gender as the sensitive attribute under $\triangle_{PPV}$, PRULE, $\triangle_{BFP}$, and $\triangle_{BFN}$.

We observe the following:

• CS generally achieves the best fairness trade-off performance across the four tested fairness notions.
• On the Adult dataset, BFN is generally minimized more effectively than BFP.
• Since BFN is related to EO, the ranking of $\Delta_{BFN}$ aligns with $\Delta_{EO}$ in Table 1 of the paper. Note that, as stated in [3], there is an inherent trade-off between BFP and BFN in practice.

[Weakness 3: Section 3 could benefit from clearer organization.] We thank the reviewer for the construction feedback. We have reorganized Section 3 in our paper.

[Question 1: Extend to broader fairness notions.] We appreciate the reviewer’s constructive feedback. Please refer to our response under [Weakness 2].

[Question 2: Why does a higher fairness coefficient cause a higher disparity in Figure 5?]

Thank you for the interesting question. A higher fairness coefficient does not always lead to better fairness performance. This is because fairness metrics such as DP and EO are based on the model's predictions $\hat{Y}$, and poor prediction accuracy can negatively impact fairness.

When the accuracy falls below an acceptable range, the fairness measures may also degrade. For instance, in our observations, increasing the fairness coefficient $\alpha$ from $0.5$ to $10.0$ caused the accuracy to drop from $85.7\%$ to $81.2\%$, on the Adult (gender), as shown in Figure 5. This is relatively low in practice and leads to poor convergence of the fairness loss, preventing it from approaching a small value close to $0$. Therefore, achieving fairness requires not only balancing the fairness coefficient but also maintaining a reasonable accuracy.

Dear Reviewer 4GYJ,

The authors have provided detailed responses to your review. Could you please indicate if the authors have addressed (some of) your concerns, and let us know if you will keep or modify your assessment on this submission?

Also, I read the three weaknesses you mentioned. The second and the third weaknesses seem minor to me. It seems to me that they are more about the positioning or the organization of the paper. May I ask whether the negative evaluation is mainly based on the first weakness, especially, the missing steps in the proof Proposition 2？If so, could you please clarify where in the proof you would like the authors to provide more details?

Thank you very much.

Area Chair

We greatly appreciate the positive impression that our paper addresses the important topic, and is highly relevant to the conference's focus areas. In response, we provided detailed point-by-point responses below.

[Weakness 1: Demographic parity and equalized odds are conflicting fairness definitions.]

Thank you for bringing this up. We agree that achieving different fairness notions, such as DP and EO, can conflict in practice. Specifically, when one fairness notion is treated as the objective, the other may not always be achieved.

However, achieving both EO and DP is still possible under specific conditions, such as when the trained model assigns predictions uniformly across different sensitive groups. In this scenario, by removing the dependency on $S$, it is clear that: $P(\hat{Y}=1 \mid S=0) = P(\hat{Y}=1 \mid S=1)$, thus satisfying DP, and: $P(\hat{Y}=1 \mid Y=1, S=0) = P(\hat{Y}=1 \mid Y=1, S=1)$, thus satisfying EO.

The goal of the CS regularizer is not to prioritize one fairness notion, such as EO or DP, as the fairness objective in training. Instead, we aim to minimize the dependency between the prediction distribution and the sensitive attribute distribution, thereby addressing fairness holistically across multiple dimensions.

[Weakness 2: Lacks explanation of how Cauchy-Schwarz divergence contributes to achieving EO.]

As explained in our response to Weakness 1, minimizing the CS divergence ensures that both EO and DP are satisfied. However, the reverse does not necessarily hold.

[Weakness 3: Theoretical Contributions]

Thank you for the feedback. Although deriving the mathematical proof is relatively straightforward, the key contribution lies in utilizing the properties of the CS divergence to tackle the fairness problem. Our contributions can be summarized as follows:

• We are the first to leverage CS divergence as a fairness regularizer. Empirically, we demonstrate that this approach effectively trade-offs among EO, DP, as well as utility.

• Theoretically, we are the first to fit the CS divergence into a fairness problem setting. Under this setting, we derive boundary proofs that provide theoretical insights into the relationships between different fairness regularizers.

• Beyond theoretical analysis, we provide visual evidence to demonstrate why the CS regularizer performs well. Through distribution plots and scatter plots results, we show that the CS regularizer better minimizes the prediction distribution gap across different sensitive groups. This directly aligns with our motivation for proposing the CS regularizer: to reduce disparities in prediction distributions across sensitive groups.

We sincerely appreciate your constructive feedback and hope our response adequately addresses your concerns. If so, we kindly request your reconsideration and support in reassessing our paper.

Thanks,

Authors

Dear Reviewer tDPa,

The authors have provided detailed responses to your review. Could you please indicate if the authors have addressed (some of) your concerns, and let us know if you will keep or modify your assessment on this submission?

Thank you very much.

Area Chair

We thank Reviewer GFco for recognizing our "evaluation and comparison" are "comprehensive and convincing". To address the reviewer's questions, we provide detailed responses below.

[Weakness 1: Unclear how CS achieves EO.]

Thank you for this question. The fairness mechanisms of DP and EO regularizers differ from those of the CS divergence regularizer. Specifically:

• DP and EO regularizers directly target achieving DP and EO fairness notions, respectively.
• In contrast, the CS divergence regularizer aims to reduce the dependency of predictions on the sensitive attribute.

Minimizing the CS divergence between predictions and the sensitive attribute helps mitigate bias, which can indirectly reduce disparities in DP and EO metrics. However, minimizing DP or EO directly does not necessarily eliminate the dependency of predictions on the sensitive attribute.

[Weakness 2: Theoretical properties come from the nature of CS divergence rather than the fairness regularizer.]

Thank you for raising this question. Addressing fairness in machine learning is often approached as a problem of reducing the dependency between the prediction ($\hat{Y}$) and the sensitive attribute ($S$). Therefore, the theoretical properties of the CS divergence naturally motivate the design of our fairness regularizer.

[Weakness 3: No theoretical link to how CS divergence addresses EO.]

Thank you for the insightful question. When CS divergence is minimized, the prediction $\hat{Y}$ becomes independent of the sensitive attribute $S$. Therefore, both EO and DP are satisfied: 1) Independence of $\hat{Y}$ and $S$ implies $P(\hat{Y} = 1 \mid S = 0) = P(\hat{Y} = 1 \mid S = 1)$, which satisfies the DP, 2) Independence of $\hat{Y}$ and $S$ also implies $P(\hat{Y} = 1 \mid Y = 1, S = 0) = P(\hat{Y} = 1 \mid Y = 1, S = 1)$, which satisfies the EO.

We sincerely appreciate your constructive feedback and hope our response adequately addresses your concerns. If so, we kindly request your reconsideration and support in reassessing our paper.

Thanks,

Authors

Thank the authors for the response. Demographic Parity assumes independence between sensitive information and the decision outcome. I believe DP and CS divergence share the same objective but achieve it through different methodologies. While minimizing CS divergence undoubtedly results in a lower DP value, EO measures conditional independence, which is related to, but distinct from, both DP and CS divergence. This connection is unclear yet. It is well known that independence does not imply conditional independence. In conclusion, my original concern remains unaddressed, and my opinion is unchanged. Thanks.