Aequa: Fair Model Rewards in Collaborative Learning via Slimmable Networks

Experimental Designs

E1: We appreciate the reviewer's effort in scrutinizing the experimental design. However, we would like to clarify an important point: there is no alternative definition of collaborative fairness in our work or in the literature - the goal is consistent across methods. Our objective function in Sec. 4.2 aims to reduce the variance in $u(\mathbf{a})$, which promotes equitable benefit distribution and is a desirable property. Even without using CGS, Aequa achieves high Pearson correlation scores, reinforcing its fairness. Our formulation is principled and aligned with Tastan et al., 2025.

E2: Thank you for this comment. Our design targets a different setting than suggested. When using contribution assessment methods like CGSV - which uses the zeroing out gradients strategy - we already test the scenario the reviewer proposes. As shown in Figs. 3 & 4, Aequa (CGSV) ranks second-best in correlation and CGS, validating our approach.

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References

We will include a discussion of Lin et al., 2023 in the final manuscript. However, that work focuses on balancing performance and collaborative fairness, unlike ours, which aims to maximize fairness directly. Their results do not match the high fairness levels achieved by Aequa (please refer to the paper), but we acknowledge its relevance and will cite it in Sec. 2.

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Weaknesses

We appreciate the reviewer's feedback but would like to clarify a key aspect that may have been overlooked. Our method explicitly assumes the use of TEEs for local client training, preventing clients from accessing full model parameters and thereby preventing free-riding concerns.

We kindly request the reviewer to revisit the paper for further clarification and consider adjusting their assessment accordingly.

W1.3: The reviewer’s understanding is correct. Eq. 5 is used in training-time reward settings for simplicity.

W2.1: Our method can incorporate contribution-based strategies. Free riders can be assigned the minimum-width model (even set to zero). The utility definition (allocation minus contribution) can also be formulated as a ratio without affecting Lemmas 3 & 4 (it is straightforward to prove).

As for the case where a client's contributions equals $u$, it represents a quantity skew - a common FL challenge. Our experiments include such scenarios. Increasing the minimum width parameter adjusts allocations for lower-contributing clients.

Aequa aligns with Lyu et al., 2020's fairness principle: allocated reward proportional to contributions.

W2.2: While we don't follow prior procedures exactly, our minimum width parameter functions similarly to the existing tradeoff parameters the reviewer mentioned.

When comparing Aequa to other works, we used their best-performing parameters, as detailed in Appendix B. For CGSV, we used the best setting ($\beta=1.0$), as justified in the original paper. Moreover, Aequa's best-performing clients end up with a comparable or better model than the model obtained by simply running FedAvg, making an explicit tradeoff parameter unnecessary - a strength of our approach.

W2.3: We would like to highlight a fundamental misunderstanding in the reviewer's assessment.

First, predictive performance is unrelated to Eq. 3, as this equation is applied post-training and does not influence the model's learning process. Any suggestion otherwise misinterprets our approach.

Second, there is no "alternative" definition of collaborative fairness. The concept remains consistent across works; the only difference lies in how it is operationalized. Our objective aligns with Tastan et al., 2025, as previously stated.

Finally, we urge the reviewer to carefully examine also the Pearson correlation results, where Aequa consistently outperforms all baselines. Our method is both theoretically justified and empirically validated, reinforcing its effectiveness in achieving collaborative fairness.

W3: Due to page limitations, we included only the most essential components of our work in the main paper while placing proofs and extended experiments in the appendix - a common practice in the literature. However, we acknowledge the reviewer's suggestion and will incorporate additional details in the main text where feasible. We will use an extra page for camera-ready for this.

Comments or Suggestions

We appreciate the reviewer’s suggestions and will incorporate them.

Questions

Q1: Yes. Based on the reward mechanism in Section 4.2.

Q4: Yes, Theorem 2 applies to the post-training reward setting, where the contributions of each participant and the allocation vector are already known. The multiple iterations are used to determine $\mathbf{a}^{\star}$ that minimizes the objective defined in Eq. 3, i.e., we run an iterative optimization algorithm to determine optimal widths for each client without updating the model and only taking client contributions as an input.

We thank the reviewer for taking the time to review our paper, for highlighting its strengths, and for their valuable feedback.

W1. Continuous model performance assumption

We acknowledge that the assumption of continuous model performance across the interval $[\ell, u]$ may not always hold perfectly in practice. This assumption was primarily introduced for theoretical rigor. Nevertheless, as indicated and supported by other reviewers, our experimental results consistently achieve fairness scores close to the perfect score (mostly exceeding 0.95). Based on these comprehensive experiments, we confidently assert that the practical discreteness of model widths does not significantly impact performance or fairness, as our fairness scores consistently remain very high, often approaching or equaling 1.0.

W2. Missing references

Please note that reference [1] is already cited in our paper (lines 40, 59, 90). Regarding [2], while we acknowledge its relevance in the broader context of fairness in federated learning, its focus differs from the core objective of our work. [2] specifically targets performance fairness, aiming to personalize models for individual clients to ensure each achieves reasonable predictive performance. In contrast, our work addresses collaborative fairness, which is concerned with the fair distribution of model rewards in proportion to each participant's contribution.

Given this, [2] is not central to our framework. Nonetheless, to ensure completeness in the extended related work discussion, we will include it in Section 2 (Fairness in FL) of the final version of the paper.

W3. Scalability experiments

In response to the reviewer's suggestion, we conducted additional scalability experiments on the FEMNIST dataset to evaluate the effectiveness of our proposed method under challenging federated learning conditions. FEMNIST is a large-scale benchmark dataset characterized by its natural non-IID distribution and extensive client base, making it well-suited for evaluating scalability.

For this experiment, we employed a custom CNN architecture composed of two convolutional layers followed by a fully connected layer. The experiment involved a total of $**3597**$ clients with partial participation, randomly sampling 10 clients per communication round. We set the number of local epochs to 1, a batch size of 16, and executed training for 500 communication rounds. The training was performed using an SGD optimizer with an initial learning rate of 0.1, employing the same settings detailed in the main paper.

The results in the table demonstrate that our method consistently outperforms the FedAvg algorithm across all evaluation metrics. This indicates that Aequa effectively addresses the scalability and heterogeneity challenges inherent in large-scale federated learning scenarios.

We also include the model width vs. accuracy plot in this link: https://ibb.co/C3jpd5Nm

We thank the reviewer for taking the time to review our work, for highlighting its advantages and for their valuable positive feedback.

The writing quality of Section 5.2 should be improved.

We appreciate the reviewer highlighting the clarity issues in Section 5.2. We sincerely apologize for any confusion caused and have carefully revised this section to improve readability and coherence. Additionally, we have ensured that all important points previously omitted are clearly incorporated into the final manuscript.

How much the subnetwork will affect the optimization?

As we show in Lemmas 1 and 2, subnetworks preserve both smoothness and convexity of the objective. Thus, the effect on the optimization is minimal, and our new formulation admits standard analysis as demonstrated in our manuscript, which is the strong point. Therefore, the main effect that we need to investigate is the final quality of models given subnetwork formulation, which consistently maintains strong performance as validated by our experimental results.

Lemma 4

We apologize for the oversight in defining the terms used in Lemma 4. We clarify that $\mathbf{c}$ represents the vector containing individual client contributions, as elaborated in Section 4.2. Additionally, $\rho$ denotes the Pearson correlation (line 281). We appreciate the reviewer's comment and will revise Lemma 4 and the explanation to incorporate these clarifications in the final manuscript.

We sincerely thank the reviewer for their valuable positive feedback.

Reliance on TEEs for security

We acknowledge the reviewer's concern regarding reliance on TEEs. This limitation is already discussed in the paper, in Section 7. While TEE is a necessity for Aequa, Aequa can also operate in a setting without trusted hardware by incorporating contribution assessment methods (e.g., CGSV). In this case, model allocations are directly tied to assessed contributions, ensuring fairness without requiring hardware-based isolation.

Thus, while TEEs provide one implementation path and theoretically fair algorithm, Aequa remains effective and secure even in their absence.

Need to choose parameters (e.g. minimum model width) carefully.

We acknowledge the reviewer's concern regarding the careful selection of parameters. However, our proposed method notably differs from existing approaches regarding hyperparameter sensitivity. Unlike existing methods, which typically rely on one or two hyperparameters that must be carefully tuned to balance fairness and utility, our method utilizes only a single parameter - the minimum model width.

Importantly, we do not treat this parameter as a conventional hyperparameter requiring extensive tuning because it has a clear, interpretable meaning: it represents the minimum model width corresponding to the lowest performance model. Furthermore, setting this parameter to its minimal feasible value does not have an effect on both predictive performance and fairness metrics, thereby significantly reducing sensitivity and simplifying practical deployment.

Further exploration of real-world overhead and security challenges

Please take a look at the response to [Reviewer wyh2, W3] to see the results on the FEMNIST dataset that effectively mimics the real-world overhead and how our method performs on all evaluation metrics. We supplement the training time per communication round in seconds from the provided table.

From this table, FedAvg completes a round in 42 seconds, whereas Aequa takes 55 seconds - an increase of 1.309 times or 30.9% over FedAvg. This overhead remains modest, not significant, and is well justified by Aequa's superior performance and fairness. As per the communication overhead, there is no overhead as we communicate the same-sized models.

Regarding security challenges, the main consideration is the requirement of TEEs on client devices, as described in Section 1. However, the training-time extended version of Aequa, described in Section 4.3, does not depend on TEEs. In such a setting, broadcasted model widths from the server are based on contribution scores, mitigating security concerns related to model access.

Questions

Q1: In this work, we specifically focus on the reward allocation mechanism, operating under the assumption that the contribution assessments provided are reliable. Addressing improvements in contribution assessment methods falls beyond the current scope of our study.

Q2: Aequa's design inherently supports mitigating free-riding even in the absence of trusted hardware (TEE). By integrating robust contribution assessment methods (e.g., CGSV), Aequa effectively addresses the issue of free-riders. In this scenario, the model width allocated to each participant is determined by their assessed contribution. Consequently, clients with lower contributions receive narrower model widths and thus lack access to the full-width model. This mechanism naturally restricts the benefits available to potential free riders and removes the explicit need for TEE.

Q3: Our approach trains only two width configurations simultaneously in each forward pass. As demonstrated above, this approach does not introduce significant overhead. Specifically, we conducted an experiment on the FEMNIST dataset with a high number of clients (3597 clients), as detailed above and in the response to [Reviewer wyh2, W3]. The empirical results confirm that the overhead remains notably lower than $2\times$ (1.309) since our training leverages subnetworks. Thus, our method remains computationally efficient and scalable even with large models or substantial client counts.