Aequa: Fair Model Rewards in Collaborative Learning via Slimmable Networks
Authors: Nurbek Tastan
Samuel Horváth
Karthik Nandakumar
Venue: ICML 2025 poster (2025)
Keywords: federated learning
fairness
collaborative fairness
Primary area: optimization->large_scale_parallel_and_distributed

Abstract: Collaborative learning enables multiple participants to learn a single global model by exchanging focused updates instead of sharing data. One of the core challenges in collaborative learning is ensuring that participants are rewarded fairly for their contributions, which entails two key sub-problems: contribution assessment and reward allocation. This work focuses on fair reward allocation, where the participants are incentivized through model rewards - differentiated final models whose performance is commensurate with the contribution. In this work, we leverage the concept of slimmable neural networks to collaboratively learn a shared global model whose performance degrades gracefully with a reduction in model width. We also propose a post-training fair allocation algorithm that determines the model width for each participant based on their contributions. We theoretically study the convergence of our proposed approach and empirically validate it using extensive experiments on different datasets and architectures. We also extend our approach to enable training-time model reward allocation.

Aequa: Fair Model Rewards in Collaborative Learning via Slimmable Networks
Nurbek Tastan' Samuel Horváth
Karthik Nandakumar
Abstract
useful in domains where data sensitivity is paramount. A special case of CL is federated learning (FL), where central server orchestrates the collaboration. However, CL/FL faces many challenges such as communication inefficiencies due to frequent exchange of updates, system heterogeneity re- sulting from different participant capabilities, and statistical heterogeneity caused by non-i.i.d data distributions across participants (Zhu et al., 2021). These issues often degrade model performance, scalability, and practical adoption.
Collaborative learning enables multiple partici- pants to learn a single global model by exchang- ing focused updates instead of sharing data. One of the core challenges in collaborative learning is ensuring that participants are rewarded fairly for their contributions, which entails two key sub- problems: contribution assessment and reward allocation. This work focuses on fair reward al- location, where the participants are incentivized through model rewards- differentiated final mod- els whose performance is commensurate with the contribution. In this work, we leverage the con- cept of slimmable neural networks to collabo- ratively learn a shared global model whose per- formance degrades gracefully with reduction in model width. We also propose a post-training fair allocation algorithm that determines the model width for each participant based on their contributions. We theoretically study the con- vergence of our proposed approach and empiri- cally validate using extensive experiments on different datasets and architectures. We also ex- tend our approach to enable training-time model reward allocation. The code can be found at
A critical yet less explored challenge in CL lies in ensuring collaborative fairness among participants. Traditional CL frameworks assume uniform contributions from all parties, but real-world scenarios involve disparities in data quality, quantity, and computational resources. For instance, partici- pants with high-quality data may receive disproportionately less rewards despite their critical role in model generaliza- tion. The interplay between faimess and incentivization is essential sustaining long-term collaboration. Without equitable incentives, participants may withhold resources or disengage entirely, leading to the "free-rider problem" where some entities benefit without contributing meaning- fully. To address this issue, recent research has explored incentivization mechanisms such as Shapley value-based reward allocation (Xu al., 2021; Tastan al., 2024a), reputation systems (Xu & Lyu, 2020), and game-theorete frameworks (Wu al., 2024). These methods aim quan- tify and reward participant contributions transparently.
1. Introduction
Achieving collaborative fairness in CL fundamentally hinges on two key sub-problems: (1) contribution assess- ment and (2) reward allocation mechanism. Contribution assessment evaluates the marginal impact (contribution) of each participant's data or computational resources on the overall performance of the global model. Recent efforts (Xu et al., 2021; Jía et al., 2019; Shi et al., 2022; Jiang et al., 2023) attempt to quantify the quality of the local model updates as a proxy for measuring their actual contribution. Once the marginal contribution of each participant is de- termined, a reward allocation mechanism is necessary for incentivizing the participants to collaborate. Rewards can be in the form of financial compensation (monetary rewards) or differentiated final models (model rewards). To ensure fairness, rewards must be commensurate with the contribu- tion. This work focuses exclusively on fair distribution of model rewards in CL.
Collaborative learning (CL) has emerged as a transforma- tive paradigm for training machine learning models across data silos while preserving data privacy. Unlike centralized approaches, CL enables participants (e.g., hospitals, finan- cial institutions, etc.) to jointly train shared global model by exchanging only focused updates rather than raw data (McMahan et al., 2017). CL mitigates privacy risks and com- plies with regulations such as GDPR, making it particularly
'Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), Abu Dhabi, UAE *Michigan State University (MSU), Michigan, USA. Correspondence to: Nurbek Tastan <nurbek.tastan mbzuai.ac.ac>.
Proceedings of the nnnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
A critical question arises: How can participants be rewarded with different models whose performance (accuracy) faith- fully reflects their heterogeneous contributions? While pre- vious work has explored this problem (Xu et al., 2021; Wu et al., 2024), they typically rely on sharing partial updates with participants, which often lacks rigorous convergence guarantees. While (Wu et al., 2024) provide convergence analysis, their approach suffers from the following limita- tions: (1) The introduction of a stochastic recovery mecha- nism necessitates occasional broadcasting of the full global model to all participants, thereby aiding free-riding; (2) Since participants begin training from divergent starting points in each round (except the initial broadcast), it leads to potential instability in model convergence; (3) Finally, sampling a subset of updates from a pool of gradients does not inherently ensure that low-contribution participants re- ceive low-quality models; they may still receive high-quality gradients originating from high-contribution participants.
optimality guarantees. (4) We also empirically validate the efficacy of our framework through experiments bench- mark datasets, highlighting balanced model performance and fairess across diverse scenarios.
2. Related Work
Fairness in FL. Fairness in federated learning has been ex- tensively studied through two primary lenses: performance fairesss (Jiang et al., 2023), which emphasizes uniform model performance across all participants, often via per- sonalization strategies such as (Li et al., 2021) that learn a -pient-ppecffcc model regularized toward a shared global model, and collaborative fairness (Lyu et al., 2020), which advocates proportionality between client contributions and rewards. Our work focuses on collaborative faimess, where clients receive model rewards commensurate with their con- tributions. The foundational work of Lyu et al. (2020) op- erationalizes collaborative fairness by assigning only the allocated aggregated updates based on their reputations. Other studies consider fairness by quantifying the impact of clients on the global model the naive choice being the self- reported dataset sizes (self-reported information) (Donahue & Kleinberg, 2021; Zhang et al., 2020), and similarly, Kang et al. (2019) employ such self-reported information to build fair scheme based on contract theory. Various approaches also assess client importance through Shapley values (Shap- ley, 1953; Ghorbani & Zou, 2019), utility games (Gollapudi et al., 2017; Nishio et al., 2020) and empirical methods (Shyn et al., 2021). For a complete taxonomy of fairness in FL., we refer the reader check (Shi et al., 2024).
To circumvent these problems, we draw inspiration from the concept of slimmable networks (Yu et al., 2019; Yu & Huang, 2019b), which were originally proposed to dynami- cally adjust the width of neural network models for efficient inference with a graceful degradation in model performance. By extending this concept the collaborativeficderiaa ting, we obtain global model with a nested structure, where subnetworks of varying widths (e.g. 0.25x, 0.5x, 1.0x) ) are embedded. Participants are then assigned subnetworks cor- responding to their contribution levels higher contributors receive wider, perhhernnr subnetworks, while lower contributors obtain narrower ones. Our approach ensures that model rewards are proportional to client contributions, achieving both high performance and collaborative fairness simultancously. However, there is one significant obstacle that needs to be surmounted. Collaborative learning in the plaintext domain exposes the intermediate models to the participants, thereby re-introducing the free-rider problem. To overcome this limitation, we assume that each participant has access to a trusted execution environment (TEE), and local training happens confidentially within a TEE. While the use of TEEs has been considered in the FL literature (Huba et al., 2022; Eichner et al., 2024; Daly et al., 2024), they are most used on the server side for secure aggregation and minimization of privacy risks. This work uses TEEs on the client side to enhance collaborative fairness.
set-
Contribution assessment. A substantial body of work has addressed the problem of evaluating individual client con- tributions in federated learning. As mentioned earlier, an initial approach to collaborative fairness (Lyu et al., 2020) employed a global validation set, applying a function sinh to the validation accuracy of each client as a penalty mecha- nism to approximate their contribution reputation. Sub- sequently, Xu et al. (2021) removed the need for global validation set by approximating game-theoreti, Shapley val- ues with the cosine similarity of shared parameter updates thereby capturing each client's 's marginal contribution. A range of follow-up studies (Shi et al., 2022; Jiang et al., 2023; Lin et al., 2023; Tastan et al., 2024a) further expanded and refined these strategies for contribution assessment.
The main contributions of this work are follows: (1) We introduce CL framework called Aequa (Latin: fair) that leverages slimmable networks to dynamically adapt model performance to client contributions, serving as a reward mechanism that complements any contribution assessment method. (2) We propose a fair allocation mechanism for post-training distribution of model rewards and then extend this approach for training-time rewards. (3) We provide con- vergence analysis demonstrating that our framework retains
Reward mechanisms. Broadly, existing incentive mecha- nisms in FL fall into two categories: post-training monetary rewards and training time model rewards. The former em- ploys frameworks such as Stackleberg games (Zhan et al., 2020), auctions (Zhang et al., 2021; Cong et al., 2020), and contract theory (Liu et al., 2022a; Yang et al., 2024) to dis- tribute monetary compensation post hoc based on client contributions. The latter focuses on model-based rewards
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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
during training, incentivizing participation by dynamically adjusting access to the model's capabilities. For example, CGSV (Xu et al., 2021) allocates sparsified model outputs to clients proportionate to their contributions, while achieving fairness. Similarly, IAFL (Wu et al., 2024) shares aggre- gated gradients based on each client's ' contribution through probabilistic sampling, thus restricting highly performing models from contriicotti clients. Another related ap- proach is proposed by Lin et al. (202)), who aim to balance collaborative fairness with long-term equality by introduc- ing an explore-then-exploi mechanism that estimates client contributions over time and gradually allocates rewards to ensure asymptotic parity across clients. While CGSV re- lies on a heuristic approach and lacks formal convergence analysis, IAFL includes a convergence proof but exhibits its own limitations. Specifically, its stochastic recovery mech- anism shares the full model updates with all participants based on a certain probability: setting this probability to zero yields higher fairness at the expense of performance, and increasing it boosts performance at the cost of fairness yet still falls short of the performance achieved by FedAvg (MeMahan et al., 2017).
Algorithm 1 Aequa: Federated optimization
Input: minimum width Pmin. maximum width Pmaxx comm. rounds T, number of participants N, randomly initialized parameters z0, number of local iterations E 1: for each round & < 0, , do
2: Server broadcasts x' to each client i [N]
3: for each participant i E [M] do
for k + 0,,,..,, E do
Local iterations
5:
Sample width P(t.)) < U[Ipmin Proax] where U([a, bi] represents uniform distribution in [a, b].
6:
Update parameters of the model correspond- ing to pmax and P(t.k)
7:
end for Send the updated parameters aft,E) to the server 9: end for
8:
1 10: Server updates x² f- N Ein ((.E)
1I: end for
number of classes. The empirical loss function for each client is defined as F, E) = [((m(z), where L is the loss function and = (2, is training sample drawn from the local dataset of participant i. Additionally, the model parameters æ must be slimmable, i.e., it should be possible to obtain a subnetwork azp)) S x by dynamically adjusting the width parameter p. such that the model performance is proportional to the width (which in turn is set proportional to the participant contribution).
Slimmable networks. The seminal work by (Yu et al., 2019) introduced the idea of training a single neural network that can operate at multiple widths, enabling dynamic trade-offs between model size and performance. This innovation led to numerous follow-up studies and applications in feder- ated learning, predominantly focused on resource efficiency (Mei al., 2022: Horváth al., 2021), communication and computational efficiency (Wang et al., 2022), and neural architecture search (Yu & Huang, 2019a). To the best of our knowledge, we are the first to leverage slimmable networks in the context of fair federated learning.
4. Proposed Solution
In federated learning, ensuring fairness in model allocation is a fundamental challenge, as clients contribute to training with varying levels of data quality, quantity, and computa- tional resources. To address this, we propose an allocation mechanism based on slimmable networks, which ensures fair model rewards by adjusting the model width assigned to each client commensurate to their contributions. This sec- tion details our approach to federated optimization (Section 4.1), the allocation algorithm (Section 4.2) and the extension to training-time model rewards (Section 4.3).
3. Preliminaries
We consider a federated learning (FL) setup with N par- ticipants collaboratively learning the parameters E Rd of a machine learning model. Each client i possesses local dataset DD; and the overall objective is to minimize sum-structured FL optimization problem, given by
N F(x)
x= +- arg min #CRd
1 N
(((1)
4.1. Federated Optimization
Slimmable networks are a class of deep learning architec- tures that allow dynamic adjustments of model width, ensur- ing that different clients can operate with models of varying capacities without training separate networks. Initially in- trodueed for efficient model scaling (Rippel et al., 2014; Yu et al., 2019; Yu & Huang, 2019b; Horváth et al., 2021; Kusupati et al., 2022; Horváth et al., 2023), we repurpose slimmable networks for fairness in federated learning, ensur- ing that model allocation reflects each client's 's contribution to training. Each slimmable network can switch between
where the local loss components F. I&d + IR are dis- tributed among N participants and are expressed in a stochastic format as F.(x)) == Eς-D. [F)
Our specific task is supervised classification, where we define a classifier C2 parameterized by æ, mapping input samples to class labels C₂2 : 2 > Y, where Z S RD represents the input space, y = (1.2...., C) denotes the label space, D is the input dimensionality, and C is the
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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
different widths p |Pinn Pmaa] where Pmax represents the full model and Puii is the smallest subnetwork. In- termediate widths allow for smooth transition based on contribution levels.
CIFAR-10
CIFAR-100
90
65
Accuracy
88
60
Incorporating slimmable networks into FL requires adapting the optimization process to ensure that all subnetworks con- tribute effectively to the learning process. In our approach, clients train with varying model widths at each local itera- tion, ensuring that all width configurations are updated. The training process follows a random-width sampling strategy, where each client trains on different subnetworks in each it- eration (one forward-backwa
86
55
84
50
0.25
0.50
0.75
1.00
0.25
0.50
0.75
1.00
Model Width
Model Width
Figure 1: Performance vs. network width (p) using CIFAR- 10 and CIFAR- datasets on ResNet-18 model.
pass), promoting a balanced optimization process across all model sizes.
2. Low variability in utility: The variability of utilities {u,(a)} should be kept low so that no client's 's benefit is disproportionnatly high or low compared to others.
The federated optimization process begins with the server initializing the global model and setting the minimum and maximum possible widths. In each communication round, the server broadcasts the full model to all clients, who then train locally using a uniformly sampled width from the allowed range [pmin: Pm Clients update both the full model parameters and the parameters corresponding to their sampled width, following strategy similar to the sandwich rule in (Yu& & Huang, 2019b), but with improved efficiency. Upon completing local training, clients transmit their up- dated weights to the server, which then aggregates them to update the global model.
3. High average gain: The average gain across clients,
1
N (a)a), should be large as possible, balanc- ing overall performance with fairness.
Definition 1 (Individual Rationality (IR)). An allocation satisfies IR if (a(a)) 2 0 for all i. This ensures that no client is worse off than their standalone performance.
The above objectives can be achieved by maximizing the average gain E(u(a) while simultaneously minimizing the variance Var[u(a)) and respecting the IR constraint. For- mally, we define the optimization problem as minimizing the cost function:
4.2. Fair Allocation Algorithm
The allocation problem. We consider federated learn- ing setting with N clients, indexed by i = 1,2,... N. Each client has an associated contribution score, represented by the vector c = (c1, . cx), where e, denotes the contribution (e.g., standalone accuracy as a surrogate) of the client i. Without loss of generality, we assume that the clients sorted such that q S 02 S.... We have access to (sufficiently large) family of models whose performances { lie within a continuous inter- val [e, uI. where u 2 cN represents the highest achievable accuracy and S C1 + (u - cx) denotes the lowest possi- ble accuracy to satisfy fairness constraints. An allocation A {az} N assigns exactly one model with a performance level a; [e, u] to each client i. The gain (utility) of client under allocation a is defined as:
B[u(a)]
f(a) = Varlu(a)] +
(3)
where f A + R, and E > 0 is a small constant intro- duced prevent division by zero. To solve this constrained optimization problem, we employ a simulated annealing algorithm (Granville et al., 1994; Bouttier & Gavra, 2019) and adapt it to our specific setting.
CN-
4.3. Extension to Training-time Rewards
(ai...., aN)
Thus far, focus has been on post-training model re- wards. However, our approach can be seamlessly extended to incorporate training-time rewards as well. The primary modification involves sending a slimmed-down network (a subnetwork) to each client during training, rather than the full model, and dynamically updating their contributions based on the quality of their shared updates.
u (a) = - Ci.
(2)
An ideal allocation algorithm must satisfy the following three objectives:
A well-established method for the evaluation of contribu- tions is CGSV (Xu et al., 2021), and our approach can leverage CGSV to evaluate the contribution of each partici- pant. Additionally, other contribution assessment methods (CA), such as FedSV (Wang et al., 2020), GTG-Shapley (Liu et al., 2022b), ComFedSV (Fan al., 2022), FedFAIM (Shi
1. Individual Rationality (IR): The gain of every client must be nonnegative, satisfying individual rationality (Definition 1) and ensuring the participation of all ra- tional agents.

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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
al., 2022), ShapFed (Tastan et al., 2024a), among others, also be integrated within this framework.
Assumption 2 (Convexity). The local objective (x))), Vi [N] both convex and L-smooth, then for all ar, 2, TR.
Since some clients, typically low-pent www-performing clients, train only on subnetworks and do not share full gradients, they instead transmit their trained subnetwork updates. During the contribution assessment phase, we evaluate the gradients corresponding to the hinimum-widhe subnetwork selected by the algorithm (e.g. 0.25x as shown Figure 1), en- suring consistency since all clients train on this common subnetwork.
Σ VV(()) - VF(')) - (VF(() - y)
F((x) S (g) + (2), x - y) l = =11?
Vi [N].
Next, we present the lemmas that establish the L-smoothness and convexity properties of the slimmed models.
Furthermore, any incentive mechanism can be employed to dynamically adjust client contributions. In our approach, since the maximum width assigned to each participant is determined on the basis of estimated contributions, we di- rectly map the normalized contributions to network widths. This ensures that the highest contributor(s) receive the full model. The update rule of contributions is given by:
Lemma Let F RR + R be T-smooth function over Rd. Consider selection of dj S d coordinates from which we denote by I E R Define
Fx = F(E,r),
(6)
where r E TRd-dd, contains the fixed (zeros) coordinates of 2. Then Fy is L-smooth in the subspace Rd with a constant L I L.
70 + (1 - 73
(4)
where y momentum parameter and è; obtained using the CA method. The reward mechanism is then defined
Proof of Lemma See Appendix A.I
Lemma 2. Let F: Rd + be a convex function,, as per
[Ma(c)); v(cs/ max C&)
(5)
Assumption 2. Consider subset of coordinates ä Rd,
(with d1 S d) of the vector E IRd, and let E RRd-d, denote the remaining fixed coordinates. Define
where v is a utility function that directly maps the contribu- tions to network widths (p:).
Fx( F(ū,r).
We present experimental results for this approach in Fig- ures 3 and 4. Further implementation details and pseudo- algorithm outlining this extension are provided in Ap- pendix C.
Then Fy is convex in Rd:
Proof of Lemma 2. See Appendix A.2
5. Theoretical Analysis
With the established L-smoothness and convexity parame- ters of the slimmed models, we can now directly derive the convergence guarantee for Local SGD. For completeness, we present Theorem 1, which provides a performance guar- antee under the given assumptions (Stich, 2018; Wang et al., 2021; Khaled al., 2020; Woodworth et al., 2020).
We now present the convergence analysis of the main algo- rithm described in Section 1. along with the convergence and fairness analysis of the allocation algorithm in Sec- tion 4.2.
5.1. Convergence Analysis
Theorem 1 (Performance guarantee). Under the assump- tions 2, 3, 4, if the participant learning rate satisfies n S TL. then the performance is bounded by
Formally, we establish the following assumptions for the convergence analysis, which pertain to the properties of local objectives namely, L-smoothness (Assumption 1), convexity (Assumption 2), and bounded variance (Assump- tion 3, Appendix A.5.1). Additionally, we account for the data similarity across participants through bounded dissimi- larity (Assumption Appendix 1.5.1).
T-1
E
ΣΣ
F( -F(x')
TT
D2
no²
+
4 Lo² + 1872
2nTT
N
Assumption (L-smoothness)) The local objective F.(x), vi [N] is L-smooth, then for all x, y Rd,
where D = ||00,00 x=1|.
L
Proof of Theorem Taking into account Lemmas 1, 2, 5, and 6, combining them, and telescoping t over T communi- cation rounds gives us the main theorem.
(()) S Fgy) + (VE(x), y - x)
y1? Vi E [N].
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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
5.2. Fairness Analysis
algorithm converges asymptotically to the global minimizer of the cost function f with probability one.
We now analyze the fairness properties of the proposed allocation algorithm, highlighting key theoretical results and their practical implications.
Theorem 2 (Asymptotic convergence). Under Assumptions 5, 6. 7, 8, 9, consider the time-inhomogeneous Markov chain } on A with transition probabilities defined by
Lemma 3. Let a (a1, "o2....... aN) be an optimal alloca- tion that minimizes fa). Then, it holds that max 0, u. In other words, the client with the highest contribution always receives the model with the maximum accuracy u.
(_fla')
-f(aa) Tk
exp
[(d') > s(a),
Ps(a - a') d')= =
f(a') S f(a),
Proof of Lemma 3. See Appendix A.3.
where Tl is the temperature parameter at iteration k.
for any global minimizer a' E A satisfying I f(a") =
Lemma 3 ensures that the highest contributor is always re- warded with the most accurate model available, satisfying intuitive fairness criteria. Furthermore, under the assump- tion of continuous availability of model accuracies within the interval [e, ud. the optimal allocation aligns perfectly with client contributions, as shown by the following lemma:
minacA f(a), the following holds:
lim -(A==a") a1
In other words, the Markov chain {Ax} converges (with probability 1) to the set of global minimizers of f.
Lemma 4. Let a' be the optimal allocation minimizing f(a), and the vector of client contributions. Under the assumption of continuous model performance uvailability, the Pearson correlation coefficient between allocated accu- racies a* and client contributions c. is exactly one:
Proof of Theorem 2. See Appendix A.6.
6. Experiments and Results
6.1. Experimental details
p(a" .c) =
Datasets and partition settings. We use the following datasets carry out our experiments (following (Li et al., 2020)): MNIST (LeCun, 1998), Fashion-MNIST (FMNIST) (Xiao et al., 2017), SVHN (Netzer et al., 2011), CIFAR-10 & CIFAR-100 (Krizhevsky et al., 2009), Stanford Senti- ment Treebank (SST) (Socher et al., 2013), and the feder- ated handwriting dataset FEMNIST (Caldas et al., 2019). FEMNIST is already partitioned by writer identities, yield- ing a naturally heterogeneous distribution that we keep unchanged. The other datasets are partitioned using the following strategies: (i) homogeneous, where each partic- ipant gets an equal number of data points per class; (ii) heterogeneous, where each client gets a varying number of data points per class based on a Dirichlet(a) distribution (concentration parameter a reflects the degree of non-i.i.d. characteristics within the dataset); (iii) quantity skew al- locates л proportion of total data points to each of the m selected participants and the remaining N - participants split the remaining data equally; (iv) label skew, denoted by #C = m, creates a label imbalance by sampling m classes for each client and then randomly distributing samples from class m among selected participants.
Proof of Lemma 4. See Appendix A.4.
Here, p(a", e) denotes the Pearson correlation between vec- tors a* and c, measuring how well the allocated rewards match the client contributions. A correlation of exactly one indicates perfect aligmment, thus confirming that the proposed method achieves the theoretical ideal collabora- tive fairness (Lyu et al., 2020; Tastan et al., 2025) under continuous model performance conditions.
In practice, however, model widths and their associated per- formance values must be discretized due to inherent com- putational constraints, introducing slight deviations from perfect correlation. Despite this practical limitation, we consistently observe near- hear-perfect Pearson correlation coeffi- cients in empirical evaluations, shown in Table 2.
5.3. Convergence of the Allocation Algorithm
As described in Section 4.2, we define cost function
f A R,
Baseline approaches. We compare our approach to a Fe- dAvg algorithm (McMahan et al., 2017) followed by an additional epoch of local training to obtain differentiated model for each participant, CGSV (Xu et al., 2021) and IAFL (Wu et al., 2024). We also evaluate standalone accu- racy (SA), where each client trains its ML model on its local dataset without any collaboration with others. For a detailed
which we aim to minimize over the space of allocations A. Under suitable conditions, we can guarantee convergence of the allocation algorithm to globally optimal solution.
We now present Theorem 2, which guarantees that, under as- sumptions 5-9 (provided in Appendix A.6.1). 1), the allocation
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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
Table 1: Predictive performance (%, higher is better) of ABQUA and baselines under different dataset partitioning regimes.
PARTITION
DATASET
FEDAVG
CGSV
IAFL
AEQUA
MNIST
98.67 0.07
90.62 :: 2.32
98.40 + 0.13
98.60 + 0.11
FMNIST
89.45 0.33
77.01 + 2.51
88.54 + 0.25
89.63 + 0.19
SVHN
90.54 0.18
77.61 + 3.16
89.64 + 0.13
90.18 + 0.15
HOMOGENEOUS
CIFAR-10
89.99 - 0.23
61.29 - 2.92
88.42 + 0.07
90.84 + 0.26
CIFAR-100
63.92 -: 0.22
35.36 = 0.77
0.77
63.23 + 0.30
67.83 + 0.32
SST
34.44 + 1.33
30.12 + 1.03
34.02 + 0.51
34.44 + 1.19
MNIST
97.38 0.62
94.06 1.79
88.45 7.87
7.87
97.30 + 0.58
FMNIST
83.32 1.78
71.51 + 8.20
66.46 + 4.72
84.60 + 1.32
HETEROGENEOUS:
SVHN
86.38 0.87
72.48 - 4.81
68.71 7.85 7.85
86.33 + 1.00
DIRICHLET (a = 0.1)
CIFAR-10
74.73 3.65
48.77 = 5.02
46.36 + 9.31
75.97 + 3.36
CIFAR-100
61.16
34.16 + 1.63
43.38 + 4.53
63.42 + 0.54
SST
32.17 + 1.60
21.54 + 1.89
27.28 + 3.06
33.54 + 1.48
MNIST
98.69 0.10
93.22 + 0.99
98.40 0.12
98.62 + 0.09
FMNIST
89.53 0.24
78.73 2.14
88.53 + 0.26
89.72 + 0.17
QUANTITY SKEW: IMBALANCED (0.15,6) 6)
SVHN
90.59 - 0.17
77.54 2.34
89.51 + 0.13
90.26 + 0.16
CIFAR-10
90.00 - 0.13
66.89 = 1.85
89.51 + 0.15
90.71 + 0.14
CIFAR-100
65.88 + 0.38
39.62 + 0.74
64.60 + 0.25
68 68.24 24 + 0.11
SST
34.26 + 0.98
29.53 + 1.02
33.70 + 0.70
34.64 + 1.01
MNIST
94.37 + 3.43
79.19 + 7.94
73.10 + 15.00
95.37 + 1.15
FMNIST
79.73 3.80
61.54 8.03
60.10 8.03
80.51 + 3.27
SVHN
79.73 + 5.89
64.07 + 7.65
55.83 + 11.89
80.69 + 6.05
LABEL SKEW: (3,
CIFAR-10
71.88 3.28
48.02 = 3.88
44.12 + 21.15
72.40 + 3.17
C00
60.95 + 1.18
35.09 + 0.42
55.26 + 3.85
62.84 + 1.18
SST
33.96 + 0.35
24.88 = 2.04
30.33 + 1.68
33.01 + 0.90
NUMBER OF TIMES THAT PERFORMS THE BEST
7/24
0/24
0/21
17/24
explanation of the baseline approaches, evaluation metries, and implementation details, we refer to Appendix B.
actual usefulness by computing the Pearson correlation co- efficient (p) between the client model accuracies achieved after T' communication rounds and their standalone accu- racies. This metric also serves as a measure of faimess, as highlighted in (Xu et al., 2021; Lyu et al., 2020). Following IAFL (Wu al., 2024), we use standalone accuracies as a surrogate for client contributions. Thus, by analyzing the correlation values, we can directly compare the incentiviza- tion effectiveness of different algorithms. A high positive p indicates that clients who contribute stronger local models are rewarded with higher final accuracies, while values near zero or negative signal poor incentive alignment.
6.2. Predictive performance
We begin by benchmarking existing algorithms (FedAvg, CGSV, and IAFL) across the six datasets introduced in Section 6.1, using the partitioning strategies detailed in Table 1. Due to space constraints, we present a summarized version of the results, while full details can be found in Table 5, Appendix D.2. We report the accuracy (balanced accuracy) of the global model as an evaluation metric.
Aequa attains the highest mean accuracy in 17 of the 24 settings, while FedAvg leads in the remaining 7 settings. In contrast, the fairness-oriented baselines CGSV and IAFL lag markedly, especially under the more challenging hetero- geneous partitioning settings (Dirichlet (a = 0.1) and label skew). Their lower scores reflect an inherent performance- fairness trade-off. By adaptively allocating capacity, Aequa mitigates this trade-off and matches exceeds FedAvg across all splits, corroborating the established performance guarantees. We also provide per-participant performance analysis on the C00 dataset in Appendix D.5.
Table 2 summarizes the results. FedAvg- fails es- tablish a strong correlation, often yielding negative values, particularly under quantity-skewed splits, while CGSV and IAFL struggle whenever standalone accuracies show little variation (e.g.. the homogeneous setting). IAFL improves under quantity skew, yet Aequa delivers both tighter and uni- formly higher correlations, ranking first in 24/24 cases. For comprehensive analysis, we refer the reader to Figure 6, Appendix D.3, where we present results in 54 different sce- narios. In particular, our method outperforms all baselines in all cases, achieving perfect score of 54/54.
6.3. Incentivization performance
Collaboration gain spread. We evaluate the collabora- tion gain spread (CGS) achieved by Aequa in compari- son to other baseline methods. As reported in Table 7 in
Correlation to contribution. Following Wu et al. (2024), we quantify how well each algorithm aligns incentives with

---

Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
Table 2: Incentivization performance comparison of our method and baseline approaches across different dataset partitions, evaluated using the Pearson correlation coefficient between the accuracies of the final model and the accuracies of the standalone model. The results are averaged over five independent evaluations. For complete results, refer Appendix D.3.
PARTITION
DATASET
FEDAVG-FT
CGSV
IAFL
AEQUA
MNIST
0.07 + 0.24
-0.30 +: 0.16
0.16 + 0.20
0.97 + 0.02
FMNIST
0.21 0.07
-0.09
0
0.98 - 0.02
SVHN
-0.12
0.01
0.07
0.98 + 0.02
HOMOGENEOUS
CIFAR- 10
0.04 0.27
0.04
0.15
0.99 0.01
CIFAR-100
-0.07 0.37
0.36
0.02
0.96 + 0.01
SST
0.06 + 0.25
0.32
1003
0.98 + 0.01
MNIST
0.39 + 0.37
0.16
0.61 0.23
0.85 + 0.03
FMNIST
0.21 0.41
0.28
0.61 0.25
0.89 0.06
HETEROGENEOUS:
SVHN
0.66 0.24
0.46
0.80 4
0.92 +
DIRICHLET (a = 0.1)
CIFAR-I0
-0.8 0.35
0.17
0.15
0.94 0.02
CIFAR-100
0.17 0.46
0.52
0.89 +
0.99 + 0.01
SST
0,20 0.35
0.54
0.91 0.09
0.98 + 0.01
MNIST
-.63 0.18
0.80
0.95 + 0.04
0.98 + 0.01
FMNIST
-0.45
0.79
0
0.98 0.01
QUANTITY SKEW: IMBALANCED (0.15, 6)
SVHN
0.76
0.775
0.99
1.00 0.00
CIFAR-I0
0.37
0.02
0.99
1.00 0.00
CIFAR 100
1.06:0.44
0.97
1.00
1.00 = 0.00
SST
0.07
0.53
0.90 0.02
0.94 + 0.04
MNIST
0.41
0.27
0.23 + 0.24
0.81 0.13
FMNIST
0.27
0.41
0.08 A08
0.99 + 0.01
SVHN
0.43
0.42
0.01
0.98 + 0.00
LABEL SKEW: #OC= (3,-=(3,5))) 30}
CIFAR-10
0.19 0.32
0.32
0.22
0.97 0.02
CIFAR-108
R-00
0.38 + 0.22
0.00 0.24
0.31 0.22
0.98 + 0.02
SST
0.45
+ 0.38
0.48
0.48 + 0.37
0.97 + 0.03
NUMBER TIMES PERFORMS THE BEST
0/24
0/24
1/24
24/24
Table Comparison to baseline methods when the contri- bution measure corresponds the participation rate.
CIFAR-10 (ResNet-18)
MNIST (CNN)
85
p 0.99 + 0.00
1.090
0.98 # 0.01
1.0
0.8
0.8
METHOD
MNIST
CIFAR 10
Accuracy
80
0.6
0.8
Participa tianioon
ACC.
ACC.
0.4
FEDAvG-FT
0.12 0.1
97.65 + 0.3
0.11+00.1
83.73 + 0.3
CGSV
0.50
96.18
- 0.7
0.1
55.97 + 0.3
0.2
IAFL
0.79 0.0
96.01 = 0.2
0.69 0.2
82.58
: 0.1
70
10
20
50
0.0
20
30
0.0
AEQUA
0.9
98.19
0.1
0.99 1: 0.0
84.59 1 0.2
Client
Figure 2: Visualization of the correlation when contribution measure corresponds to the participation rate CIFAR-10 and MNIST datasets with N = 50 participants.
Appendix D.4, Aequa consistently outperforms other ap- proaches in 42 out of 54 scenarios, demonstrating consis- tently superior collaboration dynamics.
under partial participation, rate as contribution measure. Following the experimental setup studied in (Wu et al., 2024), we define the client participation rate as r; = 0.5 x (1 + i/N), introducing a mild but controlled variation across clients, as illustrated in Figure 2.
the client participation
One scenario which Aequa underperforms relative to IAFL is in the quantity skew setting. This is due the choice of Pwin. which determines the minimum accuracy level l assigned to low-contributing clients. To address this, we conducted an additional set of experiments, adjusting Pmii to 0.1, presented in Table 8. Under this configuration, Acqua outperforms other methods in 11/12 cases, bringing the overall total 48/5 /54 cases.
We conducted experiments on MNIST and CIFAR-10 us- ing distinct architectures to assess Aequa's 's robustness. As shown in Figure 2, our method yields very high Pearson correlation coefficient, demonstrating a strong alignment between participation rates and the resulting model perfor- mance. Table 3 presents a comparative analysis with other baseline algorithms, which confirms that Aequa consistently
6.4. Alternative contribution measures
Aequa is designed to flexibly incorporate a wide range of contribution measures. To demonstrate its effectiveness
8

---

Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
Homogeneuus partition
Table 4: Predictive and incentivization performance of Ae- qua in comparison to FedAvg-FT on FEMNIST dataset.
Accuracy
Correlation
CGS
90-
1.00
0.8
0.75
Accurttty
80
Powrrin Corneet
0.50
0.6-
ALGORITHM
ACCURACY
PEARSON (p)
MCGECGS
8 0.4
8-
0.5
FEDAVG-FT
71.48
0.2904
47.02 + 5.07
0.00
AEQUA
73.10
0.9888
52.19 2.57
80
0.25
Resleg
cubV
IAFL
Acqual
Negua Acqu (Shap)
Redligg
cosv
IATU
Asqua
Aeque (5am))
redluger
cosv
Aesua (CoSV)
Augue (happ)
by +1.6 pp over the FedAvg-FT baseline but also yields markedly better fairness scores (Pearson correlation coef- ficient of 0.9888(t) and CGS of 2.57(4)), confirming its ability handle both data heterogeneity and the scale of thousands of clients.
Aeque
Figure 3: Performance comparison of our proposed methods
on CIFAR-10 under a homogeneous partitioning strategy.
CIFAR10 Quantity Skow (0.15, 6)
FEMNIST
Accuracy
Correlation
85-
0.75
12.5
70
Accuns
80-
0.50
10.0
Accuracy
0.5
15
Poarson
5.0-
65
0.00
Foolog
IAFL
Abaus
Shoo
Fosting FT
OS5V
Aequa
Aequa ghaal
0.0
O55V
Auqua Icasv Aeque bhead
60
0.25
0.50
0.75
1.00
Aequs
Model Width
Figure Performance comparison of our proposed methods on CIFAR-10 under a quantity skew.
Figure 5: NIST dataset using custom CNN architecture composed of two convolutional layers followed by fully connected layer.
Performance vs. network width (p) using FEM-
outperforms competitors in both correlation and final per- formance. In particular, IAFL the second-best method in this setting, yet it trails Aequa by a clear margin.
Figure further plots accuracy versus model width on FEMNIST, illustrating that Acqua's advantage consis- tent across a wide range of capacities and that the resulting accuracy difference is sufficiently large to discourage free- riding behavior. These findings suggest that the proposed mechanism scales to large, real-world FL deployments.
6.5. Training-time model rewards
We evaluate our method in combination with existing con- tribution assessment algorithms for training-time model re- wards, comparing against baseline methods. The results, presented in Figures 3 and 4, illustrate the performance on the CIFAR-10 dataset under homogeneous and quantity skew partitioning strategies. The plot presents all evalua- tion metrics, and the results clearly demonstrate that Aequa consistently outperforms other approaches across all cases. Among the variants, Aequa with CGSV ranks as the second- best in both settings, while Aequa with ShapFed shows comparable performance, further validating the robustness of our approach.
Computational overhead. On the same FEMNIST setup, we recorded wall-clock training time per communication round on single GPU. FedAvg finishes a round 42+2.9 s, whereas Aequa takes 55 + 3.1 s only a 1.309x (s= 30.9%) increase that is modest, not significant and well justified by Aequa's 's superior performance and fairness. Communication cost is unchanged because both methods transmit identically sized model parameters.
7. Conclusion
6.6. Scalability to large-scale FL
Using the concept of slimmable networks, we have pre- sented Aequa, framework for achieving a provably fair model rewards in collaborative learning. While we initially present TEEs as a means to ensure the confidentiality of local updates, TEEs are not a strict requirement. Aequa also operates securely in the absence trusted hardware by incorporating contribution assessment methods.
To probe the limits of Aequa under realistic federated condi- tions, we performed an experiment on the FEMNIST dataset (Caldas et al., 2019) a challenging benchmark comprising 597 naturally non-lID users and more than 800 thousand handwritten character images.
Table shows that Aequa not only improves test accuracy

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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
Acknowledgements
P., Cheu, A., al. Confidential federated computations. arXiv preprint arXiv:2404. 10764, 2024.
This material is partly based on work supported by the Office of Naval Research N00014-24-1-2168.
Fan, Z.,, Fang, H., Zhou, Z.. Pei, J., Friedlander, M. P., Liu, C., and Zhang, Improving fairness for data valuation in horizontal federated learning. In 2022 IEEE 38th In- temational Conference on Data Engineering (ICDE), pp. 2440 2453. IEEE, 2022.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
Ghorbani, A. and Zou, J. Data shapley: Equitable valuation of data for machine learning. In Chaud- huri, K. and Salakhutdinov, R. (eds.), Proceedings of the 36th International Conference on Machine Learn- ing, volume 97 of Proceedings of Machine Learning Research, pp. 2242-2251. PMLR, 09-15 Jun 2019, URL https::/ ghorbani19c.
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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
A. Mathematical proofs
A.1. Proof of Lemma
Proof of Lemma 1. Since F' L-smooth over IR for any æ, x' E gd we have
VF(x)) VF (1) - x - x'Il.
(8)
Let ". W' E Ie be two points in the reduced space of selected coordinates. We embed each into Rd by defining
a r) x' (
99)
where r is held fixed in both cases (e.g. zeros in our case). Note that
x - x'1l (ž, r) - (3' r
2'||.
(10)
Next, write the gradient of F in coordinates:
VF(x) (V=F(e)) VF(a') (V='(())) v,F(#)))
(11)
Because æ and x' differ only in the T--cordinatess (the r-part is the same), it follows that
(T) VF, V+)())> F(= VF
(12)
By the L-smoothness of F in Id and putting all together, we have
()) VF, = - V < (VF(x) VF(x')I S L| - x'1 = L ø'|l
(13)
Hence F.. (the restriction of F the subset Ø)) is also L-smooth with respect to ズ. In particular, its smoothness constant satisfies L L.
This concludes the proof.
A.2. Proof of Lemma 2
Proof of Lemma 2. Since F' is convex over Rd, for any two points y E IRd and any À E [0,
F(Ax + (1 ))y) - AF(x) (1-
(14)
Now fix Rdd-d.
Let H. W e Rdd and [0, 1]. Then,
FA + (1 (1-1)))) =(( -Q- (1-
(15)
Since r is fixed, the interpolation in Rd satisfies
X( r) (1 (1- (Az + (1 r).
(16)
By convexity of F' in Rd
F -
(17)
Rewriting in terms of Fr.
FA + ((-
(18)
Hence F, is convex in R4, This completes the proof.
13

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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
A.3. Proof of Lemma
Proof. Suppose, for contradiction, that in the optimal allocation a", we have aty < Let ai Because the set of achievable accuracies is assumed to be continuous [f, ul,, attempt shift all the assigned accuracies in a² upward by Define the new allocation (ā1, ................................ ........))))... by a a + +al A.
Then, this gives us
S(a" + +a1) Elu(a") +a B[u(a*)) Varlu[a*)] +
[(ā)
Var(u(a*)) +
f(a"),
contradicting the optimality of a". Thus, we must have and therefore ajy u.
A.4. Proof of Lemma 4
Proof. Since model performances are continuous on the interval [e, u] and that ax we define a = u cN > 0. By construction, for every client i, the allocation satisfies o; = G + a. Consequently, the Pearson correlation coefficient computed as:
c)a = plc al, = p(c,c) = =.
(19)
This proves the lemma.
A.5. Deferred Lemmas
A.5.1. ASSUMPTIONS
Assumption 3 (Expected stochastic gradient variance), The variance of an unbiased stochastic gradient in participant is 2 uniformly bounded in L2 norm, vi [N], Ak E [T]. Vt E [T].
E [=((a) [oft ( |alao ( || a S 0².
Assumption 4 (Gradient dissimilarity). The difference of local gradient V (.x)) and the global gradient FF(x)) is (-uniformly bounded in L2 norm, E [N], vk E [7) [T].
maxsup m
(F( 4.
VF SC.
A.5.2. LEMMAS
Lemma S. Assuming the participant learning rate satisfies S
I7, then
1
E
F(
F(x²))
Ft.0
<
1
x²
E
||| [200
201T
1=1
N
n1²
N
Nr
k==
where Ft. the a a-field representing all the historical information up to the start of the t-th round.
14

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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
Proof of Lemma 5.
(1(())
1 x²) (-
x²) | a² 11)
(20)
7
1
1 xx* 1|
s T
(21)
21
where (20) uses the update rule gf T4t N 2im1 g:(2) (. (21) uses the parallelogram law, which is (u, v)
[1|u||2 + ||0|| - l|u - v1l²) Vu, v Rd.
By assumptions and 2, we have:
(( S ) + (v B4
x.
+
f
(22)
T.
2
S F((x') + (2 k.1)
x²"
+
2
33,
(23)
=F((=*)+ + (VF.(2z 1 x²))
L
(wrth+1
kx
(10
2
(24)
2
F(((*) + (VF(( b. -x*)
+
L
x
k+1
Tlik
|2
2
L
1.b
T
(25)
where (22) uses the L-smoothness property (see Assumption 1), (23) uses the three points descent lemma, which holds true
when (x) both convex and L-smooth (see Assumption 2), (24) includes the addition and subtraction of xf to the third term, and (25) applies this inequality ||u + e/|2 2[[|u]|2 2 || Yu, v E F²
By combining (21) and (25), get:
1
F(
-F((")
B( - F(xx)) -F(ct))
(26)
N i=1
((l(o! - (:( )
x² ) + L glic+1 Ztink
N
Σ
Ilih
+ gx(2
1,kk x
+ 1+1 x²
(27)
N
1=1
V F3 c. - gi (zo
xh,++
x )
+ L
x +1++ - Il. |
N 1=1
||
1
z x² | - x k+ x |
x1u41 x 11)
x
k
+
(28)
2n
where (27) adds and subtracts
Σm
),
x²
(28) replaces the last term with (21).
N
15

---

Aequa: Fair Model Rewards in Collaborative Learning via Slimmable Networks
Since E VF((2,)" (xx 1.
[tc]] Ft.k)
O we have
N
E
(F((zt))) gi(at glitt x
Ft,k
(29)
N
E
1
(F((t ) 9o (a x - x
Ft.
(30)
N
1 NN s nE
Σg VF((o,
ou(af(+))
FFlh
1
E
|
 2 Fzu
(31)
4nj
m1022
E2
x
(32)
47)
1
where (31) uses Young's inequality, which is (u, u) S 2
||0 Ve > 0 and Vu, R. and (32) uses bounded
4e
covariance assumption (see Assumption 3) and independence across clients.
By plugging (32) back to the conditional expectation of (28) with S II we get:
2
2
E
[F(
- F(xx)] )]
+
E
x²
Ft,k
T
x²
2n
110"
1 L
E
Fr,h
+ I.
Σ
1.k
(33)
N
4n
N
N
L
Σ
(34)
N
where (34) holds true since 7) S 1 4E-
Telescoping from 0 to gives us:
F ( F(x*)
Ft.
S
1
x
F
20/T
11²
N
E
t.k
N
Nr
2m
k=0
which completes the proof.
Lemma 6. Assuming the client learning rate satisfies 17 S 41. then
E
Fr.0 1187312 4
where Ft.0 is the o-field representing all the historical information up to the start of the t-th round.
Proof of Lemma 6.
1
1,++1
1,++1
||
|
ptck
= R
| xn 1) (6) (Br - 92 (2( (aa)t))(( Fri
(35)
t.k
1|²
2n
v (F - (o2 stth xr (.k
2
(a - (2 dod /|
+ 222202
(36)
16

---

Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
where the last term in (36) is from Assumption for both 91 () and 9o ( Following Assumption 4, the second term of (36) is bounded as
(VR( (22(222))...
.. k.k
22
(VF(2) VF(a2"
1. L.k)
222
+ 01 - t.ti (1)
V [2] 11(k) (F(af*))
VF (x
(37)
(V VF(a5" )
+ 2
af 22
(38)
$ - ( vF(ag )
||
2
| x2
+ 2
(39)
S |F(a)*) -
VF(a2
1
1.k
022
1.4
21y7
(40)
where (39) uses smoothness and convexity properties as per Assumption 2, (40) uses AM-GM inequality (weighted sum -z
u² ev²
version), which is we < +
with
22 1.k 2 26. and c = T7T.
2e
2
Similarly, the third term (36) is bounded as
VF[ - v - (2 ( - |))
(41)
4c + VR (
+ +4< vn (ar
(42)
( - || + +66
3
(43)
u²
cv2
where (41) uses Assumption 4. (43) is obtained using AM-GM inequality on the last term of (42), which is we S with u Vii(aa - v (a2 >
+
2e 2
1
= 4C, and e 4"
Putting all these results together gives
t.++ 1
x22
1,+1
Jtik S 1
| t.k - 2i + 22 +
+ +6)2(2+21166 22)20²
(44)
|
+ 10 + 20
(45)
where (44) drops
VF
(2f
term, since the resulting term is always negative given that 1) S 4
1
Telescoping gives
(1 +)* -1 (1071732+2277)
E
k. x
x2
2
F1.
1
(46)
18 2 0
(47)
where the multiplier (46) is obtained from ΣØ=d (1 2 and its numerator is upper bounded by scalar value of 1.8.
Then, by convexity, have
2
1 Ft.0 118721252 + 4471002 (eE
x
(48)
which completes the proof.
17

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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
A.6. Convergence of the Allocation Algorithm
A.6.1. ASSUMPTIONS
Assumption (Finite state space), The state/allocation space A is finite: |A| M 20.
Assumption 6 (Bounded cost function). The objective function f A RR is bounded: 3 0 such that If(a)) ç for all E A.
Assumption (Irreducibility). For each fixed 7 >> 0, the induced Markov chain (with acceptance probabilities depending on T' is irreducible on A. That is, for any a, a' E A, we can reach a' from a with a positive probability in a finite number
of steps.
Assumption (Aperiodicity). For each fixed T 0, the chain is aperiodic: there is no integer d > 1 such that transitions occur only in multipliers of d. Equivalently, for each a E A, ged(m) PT (a a) > 0) =1.
Assumption (Annealing schedule). The temperature Ti satisfies:
1. Tk + 0 as k DC.
2. Σ exp AS/Tk) = oo, where Aʃ minaaa (ʃ(a) - s(a')l > is the smallest nonzero gap of J.
A classic example is T log(k + ko)
with Ko > 1.
A.6.2. PROOF OF THEOREM 2
Proof. Let }A&} be our Markov chain on A. Denote the transition probability at iteration k by Pa(a a'). We show that, almost surely, the chain eventually remains (or keeps returning) to global minimizer.
First, fix a temperature >> O. By Assumptions and 8, the homogeneous Markov chain with transitions
exp (-f(a)) [(a') - -(aa)
f(a') > f(a).
Pr(a Pr(a-a'' + aa) =
T
(49)
1,
f(a') S f(a).
is irreducible and aperiodic on the finite state space A. Therefore, it has unique stationary distribution тт.
Standard Metropolis-Hasting arguments show that
f(a) T
(a) x exp
(50)
As T + 0, (x)(T((a) is maximized by allocations/states a E A that minimize f. fact, a* a global minimizer (with/ f(aa) = fmin), then for any a with f(a > fmin.
क (a")
exp (
(51)
Tr a
T-
Hence, T + 0, all stationary mass concentrates the set of global minima.
However, in our algorithm, T is not fixed but varies with iteration k. Thus (Aş} is time-inhomogeneous Markov chain whose transition matrix Ps depends on Tj. The chain does not, in general, admit a single stationary distribution.
} decreases slowly enough, the chain nearly equilibrates around each temperature. This ensures we do not remain trapped in a suboptimal local minimum.
Consider suboptimal state a where f(a) > fmin. Because A is finite, there is finite path from a some global minimizer a" along which m-) is well-defined Let A(a a²) be the "energy barrier" above faa, along that path i.e., the minimal extra cost one must pay to move from eventually down to a". Formally,
A(a + a') min max [f()) min(f(a),f(aa)) 7:0- ZE?
(52)
Because f(at) < f(a), we have A(a + a²) > 0. single uphill step cost has acceptance probability of exp (-8/Tk).
18

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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
Why need Σx expl-(A/)/TA) oo: Let A max be the maximum barrier needed to reach any global minimizer from any suboptimal state:
Amax
max
A(a - + a").
(53)
)=/min a:f(a>> Smin) )
Then any upward move < max is accepted with probability at least exp (- (-Amax/Th) If
A Amx
exp
= 00;
(54)
Tk
we get infinitely many chances (with positive probability) to surmount each barrier. By the Borel-Cantelli lemma, almost surely, the chain eventually does surmount every finite barrier and thus can move from any suboptimal state to a strictly better region. Repeatedly, the chain escapes local minima with probability 1.
Once the chain hits a global minimum a" at sufficiently small Tg. any transition to higher-cost state is exponentially unlikely (with probability exp (- (-If(a') - f(a*)) As T. + 0, these moves become negligible, causing the chain to remain in (or return quickly to) a global minimizer. Consequently,
lim P(Ax E {a: :((a) = fmin = 1, R-+00
(55)
proving convergence to a global minimizer with probability 1.
This completes the proof.
B. Implementation Details
Evaluation metries. We evaluate our approach and the baseline methods in terms of both predictive performance and fairnes/innentivizationn For predictive performance, we use balanced accuracy. Fairness, on the other hand, can assessed using several metrics, including the incentivized participation rate (IPR) proposed by (Cho et al., 2022), the Pearson correlation coefficient and the collaboration gain spread (CGS) proposed by (Tastan al., 2025). However, we exclude IPR allocation algorithm inherently guarantees perfect IPR score by design. Instead, we primarily benchmark our results using Pearson correlation and CGS.
from our analysis,
Implementation details. We use cross-entropy loss for all image and language classification tasks and maintain consistent training hyperparameters across all experiments. The optimizer of choice is SGD with momentum, with default initial learning rate of 0.01. A learning rate scheduler is applied, reducing the learning rate by factor of 0.1 at rounds 50 and 75, when the total number communication rounds is set to 100. The total number communications is set as follows:
CIFAR-10, CIFAR- 100, and SST: = 100,
MNIST, FMNIST, and SVHN: T 50.
In each round, clients perform one local epoch of training. The batch size fixed at 128 across all experiments. Additionally, we specify one parameter: the minimum width of the slimmable network, which set to Pmin = 0.25 unless stated otherwise. The maximum width always kept at Pmax 1.0. All experiments were carried out on NVIDIA A A100-SXM4440GB GPUs, with each run utilizing single GPU.
Architecture details. To ensure the reproducibility of our experiments, used easily implementable model architectures. Specifically, we employed four different architectures:
1. Convolutional neural networks (CNN) - a lightweight CNN with one convolutional layer followed by two fully connected layers, totaling 0.0606M parameters. This model used for the MNIST and FMNIST datasets.
19

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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
Algorithm 2 Aequa (with training-time model rewards)
Input: minimum width Pmin+ maximum width Pmax. number of communication rounds T, number of participants N, randomly initialized weights æº, number of local iterations E, momentum factor 7 = 0.5, CA contribution assessment algorithm (e.g. CGSV, ShapFed, FedFAIM, etc.)
1: for each round 4 0, ,,...,,T''.. To
b Communication rounds
2: if = then
3:
Pmax f- Pmax:: Vi E [N]
4: Server broadcasts the full-sized model (Pmax-model) each client i [N]
5: else
Server broadcasts (f.t) submoddl to each client E [A]]
6:
7: end if
8: for each participant E [N] do
9:
for k: 4 0, do
b. Local iterations
10:
Sample width P(i,k) + (pminn Pinax) uniformly
(i.t)
11:
Update the weights of the model corresponding to pinax and P().)) widths
12:
Send the updated weights of the mmax--submode
(i.t)
13:
end for
end for
15:
Server computes the contribution of each client using gradients according to Pmin submodel via the CA algorithm
16:
Server updates the contribution of each client using Eq. 4, +- -c--- (1 7)2 if I >> 0 else of t-
17:
Server updates the participant widths using the reward mechanism in Eq. 5
D Any reward mechanism
18:
Server updates #++1 using the masked averaging (Mohtashami et al., 2022; Tastan et al., 2024b)
19: end for
2. Enhanced CNN model A complex architecture with two convolutional layers followed by two fully connected layers, comprising 2.0566M parameters. This model is used for the SVHN dataset.
3. ResNet18 (He et al., 2016) deeper model with 10.7527M parameters, used for the CIFAR- 10 and CIFAR-100 datasets.
4. Long short-term memory network (LSTM) this model includes embedding layer of dimension 300, an LSTM layer, and three fully connected layers, comprising 6. 6.146IM parameters. It is used for the SST dataset.
For slimmable model implementation, we apply width slimming to the following layers: convolutional layers, fully connected (FC) layers, batch normalization (BatchNorm), and LSTM layers. Slimming convolutional, fully connected, and LSTM layers is straightforward; however, BatchNorm layers require special treatment due to the inconsistencies between training and testing. To address this, we implement switchable batch normalization, which maintains multiple sets batch normalization statistics corresponding to different model widths. For further details, refer (Yu et al., 2019).
Baseline approaches. We describe the baseline methods used for comparison.
FedAvg-FT. The standard FedAvg algorithm (McMahan et al., 2017) returns the same model to all clients each FL iteration, making the computation of the Pearson correlation coefficient undefined. To address this, following IAFL (Wu et al., 2024), we introduce a fine-tuning step at the end of collaboration, where each client trains the global model locally for an additional round without sharing the updates with the server. This approach enables each client to receive a personalized model, allowing for valid estimation of the Pearson correlation coefficient.
CGSV. In cosine-gratee graniett Shapley-value (CGSV) (Xu al., 2021), the server estimates each participant's contribution using gradient alignment, computed as the cosine similarity between individual gradient and the aggregated gradient. During the broadcasting phase, updates are sparsified based on these estimated contributions before being sent back to clients.
IAFL. For incentive-aware federated learning (IAFL), we adopt the same setting described in (Wu et al., 2024) under the most fair scenario, which corresponds to setting the hyperparameters A = 0 (sharing parameter) and (stochastic recovery probability). Additionally, we use standalone accuracies as the contribution measure, which consistently used in (Wu et al., 2024).
20

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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
ShapFed. We incorporate Shapley-driven federated learning (ShapFed) (Tastan et al., 2024a) and CGSV contribution assessment algorithms within our method. ShapFed estimates contributions using last-layer gradients instead of full-model parameters, making it efficient and accurate approach in certain scenarios. To better align with our objective, we modify the ShapFed algorithm to use the last m layers instead of only the classification layer, as our method does not require class-specific contribution values. We set m = 10 in CIFAR-10 experiment with ResNet-18 architecture.
C. Extension to Training-time Model Rewards
In this section, we provide Algorithm which extends our proposed Aequa framework incorporate training-time model rewards. While primary allocation mechanism focuses post-training model distribution, this extension enables dynamic model adaptation during training, where clients receive real-time adjustments to their assigned model widths based on their contributions. However, our approach remains flexible and can seamlessly integrate with the primary allocation mechanism. We showcase this extension to demonstrate that our method is compatible with various reward allocation algorithms, emphasizing its adaptability.
Algorithm outlines the federated training process with contribution-awwre model scaling. The algorithm initializes all clients with the full-sized model. As training progresses, client contributions are continuously assessed using contribution assessment (CA) algorithm, such CGSV (Xu et al., 2021), ShapFed (Tastan et al., 2024a), FedFAIM (Shi et al., 2022). These contributions are then updated iteratively using momentum-basee update rule, ensuring a fair and stable estimation over miltiple communication rounds.
Based on the updated contributions, the server dynamically adjusts the model width allocated to each client, applying a reward mechanism that incentivizes global model benefit from larger subnetworks, while maintaining fairness in model distribution. The global model update is performed using masked averaging (Mohtashami et al., 2022; Tastan al., 2024b), robust aggregation technique that ensures stability across varying model widths. The full details of Algorithm 2 are presented below.
participants. This ensures that clients who contribute more the
D. A Additional Experiments
D.I. Performance vs model width
Remaining results of Figure 1. this section, we present the remaining plots corresponding to Figure from the main paper, which analyzed CIFAR-10 and CIFAR-100 on the ResNet- 18 architecture. Here, we extend the analysis by including Figure 6, which illustrates the same performance vs. model width relationship for four additional datasets: MNIST, FMNIST, SVHN, and SST and under a homogeneous partitioning strategy.
As described in Appendix B, these datasets are trained using different model architectures: MNIST and FMNIST utilize a lightweight CNN, SVHN is trained on a more complex CNN, and SST is trained using an LSTM network. Despite the differences in model complexity, the width-accuracy relationship remains monotonic, demonstrating that even small-sized network architectures exhibit clear relation between model width and predictive performance.
MNIST
FMNIST
SVHN
SST
90
90
34
Accuracy
98
88
89
33
88
32
97
86
87
31
86
0.25
0.50
0.75
1.00
0.25
0.50
0.75
1.00
0.25
0.50
0.75
1.00
0.25
0.50
0.75
1.00
Model Width
Model Width
Model Width
Model Width
Figure 6: Performance vs. network width (p E [Pmin = 0.25, Pina = 1.0]) on MNIST, FMNIST, SVHN, and SST datasets using CNN and LSTM models. This figure extends the results presented Figure 1.
21

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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
An important observation is that the model complexity increases, the range between minimum and maximum accuracy widens, making the trade-off between model width and accuracy more pronounced.
D.2. Predictive performance
Remaining results of Table 1. For completeness, we present the full results of our approach alongside the baseline methods across all partitioning strategies. Table previously reported results for subset of partitions, while the complete results are provided in Table 5.
From the overall results presented in Table 5, it is evident that our method outperforms all other approaches, including FedAvg, and significantly surpasses fairness-hased methods. Specifically, across 9 partitioning strategies and 6 datasets per partition, our approach achieves the best performance in 36 cases. In the remaining 18 cases, it ranks as the second-best, performing on par with the FedAvg algorithm.
D.3. Pearson correlation
Remaining results of Table 2. In Table 6, we present the complete results of the experiments on Pearson correlation, extending the findings of Table 2, which reported results for only a subset of partitions. The results demonstrate that our algorithm outperforms all other methods in all 54 cases.
As discussed in the fairness analysis (Section 5.2), our method consistently achieves near-perfeed correlation coefficients, typically in the range of 0.98 to 0.99. While the theoretical analysis guarantees a perfect correlation, in practice, the continuity assumption slightly violated due to the need for discretizing the accuracy range [e, ud.
Additionally, in experiments using the ResNet-18 architecture, we introduce discretization by modifying the uniform sampling strategy. Instead of continuous sampling, we define a set of model widths sampled from a bucket of predefined values, starting from 0.25, 0.3, 0.35, and increasing in increments of 0.05, up 1.0.
D.4. MCG and CGS results
In this section, we provide the detailed results for mean collaboration gain (MCG) and collaboration gain spread (CGS), expanding upon the summary presented in Section 6.3. Table 7 presents a comprehensive comparison of Aequa against baseline methods across all experimental settings. The results confirm that Aequa consistently achieves superior performance, outperforming other methods in 42 out of 54 cases.
For cases where Aequa does not achieve the lowest CGS, we highlight the MCG values in green, demonstrating that Aequa still significantly outperforms other methods MCG (by ~ x3). This suggests that even when another method achieves a comparable CGS, does so at the expense of lower mean collaboration gain, indicating a weaker overall incentivization effect. Such cases occur in extreme heterogeneous settings.
As explained earlier, another scenario where Aequa underperforms compared to IAFL is in the quantity skew setting. This performance gap is primarily due to the selected value of Pmin. which determines the minimum accuracy e &assigned to low-contributing clients. To examine the impact of this parameter, we conduct an additional set of experiments where pmiin is adjusted to 0.1, specifically for the quantity skew partition. The results, presented in Table 8, show that under this revised configuration, Aequa outperforms other baselines methods in 11 out of 12 cases, bringing the overall success rate of 48 out of 54 cases.
The partitioning strategies where Aequa consistently outperforms all methods include homogeneous, Dirichlet (a (100,2.0, 5.0)), quantity skew (0.15, 6), and label skew (C 3). In contrast, CGSV performs well in extremely het- erogeneous settings, achieving positive MCG values, but it tends to yield negative values in other settings, limiting its generalizability. FedAvg-FT, on the other hand, consistently achieves positive MCG values but exhibits higher CGS values, indicating a lack of fairness.
D.5. Per- Per-participant performance
In this section, we present per- perpparticipat performance analysis of Aequa and other baseline methods on the CIFAR-100 dataset under the following partitioning strategies: Dirichlet (a = 0.5). label skew (0.15, 6) and quantity skew (0.4, 2). From the results, we observe that IAFL exhibits high variability in the first two partitioning strategies, and in some random
22

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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
Standalone
FedAvg-F1 FT
CGSV
IAFL
Aequa
60
45
30
6
1
10
10
10
6
8
9 Quantity skew (0.4,
Dirichlet (a 0.5)
Label skew (#C-30)
Figure 7: Per-participant performance comparison of Aequa and other baseline methods CIFAR- 100 dataset under Dirichlet (a = 0.5), label skew (0.15, 6) and quantity skew (0.4, 2) partitions.
seeds, IAFL fails to achieve a positive collaboration gain. On the other hand, CGSV remains very close to standalone accuracy, significantly limiting the overall collaboration gain of the participants. Meanwhile, Acqua achieves performance comparable to FedAvg-FT, while precisely capturing the correlation between final model accuracies and client contributions, reinforcing its fairness properties.
In the quantity skew partition, observe that Clients and 2 are the highest contributors, each holds 40% of the dataset. The goal of fair allocation algorithm is to match this distribution while expanding the total collaboration gain. Both IAFL and Aequa produce similar-shaped performance plots to the standalone accuracy, consistently ensuring positive collaboration gains. However, Aequa provides even better performance to Clients and 2 compared to FedAvg-FT, while also maximizing the mean collaboration gain (MCG) across all participants, demonstrating its effectiveness fair model allocation.
D.6. Model width as reward
In this section, we explore the transferability of Aequa when the server lacks prior knowledge of each sub-model's performance (does not possess validation set). In this scenario, participants are rewarded directly based on the model width (p), rather than using explicit performance-based allocations (a).
To implement this approach, we use standalone accuracies the contribution measure, and the allocation problem becomes mapping model widths to participant contributions. We employ the same allocation algorithm described in Section 4.2, but modify the utility measure to be computed using normalized contributions, i.e. c max Ck. ensuring that both vectors share the same range of values before executing the allocation process.
Table presents the results of this experiment, covering three datasets across all partitioning strategies. The results demonstrate that the correlation coefficient between participant contributions (input) and assigned model widths (output) perfectly aligned, consistently achieving a correlation coefficient of 1.0.
Additionally, report the correlation coefficient between the contribution measure and the assigned sub-model's corre- sponding accuracy. The findings indicate that even when the server lacks knowledge of each sub-model's 's accuracy, high fairnsss performance is still achievable. Compared to the results presented in the main paper, this method still outperforms all baseline approaches, demonstrating that Aequa maintains strong fairness properties.
23

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Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
Table The predictive performance our method and other baselines using different dataset partitions. The results are averaged over five independent evaluations.
PARTITION
DATASET
FEDAVG
CGSV
IAFL
AEQUA
MNIST
98.67 + 0.07
90.62
2.32
98.40 0.13
98.60 = 0.11 0.11
FMNIST
89.45 0.33
77.01 -
88.54 0.25
89.63 = 0.19
SVHN
90.54 0.18 0.18
77.61 3.166
89.61
90.18 + 0.15
HOMOGENEOUS
CIFAR-10
89.99 + 0.23
61.29 +.99
8.04
90.84 + 0.26
CIFAR-100
65.92 + 0.22
15.36
+
67.83 + 0.32
SST
34.44 1.33
30.12 1.03
34.02 0.51
34.44 = 1.19
MNIST
97.38 + 0.62
94.06
88.45
97.30 + 0.58
FMNIST
83.32 + 1.78
71.51 8.20
66.46 4.72
84.60 + 1.32
HETEROGENEOUS: DIRICHLET (ar 0.1)
SVHN
86.38 1 0.87
72.48 4.811
68.71 7.85
86.33 1 1.00
CIFAR-10
74.73 1 3.65
48.77
46.36 9.31
75.97 3.36
CIFAR-100
61.16 1 0.25
34.16
1.53
63.42 0.54
SST
32.17 1.60
21.54
27.28
33.54 = 1.48
MNIST
98.45 + 0.13
93.18
97.69 0.44
98.29 0.16
FMNIST
87.86 + 0.42
79.24
85.42 = 1.67
88.20 = 0.49
HETEROGENEOUS:
SVHN
89.33 + 0.25
80.20 1.82
87.08 = 0.79
89.14 + 0.28
DIRICHLET (a = 0.5)
CIFAR-10
87.74 0.23
81.99 2.46 2.46
88.75 = 0.39
CIFAR-100
64.54 + 0.30
32.19
62.37 1.35
66.39 + 0.41
SST
33.52 + 0.80
2.08
29.64 2.15
34.21 + 1.08
MNIST
98.52 1 0.14
92.71 + 1.53
98.17 0.15
98.48 + 0.13
FMNIST
88.72 + 0.36
78.76
87.67 0.22
89.09 + 0.23
HETEROGENEOUS:
SVHN
89.96 0.30
79.53 2.50
88.51 + 0.26
89.75 + 0.22
DIRICHLET (a = 1.0)
CIFAR-10
88.76 0.26
65.88 5.02
84.02 + 1.12 1.12
89.67 + 0.28
CIFAR-100
65.06 0.29
33.92 1 1.32
64.37 0.68
67.06 0.44
SST
34.00 1 0.87
25.24 1 2.55
31.26 1 1.22
34.15 1 1.07
MNIST
98.58 - 0.15
92.56 1.54
98.22 = 0.13
98.50 = 0.09
FMNIST
89.12 + 0.36
4.
88.27 + 0.28
89.46 + 0.18
HETEROGENEOUS:
SVHN
90.19 + 0.10
78.16
88.91 + 0.14
89.94 + 0.13
DIRICHLET (a 2.0)
CIFAR-10
89.35 + 0.28
68.60 3.36
88.09 + 0.33
90.30 + 0.18
CIFAR- R-00
65.56 + 0.22
1.66
64.71 0.70
67.51 = 0.29
SST
33.45 + 1.32
27.53 0.62
33.25 = 0.27
34.58 + 0.49
MNIST
98.64 + 0.12
91.73 + 2.95 2.95
98.31 + 0.13
98.54 + 0.11
FMNIST
89.35 1 0.31
77.03 1
88.44 = 0.24
89.60 + 0.15
HETEROGENEOUS: DIRICHLET (a 5.0)
SVHN
90.48 1 0.16
80.52 1.94
89.32 1
90.08 +
CIFAR-10
89.56 1 0.29
68.09
88.47 + 0.50
90.35 + 0.14
CIFAR-100
65.79 + 0.29
65.61 + 0.39
67.77 + 0.16
SST
34.26 + 1.01
28.75
33.41 - 1.17
34.47 = 0.55
MNIST
98.69 1 0.10
93.22 0.99
98.40 1 0.12
98.62 1 0.09
FMNIST
89.53 + 0.24
78.73
88.53 = 0.26
89.72 = 0.17
QUANTITY SKEW:
SVHN
90.59 0.17
89.51 = 0.13 2.34
90.26 + 0.16
IMBALANCED (0.15,
CIFAR 10
90.00 + 0.13
66.89
89.51 = 0.15
90.71 71 + 0.14
CIFAR-100
65.88 + 0.38
0.74
64.60 + 0.25
68. 24 + 0.11
SST
34.26 + 0.98
29.53
33.70 0.70
34.64 + 1.01
MNIST
98.69 + 0.09
92.73
98.42 0.16
98.63 + 0.12
FMNIST
89.58 + 0.29
76.14
88.51 + 0.25
89.78 + 0.19
QUANTITY SKEW:
SVHN
90.69 + 0.15
79.11 1.33
89.49 + 0.07
90.35 + 0.17
IMBALANCED (0.4,2)
CIFAR-10
89.78 + 0.14
69.13
88.50 - 0.13
90.86 + 0.15
CIFAR-100
66.30 1 0.23
38.35 1. 1.01
63.67 1 0.10
68.80 1 0.16
SST
35.45 1 1.11
26.10 1110-00 0.65 0.65
34.65 1 0.56
35.29 1 0.76
MNIST
94.37 1 3.43
79.19 7.94 7.94
73.10 1: 15.00
95.37 1 1.15
FMNIST
79.73 1: 3.80
61.54 8.03
60.10 +: 8.03
80.51 + 3.27
SVHN
79.73 + 5.89
64.07 7.65
55.83 -: 11.89
80.69
6.05 6.05
LABEL SKEW: #OC=(3, 30}
CIFAR-I0
71.88 3.28
48.02 3.88
4.12121115
72.40 = 3.17
CIFAR-
--00
60.95 + 1.18
35.09 0.42
55.26 2.88
62. + 1.18
SST
33.90 96.96666
+ 0.35
24.88 + 2.04
30.33 = 1.68
33.01 + 0.90
NUMBER OF TIMES THAT PERFORMS THE BEST
18/54
0/54
0/34
36/54
24

---

Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
Table 6: The incentivization performance of our method and other baselines under different dataset partitions, measured using the Pearson correlation coefficient between the final model accuracies and standalone accuracies. The results are averaged over five independent evaluations.
PARTITION
DATASET
FEDAVG- FT
CGSV
IAFL
ABQUA
MNIST
0.07 0.24
-0.30 0.16
0.16 + 0.20
0.97 + 0.02
FMNIST
0.21 0.07
0.35
0.27 0.24
0.98 + 0.02
SVHN
-0.12 0.36
0.11
0.07 0.22 0.28
0.98 0.02
HOMOGENEOUS
CIFAR-10
0.04
0.04
-0.01 0.15
1.991
CIFAR-I00
-0.07 0.37
0.39
0.02 0.31
SST
0.06 - 0.25
0.32
-0.03 0.26
0.98 + 0.01
MNIST
0.39 0.37
0.188
1: 0.23
0.85 0.03
FMNIST
0.21 = 0.41
0.28
0.61 = 0.25
0.89 0.06
HETEROGENEOUS: DIRICHLET (a 0.1)
SVHN
0.66 = 0.24
0.41
0.80 = 0.18
+
CI
-0.18 0.35
0.17
0.84
0.94
CIFAR-100
0.17 0.47
0.52
0.89 0.07
0.99 0.01
SST
0.20 0.35
0.50
0.91 + 0.09
0.98 +
MNIST
0.32 0.40
0.22
0.67 0.16
0.87 0.03
FMNIST
0.67 + 0.19
0.48
0.10 0.10
0.88 + 0.05
HETEROGENEOUS:
SVHN
0.80 + 0.09
-0.18
0.59 0.41
0.90 + 0.04
DIRICHLET (a 0.5)
CIFAR-10
-0.66 0.21
0.38
0.93 + 0.03
+
CIFAR-100
0.19 0.29
0.14
0.58 = 0.23
0.99
SST
0.15 = 0.07
0.30
=
0.94
MNIST
0.55 0.20
0.13
0.48 + 0.40
0.91 + 0.03
FMNIST
0.42 0.31
0.27 0.30
1 0.69 1 0.08
0.95 1
HETEROGENEOUS:
SVHN
0.52 1 0.28
1.18
0.94 0.03
DIRICHLET (a = 1.0)
CIFAR-10
0.05 1 0.15
0.40
0.89 0.04
0.94 0.02
LIFAR-100
-0.08 = 0.35
0.53
0.65 = 0.37
0.99
SST
0.09 = 0.29
0.36
0.71 = 0.12
4.4
MNIST
0.49 + 0.32
0.07 0.57
0.37 0.41
0.96 0.03
0.03
FMNIST
0.50 0.24
0.52
0.51 0.27
0.96
HETEROGENEOUS:
SVHN
0.67 = 0.10
0.06
0.72 0.16
0.95
DIRICHLET (a = 2.0)
CIFAR-10
0.17 0.28
0.58 0.21
0.83 = 0.15
0.97
CIFAR-100
0.26 + 0.25
0.46
0.65 0.26
0.99
SST
0.19 + 0.36
0.08
0.59 + 0.28
0.95 +
+ 0.04
MNIST
0.26 + 0.32
0.40
0.39 0.46
0.96 0.03
0.03
FMNIST
0.26 + 0.32
0.22
0.12 0.29
0.97 0.02
HETEROGENEOUS:
SVHN
0.09 + 0.18
0.34
0.62 0.05
0.98 0.01 0.01
DIRICHLET (a = 5.0)
CIFAR-10
0.01 + 0.16
0.54
0.80 = 0.08
0.98 0.02
CIFAR-100
0.11 1: 0.25
0.24
0.18 1 0.50
1.99-0.00
SST
0.01 0.21
1211:0.33
0.37 1 0.36
0.01
1 0.01
MNIST
0.63 +: 0.18
0.31
0.95 + 0.04
0.98 + 0.01
FMNIST
0.45:00222
0.49
0.93 0.02
0.98 + 0.01
QUANTITY SKEW: IMBALANCED (0.15,6) 6)
SVHN
0.76 0.12
0.75
0.99 - 0.01
1.00 1.00 0.00
CIFAR-10
0.37 0.16
0.002
0.99 = 0.00
1
CIFAR- 100
0.06 + 0.40
0.03
1.00 0.00
1.00 +.000
SST
0.07 0.48
0.23 0.54
0.90 = 0.02
0.94
MNIST
00.61 0.13
0.33 0.76
0.76 + 0.04
0.96
FMNIST
00.21 +: 0.26
0.51
0.90 0.04
0.99
QUANTITY SKEW: IMBALANCED (0.4,2) 2)
SVHN
-0.54 0.20
0.79
0.93 0.01
0.98
CIFAR-10
-0.41 = 0.35
0.96
0.92 0.04
1.00
CIFAR- 100
0.05 - 0.25
0.77
0.98
1.00
SST
0.05 - 0.61
-0.97
0.90=000
0.97
MNIST
0.03 = 0.41
-0.27 0.27
0.23 = 0.24
0.81 0.13
FMNIST
10.440.27
0.11
0.08 0.26
0.99
SVHN
0.43
-0.43 0.44
0.01 0.25
0.00
LABEL SKEW: #3C=(3,
CIFAR-10
0.19 0.32
0.12
0.22 0.38
0.97
CIFAR-100 100
0.38 + 0.22
0.00 0.24
0.31 + 0.22
0.98 + 0.02
SST
0.45 + 0.38
-.49
0.48 + 0.37
0.97 + 0.03
NUMBER OF TIMES THAT PERFORMS THE BEST
0/54
0/54
1/54
54/54
25

---

Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
Table 7: The average collaboration gain (MCG) (1) and the collaboration gain spread (CGS) (4) results of our method and other baselines under different dataset partitions. The results are averaged over five independent evaluations.
FEDAVG-FT
CGSV
IAFL
AEQUA
PARTITION
DATASET
MCG
CGS
MCG
CGS
MCG
CGS
MCG
CGS
MNIST
1.58
0.29
-6.63
0.18
1.64
0.18
1.64
0.04
FMNIST
3.05
0.77
7..0
0.68
3.13
0.68
3.43
0.17
SVHN
5.08
0.81
5.26
0.39
6.57
0.39
6.68
0.08
HOMOGENEOUS
CIFAR-10
13.62
0.65
15.08
0.48
13.62
0.59
13.31
CIFAR-
31.54
0.64
2.02
0.43
31.54
0.56
31.14
0.14
SST
8.77
1.48
4.74
0.81
8.96
0.81
8.11
0.22
MNIST
29.27
14.60
43.72
12.46
16.41
11.77
47.63
12.50
FMNIST
15.56
14.75
31.43
9.80
8,48
10.21
40.89
9.07
HETEROGENEOUS: DIRICHLET (a 0.1)
SVHN
11.97
12.15
36.77
11.81
9.47
9.63
49.85
9.92
CIFAR-10
25.51
16.13
7.63
7.00
3.77
6.87
39.91
4.73
CIFAR-106
41.13
3.02
8.09
1.91
10.79
4.99
40.65
0.39
SST
9.19
3.80
-0.89
2.24
1.26
1.01
8.65
0.88
MNIST
9.48
7.37
7.27
6.00
8.93
4.59
11.56
5.23
FMNIST
6.81
4.60
6.24
5.92
5.86
6.05
14.34
4.41
HETEROGENEOUS: DIRICHLET (a 0.5)
SVHN
12.90
5.85
14.37
7.75
12.85
6.11
21.79
6.33
CIFAR
28.88
9.52
6.34
6.56
13.97
3.26
28.25
5.73
CIFAR-100
37.35
1.74
4.99
1.23
27.71
7.99
37.00
0.23
SST
9.81
2.45
0.15
2.53
3.52
1.48
9.49
1.15
MNIST
4.69
2.84
0.38
2.75
4.55
2.53
5.19
2.10
FMNIST
4.20
2.55
1.17
2.74
4.35
2.39
7.06
1.22
HETEROGENTOUS:
SVHN
8.10
7.11
4.08
4.53
9.72
3.38
13.36
3.13
DIRICHLET (a 1.0)
CIFAR-10
22.69
6.54
2.38
4.75
14.94
3.50
20.39
3.50
CIFAR-100
35.46
2.00
3.37
1.22
2.94
34.60
0.22
SST
9.48
1.97
0.77
2.12
4.74
1.73
8.50
0.88
MNIST
2.64
1.10
3.39
1.13
2.54
1.04
2.59
0.52
FMNIST
2.54
2.13
3.53
1.58
3.84
1.33
4.63
0.52
HETEROGENEOUS:
SVHN
6.97
1.94
1.63
2.48
7.36
1.72
8.79
1.15
DIRICHLET (a 2.0)
CIFAR-10
17.99
4.31
7.89
3.06
14.08
2.30
15.70
1.47
CIFAR-I00
33.68
1.92
1.12
1.24
29.58
5.11
32.89
0.22
SST
8.32
2.14
2.87
1.55
6.02
1.49
8.53
0.76
MNIST
1.84
0.40
4.23
0.37
1.78
0.32
1.65
0.09
FMNIST
3.05
2.58
6.74
1.62
3.74
1.64
4.45
0.67
HETEROGENEOUS: DIRICHLET (a 5.0)
SVHN
5.26
4.35
-1.12
1.16
6.78
0.93
7.06
0.21
CIFAR-10
15.38
3.03
11.04
2.34
9.29
7.61
13.04
0.78
CIFAR-100
32.47
1.04
0.58
0.66
31.92
1.33
32.01
0.17
SST
8.73
1.17
3.66
1.02
7.36
1.20
8.28
0.27
MNIST
2.83
2.50
2.23
2.09
2.33
1.73
2.43
1.56
FMNIST
3.96
2.85
4.98
2.17
3.59
1.42
3.72
0.49
QUANTITY SKEW:
SVHN
7.91
5.00
33.66
4.47
6.25
1.73
7.98
2.84
IMBALANCED (0.15,6)
CIFAR-10
19.88
14.02
5.85
8.96
13.67
7.01
17.62
10.31
CIFAR-100
34.13
13.63
2.87
7.93
21.23
1.26
29.56
5.07
SST
9.16
2.46
4.78
2.40
6.85
1.75
8.62
1.21
MNIST
4.60
2.38
1.13
2.33
2.58
1.58
3.44
1.58
FMNIST
5.71
3.00
5.93
2.72
3.28
1.73
3.76
1.25
QUANTITY SKEW:
SVHN
11.80
5.04
1.30
4.77
5.94
2.21
10.08
3.49
IMBALANCED (0.4,2)
CIFAR-10
29.65
14.70
2.20
10.60
13.88
8.03
24.88
11.76
CIFAR-100
41.66
19.02
6.92
11.84
10.63
5.59
30.96
12.42
SST
11.08
3.56
3.63
4.16
3.85
9.64
2.39
MNIST
18.99
10.86
50.04
0.71
17.88
11.98
64.34
0.06
FMNIST
11.61
9.64
34.63
2.35
9.58
51.80
0.20
SVHN
1.13
1.07
38.59
1.13
9.21
8.47
51.45
0.09
LABEL SKEW: #3C=(3, 30)
CIFAR- 10
24.11
6.38
5.79
4.86
5.57
11.68
39.42
0.18
CIFAR- 100
40.28
1.40
5.25
0.84
23.16
9.93
40.75
0.16
SST
8.86
1.70
0.89
1.97
5.55
1.92
7.60
0.65
NUMBER TIMES THAT PERFORMS THE BEST
1/54
0/54
11/54
42/54
26

---

Acqua: Fair Model Rewards in Collaborative Learning via Slimmable Networks
Table The average collaboration gain (MCG) (f) and the collaboration gain spread (CGS) (4) results of our method and other baselines under different dataset partitions. The results are averaged over five independent evaluations. Continuation of Table with different minimum width (pmin 0.1).
FEDAVG-FT
CGSV
IAFL
ABQUA
PARTITION
DATASET
MCG
CGS
MCG
CGS
MCG
CGS
MCG
CGS
MNIST
2.83
2.50
2.23
2.09
2.33
1.73
1.70
1.02
FMNIST
3.96
2.85
-4.98
2.17
3.59
1.42
2.82
0.45
QUANTITY SKEW:
SVHN
7.91
5.00
-3.56
4.47
6.25
1.73
5.38
0.75
IMBALANCED (0.15,6)
CIFAR-10
19.88
14.02
-5.85
8.96
13.67
7.01
13.46
3.04
CIFAR-100
34.13
13.63
2.87
7.93
21.23
1.26
24.72
1.16
SST
9.16
2.46
4.78
2.40
6.85
1.75
7.71
0.58
MNIST
4.60
2.38
-113
2.33
2.58
1.58
2.59
1.28
FMNIST
5.71
3.00
5.93
2.72
3.28
1.73
2.69
1.09
QUANTITY SKEW: IMBALANCED (0.4,2)
SVHN
11.80
5.04
1.30
1.77
5.94
2.21
5.89
1.71
CIFAR-10
29.65
14.70
2.20
10.60
13.88
14.20
6.62
CIFAR-100
41.66
19.02
6.92
11.84
10.63
5.59
21.08
7.64
SST
11.08
3.56
3.63
4.16
3.85
1.69
8.80
1.66
NUMBER OF TIMES THAT PERFORMS THE BEST
0/12
0/12
1/12
11/12
Table 9: Incentivization performance of Aequa on CIFAR-I0, CIFAR-100, and MNIST datasets under various partitioning strategies, evaluated using the Pearson correlation coefficient between (1) assigned model widths and contributions and (2)
final model accuracies and contributions. Results are averaged over five independent runs.
DATASET
PARTITION
p(c, p) (CONTRIBUTION, WIDTH) plc. a) (CONTRIBUTION, ACC.)
HOMOGENEOUS
1.00 T 0.00
0.94 1: 0.01
DIRICHLET (a 0.1)
1.00
0.91 = 0.04
DIRICHLET 0.5)
1.000
0.95 0.02
DIRICHLET (a 1.0)
1.00
0.95 0.02
CIFAR-10
DIRICHLET (a 2.0)
1.00
0 = 0.02
DIRICHLET (a 5.0)
1.00
0.92 0.04
QUANTITY SKEW (0.15,6) 6)
0.00
1.00 0.00
QUANTITY SKEW (0.4,2)
0.00
0.99 0.01
LABEL SKEW (#C- (3. 30))
0.00
0.96 = 0.02
HOMOGENEOUS
1.00
0.95 0.01
DIRICHLET (a = 0.1)
1.00
0.96 1 0.02
DIRICHLET (a 0.5)
1.00
0.92 1 0.02
DIRICHLET (a 1.0)
0.00
0.94 0.03
CIFAR-100
DIRICHLET (a 2.0)
1.00
0.95 0.03
DIRICHLET (a = 5.0)
0.00
0.96 0.02
QUANTITY SKEW (0.15,
0.00
1.00 0.00
QUANTITY SKEW (0.1,2)
1.00
1.00 = 0.00
LABEL SKEW (#C- (3, 30))
1.00 0.00
0.95 0.02
HOMOGENEOUS
1.00 0.00
0.91 + 0.04
DIRICHLET (a 0.1)
1.00 + 0.00
0.90 + 0.03
DIRICHLET (a = 0.5)
1.00
0.92 + 0.04
DIRICHLET (a 1.0)
1.00 0.00
0.92 1 0.03
MNIST
DIRICHLET (a 2.0)
1,00 +.000
1 0.94 0.04
DIRICHLET (a 5.0)
1.00
0.88 1 0.03
QUANTITY SKEW (0.15,
0.000
0.95 0.04
QUANTITY SKEW (0.4,2)
1.00
0.94 0.04
LABEL SKEW (#C= = (3,33,))) (0))
1.00
0.90 = 0.04
27


Reviews:
[{"review_id": "zG4tUvYgBf", "reviewer": "Reviewer_yhe8", "summary": "This paper introduces Aequa, a framework to ensure collaborative fairness in federated learning using slimmable neural networks. It trains a single global model whose sub-networks of varying widths serve as rewards aligned with participant contributions. Experiments on six benchmark datasets show Aequa achieves near-perfect correlation between contributions and model performance without significant loss in overall accuracy.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for their valuable positive feedback. \n\n> Reliance on TEEs for security \n\nWe 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. \n\nThus, while TEEs provide one implementation path and theoretically fair algorithm, Aequa remains effective and secure even in their absence. \n\n> Need to choose parameters (e.g. minimum model width) carefully. \n\nWe 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. \n\nImportantly, 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. \n\n> Further exploration of real-world overhead and security challenges\n\nPlease 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. \n\n| Algorithm | time (s) | \n| ---- | ----- |\n| FedAvg | 42 $\\pm$ 2.9 s | \n| Aequa | 55 $\\pm$ 3.1 s | \n| \n\nFrom 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. \n\nRegarding 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. \n\n\n> Questions \n\n**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.  \n\n**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. \n\n**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.", "claims_and_evidence": "The authors claim Aequa ensures proportional rewards, maintains strong performance, and converges theoretically. Correlation results (often above 0.95) back up the fairness claim, and tables show minimal accuracy drop compared to FedAvg. The theoretical analysis relies on standard convex FL assumptions; while the reviewer did not verify every proof step, the logic appears sound.", "methods_and_evaluation_criteria": "The methods and experimental setup appear well-chosen for the problem. The core method – using a slimmable network to enable differentiated model widths – is appropriate because it allows implementing fairness in a single training run (as opposed to training separate models per client). This design is elegant and efficient, ensuring that smaller “reward” models are true sub-networks of the larger model and thus require no additional training ￼. The paper’s federated optimization procedure (Algorithm 1) is an adaptation of FedAvg to train all widths of the slimmable model in a coordinated way; this is a sensible approach to ensure that the global model performs well across the full width range. The fair allocation algorithm (post-training) is described formally as an optimization problem with clear objectives (non-negativity of gains, proportionality, etc.), and the chosen solution method (simulated annealing-based heuristic) is reasonable for finding an approximate optimal width allocation. The evaluation criteria align with the claims: the authors evaluate global model accuracy to ensure overall performance isn’t degraded, and they evaluate fairness through the correlation between contributions and rewards (as well as a metric called collaboration gain spread). These metrics directly measure the goals of collaborative fairness and are standard in this line of work ￼. The use of six diverse datasets (MNIST, FMNIST, SVHN, CIFAR-10, CIFAR-100, SST) and multiple data partition strategies (homogeneous, Dirichlet heterogeneous, quantity skew, label skew) is commendable ￼ ￼. This covers both IID and non-IID scenarios, which is critical for federated learning experiments. The chosen baselines are appropriate: FedAvg with fine-tuning represents a naive approach to personalization, CGSV and IAFL represent state-of-the-art fairness/incentive methods ￼, and evaluating Standalone accuracy provides a reference point for each client’s individual performance. By including these baselines, the authors ensure a fair and informative comparison. The experiments were run across all methods on identical settings, and the paper mentions using balanced accuracy for class-imbalanced data which is a proper choice ￼. In terms of methodology, everything from the training procedure to the selection of metrics seems well-justified and in line with common practice, indicating the experimental design is suitable for validating the paper’s contributions.", "theoretical_claims": "The paper makes two primary theoretical claims: a performance guarantee for the federated training process with slimmable networks, and the convergence of the fair allocation algorithm. Both are presented with formal statements (Theorem 1 and Theorem 2) and proved under certain assumptions. The correctness of these claims appears plausible – the authors assume standard conditions (convex loss, L-smoothness, bounded variance in gradients, etc., as referenced in the paper) and then build on known federated optimization analyses ￼. Theorem 1 provides a convergence rate or error bound for Aequa’s training algorithm, ensuring that training on slimmable networks still optimizes the global objective within the same order of convergence as regular federated learning (given a suitable learning rate) ￼. Theorem 2 addresses the allocation algorithm, stating that the iterative simulated annealing approach will converge asymptotically to the optimal allocation (with probability 1) ￼. The proofs for these are relegated to the appendix, and the authors reference lemmas and prior work (e.g. citing known results from convex optimization literature) to support their derivations. From a clarity standpoint, the theoretical section in the main text is relatively concise – it outlines key lemmas and theorems, deferring detailed technical proofs to Appendix A. This makes it a bit challenging to verify every step without deep diving into the supplementary material, but it keeps the main content accessible. The reviewer’s confidence in the theoretical claims is moderate rather than absolute: the reasoning is sound on a high level and no obvious errors were found, but a full verification would require checking each appendix lemma and assumption carefully. The authors do clearly state assumptions (e.g., convexity and bounded client-drift via bounded dissimilarity in data, as noted in the appendix references) and the statements of theorems seem consistent with those assumptions. The convergence claim for the allocation algorithm is particularly interesting since it uses a heuristic method (simulated annealing); proving convergence in that context is non-trivial, but the authors appear to have done so under specific conditions (assuming a certain form of cost function and proper cooling schedule, presumably detailed in Appendix A.6). In summary, the theoretical claims are well-motivated and likely correct, though their practical applicability is bound by the validity of the assumptions (which is typical for FL theory). The clarity is sufficient, but readers who want full detail will need to consult the supplementary proofs, which the reviewer trusts with some caution.", "experimental_designs_or_analyses": "Experiments include varied data partitions (Dirichlet, label/quantity skew) and compare multiple metrics. Aequa consistently achieves the highest fairness correlation while matching or improving baseline accuracies. A few corner cases require tuning the minimum width parameter.", "relation_to_broader_scientific_literature": "The paper builds on existing FL incentive work (e.g., CGSV, IAFL) but uniquely applies slimmable networks to allocate capacity. It aligns with “collaborative fairness” studies, providing a more rigorous and flexible mechanism than many prior heuristic methods.", "essential_references_not_discussed": "No major omissions are evident; the authors cite relevant literature on fair FL and slimmable architectures.", "other_strengths_and_weaknesses": "Strengths include a novel architectural approach, solid empirical validation, and flexible application to different contribution measures. Weaknesses are its reliance on TEEs for security and the need to choose parameters (e.g., minimum model width) carefully. Further exploration of real-world overhead and security challenges would be beneficial.", "other_comments_or_suggestions": "Strengths include a novel architectural approach, solid empirical validation, and flexible application to different contribution measures. Weaknesses are its reliance on TEEs for security and the need to choose parameters (e.g., minimum model width) carefully. Further exploration of real-world overhead and security challenges would be beneficial.", "questions_for_authors": "(1) How does Aequa handle noisy or imperfect contribution assessments?\n\n(2) Could Aequa mitigate free-riding if trusted hardware is unavailable, or is TEE integral?\n\n(3) Does simultaneously training many width configurations introduce significant overhead for very large models or client counts?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wwSkvlXp0f", "reviewer": "Reviewer_iRNs", "summary": "The manuscript studies the question of assigning rewards to participants with different models whose performance faithfully reflects their heterogeneous contribution, and extends/repurposes the concept of the slimmable network for fairness in federated learning, so as to make sure that model rewards are proportional to client contributions, achieving both high performance and collaborative fairness simultaneously.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for taking the time to review our work, for highlighting its advantages and for their valuable positive feedback. \n\n> The writing quality of Section 5.2 should be improved.\n\nWe 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. \n\n> How much the subnetwork will affect the optimization?\n\nAs 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.\n\n> Lemma 4 \n\nWe 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.", "claims_and_evidence": "Yes. The manuscript includes theoretical convergence results and fairness analysis, as well as extensive numerical results to support the advances of the proposed method.", "methods_and_evaluation_criteria": "Looks reasonable:\n* it considers MNIST, Fashion-MNIST, SVHN, CIFAR-10 & 100;\n* it uses homogeneous, heterogeneous, and quantity skew scenarios;\n* it also considers several baselines, including CGSV, IAFL, SA.", "theoretical_claims": "* The proof of the convergence analysis looks standard, and it is unclear how much the subnetwork will affect the optimization.\n* The writing quality of sec 5.2 should be improved.", "experimental_designs_or_analyses": "Yes. The numerical experiments look strong.", "relation_to_broader_scientific_literature": "NA", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "NA", "other_comments_or_suggestions": "Lemma 4 is very hard to understand, as it does not explain the meaning of $\\mathbf{c}$ and $\\rho$.\n\n## update after rebuttal\nThe reviewer thanks the authors for providing feedback and will maintain the original score.", "questions_for_authors": "NA", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FowctFKWu3", "reviewer": "Reviewer_wyh2", "summary": "The paper introduces a framework (Aequa) for fair model rewards in collaborative learning (CL) by leveraging slimmable neural networks. The core idea is to proportionally allocate model capacity to participants based on their contributions, rather than distributing identical models to all. The method ensures that higher contributors receive better-performing models (e.g., a neural network with a bigger width), while lower contributors get degraded versions.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for taking the time to review our paper, for highlighting its strengths, and for their valuable feedback. \n\n> W1. Continuous model performance assumption \n\nWe 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. \n \n> W2. Missing references \n\nPlease note that reference [1] is already cited in our paper (lines 40, 59, 90). \nRegarding [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. \n\nGiven 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.  \n\n\n> W3. Scalability experiments \n\n| Algorithm | Acc. | $\\rho$ | MCG $\\pm$ CGS | \n| --- | --- | --- | --- | \n| FedAvg-FT | 71.48 | 0.2904 | 47.02 $\\pm$ 5.07 | \n| Aequa (ours) | **73.10** | **0.9888** | 52.19 $\\pm$ **2.57** | \n[FEMNIST experiment] \n\nIn 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. \n\nFor 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 $\\textbf{3597}$ **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. \n\nThe 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.  \n\nWe also include the model width vs. accuracy plot in this link: https://ibb.co/C3jpd5Nm", "claims_and_evidence": "supported", "methods_and_evaluation_criteria": "probably sound", "theoretical_claims": "probably sound", "experimental_designs_or_analyses": "valid", "relation_to_broader_scientific_literature": "Fair reward allocation is important in federated learning.", "essential_references_not_discussed": "References ok.", "other_strengths_and_weaknesses": "Strengths\n \n1. Integrating slimmable networks (a single neural network that can operate at multiple widths) with federated learning for fairness is unique. Unlike previous methods that use heuristic reward distributions, Aequa directly controls model width, ensuring a structured degradation in performance.\n\n2. Theoretical Convergence Analysis\nThe framework provides a comprehensive convergence analysis to ensure the optimization of the training-time reward allocation algorithm remains stable and optimal. This is a notable improvement over methods like CGSV, which lack formal convergence proofs, and IAFL, which introduces trade-offs between fairness and performance.\n\n\n\nWeaknesses\n1.   Assumption of Continuous Model Performance: The theoretical analysis assumes that model performance is continuous within the interval [ℓ, u], allowing for perfect Pearson correlation coefficients. However, in practice, model widths are discrete due to hardware and software limitations, and performance may not scale smoothly with width reduction. This discreteness could lead to suboptimal allocations, where small changes in width result in disproportionate performance drops, affecting fairness.\n\n2. There are some missing refs in related fields such as [1][2].\n\n3. Limited Scalability to Large Federated Learning Setups\nThe approach has only been tested on relatively small datasets (CIFAR-10, MNIST, etc.), which do not reflect the scale of real-world FL scenarios (e.g., federated medical imaging, large-scale NLP). Lack of experiments on larger FL benchmarks: The method should be evaluated on federated benchmarks like FEMNIST or OpenImage-FL to test scalability.\n\n\n[1] Jiang, M., Roth, H. R., Li, W., Yang, D., Zhao, C., Nath, V., ... & Xu, Z. (2023). Fair federated medical image segmentation via client contribution estimation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. 16302-16311).\n[2] Li, T., Hu, S., Beirami, A., & Smith, V. (2021, July). Ditto: Fair and robust federated learning through personalization. In International conference on machine learning (pp. 6357-6368). PMLR.", "other_comments_or_suggestions": "NA", "questions_for_authors": "NA", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jLGnE0CcnF", "reviewer": "Reviewer_gzyb", "summary": "Achieving collaborative fairness in federated learning involves contribution assessment and reward allocation mechanisms. This paper proposes a new reward mechanism that leverages slimmable neural networks (a client with lower contribution would get a neural network of smaller width and lower accuracy). Aequa can decide reward values for post-training distribution of model rewards and also be adapted as part of training time model rewards.\n\n## update after rebuttal\nThe authors have addressed my concerns and I raised my score during the rebuttal process.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Experimental Designs \n\n**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.\n\n**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.  \n\n--- \n> References \n\nWe 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. \n\n---\n> Weaknesses \n\nWe 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. \n\nWe kindly request the reviewer to revisit the paper for further clarification and consider adjusting their assessment accordingly. \n\n**W1.3**: The reviewer’s understanding is correct. Eq. 5 is used in training-time reward settings for simplicity. \n\n**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).\n\nAs 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. \n\nAequa aligns with Lyu et al., 2020's fairness principle: allocated reward proportional to contributions. \n\n**W2.2**: While we don't follow prior procedures exactly, our minimum width parameter functions similarly to the existing tradeoff parameters the reviewer mentioned. \n\nWhen 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. \nMoreover, 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. \n\n\n**W2.3**: We would like to highlight a fundamental misunderstanding in the reviewer's assessment. \n\nFirst, 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. \n\nSecond, 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. \n\nFinally, 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. \n\n**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.\n\n> Comments or Suggestions \n\nWe appreciate the reviewer’s suggestions and will incorporate them. \n\n> Questions\n\n**Q1:** Yes. Based on the reward mechanism in Section 4.2. \n\n**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.", "claims_and_evidence": "Mostly, the objectives of an ideal allocation algorithm (in Sec 4.2) should be better justified. For example, reducing the variability may actually help free riders. Equation 3 is only one way to achieve these objectives. There are others concepts that consider both fairness and efficiency such as Nash social welfare (which consider the product of utilities) and lp-norms.", "methods_and_evaluation_criteria": "Mostly, I am unsure about post-training model rewards as each client would have got the best possible full model during training based on Algorithm 1 (without TEEs).", "theoretical_claims": "I only checked the correctness of Sec 5.2.", "experimental_designs_or_analyses": "I checked the soundness of any experimental designs or analyses.\n* It is better to only change one factor at a time, e.g., replace zeroing out gradients with slimmable neural networks and see whether the trend in accuracy is better. Fix the reward mechanism that decide reward values (e.g., proportional to shapley value) and show that the resulting model reward accuracy is more correlated to the Shapley values.", "relation_to_broader_scientific_literature": "The paper proposes an alternative reward mechanism, slimmable neural networks, for federated learning. Existing works have considered sparsifying model parameters, only sharing updates from a few clients, and controlling the frequency of the client receiving updates.", "essential_references_not_discussed": "Under reward mechanisms, Lin et al, 2023 should also be discussed and compared against.", "other_strengths_and_weaknesses": "**Strengths**\n1. The paper introduces a novel reward allocation mechanism by leveraging slimmable neural networks.\n2. The paper is generally well-written and easy to understand.\n\n**Weaknesses**\n1. I assume that the fair allocation algorithm is directly applied after algorithm 1. **However, if algorithm 1 is used, each client would already get the full model parameters.** As the authors have pointed out in the introduction (line 62), this aids free-riding and does not ensure collaborative fairness subsequently. \n    * This limitation is partially addressed by TEEs\n    * In Sec 4.3, an alternative reward mechanism (Equation 5) is proposed. **Is this used in place of Equation 3 when training-time rewards are desired?**\n2. The paper is not self-contained and important details (e.g., algorithm in Sec 4.3, proofs in Sec 5, experiment results) are left to the appendix). It would be better to incorporate them in the main paper, for example, give the intuition behind some more complex proofs (e.g, Theorem 2) in the main paper.", "other_comments_or_suggestions": "* The Impact Statement is missing\n* Move algorithm 2 to the main paper. Specify the reward mechanism\n* Instead of claiming aqua to be \"agnostic to any contribution measure\",  it might be more accurate to say that it is a reward mechanism (and complements any contribution assessment step).", "questions_for_authors": "1. In Sec 6.2 and 6.3, is Aequa referring to combining Algorithm 1 and Sec 4.2? What is Aequa reward mechanism in Sec 6.3? How do you decide the model width for each client?\n2. Address weakness 1 and in particular, the bolded statements. \n4. Can you provide some intuition and description about the setting in Theorem 2? Is it still the post training reward setting? Why are multiple iterations considered?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper introduces Aequa, a framework for collaborative federated learning to improve reward fairness with respect to their contributions. The key technique is repurposing slimmable networks as the media for reward allocation. The paper conducts a comprehensive theoretical analysis of the reward allocation mechanism and empirically validates it on six diverse datasets. All reviewers are satisfied and lean towards acceptance after reading the author's response. There are still some intrinsic limitations of Aequa, e.g., requiring the TEE environment to mitigate the free-riding problems. Overall, it is a solid contribution to fair collaborative learning. Authors should incorporate the feedback in the camera-ready version, e.g., adding the FEMNIST scalability experiment.
Decision: Accept (poster)