# FADE: federated aggregation with discrimination elimination

Adda-Akram Bendoukha, Héber H. Arcolezi, Nesrine Kaaniche, Aymen Boudguiga, Renaud Sirdey, Pierre-Emmanuel Clet

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Adda-Akram Bendoukha, Héber H. Arcolezi, Nesrine Kaaniche, Aymen Boudguiga, Renaud Sirdey, et al.: FADE: federated aggregation with discrimination elimination. FAccT 2025 - ACM Conference on Fairness, Accountability, and Transparency (FAccT), Jun 2025, Athènes, Greece. pp.1-14. hal-05105146

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# FADE: Federated Aggregation with Discrimination Elimination

Adda-Akram Bendoukha

Samovar, Telecom-SudParis, Institut
Polytechnique de Paris
France
adda-akram.bendoukha@telecom-sudparis.eu

Héber H. Arcolezi

Inria Centre at the University
Grenoble Alpes
France
heber.hwang-arcolezi@inria.fr

Nesrine Kaaniche

Samovar, Télécom SudParis, Institut
Polytechnique de Paris
France
kaaniche.nesrine@telecom-sudparis.eu

Aymen Boudguiga

Université Paris-Saclay, CEA-LIST
France
aymen.boudguiga@cea.fr

Renaud Sirdey

Université Paris-Saclay, CEA-LIST
France
renaud.sirdey@cea.fr

Pierre-Emmanuel Clet

Université Paris-Saclay, CEA-LIST
France
pierre-emmanuel.clet@cea.fr

## ABSTRACT

In this work, we investigate how unfair updates with opposing biases can cancel each other out during aggregation in federated learning (FL), leading to a fairer overall model from a group fairness perspective. We analytically and empirically analyze this Federated Aggregation with Discrimination Elimination (FADE) phenomenon, considering both linear and nonlinear models. In addition, we build on this observation and introduce two novel fairness-aware FL aggregation strategies. The first strategy, FADE-OptW, uses sequential optimization to optimize weights assigned to each client based on their fairness levels. The second approach, FADE-SSP, identifies the optimal subset of clients that minimizes the weighted average fairness level at each round along the convergence path, and for a given metric. Our experiments demonstrate significant improvements in fairness, achieving up to a 60% reduction in discrimination compared to standard FedAvg-based FL. We achieve these gains while maintaining the model's predictive performance on highly heterogeneous client data distributions.

## CCS CONCEPTS

* **Computing methodologies → Machine learning; Distributed computing methodologies; Social and professional topics → User characteristics.**

## KEYWORDS

Federated Learning, Group fairness, Algorithmic Fairness

**ACM Reference Format:**

Adda-Akram Bendoukha, Héber H. Arcolezi, Nesrine Kaaniche, Aymen Boudguiga, Renaud Sirdey, and Pierre-Emmanuel Clet. 2025. FADE: Federated Aggregation with Discrimination Elimination. In. ACM, New York, NY, USA, 14 pages. https://doi.org/XXXXXXX.XXXXXXX

## 1 INTRODUCTION

Algorithmic fairness [3, 29], particularly group fairness, assesses how an algorithm treats different groups of populations. These groups are distinguished by sensitive attributes, such as gender or race, which ideally should not influence the algorithm's outputs. In practice, a well-performing machine learning model generalizes observed statistical patterns in the training data to make accurate predictions on new, unseen data. However, the generalization process can be problematic if irrelevant attributes (for the prediction tasks) are heavily weighted in the decision-making process. So, to ensure fair outcomes in machine learning, careful consideration of training methodologies is crucial, especially when addressing group fairness challenges.

Federated learning (FL) [20, 28] aggregates model updates from various decentralized clients holding individual datasets. Through repeated aggregation over multiple rounds, FL builds a global model with predictive performance approaching a model trained on all the combined data. This approach offers a powerful tool to enhance model performance by diversifying data sources, while ensuring that the data remains on users' devices throughout the process. However, achieving fairness in FL involves overcoming several issues including heterogeneous data distributions, varying participation rates, and the lack of centralized oversight. Each participating entity may have data that is limited in size and biased in unique ways, further complicating efforts to develop fair FL solutions.

Recent research on fair FL has introduced several approaches to address group fairness issues (see Section A for a full discussion on related works). For instance, Salazar et al. [32] proposed Fair-Fate, which incorporates local fairness evaluations into the global model update using a decaying momentum term. Huang et al. [19] introduced a double momentum optimization technique to equalize global model performance across clients' data. Zeng et al. [35] proposed FedFB, a synchronized federated version of the FairBatch approach [31] algorithm to improve group fairness. More recently, Ezzeldin et al. [15] developed FairFed, the state-of-the-art approach for fair FL that uses pre-processing at the client side and assigns, at the server side, weights to client updates based on their measured local fairness. However, these approaches typically assign higher weights to fairer clients, while unfair clients are penalized with reduced (or null) weights. Moreover, they require clients to

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<footnote>ACM Conference, XXX, XXX</footnote>
implement local debiasing strategies [24, 31], thus increasing the computational cost on the client's end.

**Contributions** – In this work, we first show that aggregating model updates with opposing discrimination<fnref n="1" /> can help mitigate biases and improve fairness in the global model. We refer to this phenomenon as Federated Aggregation with Discrimination Elimination (FADE), which is well-suited for real-world FL deployments where there is high heterogeneity. In other words, we positively answer the following question: “Does aggregating update *A*, which discriminates against group *g*₁ and favors group *g*₂, with update *B*, which discriminates against group *g*₂ and favors group *g*₁, result in a fairer global model *C* (towards *g*₁ and *g*₂) compared to *A* and *B*?”.

Additionally, we propose two innovative fairness-aware aggregation strategies to improve the fairness of the global model based on FADE. The first strategy relies on sequential optimization techniques to fine-tune the weights assigned to clients' updates based on their fairness contributions, referred to as FADE-OptW. The second strategy, formulated as a Subset-Sum-Problem (SSP) [4], selects the optimal subset of clients that minimizes unfairness in the aggregated model, referred to as FADE-SSP. Our experiments confirm that these strategies significantly improve group fairness metrics such as statistical parity [13] and equal opportunity [16] compared to standard federated averaging [28] and approach state-of-the-art methods such as FairFed [15] in terms of fairness benefits. We evaluated our solutions at various levels of data heterogeneity in the following benchmark datasets: Adult [12] and ACSIncome, ACSEmployment, and ACSPublicCoverage from FolkTables [11].

**Paper organization** – The rest of this paper is organized as follows. First, we review key concepts such as group fairness metrics and Federated Learning in Section 2. In Section 3, we give a theoretical and empirical analysis of how group unfairness can be eliminated in both linear and non-linear models. We then introduce our proposed FADE methods in Section 4. We describe in Section 5 the experimental setup and discuss our results and findings. Finally, we conclude this work by describing future perspectives in Section 6. The supplementary material of this paper provides a detailed related-work (Section A), additional experiments including benchmarks our solutions' computational cost (Section B), a discussion on the FADE phenomenon for non-binary sensitive attributes (Section C), and finally, experiments under other group fairness metrics (Section D).

# 2 BACKGROUND

In this section, we briefly review group fairness metrics (Section 2.1) and FL (Section 2.2). Table 1 summarizes the notations used throughout this paper.

## 2.1 Group fairness

In this paper, we focus on group fairness metrics, which assess the fairness of a classifier for different demographic groups that differ by the protected attribute (e.g., race, gender, ...). Without loss of generality, let S be a binary sensitive attribute with values {0, 1}, Ŷ be a predictor of a binary target Y ∈ {0, 1}. The metrics we use to evaluate fairness are:

<table><caption>Table 1: List of notations and acronyms.</caption><thead><tr><th>Symbol</th><th>Description</th></tr></thead><tbody><tr><td>n</td><td>Number of clients</td></tr><tr><td>[k]</td><td>Set of integers {1, ..., k}</td></tr><tr><td>n<sub>k</sub>, |D<sub>k</sub>|</td><td>Number of datapoints at client k</td></tr><tr><td>θ<sup>t</sup><sub>i</sub></td><td>Parameters update by client i at iteration t</td></tr><tr><td>f<sub>i</sub><sup>t</sup></td><td>Local fairness measured by client i at iteration t</td></tr><tr><td>g<sub>i</sub><sup>t</sup></td><td>gradient update by client i at iteration t</td></tr><tr><td>θ<sub>g</sub><sup>t</sup></td><td>Global model parameters at iteration t</td></tr><tr><td>f<sub>g</sub><sup>t</sup></td><td>Fairness of θ<sub>g</sub><sup>t</sup> measured on the union datasets at iteration t</td></tr><tr><td>h<sub>θ</sub>(x)</td><td>Inference of model h<sub>θ</sub> on input x</td></tr><tr><td>Y</td><td>Set of target values, Y = {0, 1}</td></tr><tr><td>ŷ</td><td>Prediction of binary target Y</td></tr><tr><td>ŷ<sub>ρ</sub></td><td>Probability prediction of binary target Y</td></tr><tr><td>S</td><td>Binary sensitive attribute, S = {0, 1}</td></tr></tbody></table>

* **Statistical Parity Difference (SPD)** [10, 13] measures the difference in the probability of positive outcomes between groups defined by a sensitive attribute and is defined as:

$$SPD = \Pr(\hat{Y} = 1 | S = 0) - \Pr(\hat{Y} = 1 | S = 1).$$

* **Equal Opportunity Difference (EOD)** [16] measures fairness by ensuring that the true positive rate is similar across groups defined by a sensitive attribute. Formally, EOD is defined as:

$$EOD = \Pr(\hat{Y} = 1 | S = 0, Y = 1) - \Pr(\hat{Y} = 1 | S = 1, Y = 1).$$

Note that both the Equal Opportunity Difference (EOD) and Statistical Parity Difference (SPD) metrics are designed to measure disparities between groups. In both cases, values closer to zero indicate minimal disparity, signifying a higher level of fairness in the model's outcomes. A value of zero reflects perfect fairness, where the model's decisions or predictions are unbiased with respect to the sensitive attribute being evaluated. Conversely, larger absolute values of EOD or SPD suggest greater disparities, highlighting potential bias in the model's performance.

## 2.2 Federated learning with model averaging

Federated Learning (FL) is a distributed learning framework in which different entities, including a server and multiple workers, collaborate to train a shared model. Crucially, raw data never leaves the devices. Instead, clients share model updates with a central server. Starting from an initial common model, each worker locally performs several SGD steps using its collection of data samples and offloads the updated model's parameters to the aggregation server. This server generates a unified model by averaging the weights (FedAVG) of the workers' updates [28] (see Algorithm 1). This process requires several rounds to achieve high predictive performances across all workers' data. Several key elements are known to negatively impact the FL process's convergence. For instance, data heterogeneity [18, 26, 36], clients' dropouts [34], and Byzantine behaviours from clients or server [6].

<footnote n="1">¹By "opposing discrimination", we mean complementary values (e.g., 0.22 and -0.20) that sum close to 0.</footnote>
**Algorithm 1** Federated Averaging (FedAvg) [28]

1. **Input:** Number of communication rounds *T*, number of workers *K*, local epochs *E* and learning rate *η*.

2. **Server initializes:** Global model $\theta_0$

3. **for** $t = 1, 2, \dots, T$ **do**

4. Broadcast global model $\theta_t$ to workers

5. **for** each client id $k \in [n]$ **in parallel do**

6. $\theta_k^{t+1} \leftarrow \text{LocalUpdate}(\theta^t, D_k, E, \eta)$ ▷ Run local training

7. **end for**

8. Aggregate: $\theta^{t+1} \leftarrow \sum_{k \in [n]} \frac{n_k}{n} \theta_k^{t+1}$ ▷ Weighted average

9. **end for**

10. **Return:** Final global model $\theta_T$

# 3 FADE FOUNDATIONS

In this section, we investigate how the standard federated averaging FedAvg affects group fairness. For ease of presentation, we consider two clients’ updates trained on datasets with opposing biases with respect to a binary sensitive attribute *S*.

## 3.1 Opposing unfairness cancellation via parameter averaging

A linear classifier $h_\theta$ is characterized by a coefficient vector $\theta = (\theta_1, \dots, \theta_n) \in \mathbb{R}^n$, along with an additional scalar term $\theta_0 \in \mathbb{R}$, commonly referred to as the *bias* or *intercept*.

The prediction for a given data point $x = (x_1, \dots, x_d) \in \mathbb{R}^d$ is computed as: $h_\theta(x) = \langle \theta, x \rangle + \theta_0$ where $\langle \theta, x \rangle$ denotes the dot product of $\theta$ and $x$. Data points are classified as positive (1) if $h_\theta(x) \ge 0$, and as negative (0) if $h_\theta(x) < 0$.

**Fairness of linear classifiers.** Let $x_j$ represent a binary sensitive attribute, such as gender or race, and consider two data points $x^{(0)} = (x_1, \dots, x_j = 0, \dots, x_d)$ and $x^{(1)} = (x_1, \dots, x_j = 1, \dots, x_d)$, which differ only in the value of $x_j$. If the classifier $h_\theta$ is perfectly fair with respect to the sensitive attribute $x_j$, then the predictions for these two data points should be identical: $h_\theta(x^{(1)}) = h_\theta(x^{(0)})$. This holds if and only if $\theta_j = 0$, indicating that the sensitive attribute $x_j$ does not influence the model’s decision. Therefore, the fairness of $h_\theta$ with respect to $x_j$ is proportional to the magnitude of $|\theta_j|$: the closer $\theta_j$ is to zero, the fairer the model is with respect to that attribute. If $\theta_j$ is large and positive, the group $\{x \in D \mid x_j = 1\}$ is favored, while large negative values of $\theta_j$ penalize this group, favoring $\{x \in D \mid x_j = 0\}$.

The influence of $\theta_j$ on the decision is not the sole source of unfairness. Unfair behavior can also arise from correlations between sensitive and non-sensitive attributes [21]. In such cases, non-sensitive attributes serve as statistical proxies for the sensitive attribute, even when $\theta_j = 0$. The stronger the correlation between a non-sensitive attribute $x_i$ and the sensitive attribute $x_j$, the more $|\theta_i|$ reflects the degree of unfairness in the model.

However, setting the coefficient $\theta_i$ for such proxy attributes to 0 is not a viable solution, as it would negatively impact the accuracy of $h_\theta$. This is because these attributes are often highly correlated

with the target label. For example, a zip code may act as a statistical proxy for *race*, yet it remains a critical predictor of income<fnref n="2" />.

**Aggregation using** FedAvg. Consider a linear model $h_\theta$ produced by aggregating two unfair linear classifiers, $h_{\theta^{(1)}}$ and $h_{\theta^{(0)}}$, using the FedAvg algorithm. Suppose that $h_{\theta^{(1)}}$ and $h_{\theta^{(0)}}$ exhibit opposing biases with respect to a sensitive attribute $x_j$, such that $\theta_j^{(0)} > 0$ and $\theta_j^{(1)} < 0$, and further, that $|\theta_j^{(1)}| \approx |\theta_j^{(0)}|$. These coefficients are assumed to be large enough to significantly influence the decisions made by their respective models.

The parameter $\theta_j$ in the aggregated model $h_\theta$ is computed as: $\theta_j = \omega_0\theta_j^{(0)} + \omega_1\theta_j^{(1)}$, where $\omega_0$ and $\omega_1$ are the aggregation weights assigned to $h_{\theta^{(0)}}$ and $h_{\theta^{(1)}}$, respectively. The sign and magnitude of $\theta_j$ in the aggregated model are determined by the relative values of $\omega_0$ and $\omega_1$. In particular, the sign of $\theta_j$ is dominated by $\max(\omega_0, \omega_1)$. When $\frac{\omega_0}{\omega_1}$ is close to 1, the contributions of $\theta_j^{(0)}$ and $\theta_j^{(1)}$ effectively cancel out, causing $|\theta_j|$ to approach 0. This results in a significant reduction in the unfair behavior of the aggregated model $h_\theta$ compared to models $h_{\theta^{(0)}}$ and $h_{\theta^{(1)}}$, as the influence of the sensitive attribute $x_j$ is greatly diminished.

## 3.2 FADE with deep learning models

We set two experiments to study the impact of FedAvg on the fairness of neural networks (NNs):

* **Experiment 1.** We first train a three-layer NN classifier $h_{\theta_1}$ for 3 local epochs on the Adult [12] and FolkTables datasets [11]: income (ACSIncome), health (ACSPublicCoverage) and employment (ACSEmployment). Then, we randomly flip the values of the SEX sensitive attribute for $\frac{3}{4}$ of the rows of the datasets, before using them to identically train a second NN $h_{\theta_2}$. The two models are then aggregated using FedAvg with $(\omega_1, \omega_2) \in \{(0.75, 0.25), (0.5, 0.5), (0.25, 0.75)\}$. Figure 1 illustrates the fairness metrics (EOD and SPD) of the resulting aggregated models along with the fairness levels of $h_{\theta_1}$ and $h_{\theta_2}$.

* **Experiment 2.** To grasp the general dynamics of fairness throughout the FL process, we train 10 models for 3 local epochs, with equal-size datasets (i.e., $\omega_i = \frac{1}{10}$ $i \in [10]$), and aggregate them using FedAvg. Then, we compare the resulting aggregated model’s fairness level with the weighted average of the 10 clients’ fairness levels. We present the comparison results in Figure 2.

Figure 1 depicts the FedAvg aggregation of two models with opposing unfairness for the SEX attribute on the aforementioned datasets. We observe that FedAvg can compensate for opposing unfairness in NNs, and so improve the fairness of the aggregated model (up to a 75% decrease in unfairness).

Figure 2 compares the aggregated model’s fairness level (EOD and SPD) to the weighted average of the updates’ fairness levels. These experiments show that the aggregated model’s fairness is close to the weighted average fairness of individual updates through the collaborative training rounds. This observation offers several perspectives for fairness-aware aggregation since the unfairness

<footnote n="2">²For instance, the zip code is often a statistical proxy for *race* but is highly relevant for predicting income where *race* is considered a discriminatory variable [1].</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1: Equal opportunity difference (a, b, and c) and statistical parity difference (e, d, and f) of two models, and their aggregation with FedAvg following three different weights distributions ($\omega_1, \omega_2$). Error bars indicate variations over 10 seeds.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: Comparison of 10 updates' fairness (EOD: a, b, c, d; SPD: e, f, g, h) and aggregated model's fairness against the weighted average fairness. Values are averaged over 10 seeds, and error bars represent the variations.</figcaption></figure>

level of the aggregated model at every training round can be estimated from individual updates' fairness, and therefore it can be minimized.

## 3.3 Analyzing the Underlying Mechanics of FADE

We aim to explain the observed group fairness regularity in the model averaging process. Specifically, we aim to characterize $F(\theta_1, D_1)$, $F(\theta_2, D_2)$, and $F(\tilde{\theta}, D_1 \cup D_2)$, where F is either SPD or EOD, $\theta_1$ and $\theta_2 \in \Theta$ and $\tilde{\theta}$ is any model between $\theta_1$ and $\theta_2$ in $\Theta$ (a convex combination of $\theta_1$ and $\theta_2$).

### 3.3.1 Fairness smoothness over the parameter space.
To accurately characterize the global fairness of an aggregated model

$\tilde{\theta}$ given the local fairness measures of $\{\theta_i\}_{i=1}^n$, conventional fairness measures, being discrete, cannot straightforwardly facilitate a proper continuity analysis over the parameter space. Hence, their smoothness is investigated via proxy definitions. We use continuous versions of the EOD and SPD metrics that closely align with probability-based measures but have additional continuity properties over $\Theta$:

For the continuous version of the Equal Opportunity Difference (EOD$_{\mathbb{E}}$), we define:

$$EOD_{\mathbb{E}}(\theta, D) = E_{X \sim D}[\hat{Y}_p | Y = 1, S = 1] - E_{X \sim D}[\hat{Y}_p | Y = 1, S = 0]$$

where D represents the data distribution and $\mathbb{E}$ denotes the expectation over this distribution.
Similarly, for the continuous version of Statistical Parity Difference (SPD<sub>E</sub>), we define:

$$SPD_{\mathbb{E}}(\theta, D) = \mathbb{E}_{\mathcal{X} \sim D} [\hat{Y}_p | S = 1] - \mathbb{E}_{\mathcal{X} \sim D} [\hat{Y}_p | S = 0]$$

These fairness metrics are now defined as expectations over the data distribution, making them inherently continuous with respect to the model parameters $\theta$. This is due to the continuity and differentiability of the neural network inference represented by the probability predictor $\hat{Y}_p$ (without rounding) associated with $\theta$, and the use of the expectation operator $\mathbb{E}$ which is also continuous (and linear). Therefore, we ensure the resulting fairness metrics exhibit continuity over the parameter space $\Theta$. Indeed, small changes in $\theta$ lead to small changes in the expectations, thus preserving the smoothness property. This makes the continuous representations of EOD and SPD suitable for understanding the impact of averaging over $\Theta$ on group fairness.

**Group fairness through a random walk in $\Theta$.** We start from a random initial point $\theta_0 \in \Theta$, and perform a random walk with a fixed Gaussian step. That is, $\theta_i = \theta_{i-1} + \text{Step}$, where Step $\leftarrow \frac{N(0, \sigma_{\text{step}}^2)}{\mathbb{R}^d}$, $\sigma_{\text{step}}^2 = 0.01$ and $d$ is the model dimension. We evaluate the different group fairness metrics at each step over three sets of data samples: $D_1, D_2$, and $D_1 \cup D_2$ with $|D_1| = |D_2|$. The three collections of data samples, $D_1$ and $D_2$, are obtained from the Adult dataset using a Dirichlet sampling over the gender attribute. We set the Dirichlet concentration parameters to $\alpha = \frac{1}{10}$ to introduce two distinct distributions of the sensitive attribute artificially. Results are reported in Figure 3, with the x-axis being the Euclidean distance from the starting point $\theta_0$.

Figure 3 confirms the continuity of EOD<sub>E</sub>. Indeed, fairness varies progressively throughout the random walk. At each step, the three measures remain extremely close, with differences of approximately 0.005, making them nearly visually indistinguishable. The values of EOD<sub>E</sub>(·, $D_1 \cup D_2$) consistently fall between SPD<sub>E</sub>(·, $D_1$) and SPD<sub>E</sub>(·, $D_2$). This similar pattern is observed for EOD<sub>E</sub>.

**Model averaging effect on global fairness.** Let $\theta_1, \theta_2$ be two models in $\Theta$. As previously observed, the functions EOD<sub>E</sub>($D_1, \theta$), EOD<sub>E</sub>($D_1 \cup D_2, \theta$), and EOD<sub>E</sub>($D_2, \theta$) are continuous, follow identical variations, and are approximately equal (similar observation for SPD<sub>E</sub>). When these functions cross the horizontal axis ($y = 0$) along the line segment defined by the two models $\theta_1$ and $\theta_1$ —as observed in Figure 3 at distance 25 from the origin, the Intermediate Values Theorem (IVT) provides the existence of a convex combination of the two models with perfect or almost perfect global group fairness. That is, $\exists \omega \in [0, 1]$ such that $\bar{\theta} = \omega\theta_1 + (1 - \omega)\theta_2$, and EOD<sub>E</sub>($D_1 \cup D_2, \bar{\theta}$) $\approx 0$ or SPD<sub>E</sub>($D_1 \cup D_2, \bar{\theta}$) $\approx 0$. As $\omega$ approaches 1, the fairness of the convex combination approaches that of $\theta_1$. In contrast, small values of $\omega$ (approaching 0) provide a convex combination model $\bar{\theta}$ with fairness approaching that of $\theta_2$. Determining the exact convex combination at which the continuous fairness function is nullified requires knowledge of the entire fairness landscape along the line segment defined by the $\theta_1$, and $\theta_1$. This presents two main challenges. first, the computational complexity of dealing with the infinite number of convex combinations between two or more models in $\Theta$; and second, the necessity of

having representative data at the server side to accurately compute the fairness landscape within the segment (or the convex hull when there are more than two models). However, based on our experimental observations, when the models are sufficiently close in $\Theta$, the fairness landscape can be reasonably approximated by a linear function interpolated between the values at the segment's endpoints (SPD<sub>E</sub>($D_1, \theta_1$) and SPD<sub>E</sub>($D_2, \theta_2$)). Therefore, in cases of opposing unfairness, within a small-norm segment in $\Theta$ defined by two models, the midpoint model serves as a heuristic yet effective approximation of the zero-fairness model along this segment.

# 4 OUR PROPOSED METHODS

From previous analysis (3), we established the possibility to closely predict the fairness level $f_g^{t+1}$ of $\theta_g^{t+1}$ from the fairness levels of $\{\theta_1^t, \dots, \theta_n^t\}$, denoted as $f^t = \{f_1^t, \dots, f_n^t\}$. More precisely, we have:

$$f_g^{t+1} \approx \sum_{i=1}^{n} \omega_i^t f_i^t = \langle \omega^t, f^t \rangle. \quad (1)$$

To minimize $f_g^{t+1}$ at each iteration, the aggregation server can either find the optimal weight vector $\omega_{\text{opt}}^t = \arg \min_{\omega} \langle \omega, f^t \rangle$ to reduce the aggregated model's unfairness according to group fairness metric like EOD or SPD or select a subset of updates with balancing unfairness measures for aggregation. Based on these intuitions, we present our two proposed FADE methods in the following.

**Fair aggregation via optimized weights (FADE-OptW).** To assign the aggregation weights, FADE-OptW aims to find a weight vector $\omega_{\text{opt}}^t$, at iteration $t$, that minimizes a real-valued dot product with the fairness values of individual updates. To do so, we have to satisfy two main constraints.

First, the weight vector $\omega_{\text{opt}}^t$ should correspond to a probability distribution, defined as:

$$\omega_i^t \ge 0 \quad \forall i \in [n] \quad \text{and} \quad \sum_{i=1}^{n} \omega_i^t = 1.$$

Second, the weight vector $\omega_{\text{opt}}^t$ has to be close to the uniform distribution. This prevents situations where a minority of updates are assigned the majority of the weights, causing the shared model to not benefit equally from the pool of datasets at its disposal. This can lead to an unbalanced predictive performance of the final model across the clients' datasets, raising other unfairness issues. Therefore, our optimization problem can be formalized as follows:

$$\text{minimize (with } \omega^t\text{):} \quad \sum_{i=1}^{n} (f_i^t \omega_i^t)^2 + \sum_{i=1}^{n} \left(\omega_i - \frac{1}{n}\right)^2 \quad (2)$$

$$\text{subject to: } \omega_i^t \ge 0 \quad \forall i \in [n] \quad \text{and} \quad \sum_{i=1}^{n} \omega_i^t = 1. \quad (3)$$

For this optimization task, a well-suited solver is the Sequential Least Squares Programming (SLSQP) algorithm implemented in the Scipy library [33].

To evaluate the SLSQP performance on the considered optimization problem, we first generate random vectors simulating the fairness metrics of $n$ different clients. Then, we compute the runtime required for SLSQP to find the optimal weights $\omega_{\text{opt}}$. We report runtime for different values of $n$ and the effectiveness of this method by measuring the terms $\sum_{i=1}^{n} f_i^t \omega_{\text{opt},i}$ and $\sum_{i=1}^{n} (\omega_{\text{opt},i} - \frac{1}{n})^2$.
<figure><img src="image_4.png" /><figcaption>Figure 3: Evolution of EOD<sub>E</sub>, and SPD<sub>E</sub> computed on gender through a random (fixed Gaussian step) walk over Θ.</figcaption></figure>

As reported in Table 3 in supplementary material B, the main optimization objectives are achieved for up to 1000 clients. Additionally, optimal weights are computed within a practical computational time frame. The server is therefore capable of determining a weight vector that ensures both: (1) equally distributed contributions to the aggregation among clients' updates, and (2) a group fairness-aware aggregation strategy, the effects of which are empirically demonstrated in Section 5. Algorithm 2 illustrates this approach.

**Algorithm 2 FADE-OptW.**

**Require:** The set of clients' model updates: $\theta^{(t)} = \{\theta_1^t, \dots, \theta_n^t\}$

1. Their respective local fairness metric evaluations: $f^{(t)} = \{f_1^t, \dots, f_n^t\}$

**Ensure:** Aggregated model $\theta_g^{t+1}$

2. $\omega^t \leftarrow \text{minimize: } \sum_{i=1}^n (f_i^t \omega_i^t)^2 + \sum_{i=1}^n (\omega_i - \frac{1}{n})^2$

3. Such that: $\omega_i^t \ge 0 \quad \forall i \in [n]$ and $\sum_{i=1}^n \omega_i^t = 1$.

4. $\theta_g^{t+1} \leftarrow \text{FedAvg}(\theta^{(t)}, \omega^{(t)})$

5. **Return** $\theta_g^{t+1}$

**Fair aggregation via optimized client selection (FADE-SSP).**

While a specific weight attribution strategy is implemented in the FL process, such as $\omega_k^t = \frac{n_k}{\sum_{i=1}^n n_i}$ in [28], we propose a second approach that selects a subset of clients based on their fairness evaluations $\{f_1^t, \dots, f_n^t\}$ following the observation in Equation (1). This method involves identifying the largest subset of updates such that the weighted average unfairness, determined by the chosen weight attribution strategy, is minimized. Formally, the selected subset is defined as follows:

$$ \text{Largest } S_{\text{opt}} \in \mathcal{P}([n]) \text{ s.t. } -\epsilon \le \sum_{i \in S_{\text{opt}}} \omega_i^t f_i^t \le \epsilon, $$

where $\mathcal{P}([n])$ indicates the power-set of $\{1, \dots, n\}$, and $\epsilon$ is a tolerance value defining the range of the interval around 0 where the weighted sum should be. For this task, we rely on existing algorithmic solutions for the *Subset-Sum-Problem* (SSP), which has a semantically close formulation to our optimization task. The *Subset-Sum-Problem* is defined as follows:

**Definition 4.1.** Given a set of integers $X = \{x_1, \dots, x_n\}$, and an integer target value $T$, SSP consists of deciding if there exists a subset $\tilde{X} \subset X$ such that $\sum_{x_i \in \tilde{X}} x_i = T$.

SSP, as originally defined in [22], is an NP-Hard problem or NP-Complete (when $T=0$). Variants of SSP referred to as approximate SSP [2] target an interval of range $2\epsilon$ around $T$ instead of an exact value. This means, deciding if there exists a subset $X_{\text{opt}}$ such that $T-\epsilon \le \sum_{x_i \in X_{\text{opt}}} x_i \le T+\epsilon$. In this work, we rely on state-of-the-art solution [23] to solve instances for the approximate SSP problem.

For FADE-SSP, at each iteration, the server attempts to identify a subset of updates where the weighted average fairness falls within the range of $[-\epsilon, \epsilon]$. If it fails to find such a subset, the value of $\epsilon$ is increased linearly by a factor $\beta$, allowing for greater unfairness tolerance. After a predetermined number of unsuccessful attempts to find an optimal subset, despite increasing the unfairness tolerance, the server proceeds to aggregate all the received updates. When an optimal subset is identified, the corresponding weights $\omega_{\text{opt}}^{(t)}$ are re-normalized, and the updates from the optimal subset are aggregated to update the global model. Algorithm 3 describes this aggregation procedure.

For the implementation of our aggregation strategy, we rely on the SciPy implementation of Approx_SSP from [30]. Regarding the unfairness relaxation parameters $\epsilon, \beta$, and max_attempts, we emphasize that approximate SSP algorithms are polynomial in the size of the input set and the inverse of the approximation $\frac{1}{\epsilon}$. Hence, the first attempts are the most computationally expensive but are more likely to yield the best subsets for fairness. Building on this fact, we distinguish two strategies:

(1) Impose a tight initial approximation $\epsilon$, and a large relaxation rate $\beta$, while allowing a relatively small number of attempts.

(2) Impose a loose initial approximation $\epsilon$, and a small relaxation rate $\beta$, while allowing a larger number of attempts.

Under high client data heterogeneity, the first strategy is preferred because opposing unfairness measures are more likely to appear within $f^{(t)}$, leading to higher success probabilities of Approx_SSP under small values of $\epsilon$. Conversely, when updates' unfairness measures are unlikely to compensate (which is often the case when client datasets are homogeneously distributed), small values of $\epsilon$ are more likely to induce computationally expensive failures of Approx_SSP, which degrades the aggregation efficiency without considerable fairness improvement. Table 4 in supplementary material B provides runtimes of approximate SSP search for various values of $\epsilon$ and $n$.
**Algorithm 3 FADE-SSP.**

**Require:** Set of clients' updates $\theta^{(t)} = \{\theta_1^t, \dots, \theta_n^t\}$

1: Set of clients' fairness metrics $f^{(t)} = \{f_1^t, \dots, f_n^t\}$

2: Pre-computed weights $\omega^{(t)} = \{\omega_1^t, \dots, \omega_n^t\}$

3: Unfairness tolerance parameters $(\epsilon, \beta)$ and max_attempts

**Ensure:** Aggregated model $\theta_g^{t+1}$

4: count ← 0

5: **while** $S_{opt} \leftarrow \text{Approx_SSP}(f^{(t)}, \omega^{(t)}, \epsilon) == \text{NIL} \land \text{count} < \text{max_attempts}$ **do**

6: $\epsilon \leftarrow \beta\epsilon$

▷ Linearly relax fairness constraint

7: count += +

8: **end while**

9: **if** $S_{opt} == \text{NIL}$ **then**

▷ [Failure]

10: $\theta_g^{t+1} \leftarrow \text{FedAvg}(\theta^{(t)}, \omega^{(t)})$

11: **else**

▷ [Success]

12: $\omega_{opt}^{(t)} \leftarrow \omega^{(t)}$ [indexes in $S_{opt}$]

13: $\omega_i^{(t)} \leftarrow \frac{\omega_i^{(t)}}{\sum_{k \in S_{opt}} \omega_k^{(t)}} \quad \forall i \in S_{opt}^{(t)}$

14: $\theta_{opt}^{(t)} \leftarrow \theta^{(t)}$ [indexes in $S_{opt}$]

▷ [Only optimal]

15: $\theta_g^{t+1} \leftarrow \text{FedAvg}(\theta_{opt}^{(t)}, \omega_{opt}^{(t)})$

16: **end if**

17: **return** $(\theta_g^{t+1})$

# 5 EXPERIMENTAL EVALUATION

This section presents our experimental setup to assess the proposed FADE methods, as well as the main results. In addition to accuracy convergence, we aim to track the evolution of the aggregated model's fairness throughout the collaborative training iterations for our approaches and compare it with canonical FedAvg.

## 5.1 Experimental setup

**Environment.** All algorithms were implemented in Python 3 and run on a local machine with a 2.50GHz Intel Core i9 processor and 64GB of RAM. The source code is publicly available in our GitHub repository: https://github.com/Akram275/Fairness-FL.

**FL settings.** To simulate a real-world FL process, we implement the Dirichlet sampling method that samples distributions from the probability simplex (i.e., a distribution over the distribution space) to divide each dataset into $n$ clients. This sampling strategy enables parametric control over the heterogeneity level of the $n$ distributions of the selected sensitive attribute through a concentration parameter $\alpha^3$.

* Number of clients $n$. We fix $n = 10$.

* Heterogeneity levels $\alpha$. To validate our FADE methods on different levels of heterogeneity for Adult, we set $\alpha = 0.5$ as in [15].

* Hyperparameters. We set the number of global epochs to 100 and the number of local epochs to 5. We use a consistent local learning rate of $10^{-4}$ for all iterations and across all clients.

**Datasets.** For ease of reproducibility, we use the following open datasets.

* Adult. This is a classical dataset from the UCI machine learning repository [12] with 45, 222 samples after cleaning. The goal is to predict the yearly income (binary label indicating over or under $50,000) using twelve categorical or continuous features. We set the sensitive attribute as the gender of each individual (i.e., male or female).

* FolkTables [11]. We select $n = 10$ states with at least 40, 000 data samples in their ACSEmployment, ACSIncome, and ACSPublicCoverage datasets as clients' datasets. As pointed out in [11], FolkTables datasets are well-balanced with respect to the attribute SEX. However, they exhibit different levels of discrimination due to disparate impact in the different censuses (USA states) distributions.

**Methods.** We evaluate the following approaches:

* Centralized. We start by training a baseline classifier on the union of the $n$ training datasets. This classifier represents the first baseline for centralized training, for both predictive performance and fairness measures.

* FedAvg. We then collaboratively train a second classifier using the conventional FedAvg approach, as described in [28]. This classifier represents a second baseline for collaborative training without fairness considerations.

* FairFed. We also consider the state-of-the-art method for fair FL proposed in [15]. We select the Reweighing method as the local pre-processing approach to be applied by each client and use the aforementioned FL setting for a fair comparison. The fairness budget is set to $\beta = 1$ as used in [15].

* FADE. Our proposed FADE methods, namely FADE-OptW (see Algorithm 2) and FADE-SSP (see Algorithm 3). For FADE-SSP, we set $\epsilon = 10^{-5}$, $\beta = 10$, and $\text{max_attempts} = 4$, thereby ranging unfairness tolerance from $10^{-5}$ to $10^{-1}$.

## 5.2 Findings

Hereafter, we summarize our key findings, illustrated in Figures 4 and 5, based on the ACSIncome and ACSEmployment datasets. Similar patterns are observed in our experiments on the ACSPublicCoverage and Adult datasets, with additional results provided in Appendix B (Figures 6 and 7, respectively).

**Continuous unfairness elimination improves fairness.** Balancing positive and negative discrimination throughout the training process allows for the collaborative neutralization of the negative impact of unfair updates from clients with highly biased datasets. This ongoing balancing of discrimination prevents the spread of discriminatory behaviour identified by Chang et al. [9] to subsequent updates from less biased clients. As shown in Table 2, both approaches (weight optimization and client sampling) outperform FedAvg in terms of fairness. The artificially introduced heterogeneity in clients' dataset partitioning induces several instances of opposing discrimination throughout the FL process, allowing both optimization approaches also to outperform centralized training's fairness, and occasionally outperform the approach in [15], as shown in Table 2 (Health and Employment).

**Weight optimization impact on utility.** Optimizing aggregation weights to minimize the weighted average fairness level might

<footnote n="3">³Large values of $\alpha$ induce a homogeneous sampling, while small values induce heterogeneous sampling.</footnote>
<table><caption>Table 2: Comparisons of our methods' predictive performance and fairness compared with centralized training, FedAvg, and FairFed. Arrows (↓) and (↑) respectively indicate positive and negative impacts of our methods on the final model compared to conventional FedAvg.</caption><thead><tr><th rowspan="2"></th><th rowspan="2">Measures</th><th colspan="5">Model</th></tr><tr><th>Centralized</th><th>FedAvg</th><th>FairFed</th><th>FADE-OptW</th><th>FADE-SSP</th></tr></thead><tbody><tr><td rowspan="5">Adult</td><td>Acc</td><td>0.8653</td><td>0.8408</td><td>0.835</td><td>0.8128 (↓ <b>0.028</b>)</td><td>0.7953 (↓ <b>0.0455</b>)</td></tr><tr><td>Precision</td><td>0.8712</td><td>0.8562</td><td>0.8810</td><td>0.8215</td><td>0.8601</td></tr><tr><td>Recall</td><td>0.6812</td><td>0.7631</td><td>0.7487</td><td>0.7283</td><td>0.6817</td></tr><tr><td>EOD</td><td>0.1303</td><td>0.3054</td><td>0.0441</td><td><b>0.1492</b> (↓ <b>0.1562</b>)</td><td><b>0.0917</b> (↓ <b>0.2137</b>)</td></tr><tr><td>SPD</td><td>0.1880</td><td>0.2161</td><td>0.1045</td><td><b>0.1497</b> (↓ <b>0.0664</b>)</td><td><b>0.1602</b> (↓ <b>0.0559</b>)</td></tr><tr><td rowspan="5">Income</td><td>Acc</td><td>0.7608</td><td>0.7595</td><td>0.7509</td><td>0.7403 (↓ <b>0.0192</b>)</td><td>0.7311 (↓ <b>0.0284</b>)</td></tr><tr><td>Precision</td><td>0.7531</td><td>0.7147</td><td>0.7448</td><td>0.6929</td><td>0.7300</td></tr><tr><td>Recall</td><td>0.5901</td><td>0.6100</td><td>0.6078</td><td>0.6353</td><td>0.6178</td></tr><tr><td>EOD</td><td>0.1952</td><td>0.1843</td><td>0.01247</td><td><b>0.0591</b> (↓ <b>0.1252</b>)</td><td><b>0.04104</b> (↓ <b>0.1432</b>)</td></tr><tr><td>SPD</td><td>0.2015</td><td>0.1435</td><td>0.04293</td><td><b>0.08105</b> (↓ <b>0.0511</b>)</td><td><b>0.0585</b> (↓ <b>0.0850</b>)</td></tr><tr><td rowspan="5">Employment</td><td>Acc</td><td>0.8255</td><td>0.8213</td><td>0.8183</td><td>0.8104 (↓ <b>0.0109</b>)</td><td>0.8201 (↓ <b>0.0012</b>)</td></tr><tr><td>Precision</td><td>0.8711</td><td>0.8884</td><td>0.7544</td><td>0.7871</td><td>0.8364</td></tr><tr><td>Recall</td><td>0.9530</td><td>0.9188</td><td>0.8918</td><td>0.9103</td><td>0.8909</td></tr><tr><td>EOD</td><td>0.0598</td><td>0.0551</td><td>0.0281</td><td><b>0.0203</b> (↓ <b>0.0348</b>)</td><td><b>0.0313</b> (↓ <b>0.0238</b>)</td></tr><tr><td>SPD</td><td>0.0608</td><td>0.0584</td><td>0.0331</td><td><b>0.0374</b> (↓ <b>0.0253</b>)</td><td><b>0.0401</b> (↓ <b>0.0183</b>)</td></tr><tr><td rowspan="5">Health</td><td>Acc</td><td>0.7889</td><td>0.7633</td><td>0.7588</td><td>0.7518 (↓ <b>0.0115</b>)</td><td>0.7464 (↓ <b>0.0169</b>)</td></tr><tr><td>Precision</td><td>0.7091</td><td>0.6904</td><td>0.6799</td><td>0.6871</td><td>0.5741</td></tr><tr><td>Recall</td><td>0.7909</td><td>0.8088</td><td>0.7711</td><td>0.7653</td><td>0.7936</td></tr><tr><td>EOD</td><td>0.0791</td><td>0.0614</td><td>0.0189</td><td><b>0.0518</b> (↓ <b>0.0096</b>)</td><td><b>0.0081</b> (↓ <b>0.0533</b>)</td></tr><tr><td>SPD</td><td>0.0614</td><td>0.0534</td><td>0.0238</td><td><b>0.0045</b> (↓ <b>0.0489</b>)</td><td><b>0.0107</b> (↓ <b>0.0427</b>)</td></tr></tbody></table>

induce significant weight disparities, especially when fairness levels are not sufficiently opposed (i.e., all updates' discrimination is directed towards the same demographic group). We additively introduce the second term $\sum_{i=1}^{n}(\omega_i^t - \frac{1}{n})^2$ in Equation (2), which measures the distance between our weight distribution, and the uniform distribution as a second element to minimize for the SLSQP solver. Nevertheless, under high data homogeneity, all clients' updates' discrimination is directed mostly toward the same groups. In these situations, the SLSQP solver outputs a distribution of $\omega_i^t$ where a high weight value (close to 1) is assigned to the least biased update, and low weight values (close to 0) are assigned to the most biased updates. This results in a non-negligible loss in the contributed data, which leads to a performance decrease when the global model is evaluated on the union test dataset.

**Impact of client sampling on utility.** The impact of sampling based on the approximate subset sum problem on convergence is similar to that of clients' dropout in FL, as observed in [9]. Indeed, sampling subsets of clients' updates to aggregate instead of the entire set of uploaded clients' updates results in a loss of utility when evaluating the global model on the union of validation data. Discarding updates from the aggregation process inevitably alters the resulting global model's performance on the discarded datasets. When a large proportion of updates are discarded, the utility loss is significant. This occurs when insufficient opposing unfairness

is manifested in the updates. Conversely, under high data heterogeneity, resulting in disparately directed unfairness, the subset that minimizes the weighted average fairness level is (on average) larger. This enables the integration of more updates, hence, improving predictive performance on the union of validation data.

**Unfairness elimination under homogeneous clients' data distributions.** When clients' data distributions are statistically similar, the discriminatory behaviors they induce are likely to be similar in both amplitude and sign (e.g., see Figure 2 (c) and (d)). Under these circumstances, clients' local updates are unlikely to produce opposing unfairness. Consequently, leveraging the unfairness elimination property of FedAvg identified in Section 3 to improve fairness without any local pre-processing becomes less effective. However, we highlight that our methods proposed in Section 4 are particularly well-suited for settings where clients' data exhibit significant statistical diversity, especially in terms of the biases they contain (disparate impact/treatment). This aligns well with real-world FL scenarios, which often involve highly heterogeneous data distributions among clients.

# 6 CONCLUSION AND PERSPECTIVES

In this paper, we have identified and analyzed a new approach for fair FL based on the principle of Federated Aggregation with Discrimination Elimination (FADE). Specifically, FADE enhances
<figure><img src="image_5.png" /><figcaption>Figure 4: Convergence (validation accuracy) of FADE-OptW and FADE-SSP training compared to FedAvg and centralized training.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: Fairness evolution of FADE-OptW and FADE-SSP compared to FedAvg and centralized training.</figcaption></figure>

group fairness in FL through aggregating model updates with opposing discrimination. Moreover, we proposed two innovative FADE methods: a fairness-aware weight optimization technique (FADE-OptW) and a fairness-driven update subset selection method (FADE-SSP). Our experimental results on benchmarking datasets such as Adult [12] and those from FolkTables [11] have demonstrated the effectiveness of these strategies in significantly improving group fairness metrics compared to standard FedAvg and state-of-the-art methods like FairFed [15].

While our methods show promising results, they are better suited to scenarios where clients' data exhibit substantial statistical diversity, particularly regarding biases. In settings with homogeneous data distributions, the potential for eliminating unfairness without local pre-processing is reduced. However, given the typical heterogeneity of real-world FL applications, our methods are well-positioned to address fairness challenges in practical deployments. An important perspective is the integration of privacy-preserving techniques with our FADE methods, such as differential privacy [14] and secure aggregation [7].

## ACKNOWLEDGMENT

The work of Adda-Akram Bendoukha and Nesrine Kaaniche was supported by the French ANR project ANR-22-CE39-0002 EQUIHID. The work of Héber H. Arcolezi has been partially supported by the French National Research Agency (ANR), under contracts "ANR-24-CE23-6239" JCJC project AI-PULSE, "ANR-19-P3IA-0003" MIAI @ Grenoble Alpes, and "ANR 22-PECY-0002" IPOP (Interdisciplinary Project on Privacy) project of the Cybersecurity PEPR.

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## A DETAILED RELATED WORK

Since the seminal work by McMahan et al. [28] on distributed training using the FL approach, iteratively aggregating multiple models into a single global model has attracted significant research interest. For instance, works such as [17, 18, 20, 26] examine the convergence speed (in terms of communication rounds) of the FL process under different conditions, including high data heterogeneity [18, 36], the number of participating clients [25], aggregation strategies [8], and data-feature heterogeneity [27] establishing a clear negative impact of data heterogeneity, and clients' dropout on convergence speed.

However, while the impact of non-independent and identically distributed (non-IID) data on FL has been well-studied [5, 18, 26, 27, 36], understanding and improving group fairness in FL is a more recent area of research. Recently, Chang and Shokri [9] have empirically demonstrated that FL can amplify bias and discrimination when compared to centralized training on the union of datasets. This happens because biased updates from individual clients can propagate through the aggregation process, amplifying discrimination in the aggregated model. This biased aggregated model then influences subsequent updates from clients, even if their local data is fair. As a result, even federations that begin with a majority of fair individual datasets can generate biased model updates over time, when trained with the canonical FedAvg [28] without any fairness intervention.

Recent works [15, 19, 32, 35] propose fairness-aware aggregation methods to mitigate bias and improve group fairness in FL. Salazar et al. [32] introduce FairFate, a collaborative fairness-aware training where the server updates the global model using conventional FedAvg and performs a second update using local fairness evaluations of individual updates. The global model is finally computed from both updates using a decaying momentum term. Their method outperforms FedAvg under high data heterogeneity. Huang et al. [19] introduce a double momentum optimization technique at the client level to accelerate convergence under high data heterogeneity, their method equalizes global model performance over clients' data, while achieving some degree of group fairness. Zeng et al. [35] propose FedFB, a FedAvg-based collaborative learning with a local application of FairBatch [31], a centralized fairness-aware training algorithm based on assigning weights to the losses of the model on different demographic groups. FedFB improves the global model's group fairness at the computational expense of a federated application of FairBatch.

More recently, Ezzeldin et al. [15] proposed FairFed, the state-of-the-art fair FL approach, which assigns weights to clients' updates based on their perceived fairness compared to an estimated global fairness measure. However, this approach can lead to a low trade-off between fairness and overall model performance. This happens because FairFed heavily penalizes updates deemed unfair, potentially discarding valuable data from clients with biased datasets. For instance, an update with high discrimination will be assigned a weight close to 0, even if it has the potential to strongly improve the global model's utility. However, we emphasize that even biased data can offer significant utility and unique statistical patterns that contribute positively to the global model's learning. Thus, unlike previous works that either penalize unfair updates [15, 32] or require computationally-heavy local pre-processing, or in-processing strategies [15, 35], our proposed approach FADE reconciles disparate fairness contributions to achieve a more balanced outcome using the conventional FedAvg aggregation, without any local pre-processing requirement at the client edge.
## B ADDITIONAL EXPERIMENTS

We provide in this section complementary experiments of our methods on two additional benchmark datasets. Namely, ACSPublicCoverage and Adult. Figures 6 and 7 illustrate these experiments. The experimental setting is consistent with the one used in Section 5.

### B.1 Complementary plots (ACSPublicCoverage and Adult)

### B.2 Computational cost of FADE

Our fairness-aware aggregation strategies introduce a computational overhead on the server side. This overhead is primarily due to the optimization tasks required by FADE-SSP, and FADE-OptW to either find the optimal subset or optimal weights distribution. Tables 4, 3 present runtime evaluations of the optimization tasks associated with FADE-SSP, and FADE-OptW respectively.

<table><caption>Table 3: SLSQP performance on our optimization task (server side). Dot product denotes the value of $\langle\omega_{opt}^t, f^t\rangle$, and uniformity refers to the distance $\frac{\sum_{i=1}^n (\omega_{opt,i} - \frac{1}{n})^2}{n}$. Fairness measures ($f_i$) are simulated by uniformly sampled values in $[-0.3, 0.3]$.</caption><thead><tr><th>n</th><th>runtime (ms)</th><th>dot product</th><th>uniformity</th></tr></thead><tbody><tr><td>10</td><td>3.09</td><td>2.57e-11</td><td>0.0053</td></tr><tr><td>50</td><td>5.54</td><td>2.11e-12</td><td>0.0032</td></tr><tr><td>100</td><td>16.73</td><td>2.80e-13</td><td>0.0081</td></tr><tr><td>150</td><td>21.77</td><td>8.97e-12</td><td>0.011</td></tr><tr><td>200</td><td>46.97</td><td>4.52e-14</td><td>0.037</td></tr><tr><td>1000</td><td>2097.59</td><td>1.12e-12</td><td>0.063</td></tr></tbody></table>

<table><caption>Table 4: Runtime of a single Approx_SSP in milliseconds.</caption><thead><tr><th rowspan="2">n</th><th colspan="4">ε</th></tr><tr><th>10<sup>-1</sup></th><th>10<sup>-2</sup></th><th>10<sup>-3</sup></th><th>10<sup>-4</sup></th></tr></thead><tbody><tr><td>10</td><td>0.761</td><td>1.114</td><td>3.003</td><td>3.791</td></tr><tr><td>20</td><td>22.83</td><td>31.14</td><td>42.31</td><td>67.92</td></tr><tr><td>50</td><td>220.13</td><td>320.45</td><td>432.92</td><td>631.71</td></tr><tr><td>100</td><td>2567.11</td><td>3190.32</td><td>4224.34</td><td>6238.31</td></tr></tbody></table>

## C ABOUT NON-BINARY FAIRNESS ATTRIBUTES AND MULTIPLE FAIRNESS ATTRIBUTES

This work highlights a way to improve fairness in federated learning without requiring specific pre-processing or training algorithms, assuming that not all parties are similarly biased. If this technique is proven effective for a single binary fairness attribute, what can we expect when balancing multiple fairness attributes or/and non-binary fairness attributes?

From a mathematical perspective, our technique can be easily adapted to such contexts. Non-binary attributes can be binarized and the minimization process can be adapted. In order to binarize one attribute A with n possible values, we can create n new binary attributes $A_i$ so that $A_i = 1$ if and only if $A = i$. In order to adapt the minimization process, we can replace the target ($\sum_{i=1}^n f_i^t \omega_i^t$) with ($\sum_{i=1}^n (\omega_i^t - \frac{1}{n})^2$) where $f_i$ is the fairness value of

the only attribute considered by $\sum_{j=1}^N (\sum_{i=1}^n f_i^{t,j} \omega_i^t)^2 + \sum_{i=1}^N (\omega_i^t - \frac{1}{N})^2$ where the $f_i^{t,j}$ is the fairness value at time t of the i-th party regarding the j-th fairness attribute.

**Practical challenges and adaptations.** In practice, however, the fairness values of the participating parties need to be properly balanced for this solution to be effective. As the number of attributes increases, the likelihood of achieving this balance decreases, leading to suboptimal results when applying our technique to multiple fairness attributes simultaneously. Nonetheless, we believe that our technique could be used with positive results with the use of local debiasing through pre-processing algorithms (as in [15]), or in scenarios where it is reasonable to assume that the fairness attributes are balanced across participants (e.g., in the case of medical cooperation between entities close to different ethnic groups).

**Alternative approach.** Another approach would be to compute separate weight vectors for each fairness attribute and then average them, which might yield reasonable results depending on the context.

**Discussion and open questions.** Why might the binarization of attributes fail in some cases? One possibility is that one of the sub-attributes is similarly biased across all parties, rendering the approach ineffective. Can this method still work under certain conditions? Are there properties that could predict the likelihood of success? These questions remain open for further exploration.

## D FADE WITH FNR/FPR DISPARITIES

Other commonly used fairness measures are the model's error demographic disparities. For a binary decision problem, these measures are defined as follows:

$$FPRD = P(\hat{Y} = 1 | Y = 0, S = s_0) - P(\hat{Y} = 1 | Y = 0, S = s_1)$$

$$FNRD = P(\hat{Y} = 0 | Y = 1, S = s_0) - P(\hat{Y} = 0 | Y = 1, S = s_1)$$

In this section, we evaluate FADE-OptW and FADE-SSP using FNRD and FPRD as the target optimization metrics. Similar to our analysis of EOD and SPD in Section 5, we assess the reduction of discrimination in individual updates through averaging. We then follow the same experimental setup as described earlier, using FNRD and FPRD as the group fairness metrics. We conduct similar baseline experiments and experiments with our aggregation strategies, FADE-SSP and FADE-OptW. The results are presented in Table 5. Similarly to the EOD and SPD metrics presented in the main paper, we observe similar behavior in terms of the averaging effect on opposing discrimination.
<figure><img src="image_7.png" /><figcaption>Figure 6: ACSPublicCoverage Convergence, and fairness evolution throughout FedAvg, FADE-SSP, and FADE-OptW training.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7: Adult Convergence, and fairness evolution throughout FedAvg, FADE-SSP, and FADE-OptW training.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 8: Comparison of 10 updates' fairness (FPRD: a, b, c, d; FNRD: e, f, g, h) and aggregated model's fairness against the weighted average fairness. Values are averaged over 10 seeds, and error bars represent the variations.</figcaption></figure>

From Figures 8, we observe that the discrimination compensation through models' averaging is also evident for FNRD and FPRD metrics, similar to what was observed with EOD and SPD. This further indicates that the proposed methods effectively address fairness concerns across different metrics while maintaining their convergence properties.

Figures 9 and table 5 demonstrate the performance of FADE-OptW, and FADE-SSP methods in terms of accuracy convergence and fairness metrics. As shown earlier with EOD and SPD, these methods are also effective when evaluated using FNRD and FPRD metrics. The fairness enhancement strategies produce comparable trends in eliminating bias while maintaining similar impacts on

model accuracy during the convergence process, ensuring that fairness and model performance can be optimized concurrently.

Received 20 February 2007; revised 12 March 2009; accepted 5 June 2009
<figure><img src="image_10.png" /><figcaption>Figure 9: Convergence and fairness evolution (FPRD/FNRD) throughout FedAvg, FADE-SSP, and FADE-OptW training on ACSPublicCoverage (Row 1), ACSIncome (Row 2), ACSEmployment (Row 3), and Adult (Row 4).</figcaption></figure>
<table><caption>Table 5: Comparisons of our methods' predictive performance and fairness compared with centralized training, FedAvg, and FairFed. Arrows (↓) and (↑) respectively indicate positive and negative impacts of our methods on the final model compared to conventional FedAvg and FairFed [15].</caption><thead><tr><th rowspan="2">Measures</th><th rowspan="2"></th><th colspan="5">Model</th></tr><tr><th>Centralized</th><th>FedAvg</th><th>FairFed</th><th>FADE-OptW</th><th>FADE-SSP</th></tr></thead><tbody><tr><td rowspan="5">Adult</td><td>Acc</td><td>0.8653</td><td>0.8508</td><td>0.8478</td><td>0.8388 (↓ <b>0.012</b>)</td><td>0.8478 (↓ <b>0.003</b>)</td></tr><tr><td>Precision</td><td>0.8712</td><td>0.8562</td><td>0.8360</td><td>0.8712</td><td>0.8332</td></tr><tr><td>Recall</td><td>0.6812</td><td>0.7631</td><td>0.7233</td><td>0.6799</td><td>0.7111</td></tr><tr><td>FNRD</td><td>-0.0891</td><td>-0.0912</td><td>-0.0237</td><td>0.0513 (↓ <b>0.0378</b>)</td><td>0.0013 (↓ <b>0.0878</b>)</td></tr><tr><td>FPRD</td><td>0.0341</td><td>0.0435</td><td>0.0218</td><td>0.0348 (↓ <b>0.0087</b>)</td><td>0.0338 (↓ <b>0.0097</b>)</td></tr><tr><td rowspan="5">Income</td><td>Acc</td><td>0.7608</td><td>0.7595</td><td>0.7571</td><td>0.7415 (↓ <b>0.018</b>)</td><td>0.7453 (↓ <b>0.0142</b>)</td></tr><tr><td>Precision</td><td>0.7531</td><td>0.7147</td><td>0.7339</td><td>0.7341</td><td>0.7792</td></tr><tr><td>Recall</td><td>0.5901</td><td>0.6100</td><td>0.6316</td><td>0.6084</td><td>0.5894</td></tr><tr><td>FPRD</td><td>-0.2730</td><td>-0.2458</td><td>-0.0922</td><td>-0.1176 (↓ <b>0.1282</b>)</td><td>-0.1038 (↓ <b>0.1420</b>)</td></tr><tr><td>FNRD</td><td>-0.2611</td><td>-0.2701</td><td>-0.1182</td><td>-0.1682 (↓ <b>0.1019</b>)</td><td>0.1712 (↓ <b>0.0989</b>)</td></tr><tr><td rowspan="5">Employment</td><td>Acc</td><td>0.8255</td><td>0.8213</td><td>0.8198</td><td>0.8173 (↓ <b>0.004</b>)</td><td>0.8129 (↓ <b>0.0084</b>)</td></tr><tr><td>Precision</td><td>0.8711</td><td>0.8884</td><td>0.8516</td><td>0.8163</td><td>0.8287</td></tr><tr><td>Recall</td><td>0.9530</td><td>0.9188</td><td>0.8919</td><td>0.8417</td><td>0.7989</td></tr><tr><td>FPRD</td><td>-0.0231</td><td>-0.0248</td><td>-0.0043</td><td>0.0054 (↓ <b>0.0194</b>)</td><td>0.0141 (↓ <b>0.0107</b>)</td></tr><tr><td>FNRD</td><td>-0.0259</td><td>-0.0262</td><td>0.0039</td><td>0.0170 (↓ <b>0.0092</b>)</td><td>-0.0074 (↓ <b>0.0188</b>)</td></tr><tr><td rowspan="5">Health</td><td>Acc</td><td>0.7889</td><td>0.7633</td><td>0.7576</td><td>0.7498 (↓ <b>0.0135</b>)</td><td>0.7508 (↓ <b>0.0125</b>)</td></tr><tr><td>Precision</td><td>0.7091</td><td>0.6904</td><td>0.7024</td><td>0.6524</td><td>0.6871</td></tr><tr><td>Recall</td><td>0.7909</td><td>0.8088</td><td>0.7699</td><td>0.8100</td><td>0.7713</td></tr><tr><td>FPRD</td><td>-0.0335</td><td>-0.0501</td><td>-0.0166</td><td>-0.0211 (↓ <b>0.029</b>)</td><td>0.0175 (↓ <b>0.0326</b>)</td></tr><tr><td>FNRD</td><td>-0.0315</td><td>-0.0435</td><td>-0.0037</td><td>-0.0105 (↓ <b>0.033</b>)</td><td>0.0092 (↓ <b>0.0343</b>)</td></tr></tbody></table>