#  GLocalFair: Jointly Improving Global and Local Group Fairness
in Federated Learning

Syed Irfan Ali Meerza1 Luyang Liu2 Jiaxin Zhang3 Jian Liu1
1University of Tennessee Knoxville
2Google Research
3Intuit AI Research
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Federated learning (FL) has emerged as a prospective solution for
collaboratively learning a shared model across clients without sacrificing
their data privacy. However, the federated learned model tends to be biased
against certain demographic groups (e.g., racial and gender groups) due to the
inherent FL properties, such as data heterogeneity and party selection. Unlike
centralized learning, mitigating bias in FL is particularly challenging as
private training datasets and their sensitive attributes are typically not
directly accessible. Most prior research in this field only focuses on global
fairness while overlooking the local fairness of individual clients. Moreover,
existing methods often require sensitive information about the client’s local
datasets to be shared, which is not desirable. To address these issues, we
propose GLocalFair, a client-server co-design fairness framework that can
jointly improve global and local group fairness in FL without the need for
sensitive statistics about the client’s private datasets. Specifically, we
utilize constrained optimization to enforce local fairness on the client side
and adopt a fairness-aware clustering-based aggregation on the server to
further ensure the global model fairness across different sensitive groups
while maintaining high utility. Experiments on two image datasets and one
tabular dataset with various state-of-the-art fairness baselines show that
GLocalFair can achieve enhanced fairness under both global and local data
distributions while maintaining a good level of utility and client fairness.

##  1 Introduction

Federated Learning (FL) McMahan et al. (2017) allows multiple data holders
(i.e., clients) to collaboratively train a global model under the coordination
of a central server without directly sharing their private datasets. Its
strong emphasis on client privacy has made it particularly attractive for many
privacy-critical applications, such as speaker verification Granqvist et al.
(2020), predictive typing Hard et al. (2018), image classification Chen and
Chao (2021) and healthcare Rieke et al. (2020); Xu et al. (2021). Although
providing great privacy benefits, the inherent characteristics of FL, such as
data heterogeneity, party selection, and client dropping out, will inevitably
make the resulting FL model biased against specific demographic groups (e.g.,
racial and gender groups), causing specific populations to receive undesirable
treatments. Additionally, to protect clients’ privacy, in FL the server is
prohibited from directly accessing the clients’ datasets, rendering most
centralized bias mitigation strategies inapplicable Grgić-Hlača et al. (2018);
Zhang et al. (2018); Lohia et al. (2019). Therefore, how to improve fairness
has become an open problem in this field and has gained considerable research
attention recently Kairouz et al. (2021).

Figure 1: An illustrative example of bias in classification when considering
two sensitive gender groups (i.e., male vs. female) in FL with three
participating clients. At the global scale, the model is fair due to the equal
true-positive rate (or false-negative rate). However, when considering the
model’s fairness on each client, the classification result is biased. Note
that the model performs differently between clients mainly due to the client
non-IIDness in FL, such as different distributions in racial and age groups.

Existing Efforts in Improving Fairness in FL. Initial studies Li et al.
(2020); Mohri et al. (2019); Lyu et al. (2020); Li et al. (2021); Zhang et al.
(2020b) in this field mainly focus on client fairness, which aims to promote
equalized accuracies across all clients. This is typically achieved through
reducing the client’s performance disparity Li et al. (2020) or maximizing the
performance of the worst client Mohri et al. (2019). However, under such
frameworks, the FL model can show similar accuracies on all clients while
still being biased against certain under-represented groups. To address this,
a few recent studies Abay et al. (2020); Zhang et al. (2020a); Rodríguez-
Gálvez et al. (2021); Chu et al. (2021); Ezzeldin et al. (2023); Du et al.
(2021) have proposed group fairness solutions for improving fairness across
different sensitive groups. These studies mainly utilize deep multi-agent
reinforcement learning Zhang et al. (2020a), re-weighing mechanisms Abay et
al. (2020); Ezzeldin et al. (2023); Du et al. (2021), optimization with
fairness constraints Chu et al. (2021) or the modified method of differential
multipliers Rodríguez-Gálvez et al. (2021) to achieve group fairness in FL.

Limitations of Existing Group Fairness Research in FL. While the
aforementioned group fairness approaches can improve FL fairness in certain
cases, most of them Zhang et al. (2020a); Rodríguez-Gálvez et al. (2021); Chu
et al. (2021); Du et al. (2021) only focus on achieving global fairness under
the overall data distribution, without considering local fairness on
individual clients. Due to the non-IID111IID: identically and independently
distributed. nature of the client’s data in FL, local distribution at a
specific client may not reflect the exact global data distribution. Thus,
merely achieving global fairness cannot provide guarantees of local fairness.
In practice, this could result in emergent group unfairness at individual
clients due to the client’s “non-IIDness” in FL. Figure 1 provides an
illustrative example of the difference between global group fairness and local
group fairness. To further illustrate, let’s consider a cross-silo FL scenario
where image classifiers for brain tumor detection are trained across several
hospitals with distinct patient demographic compositions. In such a case,
local fairness is more important since the model will eventually be deployed
at each client and thus would directly affect the model’s diagnosis accuracy
Cui et al. (2021). To the best of our knowledge, only a few studies Abay et
al. (2020); Ezzeldin et al. (2023) attempt to address the local group fairness
issue. However, they all require clients to share sensitive statistics (e.g.,
data sample counts per sensitive group or group-specific positive prediction
rates), which would raise privacy concerns. Additionally, they rely on either
reweighing or debiasing-based mechanisms to ensure local fairness, which will
be heavily influenced by the distribution of local datasets, thus they may
only achieve suboptimal fairness with highly heterogeneous datasets in FL.

To circumvent the aforementioned limitations, in this paper, we propose
GLocalFair, the first client-server co-design fairness framework that can
jointly improve global and local group fairness to learn a fair global model
on both local datasets of individual clients and the global dataset.
Specifically, to ensure local fairness, we pose the classification task on
each client as a constrained optimization problem with a constraint that
maximizes the true-positive rate (TPR) of sensitive groups and minimizes the
false-positive rate (FPR). We then transform the constrained optimization
problem into a two-player zero-sum game. To ensure global group fairness, we
incorporate a clustering-based aggregation method on the server side.
Specifically, we cluster the client updates based on their fairness level
measure using the Gini Coefficient Gini (1912) and calculate a weighted mean
within each cluster, based on their dataset size, to update the global model.
We believe such two-layer fairness strategies can ensure enhanced model
fairness at both global and local levels.

Our framework brings several benefits over existing methods: (1) Improved Data
Privacy: In GLocalFair, the server assigns weight to client updates using a
clustering algorithm. This way does not require the server to have access to
any information about each client’s sensitive local dataset; (2) Customizable
Fairness Configuration on Client Model: By adopting constrained optimization
at the client’s side, our framework provides the client with the freedom to
choose which fairness metric (FPR or TPR) to emphasize locally. For instance,
in the brain tumor diagnosis application, it is more expensive to falsely
classify a tumor case as healthy (false negative) than to diagnose a healthy
person as malignant (false positive); and (3) Multiple Sensitive Attributes:
Leveraging constrained optimization, our framework can be naturally
generalized to the scenario with multiple sensitive attributes by
incorporating additional constraints.

We conduct extensive experiments involving multiple federated datasets,
including image, and tabular data formats, and various state-of-the-art FL
fairness baselines, such as group-fairness studies FairFed Ezzeldin et al.
(2023) and FedFB Zeng et al. (2022), and client-fairness studies q-FFL Li et
al. (2020) and GIFAIR Yue et al. (2023). The results demonstrate that
GLocalFair can achieve enhanced fairness under both global and local data
distributions while maintaining a good level of model utility and client
fairness.

##  2 Related Work

Client Fairness in Federated Learning. The goal of client fairness is to
diminish the performance difference of models across different clients (known
as the client parity). To achieve this, Li et al. Li et al. (2020) propose
q-Fair Federated Learning (q-FFL), an algorithm that aims to encourage a
uniform accuracy distribution across all clients by re-weighting the clients
according to their local empirical loss. Mohri et al. Mohri et al. (2019)
propose an agnostic federated learning framework to maximize the performance
of the worst-performing client. GIFAIR Yue et al. (2023) leverages a
regularization to penalize spread in client loss for uniform performance.
Differently, Ditto Li et al. (2021) seeks to reduce a client’s performance
variance through personalization. In addition, there are some other studies
Lyu et al. (2020); Zhang et al. (2020b) aiming to achieve proportional
fairness that allows clients with higher contributions to be rewarded with a
better-performing local model. Despite improving client disparity, these
methods still fail to address the discrimination against certain demographic
groups in the FL scenario due to the high data heterogeneity across clients.

Group Fairness in Federated Learning. A better practice of responsible AI is
group fairness, which requires the model to treat different demographic groups
defined by the sensitive attribute (e.g., race and gender) equally. Several
recent studies have explored group fairness in the FL setting from the global
fairness perspective. In particular, FairFL Zhang et al. (2020a), AgnosticFair
Du et al. (2021), FCFL Cui et al. (2021), and FedFB Zeng et al. (2022) propose
to achieve global group fairness by incorporating fairness constraints into
objective functions. However, this requires clients to share sensitive
statistical information with the server. Beyond global fairness, only a few
studies have tackled the local fairness issue by designing fairness mechanisms
on the client side or the server side. For instance, Abay et al. Abay et al.
(2020) propose to adopt centralized fair learning strategies by applying
debiasing mechanisms locally on each client. Unfortunately, this method would
be less effective if the client’s sensitive attribute distributions are highly
heterogeneous, making it unfavorable in practical scenarios. FairFed Ezzeldin
et al. (2023) can adaptively modify the aggregation weights on the server side
according to the mismatch between the server’s global fairness measure and the
client’s local fairness measure. However, this method requires the statistics
from the client’s private datasets to be shared at the start of the training.
Differently, in this paper, we propose a client-server co-design framework in
FL for jointly improving global and local group fairness regarding multiple
sensitive attributes without requiring clients to share any sensitive
information about their local datasets.

##  3 Primer on Federated Learning and Fairness

###  3.1 Federated Learning

The goal of FL is to collaboratively train a global machine-learning model
among a large number of distributed clients under the coordination of a
central server without accessing client data. Let $\theta$ denote the learned
parameters of the global model, then the main objective of FL is to solve:

$\footnotesize\underset{\theta}{\min}\quad
f(\theta)=\sum_{k=1}^{n}p_{k}L_{k}(\theta),\quad
L_{k}=\frac{1}{d_{k}}\sum_{j_{k}=1}^{d_{k}}l_{j_{k}}(\theta),$ (1)

where $n$ is the total number of clients. For the $k$-th client, $p_{k}$
denotes the probability of its participation in the current training round
($\sum_{k=1}^{n}p_{k}=1$). $L_{k}$ is the empirical loss over its local data
where $l_{j_{k}}$ is the loss on the $j$-th sample of the $k$-th client, and
$d_{k}$ is the number of local data samples. The majority of studies solve the
optimization problem formulated in Equation 1 by choosing a selection of
clients, each with probability $p_{k}$ at each round and then executing an
optimizer such as stochastic gradient descent (SGD) for several iterations
locally at each client. FedAvg McMahan et al. (2017) is a commonly used
approach for solving Equation 1 in non-convex settings. The approach works
simply by applying $E$ epochs of SGD locally at each chosen client and then
averaging the resulting local models. Unfortunately, this approach introduces
highly inconsistent performance between different clients. Specifically, the
resulting global model can be biased toward the clients with a greater volume
of data points or those who frequently engage in the training process.
Additionally, naive averaging can result in a global model that is biased
against certain demographic groups presented in the dataset.

###  3.2 Fairness Notion

In client fairness, if the performance of model-a on the $n$ clients is more
uniform than the performance of model-b on the $n$ clients, we informally
state that model-a provides a fairer solution than model-b. For group
fairness, the model performance is measured with respect to how it performs
compared to the underlying groups defined by sensitive attribute $G$. If a
model performs equally for the privileged group $G=1$ and the underprivileged
group $G=0$, then the model is fair through the group fairness lens. Formally,
given data sample $X$ and its corresponding label $Y$, for a model with binary
prediction $\hat{Y}$, we consider the following two metrics to quantify its
group fairness:

Demographic Parity Dwork et al. (2012): If a classifier’s prediction $\hat{Y}$
is statistically independent of $G$, it meets demographic parity under a
distribution $(X,G,Y)$. This is equivalent to
$\mathbb{E}[\hat{Y}|G=a]=\mathbb{E}[\hat{Y}]$, where $a=0$ or $1$ for a binary
group. Demographic parity can also be measured by the Demographic Parity
Difference (DPD), with $Pr(\cdot)$ denoting the probability:

$\footnotesize{\textup{DPD}=\left|Pr(\hat{Y}=1|G=1)-Pr(\hat{Y}=1|G=0)\right|.}$
(2)

Equalized Odds Hardt et al. (2016): If a classifier satisfies equalized odds,
its predictions $\hat{Y}$ are conditionally independent of $G$ given the true
label $Y$ under a distribution $(X,G,Y)$. Mathematically, this can be
expressed as $\mathbb{E}[\hat{Y}|G=a,Y=1]=\mathbb{E}[\hat{Y}|Y=1]$, where $a$
represents a sensitive group. In other words, a model satisfies equalized odds
if the probability of a true positive and the probability of a false positive
are the same across different sensitive groups. Equalized odds can be
formulated as:

$\footnotesize{Pr(\hat{Y}=1|Y=y,G=1)=Pr(\hat{Y}=1|Y=y,G=0).}$ (3)

###  3.3 Global and Local Group Fairness

Clients in FL generally have non-IID data distributions. To account for this
data heterogeneity, we further consider two levels of fairness: global and
local group fairness. The global fairness performance of a given model
considers the entire dataset across the $n$ clients who participate in FL,
whereas the local fairness performance is determined by only applying the
fairness metrics described in Equations 2 and 3 to the local dataset $D_{k}$
at client $k$. Because of the non-IID characteristics of the local data
distributions, the overall global data distribution may not be well
represented by a single local distribution at any of the clients. As a result,
only mitigating the fairness issue on each client may not improve global
fairness. And vice versa, improving global fairness may not ensure local
fairness for each client. To balance local and global fairness, we propose a
method that can jointly mitigate the bias in the trained model, addressing
both the individual client’s local datasets and the full datasets spanning all
participating clients.

##  4 Design of GLocalFair

###  4.1 Design Overview

Due to the inherent data heterogeneity in FL, the distribution of local
datasets on individual clients may be highly distinct from the distribution of
the full dataset across all participating clients. Thus, merely achieving
global fairness cannot provide a guarantee of local fairness. To solve this
challenge, we propose a client-server co-design fairness framework,
GLocalFair, to jointly improve both global and local group fairness. As shown
in Figure 2, we consider $n$ clients that collaboratively train a global
model. On the client side, we use constrained optimization to ensure equitable
performance of the shared global model across each sensitive group present in
the local datasets. Specifically, in communication round $t$, all
participating clients first download the current global model
($\theta^{t}_{global}$) from the central server and then compute local model
updates on their local datasets, with constraints to minimize False Positive
Rate (FPR) and False Negative Rate (FNR) to values below preset thresholds set
by $\tau_{\textup{FNR}}$ and $\tau_{\textup{FPR}}$. Next, the clients upload
their model updates ($\theta_{k}^{t}$, $k\in\left\\{1,..,n\right\\}$) to the
server to participate in the federated training. On the server side, we employ
K-means clustering to group the client updates based on their Gini
coefficients Gini (1912), which serve as a fairness proxy. Subsequently, the
updates of each cluster are aggregated, where the weight assigned to each
update is proportional to the amount of data utilized to train the local
model. Then, the aggregated updates undergo another round of aggregation,
where a set of weights ${\hat{w}_{1},...\hat{w}_{n}}$ is assigned based on
their cumulative Gini coefficient value, favoring updates that exhibit greater
fairness. This results in a more balanced and fair distribution of updates.
The resulting aggregated update is then used to update the global model,
leading to a more equitable and unbiased model ($\theta_{global}^{t+1}$).
Finally, this updated model is sent back to the clients in the next round,
ensuring that all participants benefit from the fairness improvements gained
on both the server and client sides.

###  4.2 Local Fairness by Imposing Constraints

####  4.2.1 Local Fairness Constraints

To ensure local fairness (e.g., group fairness of the learned model
$\theta^{t}_{k}$ on the $k$-th client’s local dataset), we propose a
constrained optimization-based approach that is employed on each client.
Specifically, in GLocalFair, the fairness problem is recast as a problem of
minimizing an empirical loss while adhering to one or more fairness
constraints. We constrain the model with two fairness metrics, namely, False
Negative Equality Difference (FNED) and False Positive Equality Difference
(FPED),

Thus, the binary classification task, incorporating fairness constraints to
mitigate unintended bias, can be formulated as follows:

$\footnotesize\begin{split}&\underset{\theta_{k}^{t}\in\Theta}{\min}\quad\frac{1}{d_{k}}\sum_{X\in
D_{k}}f_{L}(X;\theta_{k}^{t}),\\\
&\textup{s.t.}\quad\max\\{\textup{FNR}_{G_{1}},..\\}-\textup{FNR}_{overall}\\\
&<\tau_{\textup{FNR}}\cdot\textup{FNR}_{overall}-\min\\{\textup{FNR}_{G_{1}},\cdots\\}\\\
&<\tau_{\textup{FNR}}\cdot\max\\{\textup{FPR}_{G_{1}},\cdots\\}-\textup{FPR}_{overall}\\\
&<\tau_{\textup{FPR}}\cdot\textup{FPR}_{overall}-\min\\{\textup{FPR}_{G_{1}},\cdots\\}\text{\footnotesize$<\tau_{\textup{FPR}}$},\end{split}\normalsize$
(4)

where $f_{L}$ is the binary cross-entropy loss that we want to minimize,
$D_{k}$ is the local dataset, and $\theta_{k}^{t}$ is the local model of
client $k$ at $t$th communication round and $\tau_{\textup{FNR}}$ and
$\tau_{\textup{FPR}}$ are the allowed deviations from overall FNRs and FPRs,
respectively, $\textup{FNR}_{overall}$ and $\textup{FPR}_{overall}$ are the
false negative and false positive rates on the entire local dataset,
respectively, and $\textup{FNR}_{G_{p}}$ and $\textup{FPR}_{G_{p}}$ represent
the corresponding rates in the group $G_{p}$.

Figure 2: Illustration of the proposed GLocalFair.

####  4.2.2 Constraint Optimization

Solving the constrained problem in Equation 4 presents significant challenges
due to the data-dependent nature of the constraints, which leads to high
computational costs for verification. Furthermore, the constraints represent a
linear combination of indicators, rendering them non-sub-differentiable with
respect to $\theta_{k}^{t}$. The problem with constraints outlined in Equation
4 can be resolved through the following alternative approach:

$\footnotesize{\underset{\theta_{k}^{t}\in\Theta}{\min}\quad\frac{1}{d_{k}}\sum_{X\in
D_{k}}f_{L}(X;\theta_{k}^{t})+\sum_{i=1}^{m}\lambda_{i}g_{i}(\theta_{k}^{t})},\vspace{-1mm}$
(5)

where $g_{i}(\theta_{k}^{{t}}),i\in\left\\{1,..,m\right\\}$ represents the
$i_{th}$ fairness constraint and $\lambda_{i}$ is a dual term that maximizes
the objective function when the constraints are violated and relaxes the
constraints when fairness goals are achieved. To solve this problem, a
gradient-based approach is not viable due to the piecewise constant nature of
the constraint functions in our scenario, resulting in near-zero gradients
across most points. Utilizing a surrogate is an alternative, where indicators
in Equation 4 are replaced with sigmoids for gradient-based optimization.
While it resolves differentiability, a new issue emerges: Nash equilibrium
aligns with solutions satisfying the sigmoid-relaxed constraint rather than
the real one. To solve these issues, we approximate the penalty function
through a maximization task. Specifically, we convert the Equation 5 into a
min-max optimization problem:

$\footnotesize\begin{split}\footnotesize{\underset{\lambda_{i}\geq
0}{max}\quad\underset{\theta_{k}^{t}\in\Theta}{min}\quad\frac{1}{d_{k}}\sum_{X\in
D_{k}}f_{L}(X;\theta_{k}^{t})+\sum_{i=1}^{m}\lambda_{i}g_{i}(\theta_{k}^{t})},\end{split}\normalsize\vspace{-6mm}$
(6)

which converts the problem into a two-player zero-sum game. One player
optimizes $\lambda_{i}$ to maximize the objective, while the other optimizes
$\theta_{k}^{t}$ to minimize it. We optimize the inner minimization by taking
the gradient to update the model weights:

$\footnotesize{\theta^{t*}_{k}\leftarrow\theta^{t}_{k}-\eta(\frac{1}{d_{k}}\sum_{X\in
D_{k}}\nabla f_{L}(X;\theta_{k}^{t})+\sum_{i=1}^{m}\lambda_{i}\nabla
g_{i}(\theta^{t}_{k}))}.$ (7)

Similarly, we update the dual term $\lambda_{i}$ through gradient ascent to
penalize the constraint violations and promote constraint satisfaction:

$\footnotesize{\lambda_{i}^{*}\leftarrow\lambda_{i}+\eta(\nabla
g_{i}(\theta^{t*}_{k}))}.$ (8)

###  4.3 Global Fairness by Fairness-aware Clustering

Regular averaging aggregation methods, such as FedAvg McMahan et al. (2017),
favor the clients that have more data points and frequently participate in the
training, which potentially leads to a biased global model if these clients
contribute biased local updates. In GLocalFair, we address this problem by
clustering client updates based on their Gini coefficients Gini (1912), which
are used for measuring inequality in the distribution of weights in the client
update. We observe that biased models often exhibit skewed or imbalanced
weight distributions due to imbalanced datasets or scenarios where specific
parameters or features hold disproportionate significance, particularly under
our local fairness constraints. As shown in Equations 7 and 8, the
introduction of fairness constraints leads to penalties when violations occur,
prompting adjustments in model weights to meet these constraints. A higher
bias in the model results in more pronounced changes in $\lambda_{i}$,
consequently leading to more significant adjustments in model weights
$\theta_{k}^{t}$ and thereby disrupting its weight distribution. Hence,
gauging model weight distributions could serve as a useful proxy for
evaluating fairness within our proposed framework.

Figure 3: Lorenz curve for different client model weights.

To better understand this, we plot Lorenz curves for four local model updates
with corresponding Equalized Odds Difference (EOD) in Figure 3. Within biased
models (clients 1 and 2), the Lorenz curves markedly diverge from the line of
perfect equality, indicating the dominance of a limited subset of parameters
in their weight distributions, while the Lorenz curve of relatively fair
models (clients 3 and 4 with relatively lower EOD) would closely follow the
line of perfect equality (empirical analysis is presented in Supplemental
Material 12.1). We employ the Gini coefficient to provide an approximate
measure of the fairness level within model updates. Subsequently, we employ
K-means clustering to group client updates based on their Gini coefficients:

$\footnotesize{\min\sum_{i=1}^{p}\sum_{j=1}^{n_{i}}||\mathcal{G}_{ij}-\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\mathcal{G}_{ij}||^{2},}\vspace{-1mm}$
(9)

where
$\mathcal{G}_{ij}=\frac{\sum_{a=1}^{w}\sum_{b=1}^{w}|W_{i,j,a}^{t}-W_{i,j,b}^{t}|}{2w^{2}\hat{W}_{i,j}^{t}}$
is the Gini coefficient of the $j$th client in the $i$th cluster.
$W_{i,j,a}^{t}$ and $W_{i,j,b}^{t}$ represents the weights in $j$th client’s
model, and $\hat{W}_{ij}^{t}$ is the mean of all the weights while $w$ is the
number of weights in the $j$th client’s model. $p$ is the number of clusters
determined by the elbow method, and $n_{i}$ is the number of clients in the
$i_{th}$ cluster. We then aggregate the updates using FedAvg with proportional
weights at each cluster. The weight assigned to each client’s update is
proportional to the number of samples they contributed during that round of
training. Those aggregated updates are further aggregated based on the
cumulative Gini coefficient of each cluster, favoring the fairer cluster. The
resulting global model $\theta_{global}^{t+1}$ can be obtained by:

$\footnotesize{\theta_{global}^{t+1}=\theta_{global}^{t}+\sum_{i=1}^{p}exp(-\gamma\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\mathcal{G}_{ij})\sum_{j=1}^{n_{i}}\frac{d_{ij}}{\sum_{k=1}^{n_{i}}d_{ik}}\theta_{ij}^{t},}$
(10)

where $\gamma$ is a parameter that controls the degree of fairness. Note that
when $\gamma=0$, the global model computed in Equation 10 is equivalent to
that computed by FedAvg. $d_{ij}$ denotes the number of data samples
contributed by the $j$th client in the $i$th cluster. By reducing the
influence of variations in updates across clients and prioritizing updates
from clients with lower Gini coefficients (i.e., more fairness), our approach
can promote fairness and reduce bias in the global model.

Table 1: Global group fairness comparison of GLocalFair with various baselines
on three federated datasets.

Method | CelebA | Adult Income | UTK Faces
---|---|---|---
EOD ($\downarrow$) | DP-Dis | Utility |  EOD ($\downarrow$) | DP-Dis | Utility |  EOD ($\downarrow$) | DP-Dis | Utility
Gender | Age | ($\downarrow$) | ($\uparrow$) | Gender | Race | ($\downarrow$) | ($\uparrow$) | Gender | Age | ($\downarrow$) | ($\uparrow$)
FedAvg McMahan et al. (2017) | 0.45 | 0.50 | 0.39 | 83.0% | 0.30 | 0.32 | 0.35 | 83.8% | 0.41 | 0.48 | 0.31 | 87.0%
q-FFL Li et al. (2020) | 0.36 | 0.39 | 0.34 | 79.3% | 0.28 | 0.27 | 0.29 | 81.5% | 0.37 | 0.41 | 0.39 | 86.0%
GIFAIR Yue et al. (2023) | 0.30 | 0.27 | 0.25 | 79.9% | 0.23 | 0.24 | 0.20 | 82.7% | 0.31 | 0.29 | 0.27 | 85.9%
FairFed Ezzeldin et al. (2023) | 0.16 | 0.20 | 0.17 | 78.6% | 0.12 | 0.12 | 0.11 | 77.5% | 0.18 | 0.22 | 0.19 | 85.1%
FedFB Zeng et al. (2022) | 0.21 | 0.17 | 0.15 | 79.1% | 0.10 | 0.13 | 0.12 | 79.0% | 0.23 | 0.19 | 0.17 | 85.3%
GLocalFair | 0.09 | 0.10 | 0.10 | 79.2% | 0.08 | 0.11 | 0.09 | 81.2% | 0.09 | 0.12 | 0.14 | 86.4%

##  5 Experimental Evaluation

###  5.1 Experimental Setup

####  5.1.1 Federated Datasets

(1) CelebA Liu et al. (2018): A collection of $200k$ celebrity face images
from that have been manually annotated. The dataset has up to 40 labels, each
of which is binary-valued. We resize each image to $128$×$128$px and create a
classification task to classify whether the celebrity in the image is smiling
or not. For CelebA, each subject’s gender (male or female) and age (young or
adult) are the sensitive attributes.

(2) Adult Income Dua and Graff (2017): A tabular dataset that is widely
investigated in ML fairness literature. It contains $48,842$ samples with $14$
attributes. We create a binary classifier to classify whether a subject’s
income is above $50k$ or not. In this dataset, race (white or non-white) and
gender (male or female) are used as the sensitive attributes.

(3) UTK Faces Zhang et al. (2017): A large-scale face dataset with more than
20,000 face images with annotations of age, gender, and ethnicity. The images
cover large variations in pose, facial expression, illumination, occlusion,
resolution, etc. We create a binary classification task of classifying young
(less than 40 years) and old (older than 40 years). Here, race (white or non-
white) and gender (male or female) are used as the sensitive attributes.

####  5.1.2 Evaluation Metric

Table 2: EOD and DP-Dis of individual clients and Discrepancy among the client
performance to measure client fairness. Lower mean/std values of EOD & DP-Dis
mean greater local group fairness. Lower DIS means higher client fairness.

Method | CelebA | Adult Income | UTK Faces
---|---|---|---
EOD | DP-Dis | DIS | EOD | DP-Dis | DIS | EOD | DP-Dis | DIS
| (Gender Group) | (Gender Group) | (Gender Group)
|  Mean($\downarrow$) |  Std($\downarrow$) |  Mean($\downarrow$) |  Std($\downarrow$) | ($\downarrow$) |  Mean($\downarrow$) |  Std($\downarrow$) |  Mean($\downarrow$) |  Std($\downarrow$) | ($\downarrow$) |  Mean($\downarrow$) |  Std($\downarrow$) |  Mean($\downarrow$) |  Std($\downarrow$) | ($\downarrow$)
FedAvg McMahan et al. (2017) |  0.43 |  0.25 |  0.39 |  0.27 | 13.5% |  0.21 |  0.15 |  0.33 |  0.26 | 34.6% |  0.32 |  0.21 |  0.35 |  0.29 | 22.2%
q-FFL Li et al. (2020) |  0.40 |  0.25 |  0.39 |  0.25 | 8.3% |  0.18 |  0.12 |  0.32 |  0.28 | 28.0% |  0.29 |  0.18 |  0.32 |  0.29 | 11.2%
GIFAIR Yue et al. (2023) |  0.37 |  0.23 |  0.38 |  0.24 | 7.7% |  0.17 |  0.09 |  0.30 |  0.27 | 11.4% |  0.27 |  0.18 |  0.31 |  0.29 | 10.1%
FairFed Ezzeldin et al. (2023) |  0.33 |  0.21 |  0.33 |  0.20 | 9.8% |  0.15 |  0.07 |  0.26 |  0.23 | 25.0% |  0.22 |  0.15 |  0.27 |  0.25 | 11.2%
FedFB Zeng et al. (2022) |  0.29 |  0.20 |  0.26 |  0.20 | 9.8% |  0.14 |  0.08 |  0.23 |  0.21 | 27.3% |  0.23 |  0.12 |  0.13 |  0.18 | 11.5%
GLocalFair |  0.17 |  0.17 |  0.21 |  0.17 | 9.5% |  0.11 |  0.05 |  0.20 |  0.15 | 17.3% |  0.14 |  0.16 |  0.11 |  0.15 | 10.1%

(1) Equalized Odds Difference (EOD): We use EOD of each sensitive group to
measure group fairness:

$\footnotesize{\textup{EOD}=|Pr(\hat{Y}=1|G_{g}=0)-Pr(\hat{Y}=1|G_{g}=1)|_{g\in
N}.}$ (11)

(2) Demographic Parity Disparity (DP-Dis): DP-Dis is another metric used for
measuring group fairness, which is calculated as:

$\footnotesize{\textup{DP-Dis}=\underset{g\in
N}{\max}\left|Pr(\hat{Y}=1|G_{g}=a)-Pr(\hat{Y}=1)\right|.}$ (12)

(3) Utility: In our experiments, we use the model’s prediction accuracy on the
testing dataset as the metric to quantify global model utility.

(4) Discrepancy (DIS): To evaluate client fairness, we use the performance
discrepancy among clients, which is measured as the difference between the
highest and lowest accuracies on the clients.

Note that EOD and DP-Dis are common fairness metrics that have been widely
used in the literature Zeng et al. (2022); Ezzeldin et al. (2023).

####  5.1.3 Parameter Selection

GLocalFair uses $\gamma$ to control the balance between fairness and utility
of the global model and $(\tau_{\textup{FNR}}$, $\tau_{\textup{FPR}})$ to
adjust the local fairness constraint at the client’s side. Although a lower
value for $(\tau_{\textup{FNR}}$, $\tau_{\textup{FPR}})$ can result in better
FPR and FNR, it would also increase the risk of overconstraining the model and
potentially converge to a trivial solution (e.g., a model that outputs same
prediction for all inputs). To address this, we conducted preliminary
experiments to explore different parameter values (presented in Supplemental
Material 12.3) and empirically set $\gamma$ to $0.6$ for all datasets and
$(\tau_{\textup{FNR}}$, $\tau_{\textup{FPR}})$ to $(0.1,0.08)$, $(0.1,0.08)$,
and $(0.06,0.06)$ for the CelebA, Adult Income, and UTK Faces datasets,
respectively, to maintain good model utility.

####  5.1.4 Baselines

We compare GLocalFair with a set of state-of-the-art solutions, including a
non-fairness method (FedAvg McMahan et al. (2017)), client-fairness methods
(q-FFL Li et al. (2020) and GIFAIR Yue et al. (2023)), and group-fairness
methods (FedFB Zeng et al. (2022) and FairFed Ezzeldin et al. (2023)).

(1) FedAvg McMahan et al. (2017): A common aggregation method for FL, where
the client updates are weighted based on the number of the data samples in
their local datasets. It does not account for any type of fairness.

(2) q-FFL Li et al. (2020): q-FFL is one of the client-fairness-based methods,
aiming to equalize the accuracies of all the clients by dynamically
reweighting the aggregation, favoring the clients with poor performance.

(3) GIFAIR Yue et al. (2023): GIFAIR aims to achieve client fairness using
regularization term to penalize the spread in the loss.

(4) FedFB Zeng et al. (2022): FedFB is a group fairness method, where they
have fitted the concept of fair batch from centralized learning into FL by
leveraging the shared group-specific positive prediction rate for each client.

(5) FairFed Ezzeldin et al. (2023): FairFed is a group fairness method that
can improve both local and group fairness. It employs FedAvg and a fairness-
based reweighting mechanism to account for the mismatch between global
fairness measure and local fairness measure.

###  5.2 Global Group Fairness

Table 1 shows the results of global group fairness (EOD and DP-Dis on the
global test datasets) on the three datasets. As FairFed is limited to only a
single binary sensitive group, to make a fair comparison, we run the federated
training twice for all three datasets (once for each sensitive group) and
report the EOD and the average DP-Dis of these training rounds. As we can see
from the results, FedAvg, q-FFL, and GIFAIR perform poorly as their objective
is not group fairness. As for the state-of-the-art group fairness solutions,
FairFed and FedFB, we find that GLocalFair can outperform these solutions in
all three datasets, with a significant global group fairness enhancement.
Specifically, GLocalFair can improve EOD of gender and age by $44\%$ and
$41\%$, respectively, and improve DP-Dis by $34\%$ compared to FedFB in
CelebA. We observe a similar fairness improvement for the Adult Income and UTK
Faces datasets. By jointly considering all the fairness metrics, GLocalFair
achieves an overall $21\%$ improvement compared to FedFB and $18\%$ compared
to FairFed in the Adult Income dataset. For the UTK Faces dataset, GLocalFair
achieves $37\%$ and $40\%$ fairness improvements compared to FedFB and
FairFed, respectively.

###  5.3 Local Group Fairness

To compare the local group fairness, we measure the mean and standard
deviation of EOD and DP-Dis on individual clients. As shown in Table 2, we
observe that the proposed GLocalFair is able to achieve a higher level of
local group fairness compared to FairFed and FedFB with lower EOD and DP-
Disparity values (in terms of both mean and standard deviation). Most
noticeably, in the CelebA dataset, GLocalFair can achieve an average EOD of
$0.15$ for the gender group, which is $48\%$ lower than FedFB with the second
lowest measured EOD. To further visualize the distribution of local fairness
measures across all clients, we compare the distribution of EOD and DP-
Disparity with other group fairness methods in Figure 4. Note that for a
completely fair model, the measured fairness metrics of clients should be of
the same value and approach $0$. We observe that the result of GLocalFair is
more centered with lower variance, indicating better local group fairness.
From the figure, we can observe that for gender groups in CelebA, almost $20$
out of $100$ clients have near-zero EOD. Similarly, for DP-Dis in the CelebA
dataset, most of the clients have less than a $0.2$ disparity. These results
demonstrate that by utilizing the proposed server-client co-design framework,
GLocalFair can achieve greater local group fairness among the clients compared
to other group fairness methods.

Figure 4: Distribution of EOD and DP-Dis on individual clients for the CelebA
datasets (Results on other datasets can be found in Supplementary Material
12.4).  Table 3: Performance under different FL data settings.

Metric | Method | CelebA | UTK Faces
---|---|---|---
$\alpha$ | $\alpha$
0.1 | 10 | 100 | 0.1 | 10 | 100
EOD ($\downarrow$) | FedAvg | 0.50 | 0.45 | 0.32 | 0.48 | 0.41 | 0.40
q-FFL | 0.42 | 0.36 | 0.28 | 0.44 | 0.37 | 0.36
GIFAIR | 0.33 | 0.27 | 0.25 | 0.33 | 0.31 | 0.29
FairFed | 0.25 | 0.18 | 0.17 | 0.25 | 0.18 | 0.16
FedFB | 0.28 | 0.21 | 0.18 | 0.27 | 0.19 | 0.16
GLocalFair | 0.11 | 0.09 | 0.08 | 0.12 | 0.09 | 0.08
DP-Dis ($\downarrow$) | FedAvg | 0.50 | 0.47 | 0.30 | 0.35 | 0.31 | 0.29
q-FFL | 0.44 | 0.35 | 0.27 | 0.39 | 0.39 | 0.34
GIFAIR | 0.32 | 0.27 | 0.22 | 0.31 | 0.27 | 0.26
FairFed | 0.21 | 0.16 | 0.15 | 0.25 | 0.19 | 0.17
FedFB | 0.22 | 0.15 | 0.12 | 0.18 | 0.21 | 0.16
GLocalFair | 0.11 | 0.07 | 0.07 | 0.15 | 0.14 | 0.12
Utility ($\uparrow$) | FedAvg | 81.7% | 83.0% | 85.0% | 84.2% | 87.0% | 88.2%
q-FFL | 79.7% | 79.3% | 81.2% | 83.3% | 86.0% | 87.1%
GIFAIR | 78.7% | 79.9% | 81.7% | 82.8% | 85.9% | 86.4%
FairFed | 76.7% | 78.6% | 80.1% | 82.1% | 85.1% | 86.9%
FedFB | 76.2% | 79.1% | 79.9% | 81.1% | 85.3% | 86.0%
GLocalFair | 78.5% | 79.1% | 80.2% | 83.9% | 86.4% | 86.6%

###  5.4 Utility and Client Fairness

From Table 1 and 2, we observe that the proposed GLocalFair offers competitive
and stable performance in terms of maximizing the utility and mitigating
discrepancy among the clients compared to other group-fairness-based methods.
FedAvg achieves relatively higher model utility (e.g., the highest utility in
the CelebA dataset) compared to other methods, which is expected as it is only
optimized for utility and does not take fairness into account. Although the
proposed GLocalFair is not optimized for client fairness, it can still
outperform other group-fairness baselines (FedFB and FairFed). Specifically,
as can be seen from Table 2, on the CelebA dataset, GLocalFair achieves an
average performance improvement of $5\%$ and $7\%$ in discrepancy compared to
FedFB and FairFed, respectively. Moreover, GLocalFair achieves $18\%$ and
$15\%$ improvement on the Adult Income dataset and $6\%$ and $9.5\%$ on the
UTK Faces dataset. These results show that GLocalFair can maintain a good
level of model utility and client fairness while providing enhanced group
fairness compared to other baselines.

###  5.5 Impact of Different FL Data Settings

To assess the robustness of our method to data heterogeneity, we perform
experiments under different data distributions, including both IID and non-IID
scenarios. We distribute the dataset among $n$ clients such that the
distribution of sensitive attributes (G) is non-IID and can be controlled
using a heterogeneity controller $\alpha$, where $\alpha\rightarrow\infty$
corresponds to IID distributions. To create this heterogeneity, we use a power
law distribution Clauset et al. (2009). The results in Table 3 show that while
data heterogeneity has a significant impact on baseline methods, our method is
immune to it due to the use of a clustering algorithm that helps to mitigate
variance among client updates.

###  5.6 Interpretability

To examine model interpretability, we consider a binary classification task
(i.e., smiling or not) on the CelebA dataset and compare the heat maps of the
models produced by different FL framework baselines. We tested $20$ randomly
selected images from the CelebA test set and generated the feature heat map
using the Grad-CAM method Selvaraju et al. (2017). A visual comparison is
provided in Figure 5, in which we observe that the heat maps of models from
FedAvg, q-FFL, and GIFAIR mostly concentrate on regions of the human face
(e.g., hair and eyes) that are closely related to sensitive demographic
information such as gender or identity, which may introduce a bias towards
certain groups. As FairFed and FedFB focus on removing bias on sensitive
attributes in the model, we can see from the heat maps that they only
concentrate on specific regions that are highly related to smiling, such as
lips. However, FairFed concentrates on the forehead and FedFB concentrates on
the hair. Emphasizing features from these areas can introduce a bias toward
the age and gender of the subject. In the case of GLocalFair, we found that
the heat map concentrates on less-biased areas of the human face, such as the
chin, which indicates better model fairness compared to other methods.

Figure 5: Comparison of feature heat maps produced by different baselines and
GLocalFair (More results can be found in Supplementary Material 12.7).

##  6 Conclusion

We propose GLocalFair, a client-server co-design framework that mitigates the
demographic biases while ensuring group fairness in FL at both global and
local levels. We utilize constrained optimization to counter algorithmic
biases on the client side and apply clustering-based aggregation on the server
side for equitable global model performance among sensitive groups while
preserving high utility. Our experiments across diverse datasets demonstrate
that our approach achieves strong group fairness results globally and locally
while also preserving utility similar to state-of-the-art fair FL frameworks.

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##  7 GLocalFair Algorithm

The detailed steps performed on the server and clients in GLocalFair are
summarized in Algorithm 1.

Algorithm 1 GLocalFair Algorithm

Server:

Initialize global model parameter $\theta_{global}^{0}$

for each round $t$ = 0, 1, · · · do

Server randomly selects a subset of $n$ clients

Server sends the latest global model $\theta_{global}^{t}$ to the selected
clients

for each client $k$ = 0, 1, · · · $n$ do

Gather local update ($\theta_{k}^{t}$)

Cluster the updates upon their Gini coefficients ($\mathcal{G}$)

$\min\sum_{i=1}^{p}\sum_{j=1}^{n_{i}}||\mathcal{G}_{ij}-\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\mathcal{G}_{ij}||^{2}$

end for

Update the global model parameter

$\theta_{global}^{t+1}=\theta_{global}^{t}+\sum_{i=1}^{p}exp(-\gamma\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\mathcal{G}_{ij})$

$\sum_{j=1}^{n_{i}}\frac{d_{ij}}{\sum_{k=1}^{n_{i}}d_{ik}}\theta_{ij}^{t}$

end for

Client (the $k$-th client):

Receive global model parameter $\theta_{global}^{t}$ from the server at the
round $t$;

Train the local model on the local dataset to obtain the updated model
parameter $\theta_{k}^{t}$

Initialize $\theta_{k}^{t,0}=\theta_{k}^{t}$ for local constraint optimization

for $j$ = 0, 1, · · $J$-1 do

Let $\tilde{\Delta}_{\theta}^{j}$ be stochastic gradient of Equation 6

Let $\Delta_{\lambda}^{j}$ be stochastic subgradient of Equation 6

Update
$\theta^{t,j+1}_{k}=\Pi_{\Theta}(\theta^{t,j}_{k}-\eta_{\theta}\tilde{\Delta}_{\theta}^{j})$

Update
$\lambda^{(j+1)}=\Pi_{\Lambda}(\lambda^{j}+\eta_{\lambda}\Delta_{\lambda}^{j})$

end for

Update $\theta_{k}^{t}$ to be the best local update $\theta_{k}^{t,j}$,
$j\in[0,...,$$J$$-1]$, with the lowest FPR/FNR

Send $\theta_{k}^{t}$ to the server

##  8 Convergence Rates

######  Theorem 1.

Define $\mathcal{M}$ as the set of all stochastic $(m+1)*(m+1)$ matrices,
$\Lambda:=\Delta^{m+1}$ as the $(m+1)$ dimensional simplex, where $m$ is the
number of functional constraints, and assume that each $\hat{g_{i}}$ upper
bounds the corresponding $g_{i}$ ($g_{i}$ represents the $i^{th}$ constraint).
The max-min formulation of Equation 6 in the paper uses a non-zero-sum two-
player game where the $\lambda$-player chooses how much the $\theta$-player
should penalize the (differentiable) sigmoid relaxed constraints but does so
in such a way as to satisfy the original constraints. Let simplify the
Equation 6 from the main paper as:

$\underset{\lambda_{i}\geq 0}{max}\quad\underset{\theta\in\Theta}{min}\quad
f_{L}(\theta)+\sum_{i=1}^{m}\lambda_{i}g_{i}(\theta),$ (13)

where $f_{L}(\theta)$ is the binary cross-entropy loss that we want to
minimize. Let denote the objective of two players ($\lambda$ and $\theta$) as:

$\begin{split}\mathcal{L}_{\theta}(\theta,\lambda)&=f_{L}(\theta)+\sum_{i=1}^{m}\lambda_{i}g_{i}(\theta),\\\
\mathcal{L}_{\lambda}(\theta,\lambda)&=\sum_{i=1}^{m}\lambda_{i}g_{i}(\theta),\end{split}$
(14)

where $\theta\in\Theta$ and $\lambda\in\Lambda$ jointly distributed random
variables such that:

$\begin{split}&\mathbb{E}_{\theta,\lambda}[\mathcal{L}_{\theta}(\theta,\lambda)]-\underset{\theta^{*}\in\Theta}{inf}\mathbb{E}_{\lambda}[\mathcal{L}_{\theta}(\theta^{*},\lambda)]\leq\epsilon_{\theta},\\\
&\underset{M^{*}\in\mathcal{M}}{max}\mathbb{E}_{\theta,\lambda}[\mathcal{L}_{\lambda}(\theta,M^{*}\lambda)]-\mathbb{E}_{\theta,\lambda}[\mathcal{L}_{\lambda}(\theta,\lambda)]\leq\epsilon_{\lambda}.\end{split}$
(15)

Suppose $\bar{\theta}$ is a random variable for which
$\bar{\theta}=\theta^{t}$ with probability
$\lambda_{1}^{t}/\sum_{s=1}^{T}\lambda_{1}^{s}$ and let
$\bar{\lambda}:=\left(\sum_{t=1}^{T}\lambda^{t}\right)/T$, then $\bar{\theta}$
represents a sub-optimal solution when:

$\mathbb{E}_{\bar{\theta}}[f_{L}(\bar{\theta})]\leq\underset{\theta^{*}\in\Theta:\forall_{i}.\hat{g}_{i}(\theta^{*})\leq
0}{inf}f_{L}(\theta^{*})+\frac{\epsilon_{\theta}+\epsilon_{\lambda}}{\bar{\lambda}_{1}},\\\
$ (16)

and it is a nearly-feasible solution when:

$\underset{i\in[m]}{max}\mathbb{E}_{\bar{\theta}}[g_{i}(\bar{\theta})]\leq\frac{\epsilon_{\lambda}}{\bar{\lambda}_{1}}.$
(17)

Note that the optimality inequality is weaker than it may appear since the
comparator in this equation is not the optimal solution w.r.t. the constraints
$g_{i}$, but rather w.r.t. the sigmoid constraints $\hat{g}_{i}$.

###### Proof.

Optimality: If we choose $M^{*}$ to be a matrix with its first row being all-
one and all other rows being all-zero, then
$\mathcal{L}(\theta,M^{*}\lambda)=0$, which shows that the first term in the
LHS of the second line of Equation 15 is non-negative. Hence
$-\mathbb{E}_{\theta,\lambda}[\mathcal{L}_{\lambda}(\theta^{t},\lambda^{t})]\leq\epsilon_{\lambda}$,
and thus by the definition of $\mathcal{L}_{\lambda}$ and the fact that
$\hat{g_{i}}\geq g_{i}$, we have:

$\mathbb{E}_{\theta,\lambda}\left(\sum_{i=1}^{m}\lambda_{i+1}\hat{g}_{i}(\theta)\right)\geq-\epsilon_{\lambda}.\\\
$ (18)

Here, $\mathcal{L}_{\theta}$ is linear in $\lambda$, so the first line of the
Equation 15, combined with the above results and the definition of
$\mathcal{L}_{\theta}$, becomes:

$\footnotesize\mathbb{E}_{\theta,\lambda}[\lambda_{1}f_{L}(\theta)]-\underset{\theta^{*}\in\Theta}{inf}\left(\bar{\lambda}_{1}f_{L}(\theta^{*})+\sum_{i=1}^{m}\lambda_{i+1}\hat{g}_{i}(\theta^{*})\right)\leq\epsilon_{\theta}+\epsilon_{\lambda}.\\\
$ (19)

Suppose $\theta^{*}$ is the optimal solution that satisfies the sigmoid
constraints $\hat{g}$, so that $\hat{g}_{i}(\theta^{*})\leq 0$ for all
$i\in[m]$, then we can have:

$\mathbb{E}_{\theta,\lambda}[\lambda_{1}f_{L}(\theta)]-\bar{\lambda}_{1}f_{L}(\theta^{*})\leq\epsilon_{\theta}+\epsilon_{\lambda},\\\
$ (20)

which is the optimality claim.

Feasibility: To simplify the notation, suppose $l_{1}(\theta):=0$ and
$l_{i+1}(\theta):=g_{i}(\theta)$ for $i\in[m]$, then
$\mathcal{L}_{\lambda}(\theta,\lambda)=\langle\lambda,l(\theta)\rangle$.
Consider the first term in LHS of the second line of Equation 15:

$\begin{split}\underset{M^{*}\in\mathcal{M}}{max}\mathbb{E}_{\theta,\lambda}[&\mathcal{L}_{\lambda}(\theta,M^{*}\lambda)]=\underset{M^{*}\in\mathcal{M}}{max}\mathbb{E}_{\theta,\lambda}[\langle
M^{*}\lambda,l(\theta)\rangle],\\\
&=\underset{M^{*}\in\mathcal{M}}{max}\mathbb{E}_{\theta,\lambda}\left[\sum_{i=1}^{m+1}\sum_{j=1}^{m+1}M^{*}_{j,i}\lambda_{i},l_{j}(\theta)\right],\\\
&=\sum_{i=1}^{m+1}\underset{j\in[m+1]}{max}\mathbb{E}_{\theta,\lambda}[\lambda_{i}l_{j}(\theta)],\end{split}$
(21)

where we used the fact that since $M^{*}$ is left-stochastic, each of its
columns is an (m+1)-dimensional mulitinoulli distribution. For the second term
in the LHS of the second line of Equation 15, we can use the fact that
$l_{1}(\theta)=0$:

$\mathbb{E}_{\theta,\lambda}\left[\sum_{i=2}^{m+1}\lambda_{i}l_{i}(\theta)\right]\leq\sum_{i=2}^{m+1}\underset{j\in[m+1]}{max}\mathbb{E}_{\theta,\lambda}[\lambda_{i}l_{j}(\theta)].\\\
$ (22)

Plugging these two results into the second line of Equation 15, the two sums
collapse, leaving:

$\underset{i\in[m+1]}{max}\mathbb{E}_{\theta,\lambda}[\lambda_{1}l_{i}(\theta)]\leq\epsilon_{\lambda}.\\\
$ (23)

Given the definition of $l_{i}$, we can yield the feasibility claim. ∎

######  Corollary 1.

Let $f_{1},f_{2},...:\Theta\rightarrow\mathbb{R}$ be a sequence of convex
functions that we wish to minimize on a compact convex set $\Theta$. Let the
step size $\eta=\frac{B_{\Theta}}{B_{\hat{\Delta}\sqrt{2T}}}$ where
$B_{\Theta}\geq max_{\theta\in\Theta}\lVert\theta\rVert_{2}$ and
$B_{\hat{\Delta}}\geq\lVert\hat{\Delta^{t}}\rVert_{2}$ is a uniform upper
bound on the norms of stochastic subgradients. Suppose that we perform $T$
iterations of the following stochastic update starting from
$\theta^{1}=argmin_{\theta\in\Theta}\lVert\theta\rVert_{2}$:

$\theta^{t+1}=\Pi_{\Theta}\left(\theta^{t}-\eta\hat{\Delta}^{t}\right),\\\ $
(24)

where $\mathbb{E}[\hat{\Delta}^{t}|\theta^{t}]\in_{t}(\theta^{t})$, i.e.
$\hat{\Delta}^{t}$ is a stochastic subgradient of $f_{t}$ at $\theta^{t}$ and
$\Pi_{\Theta}$ projects its arguments onto $\Theta$ w.r.t. the Euclidean norm.
Then with probability $1-\delta$ over the draws of the stochastic
subgradients:

$\frac{1}{T}\sum_{t=1}^{T}f_{t}(\theta^{t})-\frac{1}{T}\sum_{t=1}^{T}f_{t}(\theta^{*})\leq
2B_{\Theta}B_{\hat{\Delta}}\sqrt{\frac{1+16ln\frac{1}{\delta}}{T}},\\\ $ (25)

where $\theta^{*}\in\Theta$ is an arbitrary reference vector.

######  Lemma 1.

Suppose that $\Theta$ is a compact convex set and the objective, and sigmoid
constraint functions $\hat{g}_{1},....,\hat{g}_{m}$ are convex (but not
$g_{1},....,g_{m}$). The three upper bounds are $B_{\Theta}\geq
max_{\theta\in\Theta}\lVert\theta\rVert_{2}$, $B_{\hat{\Delta}}\geq max_{t\in
T}\lVert\hat{\Delta}^{t}\rVert_{2}$ and $B_{\Delta}\geq max_{t\in
T}\lVert\Delta_{\lambda}^{t}\rVert_{\infty}$. If we run the client side
optimization with step size
$\eta_{\theta}:=B_{\Theta}/B_{\hat{\Delta}\sqrt{2T}}$ and
$\eta_{\lambda}:=\sqrt{(m+1)ln(m+1)/TB_{\Delta}^{2}}$, then the results would
satisfy the conditions of Theorem 1 for:

$\begin{split}&\epsilon_{\theta}=2B_{\Theta}B_{\hat{\Delta}}\sqrt{\frac{1+16ln\frac{2}{\delta}}{T}},\\\
&\epsilon_{\lambda}=2B_{\Delta}\sqrt{\frac{2(m+1)ln(m+1)(1+16ln\frac{2}{\delta})}{T}},\end{split}$
(26)

with probability $1-\delta$ over the draws of stochastic subgradients.

###### Proof.

By applying Corollary 1 to the optimization over $\theta$ and with probability
$1-2\delta^{\prime}$ over the stochastic subgradients, we can get:

$\begin{split}\frac{1}{T}\sum_{t=1}^{T}\mathcal{L}_{\theta}(\theta^{t},&\lambda^{t})-\frac{1}{T}\sum_{t=1}^{T}\mathcal{L}_{\theta}(\theta^{*},\lambda^{t})\leq\\\
&2B_{\Theta}B_{\hat{\Delta}}\sqrt{\frac{1+16ln\frac{1}{\delta^{\prime}}}{T}},\\\
\frac{1}{T}\sum_{t=1}^{T}\mathcal{L}_{\lambda}(\theta^{t},&M^{*}\lambda^{t})-\frac{1}{T}\sum_{t=1}^{T}\mathcal{L}_{\lambda}(\theta^{*},\lambda^{t})\leq\\\
&2B_{\Delta}\sqrt{\frac{2(m+1)ln(m+1)(1+16ln\frac{2}{\delta})}{T}}.\end{split}$
(27)

By taking $\delta=2\delta^{\prime}$ and using the definitions of
$\bar{\theta}$ and $\bar{\lambda}$, we can yield the claimed result. ∎

##  9 Federated Datasets

We evaluate the proposed GLocalFair using the following three datasets:

(1) CelebA Liu et al. [2018]: A collection of $200k$ celebrity face images
from the Internet that have been manually annotated. The dataset has up to 40
labels, each of which is binary-valued. We resize each image to $128$×$128$px
and create a classification task to classify whether the celebrity in the
image is smiling or not. For CelebA, each subject’s gender (male or female)
and age (young or adult) are considered sensitive attributes. We sort the
images by the celebrity ID according to Caldas et al. Caldas et al. [2018].
This results in a dataset with a total number of $200,288$ images from $9,343$
celebrities. The dataset is then randomly partitioned for training and testing
with no overlapping celebrity ID, each containing $180,429$ images from
$8,408$ celebrities and $19,859$ images from $935$ celebrities, respectively.
Finally, the training dataset is randomly distributed into $100$ silos with no
overlapping celebrity ID to emulate the cross-silo FL setup. In addition, in
each silo, we randomly split the image data in a 70-10-20 manner, where $70\%$
of the data are used for training, $10\%$ are used for validating constrained
optimization, and $20\%$ are reserved for testing client fairness.

(2) Adult Income Dua and Graff [2017]: A tabular dataset that is widely
investigated in machine learning fairness literature. It contains $48,842$
samples with $14$ attributes. We create a binary classifier to classify
whether a subject’s income is above $50k$ or not. In this dataset, race (white
or non-white) and gender (male or female) are used as the sensitive
attributes. We keep $10\%$ of the remaining dataset for testing the global
model. We distribute the rest of the dataset among four clients where each
client holds one of the four possible combinations (i.e., white male, nonwhite
male, white female, and non-white female) of two sensitive groups. Clients use
the same 70-10-20 data split as the previous dataset.

(3) UTK Faces Zhang et al. [2017]: A large-scale face dataset with more than
20,000 face images with annotations of age, gender, and ethnicity. The images
cover large variations in pose, facial expression, illumination, occlusion,
resolution, etc. We resize each image to $128$×$128$px and create a binary
classification task of classifying young (less than 40 years) and old (older
than 40 years). Here, race (white or non-white) and gender (male or female)
are used as the sensitive attributes. We divide the dataset into 90% training
and 10% testing data. We randomly distributed the training dataset to 20
clients where each client’s data quantity has a mean of 900 and a variance of
250.

##  10 Model Description

We evaluate GLocalFair on three datasets, namely, CelebA, Adult Income, and
UTK Faces. The details of the FL model used for each dataset are presented
below:

  1. (1)

CelebA: We train a Resnet34 He et al. [2016] model with a single fully
connected layer, equipped with a sigmoid activation function for binary
classification (i.e., classifying whether the celebrity in the image is
smiling or not). The input layer is modified to accommodate the image
dimensions of 128x128 pixels, characteristic of the CelebA dataset to reduce
the computation cost at the clients. We use Adam optimizer with a learning
rate of $0.01$ with $10\%$ decay rate.

  2. (2)

Adult Income: We use a Multi-layer Perceptron (MLP) model containing two
hidden layers, each with 32 and 16 units, respectively, and a single unit
output layer for binary classification (i.e., classifying whether a subject’s
income is above 50k or not). We use SGD with a learning rate of $0.01$ and
momentum of $0.9$

  3. (3)

UTK Faces: We train a MobileNetV2 Sandler et al. [2018] model with an input
size of $128$x$128$. We use Adam optimizer with a learning rate of $0.05$ with
$10\%$ decay rate.

##  11 Federated Learning Settings

In our evaluation, we adjust the FL setup in each dataset to accommodate data
from different genres. For datasets with fewer clients, we choose to use a
higher client participation rate.

  1. (1)

CelebA: We use $100$ clients to train the model for $150$ communication rounds
with a client participation rate of $0.3$ and a $\gamma$ value of $0.6$.

  2. (2)

Adult Income: We use $4$ clients to train the model for $30$ communication
rounds with a client participation rate of $0.75$ and a $\gamma$ value of
$0.6$.

  3. (3)

UTK Faces: We use $20$ clients to train the model for $150$ communication
rounds with a client participation rate of $0.4$ and a $\gamma$ value of
$0.6$.

Figure 6: Relationship between Model Fairness and Gini Coefficient Calculated
on Model Weights

##  12 Additional Evaluation

Table 4: Global and Local group fairness Evaluation for Dual-fold Fairness
Strategy

Method | Global Fairness | Local Fairness
---|---|---
EOD ($\downarrow$) | DP-Dis | Utility | EOD | DP-Dis | DIS
Gender | ($\downarrow$) | ($\uparrow$) | (Gender Group)
|  |  |  Mean($\downarrow$) |  Std($\downarrow$) |  Mean($\downarrow$) |  Std($\downarrow$) | ($\downarrow$)
Local Fairness Constraint | 0.15 | 0.16 | 81.6% |  0.21 |  0.22 |  0.22 |  0.20 | 13.1%
Fairness-aware Clustering | 0.13 | 0.12 | 77.9% |  0.25 |  0.21 |  0.27 |  0.23 | 8.9%
GLocalFair | 0.09 | 0.10 | 79.2% |  0.17 |  0.17 |  0.21 |  0.17 | 9.5%

###  12.1 Group Fairness vs. Gini Coefficients

We conduct experiments on the CelebA dataset to investigate the relationship
between the fairness of client model updates and their Gini Coefficients,
which are calculated to measure the distribution inequality of their model
weights and are used as a proxy of group fairness in this paper. To
empirically show the relationship between group fairness and Gini Coefficients
of model updates, we analyze and visualize the Gini Coefficient values for
each client’s update during the $15th$ training round and the final training
round ($150th$) in relation to their EOD values (i.e., one of the group
fairness metrics we use in the paper) of each client’s model in Figure 6. Our
observations reveal a strong correlation between the Gini Coefficient and EOD
values. Specifically, clients whose models exhibited higher EOD values also
displayed higher Gini Coefficients. This alignment suggests that biased models
are typically associated with unevenly distributed model weights under our
proposed local fairness constraints. Conversely, clients whose models boasted
lower EOD values showcased lower Gini Coefficients, indicating a more
equitable distribution of model weights. As the fairness of the models
improved at the end of the training, as evidenced by lower EOD values, the
Gini coefficients of these models also reduced.

###  12.2 Benefits of Two Fold Fairness Optimization in GLocalFair

We embrace a dual-fold strategy for fairness optimization. This strategy
encompasses both local and global dimensions of fairness. The first facet of
our approach focuses on attaining local fairness by integrating a local
fairness constraint optimization mechanism at the client level. Complementing
this, the second facet of our approach addresses global fairness through a
distinctive methodology that employs fairness-aware clustering-based
aggregation on the server side. This server-level operation aspires to
harmonize the contributions from diverse clients, ensuring that the
aggregation process mitigates the amplification of any inherent disparities.
To rigorously assess the efficacy of these individual fairness enhancement
methods, we conducted an additional experiment using the CelebA dataset.
Specifically, the experiment aims to evaluate the unique contributions of each
strategy (client-level or server-level) to the overall fairness improvement.
We use FedAvg on the server for aggregation when we use local fairness
constraint optimization only. In Table 4, we show the results of local
fairness constraint optimization and the fairness-aware clustering-based
aggregation individually. As we can observe from the table, the global
fairness measured on the global model is lower when we employ only the local
fairness constraints on the client side, but it gets improved when fairness-
aware clustering is used, showing the local fairness improvement does not
fully improve the global fairness. Similarly, when the local fairness
constraints are used, the local fairness is improved, but the discrepancy
among the client’s performance is degraded due to the use of FedAvg on the
server side. Local fairness is degraded when only the fairness-aware
clustering is used on the server side, but the discrepancy among the client’s
performance is improved. When we combine both the fairness strategies in
GLocalFair, they complement each other and improve both local and global group
fairness.

Table 5: Impact of the fairness constraint $\tau_{FPR}$ and $\tau_{FNR}$.

False Positive Rate Limit ($\tau_{FPR}$) |  False Negative Rate Limit ($\tau_{FNR}$) |  | EOD
---
($\downarrow$)
DP-Dis ($\downarrow$) |  Utility ($\uparrow$)
Gender |  Age
0.08 | 0.10 | 0.09 | 0.10 | 0.10 |  79.1%
0.10 | 0.10 | 0.13 | 0.13 | 0.15 | 78.2%
0.15 | 0.10 | 0.16 | 0.21 | 0.16 | 74.9%
0.20 | 0.10 | 0.18 | 0.19 | 0.18 | 74.3%
0.08 | 0.08 | 0.55 | 0.45 | 0.55 | 67.3%
0.08 | 0.12 | 0.10 | 0.11 | 0.13 | 76.9%
0.10 | 0.12 | 0.11 | 0.13 | 0.15 | 77.3%
0.10 | 0.15 | 0.15 | 0.15 | 0.18 | 75.1%

###  12.3 Impact of Client Fairness Constraint

At each client, we use constraint optimization on top of the learning
algorithm to enforce local fairness. In each round of training, we constrain
the false positive rate (FPR) and false negative rate (FNR) and use a
misclassification penalty for constraint violations. We evaluate GLocalFair
with different $\tau_{FPR}$ values (i.e., 0.08, 0.10, 0.15, and 0.20) and a
fixed $\tau_{FNR}$ value (i.e., $0.10$) on the CelebA dataset in Table 5. We
observe that it achieves the best performance when $\tau_{FPR}=0.08$. Although
further decreasing $\tau_{FPR}$ may seemingly increase performance, in our
experiments, we find that any $\tau_{FPR}$ values smaller than $0.08$ would
result in a trivial model that outputs the same prediction for all data
samples. On the other hand, if we increase $\tau_{FPR}$ to be larger than
$0.15$, there would be almost no effect of constraint as the model will
converge with an FPR value lower than the limit. We also test a few
combinations with varying $\tau_{FNR}$. We find that with $\tau_{FPR}$ fixed,
using a smaller $\tau_{FNR}$ value can help improve model fairness. However,
if we set both $\tau_{FNR}$ and $\tau_{FPR}$ to $0.08$, it over-constraints
the model and makes it unstable. Also, any value larger than $0.15$ for
$\tau_{FNR}$ has almost no effect on the model performance.

(a) Adult Income

(b) UTK Faces

Figure 7: Distribution of EOD and DP-Dis on individual clients for the Adult
Income and UTK Faces datasets.

###  12.4 Local Group Fairness for the Adult Income and UTK Faces Datasets

Figure 7 compares the local fairness (i.e., distribution of EOD and DP-Dis
among individual clients) of GLocalFair with FedFB and FairFed on the Adult
Income and UTK Faces dataset. Note that for a completely fair model, the
measured fairness metrics of clients should be of the same value and approach
$0$. We observe that the result of GLocalFair is more centered with lower
variance, indicating better local group fairness. Especially for the UTK Faces
gender groups (Figure 7(b)), almost $17$ out of $20$ clients have near-zero
EOD. Similar results can be observed for DP-Dis in the UTK Faces dataset as
most of the clients have less than $0.3$ disparity. For the Adult Income
dataset, we also observe that GLocalFair has the highest number of clients
that achieve the lowest EOD and DP-Dis. These results demonstrate that by
utilizing the proposed server-client co-design framework, GLocalFair can
achieve greater local group fairness among the clients compared to other group
fairness methods.

###  12.5 Fairness vs. Utility Trade-off

In the design of GLocalFair, we use $\gamma$ to control the balance between
fairness and utility of the global model. A larger $\gamma$ value would result
in a fairer model with lower utility. We evaluate GLocalFair with different
$\gamma$ values on the CelebA dataset and summarize the results in Table 6.We
can observe that as $\gamma$ increases, the utility of the model decreases
while the fairness performance saturates at $\gamma=0.6$. Therefore, we set
$\gamma=0.6$ in our major experiments, which ensures the model’s fairness
while still maintaining a good level of utility.

Table 6: Global group fairness vs. model utility with different fairness level
$\gamma$.

Fairness Indicator |  EOD ($\downarrow$) | DP-Dis | Utility
---|---|---|---
($\gamma$) | Gender | Age |  ($\downarrow$) |  ($\uparrow$)
0.2 | 0.41 | 0.43 | 0.36 | 82.5%
0.4 | 0.23 | 0.25 | 0.15 | 80.6%
0.6 | 0.09 | 0.10 | 0.10 | 79.1%
0.8 | 0.09 | 0.11 | 0.10 | 78.9%

Table 7: Customizable fairness configuration on client models.

$\tau_{FPR}$ | $\tau_{FNR}$ |  Utility ($\uparrow$) | Precision | Recall
---|---|---|---|---
0.06 | - | 80.9% | 0.95 | 0.75
0.08 | - | 78.9% | 0.94 | 0.75
- | 0.06 | 78.6% | 0.87 | 0.90
- | 0.08 | 80.5% | 0.87 | 0.88

###  12.6 Customizable Fairness Configuration on Client Models

The developed GLocalFair offers customizable fairness configurations on the
client side, aiming to cater to diverse fairness requirements. To evaluate
this feature, we intentionally alternate between fixing the FPR and FNR while
relaxing the other fairness constraint, mirroring real-world scenarios where a
preference for lower FPR or FNR varies. As shown in Table 7, across all
scenarios, we observe consistently high utility scores. constraining FPR
results in high precision values, around 0.95, indicating accurate positive
predictions. However, this often leads to a lower recall due to the
unconstrained FNR, resulting in a higher number of false negatives.
Conversely, when FNR is constrained, recall is notably high, signifying
comprehensive positive predictions. Nevertheless, in these instances,
precision tends to be lower due to the unconstrained FPR, which results in a
higher number of false positives. These findings underscore the importance and
effectiveness of customizable fairness configurations in GLocalFair, as they
enable the adaptation of model behavior to varying fairness requirements and
real-world scenarios.

Figure 8: Comparison of feature heat maps produced by different baselines and
GLocalFair.

###  12.7 Additional Interpretability Results

In Figure 8, we extend the interpretability analysis of each baseline method
using the CelebA dataset. Specifically, we show three examples from both
classes (smiling and not smiling). From the visualization results, it is clear
that GLocalFair focuses on the areas that contain less sensitive attributes,
which makes it fairer over the demographic groups compared to other baselines.