Mitigating Group Bias in Federated Learning: Beyond Local Fairness
Authors: Ganghua Wang
Ali Payani
Myungjin Lee
Ramana Rao Kompella
Venue: Accepted by TMLR ()

Abstract: The issue of group fairness in machine learning models, where certain sub-populations or groups are favored over others, has been recognized for some time. While many mitigation strategies have been proposed in centralized learning, many of these methods are not directly applicable in federated learning, where data is privately stored on multiple clients. To address this, many proposals try to mitigate bias at the level of clients before aggregation, which we call locally fair training. However, the effectiveness of these approaches is not well understood. In this work, we investigate the theoretical foundation of locally fair training by studying the relationship between global model fairness and local model fairness. Additionally, we prove that for a broad class of fairness metrics, the global model's fairness can be obtained using only summary statistics from local clients. Based on that, we propose a globally fair training algorithm that optimizes the fairness-regularized empirical loss. Real-data experiments demonstrate the promising performance of our proposed approach for enhancing fairness while retaining high accuracy compared to locally fair training methods.

Published in Transactions on Machine Learning Research (09/2024)
Mitigating Group Bias in Federated Learning: Beyond Local Fairness
Ganghua Wang
wang90199hmmn.co edu
School of Statistics University of Minnesola
Ali Payani Cisco Research
apoyani@@isco.com com
Myungjin Lee Cisco Reseorch
myungjle@ciscoo.com
Ramana Kompella
rkompel@@cisco.com
Cisco Research
Reviewed on OpenReview:
https: openreview. net/ forum? id= ddnzet
Abstract
The issue of group fairness in machine learning models, where certain sub-populations or groups are favored over others, has been recognized for some time. While many mitigation strategies have been proposed centralized learning, many these methods are not directly applicable federated learning, where data is privately stored on multiple clients. To address this, many proposals try to mitigate bias the level of clients before aggregation, which we call locally fair training. However, the effectiveness of those approaches is not well understood. In this work, we investigate the theoretical foundation of locally fair training by studying the relationship between global model fairness and local model fairness. Additionally, we prove that for a broad class of fairness metries, the global model's fairness can be obtained using only summary statistics from local clients. Based on that, we propose a globally fair training amgorithm that optimizes the lairness-regulariieed empirical loss. Real-data experiments demonstrate the promising performance of our proposed approach for enhancing fairness while retaining high accuracy compared to locally fair training methods.
1 Introduction
As edge devices such mobile phones and wearable devices have been heavily involved in daily life, leveraging the enormous data collected by those devices and their computational resources to train machine learning models has attracted increasing research interest. One challenge is that datasets collected by different devices are often forbidden to be shared due to commmication costs and privacy concerns. Thus, classical centralized learning, where data is gathered and stored a central database, not suitable. To address those challenges, foderated learning (McMahan et al., 2017 Konečný al.] 2016) has been proposed to train models in a decentralized manner. In federated learning, global model distributed to multiple clients, edge devices, which update the model using their own data and send the updated model back to central server. The server then aggregates the updated models to obtain new global model and the process is repeated.
While significant progress has been made the theory and application federated learning Li et al. 2020b), most rescarch has focused on improving the prediction accuracy of the global model. As these models are increasingly being used areas that have direct impact on people's lives, such as healthcare, finance, and criminal justice [Berk] 2019 Barocas & Selbst 2016
Becker 2010), the ethical implications of these models

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have attracted lot of attention. In particular, it is crucial for the learned model to treat different groups in the population equitably. Nevertheless, it has been recognized that without careful consideration of group fairness [ the learned model may be biased (Guion 1966 Cleary 1968 Hutchinson & Mitcheell 2019) For example, the COMPAS algorithm (Angwin et al. 2016), which assigns recidivism risk scores to defendants based on their criminal history and demographic attributes, was found to have a significantly higher false positive rate for black defendants than white defendants, thereby violating the principle of equity on the basis of race. This highlights the risk of similar issues to arise in other applications such as university admissions and job screenings, which can negatively impact diversity and ultimately harm the society.
Though bias mitigation has been extensively studied in the centralized setting (Kamiran & Calders Kamishima et al.] 2012 Zafar et al. 2017 Zhang et al. 2018 Hardt et al.] 2016 Lohia et al. 2019
2012
Calon &
Haas 2020), it remains under-explored for federated learning. Many of the currently proposed a gggrrthmms
Ezzeldin ct al. 2021p, hereinafter referred as locally fair training (LFT). Because the global model the average of local models, LFT hopes the global model is fair as long as local models are fair. However, theoretical understanding of LFT is limited, such under what conditions LFT is effective. One main challenge of
try to reduce the global model's bias by minimizing the bias of local models (Abay et al.) 2020
analysis is obtaining the fairness mcasure for the global model without sharing original data across devices. In this work, we tackle this challenge for a particular class of fairness metrics.
1.1 Our Contributions
Our contributions are three-fold as summarized below.
1. We formulate the definitions of group-based and proper group-based fairness metrics. For proper group-baseds metries, the global fairness value can be expressed function of fairness-ellttdd statistics calculated by local clients solely. This property enables to calculate the global fairness value without directly accessing local datasets. In particular, those fairnes--related statistics are not local fairness values, distinct from all existing works.
2. Under our proposed group-based fairness notion, we investigate the relationship between the fairness of local and global models. We find that global fairness and local fairness do not imply each other in general. Nevertheless, for proper group-based fairness defined in Section | the global fairness value controlled by the local fairness values and the data heterogeneity level. This result explains the success of LFT methods in the setting of near-homogencous clients for common fairness metrics, such as demographic parity and equal opportunity. It supports the results in Chu et al.] (2021) that local fairness implies global fairness for homogeneous clients. Our work also provides fairness analysis based on data heterogeneity, complementing the information theory driven analysis presented in Hamman & Dutta 2023).
3.
We propose a globally fair training method named FedGFT for proper group-based metrics. FedGFT goes beyond LFT by directly solving a regularized objective function consisting of the empirical prediction loss and a penalty term for fairness. Additionally, it applies to clients with arbitrary data heterogeneity. Numerical experiments on multiple datasets show that FedGFT significantly reduces the bias of the global model while retaining high prediction accuracy.
1.2 Related Work
Fair federated learning methods. For bias mitigation methods in federated learning, one popular approach is locally fair training (LFT). For example, Abay et al. (2020) (LRW"), Chu et al. (2021). and Ezzeldin et all
(2021) ('FairFed') propose to train local models by applying centralized bias mitigation methods, such
reweighing the datasct to balance the group distribution [Kmiran & Calders 2012 and adding constraint
or penalizing term of fairness on the optimization objective function (Zafar el al. 2017 Kamishima et. al.
2012
In this paper, improving model fairness is also understood as decreasing model bias.
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Works on handling fairness in federated learning other than LFT have also been emerging. Zhang et al 2018 used a reinforcement learning approach to select clients that participate in the training with the highest local fairness and accuracy, Du et al. (2021) proposed to add a global fairness constraint to the agnostic federated learning formulation. Rodriguez- Gálvez et al. (2021) proposed to solve a crimeess-oonstainnc optimization problem. Zeng et al. (2021) ("FedFB') proposed to solve a bi-level optimization problem with the outer loop adaptively choosing fair batch representation of the training data. [Mhhrabi et al. (2022) assumed a validation dataset is available for evaluating local fairness and assigned higher weights fairer clients. [Hu et al. (2024) proposed algorithm with proved fairness guarantee under the bounded group loss, a privacy notion that can be viewed as relaxation of demographic parity or equalized odds.
Theoretical understanding of fair federated learning methods. There have been a few works to understand locally fair training recently. Zeng et al. (2021) showed that training locally fair models with foderated learning better than assembling locally fair models without iterative server-client updates, but worse than centralized training. Chu ct al. (2021) proved that for homogeneous clients and a specific fairness metric, locally fair training yields global model with a fairness guarantee.
Recently, Hamman & Dutta 2023 explore the connections between global and local fairness in FL for statistical disparity from the perspective of information theory. Specifically, motivated from the definition of statistical ical disparity, Hamman & Dutta (2023) define global disparity as the mutual information between the model response and the true label, and defines local disparity as the conditional mutual information between the model response and the true label given the sensitive attribute. Benefited from the specific form of those two definitions, Hamman & Dutta (2023) derive the sufficient and necessary condition that local disparity implies global disparity, assuming the underlying data distribution is known. In contrast, our work focuses broad class of fairness metrics (see Definition D including statistical disparity, equal opportunities, calibration, and more. We show that global fairness in general does not imply local fairness, and vice versa. Moreover, when fairness metric is proper, we provide an upper bound of global fairness by local fairness and data hetcrogencity.
Comparison with existing works. Our main distinction from existing methods lies in our theory-inspired approach to minimizing global fairness, namely solving global objective function that combines empirical loss and regularization term for global fairness. Compared to FedGFT, LFT seeks weighted sum of locally fair models, thus ils mininization objective does not exactly match the global fairness objective. As a result, LFT lacks theoretical support for achieving true global fairness, potentially leading to suboptimal outcomes. On the other hand, established approaches to directly optimize global fairness often involve solving constrained optimization problems, necessitating the implementation of bi-level optimization algorithms (e.g. Rodríguez- Galigu Gálvez et al., 2021 Zeng et al.] 2021 Chu et al., 2021 Du et al. 2021). Tn contrast, FedGFT streamlines the optimization process due to the form of our penalized objective and insights from Theorem 2 Therefore, FedGFT offers advantages such as: (1) No Bi-level Optimization. By dispensing with the need for bi-level optimization, by slightly adapting the local objective. (2) Minimal Communccttion Overhead. Our proposed algorithm incurs minimal communication costs related to fairness metrics. FedGFT only requires real numbers as additional communication cost, instead of requiring extra parameter gradients such as (Chu et al., 2021 Rodriguez-Gállee et al. 2021).
algorithm seamlessly integrates with conventional federated learning algorithms
2 Preliminaries
2.1
Federated Learning
There is a large body of literature on federated learning (Li et al. 2020a) since proposed by (Konččný et al. 2016 McMahan et al. 2017). It aims to train a global machine learning model while keeping the training data privately on edge devices, also named local clients. Suppose there are K clients in total, and the k-th
client owns ng training data (x(²) y(0 nh
Tipp where X is the predictor and Y is the response. Let I(:5-) be
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loss function, federated learning aims to solve the following empirical risk minimization problem:
K
mim
Thk (0), where Lu(0)=
((f(x :0,,
(1)
Tok
km
Here, (0) is the empirical risk of the k-th client, f(; is a parameterized model.
The original idea of federated learning training the model on each client using its local dataset for several updating steps, then aggregating the local models on the central server to obtain a global model, and repeating the above procedure until meeting the terminating conditions. More specifically, at each communication round t, the server first propagates the parameters 8 of the current global model to the clients. Then, each client will perform E' epochs of local updates to get 0% kk == = I........ K. Finally, the server will aggregate B to new global model with parameter gt+1
2.2 Group Fairness
There are many different interpretations of fairness (Caton & Haas 2020 Mehrabi et al. 2021 Li et al. 2021:
Yue al., 2023: Papadaki et al., 2022). In this paper, we focus on group fairness, which ensures that the model will not have discriminatory behavior towards certain groups. For simplicity, we consider a binary classification task with the outcome Y E {0, and sensitive group A E {0, 1}. There are two major categories of group fairness quantification /Corbett-Davies. & Goel 2018). The first category is based on the classification parity, which mcans a mcasure of the prediction error is equal across different groups. For example, statistical parity (Kamishima et al. 2012 Feldman el al. 2015 also known as demographic parity, requires that the distribution of the prediction Y conditional on the sensitive group is the same. In other words, p(Ý = 1 = 0) = P(Ý = 1|A = 1). Another example is equal opportunity HHardt et al. 2016), which requires the same true positive rate across groups, i.e. IP(Y 1|A 0, 1) IP( 1|A = 1, = =1. The second category is calibration (Pleiss et al. 2017). A model is well-calibrated or achieves test fairness if the true outcome is independent of the group given the predicted value. We note that different fairness definitions may be incompatible; actually, it is impossible to achieve multiple fairness goals simultaneously (Kleinberg et al. 2016
3 Problem Formulation
This paper considers a binary classification task with outcome Y E 10, Suppose the predictors (X1...., X,)" E Rp are p-dimensional variables. Without loss of generality, we assume the first predictor be the sensitive attribute as X1 E {0, 1}, and other predictors are non-sensitive. We consider a heterogeneous scenario that there are K clients (K > 1), and the k-th client's training data & (00 ne
Jiz is IID generated from a distribution Dz. The empirical distribution of {x(" y-( i=1 is denoted as DR. Our goal is to learn a function f : RP + [0, 1] from data, where f(X) is regarded as the predicted probability of P(Y =1 X). The accuracy of the learned function f is evaluated by the prediction risk B(I(f(X,, Y)), where E denotes expectation, and I(-.-) is a loss function, such as the cross entropy loss. As for the fairness measure, we define the following group-based fairness metrics.
Definition 1 (Group-based fairness metrics). F(J,D) D) is group-based fairness metric if it is in the form of
a(f.D) c(f.D) b(SD) d(S,D)
F(JD)-
where a(f, D) and b(f,D) are some expectations on the event {A= =}, 0}, c(f,D),
()),d( d( D) are some expectations on the event {A = 1}. Moreover, we have the range of a, b, c, d, a/b and c/d be [0, where a, b, c, d stands for four functions omitting the arguments.
The concept of group-basod fairness metrics are motivated by the observation that many widely used group fairness metrics adopt the form of the disparity of model performance among different groups, such as the confusion-matrix based probabilities Catonn & Haas 2020: Kim et al. 2020). Taking statistical disparity
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Table 1: The associated functions of three fairness metrics.
Metrics
a(f, D)
b(f, D)
D)
d(f.D)
Statistical Parity
PP( = =
P(A = 0)
P(V = 1,A-=)
P(A -11=10 1)
Equal opportunity
P(P =
P(Y = 11=0,
P(Y=1,y -1AY=1,111]]
1,4=1) - P(Y = 1,
Well-Calibration
P(Y 1, 1,A=0) 1,A=,
P(Y= =1 1,A=1)
for example, it is the difference of the probability being predicted positive for individuals from different groups. In other words, when A is binary group variable laking values in 0 and 1, the statistical disparity is measured by IP(Ý = 1|A 0) P( = 1|A = 1)1, where Y is the prediction by model f. We thus propose that the model performance given a specific group s can be typically expressed as a conditional expectation on the event that = 8, namely E( = s) for any evaluation function g. In the case of statistical disparity, ,{Y) = - and (()) is the indicator function.
Then, why our Definition D takes form of the difference of two ratios? There are two critical reasons. First, the Bayes theorem shows that F(g( = s) can always be written as the ratio of two expectations, therefore group fairness metric can be naturally written as the form in Definition 1. Second, the increase in the degree of freedom (the additionally introduced b and d in Definition D enables finer grid analysis the relationship between global and local fairness. Tn particular, we will show in the following sections that while global fairness and local fairness are not aligned for general group-fairness metric, they are closcly related if b and d are irrelevant to the model f.
In short, it is the first time a theoretical formulation is given to group-based fairness metrics. Clearly, a smaller F(f, D) indicates higher model fairness and smaller model bias. Definition inchudes many common measures, such as the following three. We can verify this by checking Table with full details in supplementary document.
Statistical Parity (SP). It is defined as F(f. D) = IP( = 1/4 = 0) - P( = 1/4 = 1)].
Equal Opportunity (EOP). F(f, P 1 0,Y 1) P( 1|A 1, = 1)).
Well-Calibration. F(f. D) = IY - 0, = 1) 1|A 1 1)}-
In practice, since the true underlying distribution is typically unknown, we take the empirical estimation DR) surrogale for the local fairness of the k-th client, and use F(J, D) as the global fairness, where D D=Σ EK Dy wk ng/n2, n DE 7ik
Recall that the learned function f is expected to be both accurate (with respect to the classification task) and fair (with respect to the sensitive group A). Locally fair training is one approach to extend bias mitigation methods from the centralized setting to the federated learning setting. Essentially, it minimizes the bias of each local client at each communication round and expects that the aggregation locally fair models will yield globally fair model. To better understand the effectiveness of locally fair training methods, we are going to study the following two fundamental questions next scctions:
1. What is the relationship between fairness of local models and the global model?
2. Is there an algorithm that directly targets improving global fairness?
The answer to the first question is local fairness does not imply global fairness in general. Nevertheless, for a proper group-based fairness metric, which is defined in Section 4 we show that global fairness can be controlled by local fairness and data heterogeneity. To our best knowledge, this the first work systematically study the relationship between local and global fairness. As for the second question, we propose such algorithm called FedGFT in Section 5
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4 Locally Fair Training
This section explores the relationship between local and global fairness, which helps us in analyzing the locally fair training methods. The idea of minimizing the biases of local models is appealing at the first. glance, based on an intuition that global fairness will be guaranteed all local models are fair. Chu et al. (2021) proved that this intuition true for homogencous clients and a special fairness metric named accuracy disparity. However, we show that it does not hold in general. All proofs are included in Appendix.
Theorem (In general, Global + local). Suppose F is a group-based fairness metric. For any 0 C there exist model and local data distributions {'Dx, k: = 1..., K} such that F(J; Dx) = 0 for all k, and F D) 2 C. Conversely, for any 0 S C S 1, there also erists amother set of S and (D k I,..., that F(f.D) = O, and Fff, D&) C' for all k. Corollary Suppose F is group-based fairness metric, then FfJ,D) connot be written as a linear combination of k=
,.....,K) K} such
Theorem means that locally fair models do not imply fair global model and vice versa. Therefore, locally fair training methods are not always effective in general. Additionally, Corollary 3 indicates that global fairness cannot be obtained from a simple average of local fairness. Simpson's 's paradox (Blyth 1972 Bickel et al. 1975 is excellent example to illustrate that the local property cannot represent that of the global, as shown in Table g Suppose college has two departments, A and B, which accept applications from high school students. Here, gender is the sensitive group, and each department is considered a client. Although the acceptance rate is the same between males and females (i.e., SP is zero) for both departments, the overall acceptance rate is significantly biased toward males.
Table 2: The admission example of gender bias. 'APLL' and 'ACPT" stands for applicants and acceptance rate.
Female
Male
Dept.
APPL
ACPT
APPL
ACPT
A
90
20%
10
20%
B
10
80%
90
80%
Total
100
26%
100
74%
The major factor that may lead to the failure of LFT is data heterogeneity, as revealed in the following two theorems.
Theorem 4. Suppose F is a group-based fairness metric, f is non-degenerated, and 'D = ΣK wi Dr.k, where We is the aggregation weight. necessory and sufficient condition for F(J.D) = holds for anyy wa is that there ezists constant C such that ax/bb- cx/ d C for all k, where gk g(f Dx) for g E (a, b.
Theorem iimplie that if the global model is perfectly unbiased regardless of sample sizes of clients, then all local models are also unbiased. Note that this is assured if the data distributions of different clients are homogencous. Inspired by Theorem 4 natural idea to evaluate data heterogencity is the maximum difference of ax/bı (also cx/da) among all clients. However, those quantities involve the global model f. which is unknown before the training. Thus, we introduce the following concepts to decouple with f.
Definition 5 (Proper Group-based based fairness metrics). A group- fairness metric F(f,D) is proper if the corresponding b(f, D) and d(f, D) are degenerated with respect to f. In other words, there exist function b' such that b(f. D) = (D),,, and similarly for d(f, Definition 6 (Data heterogeneity with respect to F). For proper group-based fairness metric F, let b E. wbbr and d = Σx wpdy. The data heterogeneity coefficient is defined
b
DH( k =1 ......K)
=,...., K}) = max
b dz
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Many fairness measures proper such as SP and EOP, while calibration is not proper, indicated by Table 1 For proper metrics, DH measures the relative variation of two data-determined statistics by and da. hence reflects the influence of data heterogeneity, More importantly, DH relates the global and local fairness as follows.
Theorem 7 (Near IID, local implies global). Suppose F is proper, the dala helerogeneity coefficient of clients' data is 3, and F(f, Dx) S for all k, then F(f.D) S +
Theorem shows that the global fairness is upper-bounded by the local fairness and data heterogencity level for proper group-based fairness metries. This upper bound is light when data heterogeneily level / is small. On the one hand, it justifies the success of locally fair training methods in the region of hongeeooogsssuuss situations; on the other hand, it implics that locally fair training may fail when data distributions are highly different. Thus, together with Theorem 2 we provide a fundamental understanding of the first question asked in Section g Furthermore, the proper group-based fairness metrics provide the possibility calculate the global fairness value using information from local clients. In the next section, will utilize this observation and propose globally fair training algorithm, which answers the second question in Section 3
5 Beyond local fairness
Recall the ultimate goal is to obtain a fair and accurate global model. In the centralized setting, it is standard to minimize the empirical loss with fairness regularization (Bellamy et al. 2019 Berk et al. 2017) follows:
min L(0) T Lx() (JF((((;)))
where J() is regularization function and À is penalty parameter for the fairness term. Without the fairness regularization, Eq. 2 is reduced to Eq. [ where the gradient of the global objective function be calculated or estimated by aggregating the gradients of local objective functions Lg- FedAvg (McMahan ct al.) 2017] is inspired by this observation, which performs the gradient descent algorithm each local client and then aggregates local models. Thus, to generalize federated learning algorithms to fairness-regularizeed objective Eq. 2 the challenge is how to obtain the gradient global fairness using summary statistics from local clients. However, as we showed Corollary 3 the global fairness value cannot be simply represented by local fairness in general.
Fortunately, the next theorem shows that for proper group- group-based based fairness metric F, the gradient of Eq. 2 can be calculated from the gradients of fairness-specific local objectives.
Thcorem 8. Let b = Σx wybk, d = Ex wark; and FR ok/b bb cx/d. For a proper group-based fairness metric F, have
K
K
F
Fr, = VoJ(F(I.)) = C. C Σ VoFK, k=1
where =ignign( (FV is a consland of Fg 's.
Theorem indicates that we can apply the gradient descent algorithm to minimize Eqq 2 similar to the centralized setting. Specifically, at cach round t, the local client should optimize the following fairnoss- augmented objective
min (0) + AC 0
(3)
then the aggregation of local models will give the correct gradient descent update of the global objective function.
Motivated by Theorem we propose a globally fair training method named FedGFT and summarize it in A Algorithm We note that FedGFT can incorporate most existing FT. algorithms. While the aggregation
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Algorithm (FedGFT) Federated learning with globally fair training
Input: Communication rounds T, learning rate 7) local training epochs E, batch size B, penalty parameter
System executes:
Initialize the global model paramcters 80
for each communication round / = 1.2...... do Sample a subset S, S {1..... .K
Update the constant Cge-t + C-Csott (oontUppdtte(f-)) for each client & E St in parallel do
Receive the model parameters 0% 0t from the server 0% + Client (udene(0 CCg-1)
end
Server update global model Ø' by aggregating 0% E St, using any FL algorithm end
Return the final global model f(:30t ConstUpdate (6*):
for each client & E St in parallel do
Calculate (0')
end
F+ EKES, Wi Fs
Return (())(())
ClientUpdate (0% Cgr-1):
for each local epoch from to E do
for each batch 6 from 1 to B do
Update model parameters by any FI. algorithm with local objective Eq.
end end Return FF
method on the server side and the optimization tool on the client side remain the same, FedGFT adapts the local objective function to the fairness regularization. Morcover, FedGFT also applics to the situation where clients are purely from one group (Tor example a client with all points from Group A, and another client with all points from group B), which not allowed for LFT.
Remark 9. Many fairness metrics not differentiable. D f(x())>0.5 Au 00 is not differentiable. To employ bradient-based optimization method, a common strategy is using a surrogate, such as the softmax score E f(X &
Taking SP for example, a = P( 1)
Furthermore, prove that FedGFT will converge to a stationary point when use gradient-based optimization tools. The complete statement and proof are included in the supplementary document.
Theorem 10 (Covergence analysis). Suppose the local clients apply one-slep slochastic gradient descend to optimize Eq. 3 and the global server updates the global model by averaging a random subset of local models. Let Ht be the parameter of the global model round t. Under mild assumptions, for step-size sequence {nc, 0, o....,T 1}, have
T-1 ===
ni
min B(IVL((0))))
SC
T-- Et=
-1
- 1/:
where C' is constant independent T and Corollary 11. The choice of n. = (1/tt), > yields min B B < ((1/log(T), where O is the big-O notation. The choice of n/. 1/2) gields min E(| (L())) (loog(TTT-ll Furthermore, if the gradient descent algorithm used for optimization instead of stochastic gradient descent, then choosing nt do yields a faster rate: min T-1 (([VL[(0))12)) O(T-1).

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Remark 12 (Analysis of additional computation and communication costs of FedGFT.). Compared to classical federated learning algorithm, FedGFT involves an additional step of updating a fairness-related constant at the beginning of each federated round, as shown in Algorithm 1. In particular, this constant update step calculates Fs = ax/b- c for each client k where aş and C are specified by the fairness criterion and the local model, while and d are two universal constants. Moreover, Ok and 4 can be obtained via forward propagating the local model on the local dataset for once. As result, the additional computation cost is one round of forward propagation step, while the computation cost for typical federated learning algorithm at each federated round is E rounds of forward and backward propagation steps, where E is the number of epochs.
As for the communication cost, our originally proposed Algorithm has an addition round to synchronize the constant, as pointed out by the reviewer. Nevertheless, this communication round only transmits a single scalar, therefore the cost much smaller than transmitting model parameters. Furthermore, possible way climinate this additional communication round synchronizing the constant the same round as model parameters. Specifically, the local clients can caleulate an approximation of Fk at the end of their local training round using the local model, and pass it to the global model. In this way, do not need separate round of constant update, at the cost of inaccurate calculation of Fg.
Remark (Extension to multi-class cases). For multi-class group variable, say, (1...., C} for some positive integer C 2 2, one possible direction to generalize the definition of group- fairness metrics is considering
Σ a.(f.) af(f.D Isijss be()D) by(l.D)
F(J.D)=
where as D) and be(S, D) are some expectations on the event (A= i). Accordingly, a proper group-based fairness metric is one such that 6, is are degenerated with respect to f. We believe that by using the same techniques as the current work, we can derive similar results Theorem 2 and 7
If there is more than one sensitive attribute, say, A, E (1,...,C.) for 1,..., S, we can analogously consider
af(.D) a5(ID) 180581SKCjSC (((,D) (G
F(F.))
E Σ
There are also other ways formulate the fairness metric for multi-class attributes, such as the variance of ratios nz/b,, instead of pairwise differences. Nevertheless, the proposed FedGFT can be extended and applied to those scenarios long as a/b can be obtained using local statistics.
6 Experiments
6.1 Performance under Different Data Heterogeneity levels
Datasets. We use the following three datasets in this subsection.
1. Adult dataset Duua & Graff 2017) This dataset contains the income level and demographic attributes of 48842 people. We train a logistic regression model to predict a binary response 'Income' (high or low) with 14 continuous and categorical predictors. The predictor 'Race' (white or non-white) is considered the sensitive group.
2. COMPAS dataset (Angwin et al. 2016 This dataset includes ten demographic attributes of 6172 criminal defendants and whether they recidivate in two years. A logistic model is trained to predict recidivism, and gender is the sensitive variable.
CclebA datasct [Liu et al. 2015). This dataset contains 202, 599 face images, and each image has binary attributes. We train ResNet18 model (He al. 2016) targeting at classifying the 'Smiling' attribute (yes no); and take 'Male' (yes or no) the sensitive attribute. To speed up the training
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Table Hyper-parameters used in our experiments.
Dataset
Adult
COMPAS
CelebA
Architecture
Linear
Linear
ResNet18
Number of clients
10
10
10
Communication round
50
50
50
Batch size
256
256
64
Epoch
3
1
Optimizer
ADAM
ADAM
ADAM
Learning rate
0.002
0.01
0.001
Scheduler
N/A
N/A
MultistepLR
Weight decay
N/A
N/A
0.1
process, we randomly select 10, 000 images for training and 6, 000 for testing in each independent experiment.
Setup. There are six methods in comparison: baseline method 'FedAvg' McMahan et al. 2017). state-of-the- art fairness-aware FL 'FairFed' Ezzeldin et al. 2021 and 'FedFB' ZZnn et al. 2021), LFT methods locally reweighing 'LRW" (Abay et al.) 2020 and 'FedLFT', and our proposed globally fair method 'FodGFT' For each dataset, we first randomly split. the original dataset into three parts, training, validation, and Lest dataset. The training dataset is further split into disjoint subsets, serving as local datasets of clients. The steps of dividing the training dataset are detailed as follows. First, generate the proportion of each combination of the group variable A and response variable Y for each client from a Dirichlet distribution Dir(a). A larger a implies more homogeneous clients. Then, we randomly assign the corresponding proportion of data points to each client. Throughout this section, a takes values in 0.5, 5, and 100. After training is done, the test accuracy (using zero-one loss) and fairness metric of the global model is calculated on the test dataset. Note that we conduct the experiments using both SP and EOP as the fairness metric, respectively. For each combination of dataset, learning method, and a, we run 20 independent experiments.
The default hyper-parameterrs are listed in Table 3 The penalty parameter for 'FairFed' is chosen from {0.1, 1, 10} with cross-validation and for 'FedGFT" is chosen from {10, 20, 50}. Also, the regularization function used by FodGFT is J(z) =2
Experiment results are summarized in Table 4 We can see that FedGFT has the smallest bias among almost all situations, while the accuracy drop by using FedGFT negligible. Notably, 'FeLL'T' and 'FedFB' achieve similar fairness performance as 'FodGFT' but experience markedly degraded accuracy. 'LRW" is worse than 'FedGFT' except for EOP in the CelebA dataset. While the accuracy of FedGFT is within two standard errors compared to the best method, FedGFT significantly decreases the bias even in the highly heterogeneous case.
Results.
Discussion on the choice of hyper-parameters for FedGFT. The three datasets used in this paper differ in learning difficulty, which influences the choice of proper hyper-parameters. We also conduct the experiments that vary each hyper-parameter with communication round T 50, which ensures that the model achieves convergence at the end of training. On the COMPAS dataset, we observe that the learning rate and number of epochs will jointly determine the performance of FedGFT. When the learning rate and number of epochs is increased, it will first accelerate model convergence (less communication round), since there are more local updating steps in each round and the step size is increased. However, when the learning rate or the number of epochs is too large, will impede model training instead. In particular, a large learning rate will cause the local model to oscillate around the minimizer and hence hard converge, which is a known issue of the classical neural network training using an optimizer with a fixed learning rate. On the other hand, large number of epochs amplify the difference among local models at each communication
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Table 4: The average accuracy and bias (standard error in parentheses) on three datasets, under three
heterogeneity levels and two fairness metrics. The proposed method is marked by f.
Dataset
Adult
COMPAS
CelebA
Method
Acc (f)
SP (J)
EOP (4)
Acc(t)
SP (U)
EOP (-)
Acc (t)
SP (4)
EOP (I)
FedAvg
81.8 (0.4)
5.7 (1.0)
18.3 (3.3)
64.2 (0.5)
12.8 (1.8)
15.2 (2.0)
91.4 (0.7)
14.7 (3.2)
7.5 (2.7)
LRW
81.1 (0.5)
19 (0.4)
118 (0.5)
60.9 (1.3)
5.1 (0.8)
(1.4)
91.9 (0.4)
13.7 (1.0)
1.3 (0.7)
0.5
FairFed
80.8 (0.4)
2.0 (0.4)
10.8 (0.5)
61.2 (1.0)
5.2 (0.7)
(0.6)
91.7
13.8 (3.0)
7.7 (2.9)
FedLFT
79.3 (0.3)
3.4 (0.7)
(0.4)
61.6 (0.9)
2.3 (0.2)
(0.4)
86.2 (7.9)
6.4 (3.3)
3.7 (2.5)
FedGFT!
81.8 (0.4)
0.8 (0.1)
1.3 (0.3)
63.6 (0.6)
0.9 (0.1)
1.3 (0.2)
88.9 (3.6)
(5.1)
1.7 (2.3)
FodArg
83.3 (0.1)
6.2 (0.2)
3.6 (0.4)
66.7 (0.2)
12.4 (0.6)
12.0 (0.5)
92.0 (0.3)
13.6 (0.4)
5.8 (0.4)
LRW
83.4 (0.1)
2.2 (0.1)
10.2 (0.1)
66.3 (0.3)
3.6 (0.2)
3.8 (0.3)
91.8 (0.3)
13.8 (0.2)
0.4 (0.2)
5
FairFed
83.4 (0.1)
2.0 (0.1)
10.3 (0.1)
65.9 (0.3)
3.8 (0.4)
3.6 (0.4)
91.9 (0.3)
13.8 (0.3)
6.0 (0.5)
FedLFT
82.4 (0.2)
2.4 (0:2)
2.6 (0.2)
65.3 (0.3)
1.5 (0.1)
(0.2)
91.7 (0.4)
6.0 (1.0)
5.3 (1.3)
FodGFT!
83.1 (0.1)
0.5 (0.0)
0.5 (0.1)
66.1 (0.2)
0.5 (0.0)
0.7 (0.0)
90.7 (0.5)
4.9 (1.6)
0.7 (0.4)
FodAvg
83.3 (0.1)
6.9 (0.1)
2.3 (0.1)
67.1 (0.2)
13.4 (0.2)
13.5 (0.2)
92.0 (0.3)
13.6 (0.2)
5.7 (0.3)
LRW
83.4 (0.1)
2.4 (0.0)
10.0 (0.1)
65.9 (0.2)
2.9 (0.1)
2.8 (0.1)
91.9 (0.3)
13.8 (0.1)
0.2 (0.1)
100
FairFed
83.1 (0.1)
2.5 (0.1)
9.8 (0.1)
65,9 (0.2)
2.9 (0.2)
2.7 (0.2)
91.4 (0.4)
13.7 (0.1)
5.8 (0.3)
FedLFT
83.2 (0.1)
2.0 (0.1)
2.9 (0.1)
65.8 (0.2)
1.4 (0.1)
2.1 (0.2)
91.4 (0.4)
6.0 (0.4)
5.4 (1.2)
FedGFTt
83.3 (0.1)
0.5 (0.0)
0.6 (0.1)
66.3 (0.2)
0.5 (0.0)
0.5 (0.0)
90.8 (0.7)
6.4 (3.3)
0.3 (0.3)
FedFB
76.6 (0.3)
1.6 (1.5)
NA
54.2 (0.0)
0.1 (0.0)
NA
NA
round, causing slow convergence for the global model. Nevertheless, according to the above analysis, we may accommodate the choice of hyper-parameted by using optimizer with shrinking learning rate and an objective that can address data heterogencity. Morcover, experimental results with different combination of lyper-parameters consistently show that FedGFT outperforms other methods. Lastly, is recommended lo choose those hyper-parameters through cross-validation.
Discussion on the required conditions of FedGFT. In our originally proposed FodGFT, there are two assumptions: (1) the fairness metric is proper, (2) all clients are involved in each federated round. Here, proper group-based fairness metric plays essential role for FedGFT, while the second condition be relaxed. Specifically, for cross-device federated learning, we apply FedGFTT as long clients are independently sampled in each round. Recall that our ultimate goal is optimizing the penalized objective Eq. (2), and FedGFT shows that its gradient can be written as the linear combination of gradients from local clients. When clients are sampled independently, then we can replace the whole gradiont with its stochastic counterpart, namely the combination of gradients from those independently sampled clients. This is akin to the relationship between gradient descent algorithm and stochastic gradient descent algorithm.
6.2 Performance under the Extreme Case of Data Heterogeneity
In this subsoction, we performed experiments under the sccnario where cach client's data are purely from one group. In this case, local fairness is not well-defined. Therefore, locally fair training is not applicable, and our proposed FedGFT is preferred instead. With other settings the same as above subsection, we conduct experiments on the COMPAS dataset to corroborate our algorithm's effectiveness. The results are summarized in Table 5 From the results, FedGFT still mitigates the bias compared to FedArg, though the accuracy-fairness trade-off is worse than the situation where the clients have data from both groups
6.3 Performance under Different Client Heterogeneity Levels
In this subsection, we investigate the learning algorithm's performance given the following two scenarios: 1. there are many more clients. 2. The number of data points owned by each client is highly unequal. We conduct experiments on the COMPAS dataset and compare all six methods except FedFB.
For the first scenario, with other setting the same in Subsection 6.1 the number of clients is 100. That is, each client has tens of data points average. The results reported in Table Compared to Table
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Table The average accuracy and bias (standard error in parentheses) on the COMPAS dataset under two fairness metrics. Pure group represents the situation where clients are purely from one group; mixed group represents the case where clients have data from both groups; and X is the penally parameter.
Method
Acc
SP
EОP
FedAvg (Mixed group)
66.7 (0.2)
12.4 (0.5)
12.0 (0.5)
FedGFT (Mixed group, À 20)
66.1 (0.2)
0.5 (0.0)
0.7 (0.0)
FodAvg (Pure Group)
66.6 (0.2)
6.9 (0.5)
6.2 (0.4)
FedGFT (Pure group, À 10)
65.5 (0.3)
2.4 (0.3)
2.2 (0.2)
FedGFT (Pure group, À = 20 )
64.4 (0.4)
1.7 (0.2)
1.4 (0.1)
FodGFT (Pure group, À 100
59.0 (0.8)
1.0 (0.1)
0.5 (0.1)
Table The average accuracy and bias (standard error in parentheses) on COMPAS datasets when there are 100 clients.
Method
Ace (t)
SP (1)
EOP (J)
0.5
FedAvg
66.0 (0.2)
9.0 (0.8)
9.3 (2.0)
LRW
66.9 (0.4)
9.6 (0.5)
6.5 (1.3)
FairFed
61.9 (1.7)
5.3 (0.5)
3.9 (0.8)
FodLFT
61.6 (0.9)
0.9 (0.1)
1.7 (0.3)
FedGFT
62.2 (1.6)
0.8 (0.1)
1.1 (0.2)
5.0
FedAvg
67.4 (0.3)
12.9 (0.2)
13.0 (0.2)
LRW
66.3 (0.3)
3.7 (0.2)
3.5 (0.2)
FairFed
66.7 (0.3)
3.6 (0.2)
3.0 (0.2)
FedLFT
66.2 (0.4)
1.3 (0.1)
1.0 (0.1)
FedGFT
66.0 (0.3)
0.5 (0.0)
0.7 (0.0)
100.0
FedAvg
67.6 (0.4)
14.2 (0.1)
14.1 (0.1)
LRW
66.9 (0.2)
2.5 (0.1)
2.6 (0.1)
FairFed
66.4 (0.3)
2.3 (0.1)
2.3 (0.2)
FedLFT
66.5 (0.2)
1.1 (0.1)
1.1 (0.1)
FedGFT
66.8 (0.2)
0.5 (0.0)
0.5 (0.0)
where only 10 clients are involved, FedGFT is still the algorithm with the smallest bias. Morcover, the accuracy drop by FedGFT is similar smaller to other fairness-aware algorithis.
Next, we consider the scenario where each client can have different number of data points. With other setting the same as Subsection 6.1 we modify the data spliting step as follows. First, we generate the proportion of data points for each client from Dirichlet distribution Dir(a:. Then, for each combination of sensitive attribute and response Y, we randomly assign the corresponding proportion of data points each client. It means that the data completely homogeneous each client, which is the most favorable scenario for locally fair training methods. The parameter a takes values in 0.5, 5, and 100, and higher indicates more evenly distributed numbers of data points. For example, the median ratio of the largest and smallest number of data points is around 400 when a = 0.5, while this ratio is 1.3 for a 100.
The results of clients with varying numbers data points are reported in Table 7 Clearly, FedGFT has the smallest bias even in the homogencous setting, with comparable accuracy to FodAvg. The results, together with the experiments under different data heterogeneity levels, demonstrate that FedGFT robust and effective fairness-aware federated learning algorithm.
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Table 7:
The average accuracy and bias (standard error in parentheses) on COMPAS datasets when each client's number of data points can vary.
Method
Acc (t)
SP (4)
EOP (4)
0.5
FodAvg
65.7 (0.2)
22.4 (0.9)
23.3 (0.8)
LRW
66.4 (0.3)
3.6 (0.7)
2.6 (0.3)
FairFed
65.8 (0.3)
2.2 (0.6)
1.8 (0.4)
FedLFT
64.8 (0.4)
1.4 (0.4)
3.8 (0.8)
FedGFT
65.4 (0.3)
0.3 (0.0)
0.4 (0.1)
5.0
FedAvg
66.3 (0.3)
27.1 (0.2)
26.4 (0.4)
LRW
67.0 (0.2)
5.4 50.40..)
3.1 (0.3)
FairFed
65.9 (0.4)
4.3 (0.8)
0.9 (0.1)
FedLFT
65.3 (0.3)
0.9 (0.1)
1.4 (0.3)
FedGFT
65.3 (0.2)
0.3 (0.0)
0.4 (0.0)
100.0
FedAvg
65.5 (0.3)
26.0 (0.1)
26.5 (0.1)
LRW
66.4 (0.3)
3.4 (0.1)
3.3 (0.2)
FairFed
65.6 (0.3)
2.7 (0.4)
2.5 (0.3)
FedLFT
65.9 (0.3)
1.0 (0.2)
0.7 (0.1)
FedGFT
65.4 (0.2)
0.3 (0.0)
0.3 (0.0)
7 Conclusion
In this work, we proved that the fairness of the global model federated learning is upper-bounded by the fairness of local models and the data heterogeneity level for proper group-based based fairness metrics, thus providing theoretical support for locally fair training methods. Nevertheless, locally fair training may fail in highly heterogeneous cases. We also proposed globally fair training method called FedGFT for proper group-based fairness metries, which directly minimizes the prrresss-pnnaiiee empirical loss of the global model and can be easily incorporated with existing FL algorithms. Experiments on three real-world datasets showed that the proposed mothod can significantly reduce the model bias while retaining a similar prediction accuracy compared to the baseline.
Limitations There are several problems not fully addressed and will be interesting future work. For example, the calibration is not a proper fairness metric, thus, how to generalize results to calibration, more generally, group-based metrics, is of interest. It is also worth thinking about how to generalize the proposed method regarding multiple group variables. Moreover,
examination of the proposed method on large-scale datasets desired. For instance, Ding et al. (2021) proposed to retire the Adult dataset and use suite of larger datasets derived from US Census surveys for fair machine learning research.
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A Fairness metrics
In this appendix section, we validate that three fairness metrics (SP, EOP, and Calibration) satisfy our Furthermore, SP and EOP proper group-based fairness metric Definition
Definition |
Statistical Parity. Recall that F(J,2) = IP( = 1 = 0) (P(( 1/4 1)}. By Bayes' Theorem, we have
PPÝ = 1|A P(Ý = -,
PAA 0)
Therefore, let D)
= 1,A D(, = P(A 0), then = 1] =)))=))==(()))
Similarly, we have P(Y =|A41) c(f.D)/dff.D) where D) 1( 1, 1), d(f. 1). Thus, SP is group-based fairness metric. Furthermore, it is clear that D) and d(f, are independent. of f, hence SP is also proper group-basod fairness metric.
1/4 Y, 1)/. Using Bayes' Equal Opportunity. For EOP, F(f D) = IP( 1/4 = 0. = 1) -, Theorem again, we know
= 1, -, P( - 0
P(P- = 0,Y
P( -(=1,A=1
1, = P(P 1 1,Y
P(A=1,Y=
which aligns with Table Well-Calibration. In this case, F(JD) - IP(YY 1/A 0, 1) P(Y 1/A 1, = =1)], and
P(Y 1/A - P(Y - =)
P,A=0, 1)
P(Y = 1/A 1, = P(Y 1,,
P(A= 1-1-11 1)
We note that for calibration, D) = PP(A - 0, 1) and d(f,D) = P(A 1, 1, ==11 =1), which are functions of both function and distribution D.
B Missing Proofs in Section
Proof of Theorem 2 We first prove that fair local models do not imply a fair global model. Let ge and g be the abbreviations of gf,, Dz) and g(f, D) for E {a, b, c, (see Definition L respectively. Note that the summation in g (f. Ex Dx) can be taken out because is an expectation by Definition [ For example, a(f, D) can be written as J(x,y>< h(f, X, Y)dXaY for some function Specifically, a), = SX ) for statistical disparity, where 1,3 is indicator function. Now, the basic
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property of integral gives that
f. Σ wpDss)
Mf. YdxxaY
WeDe Σ wh I (XYYDD MU,X,Y)dxalk
Σ wrals, Db).
As a result, we only need to show that there exist a f and data distributions Dr s such that
Σx waak
Σn WKCK
(U, Σ wk Dx)
C,
(4)
En webr
Σn axde
where wk = nx/nn.
Note that wi can take an arbitrary value in [0, 1] as long as D3's are properly chosen. Furthermore, according Definition D Us 's are arbitrarily manipulable as well. Next, we will construct yı and wn 's that satisfy Eq. In particular, we consider quantities with 01/b1 = / 1, wi (1+ C)/2, and az/by cx/ds and w2 = w, //K - 1) for k = 2....,.. K. By simple calculation, when b,, d2,.... dg converge to and d1, .... ... bx converge to zero, F(J,D) converges to W1, which larger than C. It. immediately implies that there exists a proper choice satisfying Eq.
As for the converse result, the following choice suffices:
/b2 = C,02+1/0214 = 0,
/ /22 = 0, 02.2211/00 /22211111111 = C,
[(K + 1)/2]
W21 = 2 2[K/2] + 1)/2l'
[K/2]
W21+1 2| L + +1/22 ,1= o........ [K/2].
where [x]] means the floor of a number x.
Proof of Corollary 3 If the claim is false, then there exists a sequence of constants {0k, 1,..., K), such that F(J.D) D) = Σ ug F(, Dx) always holds. Now, evoking Theorem we know it is possible that F(f, D) with F(f.Da) for all k, which contradiction, and thus concludes the proof.
Proof of Theorem 4. Recall that F(S,Da, lax/be ck/dgl- From Eq. we know that
Σ. wRaK Σx WiCk
F(f. D) =
Σn wpbb
Σx wadk"
Multiplying (bd)) wxdx) the both hand sides and rearranging the above equation yields that w"Mu Mw = =0, where w (wi,...,wxy and M E R is matrix with (i. j)th element Mi a,d, begy- Thus, Mu = 0 for any w is equivalent to that a,d - big for all S i.j S K, which completes the proof.
Proof of Thcorem The local fairness condition F(f. Dx) s a gives that lax/br - cx/dal S a, thus On S (a (a++0 bb/ and we have
Waak Es whck Ex weds
I a En Wrcaba/de Ee whbn
Ex Sat Ee whdn
Ex WaCk ( (d -1 S aatB B.
Ex wybr
Ee whdy d
The last step is due to cx/da < 1 and the definition data heterogeneity coefficient. Similarly, we have a 2 (ct/dx- abbb and
Σg whak
Es week
Ex wkCebe/dk -
WKCK
Σ wpbb
Σx wedk
Σx wybb
Σx whds
+ Σx whCk ( /d by
En widi t dd -1) 2 -(a + B),
17

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Published in Transactions on Machine Learning Research (09/2024)
which concludes the proof.
C Missing Proofs in Section 5
C.1 Proof of Theorem B
For proper group-based fairness metric F, we have
F(J,D;) =
aj
Ck
bR
dx
F(S D) =
WK Okk
Ex webb
wedk
where ag and eg are functions of and DRy and by and de are functions of Dr only. Recall that = En. wn bb, d Ex wydk, and F6 ax/b /x/ dd it is straightforward verify that
Es Wkak
Es waek
F(F F1. F
Σ WkFF. wxyk
En wkbk
Therefore,
Σx wh Voak Σx wabs
Σx wh Vock
sign
sign
Σn wxdk
Voak
Vack
sign (F),(F())
Σ
Wk
we sign (ig
Finally, by chain rule, we have that
-
Σ we Fo
which completes the proof.
C.2 Convergence Analysis
We first restate the problem setup for clarity. Recall the global objective function Eq. equation
K
min L
Σ
n LA ((((((:))))
n
And the local objective functions
min (+) (0) + XCg-1FF(0).
Next, we state the training procedure with random client selection and stochastic gradient descent optimization. In particular, at the communication round t, have
g = 0¹ (0' 1 e St.
g++1 = K
01+1
KES.
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Published in Transactions on Machine Learning Research (09/2024)
where (6 [) is the stochastic gradient of Hx. & represents the stochastic batches of datasets, St is randomly selected subset of clients with cardinality M (in which client. & is selected with probability ng/n), and 7/2 is the step size.
We make the following technical assumptions often used in the optimization literature, e.g. Li et al. 2020c :
Wang et al. (2020) and the references therein. For two vectors u and v, (u, v) = u² the inner product of and v, and ||||| = (u" is the &2 norm of The gradient operator V is with respect to the model parameter 0 throughout this subsection.
Assumption 14 (Smoothness). The gradients of Lą's and J are L-Lipshitz continuous. A function f(-) is I-Lipshitz continuous for any æ; we have
[V/(()) - V S L||a - yyl-
Assumption 15 (Unbiasedness). The stochastic gradient unbiased for all clients, that is, Rgg (ax (0' {)} = VH,("") for all k = I. K.
Assumption 16 (Bounded variance). The stochastic gradient has bounded variance for all clients, namely (f E) (H)()'))) o², for all I,...,K K.
Assumption 17 (Bounded dissimilarity). There exist constant B such that for all =k=1 Wh 1, wv 2 0, have
K
K
wk V
- B
Wk V HR (0)
Assumption 18. The objective function lower bounded, L := infø L(0) > -00.
Remark 19. Assumptions 14 15 and 16 are standard in optimization literature, which ensure that the SGD update produces sufficiently large decrease in the function value, leading to the convergence. Assumption 17 ensures the convergence with data helerogeneily. Larger B indicate more severe data heterogeneily, and B = corresponds to the homogeneous case.
Remark 20. If we use GD instead of SGD, then the update of local clients will be
o 07+ 176 H (0') E St,
and the Assumptions 15 and 16 are automatically satisfied with o = 0.
Now, we restate and prove Theorem below.
Theorem 10 (Convergence result) Under Assumptions 1418 18
when the step-size sequence {ne,
0, O...., T satisfies Co 2 170 2 n we have
2t o²
min B(IVL()')) S ET- Tt TT- 1/:
+
where Co and are two constants independent of and (n,, = 0, ..., T 1}.
Proof of Theorem 10 We assume g is fixed for now and denote ge+1 = V VL Note that
E(g(+1) g++1 by Theorem By Assumption 14 we have
E[L( < L(0") + E((VL(0'), (0 gt+1 00)7 + E T. 110tt2 -19+) -11)))-001113
S <((0') + (VVL 00+1 - r.)
- @ -
-(0') (1 -m) VL)))) 21 00+1112).
(5)
19

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Published in Transactions on Machine Learning Research (09/2024)
Since all clients are independent of each other, we have
B(1Ø++ 0++1111 112) Eg (Es (110 - 100+ S Es { M
Ex -110:t2-00-111 -0-1113 -101(10tt00
v
(A(1I=((0'EE)) - VI(00111
Es
M
2n
(V))))))))
M
Ane
E& ( ( +10))11+++++1)))1111 + +02
M
4nj²
(B² + 1 +02
(6)
M
where the second and third inequalities is due to triangle inequality and Assumption 16 and the last inequality is due Assumption Plugging Eqs. equation 6 into Eq. equation 5
we have
E(L(0**)) } s L(0") ne(1 - Lm) VL((')II + 4M ( ())) +
=(00)) 4M - - (4 4M 4M''L(B' L(B ())))
+1+
= L(00) mc(1 - [_ c2mh
where q [41 + 4M1+4M-L 1)] and 0 4M2 Next, we take expectation on Eq. equation reorganize and sum it from t 0 to to 1, obtaining
T-1
T-1
Σ
ne m(--c))E((V<(1)))) (B(((P))))))))))))))))))) - I(O++1)) +
Σ c2
For sufficiently small sequeeee such that Nt 1 1/(2c) for all t, we have
T-1 E ((IVI((0))) 1-0....I-111
min
n "h ((VI(()))) B(T(OP) I((0))
Σ
c2M
t=
t=
As a result,
2E(L(80) - -1'
202 Σ
min B(IVL(0))13) <
1=0 qi
ΣlD 11 Σ 7
which concludes the proof.
Proof of Corollary 11 The first. two statements regarding the special choices of the step-size sequence
are directly obtained from Theorem 10 As for the gradient descent, when o = 0, the constant C2 in the proof
of Theorem 10 disappears. This leads
min B(IVL(0))P) < 2E(L(W)) - -L*}
IT-1 1t
which completes the proof.
D Experimental Details
For completeness, we state the algorithms of 'FedAvg', FairFed', and 'LFT' in Algorithm 2 Algorithm 3 and Algorithm 4 respectively. The hm of 'FedLFTT is the same as 'FedAvg' except that the local client performs gradient descent according to the following equation:
00 0- o (Lx
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Published in Transactions on Machine Learning Research (09/2024)
Algorithm ('FedAvg') Federated Average
Input: Communication rounds T, learning rate 7) local training epochs E.
System executes:
Initialize the global model parameters 8 for each communication round t 1.2.... do
for each client k = .,..., K in parallel do Receive the model parameters 0% 0t~1 from the server O.. ClientUpdatee(( (0:00 ,2) &
end
Server update global model 0r Σx wx
end
Return the final global model (1) ClientUpdate (0% Z):
for each local epoch from to E do
Perform gradient descent 0% + 0 0-e-1 1/ Vote-ilxx
end
Return B
Algorithm ('FairFed') Fairness-aware Federated Average
Input: Communication rounds T', learning rate 7), local training epochs E, ryper-parameter B.
System executes:
Initialize the global model parameters /
for each communication round f = ..2....T do
for each client i 1,.... in parallel do Receive the model parameters 0%.0 = 0t-1 from the server
0%. Fr. m, Client (pdate(O, Z)
end
Calculate global fairness F² + En Wamn
Aggregation weights w + exp( exp(-33FF F F(l)ur Server update global model 0r +- En w/ end
Return the final global model f(; Client (0%.0) Z):
for each local epoch from to E do
Perform any bias mitigation algorithm to this local client
end
Calculate the local fairness Ft + F(J(; R, D.) Calculate the global fairness component m /* See Ezzeldin et al. (2021
Return Or.FE Fggm,, m
21

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Published in Transactions on Machine Learning Research (09/2024)
Algorithm ('LRW" Locally reweighing
Input:
Communication rounds T, learning rate local training epochs E, penalty paramoter X.
System executes:
Initialize the global model parameters 0 for each communication round 2. T do
for each client & = ..... K in parallel do
Receive the model parameters 0 6t-1 from the server 0.E - ClieCttinttttt COiiteUp (0 2
end
Server update global model 0c Σx wxO.f.
end
Return the final global model f²)
ClientUpdate (04.0 Z):
for each local epoch e from to do
Assign each data point a score associated with its sensitive attribute See Abay et al. (2020
*/
Perform ordinary gradient descent on the weighted loss function end
Return B%
22


Reviews:
[{"review_id": "N2m6MgDYjR", "reviewer": "Reviewer_sdLf", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The relation between local and global fairness in the context of federated learning is interesting and important.\n- The proposed method is simple and seems effective.\n\nWeaknesses:\n- Proof of theorem 2 is not straight forward to me. Specifically, in equation 4, it is not obvious to me why the summation could be taken out. Basically, why is it true that $a(f,\\sum_k w_kD_k)/b(f,\\sum_k w_kD_k)=\\sum_kw_ka(f,D_k)/\\sum_kw_kb(f,D_k)$. If the authors could provide detailed derivation that will be very helpful.\n- On the other hand, consider the case when $K=1$, then $D_k=D$. Theorem 2 seems to fail under this scenario. Am I missing anything?\n- Convergence analysis seems to apply only in the setting where $E=1$.\n- As the authors mentioned in Remark 9, the authors are using the loss as the surrogate for parity. That means what is actually optimized is not the fairness metric mentioned earlier (e.g. DP, EO). As a result, the convergence bound does not directly transfer to fairness guarantee.  Hence, it's unclear how well the proposed optimization does in terms of provable fairness guarantee.\n- Insufficient comparison with prior works. The authors should add comparison with a few prior works[1,2,3] that empirically evaluate on DP and EO in FL settings.\n- Zeng et al. present a similar result between local vs global fairness. Could the authors comment on that?\n\n[1] S. Cui, W. Pan, J. Liang, C. Zhang, and F. Wang, “Addressing algorithmic disparity and performance inconsistency in federated learning,” Advances in Neural Information Processing Systems, vol. 34, 2021\n\n[2] Hu, S., Wu, Z. S., & Smith, V. (2024, April). Fair federated learning via bounded group loss. In 2024 IEEE Conference on Secure and Trustworthy Machine Learning (SaTML) (pp. 140-160). IEEE.\n\n[3] Chu, L., Wang, L., Dong, Y., Pei, J., Zhou, Z., & Zhang, Y. (2021). Fedfair: Training fair models in cross-silo federated learning. arXiv preprint arXiv:2109.05662.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this work, the authors present theoretical analysis on the difference between local and global fairness. In addition to that, the authors propose a new fair FL method to optimize the global fairness constraints. Empirical evaluations on three common group fairness benchmarks seem to suggest that the proposed method is effective.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "See weaknesses above, specifically:\n- Explanation of proof of Theorem 2.\n- More experiments and comparison with prior works.\n- Discussion with Zeng et al.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "21CkDGLKxv", "reviewer": "Reviewer_wLz4", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths:**\n* Studying and formalizing how local and global fairness notions connect/depart is an important area in federated learning literature which has attracted much interest.\n* The proposed objective and solution are simple and can be easily integrated into other approaches (please see some concerns about this below).\n\n**Weaknesses:**\nThe experimental results seem to require substantial improvements and some claims throughout the paper need more clarification and/or adjustments to be valid. Also, discussion and comparison to some relevant works are missing. \n\nIn particular:\n \n*(**Critical**) Fairness and validity of empirical comparisons/results:* \n\n * Figures $1$ and $2$ use a learning rate equal to $0.002$ but *Figure $3$ proves that this learning rate is not good enough* compared to other options. This option seems to be used for all baseline methods in Table 1 as well, which raises the questions of why this was selected and whether this option is good for training other methods.\n\n  * The communication rounds are set to $20$ for each method across all datasets. Again, it is not clear whether this is sufficient for each method to converge given that also the local epochs for all approaches are set to $1$ and the learning rate is small.\n\n * In Figure $1$, for a fixed number of rounds, the largest number of local epochs seem to perform best in most cases (especially wrt average accuracy), even for high heterogeneity which is unusual.\n \n*Clarifications on claims:*\n * \"Note that each epoch will divide local datasets into several batches and thus perform multiple steps of local update.\" Is this the correct setting for all approaches? \n\n* FedGFT adapts the local objective which indeed can be incorporated into various existing algorithms. However, this doesn't necessarily mean that fairness guarantees will automatically apply since it depends on the optimization objective of the algorithm it's added to.\n\n* \"To speed up the training process, we randomly select $10,000$ images for training and $6,000$ for testing in each replicate.\" Does this mean that the proposed approach scales badly with the data dimensionality?\n\n* The utility of applying FedGFT in situations where clients comprise a single group is unclear. (Table $6$): since FedGFT aims for global group fairness shouldn't the results for FedGFT with mixed groups align with at least one of the pure group versions? \n\n*Relevant related works:*\n * The authors should discuss how their findings compare to [1], which also formally explores the connections between global and local fairness in FL. \n\n * How does FedGFT empirically compare to [2] that can recover demographic parity or equal opportunity fairness notions?\n \n [1]  Faisal Hamman, Sanghamitra Dutta, Demystifying Local \\& Global Fairness Trade-offs in Federated Learning Using Partial Information Decomposition.\n\n [2] Shengyuan Hu, Zhiwei Steven Wu, Virginia Smith, Fair Federated Learning via Bounded Group Loss.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper explores global group fairness in federated learning, focusing on statistical parity and equal opportunity metrics. It proposes a regularised learning objective for achieving global fairness, a FedAvg-inspired algorithm to solve the objective and its convergence guarantees for a single local epoch. It also analyses the differences between local and global model fairness notions in FL settings.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see the weaknesses section.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "DDhwjcrfwn", "reviewer": "Reviewer_eYUo", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n* Well-written with sound reasoning that is easy to follow. Includes discussion of theoretical proofs with implications of each result. For example, in addition the the proof provided, demonstrates via a simple illustrative example that local fairness does not imply group fairness.\n* Proves that locally fair training and heterogeneity upper-bound the resulting global model fairness given by a proper group-based fairness metric.\n* Presents an algorithm FedGFT for globally fair training with a proper group-based fairness metric, which uses strictly local summary statistics to minimize the fairness-penalized empirical loss of the global model.  \n* Strong theoretical support for the globally optimal fair federated training algorithm, FedGFT.\n* FedGFT is applicable more generally to any proper group-based fairness metric, beyond those presented in the paper.\n* Integrates well with existing federated learning algorithms.\n* Does not require bi-level optimization, as is required by several other fair federated algorithms.\n\nWeaknesses:\n* For every federated round, FedGFT requires another round of client computation and communication to compute the constant C, which is then used in the fairness-regularized client update. This is 2x the communication steps (albeit much smaller packets being sent).\n* Experimental setup details missing:\n    * What are the client and server optimizers used for FedAvg and variants?\n    * What cohort size is used in each federated round (all clients, or some sampling)?\n    * How are evaluations carried out? Is evaluation done on just the test portion of each dataset, or is each test dataset partitioned with $\\alpha$ so that federated evaluation can be performed?\n* Lacking empirical analysis:\n    * Very little discussion of the experimental results.\n    * Ablations presented in the main text are simply a hyperparameter search and do not provide support for the method presented.\n* While FedGFT is applicable to the cross-silo federated learning setting, it is unclear how FedGFT might perform in the cross-device federated learning setting, where there is likely to be partial client participation and it may not be possible to sample the same set of clients for computing C as for computing the client updates.\n* How might FedGFT extend beyond binary classification with a single sensitive attribute?\n* How should $\\lambda$ be chosen? Is there any relation to the underlying data heterogeneity $\\alpha$?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents theoretical foundations for fair federated training. The authors categorize different fairness objectives and define the class of proper group-based fairness metrics that can be optimized for locally. They prove that global model fairness using a proper group-based fairness metric is upper bounded by local fairness and data heterogeneity across clients, which gives support for the success of locally fair training in homogeneous settings. They present an algorithm, FedGFT, for globally fair federated training achieved by augmenting the local client objective function to include a fairness regularization term. The paper includes some empirical results on a few datasets commonly used in the fairness literature.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Critical changes:\n*  While the theoretical sections of the paper are especially strong, the paper lacks empirical analysis that could greatly strengthen the work. The experimental section and discussion could be bolstered by the following changes:\n    * State how evaluation is done.\n    * State the default hyperparameters used in the results presented in Table 3 in section 6.2.\n    * The discussion in 6.3 is significantly lacking, does not reflect what is shown in the figures and does not provide the strongest experimental support for the method presenting. For example, Figure 1 demonstrates that accuracy and bias improve with increasing number of epochs. This is not reflected in the text under 6.3 “Number of epochs.” Furthermore, there is no baseline for comparing how varying the number of epochs impacts performance (i.e. in this setting does more epochs of training also benefit FedAvg?).\n    * Pages 9-11 only include figures, without enough discussion or analysis warranting their inclusion in the paper. Consider including just a few plots that are strictly necessary for assessing the performance of FedGFT across settings of interest. Arguably what is shown in Figures 1, 2 and 3 is simply the hyperparameter search, which belongs in the appendix.\n    * Far more interesting and informative ablations to depict and discuss in the main text include: how does FedGFT handle extreme forms of client heterogeneity (pure clients from a single group)? How does the data partitioning (to many more clients, to different types of clients, to uneven numbers of examples per client) change the results across all methods of comparison?\n    * Expand 6.2 to discuss the characteristics of settings in which FedGFT succeeds.\n* Discuss the added computation and communication costs of FedGFT.\n\nMinor changes:\n* Bold the best value for each setting in the table so that it’s more readable and clear how the methods compare.\n* Specify which of the methods discussed in Section 7 “Past works” are included in empirical comparison. Include the names in the table used to refer to these methods, and bold them in the text.\n* Table 6: include results for FedAvg on the pure group.\n* Include discussions about what assumptions are made of the federated setting in which FedGFT is applicable and has been applied to in the experiments done.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "qCW261Kni0", "reviewer": "Reviewer_qxFB", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n+ Group fairness in Federated Learning (FL) is a very important and often underestimated area. This paper makes significant progress by addressing group fairness in FL.\n+ The theoretical results presented in this paper are interesting and sound.\n\nWeaknesses:\n\n- Definition 1 is not very clear. Why is a ratio used?\n- The literature review is not thorough. This paper omits some important references or benchmark models, including but not limited to:\n[1] Li, T., Hu, S., Beirami, A., & Smith, V. (2021, July). Ditto: Fair and robust federated learning through personalization. In International Conference on Machine Learning (pp. 6357-6368). PMLR.\n[2] Yue, X., Nouiehed, M., & Al Kontar, R. (2023). Gifair-fl: A framework for group and individual fairness in federated learning. INFORMS Journal on Data Science, 2(1), 10-23.\n[3] Papadaki, A., Martinez, N., Bertran, M., Sapiro, G., & Rodrigues, M. (2022, June). Minimax demographic group fairness in federated learning. In Proceedings of the 2022 ACM Conference on Fairness, Accountability, and Transparency (pp. 142-159).\n[4] Zeng, Y., Chen, H., & Lee, K. (2021). Improving fairness via federated learning. arXiv preprint arXiv:2110.15545.\n\nFurthermore, the paper's formulation resembles that of [4]. The authors should elaborate more on the differences. Additionally, it is recommended that the authors include more references related to group fairness in federated learning.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a framework to mitigate bias at the level of local clients before aggregation. The goal is to mitigate the group bias. Detailed theoretical analysis is provided to support the foundation of the algorithm. Overall, this paper is good.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "+ Elaborate on Definition 1.\n+ Add more references related to group fairness in federated learning.\n+ Clearly state the difference between the proposed work and [4].\n+ Highlight the best performed models in Table 3 for clarity.", "broader_impact_concerns": "Not applicable.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper studies group fairness in federated learning, and explores the relationship between local group fairness (focusing on statistical parity and equal opportunity notions) and global group fairness. The main contribution of the paper is that it proposes a regularized objective for achieving global fairness, an algorithm for solving it with a single local epoch, and analyzing its convergence guarantees. The paper also provides results on analysis of differences between local and global model fairness notions.

The paper was reviewed by four experts on group fairness in federated learning. During the review process, key pieces of related work were identified by reviewers that had not been sufficiently discussed in the paper. After major revisions, all reviewers believe that the paper could be published given the empirical and theoretical evidence. However, the writing needs to still be revised to put the work in better perspective with respect to the related work. In particular, the work [1]-[4] needs to be discussed up front in the introduction, and the claims in the introduction, such as "We study the relationship between fairness of local and global models, for the first time revealing their underlying theoretical connection" need to be removed and replaced with statements that connect with these pieces of related work.

[1] Hamman, F., Dutta, S. Demystifying Local & Global Fairness Trade-offs in Federated Learning Using Partial Information Decomposition, 2023.

[2] Zeng, Y., Chen, H., & Lee, K. Improving fairness via federated learning, 2021.

[3] Hu, S., Wu, Z. S., & Smith, V. Fair federated learning via bounded group loss.

[4] Chu, L., Wang, L., Dong, Y., Pei, J., Zhou, Z., & Zhang, Y. Fedfair: Training fair models in cross-silo federated learning.

Additionally, Reviewer wLz4 has identified the following important issue with the training that needs to be addressed and/or acknowledged in a limitations section: "the new training parameters for all approaches seem inconsistent and irrational. For example, for the smallest dataset (COMPAS, with approx. 7K data points), the local updates and learning rate being used are quite large compared to the other datasets. I also note that in general, the proposed approach is evaluated on datasets that are very small and/or have been retired [5]."

[5] Retiring Adult: New Datasets for Fair Machine Learning. Frances Ding, Moritz Hardt, John Miller, Ludwig Schmidt

The paper is recommended to be accepted subject to the authors making these critical changes to their camera-ready revision and providing a point-by-point response to this letter. Congratulations!
Decision: Accept (poster)