FedFACT: A Provable Framework for Controllable Group-Fairness Calibration in Federated Learning
Authors: Li Zhang
Zhongxuan Han
XiaoHua Feng
Jiaming Zhang
Yuyuan Li
Chaochao Chen
Venue: NeurIPS 2025 poster ()
Keywords: Federated Learning; Fairness; Multi-Class Classification
Primary area: social_and_economic_aspects_of_machine_learning
TL;DR: A controllable federated group-fairness calibration framework that achieves global and local fairness in multi-class classification with theoretical guarantees.

Abstract: With emerging application of Federated Learning (FL) in decision-making scenarios, it is imperative to regulate model fairness to prevent disparities across sensitive groups (e.g., female, male).
Current research predominantly focuses on two concepts of group fairness within FL: *Global Fairness* (overall model disparity across all clients) and *Local Fairness* (the disparity within each client).
However, the non-decomposable, non-differentiable nature of fairness criteria pose two fundamental, unresolved challenges for fair FL: (i) *Harmonizing global and local fairness, especially in multi-class classification*; (ii) *Enabling a controllable, optimal accuracy-fairness trade-off*.
To tackle the aforementioned challenges, we propose a novel controllable federated group-fairness calibration framework, named FedFACT.
FedFACT identifies the Bayes-optimal classifiers under both global and local fairness constraints in multi-class case, yielding models with minimal performance decline while guaranteeing fairness.
To effectively realize an adjustable, optimal accuracy-fairness balance, we derive specific characterizations of the Bayes-optimal fair classifiers for reformulating fair FL as personalized cost-sensitive learning problem for in-processing, and bi-level optimization for post-processing.
Theoretically, we provide convergence and generalization guarantees for FedFACT to approach the near-optimal accuracy under given fairness levels.
Extensive experiments on multiple datasets across various data heterogeneity demonstrate that FedFACT consistently outperforms baselines in balancing accuracy and global-local fairness.

FedFACT: A Provable Framework for Controllable Group-Fairness Calibration in Federated Learning
Li Zhang', Zhongxuan Han', Xiaohua Feng', Jiaming Zhang', Yuyuan Li², Chaochao Chen
'Zhejiang University, HHangzhou Dianzi University 2hang1iz180@gmalle com, (zxhan, fengxiachua, 223213501 ju.u.e edu. cn y21i edu. en, juuccczzj.. edu. cn
Abstract
With the emerging application of Federated Learning (FL.) decision-making scenarios, imperative regulate model fairness to prevent disparities across sensitive groups (e.g., female, male). Current research predominantly focuses on concepts of group fairness within FL: Global Fairness (overall model disparity across all clients) and Local Fairness (the disparity within each client). How- ever, the non- non-decomposable. non- dinddfffernntablle nature of fairness criteria poses two fundamental, unresolved challenges for fair FL: (i) Harmonizing global and local fairness, especially multi-class setting; (ii) Enabling a controllable, opti- mal aecuracy-fairnent trade-off. To tackle these challenges, we propose a novel controllable federated group-fairness calibration framework, named FedFACT. Fed- FACT identifies the Bayes-optimal classifiers under both global and local fairness constraints, yielding models with minimal performance decline while guaranteeing fairness. Building the characterization of the optimal fair classifiers, refor- mulate fair federated learning as personalized cost-sensitive learning problem for in propoeessing and bi-level optimization for post- processing. Theoretically, we provide convergence and generalization guarantees for FedFACT to approach the near-optimal accuracy under given fairness levels. Extensive experiments multiple datasets across various data heterogeneity demonstrate that FedFACT con- sistently outperforms baselines in balancing accuracy and global-local fairness.
Introduction
Federated learning (FL) is a collaborative distributed machine learning paradigm that allows multiple
clients to jointly train shared model while preserving the privacy of their local data (55)
As
FL is increasingly adopted in high-stakes domains-healtheer
[80 65 62] finance [5.
72).
pattern recognition [52] 63 94| 86 19], and recommender systems [TO]
35
73
ensuring
fairness is imperative
prevent discrimination against demographic groups based on sensitive attributes [32 90 88 64), such as race, gender, age, etc. Although a rich literature addresses group
fairness in centralized settings 22 3 [3) these methods depend full access to the entire dataset and centralized processing, imposing excessive communication overhead and elevating privacy concerns when directly applied in the FL context.
To provide faimess guarantees for federated algorithms, recent works have concentrated on two group-fairness concepts in FL.: Globul Fairness and Local Fairness 33
25
95
18
31). Global
fairness aims to identify model that provides similar treatment to protected groups across the entire data distribution. Local fairess concerns models that mitigate disparities and deliver unbiased predictions for sensitive groups within each client's 's local data. Previous work [33 thcoretically demonstrated that, under statistical heterogeneity across clients, global and local fairness can differ,
"Corresponding author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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and both entail an inherent trade-off with predictive accuracy. As global and client-level biases can induce heterogeneous treatment disparities among sensitive groups, concurrently mitigating global and local disparities is vital for achieving group faimess in FI.. For example, in constructing a federated prediction model for clinical decision-making within hospital network [48] achieving global fairness substantially enhances performance for disadvantaged subgroups. while fairness at each hospital also carries heightened significance due to local deployment and legal requirements [21].
However, existing methods face certain challenges in controlling group fairness within FL: (i) Har- monizing global and local fairness, especially multi-class classification. Divergent sensitive- sensitive-group group distributions from client heterogeneity separate global and local faimess, thereby imposing an intrinsic trade-off [33 (25] Most fair FL approaches focus exclusively on either global or local fairness [1 18 93 23 85], thereby inevitably sacrificing the other objective and impeding the realization of both fairness criteria. Moreover, this research predominantly addresses fairness the binary case, despite the ubiquity of multiclass tasks in practical FL. scenarios. (ii) Enabling a control- lable, optimal recuracy-fairnes, trade-off with theoretical guarantee. The con-dccomosable, non- differentiable nature of group-fairness measures poses significant optimization challenges [56] 17]. Existing frameworks typically rely on surrogate fairness losses [85 31
93 18 54), yet the inevitable surrogate-fairness gap [8] [53] produces suboptimal performance and undermines convergence stability, thus hampering the controllability of the securacy-fairese trade-off.
To address these challenges, we propose novel Federated group- Saup-FAinness CalibraTion framework, named FedFACT, comprising pr-processing and post-pooessiigg approaches. Our framework is capable of ensuring controllable global and local fairness with minimal accuracy deterioration, underpinned by provable convergence and consistency guarantees. To harmonize global and local fairness, we seek to find the optimal classifier under both dual fairness constraints in the multi- class case. To this end, specific characterizations of the federated Bayes-optimal fair classifiers are established for both the in-processing and post-processing phases in FL. Building on the Bayes- optimal fair classifier's s structure, we develop efficient, strategies that realize controllable and theoretically optimal carmess--acurral trade-off. In detail, FedFACT reduces the in propcesss task into a series of personalized cost- sensitvve classification problems, and reformulates post-processing as a bi-level optimization that leverages the explicit form of the federated Bayes-optimal fair classifiers. We further derive theoretical convergence and generalization guarantees, demonstrating that methods achieve near-optimal model performance while enforcing tunable global and local fairness constraints.
federated optimization
Our extensive experiments on multiple real-world datasets verify the efficiency and effectiveness of ACT highlighting that FedFACT delivers a superior, controllable accuracy-fairnes balance while maintaining competitive classification performance compared to state-of-the-arr methods.
Our main contributions are summarized as follows:
To the best of our knowledge, we are the first propose multi-class federated group- fairness calibration framework that approaches the Bayes-optimal fair classifiers, explicitly tailored achieve a provably optimal and controllable balance between global fairness, local fairness, and accuracy.
We further develop efficient algorithms to derive optimal classifiers under global-local faimess constraints at in-processing and post-processing stages with provable convergence and consistency guarantees. The in in-processing fair classification is reduced to a sequence of personalized cost-sensitive learning problem, while the post-processing formulated as bi-level optimization, using the closed-form representation of Bayes-optimal fair classifiers.
We conduct extensive experiments on multiple datasets with various data heterogeneity. The experimental results demonstrate that FedFACT outperforms existing methods in achieving superior balances among global fairness, local fairness, and accuracy. Experiments also show that FedFACT enables the flexible adjustment of the accuracy-fairnes, trade-off in FL..
2
Related Work
Group Fairness in Machine Learning. As summarized in previous work [6 group fairness is broadly defined as the absence of prejudice or favoritism toward a sensitive group based on their inherent characteristics. Common strategies for realizing group faimess in machine learning can be

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classified into three categories: pre-, in-, and post- proseesonss methods. Pre-processing 47
42 43
approaches aim to modify training data to eradicate underlying bias before model training. In- processing 45 50 2
84 methods are developed to achieve fairness requirements 46 methods adjust the prediction results generated by a given model to adapt to fairness constraints after the training stage. However, because these methods require access to the full dataset, they are confined to mitigating disparities only at the local level.
87
by intervening during the training process. Post-processing [3] 20
Group Fairness in Federated Learning. Current methods primarily utilize in- pr-peeseesnng and post-processing strategies to address global or local fairness issues. Concerning local faimess, prior work [12] highlights potential detrimental effects of FL on the group fairness of each clients, while [8] and [85] employ unified and personalized multi-objective optimization algorithms,, respec- tively, to navigate the trade-off between local faimess and accuracy. Concerning global fairness, two main approaches are adaptive reweighting techniques [58 31 93 1 and optimizing relaxed fairness
objectives within FL. [3 75 26 generally replacing the fairness metries with surrogate functions.
Furthermore, previous work 333 offered theoretical study elucidating the divergence between local and global fairness in FL, while revealing the intrinsic trade-off' between these fairnsss objectives and accuracy. [25) formulates local and global fairness optimization into a linear program for minimal fairness cost, but does not realize the sayes-optimal balance between accuracy and fairness. [95] derives the Bayes-optimas classifier and decomposes the overall problem into client-specific optimizations, yet their approach applies only binary classification
protessocss.ing. Persistent challenges include inadequate accuracy-fairnes flexibility and limited convergence guarantee in mitigating disparities within multi-class classification across various stages of FL. training.
3 Preliminary
Notation. Denote by 1m the m-dimensional all-ones vector and by 1 1mxm the m x m all-ones matrix. Write the probability simplex as Am. = { E [0, 1Jmh |glli = 1}. and let be the i-th standard basis vector. Throughout this paper, bold lowercase letters denote vectors and bold uppercase letters denote matrices. For two equally sized matrices A and B, their Frobenius inner product is B) = Ex, 0ij bij. For positive integer n. [n] = (1).
Fairness in classification. Let AX,A, be a random tuple, where X E X' for some feature space X' C Rdd labels Y y [m]] and the discrete sensitive attribute E A. Given the randomized classifiers : X' x A + Am. the prediction Y is associated with the random outputs of li defined by PP
x) = (x). In this work, we generally focus on three popular group- fairness criteria- -emoorapphi Parity (DP) (27) Equalized Opportunity (EOP) [37 and Equalized Odds 337 in multiclass classification tasks with multiple sensitive attributes, as defined in prior
works 20 78 79.
Group fairness in FL. A federated system consists of numerous decentralized clients, so that we consider the population data distribution represented by jointly random tuple (X, A, Y, K) with total N clients. The k-tth client possesses a local data dataset D,, & E [N]. Each sample in Dj is assumed to be drawn from local distribution, represented the number of samples for client k. The Bayes score function is commonly used to characterize
{< OM,, yk. } where nig represents
the performance-optimat classifier under fairness constraints [91 13 78 20), which possesses
natural extension in the federated setting: (z, a, c) == P (Y
X = x, A = a. K k). We are
interested in both local and global fairness criteria in the FL context following [33 25 95
Definition 1. (Global Fairness) The disparity regarding sensitive groups aroused by the federated model in global data distribution P(X, A, Y) across all clients.
(Local Fairness) The disparity regarding sensitive groups aroused by the federated model when evaluated on each client's data distribution P(X, A. K).
Definition 2.
Confusion matrices. Confusion matrices encapsulate the information required to evaluate diverse
performance metries and assess group fairness constraints in classification tasks [8] 60, 46). The population confusion matrix is C E [0, 1mm with elements defined for ú, E [m] as C.,, =
P(Y i, Y = j). To capture both local and global fairness across multiple data distributions within
3

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Table 1: Example of Matiuusiion-Mtti----aaaa Group Fairness Constraints in Centralized & Federated Learning.
Faireess Criterion
Demographic parity
Equal Opportunity (EOP)
Group Constraints (Centralized)
PPP A d') P(P 1t)
P(P A d' Y P(P y
v)
Va' E A. y
Matrix Notations (Centralized)
L 2. (Ifa = - -Po)
Ln ( Ner I|a = d| 1 C2
Pvr
Na A. EY
Val É A. e
Global Fuimess (Federated)
Ee En Ei (putor = C
E. En (poa = al 1
Va C C.V.E
No C A. Vy Ey
Local Fairmess
E. E, (1aa = a'] Pa1))
E. Ex Puline
Il of)
Pot
(Federated)
Va' E.V.Ey
Va' A. E
FL, we propose the decentralized group-specific confusion matrices C E A, E [N]. with elements defined for j [m]] as C (h) == P(Y i, Y = a k).
Fairness and performance metrics. As presented in Table (EO criterion and notation explanations Appendix A), the global fairness constraints typically can be expressed by ! (h) 5", "
Ug. where 24, (h) = E. >x (Du, (h.) represents the constraints required to achieve the global fairness criterion. Similarly, the local fairness constraints are 1 uy. (h)] S c*, uh Uk &
[N], with "44 (h) = Ea DD.. C Ca)*())). For performance metries, we consider a risk metric expressed as linear function the population confusion matrix, i.c. R(h)
(R. C(h)
E. Σn P RR (a.)( This formulation has been explored in multi-label and fair classification contexts 776 81], and encompasses a variety of performance metrics, such as average recall and precision. In this paper, primarily focus standard classification error to set R mmxmm I.
4 Methodology
4.1 Federated Bayes-Optimal Fair Classifier
To investigate the optimal classifier with the group fairness guarantee within FI., we consider the situation that there is unified fairness constraint at the global level, and each client has additional local fairness restrictions in response to personalized demands. Therefore, is appropriate to consider personalized federated model to minimize classification risk and ensure both local and global fairness. Denoting the set classifiers as = {h: X + Am the federated Bayes-optimal fair classification problem can be formulated as
min R(h), HEM
(2) se",, k [N s.t. (29 S E9,
where e* 59 are positive bounds, (h) == (2 [h),,, 9e (h) = { (2h, us (h)) Elt. and the inequality applies element- wise. FL model = (hi,..., NN) comprises N local classifiers.
Before delving into the optimal solution for Problem T we present formal result on the structure [
of the federated Bayes-optimal fair classifier. Proposition indicates that the Bayes-optimal classifier can be decomposed into local deterministic classifiers for all clients. This observation provides valuable insights for the subsequent algorithm design. The proof and the discussion on feasibility are given in Appendix B.J
Proposition 1. If ( is feasüble for any positive gk and ç9, the client-wise classifier hyg in federated Bayes-optimal fair classifier h" = (hi,....h's can be expressed as the linear combination of some deterministic classifiers (h/ Vde i.e., (z(z)) = ao. (x), OA Ady-
4.2 In-processing Fair Federated Training via Cost Sensitive Learning
In this section, aim seek for the optimal solution of h (hi...., hy), where each local classifier is attribute-bling and parameterized by dk. A direct approach to solving I is to formulate
an equivalent convex-concave saddle point problem in terms of its Lagrangian L x))
N
R(h) + (1 (h) (2 +
(n
( (h) (n()) + (1)

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where À (1(1))10 and μ {uk² : Hk k= are the dual parameters. Let A := {} e R 3
(1)
(2)
]
||a||2 & Ba} and M := {u R. D M4l || _ Ba). Since random classifier, by Sion's
minimax theorem [66], the primal problem can be written as saddle-point optimization
max min L(h, p) min max X hEH
C(,,A,11)
(2)
hE AEA.IEM
The boundedness of optimal X", ur will be shown later. To derive the representation of optimal saddle point, initially focus the inner minimization optimization task, namely minnen [(h, X, jt). Proposition 2. Given non-negative X and p. then an optimal solution h- (hi..... hīy) ) the inner problem min L(h, X, 14) is realized by local deterministic classifiers (x),,, k E [N] satisfying (x) = Ey. y E arg mag JE(m]
Σ P(A = alx, k) [M* ka, a(2x,
33)
acA
where M Ma.H(a, k) = Paai En, E
(10g Aan Day - Enney, (M
(2) Mk Dakk
The proof of Proposition 2 is given in Appendix B.2 Notice that the optimal solution in remains computationally intractable, because the point- wise distributions P(A a xr, k) and the Bayes- optimal classifier y) are unknown. We reformulate the task of solving h" within a cost-sensitive learning framework, by designing sample-wise calibrated training losses for each hī (z) that can yield an equivalent objective.
Proposition 3. Let the personalized cost-sensitive loss for client be defined by m M (a, k) log ΣT- exp((s(x))))
cxp((s(x)]{)
Ex (x(y, (())
where X -> Ih is the scoring function, and M 1l (a., k) =M>"(a, + mmmm with k chosen to ensure all matrix entries are strictly positive. Denoting the optimal scoring function to minimize lu over the local data distribution as st (z), then the loss Ex is calibrated for the inner problem minneH C(h, X, p)) i.e., hī(z)) Ey, y E arg maxje[m] (s)(()))) (4)l; is equivalent to that in (
In practice, s is parameterized by OE for l [N], formulating the personalized optimization objective Le(fez) = >>> ri=1 (3 S( 0k), ak.i) the FL setting. Appendix B.3 provides the proof of Proposition 3 and further presents that the loss la in (4) is also calibrated for the unified federated Bayes-optimal fair classifier. Inspired by this property, we propose an efficient in-processing algorithm for group-fair classification within FL, as detailed in Algorithm |
At each iteration t, the personalized classifier hy, is obtained by ensembling the unified model ø with the local model oi. The ensemble weight w and its update rule balance the contributions of the unified and local models. The following theorem establishes the personalized regret bound w.r.t. the best model parameter, and further demonstrates that our algorithm achieves c-approximate stochastic saddle point.
Theorem 4. Under mild assumptions, there exist constants BR, Br, such that the following cumula- tive regret upper bound guaranteed for the ensemble personalized models:
T
11 nRT
log(2)
Σ
E(Li(hheens) 110--1,.fee [f
+ nB ThT BL
(5)
Furthermore, suppose that personalized models achieve a p'-approximate optimal response at iteration I, namely L(h', ,²) S min, L(h, X², ,x)) + p², denoting oft -Ei, p¹ /TT then the sequences of model and bounded dual parameters comprise an approximate mixed Nash equilibrium:
I T
max X .
inf h
(6)
The complete Theorem 4 with its proof is provided the Appendix B.4 The regret bound yields a convergence rate of ((1/v VT) by appropriately choosing the learning rate, reflecting the stability of the proposed algorithm. Moreover, as p' decreases with t, the algorithm will gradually converge to the optimal equilibrium. For instance, if p² x C/VI, e will also exhibit an (1/vTT) convergence rate. The generalization error between the optimal solutions of the empirical dual and primal problems under finite samples is given in Appendix B.5
5

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Algorithm 1: FedFACT (In-processing)
Input Dataasets (a ik ax. J from client k. k E K; Communication round T'; Local round Rt; Initial parameters X², ne, 0², &b wh k E K; Learning rate {1) 'dd The )
for 1 0....... do
Each client E K. in parallel do:
Ensemble unified and local model: 11 Ey. y arg mf (1-w
(x)); where Jik ff fo (z) + (1 - ff (I) and folæ) == softmax(s(x;;));
Update calibration matrix M Mi MAM kl mxmi
Update the weight wi [+1
()( 1-w exp( exp(-nnlLLa (Foz) Lx(Sor)
(=
w
Calculate update of global dual parameter Axpt1 and update local dual parameter Mk t+1
A A (f),++1 (3 21) E (Da.k " (aA) - 69, E [1,2],
, U E Mg.
(0,441 11 (3 2i) EnEA (Da.k (a())))))) - <*), iE [1, un E Uhi kime
Perform R local-batch update of le and oi guided by loss La with M .uh.
= 0 0.--1
7h V La (0**;; :B"") strtl for
of The V La (Øt+;; (B, 0, ..... ... R 1;
Send last update 20g- 00 Hf to the server;
Server do:
Server aggregates (
+ J: at+1 = 0c + Σx= Ps A
Update global dual parameter: A++1 = ll na km Ax
Send gt+1 1 to clients;
end
Return Personalized classifier h = (h1, hy), where h3 = T Ei=r
4.3 Label-Free Federated Post-Fairness Calibration based on Plug-In Approach
This section introduces post-hoc faimess approach that calibrates the classification probabilities of a pre-trained federated model. We formulate an closed-form representation of the federated Bayes optimal fair classifier under standard assumptions, and then derive the primal problem into bi-level optimization through the plug-in approach. To begin, we introduce the following assumption.
Assumption 1. (n-cooninuity). For each client k; denoting P. == IP(X A a, K k), let the put forward distribution Ta,k := n;Pay a E be absolutely continuous with respect to the Lebesgue measure restricted to Am-
Assumption T. which be met by adding minor random noises Ta,k- commonly used in the literature on post-processing fairness [3] 81 20 16). Next, we derive more explicit characterization
of federated Bayes-optimal fair classifier.
Theorem Under Assumption ] suppose that the primal problem T is feasible for any E9, çk > and all non-zero columns of each D are distinct, the attribute aware personalized classifier (ht ME(N Bayes- cotte-l under local and global fairness constraints, if hi( = ey. E
arg maxje[m] (IMA .²" (a, k)] "1(,, a, ki); where the dual parameters are determined from
(M, arg minx, H)),))).
H(,, Σ Σg KEIN aEA
Po. E
max ([M*'(a,k)) a(x,a,)) ( + + + [E(m)
Σ RE[N]
ShIlpeell
XxPx 0.0
(7)
The optimal dual parameters (X*,p) are bounded, and the optimality of X and μ respectively guarantee global fairness and local fairness.
Proof of Theorem is given Appendix B.6 Since the dual parameter 1s only related to local fairness constraints, each clients can finish update of this parameter without global aggregation. Therefore,

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the federated opyemal classification problem can be reformulated into bi-level optimization:
Σ DaPa Palflii Pa(A):= min (1, pk): PeEMs
min AEA
F(A)
(8)
where (2, == 1
nuy ii= maxye[mi ([M~* k)) "( ai k)), + E + Pu I1 ll is
the plug-in estimation of ( Considering that the nonssmootsssss of the optimization objective may lead to convergence issues in federated optimization [89]. we replace the maximum operation exp(=:/0)
in H), p&) with soft-max weight function (x)x)) = ΣE=1 Σ exp(2,/8) 2s, which reduces to the
hard-maximum temperature / 0. Denoting the relaxed local objective as F7 (X, (kk), we propose Algorithm D to solve the federated Bayes-optimal fair classifier.
Algorithm 2: FedFACT (Post-processing)
Input Datasets D; {JR.. Mr,i; on,4 ) from client & E [N]; Communication round T; Local round R: Initial parameters Xo, ,n,, & E [N]: Learning rate TJ, Pre-trained îj for f 0, 1,...... do
Each client is E [N] in parallel do:
Perform R local-batch update of pdg 1,2 with Hy (A, Mx):
14 crr+1 11,e (uk² for na M px)), r = 0, ... R
Set 1t [+1 = Mk and calculate update of X:: A V2Hrg(A ((KK,,,, (11).
t.
Server do:
Server aggregates J 1+1 11 nl Σ P Send X++1 to clients;
end
Return Classifiers fhi...., hNl. (, a) == arg maxje(m) (M** (" (a, Mx, a, k));
(
Proposition 6. The bi-level objectives ( pk), & [N] are convex and I-smooth.
Proof of Proposition 6 is given in Appendix B.7 Existing research in FL [74] shows that the L-smoothness of the local objective suffice for Algorithm to achieve O(I/VT) convergence rate. Moreover, owing to the equivalence of nested and joint minimization under convexity [66] 36), the corresponding bi-level optimization problem. Consequently, the remaining error arises from the generalization risk induced by finite sampling, which explored in Appendix B.8
approach the optimal solution of the empirical primal
4.4 Discussion
In- versus post-processing. In- and post-prose post-processing interventions play complementary roles: the former removes bias representations during training (incurring higher computational cost), and the latter adjusts faimess on model outputs with low overhead, unable to debias learned representations. Both of them support adaptable fairness calibration in resource-limited heterogeneous FL. Note that combining the in-processing and post-processing methods is theoretically unjustifiable, as the in-processing classifier is not designed to approximate the Bayes score function.
Efficiency & Privacy. Each iteration of our in- and post-processing methods requires only a single client-server interaction and is supported by convergence guarantees that demonstrate our algorithms' efficiency. This will be empirically validated in our experiments; FedFACT is also arivacy-friendde pn-processing requires sharing À alongside the unified model 6, while post- post-processing involves sharing only X. These exchanges conform to standard FL [55) and preserve data confidentiality. Furthermore, differential privacy 28 or encryption schemes can be applied to further reinforce privacy.
Experiments
To comprehensively assess the proposed FedFACT framework, we conduct extensive experiments on four publicly available real-world datasets to answer the following Research Questions (RQ): RQ1:
7

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Table 2: Overall Experimental Results.
Dataser
Compas
Adult
CelcbA
ENEM
Partiton
Method
Ace
Acc
gpalstel
Ace
glvol
Aec
g/
FodAvg
69.73
0,2766
0,3590
84.52
0.1765
0.2310
89.14
0,1435
0,1308
67.56
0,2620
0.2162
FairFad
59.29
0.1008
0,1022
80.73
0,0983
0.1454
51.85
0.0704
0,1058
60:99
0.1165
0.1733
FedF8
58,09
0.0879
0,0985
81.85
0.7751
0.1165
35.32
0.1188
0.0949
64,35
0.0814
0.1326
PCFT.
$6.53
0.0646
0,0614
0.0645
0.1455
53.83
0,0744
0,1090
61.07
0.1260
0.1189
praFiRd
59.93
D0824
0.0968
80.96
0.0911
0.0763
85.45
0.731
0.0862
62.60
B.73%
0.0805
7 0.5
Cost
64.51
0.0585
0.0941
81.09
0.0262
0.0590
$5.60
0.0314
0.0577
65.79
0,0457
0.0674
FedPACT, (1n)
61.19
0.0944
0.0761
82.05
0.0015
0.0488
$6.49
0,0205
0.0844
63.79
0.0493
0.0568
FedFACT, (Im)
61.81
0.0600
0,0636
82.44
0.0140
0.0508
85.90
0.046
0.0312
63.95
0.0434
0.0187
FedFFCTT (In)
61.17
0.0407
0,0732
82.04
0.0014
0.0401
86.15
0,0822
0,0475
62.51
0.0366
0.0413
FedEACT, (Post)
67.27
0.0128
0,0660
82.83
0.0173
0.0276
$7.25
0.0899
0.0253
66.15
0.0175
0
FedFACT: (Pos)
0.0315
82.29
0.0154
0.007>
$7.36
00127
0.01634
66.54*
0,0197
0.0240
FodPACT you (Post)
62.33
0.0139
0.0641
82:74
0.0134
0.0274
87.06
0.0093
0.0122
06.52
0.0162
0.0184
FelAAg
69.12
0.2513
0.4044
85.34
0.1596
0.5358
89:85
0.1360
0.1742
67.45
0.2037
0.3068
FairFed
61,87
0.1825
0.2448
81.85
0.1074
0,1283
82.50
0,0622
0,1415
59.81
0,0949
0.1024
FelFB
60.16
0.1284
0.1332
81.14
00949
0.0005
56.00
0.1121
0.1284
63.68
0,0780
0,2019
FCFF.
59.96
0.149%
0.1507
79.10
00528
0.99
$4.50
0.067
0,1458
61.05
0.1134
0.1425
PrafFed
60.12
0.0902
0.1062
80.12
0.0523
0.0806
85.13
0,0892
0.1152
60.84
0.0932
0.1246
Hetero
Cost
0.0773
81.04
0,0236
0.0567
$6.28
0.0495
0.0761
62.11
0.0315
0.0730
FedFACT, (In)
0.0665
$2,09
0.0121
0.0962
86.18
0.088
0.0731
62.05
0:0380
0,0577
FodFACT, (In)
60.22
0.0730
0.0753
81.19
0.0308
0.0350
86.35 58
00124
0.0426
61.96
0.947]
0.0473
FedPACT, ya (ln)
61.44
0.0676
0.0789
81.19
0.0055
0.0239
56.38
0.0355
0.0635
61.48
0,0322
0.064
FedEACT, (Post)
64.36
0.039g
0.0699
81.09
0.0017
0.0257
$7.95
0.0094
0.0384
65.32
0
0,0043
FedFACT, 1 (Post)
64.28
0.0479
0,0740
81.27
0.0049
0.0906
53.05
00112
0
65.68
0.0246
0
FedeACT,.s (Post)
61.41
0.0408
0
$1.31
0.0053
0.0293
$7.25
0.0088
0.0247
63.19
0.0201
0.0131
The best results are marked with * The second-best results underlined.
All results are the average of five repeated experiments.
We use FedAvg as the baseline for optimal accuracy, without comparing its accuracy-faireess trade-off
Does FedFACT outperform the existing methods in effectively achieving a global-local accuracy- fairness balance? RQ2: Is FedFACT capable of adjusting the trade-off between accuracy and
influence the performance of FedFACT? RQ4: How about the communication efficiency and scalability of FedFACT?
global-local fairness (sensitivity analysis)? RQ3: How do important
5.1 Datasets and Experimental Settings
Due to space limitations, the detailed information in this section is provided in Appendix C
Datasets. Experiments are conducted on four real-world datasets: Compas 22], Adult CelebA 166. and ENEM (40, which are well established for assessing faimess in FL BI
95
Baselines. For binary classification, experiments are conducted on all four datasets. We compare our method with traditional federated baselines FedAvg (55) and five state-of-the-as methods tailored for addressing global and local fairness within Fl., namely FairFed [31], FedFB [93], FCFL [8] FraFFed [85) and the method in [25] denoted as Cost in our experiments. The reason for we did not include the experiments with [95] explained in Appendix C.2 For guoui or multi- class classification, the experiments are implemented on CelebA and ENEM datasets due to label
limitations.
Data distribution. To model the statistical heterogeneity in the FL context, we investigate two data partitioning strategies: (i) Dirichlet partition: we control the distribution of the sensitive attribute at each client using a Dirichlet distribution Dir(y) as proposed by [31]. A smaller indicates greater heterogeneity across clients. (ii) Heterogeneous split: Inspired by B3), we propose partitioning method that introduces heterogeneous correlations between the sensitive attribute A and label Y. The correlation between and Y for each client is controlled by a parameter randomly sampled from [0, 1]. as detailed in Appendix
Evaluation. (i) Firstly, we partition each dataset into 60% training set and the remaining 40% for test set. (ii) Secondly, the number of clients is set to 2 in Compas, and 5 in other datasets to ensure sufficient samples for local fairness estimation. (iti) Thirdly, we evaluate the FL model with Accuracy (Acc), global fairesss metric (gglabol), ) and maximal local faimess metric among clients (glocal) with smaller values of fairness metrics indicating a fairer FL model.
5.2 Overall Comparison (RQ1)
We perform extensive experiments comparing FedFACT against existing fair FL baselines under varying statistical heterogeneity, We set ş gk 0.01 for FedFACT. The subscript g and (
8

---

Table 3: Accuracy-Faimme Balance (Sensitivity Analysis).
Dataset
Compas (In-)
Adult (In-)
Compas (Post-)
Adult (Post-)
(5".') .g))
Acc
gpotobol
grocal
Acc
goolobol
glou!
Acc
Stalobol
golood
Acc
Salooul
grood
(0.00.0.00)
61.17
0.0407
0.0732
82.04
0.0014
0.0401
67.33
0.0139
0.064]
82.74
0.0134
0,0274
(0.02.0.00)
61.39
0.0548
0.0848
82.37
0.00288
0.0458
67.49
0.0315
0.0552
82.75
0.0154
0.0255
(0.04.0.00)
61.81
0,0600
0.0836
82.44
0.0140
0.0508
67.92
0.0557
0.0692
82.83
0.0173
0.0276
(0.00.0.02)
61.23
0,0418
0.0732
82.04
0.0018
0.0409
67.40
0.0134
0.0658
82.76
0.0139
0.0278
(0.02.0.02)
61.50
0,0569
0.0895
82.41
0.0056
0,0450
67.93
0.0558
0.0624
82.77
0.0166
0.0262
(0.04.0.02)
61.63
0.0665
0.0933
82.52
0.0080
0.0479
67.95
0.0623
0.0598
82.81
0.0150
0.0256
(0.00.0.02)
61,23
0.04188
0.0732
82.04
0.0014
0.0408
67.27
0.0128
0.0660
82.79
0.0154
0.0267
(0.02.0.04)
61.66
0.0556
0.0919
82.57
0.0089
0.0442
67.95
0.0536
0.0644
82.81
0.0174
0.0283
(0.04.0.04)
62.39
0.0720
0.1105
82.66
0.0223
0.0449
68.03
0.0645
0.0598
82.81
0.0185
0.0278
represent the presence of global and local faimess constraints, respectively. It essential to note that there is an inherent trade-off between accuracy and global-local fairness.
Binary classification results. We compare FedFACT with benchmarks tailored to binary classi- fication in terms of the binary DP and EOP criteria on the four datasets. The results of DP are presented in Table 2 We also report the Pareto frontier in Appendix D.I to evaluate the ability of FedFACT to strike a securacy-fimmess balance, along with the Pareto results of binary EOP criterion. Overall, when compared to existing SOTA methods, FedFACT demonstrates superior performance in achieving a balanced trade-off between accuracy and fairness.
Multi-Class results. We illustrate how FedFACT performs multi-class prediction using CelebA and ENEM with DP and EOP constraints. As there are no established methods addressing multi-class fairness in federated learning, we conduct comparisons only between FedAvg and our proposed in- and post-processing approaches, as shown in Figure] |
More experimental details are in Appendix D.2
FolAxx
PLEPACTVPNA)
CelebA
ENEM
CelebA
ENEM
Dilfeeence
Aee Differanne
Ace Differsuce
Diffecerce
-0.06
-2.05
-0.00
-0.10 0.30
0000
-6.1
-0.8
-0.40%
Scheal
hp
FelAvg
ReEPACTYPOA)
Celeh
ENEM
CelobA
Dilference
11:0
Avc Differaace
3.00
Arc Differsare
0.00
Differeate
0.00
8000
-000
-2.01
0.25
6.66
0230
0.33
Avnd
SOP
Figure 1: MMMMMMlCCiaa Class Fair Classification Results. The top line depict global and local multiclass Demographic
Parity (DP) results, while the bottom line show global and local multiclass Equal Opportunity (EOP) outcomes.
The outcomes of the multiclass experiments do not parallel those in the binary setting, where post- processing methods vastly outperform alternatives; instead, performance is comparatively lower. This can be attributed to the paucity of local samples for fairness evaluation at individual clients, which causes post-processing under joint global and local fairness constraints to incur significant generalization error and thus fail to precisely enforce local fairness. Under these conditions, the in-training approach, leveraging globally aggregated data, offers superior fairness calibration, thereby underscoring the complementarity of the two methods we introduce.
5.3 Flexibility of Adjusting Accuracy-Fairness Trade-Off (RQ2)
To investigate the capability of FedFACT in adjusting accuracy- trade-off, we examine the Acc, galobal and fooal under different fairnsss relaxation of (g",g") with y = 0.5 on Adult and
Compas in Table 3 Here we set the local fairness levels ç* for each client to the same value, denoted as ęL. More experimental results are presented in Appendix D.3 D.3

---

Sensitivity Analysis. Table shows that, for fixed global constraint ç", reducing e diminishes both accuracy and local faireess failyss- that stricter local fairness comes at the cost of overall performance. Conversely, by keeping & constant, one can modulate global faimnsss via adjustments to 54. Note that the difference between the constraints and the fairess metrics arises due to the unavoidable generalization error with finite samples. In general, these findings substantiate our claim that FedFACT enables flexible control over the eecurcc-fairrees trade-off in FL.
5.4
Experiments (RQ3)
There no tunable parametrr in our proposed method except for the number of deterministic classifiers utilized to construct the weight classifiers. We gradually raise the number of classifiers forming the weighted classifier, starting with the most recent one and extending to the previous 10 classifiers. The detailed experimental results provided in Appendix D.4
5.5 Efficiency and Scalability Study (RQ4)
In Appendix D.5 D.5 we undertake extensive experiments to empirically demonstrate the communication efficiency and scalability to client number of the proposed method FedFACT.
6 Conclusion
This paper introduces novel controllable Federated group-FAirne FAirness CalibraTion framework called FedFACT, to ensure both group and local fairness within FL. FedFACT is proposed to learn the federated Bayes-optimal fair classifier in both in- and post-processing stages, which achieves a theoret- ically minimal accuracy loss with both fairness constraints. We developed efficient algorithms- with convergence and consistency guarantees- -that reduce fair classification to personalized cost-sensitive learning for in-pocessing and bi-level optimization for post-processing. Extensive experiments four publicly available real-world datasets demonstrate that FedFACT outperforms SOTA methods, exhibiting remarkable ability to harmonious balance between accuracy and global-local fairness.
Reproducibility Statement
Details for the experimental setting are provided the beginning of Section andd Appendix C and the code can be found at http:///gthbbbccom/iiihng/ff/f/f
Acknowledgments
This work was supported in part by the Hangzhou Key Scientific Research Plan (No. 2024SZDIA20) and National Natural Science Foundation of China (No. 62402148).
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A Fairness Criteria in Centralized and Federated Learning Setting
In this section, we provide supplementary discussion of the fairness criteria and their corresponding confusion-matrix formulations under both centralized and federated learning settings, First, in addition to the demographic parity (DP) and equal opportunity (EOP) notions introduced above, we here present the definitions of equality of odds (EO) along with their confusion-matrix representations. Next, we clarify how these fairness notions are formalized within FL, specifying the distinct faimess metrics employed at both the global and the local levels. Note that this paper adopts a subgroup- like fairness metric 81 14 to reduce the number of constraints, while our confusion-matrio representation is also applicable to the group-wise definitions of these fairness metrics 77 79].
A.1 Group Fairness Criteria
Probabilistic notations. We elucidate some probability notations in the Preliminaries and Table
Here, we use ps to denote the probability of event 5 occurring. For example, Pa := P(A = a). Py P(Y y). Pa.k == P(A a, K k),, Px = P(K Pa.y P a, Y y) Py. P.Y y, K k). and Pa.y,k IP(A a, Y y. K k).
A a). Palk == P(A
K = k).
faarreess notations. For random tuple (X, A). the prediction of the (attribute-awary classifier is defined as Ý = h(X, A). One may simply choose Ý = h X to consider the attribute- blind setting. To represent group fairness constraints, previous works [81 60) introduce the group-specific confusion matrices C", a E A to characterize the faimess constraints,
Confusion- cenfusion-matrix-
where Ci := P(Y = = j A = a). Example 1. For DP criterion,
DDP = max max vE(m) d'EA
IPY = -v/A= A= -PP = v)
where P(Y = 1/ A d') Eie(m] P( = =, =i =i Eie((m) Cg and IP = =y)
EaEA P(A a) Eig[m] p(δ y, Y =i A aa) =EOEA Eie(m) P(A Cr.y
Hence, we have
DDP max max
Σ Σ (1la= d'p P(A = a)) C.
= max max
Σ aEA
D a' .
C
ym C'EA
( a'EA
iE(m
where w E Rmxm and the y-th column elements of Da, a' are I[a a') - P(A = a) with all
other elements set to O.
Example 2. For EOP criterion,
A= DEOP max max IP(Y- - - -d,- d,Y -) -P(? = =yl = =n)
gE[m] C'EA
where P-Y- = y =11== aY, y)
Po' Ov'
and PPP = y Y = y) = Ened
Pa Ca
Pa vi
Ps y.M
Hence, we have
DEOP
max max
Σ
Po' Ia-
o']
Pe
C
max max VE(m)
EiDe y C9a)
NE(m) A'EA
Pa
Py
v.M
a'EAA
aEA
aaA
where D Dy E Rmxmm and the entry in the y-th row and y-th column is Put Iaa = a') Pe
Ea
with all
Py
other elements set to 0.
Example 3. For EOP criterion, we follow 33 to introduce the mean equalized odds (MEO) constraint, and consider its subgroup- like representation:
LEo max max. (a) - -TPR, + (a) - FPR..))
gelml e'EA? 2
where TPR (a) = P( = y Y = a), TPR, = P(Ý = Y = y) and FPRy(a) =
F(Ý = Y # y, A = FPR, = = = v).
24

---

It shows that
((TPRy())
T + (FPRy(a) - FPPR1))
2
1
Po' Ila = a')
C
+
Σ
Pa
I1a
a²
C
2
Pal.y
Py
2.
Ey Pa
Ens Fv Pus
acA
DEA vo
1
Da.
d'.y'
COa)
+
Σ
(Dad
a' Cm)
CO)):
2
aEA
aEA
where the entry in the y-th row and y-th column Po' I(a a¹
Pa
with all other elements set to
Pal.y
Pv
O for d' E [R and the entry in the y-th column is Par
1|a
a']
Pe
except
2v Patius
2ny Po;
for the y-th row with all other elements set to O for D
a' E mRmx..
A.2 Group Fairness notations in FL
As noted in the main text, fairness at the level of each client's dataset (local fairness) differs from faireess across the aggregate dataset of all clients (global, fairness). Local fairness is defined with respect to each client's individual data distribution PP(X,Y, Y K),, whereas global fairness is defined over the overall (aggregate) distribution P(X, Y, A). Motivated by approaches that employ group-specific confusion matrices for fairness [81] 60]. we propose the decentralized group-specific
confusion matrices Ca.k, a E A, [N] capture both global and local fairness across multiple data distributions within FI., with elements defined for i E [m/] as C (h) = IP(Y A = a, K = k).
Example For DP criterion, the global DP fairness metric is defined as
Dp max max yE(m) d'EA
IP -IA= P(P = y) =y)
where P(Y- = y A = d) = ΣKEIN) Eigim Pkola
Co' ** iyy
(hx). and P(γ- = 8) =
EGEA ExE(N] Eie(m) Pa,k Ca.k (hx). Hence, we have
2bP max max yE/m] d'EA
ΣΣ aEA | SE/m]
E
(Pajo- a'1 Pa. C
Σ Σ GEA
Da.. Dacy
Ca.k
max max yE[m] A'EA
where d' E Rm and the y-th column elements of a' are P(K k &IA= = == =d][[a-
a') d'A-P(A= K k) with all other elements set to O.
The local DP fairness metric for k-th client is defined as
2 max max P(γ = A ==a', K', P(Ý = y K 1b)
ME|m E'EAA
where P(Y = A = d'K K = k) = Eie(m) C and P(Ý = y K = k) =
EOEA Eie(m) P(j = y. = 1 A = a, K = k). Hence, we have
2 max max
Σ Σ (Ta= a'] Palk) Ca (hw)
Σ
D Da. d'
C (px))
max max
WEmm d'EA
VElm| d'EA
mEAsE|m]
aEd
where D d'
Rmxm and the y-th column elements of a are I|- a' P(A K k) with all other elements set to O.
25

---

Example 5. For EOP criterion, the global EOP fairness metric is defined as
Øoo- = max max ge(m] d'EA
TP(Y= y Y = y, -=A=d)- PP Y -
=1) ww)
where IP(Y- = y Y = y. A a') = ENE(NT
C Pol.t iiy (mx)), and PpO = y Y = y) =
Pal.v
EaEA ENEINi
Park C (hx). Hence, we have
By
2 Dr = max max yE(m] d'EA
Σ
Pa Maa = al)
Pa.k
Cad (hac)
Pa
Py
REN
Σ KEANE[N]
Dark a'y
Ca (aak()
max max yE[m] C'EA
where D ?m and the entry in the y-th row and y-th column is all other elements set to O.
Pe'.h: 1[a a'} Pa'.x
Paak with Py
The local EOP faimess metric for k-th client is defined
DBOP= max max PP(P- = y A = d.Y = =v v. -P) = - Y = y, K =1)
ye ml a'EA
where PP--- = y A = Y = y: K = k:) = Palik
Pul.usk by , and P(P = Y = y, K = h:) =
EAEd Pe. y, Y =i A a, K = k). Hence, we have Py.k
max max PEOP-
Σ
P l I[a a'l
Pa.k Py,k
C7,y Cit (hx)
Σ (D aEA
C (ha)) Caak((s)))
max DElm)
WE(m) R'EA
Po' y.kk
EEA
aEA
where Da E RRmxm, and the entry in the y-th row and y-th column is Pelkk I[a = a²) Pal.z.k
Po.k with Pa.k
all other elements set to O.
Example 6. For EOP criterion, the global EOP fairness metric is defined as
1 |PI( (1
26.0 = max max- (m(m] E'EA 2
Y y.A a') PP = Y = m)
P - y Y + +1) 1)))
+ +|p(88 = 1 / ッル,
where P(Y = 1 Y # A = a²) = EbEIAT Eysru Enszy Pa'in
Pulk
Cta Cill (hx), and PPPP = V #
Co (hx)). Hence, we have v) EGEA EMEINI Eyety Eusee Poy
Park
Day C (aa)))
0
1
Σ Σ REIN
D a. 2 Ca Caak(
Σ Σ aEA REINT
EO
max max yε[m] d'EA 2
where the entry in the y-th row and y-th column is d'sk Ila [1 aa] 0 for Da. E =mmxm and the entry in the y-th column is Eujey Pal.
Park with all other elements set to
Pal.x
Pv
Ila = a')
Enye, Pvj
Po.k
except
for the y-th row with all other elements set to 0 for Da E Rm
The local EOP fairness metric for k-thh client is defined
1 (
(PPC8 = u Y = y. 4.A= d', K = PP = Y = =y =x)) k))
BO
= max max [m[m] n' E.A
2
+ |P( = y Y + y, = P = Y t K = =k))) 1:1).
26

---

where P( y 1 A # y. K = k) = =Ey Enjzy Pal.ug.
Pul.
C and P( y 1 Y x
K
=)= = =REEA Engzy Puz.
Po.t
P(Y y,y = = K k). Hence, we have
Da C Ca (
2
EO
max max
Σ
D
a.k. d. cann(h+))
+
Σ aEA REIN!
Σ
yelm) d'EA 2
REAREIN
where the entry in the y-th row and y-th column is d' ma= a')
Palk with all other elements set
Pal.u.k to for D Da Rmxx, and the entry in the y-th column is Sut except for the y-th row with all other elements O 0fror Dacki E <Rmx..
Polk
Par.k
I/a ad)
Pa.k
2 2 Poy.
A.3 Other Fairness Notations
Fairness notions formulated as ratios metrics can be converted into linear constraints under cer- tain conditions, and our framework is well suited to enforce these faireess constraints in feder- ated learning environments. Specifically, the ratios metric or constraints, which are formulated
(D' (k)) S & with constant metrix D Go and group-specific confusion matrix (h)h depend on classifier h, can certainly be transformed into multiple linear constraints the sign of Ea G (h)) is unchanged for any h. When the denominator's sign is uncertain, the feasible domain of (h()) is non-convex, precluding expression
as
[G*,)"([))
linear constraints. fact, since each entry of C (h) lies in [0,1]. whenever the entries of Go are sign-consistent, the corresponding ratio constraint admits lincar-constraint representation. For example, the Calibration within Groups (CG) which was proposed in [5] and further explored in [46), is fairness metric in binary classification TP
and can be formulated as FN=-TN= voi TP" FN=
U1. where T Ta Pa FN TN are derived from binary group-specific confusion matrix C and 0 S < S and have no implicit dependence on any entries of the fairness-confusion tensor. Because this fairness criterion appears as ratio metric and every element of corresponding Go is non-negative, it admits linear-constraint representation and can be realized in our proposed distributed framework. Moreover, the ratio metrics presented in 3 also can be formulated into multiple linear constraints based the above analysis.
27

---

B Proofs and Discussion in Section 4
B.1 Proof of Proposition
This section provides the proof of Proposition The proof is primarily inspired by the characterization of the Bayes-optimal fair classifier in the centralized fair machine learning literature (e.g. Theorem 3.1 of [81] Proposition 10 of (60).
Proof. We begin by casting the primal problem into optimization problem defined the Cartesian product confusion matrices. Consider the the set achievable confusion matrices:
ClAIx = (ClAIX == (C (CC")(()))) : H.,,
where ClAIxN be the product space of all confusion matrices C corresponding to sensitive group a A and E [N] associated with given instance E H. of the problem. is clear that the performance metrie R and faimess metries 98, 9 k [N] are continuous and bounded to CLAiXN (h) == {Ca*
Convexity of ClAIx,, Let VC,, C2 ClAixN be realized by classifier tuples h1 h2. For any [0, 1]. define the mixed classifier h' = w whi + (1 w) h2 By linearity of performance and fairness metrics, its confusion matrix satisfies
C(b') = w C(b,) + (1 w) C(b2) = = C + (1- = Cw.
Thus every convex combination of C₁ and C2 lies in ClAix,, establishing convexity.
Deterministic classifiers. It can be seen that, for any linear objective éτ (ClAIxX (h)) EaEA E (Lak, C (h,))) there is a deterministic classifiers h' = (hi, ... hx) that is optimal for Φr. (see proof in B.2). By the supporting-hyperplantt theorem 9 for compact convex sets, for each point C2 = {C A.ENN E aclAAx,,, there exists nonzero collection of matri- ces Lb (I, JOEA, kE(N] constitutes a hyperplane, such that for every C = (C***) E ClAIxxN have EeEA Ex=n (Lg Cg < E "L,.k Co.k) which is precisely the desired C.. In other words, we arrive at the conclusion that each boundary point of Cl-A) xN can be achieved by deterministic classifiers h (M), Nyx.
supporting hyperplane condition at supportigg-hyperplaee
Combination of deterministic classifiers. Since C AAIx is compact and convex, we know that its extreme points fall in its boundary. By the Krein-Milman theorem [6], have that CI-IxxN is equal to the convex hull of its extreme points. We further have from Caratheodory's theorem 99] that any C E ClA|xNN can be expressed as a convex combination of d2 = Nm? points in the extreme point set, where each extreme point can be characterized by deterministic classifiers. Hence, we have proved that the optimal solution h can be represented by the convex combination of deterministic classifiers.
Discussion on feasibility, The only condition for the above theorem to hold that the feasible set is non-empty, which is clearly satisfied by the mentioned faireess constraints, DP, EOP, and EO. For these fairness criteria, the classifier that always predicts single, fixed label y' trivially meets <9 E 0, & E [N]. and hence satisfies the faimness constraints.
The number of deterministic classifiers. As for the number of deterministic classifiers required, the parameter d' in the proof scales with the number of nonzero entries in the linear performance and fairness constraints [60]. Since each matrix D in our fairness formulation is zero except for one column, we in fact need far fewer than Nm? classifiers. Moreover, under the continuity assumption [ this number can be reduced even further [8].
B.2 Proof of Proposition|
Proof. We denote Pa == P(A a). Ph == P(K = k),, Po.k == TPAA= a, K k), and P P(X K k). Consider the form Lagrangian function of federated Bayes-optimal fair classification
28

---

problem 1
C(b, >, p))
N
N
E
Pa (1 1, Ca. +
Σ (1 1
a.k
Ca.k (hx))
Ug
k=1aEA
KyEM
k=1aEA
N
(1)
- (2)
(Dui (ax)()x))) Mk
- Σ
( + )
PRiug
k=
aEA
Mg E1y
Σ
(kka + Mkuu )5t
(1)
(2)
kE[N] WEEl
N
Po.k (1-111111 (1-1) + Σ
(2
Duk +
Σ
(1)
(2) MM,un
Dack Ca C%.((1a)
uo
k=
My
EyE
MEEU
Σ 3g EuEu,
E
Σ
+ 1 m&uuss S5*
RE[N] ExE
The inner problem of Lagrangian dual ask we to solve minner C(h, X, n)) given element-wise non-negative dual parameter X and μ. which can be formulated as
N
max V(h, A,) DEAN
ΣΣ Pack (Max'(a, C*
k=1EEA
where (a,
(2) Pa.k En, (20) Xag Dask k Enyeu (H Mk
(2)
Dark
uk
The next step is to derive the optimal solution of mN V(h, A, n). For this purpose, we perform manipulations of H to reveal its clear relationship with the personalized classifier = (h1..... hy). Denote the condition distribution of X given sensitive attribute A client K l as P a.k. i.e.
P.X == P(X A = a, K = k), we have
N
V(h, A.) =
Pa.k (a, (hh))
#=10E
Σ Pa.k / fx (a
k=1aEA
Po. Pa BUAR - (a
k=EaEA
EA,
EXIA,K [m(XX M (A,
=Ex.A.K
[n(X A,K]] M "A,
FX.,
FAIX.K [n(X, A, K)] M (A,K()((X)]]]
-Ex,K
P(A - Xx Xx,)))((X,, [ (AA),/"
aeA
Σ P&
Σ LaEA
a, k)) Ma (2)x
1=1
To derive the optimal solution of the inner optimization problem, suffices to perform a pointwise maximization of the above objective: for fixed v. k the classifier h selects the label that maximizes the term inside the expectation, i.e.,
Σ A) 711 ak),
hg (x) Eyg y E arg max
jc[m]
Thus, we have finished the proof of Proposition 2
29

---

B.3 Proof of Proposition 3 and Further Exploration
In this section, we prove that the representation in (4) is calibrated for both unified and personalized inner optimization problem. We begin by presenting the following lemma.
Lemma B.1. For any categorical distribution characterized by p E Am. the minimizer of the expected risk
17
Eypp lol (qu))
Σ pi log (q:))
over all q E Am is unique and achieved p p= q.
This lemma is commonly used in the design of multiclass loss functions 77 57 59
B.3.1 Proof Proposition 3
Proof. We aim to prove that for any fixed x E X, E [N], the optimal personalized scoring function St. X + R" that minimizes the expected loss E s(z),a) over the local data distri-
bution P(X, A, Y K k) recovers the personalized federated Bayes-optimal classifier hü (2) in
Proposition 2
is equivalent show that, for any z::
arg max [s (z)l, S arg max jE(m]
Σ P(A== al2, (M k)] n(z,a,)k), REAA
jEIm]
To this end, by leveraging the properties of conditional expectation, the cost-sensitive loss is reformu- lated as function of the marginal distribution (X, K):
My (A, log exp([s(X)]
E [(a.aa)))(Xxx, s(x), -Ex,Y,
Σド= exp( (x)))
exp((s(X))1)
=-Ex,A
BYIxAA
My
Min (A, K) log
ンド=1 exp (s((()))))))
M yi (A, K) log exp[[s(X]] (1)
--Ex,
Σ P(Y y X, A, K
ES exp( (s{ (X)],)
=E(m)
m
M n.d. vid (A, K) log exp((s(X))})
-Exx
BAIx,
Σ
Ty A,
Ej= exp((s(X));)
yE(m)
i=1
m
M M (a, k)] n(o,a, log exp([s(x))1)) Σn exp([s(x))
Σ
P(A = k)z
Σ
-
aEAA
1=1
((M²)
(M (a, (c) naa, ki). we have
Denoting vx(x,k)= = EnEA IPAA = -(xx,)
vi(X, K)
log cxp((s(X)}{)
(x.xx((((U.s(().a))] = =xx,
CX.K
#= Em (X
Σ exp((s(X)],)
where Cr =)jE(m) Vz k)) can be treated as a constant for fixed x. k. According to Lemma given fixed x, kt an optimal personalized classifier s (x) minimizing the cost-sensitive loss point-wise satisfies
(;(2, k) Eye (2 Σn (x)(();()))))
(x))
Vi [m].
30

---

It presents that, for all i E [m]], since Eie(m) ak) 1,
- v((,,)))
Σ
P(A a/2, 1b)
M MM (a, a(z),,
aEA
Σ
PPA al2, k)
M (a, k) almmm a(r,a,
aEA
Σ
P(A = a|2,
MM (a, k)] 71 a, k)) +k)
.EAA
Σ .EA
P(A = a/2; xx)
a(),a,1)) + ( k))
(a,
Hence,
arg (x)) arg max mE[m]
Σ
P(A k)) (M
(a, k)] a
WE ml
The personalized classifier hz(x)) arg mingelm) recovers that in Proposition We finish the proof.
B.3.2 Exploration of Calibrated Loss for Unified Bayes-Optimal Classifier
We start from the inner optimization objective V(h, X, pt).
Vh,
Σ Σ Pa.k S (M> (hA))
k=EREA
Ex,AA [M(
= =Ex BAAIX N
A, (
=E, =Ex
ΣΣ
P(A= a, K a) MM (a)
-EAA=1
To derive the optimal solution of the inner optimization problem, it suffices to perform a pisnt-wise maximization of the above objective: for fixed æ, the classifier h(x) selects the label that maximizes the term inside the expectation, i.e.,
N ΣΣ P(A = a, K= xX)XXXXXXX (a,(a,))(())
h" (xx) = Cy, E arg max
jE[m]
aEAk
31

---

Consider the calibrated loss function in
m
My (A, log exp((s(X))
M Ail
Resga.NhAC.r.A///u/u/sss... = EX.Y
- exp exp(1s( (()())))))
exp([s(X))s (x)
=-Ex,,
EYIx,A,
M (A, K) log
2ド=1 exp((s(X)))
m
M Mid (A, K) log exp((s(X)]})
--Ex,
Σ P(Y = X, A,
Σ exp (xp((s(X)) (xx)),
VEim
log exp((s(X)]s)
-xx EAKKI
Σ
7p (X, K)
M o. (A, K)
Σe= exp((s(X)]))
DE(m
log exp([s(x)]} Σラ exp( (z())),,
B
ΣΣ
P(A 0, K x))))
M U. (a, k))
a
GEAk=1
By leveraging Lemma B.1 and employing an approach analogous to that used in the proof of Proposition it is clear that we can obtain
N
(x)) Σ
P(A= a, = (10)([MM k)] a(x,a, + a.
ac A 1=1
Hence,
arg max [s* (x)ly S arg max yE|m]
ΣΣ dEAk=1
P(A= k/a) (a k)] M m(x,,,k))
yE/m]
The unified classifier h² (x) E arg min (s))) recovers that in Proposition We have shown that the loss Eg in is also calibrated for the unified federated Bayes-optimal fair classifier.
D
B.4 The Complete Formulation of Theorem 4 with Its Proof.
In this subsection, we fully articulate Theorem through Theorem and Theorem 8 which together form an extended version of the result in Theorem Before proceeding, we first clarify some notations and assumptions.
With a little abuse of notation, let fh,ens (x) = f(x; of) f(x; ()) o() softmax(si Φ1))
and (x; 0*) softmax(ss 0*)). The local objective for
LL(f(x;0)) = Σ
Mi'a' you (a k) log(((;;)(;; ke [N],
x1=1
which is similar to [)( (() and (Jh.eennsss)) Assumption 2. The local loss function I4,"'. Tly are convex, 3-smooth and bounded by B,, to model parameters Φ& and 0, [T].
Assumption 3. Let B be sampled from the k-th device's 's local data uniformly at random. The variance of stochastic gradients in each client is bounded:
VoLi -
for k E [N]. E [7].
Assumption and are standard in the convergence analysis of federated model [1
49 74 Now
present Theorem and Theorem g which together constitute extended form of Theorem 4
32

---

Theorem 7. Under assumptions Lnd 3 for the ensemble personalized models, denoting 0" == arg ming Et= Ex= (f 0)) and Ba R3B 3 + o²0 the following cumulative global regret upper bound of all clients is guaranteed:
10 RT, BB
log(2) nw
PE I [1;[(1;;0)) &
q/u Bt.
while denoting &% == arg minos T vx)), the k-th client achieves the following person alized regret upper bound:
T
|0 + nB +
log(2) + T/u BL.
T B[LA(fk,nnd - LA (z: +1)) -
2nRT
nheT
Theorem 8, Suppose that personalized models achieve a appppooxieaaee optimal response at iteration t, namely L(h', X x.p)) S ming E(h, x²,x') + Pt, denoting p -ET Pt/T, then the sequences of model and bounded dual parameters comprise approximate mixed Nash equilibrium:
inf h- T (h' Se=pt 1653 VO /
T
max A A
Σ
(9)
B.4.1 Proof of Theorem
The proof of Theorem 7 comprises proof' of the global regret bound, and the local regret bound.
(1) Global regret upper bound. In Algorithm 1 the model parameter is updated for R iterations locally. Therefore, for any 0 E e,
Pul 102 0 |
B110+11 - 011 01 E
p 0
(10)
Denoting 9k f.r VLL 02')) and T
LV B) the local update be written
as
E
1 k
0
B 1100%
" -012
E 1100% O| 27 0 0) 00' 2 G
< E 10 oll Bn of 0)) (B 1-' 1 -2²)
Summarizing the inequality for r 0, ..... R 1, shows that
E
k
R
0
E
||
0r 812 2m
llgh 0% 0)] + n² >
(11)
T=0
By convexity, we have
Σ
10k
1r 000""
-
(1; -1(((=)))
R- 1
Σ Lx 0 (1( (1) + LI(I(x;00)) - L;(((;;0) (12)
By the 3-smoothness, indicates that [1o 1. < 23Br, and then
B[Li(((2; (()+1))]]= ELLiS(( 0 0())]] nolk gart1 06:01
3
- 1100tt1 00.011
B17 2
-ETLU((0; Bl G.
|1G 1/2
1 Aor
1
(11
1. 1 + o²)
2
3112
23B
o²
2
2
33

---

Summing up over r = ,,... r,, it presents that
(f +x))) })] (f(;; 0)) -2 (2+
1
2
RR- - Hence, summing up over r' =0, R
again, we have
1
R 1)
([LA(((a;00))) -1 (f(x;;;; 2 (22 (2+
(13)
2
Combining
12p and
13) and let 1) S B we obtain
E
0
E 0² 0||² - 2n ()(f(x:0))
21 3n + Ro²
3
(14)
From
10
we know that
N 2 Σ k=1
E g
-112
s E
- 0|l 0
Po.k [I;(f(x; - [=(((;;0))
+ 2nj² R² 3BL + R3BL + 2 Ro²
Summing over time and dividing both sides by 2Λ obtain
N
(=(f(=;0)))
E
0 0||2-R
0T 0|2
+ nRBBB + BB + o0))
24/T
Plugging in 0 = 0* and 00 = and considering the fact that 0T+1 - 0 2 0, the result turns to
10-1
3 ;/0 BBL + 00)) (15)
3
P ELLL -1((
2nRT mRBBLL
T
1=1
==
Consider the update rule of ensemble weight w in Algorithm
w exp( (h. + (wn) w exp(-"m.L1 + - (1-w)) expl th The Li (et))
w +++
Here, the update can viewed as exponentiated gradient descent on the normalized weight vector wf = (wh, wx,2) A2, and wf,, & exp (-m) i 1,2, where 2f,1 (f(:;; 8*)), 2f,2
(f(x of). A well-known regret bound in online learning 770 7T] shows that, for any (u,, u2) A2,
Σ
E LLL + (1- Σ (ui +
log(2)
+ nw
TBL
(16)
new
By the convexity of L we have Et Li
S ΣT
Plugging in #1 = 1, U2 = 0. presents that
log(2)
ELL -L,(((;;))11
+ nen T TB.
(17)
1/w
34

---

Weighted summing
over all clients and dividing both sides by T, we obtain
T
P (fk,ess) (x((((;;))) - log(2)
+ BL.
(18)
T
nw
Combining
(88 and
15
and denoting Bk = RR3B + 2 + o² obtain
N
110-12 2T
log(2)
BB +
+ mh BL.
(19)
nu
Thus, finish the proof of the global regret upper bound.
(2) Local regret upper bound. Plugging in = 0, U2 =
[66. presents that
log(2)
Σ BiLL(Kk,ens) L% (f(x; oz)1
nB.
(20)
nw
Following the proof technique of global regret upper bound, from making S 3R. we have for any Ob.
14 since o 0% and
E 1 dxll
< E 0 ox 2 2f ol) -
2 R² BB, + 3n/ R3B, + 3 ""/ Ro².
(21)
Combining (20) and 2T. and plugging Φx o7 and de Odenoting Ba R3B 2 22
the result turns to
1
log(2)
T - (z: oi))
2n nB +
nho' + my BL.
(22)
Thus, we finish the proof of the local regret upper bound.
B.4.2 Proof of Theorem
The proof of Theorem relies Lemma B.2
Lemma B.2. 170) Let fl. ff..... A + R be a sequence of convex functions that we wish to minimize on a compact convex set A. Define the bound of the convex set Ba 2 maxλeλ ||A and BG 2 ft (X'llz is uniform upper bound on the norms of the subgradients. Suppose that we perform T iterations of the following update, starting from ( argminnes ||A|]::
A = (2 (x -V5 (X))
where (X) E Of (X)) is subgradient of ft at (A, and II, projects its argument onto A w.r.t. the Euclidean norm. Then:
T
B²
M T
Σ ft (x*)
+ 1/I B
T
2m
where an arbitrary reference vector.
Proof of Theorem
Consider the empirical form of the Lagrangian function L(h, X, p),
N
6(, n) = R(h) - (2 -x0)) {9) + E(2h 222) T (B))((((())) (h) )*).
A=1
T
M MaM (xx.)))
k=1 ye,
35

---

where MAi ka 1- clear that the inner problem is linear to classifiers in the empirical case.
Pa.k Eu,eu, (2 -10)) uy -Ense E24, (
Pk ) Dak) It
From the definition in Section 4 we have ||Al|1 Bd, ||alla < Bgg Since the norm of fairness metrics
less than 2, setting the step size TJ Ba/VT. by Lemma B.2
max A T E
T
T 2 X (²) =
T Σ
2B
+ 4maT = 16B2
T
(23)
7)
T
where x", 14" are the optimal dual parameters satisfying ||A||1 S Ba, |1l4|2 S Bd-
On the other hand, according to the sub-optimal assumption on the classifier h, we have
T Σ 2 ( inf 1 E
1
1 Σ (b',"',
1 E inf i S P.
X
(²)
T
T
(24)
where x= Et= Pt/T. Combining 23 and
24
the result shows that
1
max 2
inl
C (h' X Sp+16B2
(25)
p T
T
[N]. and let À := + denote the point-wise average of dual parameters. Therefore, due to the linearity of the empirical Lagrange function to classifiers and dual parameters, (25) can be formulated as
h= h' with h4 == + Kg E
n²
max C X =u
- inf h'
s 16B2 T
I
(26)
which presents the approximate mixed Nash equilibrium of the stochastic saddle-poid problem. 0
B.5 Generalization Error For In-processing Algorithm
We begin by introducing some notations and simplifications, which are commonly employed in generalization analyses of FL [39 [8]. Without loss of generalization, let n ne present the sample number in local datasets. For any class H = (h - [m]] denote Hl [[[((x)== = = h E H} and the maximal Sapnik-Chervoonnki dimension [69] VC(H) =
maxye[m) (C(H)))
Theorem 9. If classifiers h = (hi,...,āus, hw) with dual parameters (7, form a e-saddle point
of empirical Lagrangian L(h, M), and optimal solution h- E H satisfies both global and
local fairness constraints, denoting v(n, H. 5) 2v /2VC())) log(n+1) + N2)8( Bg
R
|D 11, = max Daa.k Ba.k Ilocr Dr = Ei Pk 1 Pxl and BB Mg
m.X
max.EA
Da ug 1, 2 max.EA
Da m6
Da boa [N], then with probability at least 8,
1 2e
9 h S + v(n H. 6, Ω8 +
Ba
1 2c
9 (h) < gk + vnn H. N6 + 2 +
Ba
R(h) < R(h") + 2m52 + 2m H. 5/2) + 2e.
The proof of Theorem 9 relies on the following lemma.
Lemma B.3. Het::: X [m]. D a distribution over X' x A m of which {x,, yi 3i are i.i.d samples.
Denoting Hy = [[((h())= = y} h E H} and VC(H) = maxyelm) VC((Hy), then with probability
at least 1 - for j E [m].
(2VC(H) log(n + 11)
2 1
(h,s(h) - C,,((a)]= $2
sup
hef
n
where Fx := (f(z) = E (x) a AN. E H.j [m]].
36

---

Proof of Lemma
B.3
Let l, (z, I(y = i j). Then we have C,y(h)
E yj fi] and (h) - (zgs yzi h.). Hence, according the classical result with
respect to cost sensitive binary classification [8] with probability at least 1 5,
2V 2VC(H;) log(n 1)
2 log !
(h) (h) s 2
sup
kEFx
n
7
By the definition of VC(H), achieves the generalization bound.
B.5.1 Proof of Theorem
Let the optimal solution h" minimize the risk R(h) subjected to global and local fairness constraints
| h 5 5",
[h") 12
S ghdd With the properties of the saddle point, it is clear that
2 h A, p) S (b. t i h x p) 2 (b. -,
vh E H.
(27)
A || lelll S Ba-
(28)
Considering the global faimness constraints, we first explore its concentration, for any E H.
29, (h)
29, (h) Σ
Do. My
(a*((ha)) - Ug (aa(())))
Mgg
k= AEA
N
Do.k (aa((ha)) (aa((1))) + (Da Bunk (aak(h))))
k= aEA
D || ([ - Ea. (ha))lo + I|D -Ba.
my 1o (aa)))))
k=EEEA
The last inequality by the Holder's inequality. Let e.j
"0, (a, y, az = -ly-llyy-nn jjAaa)
Then we have C (h) Bieg A (ax, y, a; h0] and C (h) Ei= lig M65 h). By taking
union bound in Lemma B.3 have that with probability at least
2V) + +
log(m2 |
sup max max (h) (aa)))))))))))) (h)(l 22 NEFH KE[N] aEA
Since (hx))( 1, taking the union bound again, denoting vnn, H. = 2
2V C(H) log(n+1)
logg(n N/5))
it turns out that with probability at least 1 - 5,
29 (h) (h)
29 h
v(n, H1 (1//AA))I | Da.k + Da.k Da.k 1lo0 my mg
(29)
kEEEEA
S (nn H. H,86 // + 24
(30)
Next, we consider the optimality. Denoting u == arg maxu, Eld, 19 (h)l. then we have
Ba Ba h) 5") 2(T, Bely. 0) R < L(b, xT) R(h) + where e defines as the basis vector with at the position of \by- Let h satisfy the fairness constraints. With (27). we obtain
(31)
6( A, p.)
R( s 20, Rh + s R(h) R(h) +c.
(32)
Combining 31 and 32 it shows that
R(h) R(h) 2
1 2c
(h) -
(33)
Ba
Ba
37

---

Therefore, the result shows that max elt,
1 (b) - _
1+2c
Now we consider the generalization error for the empirically optimal classifier h, with probability at least 1 - 5,
P (h) 5a Lg (h)
Lg,
(h)
94, Mgg (h)l 5"
(34)
1 + 2c
(n, H. A/ + S2 +
(35)
Bd
Taking the union bound over Ug Ug. we have that with probability at least 1-66
1+
90
(h)
- S (nn H, 6/ D/L -2
(36)
Ba
For local fairness constraints, xx S gx following the similar proof procedures as local fairness constraints, we have that
1 2e
(b) (h)l - S (1 + +
(37)
Ba
For risk metric R(h), it presents that
R h R(h") = R() R(() R(h) R(h") R(hh) R(hh).
(38)
By 27 and 28
R(h) - R(h") < (h,,0,00 -(hh, x, < E (h, +e 2 +e = 2E.
(39)
Since we have R((h) = 1 ΣN
Po (I, C* (hx)), it presents that
N
R(h) - ((()) = Σ
pr (bx) - P C* (lx))
k=1
Σ
(pk (I, (ha))) + Σ (hau
k=1
N
N
S m
lpk - pxl Σ
pamil
k=1
By taking union bound in Lemma B.3 we have that with probability least - 6,
sup max || (h) - (* (h) s 22 2VC(H) log(n + 1) + Blog(m2N/))
BEJN RE[N]
n
Hence, denoting QP = Σie 1pA pxl. we arrive that, for any h E Fr.
N
R(h) - R(h)= < N max Pp P&l + Σ pam C (hx KE[N]
S m22 + (n, H)
(40)
Therefore, combining 38
39 and 40),
obtain
R(h) R(h") S 2m12 + 2mv(ne, H,,82) + 2e.
(41)
This completes the proof.
B.6 Proof of Theorem
We begin by introducing some definitions and lemmas, which are useful in the proof of Theorem Definition 3. Let V be a real vector space and let A, C V. The sum of A and B is defined by
A (a+bb A. be B}.
38

---

Lemma B.4. [6] The subdifferential of the function F(x) = [ff(w,))) at point x is given by
0F(x) B(0f(z,w))
where f(,w) is a real-value corvex function and the set (x) is defined as
B(Of(r,w)) [ (d(x)
=fx-
= x² E Rd | +======>-
x² (w))P)(()) ()) measurable, x² (w) E (f(r,w) ae
Lemma B.5. 1661 Lel Ji,..., Sm R + +000 be convex functions. Define f(x)
max (z). S (x)), Va E RR. For ao E ni=1 dom define I (20)) (i fs(so) = [(5o))
Then (x) conv User(z)) 0f((zo).
Lemma B.6. [66] Let f IR - RR be a corvex continuous function. functtion, We consider the minimizer x²' of the function f over the B. Then for x² to be locally optimal it is necessary that
0f(x") + +N((x") 3 30,
where NB denotes the normal cone of set B. If B R. ler K. = [k E [d, + 0}. Then there exists subgradient Of (x*), such that for all l E [d] we have & and vk K. Ea 0.
Proof of Theorem From the above analysis, follows that the Lagrange function can be written as
N
C(h, A, u) - Σ E Paçk
(M,"(a, Ca Cat
Σ UgEl,
(A + gg)
k=1EEd
(1)
+ MR.ue )
(2)
(K
KE[N] te El1
We first consider the inner optimization problem minne L(h, X, u)), which is equivalent to opti- mize
N Σ
max
V(h,
Pa.k
(M)i(a, (hr(
DERN
k=1EEA
where M (a, k) =
En (1
X
Dug - Emceth (P - M az Con-
(2)
Pa.k
sidering the personalized attribute-awan classifier he(s, a), [N] post-processing, the inner function turns to
V(h, A.) ΣV Σ Pa.k
Co (or(())))
k=1aEA
N
Pa.k / [nna,, MM (hhhee a).P.
k=E
An explicit optimal solution of personalized classifier is that
Adl' (x., a) := arg max
(IM* (a, k)] nT ((,,,,,))) k,),; [E(m)
If the maximum entry of the output vector occurs multiple indices, of them is randomly selected as the predicted class. Thus, the dual problem can be formulated as
min H (A. Aypt
E
Σ
Pa.k E X-PX a.k
mxx max (IM>) k)] mxx, a, k)) 5 E (22 [u(mm
KE[N aEA
My MyEl,,
Σ BA, Eldi
(u + (k.a))) ).
(42)
39

---

Before exploring the optimal solution of outer optimization, we first prove that the optimal dual parameter X E 2 11" E IR 2 Ri is bounded. Define the Hilbert space on F == {
] with inner product (f.g) - [x f gdP(x)) Then the classifier space H. X + Am a convex subset of F. Therefore, we can also consider the topology structure on H. or H Since we assume that VV", 5* O, the feasible set of the primal problem is non-empty, it indicates that the feasible set of the primal problem has non non-empty interior for any positive ç", gh clear that for V > the dual problem
min C(h, 11) 1 H() S R(h) S Rmax Tmaxx(/at) fair)
where / denotes a classifier that satisfies fairness constraints for given 5 5h > 0. Hence, we arrive at
(), 2 Rmax (h fair)
(43)
holds for all A. 2 0. Notice that given X, 2 O, this inequality holds for any 5" > 0. Let
5a - 5* - 0, combining (42) and (43) gives that
Pa.k E
max yE|m]
(a, a(x,,, a,), 00
(44)
X~PX a.k
KE[N] AEA
Therefore, the dual problem has a lower bound
H(A,)) 5
Σ Ug Elda
(10g +
(2) 2a
+ Σ kE(N]
ck Σ u2
+
(2) Pk.us
(45)
Eldz
presents that, as |Aala 00 || - oo, there must be (A, + co, which conflicts with the dual problem minx H (A, pJ). Hence, the optimal X" ,r of dual problem minx, H 1)) must have bounded norms, denoting as ||^l] S Bd, ||ll|a Ba.
Now we consider the differential of H(), 14). It is clear that (S, = fa E X: he(r, = v), E
[m]] constructs partition of the feature space X. Hence, for dual parameter x! since the outer objective H is convex to À and μr, by the additivity subgradients and Lemma B.4 the differential ((A, 14) ex!e
can be formulated as
0
H(A RE[N] aEA
Σ
Σ Pa.k E
a
max (IM)P((,))) n(X, a)) y J gEbm)
+ te. (46)
X-PN. a.k
With a slight abuse of notation, let score function f(z,a,k) = [M^*(a, a( by Lemma B.5 we have
E
max (IM*)(a,A)]
n(xx(XX, (k)
(47)
XaPX
VE[m)
Σ
θ max
[m(m) (IM (a, ((),)))((( ( (dpxx
(48)
De (asti""
@xuy (1)
Σ Pa,k ve(m)
U
-In(x, a, k))= Dank ey
dp (x) (Ppxx(())
conv
(49)
Σ L (
[- a, k)] Daykee en dp(xex))
Pa,k ve[m)
+ / B
b-bll((,, a, D (es ew) aPPra(o)
(50)
where B3 = (31 + y. [i a, k) 2 fala,, a, k). Vi E [m]: fe(a,, a, = [a a, k)) with E [o, 1). Since the convex hull is interval here, by Caratheodory's theorem, it can be characterized by
40

---

two point here (the initial point ey and another point in the convex hull). Without loss of generality, we assume the existence of one e, such that fa a, k) = fy(2z, a, k) here. We know that (1, a, k) - fu(æ, a, k) = [n( a, k)] MM (a, (e e²). With Asumption I we obtain that the measure of B is O, unless the t-th and y-th column of M* k) are equal. An effective simplification is to exclude all X; 1z that cause M (a, k)(c* - e²) = O. Since we suppose that the non-zero columns of each Da. are distinct, the dual parameter X' ,' E St.y. such that M H/ (a, (e' - e²) O, constructs the empty relative interior in the dual parameter space. By the convexity of the objective function, we have inf S H(), P)) = minx, H(2, Jr). due to the density of (A, (4) & Sty.
Overall, under the assumptions of the theorem, we have that By has measure of zero. It follows that dPp (x) + 8
① H)) (1) aA.0
ΣEΣ
J. (mnndr [- [Dae] T M(z ki) y
KE[N] A (m(m)
= Σ Σ
-In(x a. [.]] T Dayan 2 (x, a)dPa.(r)) + +0"
N = 22 (h^ + 5
KEIN] aEAA
(51)
In similar manner, we can derive
0
H(),))
E Σ KE[N) DEA
/
[n(e, a, k)] (x, + g
(2)
9. ( +0
(52)
Considering paired optimal dual parameter (1)) ,i = 1, by Lemma B.6. if Aay x12) > 0, we
have
uy (h> 11) = -6" 2 (h (m)) = E
which leads to contradiction. If (1)+ 0, Xag
(2) 0, we have
Aug
2 (h* x" 1 2 -
2 (h* #) 50
Mg
(1)- If Xag 0, 2.' > O, we have
20 Mg
(h X my 2 --, " (h x" #)) = 8
If
(1)+ x1lp > 0, All (2)+ = 0, we have
(h x x0 - -5, "9 (h>" ) E " Ug
Overall, we have shown that for all Ug E Uy 199, (h> *) 59
The local faireess guarantee also can be derived from the optimality of n²" The proof techniques are extremely similar to our proof with respect to X Hence, we omit the proof of the local fairness guarantee here. The result turns out that 1 (h> Aixi") S E* K [N].
The next step is to prove that the classifier A is the optimal solution of the primal problem (I) From the proof above, we can obtain that, for Vug E Ug.
(A 27)
(h>
(A
Ug x2)) = 0,
which satisfies the optimality conditions for the dual solution of the constrained optimization problem. The same holds for the local fairness constraints 2* (h> .²). Consequently, the Lagrangian function equals to risk function when plugging in optimal classifier, [(h>* "²" X",H"") = Rhh "²)). For any other classifiers h' that satisfies the global and local fairness constraints, denoting its corresponding dual parameter to maximize the outer problem as X, N,pt p,, it can be deduced that
C( sa" ,²,,²²) < L(h', X R,) < R(h').
Therefore, we arrive at
R(h -(h **" ,x,,²²) < R(h').
This completes the proof.
41

---

B.7 Proof of Prosition 6
Note that À E R200 and Pk E R 2
the operator
la is linear in dual parameters' domain. We
20
can just write
1 Th
(A) = Tok 1=1
Σ
M [M^ (as, (12,, ai, ki) + 5a Σ
og
MgEMl
gk
+
Σ
(W + +
Pk
Me Edde
where M M*H(u,, = Po,k
Eu, Eu, (2
Aag Du - Emyenn (Kkm
(2) Mk,as Daat and
0o(x)= = Ei= exp(a,
Convexity. The M (a k)) is linear to À and μi,, and the soft-max operator is convex. Since the composition of an affine mapping and a convex function preserves convexity, Hy (A, (&x) is convex to À and Pkk-
Smoothness. Consider the soft-max weighted sum 0g(x) = Dj=1 exp(a; /8)
21=1 exp(x1/3) T.j. and its
Hessian matrix is given by Ha == Vorz(1), [I)()) = # (2+ 1 [(i= pj 2-
2,4,, 0
]. For Vi,j E m. if ||>|| I R.
H (x)lssl S 3
1 } (22 28t
+ (2+ m1)
4/ 6R
Ba
Hence, its spectral norm is bounded,
43 6R
(H)(1)) S [|H S
Σ
(Holi))
m
B²
Then, there exists finite constant Ly m For each sample i 1,..., nk. define the affine map
48+6R g
such that ( S Lo.
3j 2 (n) ai k)] Dark for (ii) j 1,2.
(A) = A by [Alls] Pu,k
"
Set f(()) = (A)). and let f(A) (z(2((A)), op(x) E/=
e35/0
By the chain rule and second-order derivatives, (Afi(A) = AT ( Σn ctil Ij, Vifi(A) =
A, [V2og (za()) Hence, due to the boundedness of [Аll,, the inside 2(A) is bounded, set- ting the upper bound as R for simplification here, (A) < || sup_|\V2og<<>]|| = ||A 1| L., showing fo is || smoothh. The linear term in À has zero Hessian. Therefore, since the average of smooth functions is smooth with averaged constants, the function fi pa) L- nu
smooth in À with L Σ |A:1 La. Following the similar proof procedure, we can obtain the
smoothness of HA, px) uk.
B.8 Generalization Error For Post-Processe Algorithm
We begin by introducing some notations and simplifications, same as the proof of Theorem 9 Without loss of generalization, let n ni
n& present the sample number in local datasets. Denote Palk := (PA = alK- = k), Pmii == Aikln Pajk- Assume nmmn 2 1 denotes the sample size of the sensitive group with the fewest observations across all clients.
Theorem 10. If classiffiess
hi (fi....,fig Trz) with dual parameters (X*. [ an optimal solution
of the empirical plug-in in estimation of 7 denoting p(n,, 6) /9AAmm² log(n++1)
2 2logmm m=2 (ANN/5)
B, maX
D ||, B, maxaEAAbE[N] [Da. 11. 22 maXo6AAbE(N) D
42

---

Daj loo, and B₁ max.ed [D 11,, BB Dack uk loa, k E [N].
max [D 11,
maX.eA D
(e) < 8 < 1, suppose that nt > 2LAIN Br
log g. then with probability at least 1 - 2|A|6,
Pootn E
2pmm
|AIN
2 S + OOIANNIANNOO (..0(,,6/AAl44 5/1AI))))) + 2. +
Pemin
Let <8 8 1, suppose that
2AA
log f. then with probability at least - 2|AA66.
Pming
2ymin
LA
h
< 5h (I - S
A: E [N].
A0min
The proof of Theorem 10 needs the following lemma.
Lemma B.7. Let X1,..., Xn be independent Bernoulli(p) random variables and define Sn
Σx= X5. Fix any M E (0, np) and confidence level d E (0, 1). If the sample size satisfies 2
2M + 1 log 6: then we have P(S,, > M) 2 1 1- 5.
P 2pa
Proof of Lemma B.7 By Hoeffding's ' inequality, for any t > 0. -B[S]] S -t) 1 exp(- 21
Since E[S,,] np, set np M. Then
P(S. S M) = (P( - np S (np- M)) s exp( 2 (np M)
To guarantee P(S, S M) S 6, it suffices that 2(m
logg Substitute n
2M 3/2 log
"
p 2p
Then np- -M- = 2M + zp= log } ))pp M = M +
1
log 1 and one can check
2p
(mp M2
2(M + 2p
log g))
> log 8
2M
+ 2p
log 1
F
Hence P(S,, & M) < i.e. P(S, > M) 2 - 8.
B.8.1 Proof of Theorem 10
We first consider the generalization error of the fairness constraints. Without loss of generalization, here only prove the generalization error for global fairness constraints and corresponding parameter A. The proof technique for local fairness constraints and corresponding parameter is extremely similar that for global fairness constraints.
We know that the personalized attribute-aware empirical classifier can be written as
1 X- (I, a) = arg max M . k)] MT, a, yE(m]
(53)
As h depends on the Bayes score function 1,, we can consider the input as (me. 7( ak.s: kt), an.s. Vk.)). Let ( a, a,:,, = Ily = i h(n) = j ja a'). Then we have C t (b) = EleT)"
it (1, v, az h)) and (M) = ΣT 2g (nH.2, dx,2, )1.2). Then we turn to consider the VC dimension of the function class Hijja' = {h (æ, a, v) + 1(y = A h(n) = j a')}. Thanks to the classifier's specific structural form (53). we can directly state an explicit upper bound on its VC dimension: for given class j.
a))(o.a==)) = +> [n(x, k)]T
(MM # (a, (a,
M I' ka)))) 2 0, Vi + [m].
which can be regarded as the intersection of m 1 half-spaces given 7k,i, Aj.j. A single halfspace function class can be viewed as the class of linear classifiers, possessing a VC dimension of m. By the additive property of VC dimension, for function classes )m VC (A G.) Σ (G.).
43

---

the function class Hi has VC dimension at most O(IA)m²²). By taking union bound in the Lemma B.3 we have that with probability at least 6,
-m) so (s
< log(n +
/8/A|m2
2 (||(()
sup max max - NeFu ke[N] aEA
+
== O
Hence, for the global fairness constraints D4 with the empirical optimal solution hi, by the generalization bound (29). we have that,
3 h
h S Op(nn, A/ + +
Now we consider the bound on empirical Si, (h*). The empirical optimal dual parameter X and ji are obtained by the empirical dual function:
Σ
Σ
(M^i'
(a, k)] [i,, a,) ,) t0 Σ nyEu,
(111) (!e
PORk
we(m)
KEIN
aEA
ck
(pleuue + +H p.pe
(54)
[E[N]
This representation fully consistent with that given in (8) restricting to group-a observations within the k-th client's 's data, where & denotes the sample number of group in client k. Considering the subgradient of the empirical dual function w.r.t. Xag by the additivity of subgradient,
① (A, (1) Ox0
Σ Σ Park
Ba.k
max WE m (M^> k)] Ma,, ak) , + (55)
RE(N)
aEA
Denoting empirical score function f(1, a, k:) = [M (TT,)))]TTe,,
M5. a, k), by Lemma B.5 we have
Ba.h
i
[M..'(,,))) ((1 y.
max yE|m]
(1)
no.k
(h, = =y]
(MM)"(( ( 500
max ym(m)
y ME(m)
Buck
U
- -17 k)] Day ei
conv
Pa,k DE(m)
iEarg max)e((m (s(,aa
nu.k
Σ
{
- (F( (2,, a, kj Fj.J (ml) Ma,. a k)] Danke
Pa.k 1=1 mm(m!
+ II (Ix E BB)
[- bb (7,,, k)] Daj -0) {
where B² = (r y. Ft k) 2 F((2, a, k), E [m]; Jelx k, = Py a, k)] and b. E [0, 1]. According to Carathéodory's theorem, the subgradient interval can still be represented by two points. According our assumption, the plug-in estimator î still meet the continuity assumption and we exclude singular X' n². Therefore, we know that
P
Σ Il 3t + 4y, Fels,, a, M(,, a, a,, Vi
[m]: Felx,, a, Fy(rn,))) s =
(56)
44

---

Hence, the subgradient falls into an interval. Since Ma; a, k)] Duke - [1, a, k)] T))TDablll - By, denoting Amin == minge Ma.k and By. we have that
0 11), S
Σ
Ba.k
[i,,, a, B Dy ha(
Ba a) + (57)
Daak
[N]
aEA
Σ
Bg
(58)
kE(N] aEA
(h) +
(59)
nmin
On the other hand,
1 H()11)
Σ
[M k)] Ta, he (1 a)] g9 "y
(60)
aNug
(1)
Bo,
[e[N] aEA
Σ
Bg
(61)
naakk
kE|NJaEA
LANB g
29
(h) 5
(62)
30
Demin
Hence, we obtain that
á A(.. p)) - E9
AND
|A|N B.
2g Myy
(h²)
Gg b/g
(hi)
Rmmin
Amin
similar manner, we can derive the range of subgradient for x!?))
i AT p1) - -8 c
LA
A
@0 (h²)
24
(h') + ug
Pemin
Amin
Since assume that last [A18,
2 2LAIN Piiin
log 1.- by Lemma B.7 have that with probability
2p
LAIN
A
Qmin
E9
E9
Pmii
Consider the optimality of (1)) (2)) by Lemma (x* 1),06 E A (X*,,10 nn). Thus,
B.6 if (1)
> 0, 2123
0, we have 0
LA By
9g Mg (h²) -
Umin
LAIN B
a, (h') + 50
Denin
which leads to a contradiction. For other cases, such as
20
( = 0; X
> 0, X12 = 0, and
20) 0, X(2) 0, as discussed in the proof of Theorem 5 turns that
S + LAN
[9 (h*))
Dmin
45

---

By taking union bound, we obtain that with probability at least 1 - 2|,A|o,
|
( 5 + +00p0000//111444))44N,,,, + -2 +
Romin
For local fairness constraints 2 (h"), following the same proof procedures, we arrive at that with probability least 1 2).A(o.
LAlB
19*(h")) S E + N1 + +
Almin
46

---

C Additional Datasets and Experimental Setting
C.1 Datasets and Experimental Details
C.1.1 Datasets
The Compas dataset 22 comprises 6,17 6,172 criminal defendants from Broward County, Florida, between 2013 and 2014, with the task of predicting whether a defendant will recidivate within two years of their initial risk assessment. We consider the race of each individual as the sensitive attribute and train logistic classifier as our prediction model.
The Adult dataset 4 comprises more than 45000 samples based on 1994 U.S. census data, where the task is to predict whether the annual income of an individual is above $50,000. We consider the gender of each individual as the sensitive attribute and train the logistic regression as the classification model.
The ENEM dataset 440 contains about 1.4 million samples from Brazilian college entrance exam scores along with student demographic information. We follow [3] to quantized the exam score into 2 or 5 classes as label, and consider race as sensitive attribute. As [ used a random subset of 50K samples, we instead sample 100K data points to construct our federated dataset. We train multilayer perceptron (MLP) as the classification model.
The CelebA dataset [96] is facial image dataset consists of about 200k instances with 40 binary attribute annotations. We identify the binary feature smile as target attributes which aims predict whether the individuals the images exhibit a smiling expression. The race of individuals is chosen sensitive attribute. We train Resnet18 [38] on CelebA the classification model.
The determination of sensitive attributes and labels on three datasets has been verified significant in previous research
C.1.2 Baselines
We compare the performance of FedFACT with traditional FedAvg 55 and five SOTA methods tailored for calibrating global and local faimess in FL., namely FairFed B1. FedFB 193. FCFL (18
praFFL 85), and the method in denoted as Cost in our experiments.
FedAvg serves core Federated Learning model and provides the baseline for our experiments. It works by computing updates on each client's local dataset and subsequently aggregating these updates central server via averaging.
FairFed introduces an approach to adaptively adjust the aggregation weights of different clients based on their local fairness metric train federated model with global fairness guarantee.
FedFB presents FairBatch-base approach [67] to compute the coefficients of FairBatch param- eters on the server. This method integrates global reweighting for each client into the FedAvg framework to fulfill fairness objectives.
FCFL proposed two-stage optimization to solve multi-objective optimization with fairness constraints. The prediction loss each local client is treated as an objective, and FCFL maximize the worst-performing client while considering fairness constraints by optimizing surrogate maximum function involving all objectives.
praFFL proposed preference-aweee aware federated learning scheme that integrates client-specific preference vectors into both the shared and personalized model components via hypernetwork. is theoretically proven linearly converge Pareto-optimal personalized models for each client's preference.
(25] proposed a convex convex-programinn programming- post-processing framework that characterizes and enforces the minimum accuracy loss required to satisfy specified levels of both local and global fairness constraints in multi-class federated learning by approximating the region under the ROC hypersurface with simplex and solving linear program, denoted as Cost in our experiments.
Meanwhile, we adapt FedFACT to focus solely on global or local fairness FL., denoted as FedFACT, and FedFACT, FedFACT, gad indicates the algorithm simultaneously achieving global and local fairness. The FedFACT (In) presents the in-processing method and FedFACT (Post) presents the post-procsssing method.
47

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C.1.3 Parameter Settings
We provide hyperparameter selection ranges for each model in Table 4 For all other hyperparameters, we follow the codes provided by authors and retain their default parameter settings.
Table 4: Hyperparameter Selection Ranges
Model
Hyperparameter
Ranges
Learning rate
{0.0001, 0.001. 0.003,
0.005, 0.01. 0.03 ,0.05)
Global round
{20, 30, 50, 80]
General
Local round
{10, 20, 30, 50)
Local batch size
{128. 256, 512]
Hidden layer
{ [16, 32, 64)
Optimizer
(Adam, SGD)
FedFB
Step size (a)
(0.005, 0.01, 0.05, 0.3)
Faimess budget (3)
{0.01, 0.05, 0.5, 1)
FairFed
Local debiasing (a)
{0.005, 0.01, 0.05)
FCFL
Faimess constraint (c)
{0.01, 0.03, 0.05, 0.07)
praFFL
Diversity (Tp)
{10, 15, 20)
Classifier number
FedFACT (In)
wf. learning rate (7w) Dual parameter bound
(0.03, 0.3)
Temperature 3
0.1
FedFACT (Post)
Dual parameter bound
For the fairees--cnnrrl parameters, e,g., the parameter À in praFFL 885 and the global and local fairness constraints in Cost 25], we impose stringent fairness requirements on the model in our overall comparative experiments, and we adjust the parameters governing the fairness metrics in the Pareto-curve experiments.
C.1.4 Experiments Compute Resources
We conducted our experiments on a GPU server equipped with CPUs and two NVIDIA RTX 4090s (24G).
C.2 Discussion about FedFACT and LoGoFair
95
LogoFair (95 is designed for binary-classification in federated learning under both global and local fairness constraints, seeking the Bayes-optimal classifier. By deriving a closed-form solution for the fair Bayes classifier, LogoFair reformulates the post-processing fairness adjustment optimization problem jointly solved by the server and clients, which is an approach conceptually analogous to our post-processing framework. In binary classification,, FedFact and LogoFair both target Bayes-optimal classifiers under constraints disparity metrics expressed in linear form. Theoret- ically, for an identical fairness metric, our Bayes-optimal fair classifier characterization covers that of LogoFair. Consequently, we refrain from performing a comparative evaluation of the two approaches.
bilevel
Our method differs by defining the loss at the client level, thereby achieving lower estimation error than the local group-specific objective in [95]. Crucially, by formulating the post-processing model the probabilistic simplex instead of restricting outputs the unit interval [0. in the binary case, our framework achieves enhanced scalability and naturally adaptable to multi multi-group. multiclass settings.
Note that, whether for binary or multiclass settings, our implementation of FedFACT is based on calibrating confusion matrices over the multi-dimensional probabilistic simplex.
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C.3 Heterogeneous Split of Client Distribution
We propose a partitionnnn method that introduces heterogeneous correlations between the sensitive attribute A and label Y,, thereby further elucidating the trade-off between global fairness and local fairness 33..
Heterogeneous Split. We assume a dataset D of n. samples, each with binary attribute A and binary label Y. We denote by 7kj = If Zig at Ve : (at = i, Ye = })} the number of samples in joint class (i for j E {0, 1}. Our goal is to partition D into N disjoint subsets (one per client) such that client & E [N] the correlation between and is controlled by a target parameter The E [a, b] S [0, 1]. To achieve this, we first assign each client k weight 7/
(1-) wkk
Ykk [1
(i.5) E {(0, 0), (1) (1,5) {(1,0), (0.
Tke
Then for each joint class (i.j) we compute the total weight W = Ek= wf
(6.)) and assign to
client a preliminary count & (i:5) = [(wlt>) /W(L.) may]- Any remaining samples are distributed one by one to the clients with the largest fractional remainders, that Σm % (t.)) Ray- Finally,
for each class (i, we shuffle its nij sample indices and slice them into blocks of size (K5) Client then collects all its four blocks across (i,j), yielding a partition that in expectation realizes the desired within-client correlation Y' between A and Y.
This approach can be regarded as a generalization of the synergy-level -evel-baedd heterogeneous split in [33] to the multi-client setting. where the A-Y correlation for each client is governed by parameter randomly drawn from |a, bj S [0, 1], thereby yielding a more pronounced balance between global fairness and local fairness. Throughout the experimental evaluation, we set ч E [0.2, 0.8] to guarantee that every client has sufficient number of sensitive group samples to assess local fairness.
D Detailed Experiments Results
D.1 Comparison Result and binary EOP criterion
Parato Curves of DP. We have already presented the numerical comparison between our proposed method and the baselines in the main text; here, we report the Pareto eurves illustrating the trade-off between global fairness and accuracy. More precisely, we compare the trade-off between accuracy and the global fairness measure, well as the trade-off between accuracy and the local fairness measure, as a function of the fairness constraint.
The Pareto curve for the global DP criterion is shown in Figure 2 where the horizontal axis denotes accuracy and the vertical axis represents the faimeess metric. Consequently, models located closer to the upper-right corner exhibit superior cecurrcy-fairmess trade-offs. As illustrated in Figure 2 our method outperforms all existing state-of-te saat--of-the-rr art approaches when comparing accuracy against either global fairness in isolation.
F
FeIFACT//Po))
ReIFR
Cost
FaiPPel
prfF
Compas
Adult
CelebA
ENEM
0.68
0.38
0,00
0,55
0.50
0.30
0.50
0.32
0,02
0.30
0.25
11.3
1000
0.10
0..00
0000
0.00
10055
1,,00)
011225
00000
0,300
000
Figure 2: The Pareto frontier on Compas, Adult, CelebA and ENEM datasets. The curve closer to the upper right corner indicates hetter trade-off between accuracy and fairness.
This result not only demonstrates that our model achieves more favorable accuracy-fairees balance but also highlights its controllability: by tuning the fairness constraints, one can satisfy diverse fairness requirements.
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Parato Curves of EOP. In Figure 3 we illustrate the Pareto curve for the Equalized Odds (EO) Because EOP enforces tighter constraints than DP. precise adherence in federated context requires large per-group sample counts at each client. Hence, we also compare the global EOP here. Our framework still exceeds all state-of-the-ar baselines in trading off accuracy against fairness.
Pr
TrdFF
Chot
FaitFo
Compas
ddult
ColebA
ENEM
0.34
0.50
0,58
0.00
Ace
0.54
0.50
0.52
0.35
000
EOP
20P
EOP
EOP
Figure 3: The Pareto frontier on Compas, Adult, CelebA and ENEM datasets. The curve closer the upper right corner indicates better trade-off between accuracy and fairness.
D.2 Details for Multi-Class Classification
Multi-Class fair datasets. We illustrate how FedFACT performs on multi-class prediction using CelebA and ENEM. For CelebA, with 'Gender' still serving as the sensitive attribute, We employ the binary attributes "Smile" and "Big_Nose" to construct a multiclass task by mapping their joint values {0,1) {0, 1} onto a four-class label set {0, 1, 2,33, thereby formulating a multielass classification problem on the CelebA dataset. These attributes are commonly used in centralized machine learning construct fairness-aware classification tasks. For ENEM, follow [3 to quantize the Humanities exam score to 5 classes. In order to guarantee adequate per-group sample sizes at each client in heterogeneous settings for fairness evaluation (or some clients only hold less than 10 samples for specific group under heterogeneous partitioning), we adopt the four race labels "Branca," " "Preta," " "Parda," and "Amarela" from the Race attribute as the sensitive groups. These datasets are partitioned into five clients under heterogeneous split with y 1.
literature IK 97
Evaluation. In terms of baselines, only the Cost (2] algorithm is theoretically applicable fairness optimization in multiclass federated learning scenarios. However, their experiments and code are limited binary classification, and have already been used binary baselines for comparison with our method. Consequently, we focus exclusively on reporting FedFACT's performance along with FedAvg in multiclass fairness, establishing it as a pioneering approach in this setting.
D.3 Additional Experiments for Adjusting Accuracy-Fairner Trade-Off
In Table we present additional experiments on the Compas and Adult datasets under the heteroge- neous split to illustrate the adjustment of the accuracy-fairees trade-off. Compared to the results in the main text, this partitioning yields a more pronounced trade-off between global and local fairness.
Table 5: Additional Accuracy- Faimnes Balance.
Dataset
Compas (In-)
Adult (In-)
Compas (Post-)
Adult (Post-)
($",<))
Acc
gegloiel
gloua
Acc
galoool
glond
Acc
galoooi
galoos
Acc
galsbal
golood
(0.00,0.00)
60.22
0.0404
0.0745
80.99
0.0021
0.0407
64.56
0.0083
0.0075
81.25
0.0139
0.0275
(0.02.0.00)
60.61
0.0436
0.0734
81.04
0.0021
0.0423
64.78
0.0091
0.0099
81.56
0.0146
0.0285
(0.04.0.00)
60.90
0.0490
0.0737
81.09
0.0039
0.0446
65.04
0.0123
0.0099
81.62
0.0147
0.0285
(0.00,0.02)
60.80
0.0499
0.0744
81.18
0.0046
0.0411
64.94
0.0146
0.0214
81.82
0.0238
0.0381
(0.02.0.02)
61.03
0.0503
0.0726
81.64
0.0315
0.0463
65.12
0.0311
0.0306
82.04
0.0240
0.0381
(0.04.0.02)
61.32
0.0555
0.0774
81.65
0.0318
0.0467
65.57
0.0378
0.0371
82.16
0.0257
0.0397
(0.00.0.04)
61.18
0.0581
0.0804
81.31
0.0053
0.0444
65.16
0.0294
0.0517
82.46
0.0350
0.0492
(0.02.0.04)
61.39
0.0644
0.0753
81.67
0.0177
0.0452
65.16
0.0412
0.0419
82.49
0.0346
0.0497
(0.04.0.04)
62.39
0.0878
0.0966
82.14
0.0486
0.0497
65.82
0.0507
0.0574
82.63
0.0350
0.0518
Note that the gap between the imposed constraints and the observed fairness metrics stems from
the inevitable generalization error incurred with finite local samples. Consequently, global fair-
ness exhibits greater controllability than local faimess. In practice, FedFACT remains capable of
50

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tuning the accuracy faimesss balance according to the specified fairness constraints, highlighting the controllability inherent in our approach.
D.4 Hyper-Parameter Experiments
In this subsection, we examine the impact of the number of classifiers in the in- processsig method. Specifically, we incrementally increase the size of the weighted ensemble-from using only the most recently trained classifier up to including the ten preceding classifiers. Let N₂ represent the number of classifiers comprising the weighted ensemble. As reported in T we observe that augmenting the ensemble with multiple classifiers yields negligible improvements and can even degrade performance when earlier classifiers have not been fully trained. Consequently, in light of these empirical findings, all in-processing experiments in this work utilize only the single most recently obtained classifier.
Table 6: Hyper- Parameter Experimental Results.
Compas
Adult
CelebA
ENEM
Nk
Acc
galubal
grooa
Acc
galobul
Glueni
Acc
gulobal
Glocu!
Acc
Dalobal
Giluoal
61.17
0.0407
0.0732
82.04
0.0014
0.0401
86.15
0.0382
0.0473
65.33
0.0293
0.0387
61.29
0.0408
0.0731
81.24
0.0015
0.0416
85.54
0.0382
0.0482
65.54
0.0285
0.0392
61.18
0.0410
0.0723
81.63
0,0032
0.0397
85,91
0.0377
0,0472
65.41
0.0307
0.0390
10
61.14
0.0404
0.0736
81.91
0,0048
0.0399
86.59
0.0384
0.0471
65.11
0.0398
0.0383
D.5
Efficiency and Scalability Study
In this section, conduct experiments with DP criterion examine the communication cost and scalability of FedFACT.
Efficiency. We evaluate the communication efficiency of FedFACT by monitoring its performance across varying numbers of communication rounds T. As illustrated in Figure 4 the post-processing method, built upon fully trained pre-trained model, consistently achieves convergence in fewer than 10 communication rounds, underscoring its high efficiency. The in- in-processing method likewise converges in under 40 iterations; given that it requires training the federated model from scratch, this performance is comparable to the convergence speed of FedAvg, making it highly effective compared existing federated learning algorithms.
Aec
nloohll
S foca!
0.8
05
6.73
Ace
6093
0,52
6,05
0.20
nusb
6.00
100
Compas
Adult
CelebA
ENEM
Acc
global
glocal
0.55
Ace
0.60
0..0
0,50
0.72
0,50
0.76
Compas
Adult
CelebA
ENEM
Figure 4: Communication Effectiveness Analysis. The convergence rates both in-processing (top row) and post-processing (bottom row) methods with respect to communication rounds on Compas, Adult, CelebA, and ENEM datasets.
Overall, whether employing the in in-processing or post-processing method, all three performance metrics rapidly converge to stable values across each the four datasets, empirically confirming both the communication efficiency and the overall effectiveness of FedFACT.
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Scalability. We evaluate FedFACT's performance as the number of clients varies from 2 to 50 on all four datasets, with heterogeneity parameter y 5 to ensure that each local client has adequate samples for assessing local fairness. The results, shown in Figure 5 indicate that each dataset, there is an upward shift the metric as the client count increases. Enforeing fairness constraints, especially via the in processeiin method, sometimes necessitates a modest loss in accuracy, and the post-processing approach on the Compas dataset exhibits pronounced fairness fluctuations due to substantial generalization error when sample sizes are small. Aside from this, our method reliably bounds the model's fairness, underscoring its robustness to variations client population.
E Broader Impacts and Limitations
Broader Impacts. This paper addresses critical fairness issues in FL. By embedding fairness constraints at both the global and client levels, our framework delivers models that distribute accuracy confidence and mitigating bias amplification. The contributions of
more equitably, bolstering this research enhance user satisfaction and promote social equity. This fairness-aware approach extends readily to high-stakes classification tasks beyond FL: for instance, clinical decision support in hospital networks, vision-based detection systems, and financial fraud alerts. Integrating faimess into decentralized model training promotes srivacypreeeervin equitable AI, helps satisfy emerging regulatory requirements, and encourages broader adoption of responsible machine learning across diverse application domains.
Limitations. The primary limitation of FedFACT is the fairness representation, which contains the linear disparities such communly used DP, EOP and EOP criteria, but it excludes some nonlinear formulations of fairness, e.g. Predictive Parity [22) and individual fairness [29]. Moreover, based on our generalization-erro analysis, although the proposed method enables controllable accuracy- fairness trade-off for given fairness metric, it still requires a sufficiently large local sample size to accurately estimate local fairness (whereas global fairness demands only an adequate overall sample size). While our empirical results compare favorably against existing approaches, exploiting dataset characteristics optimize fairness may reduce the sample complexity needed for local fairness optimization. Addressing these limitations remains an important avenue for future work.
Acc
global
D local
0,200
1200
100
Acc
0.2
082
0.00
065
8.20
Compas
Adult
CelebA
ENEM
Acc
salobal
glocal
R166085
0.57
Acc
0.05
0.50
1116
0.009
0200
00
300
0.54
Compas
Adult
CelebA
ENEM
Figure 5: Scalability Analysis. The behavior of both the in-processing (top row) and post-processing (bottom row) methods as the number of clients increases from 2 to 50 across Compas, Adult, CelebA, and ENEM datascts.
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Reviews:
[{"review_id": "l9SFTS5ZVV", "reviewer": "Reviewer_eRHZ", "summary": "* The paper proposes a federated fair learning method that takes into account local and global fairness constraints in multi-class classification settings.\n* The convergence and consistency guarantees are shown under smoothness assumptions for both in-processing and post-processing settings\n* Empirical comparisons across 4 public benchmarks demonstrate better accuracy-fairness tradeoffs across both local and global constraints", "questions": "Typos in Table 3 (line 7) and elsewhere in the results section's writeup reduces readability of the paper", "limitations": "Limitations around gradient conflict, and other sub-optimal optimization procedures need to be addressed beyond the Bayes-optimal fair classifier settings to bridge the gap between theoretical guarantees and practical applications.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n* The method proposed is able to provide fine-grained sensitivity based on local and global fairness constraints of federated equality of opportunity and demographic parity based on decentralized confusion matrices, without regressing on accuracy\n* The algorithm proposes to learn Bayes optimal accurate classifiers under both local and global fairness settings, which is a novel contribution\n* The usage of weighted classifiers in both in-processing and post-processing settings with minimal parameter exchange in each training rounds ensures that the remaining generalization error can be mitigated by increased sampling and training rounds (T).\n\nWeaknesses:\n* The paper is very dense to read - with the implications of the theoretical results hard to parse. The bulk of the empirical results related to multi-class classification has been relegated to the Appendix. This is a bit concerning as the main claim of the paper is that it addresses global and local fairness in federated learning in multi-class settings. The difficulty of extending prior work beyond binary classification is not explained in the paper. Further, adaptations of prior work to multi-class settings are not provided or explained as to why it is difficult to obtain.\n* The aggregation mechanism at the server adopts the aggregation mechanism as proposed in FedAvg, and does not discuss the challenges of gradient conflicts while optimizing for local and global fairness constraints.\n* Lack of confidence intervals in the results in-general reduce the statistical significance of the results. This is particularly concerning since 2 of the 4 datasets (presented in the main paper) have small evaluation dataset sizes, which needs further investigation.", "was_revised": true, "final_justification": "Update after the author response: The response has addressed my concerns, and I increase the score as it provides a sound approach at improving fairness-accuracy tradeoff in multi-class classification under federated learning settings.", "rebuttal": "### For Weaknesses\n\n> **W1:** Implications of the theoretical results: relegation of most empirical results related to multi‑class classification to the Appendix; lack of explanation regarding the difficulty of extending prior work; absence of adaptations of prior work to multi‑class settings.\n\n**Response:** We greatly appreciate your careful reading and for raising these insightful feedback. We apologize for the dense presentation of our theoretical section. Below, we provide responses to each issue highlighted here.\n\n**(i) Theoretical implications.** Overall, we derive the explicit representation of the optimal probabilistic classifier under joint local-global fairness constraints and propose theoretically grounded, unbiased learning algorithms at in- & post-processing phase to learn the optimal classifier. Our convergence and generalization analyses implicate that, as the number of iterations and sample size increase, the method converges to the optimal accuracy-fairness trade‑off. To the best of our knowledge, we are the first to derive the Bayes‑optimal fair classifier in multi‑class federated learning settings and to propose both in- and post-processing optimization methods with rigorous theoretical guarantees.\n\n**(ii) The multi-class classification results.** We fully acknowledge your feedback and apologize for delaying the multi-class classification results to the Appendix due to space limitation. **In the revised version, we will move key multi-class results with a specific analysis into the main text** to ensure that the multi-class results are given adequate attention and context.\n\n**(iii) The difficulty of extending prior work beyond binary classification.** From a technical standpoint, fairness calibration techniques tailored for binary classification in federated learning could not scale to multiclass problems due to the significantly greater complexity induced by multi-dimensional model output and multi‑class fairness constraints. For in‑processing approaches, most prior works use a differentiable surrogate to approximate the fairness metrics [4,5,6], but multi‑class fairness entails multiple constraints with cross‑dependencies, making surrogate construction challenging. In post‑processing, fairness correction typically involves setting group‑specific thresholds on output scores [7,8]. However, in multi‑class scenarios, where model outputs lie on the probability simplex, a single threshold or even a simple set of thresholds cannot capture the $\\arg \\max$ decision boundary, thereby impeding the feasible trade‑off between fairness and accuracy.\n\n\n**(iv) Supplementary experiments.** We extend the method proposed in [8] (denoted as Cost) to the multi‑group, multi‑class scenario. Since it is designed for binary DP, EO, and EOP, we did not compare it in the original paper. We compare this method with ours in the multi‑class experimental setup (Appendix D.2), with the results  presented below.\n\n|Metric|Dataset|Method|Acc|$\\mathscr{D}^{global}$|$\\mathscr{D}^{local}$|\n|---|---|---|---|---|---|\n|**DP**|**CelebA**|Cost|58.28|0.0832|0.1241|\n| | |FedFACT(In)|57.32|0.0727|0.0541*|\n| | |FedFACT(Post)|60.10*|0.0337*|0.0810|\n| |**ENEM**|Cost|37.32|0.0786|0.1050|\n| | |FedFACT(In)|38.86|0.0402|0.0651*|\n| | |FedFACT(Post)|41.85*|0.0186*|0.0887|\n|**EOP**|**CelebA**|Cost|59.96|0.0814|0.1551|\n| | |FedFACT(In)|60.32|0.0430*|0.0559*|\n| | |FedFACT(Post)|60.55*|0.0637|0.1240|\n| |**ENEM**|Cost|40.62|0.0980|0.1262|\n| | |FedFACT(In)|40.82|0.0619|0.0409*|\n| | |FedFACT(Post)|41.85*|0.0480*|0.0987|\n\nThe results clearly show that our model outperforms the Cost model, as our approach can approximate the optimal accuracy‑fairness boundary.\n\n> **W2:** The challenge of gradient conflicts.\n\n**Response:** Thank you for pointing out the valuable issue regarding the aggregation mechanism and the potential challenges of gradient conflicts. We discuss this challenge in two aspects:\n\n**(i) Motivation for adopting FedAvg:** In this paper, we underline group-fairness calibration and propose a general framework that leverages cost-sensitive loss and personalized optimization to achieve accuracy-fairness balance, **without relying on specialized aggregation schemes.** The FedAvg aggregation mechanism is adopted here due to its simplicity and effectiveness in heterogeneous federated learning scenarios. This allows us to focus on fairness calibration without introducing additional complexity in the aggregation process.\n\n**(ii) Discussion on gradient conflicts and additional experiments:** Gradient conflicts are ubiquitous in heterogeneous environments [10,11]. During the parameter aggregation stage of FedFACT, when each client sends gradients that push toward its own local fairness target and diverge the global fairness objective, e.g., improving local parity on one client may hurt overall parity, naive averaging can decelerate model convergence or produce sub-optimal trade‑offs. However, **it should be noted that our learning objective constitutes an unbiased estimate of the Bayes‑optimal fair classifier that balances global and local fairness, thereby mitigating the conflicts caused by optimizing global and local fairness to some extent.** Moreover, as discussed above, our model does not rely on any specialized aggregation scheme. Therefore, aggregation methods developed to mitigate gradient conflicts (e.g. [11]) can be directly adapted to our framework to further address this challenge.\n\nIn conclusion, we respectfully note that, our framework emphasizes group‑fairness calibration and utilizes FedAvg for simplicity, while our proposed learning objective inherently harmonizes the conflicts caused by optimizing global and local fairness.\n\n> **W3:** Lack of confidence intervals in the results.\n\n**Response:** We appreciate your attention to the details of our experimental section and for noting the omission of confidence intervals. Space limitations led us to present only the averaged metrics of multiple experiments in the main text. We fully agree that, particularly in datasets with small data sizes, including confidence intervals is vital for demonstrating the robustness of our conclusions. To enhance the persuasiveness of our experiments and address your concern, **we further provide additional experimental details with 95% confidence interval on Compas and Adult dataset mentioned here.** Due to character limitation, we consider scenarios under the Dirichlet heterogeneous split ($\\gamma=0.5$). The results are presented in the Table below. \n\n|Dataset| |Compas| | |Adult| |\n|---|---|---|---|---|---|---|\n|**Method**|**Acc**|$\\mathscr{D}^{global}$|$\\mathscr{D}^{local}$|**Acc**|$\\mathscr{D}^{global}$|$\\mathscr{D}^{local}$|\n|**FedAvg**|69.7±0.9|0.276±0.024|0.359±0.031|84.5±0.2|0.177±0.019|0.231±0.028|\n|FairFed|59.4±1.8|0.103±0.025|0.111±0.037|80.2±0.4|0.098±0.034|0.143±0.049| \n|FedFB|58.3±1.5|0.089±0.024|0.098±0.028|81.9±0.5|0.075±0.017|0.116±0.021| \n|FCFL|56.5±0.8|0.065±0.019|0.061±0.011|81.5±0.7|0.085±0.010|0.126±0.039|\n|praFFed|59.9±0.9|0.082±0.023|0.097±0.026|81.0±0.6|0.059±0.017|0.076±0.023| \n|Cost|64.5±0.4| 0.059±0.008|0.094±0.011|81.1±0.2|0.026±0.005|0.059±0.008| \n|FedFACT$_g$ (In)|61.2±0.7|0.034±0.006|0.076±0.024|82.1±0.3|0.002±0.001|0.041±0.003| \n|FedFACT$_l$ (In)|61.8±0.6|0.060±0.015|0.064±0.016| 82.4±0.4|0.014±0.004|0.051±0.005|\n| FedFACT$_{g l}$ (In)|61.2±0.7|0.041±0.008|0.073±0.022| 82.0±0.2|0.001±0.000*|0.040±0.003|\n| FedFACT$_g$ (Post)|67.3±0.3|0.013±0.003*|0.066±0.006|82.8±0.1*|0.017±0.003|0.028±0.004|\n| FedFACT$_l$ (Post) |67.5±0.4*|0.032±0.007|0.055±0.008*|82.8±0.1*|0.015±0.002|0.027±0.003*| \n| FedFACT$_{g l}$ (Post)|67.3±0.3|0.014±0.004|0.064±0.009|82.7±0.1|0.013±0.002|0.027±0.003*| \n\nThese results further demonstrate that our model outperforms all baselines, achieving a better balance between accuracy and fairness.\n\n### For Questions\n>  **Q1:** Typos in Table 3 (line 7) and elsewhere in the results section.\n\n**Response:** Thank you for your careful reading of our paper. We apologize for having these typos and we promise to correct them in the revised version. We will also review the paper in detail and revise other typos to enhance the paper’s readability.\n\n> **Limitations:** Limitations around gradient conflict, and other sub-optimal optimization procedures need to be addressed.\n\n**Response:** We sincerely appreciate your feedback. In response to Weakness 2, we discuss the challenge raised by gradient conflict. We acknowledge that certain sub‑optimal optimization procedures may impact model performance. However, **our method has demonstrated relatively significant improvements to prior methods, owing to our in‑depth theoretical investigations.** Moreover, as the proposed framework is already moderately complex, and to prevent further increases in time or memory complexity, we leave these optimization refinements for future work.\n\nSincere appreciation for your valuable feedback. We will integrate the aforementioned analyses, discussions and additional experiments into the paper’s revised version. We hope this resolves your concerns.\n\n[1] Post-hoc Bias Scoring Is Optimal For Fair Classification, ICLR 2024.\n\n[2] Fairness Guarantees in Multi-class Classiﬁcation with Demographic Parity, JMLR 2024.\n\n[3] Fair Bayes-Optimal Classifiers Under Predictive Parity, NeurIPS 2022.\n\n[4] Addressing Algorithmic Disparity and Performance Inconsistency in Federated Learning, NeurIPS 2021.\n\n[5] FairFed: Enabling Group Fairness in Federated Learning, AAAI 2023.\n\n[6] PraFFL: A Preference-Aware Scheme in Fair Federated Learning, KDD 2025.\n\n[7] LoGoFair: Post-Processing for global and local Fairness in Federated Learning, AAAI 2025.\n\n[8] The Cost of global and local Fairness in Federated Learning, AISTATS 2025.\n\n[9] FACT: A Diagnostic for Group Fairness Trade-offs, ICML 2020.\n\n[10] SCAFFOLD: Stochastic Controlled Averaging for Federated Learning. ICML 2020.\n\n[11] Federated Multi-Objective Learning, NeurIPS 2023."}, {"review_id": "McH0kRcFXH", "reviewer": "Reviewer_LAAJ", "summary": "The paper introduces FedFACT, a unified framework for controllable group fairness calibration in federated learning. The proposed method optimizes both global and local level fairness for multi-class problems. They propose two algorithms, in-processing and post-processing based methods, that approach Bayes-optimal fair classifiers. Both algorithms come with provable convergence and consistency guarantees. The authors compare the proposed algorithms with existing baselines that are targeted for only global or local fairness. The numerical results on various datasets show that the proposed methods enjoy a better tradeoff than existing methods between global-local fairness and accuracy.", "questions": "Please see weaknesses. Beyond that:\n- How would the proposed methods behave if some clients are adversarial?", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths\n- To my knowledge, the paper is the first joint consideration of global and local fairness in federated learning. It is a novel contribution. \n- The proposed methods are practical, each communication round requires only one client-server communication, and the post-processing method doesn't require labels. \n- The experiments section is quite comprehensive, authors consider multiple datasets, baselines, and data split methods. \n\nWeaknesses\n- The framework supports only fairness constraints that are linear in the confusion matrix (DP, EO, EOP); important nonlinear notions such as predictive parity are out of scope. Can the confusion-matrix formulation be extended to ratios or conditional metrics such as predictive parity via a surrogate function that is linear in terms of the confusion matrix?", "was_revised": true, "final_justification": "I think that the authors clarified all the questions raised by me. They explained how the proposed method can be extended for fairness notions that are not linear in the covariance matrix. They presented new results under adversarial clients and how to mitigate their effects.", "rebuttal": "### For Weaknesses\n\n> **W1:** The framework supports only fairness constraints that are linear in the confusion matrix (DP, EO, EOP); important nonlinear notions such as predictive parity are out of scope. Can the confusion-matrix formulation be extended to ratios or conditional metrics such as predictive parity via a surrogate function that is linear in terms of the confusion matrix?\n\n\n**Response:** Thank you for your detailed attention to the scope of our framework and for providing such valuable feedback. We fully agree with your viewpoint and endeavour to broaden our scope in the following two aspects, while evaluating the feasibility of covering predictive parity to address your concerns.\n\n**(i) Ratios Metrics.** We indeed find that **fairness notions formulated as ratios metrics can be converted into linear constraints under certain conditions**, and our framework is well suited to enforce these fairness constraints in federated learning environments. Specifically, the ratios metric or constraints, which are formulated as $\\left|\\frac{\\sum_a\\langle \\mathbf{D}^a, \\mathbf{C}^{a}(h) \\rangle}{\\sum_a\\langle \\mathbf{G}^a, \\mathbf{C}^{a}(h) \\rangle}\\right| \\leq \\xi$ with constant metrix $\\mathbf{D}^a, \\mathbf{G}^a$ and group-specific confusion matrix $\\mathbf{C}^{a}(h)$ depend on classifier $h$, can certainly be transformed into multiple linear constraints if the sign of $\\sum_a\\langle \\mathbf{G}^a, \\mathbf{C}^{a}(h) \\rangle$ is unchanged for any $h$. When the denominator’s sign is uncertain, the feasible domain of $\\mathbf{C}^{a}(h)$ is non‑convex, precluding its expression via linear constraints. In fact, since each entry of $\\mathbf{C}^{a}(h)$ lies in [0,1], whenever the entries of $ \\mathbf{G}^a$ are sign‑consistent, the corresponding ratio constraint admits a linear‑constraint representation. For example, the **Calibration within Groups (CG)** which was proposed in [1] and further explored in [2], is a fairness metric in binary classification and can be formulated as $\\frac{FN^a}{FN^a + TN^a} = v_0; \\frac{TP^a}{TP^a + FP^a} = v_1$, where $TP^a, FP^a, FN^a, TN^a$ are derived from binary group-specific confusion matrix $\\mathbf{C}^{a}$ and $0 ≤ v_0 < v_1 ≤ 1$ have no implicit dependence on any entries of the fairness-confusion tensor. Because this fairness criterion appears as a ratio metric and every element of corresponding $\\mathbf{G}^a$ is non‑negative, it admits a linear‑constraint representation and can be realized in our proposed distributed framework. Moreover, the ratio metrics presented in [3] also can be formulated into multiple linear constraints based on the above analysis.\n\n**(ii) Predictive Parity.** In binary classification, Predictive Parity [4] is a typical conditional metric states that the likelihood of being in the positive class given the positive prediction is the same for each group: $|P(Y = 1|\\widehat{Y} = 1, A = 1) - P(Y = 1|\\widehat{Y} = 1)| \\leq \\xi$. Prior work [2] explored this form of constraint, demonstrating that **this notion of fairness is not linear in the group-specific confusion matrix** and presenting a quadratic-form representation based on the group-specific confusion matrix. Additionally, the other research [5] showed that the Bayes-optimal classifier under Predictive Parity constraint even cannot be achieved by threshold calibration. In multi-class classification, we did not find any existing literature that defines a multi‑class formulation of Predictive Parity. A possible natural extension could be expressed as $\\max_{y,a}|P(Y = y|\\widehat{Y} = y, A = a) - P(Y = y|\\widehat{Y} = y)| \\leq \\xi$. Following the same derivation in [2], it is obvious that Predictive Parity also cannot be expressed in multiple linear or ratio constraints under multi-class circumstances.\n\nOverall, certain ratio metrics can be reformulated as linear constraints and enforced within our framework. However, existing research indicates that conditional metrics like predictive parity cannot be represented in the linear form of group-specific confusion matrix.\n\n### For Questions\n\n> **Q1:** How would the proposed methods behave if some clients are adversarial?\n\n**Response:** Thank you for raising this important question regarding the behavior of our method under adversarial clients. We respectfully note that this paper emphasizes group-fairness calibration in federated learning and proposes a general framework that achieves accuracy-fairness balance. Meanwhile, we acknowledge that the presence of adversarial clients can interfere with the achievement of fairness in federated learning. Thus, we will investigate the presence of adversarial clients in two respects and provide a detailed response to your question.\n\n**(i) Applicability of robust aggregation methods:** we aim to implement group-fairness calibration and propose a general framework that leverages cost-sensitive loss and personalized optimization to achieve accuracy-fairness balance, without relying on specialized aggregation schemes. While FedAvg is employed here due to its straightforward implementation and effective performance under data heterogeneity, adversarial-resilient aggregation techniques (e.g., $(f, \\kappa)$-robust aggregator [6]) can also be integrated into our framework. Moreover, the personalized procedure in our framework inherently enhances the model’s robustness [7].\n\n**(ii) Experimental evaluation with adversarial clients:** We present experimental results demonstrating our model’s performance in adversarial‑client scenarios, as well as its behavior when employing defense mechanisms such as Byzantine‑robust aggregation methods CM and Nearest Neighbor Mixing (NMM) in [6]. We partition the Compas and Adult datasets into 5 and 10 clients, respectively, under a heterogeneous environment $(\\gamma=1)$, and employ **Label Flipping** as the attack method, with adversarial clients constituting 40% of the total. Results are shown in the following table. All results are the average of three repeated experiments.\n\nThe model performance in the absence of adversarial clients:\n\n|Dataset|Method|Acc|$\\mathscr{D}^{global}$|$\\mathscr{D}^{local}$|\n|---|---|---|---|---|\n|**Compas**|FedFACT(In)|60.13|0.0589|0.0738|\n| |FedFACT(Post)|67.20|0.0344|0.0614|\n|**Adult**|FedFACT(In)|82.39|0.0180|0.0499|\n| |FedFACT(Post)|81.87|0.0191|0.0223|\n\nThe model performance with adversarial clients and defense methods:\n\n|Dataset|Method|Acc|$\\mathscr{D}^{global}$|$\\mathscr{D}^{local}$|\n|---|---|---|---|---|\n|**Compas**|FedFACT(In)|56.33|0.1143|0.2224|\n| |FedFACT(In)+CM|58.18|0.0670|0.1032|\n| |FedFACT(In)+CM+NMM|59.25| 0.0809|0.0901|\n| |FedFACT(Post)|63.33|0.0632|0.0830|\n| |FedFACT(Post)+CM|65.56|0.0425|0.0749|\n| |FedFACT(Post)+CM+NMM|66.43|0.0399|0.0685|\n|**Adult**|FedFACT(In)|81.29|0.0855|0.1271|\n| |FedFACT(In)+CM|82.27|0.0389|0.0709|\n| |FedFACT(In)+CM+NMM|82.33|0.0217|0.0652|\n| |FedFACT(Post)|80.18|0.0393|0.0451|\n| |FedFACT(Post)+CM|80.89|0.0229|0.0346|\n| |FedFACT(Post)+CM+NMM|80.95|0.0206|0.0269|\n\nExperimental results indicate that, under adversarial‑client conditions, our model’s accuracy and fairness experience only modest declines rather than severe degradation, demonstrating that **the personalized procedure in our framework inherently enhances robustness**. When employing the Byzantine‑robust aggregation methods [6], both accuracy and fairness further recover to some extent, supporting our prior analysis.\n\nWe appreciate your insightful feedback once more. The above discussions and experiments will be incorporated into the revised version of the paper.\n\n[1] Inherent trade-offs in the fair determination of risk scores, ITCS 2017.\n\n[2] FACT: A Diagnostic for Group Fairness Trade-offs, ICML 2020.\n\n[3] Beyond Adult and COMPAS: Fair Multi-Class Prediction via Information Projection, NeurIPS 2022.\n\n[4] Fair prediction with disparate impact: A study of bias in recidivism prediction instruments, Big Data 2017.\n\n[5] Fair Bayes-Optimal Classifiers Under Predictive Parity, NeurIPS 2022.\n\n[6] Fixing by Mixing: A Recipe for Optimal Byzantine ML under Heterogeneity, AISTATS 2023.\n\n[7] Ditto: Fair and Robust Federated Learning Through Personalization, ICML 2021."}, {"review_id": "jJHvXcQahE", "reviewer": "Reviewer_facd", "summary": "This paper introduces FedFACT, a novel framework for achieving controllable group fairness in federated learning under multi-class classification. Addressing key challenges—harmonizing global and local fairness, and enabling an optimal trade-off between fairness and accuracy—FedFACT identifies Bayes-optimal classifiers under fairness constraints. It reformulates fair FL as a personalized cost-sensitive learning problem (for in-processing) and as a bi-level optimization problem (for post-processing). The authors provide theoretical convergence and generalization guarantees, and experimental results show that FedFACT outperforms existing methods in maintaining both accuracy and fairness across heterogeneous datasets.", "questions": "1) Integration of in-processing and post-processing methods: While the paper evaluates the in-processing and post-processing algorithms independently, the discussion suggests that they are complementary. However, it remains unclear whether and how these two approaches can be effectively combined. Could the authors clarify if a combined approach is feasible, and if so, provide empirical evidence or discussion on its performance? Demonstrating synergy between the two could strengthen the practical utility of the framework. \n\n \n2) Scalability to a larger number of clients: The current evaluation is conducted with a very limited number of clients (2 for COMPAS and 5 for other datasets). This raises concerns about the generalizability of the proposed approach to more realistic federated learning scenarios involving dozens or hundreds of clients. Could the authors provide insights into: \n\n* How the method scales as the number of clients increases? \n\n* Whether the fairness-accuracy trade-off holds under larger, more heterogeneous client populations? \n\n* Any plans or limitations in scaling up the evaluation? \n\n3) Sufficiency of evaluation samples: Relatedly, the paper does not clearly justify whether the current number of clients and data samples per client is sufficient for meaningful statistical evaluation. Could the authors elaborate on the sampling criteria and suggest whether additional clients or data partitioning strategies might yield more reliable insights?", "limitations": "yes", "rating": 5, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths\n\n* Strong empirical performance: The proposed FedFACT framework demonstrates outstanding empirical results, consistently outperforming baseline methods in both accuracy and fairness trade-offs across a variety of experimental settings. The ability to balance global and local fairness without significant performance degradation is a notable contribution.  \n\n \n* Theoretical grounding: The paper provides solid theoretical guarantees, including convergence and generalization bounds for the proposed framework, which adds rigor and credibility to the approach. \n\n \n* Novel formulation: Reformulating fair federated learning as a cost-sensitive learning problem and introducing a bi-level optimization framework is an original and intellectually interesting contribution. This provides a principled path for balancing fairness and accuracy. \n\nWeaknesses\n\n* Limited evaluation scale: The experimental evaluation appears to be conducted on relatively small datasets or a limited number of clients. This raises concerns about the scalability and generalizability of the approach to large, heterogeneous real-world deployments. \n\n\n* Practical usability and parameter tuning: The framework relies on tuning fairness relaxation parameters (ξg ,ξl ), but the paper does not provide a systematic or automated way to set these values. As a result, achieving the desired trade-off between accuracy, global fairness, and local fairness appears to require manual trial-and-error. This could hinder the practical adoption of FedFACT in real-world applications as such tuning can be tedious.", "was_revised": true, "final_justification": "I thank the authors for their thorough rebuttal, which clarifies design choices, provides large-scale ACSIncome and 100-client results, and details parameter tuning. The explanation of integrating in-processing and post-processing methods is clear and convincing. Overall, the responses reinforce the robustness, scalability, and soundness of the approach, and I have no remaining concerns regarding the work.", "rebuttal": "### For Weaknesses\n\n> **W1:** Limited evaluation scale.\n\n**Response:** We appreciate your attention to the details of our experimental section and your valuable question. We answer your question in the following two aspects.\n\n**(i) Scale of data:** We apologize for any misunderstanding. We evaluate our method on four available real-world datasets, with sizes ranging from approximately 6K to over 100K. Notably, both CelebA and ENEM comprise over 200K samples, and according to existing benchmark studies [1], they are almost the largest publicly available datasets in the fairness domain in terms of sample size.\n\n**(ii) Large, heterogeneous real-world deployments.** To demonstrate the real-world deployment of our method, we conduct experiments on the **ACSIncome dataset** [2] with 1,664,500 datapoints, divided into **50 clients representing U.S. states**, using race (white/non-white) as the sensitive attribute. The fairness metric is demographic parity (DP), with $\\xi^g=\\xi^l=0.01$ for both algorithms.\n\n|Method|Acc|$\\mathscr{D}^{global}$|$\\mathscr{D}^{local}$|\n|---|---|---|---|\n|**FedAvg**|79.80|0.1032|0.1857|\n|FairFed|75.22|0.0883|0.1460|\n|FedFB|76.53*|0.0676|0.1148|\n|FCFL|71.95|0.0651|0.0832|\n|praFFed|74.48|0.0733|0.0914| \n|Cost|75.40|0.0290|0.0642| \n| FedFACT$_{g l}$ (In)|76.39|0.0232|0.0360|\n| FedFACT$_{g l}$ (Post)|76.50|0.0108*|0.0249*| \n\nThe results show that FedFACT significantly improves the accuracy‑fairness trade‑off compared to other methods in real‑world deployments.\n\n> **W2:**  Practical usability and parameter tuning.\n\n**Response:** Thank you for your valuable question on parameter tuning. **The parameters $(\\xi^g, \\xi^l)$ control the tolerance for global and local disparity, and should be set based on real-world needs.** Our experiments show that tuning these parameters effectively maintains fairness within the desired thresholds, particularly when using the post-processing method.\n\nFurthermore, we provide a more fine‑grained parameter tuning strategy: \n\n(1) To minimize global or local group‑wise disparity, one can set the parameters $(\\xi^g, \\xi^l)$ to values close to zero (e.g., 0.01 or 0.001). \n\n(2) Considering that there exist practical fairness tolerance $(\\kappa^g, \\kappa^l)$. First, initialize $(\\xi^g, \\xi^l)$ to near‑zero values $\\rho$. After running the algorithm, the observed fairness gap $(Gap^g, Gap^l):=(\\mathscr{D}^g-\\rho, \\mathscr{D}^l-\\rho)$ predominantly reflects the generalization error due to finite sample size, and independent of the settings of $(\\xi^g, \\xi^l)$ according to our generalization error analysis (Theorems 9 and 10 in Appendix). One can set $(\\xi^g, \\xi^l) = \\big(\\max(\\rho, \\kappa^g-Gap^g), \\max(\\rho, \\kappa^l-Gap^l)\\big)$ to meet real‑world fairness tolerance.\n\n### For Questions\n\n> **Q1:** Integration of in-processing and post-processing methods.\n\n**Response:** We appreciate your attention to the details of our experimental section and your valuable question. We answer your question in the following two aspects. \n\n**(i) Theoretical Unsoundness.** Combining these two methods can only be achieved by using the in‑processing classifier as the pre‑trained model for post‑processing. However, this is theoretically unjustifiable. Note that post‑processing requires the pre‑trained classifier to approximate the ground‑truth Bayes score function [3,4]. While the in‑processing method is designed to learn a group-fairness calibrated classifier [5,6], without aiming to learn the Bayes score function. Consequently, integrating the two approaches does not enhance performance; instead, it magnifies the errors introduced by the pre‑trained model, leading to performance decline.\n\n**(ii) Experimental Evaluation.** We conducted comparison experiments to provide empirical support. We set $(\\xi^g,\\xi^l)=(0.05,0.05)$ for in‑processing and $(\\xi^g,\\xi^l)=(0.01,0.01)$ for post‑processing to construct FedFACT(In & Post)$_1$ and $(\\xi^g,\\xi^l)=(0.01,0.01)$ for both methods to construct the method FedFACT(In & Post)$_2$, with $\\gamma=0.5$.\n\n|Dataset|Method|Acc|$\\mathscr{D}^{global}$|$\\mathscr{D}^{local}$|\n|---|---|---|---|---|\n|**Compas**|FedFACT(In)|61.17|0.0407|0.0732|\n| |FedFACT(In & Post)$_1$|62.08|0.0334|0.0719|\n| |FedFACT(In & Post)$_2$|60.81|0.0275|0.0613*|\n| |FedFACT(Post)|67.33*|0.0139*|0.0641|\n|**Adult**|FedFACT(In)|82.04|0.0014|0.0401|\n| |FedFACT(In & Post)$_1$|82.32|0.0205|0.0386|\n| |FedFACT(In & Post)$_2$|81.72|0.0088*|0.0331|\n| |FedFACT(Post)|82.74*|0.0134|0.0274*|\n|**CelebA**|FedFACT(In)|86.15|0.0382|0.0473|\n| |FedFACT(In & Post)$_1$|86.74|0.0297|0.0363|\n| |FedFACT(In & Post)$_2$|85.80|0.0242|0.0267|\n| |FedFACT(Post)|87.06*|0.0093*|0.0172*|\n|**ENEM**|FedFACT(In)|68.33|0.0493|0.0487|\n| |FedFACT(In & Post)$_1$|68.41|0.0252|0.0206|\n| |FedFACT(In & Post)$_2$|67.98|0.0127|0.0213|\n| |FedFACT(Post)|71.21*|0.0056*|0.0198*|\n\nThe results indicate that while combining post‑processing and in‑processing, the accuracy-fairness balance of FedFACT (In & Post) is significantly inferior to FedFACT (In) or (Post) in most cases. \n\n> **Q2:** Scalability to a larger number of clients.\n\n**Response:** We appreciate your thorough review and the valuable question regarding scalability. We fully acknowledge your feedback. Due to space limitations, **detailed evaluations of model stability in terms of global fairness, local fairness, and accuracy with 2 to 50 clients are provided in Appendix D.5.** To strengthen our experiments, we have conducted additional tests on four datasets here, scaling up to 100 clients. \n- **Limitations in scaling up the evaluation.** As data heterogeneity and client number increase, local datasets exhibit pronounced group or label skewness and vacancies, such as missing multiple groups or classes, making local fairness evaluation infeasible. \nTherefore, to scale our evaluation appropriately, we conduct additional experiments focusing on the stability of accuracy and global fairness with client numbers ranging from 50 to 100, extending the experiments in Appendix D.5.\n\n| |Dataset|Compas| |Adult| |CelebA| |ENEM| |\n|---|---|---|---|---|---|---|---|---|---|\n|**Clients**|**Method**|**Acc**|$\\mathscr{D}^{global}$|**Acc**|$\\mathscr{D}^{global}$|**Acc**|$\\mathscr{D}^{global}$|**Acc**|$\\mathscr{D}^{global}$|\n|**50**|FedFACT(In)|60.33|0.0704|80.04|0.0104|84.53|0.0438|62.72|0.0607|\n|**50**|FedFACT(Post)|65.81|0.0447|81.40|0.0429|86.06|0.0338|66.05|0.0503|\n|**75**|FedFACT(In)|59.28|0.0848|79.65| 0.0189|82.76|0.0652|61.53| 0.0742|\n|**75**|FedFACT(Post)|64.33|0.0610|80.35|0.0572|84.34|0.0473|65.42|0.0588| \n|**100**|FedFACT(In)|56.66|0.1017|78.10| 0.0428|81.68|0.0789|60.05|0.0823|\n|**100**|FedFACT(Post)|63.85|0.0712|79.8|0.0812|83.06|0.0650|65.29|0.0877|\n\n**Method performance and accuracy-fairness trade-off.**  Overall, the results show that while the model’s performance declines slightly with an increase in the number of clients, it remains stable in terms of fairness, demonstrating its robustness. Specifically, although the model's accuracy is inevitably impacted as the number of clients grows, the global fairness metric does not experience a sharp decline. This indicates that the model can maintain a balance between accuracy and fairness even with a large number of clients, highlighting the scalability of our approach with respect to client number.\n\n> **Q3:** Sufficiency of evaluation samples. \n**Response:** Thank you for your attention to the details of our experimental section and your valuable question. We answer your question in the following three aspects.\n\n**(i) The sampling criteria and Data partitioning strategies.** Our experiments utilize sampling schemes based on two data-partitioning methods.(1) **Dirichlet partition** [7], where a proportion vector $\\alpha$ is generated from a Dirichlet distribution $Dir(\\gamma)$, and samples are allocated to clients based on these proportions; (2) **Heterogeneous correlation split**, a novel method that introduces random heterogeneous correlations between $A$ and $Y$ as detailed in Appendix D.3. We employ these methods to introduce heterogeneity and decouple global and local fairness, since significant differences can be observed only under a certain degree of group and label heterogeneity, as shown in [8]. Thus, our experiments explore the accuracy-fairness balance under such conditions, ensuring representativeness.\n\n**(ii) Client number:** We apologize for any misunderstanding. We restricted our comparative experiments to 5 clients in order to ensure that each client retains enough samples from every subgroup to reliably assess local fairness under heterogeneous data partitions. This is consistent with prior federated learning fairness research [7,9,10], which generally conducts comparisons with a limited number of clients (less than 4). Furthermore, In the scalability study (Appendix D.5), we have conducted detailed experiments on the impact of the number of clients on the model. \n\nWe sincerely appreciate your valuable questions. The discussions and experiments mentioned above will be incorporated into the revised version of the paper.\n\n[1] FFB: A Fair Fairness Benchmark for In-Processing Group Fairness Methods, ICLR 2024.\n\n[2] Retiring Adult: New Datasets for Fair Machine Learning, NeurIPS 2021.\n\n[3] Post-hoc Bias Scoring Is Optimal For Fair Classification, ICLR 2024.\n\n[4] Fairness Guarantees in Multi-class Classiﬁcation with Demographic Parity, JMLR 2024.\n\n[5] A Reductions Approach to Fair Classification, ICML 2018.\n\n[6] Fair Bilevel Neural Network (FairBiNN): On Balancing fairness and accuracy via Stackelberg Equilibrium, NeurIPS 2024.\n\n[7] FairFed: Enabling Group Fairness in Federated Learning, AAAI 2023.\n\n[8] Demystifying Local and Global Fairness Trade-offs in Federated Learning Using Partial Information Decomposition, ICLR 2024.\n\n[9] Addressing Algorithmic Disparity and Performance Inconsistency in Federated Learning, NeurIPS 2021.\n\n[10] PraFFL: A Preference-Aware Scheme in Fair Federated Learning, KDD 2025."}, {"review_id": "aM2lxRSCOX", "reviewer": "Reviewer_vZsk", "summary": "The paper FedFACT jointly enforces global and local group fairness in multi-class classification settings. It introduces both in-processing and post-processing methods that approximate Bayes-optimal fair classifiers by reformulating the fairness-constrained learning problem into personalized cost-sensitive classification and bi-level optimization. The authors support their proposed algorithm with both theoretical analysis and empirical evaluation.", "questions": "1: The paper is difficult to follow, particularly in Section 4.3, due to the heavy use of subscript and superscript notation. With clearer explanations and better organization of the derivations and formulations, this section would be significantly easier to understand.\n\n2: The in-processing approach produces personalized classifiers for each client, while the post-processing method enforces fairness constraints. However, in the experiments, the authors evaluate FedFACT (In) and FedFACT (Post) separately. Why not consider a combined approach—FedFACT (In & Post)? Would this integration potentially yield better performance than using either method alone? If not, what are the limitations or trade-offs that prevent such a combination?", "limitations": "NA", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1: The paper provides a unified framework for addressing both global and local group fairness in multi-class federated learning.\n\n2: FedFACT approximates the Bayes-optimal fair classifier in both the in-processing and post-processing stages by reformulating the problem as a personalized cost-sensitive learning task and a bi-level optimization, respectively.\n\n3: The authors offer a theoretical analysis to support the design and effectiveness of the proposed FedFACT algorithm.\n\nWeakness: \n\nSee the Questions section", "was_revised": true, "final_justification": "The authors have thoroughly addressed my concerns in the rebuttal. Therefore, I am raising my score to 4.", "rebuttal": "### For Questions\n> **Q1:** The paper is difficult to follow, particularly in Section 4.3, due to the heavy use of subscript and superscript notation.With clearer explanations and better organization of the derivations and formulations, this section would be significantly easier to understand.\n\n**Response:** Thank you for thoroughly reviewing our paper and for pointing out the heavy use of notation. We apologize for the density of notation, since the inherent complexity of the multi‑class fairness problem necessitates a certain level of notational detail. Meanwhile, we fully agree with your feedback and will improve the notation and explanations in the following three aspects to enhance the paper’s readability and organization.\n\n**(i) Notation Table.** To improve clarity, we plan to introduce the following notation table, which summarizes **the main symbols used throughout the paper**, particularly in Section 4 (Methodology), along with their descriptions. Note that, to avoid redundancy of including all symbol definitions, we have limited this table to the key symbols that appear frequently in the paper. Readers can refer to it as needed to quickly review the meaning of these notations utilized in the subsequent derivations.\n\n| Notation | Description | Notation | Description |\n| ------ | ------ | ------ | ------ |\n| $(X,A,Y,K)$ | Joint Distribution (Data, Group, Label, Client) | $h$ | Random Classifier |\n| $(\\mathcal{X},\\mathcal{A},\\mathcal{Y},\\mathcal{K})$ | Corresponding Sample Space | $\\mathcal{H}$ | Set of Random Classifiers |\n| $N$ | Number of Client | $p_\\delta$ | Probability of Event $\\delta$ (Detailed in Appendix 1.1) |\n| $n_k$ | Local Sample Size | $\\Delta_m$ | Probability Simplex |\n| $\\mathcal{D}_k$ | Local Dataset $\\{(x_{k,i}, a_{k,i}, y_{k,i})\\}_{i=1}^{n_k}$ | $\\widehat{Y}$ | Model Prediction $h(x)$ |\n| $\\eta$ | Bayes score function | $\\mathbf{C^{a,k}}(h)$ | Decentralized Group-Specific Confusion Matrix |\n| $\\mathbf{D}^{a,k}_{u_g}$ | Global Fairness Constraint Matrix (${u_g}\\in \\mathcal{U}_g$) | $\\mathbf{D}^{a,k}_{u_k}$ | Local Fairness Constraint Matrix (${u_k}\\in \\mathcal{U}_k$)|\n| $\\mathscr{D}^{g}_{u_g}(h)$ | Global Fairness Function $\\sum_{a,k} \\langle \\mathbf{D}^{a,k}_{u_g}, \\mathbf{C}^{a,k}(h) \\rangle$ | $\\mathscr{D}_{u_k}^k(h)$ | Local Fairness Function $\\sum_{a,k} \\langle \\mathbf{D}^{a,k}_{u_k}, \\mathbf{C}^{a,k}(h) \\rangle$ |\n| $\\mathcal{U}_g$ | Index Set for Global Fairness Constraints | $\\mathcal{U}_k$ | Index Set for Local Fairness Constraints |\n| $\\xi^{g}$ | Global Fairness Bounds | $\\xi^k$ | Local Fairness Bounds ($k \\in [N]$) |\n| $\\lambda$ | Global Dual Parameter $(\\lambda^{(1)},\\lambda^{(2)})$ | $\\mu$ | Local Dual Parameter $\\bigcup_{k=1}^N \\mu_k, \\ \\mu_k=(\\mu^{(1)}_k,\\mu^{(2)}_k)$ |\n| $\\Lambda$ | Value Domain of Global Dual Parameter | $\\mathcal{M}$ | Value Domain of Local Dual Parameter |\n| $\\mathbf{h}$ | Federated Model $(h_1,\\cdots,h_N)$ | $\\mathcal{L}(\\mathbf{h},\\lambda,\\mu)$ | Lagrangian Dual Function |\n| $\\Pi_{\\mathcal{X}}(x)$ | Projection Operator | $\\widehat{\\mathcal{L}}(\\mathbf{h},\\lambda,\\mu)$ | Empirical Lagrangian Dual Function |\n| $\\mathbf{M}^{\\lambda,\\mu}(a,k)$ | Fairness Calibration Matrix | $\\mathcal{P}^X_{a,k}$ | Probability Measure $P(X \\mid A=a, K=k)$ |\n\nNote that we improve the subscript or superscript $(l,k)$ to $k$ for more explicit formulation.\n\n**(ii) Clearer Presentation of the Formulas.** To enhance clarity, we will further make several modifications to the presentation of the formulas. First, we will revise certain notations to enhance readability, ensuring that the symbols are more intuitive and consistent throughout the paper (e.g.,   $\\mathbf{M}^{\\lambda,\\mu}(a,k) \\to \\mathbf{M}^{a,k}(\\lambda,\\mu)$ and the notation $(l,k)$ mentioned above). Some typos in Algorithm 2 will be corrected. Moreover, we also find that some of the more complex equations can be simplified. For example, the lengthy equation (7) can be simplified by using 1-norm $\\| \\lambda\\|_1$ and $\\| \\mu_k\\|_1$.\n\n**(iii) Intuitive Explanation:** We will insert an concise, intuitive overview of the theoretical results in Section 4.3 to guide readers in understanding the subsequent analysis: The optimal classifier under fairness constraints from a probabilistic perspective (the Bayes-optimal fair classifier [1,2,3])  in post-processing is derived in Theorem 5. We then propose a practical, unbiased objective in equation (8) to learn this optimal classifier. Proposition 6 and Theorem 10 in Appendix B.5 provide the theoretical evidence that the post‑processing method will approximate the optimal accuracy‑fairness trade-off as the number of iterations and sample size increase.\n\n> **Q2:** The in-processing approach produces personalized classifiers for each client, while the post-processing method enforces fairness constraints. However, in the experiments, the authors evaluate FedFACT (In) and FedFACT (Post) separately. Why not consider a combined approach—FedFACT (In & Post)? Would this integration potentially yield better performance than using either method alone? If not, what are the limitations or trade-offs that prevent such a combination?\n\n**Response:** We appreciate your attention to the details of our experimental section and your valuable question. In fact, combining in‑processing and post‑processing is theoretically unsound. In the following, we clarify the inherent theoretical limitations and support our claim through empirical results. We hope this resolves your concerns.\n\n**(i) Theoretical Unsoundness.** Combining these two methods can only be achieved by **using the classifier learned from the in‑processing phase as the pre‑trained model for the post‑processing method**, which then performs further fairness calibration. Nevertheless, this approach is theoretically unjustifiable. Note that **post‑processing requires the pre‑trained classifier to approximate the ground‑truth Bayes score function** [1,2,3,4]. A widely adopted approach in fairness post‑processing is to train probabilistic classifiers aimed at maximizing accuracy without fairness calibration, typically using cross‑entropy loss,  and to treat the model’s probabilistic output as an estimate of the Bayes score function. While **the in‑processing method is designed to learn an optimal classifier that fulfills fairness constraints [5,6], it does not aim to learn a probabilistic classifier that accurately approximates the Bayes score function.** Intuitively, the objective of the in‑processing method is to approximate Equation (3) in our paper, rather than the ground‑truth Bayes score function $\\eta(X,A,K)$. Consequently, integrating the two approaches does not improve the model’s performance; instead, it magnifies the errors introduced by the pre‑trained model, leading to a reduction in performance.\n\n**(ii) Experimental Evaluation.** We conducted comparative experiments to provide empirical support for the aforementioned conclusion. The experiment is conducted on all four datasets, using a Dirichlet partition $\\gamma=0.5$ to construct the heterogeneous data. We set $(\\xi^g,\\xi^l)=(0.05,0.05)$ for in‑processing and$(\\xi^g,\\xi^l)=(0.01,0.01)$ for post‑processing to construct FedFACT(In & Post)$_1$ and $(\\xi^g,\\xi^l)=(0.01,0.01)$ for both methods to construct FedFACT(In & Post)$_2$, with all other settings identical to those in the original paper.\n\n\n|Dataset|Method|Acc|$\\mathscr{D}^{global}$|$\\mathscr{D}^{local}$|\n|---|---|---|---|---|\n|**Compas**|FedFACT(In)|61.17|0.0407|0.0732|\n| |FedFACT(In & Post)$_1$|62.08|0.0334|0.0719|\n| |FedFACT(In & Post)$_2$|60.81|0.0275|0.0613*|\n| |FedFACT(Post)|67.33*|0.0139*|0.0641|\n|**Adult**|FedFACT(In)|82.04|0.0014|0.0401|\n| |FedFACT(In & Post)$_1$|82.32|0.0205|0.0386|\n| |FedFACT(In & Post)$_2$|81.72|0.0088*|0.0331|\n| |FedFACT(Post)|82.74*|0.0134|0.0274*|\n|**CelebA**|FedFACT(In)|86.15|0.0382|0.0473|\n| |FedFACT(In & Post)$_1$|86.74|0.0297|0.0363|\n| |FedFACT(In & Post)$_2$|85.80|0.0242|0.0267|\n| |FedFACT(Post)|87.06*|0.0093*|0.0172*|\n|**ENEM**|FedFACT(In)|68.33|0.0493|0.0487|\n| |FedFACT(In & Post)$_1$|68.41|0.0252|0.0206|\n| |FedFACT(In & Post)$_2$|67.98|0.0127|0.0213|\n| |FedFACT(Post)|71.21*|0.0056*|0.0198*|\n\nThe results indicate that while combining post‑processing and in‑processing can improve fairness at the cost of accuracy, the accuracy-fairness balance of FedFACT (In & Post) is significantly inferior to FedFACT (Post) in most cases. Additionally, on the Adult dataset, the global fairness metric of the in‑processing method initially outperforms the post‑processing method, but when combined with post‑processing, global fairness is weakened. These findings provide additional support for the conclusions of our previous theoretical analysis.\n\nWe sincerely appreciate your valuable questions. The modifications, discussions, and experiments outlined above will be incorporated into the revised version of the paper.\n\n[1] Post-hoc Bias Scoring Is Optimal For Fair Classification, ICLR 2024.\n\n[2] Fairness Guarantees in Multi-class Classiﬁcation with Demographic Parity, JMLR 2024.\n\n[3] Fair and Optimal Classification via Post-Processing, ICML 2023.\n\n[4] Fair Bayes-Optimal Classifiers Under Predictive Parity, NeurIPS 2022.\n\n[5] A Reductions Approach to Fair Classification, ICML 2018.\n\n[6] Fair Bilevel Neural Network (FairBiNN): On Balancing fairness and accuracy via Stackelberg Equilibrium, NeurIPS 2024."}]

Author rebuttal:

Meta-review: This paper presents a novel controllable federated group-fairness calibration framework called FedFACT. It identifies the Bayes-optimal classifiers under both global and local fairness constraints in multi-class case. It balances fairness and accuracy by reformulating fair federated leanring as personalized cost-sensitive learning problem for in-processing, and bi-level optimization for post-processing. Convergence and generalization guarantees for FedFACT are developed. Extensive experimental results show that FedFACT consistently outperforms baselines in balancing accuracy and global-local fairness.

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Overall, the work is solid and carries substantial merits. All reviewers are on the positive side. I agree with their assessments. 

A few issues are raised in the reviews, and authors are encouraged to address them in the final version. In particular, authors are encouraged to improve the readability of certain sections, and discuss possible adaptions of related works on binary classification to the multi-class setting studied in this paper (e.g. what are the challenges, how would the adapted results compare to the present results). It would also be nice to discuss possible integrations of the proposed global fairness and local fairness solutions.
Decision: Accept (poster)