Voronoi-grid-based Pareto Front Learning and Its Application to Collaborative Federated Learning
Authors: Mengmeng Chen
Xiaohu Wu
QIQI LIU
Tiantian He
Yew-Soon Ong
Yaochu Jin
Qicheng Lao
Han Yu
Venue: ICML 2025 poster (2025)
Keywords: Hypernetwork
Multi-objective optimization
Hypervolume
Collaborative federated learning
Primary area: applications->everything_else

Abstract: Multi-objective optimization (MOO) exists extensively in machine learning, and aims to find a set of Pareto-optimal solutions, called the Pareto front, e.g., it is fundamental for multiple avenues of research in federated learning (FL). Pareto-Front Learning (PFL) is a powerful method implemented using Hypernetworks (PHNs) to approximate the Pareto front. This method enables the acquisition of a mapping function from a given preference vector to the solutions on the Pareto front. However, most existing PFL approaches still face two challenges: (a) sampling rays in high-dimensional spaces; (b) failing to cover the entire Pareto Front which has a convex shape. Here, we introduce a novel PFL framework, called as PHN-HVVS, which decomposes the design space into Voronoi grids and deploys a genetic algorithm (GA) for Voronoi grid partitioning within high-dimensional space. We put forward a new loss function, which effectively contributes to more extensive coverage of the resultant Pareto front and maximizes the HV Indicator. Experimental results on multiple MOO machine learning tasks demonstrate that PHN-HVVS outperforms the baselines significantly in generating Pareto front. Also, we illustrate that PHN-HVVS advances the methodologies of several recent problems in the FL field. The code is available at
https://github.com/buptcmm/phnhvvs.

Voronoi-grid-baseed Pareto Front Learning and Its Application to Collaborative Federated Learning
Mengmeng Chen 1
Xiaohu Wu 1
Qiqi Liu
Tiantian He Han Yu+
Yew-Soon Ong Yaochu Jin Qicheng Lao
Abstract
1. Introduction
Multi-objective optimization (MOO) is a critical area of research in machine learning, addressing problems that in- volve optimizing multiple conflicting objectives simulta- neously. In real-world applications such engineering design (Nedjah & Mourelle, 2015; Yi et al., 2021), financial planning (Nobre & Neves, 2019; Doumpos & Zopounidis, 2020), and resource allocation (Gong et al., 2019; Wu et al., 2019a; 2025; Ma et al., 2023), MOO problems are ubiqui- tous, where solutions need balance trade-offs between competing goals. Pareto- Front Learning (PFL) has emerged effective technique address these problems, utilizing Pareto HyperNetworks (PHNs) to generate solutions along the Pareto front. PFL allows the learning of a mapping function that takes given preference vector as an input and outputs the corresponding solution on the Pareto front, pro- viding a principled way to obtain optimal trade-offs between different objectives (Navon et al., 2020).
Multi-objective optimization (MOO) exists exten- sively in machine learning, and aims to find a set of Pareto-optimal solutions, called the Pareto front, e.g., it is fundamental for multiple avenues of research in federated learning (FL.). Pareto- Front Learning (PFL) is powerful method imple- mented using Hypernetworss (PHNs) to approx- imate the Pareto front. This method enables the acquisition of mapping function from given preference vector to the solutions on the Pareto front. However, most existing PFL approaches still face two challenges: (a) sampling rays in high-dimensional spaces: (b) failing to cover the entire Pareto Front which has convex shape. Here, we introduce novel PFL framework, called as PHN-HVVS, which decomposes the design space into Voronoi grids and deploys genetic al- gorithm (GA) for Voronoi grid partitioning within high-dimensional space. We put forward a new loss function, which effectively contributes to more extensive coverage of the resultant Pareto front and maximizes the HV Indicator. Experi- mental results multiple MOO machine learning tasks demonstrate that PHN-HVVS outperforms the baselines significantly in generating Pareto front. Also, we illustrate that PHN-HVVS ad- vances the methodologies several recent prob- lems in the FL field. The code is available https://gihub..oo/bbtttmmtt http://ttco/////hb//eee..
Despite the promising potential of PFL approaches, two significant limitations persist in existing methods: (a) dif- ficulties in efficiently sampling rays in high-dimensional spaces, and (b) challenges in covering the entire Pareto front, which is essential for obtaining a comprehensive set of solutions. These limitations have been widely acknowl- edged in prior work. For instance, the difficulty of sam- pling in high-dimensional spaces has been highlighted in (Hoang et al., 2023). Similarly, the issue of inadequate Pareto front coverage has been extensively studied in (Hua et al., 2021; Zhang et al., 2023; Hoang et al., 2023). To ad- dress these issues, we propose a novel framework for PFL, termed PHN-HVVS. This framework novelly decomposes the design space into Voronoi grids (Aurenhammer & Klein, 2000), with the help of a genetic algorithm (GA) (Holland, 1992) for nigh-dimensiooal Voronoi grid partitioning. This partitioning technique allows for more effective exploration of the Pareto front.
"quual contribution 'Beijing University of Posts and Telecom- munications, China 'School of Engineering, Westlake University, China 'Centre for Frontier AI Research, Institute of High Perfor- mance Computing, Agency for Science, Technology and Research, Singapore 'College of Computing and Data Science, Nanyang Technological University, Singapore. Correspondence to: Han <uan. yu@ntu couunnu..cdu.... edu.sg>, Xiaohu Wu <xiaohu wu w bupt.edu.en>>.
Furthermore, we propose a new loss function within this framework, which significantly enhances the coverage of the Pareto front and maximizes the Hypervolume Indicator (HV) (Zitzler al., 2007). HV metric simultaneously evaluate the convergence and diversity of set of solutions,
Proceedings of the 4dnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Voronoi-grid-hee based Pareto Front Learning
which refer to how closely the solutions approximate the true Pareto front (PF), and how well the solutions are spread across the entire PF, respectively. Through extensive ex- periments on various MOO machine learning tasks, we demonstrate that PHN-HVVS outperforms existing base- lines by generating more well-balanced and diverse Pareto fronts, especially in convex problem scenarios. Our ap- proach not only improves the coverage of the Pareto front but also enhances the computation of benefit graphs (Cui et al., 2022; Tan et al., 2024; Chen et al., 2024; Li et al., 2025), delivering notable improvements compared to tra- ditional methods. This work lays the foundation for more effective and scalable MOO solutions in machine learning.
multaneously using single hypernetwork. PHN-LS and PHN-EPO cover the Pareto front by varying the input pref- erence vector. PHN-LS uses linear scalarization with the preference vector as the loss function. PHN-EPO uses the EPO (Mahapatra & Rajan, 2020) update direction for each output of PHNs. Co-PMTL (Lin et al., 2020) uses Pareto MTL to connect the preference vector and corresponding Pareto solution. HV (Zitzler et al.. 2007), an important in- dicator for comparing the quality of different solution sets, is also utilized to optimize the hypernetwork. SEPNET (Chang et al., 2021) expresses PFI. as a new MOO prob- lem and then defines the fitness function as HV. HVmax (Deist et al., 2021) uses a dynamic loss function for neural network multi-objective training, optimizes dynamic loss with gradient- HV maximization, and performs well in approximating the Pareto front. PHN-HVI (Hoang et al., 2023) employs multi-sample hypernetwork and utilizes HV for hyperparameter optimization. PSL-HV1 and PSL.- HV2 (Zhang et al., 2023) address the problem of PFL from the geometric perspective. Different from the method of this paper that focuses on calculating the gradient of HV, Zhang et al. (2023) adopt the polar coordinate system and utilize the R2 indicator to obtain the Pareto front. Despite the significance of the algorithms above, there is room to improve the coverage of the entire Pareto front since they typically cluster around the middle and neglect the boundary solutions, and only identify partial solutions.
Organization. Section 2 provides an overview of the related works. In Section 3, we introduce the related techniques such as Voronoi Diagram and Pareto front learning for MOO. In Section 4, we describe the proposed PHN-HVVS frame- work in detail. In Section 5, we experimentally validate the effectiveness of PHN-HVVS. Finally, we conclude this paper in Section 6.
2. Related Work
Applications of Voronoi Diagram in Multi-Objective Op- timization The Voronoi Diagram is a pivotal tool in MOO, facilitating the partition of spaces to enhance solution dis- tributions and convergence. Its applications span various domains, including estimation of distribution algorithms (EDAs). airspace sector redesign, and zoning design. The Voronoi-based EDA (VEDA) (Okabe et al., 2004) lever- ages Voronoi cells to model solution distributions. It ap- plies PCA to project high-dimensional spaces data into low-dimensional space and then uses Voronoi grids. This approach outperforms traditional methods such as NSGA-II (Deb et al., 2002) in terms of adaptability and search effi- ciency. Combined with genetic algorithms (GAs), Voronoi Diagram partitions airspace sectors in 2D space to mini- mize multi-objective costs (Xue, 2009), achieving balanced airspace management. Similarly, in zoning design of 2D space, Pirlo& & Impedovo (2012) utilizes Voronoi Diagram alongside multi-objective GA to determine optimal zone numbers and boundaries. These instances underscore the versatility and efficacy of Voronoi Diagram in addressing complex optimization challenges across diverse fields. How- ever, applying Voronoi grids directly to high-dimensional MOO remains underexplored.
Federated Learning Federated Learning(FL) is a highly important paradigm of distributed machine learning that allows multiple FL participants (FL-PTs) to collaborate on training models without sharing their private data (Kairouz et al., 2021; Yang et al., 2020; Guo et al., 2025). This is particularly true in the era of foundation models (Ren et al., 2025; Fan et al., 2025; Yi et al., 2025; Wang et al., 2024). It is often viewed as collaborative network of FL-PTs where a FL- can be complemented by other FL-PPTs with different weights and a personalized model is trained for each individual FL-PT (Tan et al., 2022; Li et al., 2025; Miao et al., 2023; Williams, 2023). From some perspective, FL can be viewed as a type of multi-task learning. which is further type of MOO (Sener & Koltun, 2018; Marfoq et al., 2021; Bai et al., 2020; 2024). There are n FL-PTs, denoted by V= {U1 U2 ...... } here n is also the number of learning tasks. PFL can generate a Pareto front from which we can choose a n-dimennsional preference vector s** for each FL-PT o, such that the model performance of this FL-PT is the best when applying this vector. Then, yd* can be used to characterize the data complementarity between FL-PT's and quantifies the importance/weight of other FL-- PTs V - {w to U, All such preference vectors {r*** H can be used to form a directed weighted graph benefit graph G, among FL-PTs, which has played a fundamental role in guiding the design of collabrative FL network in many use
Pareto front learning for Multi-Ohjective Optimization In (Navon et al., 2020), the author proposes the new term Pareto-Front Learning (PFL) and describes an approach to PFL implemented using HyperNetworks which is termed as Pareto HyperNetworks (PHNs). PHNs is a more efficient and practical way since it learns the entire Pareto front si-
2

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Voronoi-grid-base based Pareto Front Learning
cases (Cui et al., 2022; Wu & Yu, 2024; Ding & Wang, 2022; Bao et al., 2023; Pan et al., 2016; Tan al., 2024; Chen et al., 2024; Li et al., 2025; Zhou et al., 2025). Some works assume that the values of {r'* are known, and study how to determine a subgraph Gu. of G. that satisfies some desired properties to form stable coalitions, avoid conflicts of interests, or eliminate free riders. In Gu, there a direct edge from by to U vy will make a contribution to v, in the actual FL. training process, and it defines the collaborative FL network. The work of this paper aims to achieve better coverage of the Pareto front and thus we can choose a more precise preference vector for each FL-PT. It can ad advance all the methodologies in these use cases when Hypernetworks used to obtain the benefit graph.
Definition 3.1 (Dominance principle). For any 01, 02. we say that 0 dominates 02, denoted as 01 02, if and only if
(v, (x)) = (v, (v, f(z, 02)), ViE (1.2......A) ................................ .N (3) and Ely, f(x 01)) + Cyy, fzz 02)).
Definition 3.2 (Pareto set and Pareto front). The solution 8; is called Pareto optimal when it satisfies: 70, s.t. 0j - 0;. The Pareto set is defined as:
T = {0, E θ p0 s.t. 02 - 0.}.
(4)
The corresponding images in the objectives space are Pareto front T, = Ely. ())
HV is a commonly used metric to evaluate the quality of a Ty. The HV of set A is the J-dimensional Lebesgue measure of the region dominated by Ty and bounded from above by reference point.
alffrrt campling sooledds accerdlog dlescuuss
Af 0) geeeatts weigee for suget reooods
Definition 3.3 (The hypervolume indicator). The HV indi- cator of set A is defined as:
0.
6x)
H. (4) A((a' Ea E A:aaa a' and a' N R])
(5)
e
where A() denotes the Lebesgue measure, and R E R is reference point.
min maper Q(C(O, ui) + Σ P(r.e) (2a))-0
Definition 3.4 (Voronoi Diagram). Let X' be a metric space and P = {PI, ................................ ,NN) } be set of points in X. The Voronoi grid (p:) associated with a point pt is the set of all points in X' that are closer to pt than to any other point in P. Formally,
Figure I. Multi Multi-Sample Hypernetwork framework
3. Preliminary
The goal of Multi-task learning is to find 0* E θ to optimize
defined as:
J loss functions (Sener & Koltun, 2018; Gupta et al., 2022):
V(p.) = far E X s(o.))) + #
(6)
0" arg min (x fly, f(;
(L)
where d(x, p.) denotes the Euclidean distance between point or and point pi. The point p; is referred to the site of Voronoi grid (p.). The Voronoi Diagram is the union of all such grids, that is ((P) = U V (p..).
where Ely, f(;; 0)) (y (x; 0), ..................... Cyl, f(;; 0))}
PD denotes the data distribution, E₁ yxyy - R denotes the j-th loss function and f(x; 0) X x 2 + y denotes the neural network with 0. PFL extends this framework to learn the complete Pareto front. A hypernetwork is a deep model that generates the weights of the target network, limited by its input. Let h(r; Φ) represents the hypernetwork with parameter Φ. and let f represents the target network with parameter 6, specially, h(r;)) employs Multilayer PcrCeptronnMLPP) to map the input preference vector to higher dimensional space to construct shared features. These features create weight matrices for each layer in the target network through fully connected layers. We are trying to find the best parameter 6":
eise
Hyperokure
Fasto Sctatves Duskand Saksea
Bideees Paidd
gill
f
(a) Hypervolume
(b)Voronoi Diagram
Figure 2. Hypervolume (left) and Voronoi Diagram (right)
arg min Eyppge Eyopgs (p F( r) (2) DER"
Figure 2(a) depicts the HV of the two-objective optimization problem, and Figure 2(b) depicts the Voronoi Diagram in the case of two-dimensional space.
where -.) is an extra criterion function such as LS or EPO function.
3

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Voronoi-grid-bant based Pareto Front Learning
4. Algorithm
distances, i.e.,
4.1. Multi-Sample Hypernetwork
sm E (px) if d( pk) min d(sms Pt)
(9)
i= N N
In this subsection, we introduce the existing challenge in multi-sample hypemetwork and propose solution to solve it efficiently.
In order to improve the algorithm efficiency, we further introduce the KD tree (Bentley, 1975). Let T be the KD- Tree constructed from the P. to find the nearest neighbor of simulation point sm we use the KD-Tree search algorithm, which we denote as N (sm; T), and the nearest point pv of Sm using the KD-Tree can be written as: Pk NNN(sm T)). So Eq. (9) can be transformed into
4.1.1. EXISTING CHALLENGE
We introduce the efficiency issue in multi-sample hyper- network identified in (Hoang et al., 2023). Specifically, multi-sample Hypernetwork samples N preference vec- tors r' and use h(r; σ) to generate N target networks far i E [1...., N) where 02 = M(r', (). We need to evenly divide the space into N sub-regions and collect vectors r' from these sub-regions. In J- J-dimensional space, the hyperplane H can be fully described as:
5m E (Px))
NN T)
(10)
Here, based on the geometric distribution, we define uni- formity as the situation where the number of points falling into each grid is equal. We use GA to find a more uniform partition of the space, the aim is to maximize the objective function O. We set the objective function as follows:
....................... x.) H. ((EI.)...,.,)
E | +2 1}
(7)
where 1 2 xj 2 0, a partition with N sub-regions should satisfy:
N Σ
==
(c((()))) - =z)
(11)
N U = H and O Oiags Ω2 0 i=1
(8)
O 1/(1+p)
(12)
where (p(())) is the count of points in grid (px:) and 7
In the two-dimensional space, the polar coordinate is used for sampling to evenly divide the angle in 2N 1,....N N.
is the average quantity of points per grid. When Var - 0,
O - 1; and when Var -> oo, O -> 0. An ideal partition should satisfy (8) and the value of O in Eq.(12) is 1, where the number of points contained in each grid is equal. Algorithm finally generates a fixed Voronoi partition of the hyperplane which has the maximum value of O. Figure shows the grids generated by 3D Voronoi decomposition.
The high-dimensional uniform partitioning algorithm pro-
posed by (Das & Dennis, 1998) encounters significant chal- lenge when dealing with points in J-dimensional (J >= 3) space: its computational complexity grows exponentially with the increase in dimensions, the number of points in the space ' and k = 1/2 is (J+ computational load that restricts the arbitrary setting of the number of rays.
& ") leading to an immense
4.1.2. SOLUTION: VORONOI SAMPLING
To decompose the high-dimensional design space H into separate regions and effectively cover the high-dimensional space with an arbitrary number of rays, we propose a ge- netic algorithm (GA)-based Monte Carlo Voronoi struc- ture for high-dimensional region decomposition (Auren- hammer & Klein, 2000; Rubinstein & Kroese, 2016) as shown in Algorithm 1. The time complexity of Algorithm I is shown in Appendix A. Let P = {pI, ... }NN be the set of Voronoi site and S = {si,
41
SM} be the set of simulation points in H. For each simulation point 5m = =(mm,1, Sm,2: ... sm.J) E S, satisfing the constraint Ej=1 8mj we need to calculate the Fuclidean distance d(sm.. P.) between Sm and each site Pt, then we sort the distances d(sm=P) (immPi)).e PI), d(smm (mm.P))..... d((m= PN) for each sm in ascending order. The simulation point sm belongs to the grid of Pke where d(s,, Pk)) the minimum among all
1253
Figure 3. Voronoi Diagram with J ==3 3, N = 16, M 100000
4.2. PHN-HVVS
The Voronoi Sampling method enables sampling rays across the entire H. Building on this, we propose a new objective
4

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Voronoi-grid-ben based Pareto Front Learning
Algorithm 1 Voronoi Sampling
set of solutions reaches its maximum, then these solutions are the Pareto front (Fleischer, 2003). The HV term im- proves overall solution quality by pushing the front closer to the true Pareto front, while the penalty term (D (r', (*)))) en- sures that the resulting distribution covers the entire Pareto front, regardless of its shape. This prevents solutions from concentrating in specific areas, a common issue the con- vex part of Pareto optimization. Traditional MOEAs rely the diversity of evolutionary population to search PF. The MOEA/D(Zhang & Li, 2007) and NSGA-III(Deb & Jain, 2013) variants can address convex problems by adopt- ing adaptive reference vectors. Differently, PFL typically uses a hypernetwork to approximate the PF. Existing PFL methods employ gradient optimization to maximize HV by approximating gradients with HV contributions. However, as shown in (Zhang et al., 2023), convex PF boundary so- lutions suffer weight decay: intermediate solutions have larger HV gradients, causing preferential fitting of central regions. In traditional MOO, such weight decay does not need to be addressed, and MOEAs directly search boundary solutions through population diversity maintenance, with- out gradient reliance in PFL. Unlike cosine similarity, the distance-based penalty directly controls the geometric re- lationship between solutions and preferences, enabling the discovery of prefeeeee solutions to achieve com- plete coverage of the Pareto front. By working together, the HV term expands the front globally, and the distance-based penalty term refines its distribution, achieving a high-quality and diverse Pareto front. The entire algorithmic process is presented in Algorithm
the number of species num_speccies, the num- ber of points per species N, dimension J. Monte Carlo simulation number M and num-geneerakons
Input:
Initialize population s.t. Σj= Pij L,Pis E [0, 1]. repeat
Monte Carlo rand points set S = {si, S2...., SM where 8m (Sm,1; 2m,2: ... Sm, - Smj
in H.
Voronoi Decomposition by Eq. (10).
Compute fitness by Eq. (12). Tournament selection: Choose the individual with the highest fitness as the parent.
Crossover: child
parenti + (1 a)
parent,, a E [0, 1].
Mutation: child = child + e, (o0 (0.05)²). Update the fitness of the child.
Select the individual with the highest fitness as the basis for the next generation.
until epoch > num-ggeneet num_generatioos
Sample rays in the finally generated Voronoi grids.
function to better explore the solution space and achieve complete Pareto front.
Pareto HyperNetworks with HV maximization via Voronoi
Sampling (PHN-HVVS) is designed to solve the following objective:
N
More detail can be seen in A Appendix A.4. Our goal is optimize the sole parameter to find the Pareto front.
min E r-pgo x(L( + + Σ F(r + (x.))~p
(13)
i=1
where
= (e,,..., C satisfies VQ( x v)) > O. F is defined as D(r (t), D(r", e) represents the Euclidean distance from &i (fi.e...... ...,... Eg) point p' (ri.r2..... r.) along the line with direction vector u (1,.... 1)1x,. The vector re = l' - (ci ri, - -))r)) We project the vector r' onto the direction vector v. and the projec-
x. y)).
f
f
(1, -1f--i)) tion coefficient t. where -p Σi Ej nu
Eq. (14) is the distance from rd to the line which can be computed by the magnitude of the vector r tu.
f
Figure 4. D(F', with the direction vector (1,1)
e( =
CG) (r) + +t,)))²
(14)
5. Experiments
J E (C) - -r))
In this section, we will compare the performance of PHN-HVVS with other baselines PHN-HVI (Hoang et al., 2023), PHN-EPO, PHN-LS (Navon et al., 2020), COS- MOS (Ruchte & Grabocka, 2021) and PHN-TCHE (Lin et al., 2022) in terms of the HV evaluation metric (Zitzler et al., 2007). This metric is crucial as it can comprehen-
E/=1 u3
Figure 4 depicts D( e in 2D space.
The combination of HV and the penalty term optimizes both the quality and diversity of the Pareto front. If the HV of
5

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Voronoi-grid-bant bareid Pareto Front Learning
Algorithm 2 PHN-HVVS
Problem 2 (Lin et al., 2019) with two Objective Functions
1: while not converged do
2: if J = 2 then
3: r² ..... N partition sample
(1()) = 1 exp
0
vd
4: else
5: r'. P. ~ Voronoi Sampling by Algorithm 1
(16)
6: end if
(2( = 1 exp
exp
0+ vd
7: compute 0; = h(r- ②) for i = -,,2,.. N Di=l E==1
s.t. 0 R d 100
2HV 8f, (0...))))) (
8: compute
9: Supdate == g A A 00 D'rr('0' (r),e)
Problem 3 DTLZ2 (Zitzler et al., 2000) with three Objec- tive Functions
10: Φ 5 Φ- 17 Supdate
11: end while
12: return
10
(0) cos
π 0
cos
02
Σ (0:
0.5) +1 1
2
10
(2( = cos (1 क
sin
. (022)
Σ
(0, - 0.5)² +1
11=3
10
sively reflect the quality of solutions in MOO problems. In high-dimensional space, HV can be effectively approx- (Bader Zitzler, 2011), and there standard in- dicators .hv in Python's is Pymoo library used extensively for compute HV. For the error of HV, when J > 3, the module in the library uses Monte Carlo method to sample about 10000 points to estimate the HV value. The error rate decreases with the square root of the sample size, and the actual error can controlled within the range of 1% to 5%. In this paper, the value of this HV is only computed once and used for the final evaluation of algorithm performance. We do not calculate the specific value of HV every round, but instead use gradient descent method to obtain the & that minimize (13). By analyzing their respective HV values, we can clearly understand the effectiveness of our approaches obtaining a diverse and high quality set of solutions.
(x( sin (012)
T
Σ
(0: 0.5) -1
1=3
s.t. 0 E R 0
01
3
(17)
Problem 4 DTLZ4 (Zitzler et al., 2000) with three Objec- tive Functions
10
(0) cos
(8p
cos
0 2
Σ (0 0.5) +1
10
(0) = cos (00
sin
00 2)
Σ
(0 - 0.5) 0.5)2++1
1=3
10
(0) = sin (or क
Σ 1=3
(0 - 0.5) -0.5)++1 +1
2
All experiments are trained on the NVIDIA GeForce RTX3090, the installed CUDA version is 12.2.
s.t. 0 E TM 0 < 0: < 1,a 100
(18)
5.1. Toy Examples
Problem 5 ZDTI (Zitzler et al., 2000) with two Objective Functions
For toy examples, the hypernetoork uses 2-layer hidden MIP to generate the parameters of the target network, which consists of a single linear layer. For consistency, all meth- ods share identical hyperparameters and run for the same number of iterations on the same problem.
min (0,)) 01
(01)
min (2(
1
Problem (Liu al., 2021) with two Objective Functions
(19)
m
0g
(0) 1+1
(m
1)
s.t. 0 < 0, < 1, i 1, M, m = 2
(0) = (0)2, (0) = (0 - ()) s.t. DR
(15)
Problem 6 ZDT2 (Zitzler et al., 2000) with two Objective
6

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Voronoi-grid-bant based Pareto Front Learning
Functions
5.2. Multi-Task Learning
min (01) = 01
Datasets We conduct experiments on datasets with different tasks. The datasets for the two tasks include MultiMNNST (MM.) (Sabour et al., 2017), Multi-Fashion (MF., Multi- (Fashion+MNIST) (FM.), and Drug Review (Drug) Dataset (Gräßer et al., 2018). MM. is constructed by randomly se- lecting two images from the original MNIST (LeCun et al., 1998) dataset, placing one in the top-left corner and the other in the bottom-righ corner of new image. Each digit can be shifted by up to 4 pixels in any direction. Similarly, the MF. dataset is created by overlapping FashionMNIST items (Xiao et al., 2017), as well as FM. dataset by com- bining overlapping MNIST and FNSHI items. The training, validation, and testing ratio for each dataset is 11:1:2. The Drug Review dataset is examined by two tasks: (1) Prediction of the drug's rating through regression- baseed methods; (2) Classification of the patient's condition. The training, validation, and testing ratio for each dataset is 0.65:0.10:0.25.
min (2(0) = =g(0) h(4 (01),,
m
g(0) 1+ m -
Σ 02
(20)
2=
((E( =1 e (01)
g(0)
s.t. 0 5 0, 1, m,
m 2
Problem 7 VLMOPI (Van Veldhuizen & Lamont, 1999) with two Objective Functions
1
(0) -
|012
4n
(0)
-10 10 2
1
(21)
4n
s.t.n = 30
Problem 8 VLMOP2 (Van Veldhuizen & Lamont, 1999) with two Objective Functions
The Jura dataset (Goovaerts, 1997) has 4 tasks. In the Jura dataset, the objective variables are zinc, cadmium, copper, and lead, representing 4 tasks. The dataset is di- vided into training, validation, and testing with a ratio of 0.65:0.15:0.20.
1
(0) = 4n 1012
(0) 00 2
1
(22)
For higher-dimensional data, the SARCOS (SAR.) (Vi- jayakumar, 2000) dataset is used, which has 7 tasks. The SARCOS dataset aims to predict 7 joint torques, repre- senting 7 tasks, from a 21-dimensional input space, which includes joint locations, 7 joint velocities, and joint ac- celerations. Additionally, 10% of the training data is set aside for validation.
4n
s.t. 2 < 8, < 2, n = 10
The Pareto fronts of Problems 1, 5, and 7 are convex, while those of Problems 2, 3, 4, 6, and 8 are concave.
The HV values of the results can be found in Appendix B.3. By carefully observing the results presented in the figure, we can clearly see that the Pareto front obtained (denoted by the blue points) by method, completely covers the entire true Pareto front. This demonstrates the robustness and adaptability of our approach in handling diverse Pareto front shapes. Compared to baseline methods, our approach achieves the best performance in terms of both diversity and quality of solutions.
Model Architecture We construct the hypernetwork for all datasets that employs 2-layer hidden MLP to generate the parameters of the target network. For image classification, multi-LeNet (Sener & Koltun, 2018) is used as the target network. For text classification and regression, the target network is TexICNN (Rakhlin, 2016). The target network in Jura and SAR. is Multi-Layer Perceptron with 4 hidden layers containing 256 units.
It is obvious that PHN-HVI can effectively cover the con- cave portion of the Pareto front. Nevertheless, when it comes to the convex Pareto front, the solutions tend to be concentrated in the middle part of the front. PHN-EPO performs well in Problem 2 but finds the part of true Pareto Front in other Problems. is well-recognized that PHN-LS effectively covers the convex part of the Pareto front. How- ever, for concave Pareto fronts, such as in Problems 2, 3, 4, 6, and 8, it tends to gravitate toward the two ends (Boyd, 2004). The combined use of LS and cosine similarity in COSMOS leads to poor training performance when their optimization directions conflict. Similar to PVN-HVI, un- der mild conditions, PHN-TCHE aggregates in the middle region of the convex Pareto front.
Evaluation Every method uses 25 preference vectors to eval- uate each method. When dealing with two-task datasets, the reference point is set to (2, 2). and for multi-task datasets, is set to (l..... 1) in accordance with the settings in (Hoang et al., 2023). Table presents the HV values under the given reference point setting in the form of meantstandard deviation(std), which are the results of five independent runs.
The Pareto front of PHN-HVVS is widely dispersed. PHN- HVVS has greater HV than previous methods. Our method has ability to effectively navigate the solution space and identify the Pareto Front. Additionally, our method allows
7

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Voronoi-gre coannoi-gri--uuee bassid Pareto Front Learning
F6.00
Figure 5. Results comparison on different Multi Task Datasets
Table
Results compared to the statc-of-the-ar methods on Hypervolume.
PHN-EPO
PHN-LS
PHN-TCHE COSMOS
PHN-HVI
PHN-HVVS
MM.
2.97480.004488 2.976=0.007291 2.969=0.010703 2.977=0.014792 2.993=0.017194
3.008:0.016559)
MF.
2.272+0.0044666 2.283=0.030211 2.275+0.023285 2.274#0.039690 2.303=0.01699 2.328+0.033270
FM.
2.887 0223533 2.881400027041 2.905+0.014980
2.857#0.019973
1.900=0.027066
2.947=0.020800
Drug. 1.19410.005474 005474 1.174=0.010247 1.206+0.004243
NA
1.308=0.0813000
5520=0.003622
Jura 0.876#0.041177 0.897=0.045139 0.88340.044926
0.892#0.041800
0.922#0.0444 0444198
193550.012844
SAR.
0.85240.071599 0.85840.070854 0.726#0.063612 0.865+0.068888
0.929#0.031036 0.93940.026444
us to freely specify the number of rays regardless of the number of tasks, providing greater flcxibility in handling complex optimization problems. In the drug review exper- iment, COSMOS fails to converge due to its mapping of the preference vector into an excessively high-dimensional embedding space. Figure 5 displays the results of one of the five runs, highlight the practical applicability of our method to complex multi-objective optimization tasks across various domains.
leterogneety/oom/eeennaary between FL-PPTs. Cui et al. (2022), Tan et al. (2024), and Chen et al. (2024) all use the hypernetwork technique to generate the benefit graph Go. and the effectiveness of using Hypernetwork to generate G, is also validated in (Li et al., 2025). Specifically, there are n FL-PTs. Each FL-PPT o; has local dataset DD and a risk/loss funetion L': R - R.,. Given learned hypothesis q, let the loss vector E(g) [e] ;.... (d) represent the utility loss of the FL-PTs under the hypothesis q. We say is Pareto Solution if there is no hypothesis q' that dominates q. Let = (ri,....rh) ~ Dir( E Rn denote a preference vector. Each vector represents the weight the objective local model loss which is normalized with Σn_ r = 1 and r 2 0, Vk E [I.....n] n]. The hypernetwrkk HN takes input and outputs Pareto solution g, i.e.,
5.3. Federated Learning
In this subsection, we experimentally illustrate that the pro- posed method of this paper advances the methodologies of several problems in the FL. field.
Datasets and data heterogeneity CIFAR-10 dataset (Krizhevsky et al., 2009), a benchmark for image classi- fication, comprises 60,000 natural images evenly distributed across 10 object categories (6,000 samples per class). 10% of the training data are used for the validation split. The
g +- HN()rr).
(23)
where & denotes the parameters of the hypernetwork. For each FL-PT vú linear scalarization can be used. An optimal data heterogeneity is simulated via two typical approaches: pathological distribution (PAT.) (McMahan et al., 2017: Collins et al., 2021; Cui et al., 2022) and Dirichlet dis- tribution (Dir.) (Tan et al., 2022; Ye et al., 2023), with / = 0.5. particular, eICU (Pollard et al., 2018) is dataset that gathers electronic health records (EHRs) from numerous hospitals across the United States for patients admitted to the intensive care unit (ICU). The objective is to forecast mortality during hospitalization. The training, validation, and testing sets are in a ratio of 0.::0.15:0.155 The number of FL-PT's is set to 10.
....,,"h preference vector p = (rf* ,r1",""" ......... ,.h is determined to generate the hypothesis qd* that minimizes the loss with the data D,. This is expressed
g-- = HN where p"* arg mnx (ó)(
(24)
where Per (e,(H)() denotes the performance of HN r") on the validation data of u, For each FL-PT es the value of rr is used as an estimate to the weight of uJ
to 1/+ {r'*} ] defines a directed weighted graph, i.e., the benefit graph G..
Based on the number of FL-PTs, which corresponds to the dimension of the objective functions, different sampling
Benefit Graph The benefit graph G, characterizes the data
8

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Voronoi-grid-bon based Pareto Front Learning
methods adopted according this paper. Meanwhile, the objectives transformed into Eq. (13). More details can be seen in Appendix A.5.
Table 5. Performance comparison CIFAR-10 under Dirichlet distribution (3 = 0.5) with different compete ratio (MTA, FedE. goists).
Metries MTA represents the mean test accuracy. We use MTA to measure the performance of different methods on CIFAR-10. address the severe class imbalance the eICU dataset (negative samples over 90%), we adopt the AUC (Area Under the ROC Curve) metric for performance evaluation, it is statistically robust to skewed label distri- butions. The results are presented in the form of mean + std under five runs.
Dir(3 = 0.5)
FedEgoists
,+HVVS
0.05
62.16::.722
62.94
0.1
61.93
63.59-11088
0.2
62.39=1.26
6.17
0.3
62.47:0.59
62.5
0.4
61.21+0.80
62.00=2.40
In Table 2, we carry out experiments based on the work of Cui et al. (2022). We can see that better results are obtained by adding our method.
Table 6. Performance comparison on CIFAR-10 under pathological distribution with different compete ratio (MTA, FedEgoists).
PAT.
Fedigs
+HHVS
Table Performance comparison on CIFAR- 10 under different data heterogeneity (MTA, CE).
0.05
80.60=1.67
80.41
0.1
80.87+1.01
81.16
0.2
80.85+1.07
81.76=0.53
CE
,+HVVS
0.3
11.49:1..46
81.53
PAT.
86.88 Dir(3 = 0.5) 47.13+2.003
74.00
0.4
81.37=1.011
81.63=2.29
28.35
PHN-HVVS into these methodologies in FL has brought significant improvements.
Table Performance comparison CIFAR-10 under different data heterogeneity (MTA, FedCompetitors)
6. Conclusion
FedCompetitors
+HHVVS
In this paper, we propose the PHN-HVVS algorithm that leverages Voronoi Sampling in high-dimensional spaces. Additionally, we introduce a novel objective function to tackle the PFL problems. Our approach is capable of delv- ing deeper into the Pareto front space, thereby obtaining broader range of Pareto front that can better approximate the true front. This provides a more accurate representation of the optimal trade-offs in the PFL problems. Moreover, we apply the proposed method to federated learning. Extensive experiments show that the proposed method can adaptively work well under federated learning setting. This work can inspire further developments in the application of MOO techniques to federated learning.
PAT.
82.19 0.41
82.44 0.35
Dir(3 0.5)
47.96 0.75
51.72 1.39
Table 4. Performance comparison eICU (AUC, FedEgoists).
Fedigossts
,HHVVS
U0
66.36:119.28
12.18.45
U1
81.58=6.65
86.04:1..41
U2
66.04133.21
71.9716.54
1/
84.40=5..76
15.39
11.84
12.89
V5
68.41=5.60
66.8018.14
U6
56.86=7.552
44.90
U7
17.97+14.94
91.77=2.116
A Acknowledgements
U8
90.50+
100.00++0.00
This work is supported in part by the National Key R&D Pro- gram China (Grant No. 2024YFE0200500); the National Natural Science Foundation of China (Grant Nos. 62327801 and 62302147); the International Collaboration Fund for Creative Research of the National Science Foundation of China (NSFC ICFCRT) (Grant No. W244 the Min- istry of Education, Singapore, under its Academic Research Fund Tier (RG101/24); the National Research Foundation, Singapore, and DSO National Laboratories under the Al Sin- gapore Programme (AISG Award No. AISG2RP-2020-011
Ug
79.88-8.29
87.18
Avg 74.79
79.80
FedCompetitors (Tan et al., 2022) and FedEgoists (Chen al., 2024) considered the competing relationships between FL--PTs or the self-interest of each FL.-PT. The competi- tion rate in the experiment of FedCompetitors is set to 0.2. We also test the performance of FedEgoist under different competition rates and different data distributions. Over- all, PHN-HVVS plays fundamental role and integrating

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Voronoi-grid-bane -aseed Pareto Front Learning
and the MTI, Singapore, under its AT Centre of Excellence for Manufacturing (AIMfg) (Award W25MCMF014).
Das, and Dennis, E. Normal- Normal-boundary intersection: A method for generating the pareto surface in nonlinear multicriteria optimization problems. SIAM journal on optimization, 8(3):631-65)) 1998.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Deb, K. and Jain, H. An evolutionary many-objective opti- mization algorithm using reference- point- baitt nondomi- nated sorting approach, part i: solving problems with box constraints. IEEE transactions on evolutionary computa- tion, 18(4):577-601, 2013.
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Zitzler, E., Brockhoff, D., and Thicle, L. The hypervolume indicator revisited: On the design of pareto-compliant in- dicators via weighted integration. In Evolutionary Multi- Criterion Optimization: 4th International Conference,
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Voronoi-grid-basee Pareto Front Learning
A. Theoretical Analysis.
A.1. The Time Complexity of Algorithm 1
Initializing the population involves assigning values to an array of size N x J. Since there are nummsspccies species in total, the time complexity is ('nuu_sspciin x N x J). The time complexity of generating M random points through Monte Carlo simulation is ((M x J). When performing Voronoi decomposition, the complexity of building KD-tree is O(N (N),,)), and the complexity of querying the nearest neighbors is (M log(N)), Therefore, the total time complexity of decomposition is O((N + M log(()) (N)). The time complexity of calculating the fitness is Ofuumssppeiie x N). The time complexity of crossover and mutation is Onnum seeeicssseessss x N x J). Since the main loop is executed num num- generationssss times, the overall time complexity of the Algorithm is Ofmum-generralon x (N + M) log(N))
A.2. The details of Algorithm 1
Algorithm 1 uses Monte Carlo simulation to generate points located on hyperplane H. in each round, and then searches for the nearest Voronoi site pn. for these M points. The genetic algorithm is applied to optimize the objective function in (12). In genetic algorithm, we randomly select three individuals and then choose the optimal individual from them. The crossover rate a is uniformly distributed within the range of [0.1]. The mutation method is to randomly perturb a certain point, and the range of the mutation is controlled by the parameter mutation_ste In this article, mutation std 0.05. When a newly generated point exceeds the valid range [0, I] in any dimension, the algorithm calculates scaling factor to project the point onto the nearest boundary while preserving its original direction, provided that the directional variation component in that dimension is non-zero. This ensures that the mutated points are within a reasonable range of values.
A.3. The relationships between Algorithm and 2
Algorithm 1 optimizes the objective function in Eq.(12) using genetic algorithm, ultimately yiclding N fixed Voronoi grids. In each grid, the Voronoi sites P = PP,,P,.... PPN) } and simulation point set S = {s1. s., (M } are stored, with each 8m S assigned a label indicating its partition. Algorithm 2 calls Algorithm at line and directly leverages the generated Voronoi grid structure thereby to perform fast sampling in each round within these grids (simulation point sets with the same label) without requiring recalculation. Each grid V (p:) is equivalent to a subregion 22', As detailed in (Hoang et al., 2023), adopting partition can enhance the performance of the HV and penalty functions. The round-by by-round sampling in the fixed Voronoi grids aims to explore the entire space and obtain complete PF. The Voronoi partition guarantees the coverage of each region during the sampling process, facilitating efficient and comprehensive exploration.
A.4. The process of HvMaximization
The HvMaximization is MOO algorithm that calculates dynamic weights for optimizing HV using a method based on HIGA-MO (Wang et al., 2017). The operation begins by performing non-dominated sorting (Deb et al., 2002) on the provided multi-objective solutions to create multiple Pareto fronts. Next, dynamic reference point is calculated by scaling the maximum values of the objectives with factor of 1.1. The dynamic reference point is updated to ensure that it remains outside the current solution space. Once the fronts are identified and the reference point is adjusted, the algorithm calculates the HV gradients, which represent the rate of change in HV with respect to each solution in the front, through Algorithm 3 (Emmerich & Deutz, 2014). To ensure consistency the objective space, the gradients are normalized. If any gradient has a norm close to zero, it is adjusted to avoid division by zero errors. After normalization, the gradients are used to compute dynamic weights for each solution, which reflect how much each solution contributes maximizing the overall HV. The overall goal is to maximize the HV by assigning appropriate weights to each solution, allowing for better exploration and exploitation of the solution space. Algorithm 3 illustrates the detailed steps of the GRADMULTISWEEP function.
A.5. The generation of benefit graph G.
In FL, the n FL-PTs correspond to learning tasks. The heterogeneity of data across FL-PTs entails evaluating the complementarity of data between FL-PTs, where a benefit graph arises. In multiple works (Cui et al., 2022; Tan et al., 2024; Chen et al., 2024), the benefit graph is assumed to be known, and estimated by PFL schemes. Optimization process is further taken in these existing FL algorithms for different purposes. The preciseness of these parameters directly affects the performance of these FL algorithms where the PFL scheme used previously is the one (i.e., PHN-LS) in (Navon et al., 2020). However, it can also be the scheme (i.e., PHN-HVVS) proposed in this paper. With -VVS, all these previous FL
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Voronoi-grid-bane barddd Pareto Front Learning
Algorithm 3 GRADMULTISWEEP
Require: Vector Y with subvectors, reference point
Ensure: Partial derivatives for each subvector component
1: Divide subvectors into sets Z, U,
2: if U j 0 then
3: output One-sided derivatives in U
4: end if
5: Set derivatives of Z subvectors to 0
6: Compute derivatives for E' subvectors
7: for each dimension j do
8: Project P subvectors to (J 1)-dim, omit j-th coordinate.
9: Sort projected subvectors by j-th coordinate, add queue Q Initialize tree T for non-dominated points
10:
11:
while Q is not empty do
12:
Remove max element q from Q
13:
Compute HV change (q, T)
14:
Set derivative Tytita = (q, , T), where i(q) is the index that corresponds to the index of the original subvector
8H
in Y of which is the projection.
15:
Add q to T, remove Pareto dominated points
end while 17: end for
algorithms achieve a better performance since the PF better covered.
Below, we introduce the three relevant works. First, Cui al. (2022) study the collaboration equilibrium in FL where FL-PTss are divided into multiple groups coalitions. Let (i)) denote the unique coalition to which FL-PT belongs. Cui et al. (2022) study how to form core-stable coalition structure æ, which guarantees that there is coalition C such that every FL-PT i E C prefers C over T(i). Only FL-PTs within the same coalition contribute to each other, forming subgraph of Go. Second, in cross-silo FL, FL-PTs are typically organizations. FL-PTs in the same market area may compete while those different areas are independent. Tan et al. (2024) extend the principle "the friend of my enemy is my enemy" to prevent FL.-PTs from benefiting their enemies in collaborative FI., forming a subgraph of G6- Third, Chen et al. (2024) address self-interested FL-PTs in cross-silo FL and propose framework to simultancously eliminate free riders (who benefit without contributing) and avoid conflict of interest between FL-PTs, also forming subgraph of Go.
In this paper, we employ PHN-HVVS to generate the benefit graph G.. introducing modifications to the sampling distribution and solver method. Initially, preference vectors of equal length were generated according to the number of FL--PTs. These preference vectors could follow uniform distribution, Dirichlet distribution, or the novel Voronoi distribution proposed in this study. Subsequently, these preference vectors served as inputs to the hypernetoork, which then generated the weights for the target network. The target network processed image features to produce prediction results, which were compared with the ground-truth labels compute the loss. Following this, solvers, including the LS and the newly proposed HVVS in this paper, utilized to address the total loss and conduct gradient optimization. In contrast to traditional FL algorithms, this approach innovatively integrates the hypernetwork architecture with multi-objective optimization, thereby offering novel solution for generating the benefit graph.
B. Experimental Details
B.1. The notation table
To improve the readability of our paper, we've provided a summary of key notations in Table 7.
B.2, The objective of baselines
PHN-EPO The PHN-EPO optimizes ming Erra the expected value of EPO loss. PHN- EPO requires solving each EPO subproblem.
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Voronoi-grid-bant based Pareto Front Learning
Table 7. The notation table.
Variable
Definition
The set of FL.PPTs
The i-th FL-PT
The preference vector/ray
R
The reference point
The number of FL-PT's
The parameter of hypernetwork
The parameter of target network
((.-)
The loss functiin
h(r: (VHN(():)
The hypernetwork
(a;
The target network
d(s-)
The Euclidean distance between different points
The direction vector (I...., x)
D(r,,1
The Euclidean distance from E to point r' along the line with direction vector u The set Voronoi sites The set simulation points The i-th Voronoi site The m-th simulation points The number of Voronoi grids
P
s
Ps
3m
N
M
The number of simulation points
J
The number of objectives The Pareto front
T
T
The KD-Tree
(P)((p)))
The Voronoi grid with the Voronoi site p:
The hypothesis/The pareto solution
PHN-LS The PHN-LS optimizes ming (t. Ill,, 0))r), this method can not find the concave part of the Pareto front.
PHN-TCHE The PHN-TCHE optimizes ming Er-pgs (2.9)-p max(C(y.ffr, 0)), r), this method can not find the concave part of the Pareto front.
COSMOS The COSMOS optimizes ming (as)s (ny) E(ly,, ()) A cos(r, L), this method combines LS with consine similarity.
PHN-HVI The PHN-HVI optimizes min.- - (2.)) (e find the convex part of the Pareto front.
cos (+t, (), this method can not
B.3. The Hyper Volume of Toy Examples
The reference point for the objectives-functions is (2,2), for the three abjectives-functions is (2.2.2). higher HV indicates better-quality Pareto front. The Pareto front generated by our method exhibits the largest HV among all the compared baselines. This remarkable outcome not only clearly shows its ability accurately approximate the true Pareto front but also vividly demonstrates its superiority in MOO. The results are presented in Table 8 and Figures 10-16, where the table reports the mean and standard deviation across five independent runs, while the figures illustrate the outcomes of one run among these five.
B.4. Different sampling techniques comparison
We conduct extended experiments on the Jura and SARCOS datasets, comparing our proposed method against multiple naive sampling techniques, including: Uniform random sampling, Latin hypercube sampling, Polar coordinate sampling, Dirichlet distribution sampling. K-means clustering- based representative selection. The sampling points within each Voronoi region are closest to the site of that region, which naturally avoids the problem of local aggregation or omission in the
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Yoronoi-grid-bant based Pareto Front Learning
Table Results comparison different problems.
PHN-EPO
PHN-LS
PHN-TCHE
COSMOS
PHN-HVI
PHN-HVVS
Pro.1
3.790=0.007230 3.832+0 000526 3.813+0.005223
3.210H0.736292
3.791=00002238
3.833200000026
Pro.2
3.304=0.063633 3.36240.016258 3.32140.086994 3.34420.043419 4.374#0.011347 3.38720.007168
Pro.3(DTLZ2)
7.299=0.010924 6.015#0.084161
7.303=0.011461 7.345#0.019812
03885400006700
7.38820.001225
Pro.4(TL224)
7.380+0.010336
4020=#1096995
7.267=0.010933
7.236:0.010072
7.383#0.0055 005471
7.386+0.003612
5ZDD5(
3.651+0.075391
3.65410.085641
NA
3.631#0.036321
3.520#0.01
3.735:0.001754
Pro.6(ZDT2)
2.047 1.044599
2.0=0
2.800=0.652911
3.057+0.528551
4.321=0.022386 3.32340.004026
Pro. Pro.((VLMPP))
3.810#0.00366
3.825
3.816
3.747+0.012842
3,782+000005233 3.829400000029
(ro.8VLLMP22
L.308+0.0207777
3.31 3.329 3.2000433300000 3.299*0.014255 3.335#0.001013
3.33940.000001
RdComputiors
+WWS
645
030
506
0.06
502
30
0.2
6.02
Figure 6. Pathological distribution Benefit Graph
sampling distribution. Voronoi partition method ensures effective exploration of the global PF, As shown in Table 9, the proposed approach outperforms other strategies.
Table 9. Comparison of different sampling techniques Jura and SARCOS
Random
Latin
Polar
Dir.
K-means
Voronoi
Jura SARCOS 0.884
0.928
0.922
0.928
0.923
0.925
0.935
0.883
0.881
0.888
0.877
0.949
B.5. The details of the federated learning experiment
For the CIFAR-10 dataset, the hypernetwork leverages a 2-layer hidden MLP to generate the parameters of the target network
(He et al., 2024; 2021; Shang et al., 2023; Han al., 2023; Wu et al., 2019b; 2021: network is the For the eICU dataset, construct single-layer hidden MLP hypernetwork. The target network utilizes Transformer classifier with Layer Normalization. The reference point is set as (I, 1,). We integrate the proposed method in this paper into CE, FedCompetitors and FedEgoists. As can be seen from the table 2-6, the proposed method has brought about some degree of improvement.
Wu Loiseau, 2023; 2024). The target as (Cui al., 2022; Tan al., 2024; Chen et al., 2024) and the reference point is set (3, (3.....33 3,).
In the experiment of FedCompetitors, we set up a total of FL-PTs. domly assign two categories of CIFAR-10 (5,000 images in each category) to each FL-PT. In the Dirichlet distri- bution, we 3 0.5 and the distribution vector is drawn from Dir(3) for each FL-PT. We can get the Ben- efit Graph shown in Figure 6 and 7. We the competition rate of the competition matrix to 0.2. This means that the probability of any two FL-PTs being competitors is 0.2. We randomly generate the competition graph in five runs and then observe the performance of the algorithm. At certain run, the competing FL-PTs are (vo, Ug), (2), 02), ( (U2, U4), (Va, 04), (Va, U6), (V4, va). (U4, 27), (24, 0g) the Pathological distribution case. The accuracy of FedCompetitors is 82.15 and the accuracy of.+ is 82.23. This is illustrated in Figure 8, In the Dirichlet distribution case, the competing FL-PPTs are (Vo, 02), (vi, v2), (1i, 18), (V2, us), (V2, 07), (0y, v4), (W3, 0s), (Vo, Ug). The accuracy of
the Pathological distribution, we ran-
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Voronoi-grid-bant baredd Pareto Front Learning
RedCompetiors
.WHWS
0.5
038
0.00
0.04
0.22
0.22
Figure 7. Dirichlet distribution Benefit Graph
12
17
V
17
170
17
17g
17g
1/
17
V5
1/
V/
17
1/
177
V7
V6
competition graph
data usage graph((FeCCompetttrs)
data usage graph(.+HH +HVVS)
Figure 8. The Competition Graph and Data Usage Graph under Pathological distribution
FedCompetitors is 48.28 and the accuracy of.+HVVS is 53.54. This illustrated in Figure We can find that the proposed method upplication can better identify the relationships that align with the actual situations of FL-PTs, thus leading to better experimental results.
B.6. Computer Resources
We utilize 8 NVIDIA GeForce RTX 3090 with 24GB of memory. The installed CUDA version is 12.2, and the graphics driver version is $35.104.05.
122
173
0g
17
17g
17
1/
VS
Vg.
V8
V6
competition graph
data usage graph(FedCooppttion))
data usage graph(,+Hi) +HVVS)
Figure 9. The Competition Graph and Data Usage Graph under Dirichlet distribution
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Voronoi-grid-ban based Pareto Front Learning
Figure Results comparison on Problem!
Figure 11. Results comparison Problem2
Figure 12, Results comparison on Problem3
Figure 13. Results comparison on Problem4
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Voronoi-grid-bant based Pareto Front Learning
Figure Results comparison on Problem5
Figure 15. Results comparison on Problemó
Figure 16. Results comparison on Problem7
Figure 17. Results comparison on Problem8
20


Reviews:
[{"review_id": "wHEoUTVNx0", "reviewer": "Reviewer_mv1y", "summary": "This paper proposes a method for sampling reference points (rays) from the unit simplex based on Voronoi tessellation and a genetic algorithm. Furthermore, the addition of the Hypervolume (HV) indicator to the objective function of the PHN further improves performance. The algorithm's capabilities are evaluated on synthetic functions and Federated Learning.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Notes**: All the tables and figures can be found at https://anonymous.4open.science/r/icml_rebuttal-E75A/rebuttal.pdf.\n\n**Response to W1 \\& W2**:  We appreciated your insightful questions, which ever motivated us to develop the solution of this paper. \n\nSpecifically, there is a mapping of rays to solutions. We aim to find a set of Pareto-optimal solutions that can well cover the entire Pareto front (PF). We fully agree on that the Riesz s-energy based iterative optimization method in [1] can generate uniformly distributed points or rays in high-dimensional spaces. However, a set of uniformly distributed rays does not necessarily lead to a set of uniformly distributed solutions, as shown in [A1] below and Figure 1. When rays are generated in advance, using such fixed rays generated by [1] may fail to cover some parts of the PF. \n\nThus, we propose to dynamically sample the preference vector during the optimization process, so that the hypernetwork technology can continuously explore the uncovered areas of the target space. The method proposed in this paper does not directly generate a globally uniform point set, but instead performs preference vector re-sampling based on local units of Voronoi grids generated by Alg. 1. Even with a fixed set of points generated by [1], generating preference vectors within the cell formed by these points still faces challenges, such as the geometric complexity problem in high-dimensional space [A2]. \n\nAbove, we explain why dynamic sampling of rays is used. In the ray sampling process, optimization is also taken in Alg. 1 and 2 to avoid recalculation, like your suggestion. Specifically, Alg. 1 uses Monte Carlo simulation to generate m simulation points S located on hyperplane $\\mathcal{H}$ at each iteration:  $\\mathcal{H} =${$(x_1, \\ldots, x_J) \\in \\mathbb{R}^J \\mid x_1 + \\ldots + x_J = 1$ } and then searches for the nearest Voronoi site  $p_i \\in P =$ {$p_1, p_2, \\ldots, p_n $} for these m points. The genetic algorithm is used to find the partition method with the maximum objective function, ultimately obtaining n Voronoi grids (while storing the simulation point sets of each grid). Alg. 2 directly utilizes the Voronoi grid structure generated by Alg. 1 and performs fast sampling directly in each round of these grids (simulation point sets) without the need for recalculation.\n\n\n[A1] Multi-objective deep learning with adaptive reference vectors, NeurIPS'22\n\n[A2] An optimal convex hull algorithm in any fixed dimension, DCG'93\n\n**Response to W3**：We have added experiments on the DTLZ1 benchmark problem [A3]: we measured the results for 4, 5, and 6 objectives, respectively, where the reference points were set as (2,..., 2). It can be seen from Table 1 that our method achieves the best result.\n\n[A3] Deb et al. Scalable multi-objective optimization test problems\n\n**Response to W4**：In high-dimensional space, HV can be effectively approximated [A4], and there is a standard indicators .hv in Python's Pymoo library used extensively for compute HV. For the error of HV, when J>3, the module in the library uses Monte Carlo method to sample about 10000 samples to estimate the HV value. The error rate decreases with the square root of the sample size (i.e. $\\frac{1}{\\sqrt{N}}$), and the actual error can be controlled within the range of 1\\% to 5\\%. \n\nIn this paper, the value of this HV is only computed once and used for the final evaluation of algorithm performance. We do not calculate the specific value of HV every round, but instead use gradient descent method to obtain the $\\phi$ that minimize Eq. (13). The HV gradient computation of this paper follows (Hoang et al., 2023,Wang et al., 2017, Emmerich & Deutz, 2014). \n\n[A4] HypE: An algorithm for fast hypervolume-based many-objective optimization, EVCO'11\n\n**Response to W5**: To address your concern, we will better clarify in the final version that this paper focuses on the emerging paradigm of Pareto Front Learning (PFL). Compared with traditional MOEAs, PFL has its own challenges to be addressed specially. \n\nSpecifically, traditional MOEAs rely on the diversity of evolutionary population to search PF. Variants of MOEA/D and NSGA-III can address convex problems by adopting adaptive reference vectors. Differently, PFL typically uses a hypernetwork (HN) to approximate the PF. Existing PFL methods employ gradient optimization to maximize HV by approximating gradients with HV contributions. However, as shown in (Zhang et al., 2023), convex PF boundary solutions suffer weight decay: intermediate solutions have larger HV gradients, causing preferential fitting of central regions. In traditional MOO, such weight decay does not need to be addressed, and MOEAs directly search boundary solutions through population diversity maintenance, without gradient reliance in PFL.\n\n\n**Response to W6**: We have optimized the visualization; please see Figure 2 in the link.  \n\nIn the final version of this paper, we will better clarify the above contents.", "claims_and_evidence": "Please refer to \"weaknesses\".", "methods_and_evaluation_criteria": "Please refer to \"weaknesses\".", "theoretical_claims": "This paper does not include theoretical claims.", "experimental_designs_or_analyses": "Please refer to \"weaknesses\".", "relation_to_broader_scientific_literature": "N/A.", "essential_references_not_discussed": "Please refer to \"weaknesses\".", "other_strengths_and_weaknesses": "Strengths\n\n1. This paper is well-written and easy to follow.\n\nWeaknesses\n\n1. The paper claims to address the challenge of sampling rays in high-dimensional space for multi-objective optimization. However, this challenge has been tackled by existing methods, such as the energy minimization approach proposed by [1], which can generate any number of rays in arbitrary dimensionality.\n2. According to Algorithm 2, the rays seem to be re-sampled in each iteration. According to Algorithm 1, the generated rays are solely dependent on the dimensionality (J) and the desired ray size. Consequently, these rays could be pre-computed and reused across iterations, or even across different optimization tasks.\n3. The empirical study primarily focuses on 2-3 objective optimization problems. Since the proposed sampling method is particularly applicable to high-dimensional situations, it is suggested to include more high-dimensional problems.\n4. The combination of the proposed ray sampling method with the  HV does not seem reasonable. HV can not be accurately calculated in high-dimensional spaces.  While the proposed ray sampling method aims to address challenges associated with high dimensionality, the reliance on HV as a performance metric negates its potential benefits. Therefore, it appears that at least one of the contributions, either the ray sampling method or HV is useless.\n5. The authors claim a challenge exists in covering convex Pareto Fronts. However, this issue has been well-addressed for decades [2], and this paper does not propose new methods for this challenge.\n6. In Figure 5, the points are overlapping and difficult to distinguish.\n\n[1] Generating well-spaced points on a unit simplex for evolutionary many-objective optimization, IEEE TEVC, 2020.\n\n[2] MOEA/D: A multiobjective evolutionary algorithm based on decomposition, IEEE TEVC, 2007.", "other_comments_or_suggestions": "Please refer to \"weaknesses\".", "questions_for_authors": "Please refer to \"weaknesses\".", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "QZ2PONVDwx", "reviewer": "Reviewer_xFzh", "summary": "The paper proposes a novel sampling approach for pareto front learning and federated learning. The main idea is to use genetic algorithms to sample in a way that covers the space better. They then experiment on several MOO and FL benchmarks.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Notes**: All the tables can be found at https://anonymous.4open.science/r/icml_rebuttal-E75A/rebuttal.pdf.\n\n**Response to Claims and Evidence**: \n\nIn the context of this paper, a special metric, namely Maximum Spread (MS),  can play a role similar to the standard deviation to better evaluate the solution's robustness [A1] below. We also carried out experiments to show the effectiveness of PHN-HVVS in terms of MS. \n\nSpecifically, as done in (Hoang et al., 2023), the Hyper-Volume (HV) metric can simultaneously evaluate the convergence and diversity of a set of solutions, which refer to how closely the solutions approximate the true Pareto front (PF), and how well the solutions are spread across the entire PF, respectively. For the MS metric, a larger value indicates a wider coverage of the entire PF, reflecting superior diversity in the solution set.  Please refer to [A1] for the explanation of MS.\n\nTable 2 presents the values of MS across 6 problems. Overall, PHN-HVVS has the best performance and can better cover the entire PF. Table 3 presents the values of MS across toy examples. On convex PFs, our method achieves the largest MS. \n\n[A1] Comparison of multiobjective evolutionary algorithms: Empirical results,  EVCO'00\n\n**Response to 'how do you sample point set s from it?'**: \n\nAlgorithm 1 applies a genetic algorithm to optimize the objective function in Eq. (12). The final partition approach yields n Voronoi grids and stores the Voronoi site P and simulation point set S in each grid. At this point, each s has a label for which partition it belongs to. Algorithm 2 directly utilizes the Voronoi grid structure generated by Algorithm 1, and then performs fast sampling in each round of these grids (simulated point sets with the same label) without the need for recalculation.  \n\n**Response to 'baselines for comparison'**: \n\nWe conducted extended experiments on the Jura and SARCOS datasets, comparing our proposed method against multiple naive sampling techniques, including: Uniform **random** sampling, **Latin** hypercube sampling, **Polar** coordinate sampling, **Dir**ichlet distribution sampling, **K-means** clustering-based representative selection. As shown in Table 4, the proposed approach outperforms other strategies.  \n\n**Response to FL Part**:  \n\nWe appreciated your suggestions, which help enhance the paper quality. Roughly, some parameters are assumed to be known in FL, and are estimated by Pareto front learning (PFL) schemes. Optimization process is further taken in these existing FL algorithms for different purposes. The preciseness of these parameters directly affects the performance of these FL algorithms  where the PFL scheme used previously is the one  (i.e., PHN-LS) in (Navon et al., 2020). However, it can also be the scheme (i.e., PHN-HVVS) proposed in this paper. With PHN-HVVS, all these previous FL algorithms achieve a better performance since the PF is better covered.\n\nSpecifically, in FL, the $n$ clients correspond to $n$ learning tasks.  The heterogeneity of data across clients entails evaluating the complementarity of data between clients. While PFL schemes are applied, there exists an optimal preference vector $p_{i}^{\\ast}=${ $p_{i,1}^{\\ast}, \\cdots, p_{i,n}^{\\ast}$} such that the model performance of $i$ can be maximized. $p_{i,j}^{\\ast}$ can be used to evaluate the weight of the client $j$'s data to the model performance of client $i$. These weights define a benefit graph $G_{b}$, where there is a direct edge from $j$ to $i$ if and only if $p_{i,j}^{\\ast}>0$. Several works assume that the values of $p_{1}^{\\ast}, \\cdots, p_{n}^{\\ast}$ are known, and study how to determine a subgraph $G_{u}$ of $G_{b}$ that satisfies some desired properties to form stable coalitions, avoid conflicts of interests, or eliminate free riders. In $G_{u}$, there is a direct edge from $j$ to $i$ if $j$ will make a contribution to $i$ in the actual FL training process, and it defines the collaborative FL network.\n\nExamples of questions studied in the previous FL works are introduced in Table 5. Definitions of $G_{b}$ and  $G_{u}$ can be found in (Tan et al., 2024; Chen et al., 2024). The way of generating $p_{i}^{\\ast}$ is given in Section 5.3 of this paper. In the final version , we will better clarify the FL part, including Section 2. \n\n**Response to small comments**:  \n\nWe will rewrite the content in the beginning of Section 3 as you suggested. Eq. (1) presents the standard optimization process. We will better state Eq. (2). The hypernetwork is explicitly described in Section 4 of (Navon et al., 2020); we will add a rephrasing  explanation in the appendix. Simulation points are introduced in Algorithm 1 of the paper. \n\nA good ideal partition should be equivalent to Eq. (8) and f=1 in the Eq. (12) where the number of points fall in each grid is equal [A4].\n\n[A4] De Berg M. Computational geometry: algorithms and applications\n\nWe will update the paper according to your suggestions above.", "claims_and_evidence": "There are some issues with the evidence in this paper. Mainly, they do not report any standard deviations and when the difference is quiet small, e.g. table 7,8 and some columns in table 1, it is not clear how significant is the benefit of the method proposed here.", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "NA", "experimental_designs_or_analyses": "The FL section is very unclear so it is hard to evaluate the soundness of the expriments.", "relation_to_broader_scientific_literature": "It is an incremental work on preto front learning", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "The paper offers some interesting ideas but has a few significant flaws. \n\nThe main one is the writing, specifically how the main ideas in the paper are presented. \nI am still not 100% sure how alg. 1 works - once you have your optimal partition p, how do you sample point set s from it?\nAlso the FL part was described as a major part of the work, even part of the title, but was only described in the experimental section and was very unclear. This part should be rewritten as it is not clear at all what you are doing. \n\nAnother issue is baselines for comparison. The main point of this paper is using PFL with a modified sampling technique. However, you only compare it to the standard random sampling. To show the merit of the Voronoi sampling, it would be informative to compare to other, naive, sampling techniques. For example, sampling a large number of rays and using k-means clustering to get a small representative set.", "other_comments_or_suggestions": "Small comments:\n- Explanation in beginning of Sec. 3 needs rewriting. Eq. 1 is standard optimization not PFL and it would be better to state eq. 2 in a way that would make the use of the HN clearer.\n- Eq. 8 is not a good partition, this is a definition of what a partition is. \n- you use \"simulation points\" but don't explain what they mean", "questions_for_authors": "Mainly how the FL is used, do you just take an existing FL algorithm and replace the sampling?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7EA9v2BwHi", "reviewer": "Reviewer_J3SQ", "summary": "This paper studies an interesting and important question, which is about the use of hypernetworks to efficiently approximate the Pareto Front. The proposed approach, PHN-HVVS, addresses multi-objective optimization (MOO) tasks in machine learning by novelly designing a novel loss function and sampling rays from Voronoi grids in high-dimensional spaces. The authors validated the effectiveness of the proposed solution on multiple typical examples, benchmark datasets and real-world datasets. The proposed approach of this paper has a fundamental role in many applications, as illustrated in the experimental part.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Comments 1**. Eq. (14) defines the distance metric between the solution and the preference vector along the given direction. Although it is correct, the authors can explain this equation a bit more.\n\n**Response**.  Thanks for your suggestions. In the final version of this paper, we will formally show the derivation of Eq. (14). Specifically, we denote two points on the line by $l^i = (l_1^i, l_2^i, \\ldots, l_J^i) $, and $r^i = (r_1^i, r_2^i, \\ldots, r_J^i) $, respectively. The vector  $\\overrightarrow{r^i l^i} = l^i - r^i = (l_1^i - r_1^i, l_2^i - r_2^i, \\ldots, l_J^i - r_J^i) $. Project the vector $ \\overrightarrow{r^i l^i} $ onto the direction vector $ \\mathbf{v} $, and the projection coefficient is $t$, where:\n$t = \\frac{\\overrightarrow{l^i r^i} \\cdot \\mathbf{v}}{\\mathbf{v} \\cdot \\mathbf{v}} = \\frac{\\sum_{j=1}^{J} (r_j^i - l_j^i) \\cdot 1}{\\sum_{j=1}^{J} 1^2} = \\frac{\\sum_{j=1}^{J} (r_j^i - l_j^i)}{\\sum_{j=1}^{J} 1}$. Eq. (14) is the distance from $r^i $ to the line, which can be computed by the magnitude of the vector $\\overrightarrow{r^i l^i} - t\\mathbf{v} $.\n\n**Comments 2**. Specifically, the authors can give more details about Algorithm 1, such as the number of parents involved in the tournament selection, the mutation rate, and how the mutation steps are handled if the offspring exceed the predefined parameter range.\n\n**Response**. Algorithm 1 uses Monte Carlo simulation to generate $m$ points located on hyperplane $\\mathcal{H}$ in each round, and then searches for the nearest Voronoi site $p_i$ for these $m$ points. The genetic algorithm is applied to optimize the objective function in Eq. (12). In genetic algorithm, we randomly select three individuals and then choose the optimal individual from them. The intersection rate $\\alpha $ is uniformly distributed within the range of (0,1). The mutation method is to randomly perturb a certain point, and the range of the mutation is controlled by the parameter mutation\\_std. In this article, mutation\\_std is 0.05. When a newly generated point exceeds the valid range [0,1] in any dimension, the algorithm calculates a scaling factor to project the point onto the nearest boundary while preserving its original direction, provided that the directional variation component in that dimension is non-zero. This ensures that the mutated points are within a reasonable range of values.", "claims_and_evidence": "yes", "methods_and_evaluation_criteria": "Yes, the proposed methods and evaluation criteria make sense. Like what is commonly done, the authors use Hypervolume as a metric to measure the quality of a set of solutions in the Pareto front; here, a larger hypervolume indicates a better performing set of solutions. The authors explore high-dimensional sampling by novelly using Voronoi grid and designing a new penalty term; here, the integration of Voronoi grid with genetic algorithms addresses sampling rays in high-dimensional spaces effectively and the novel loss function design balances Hypervolume optimization and Pareto front diversity. Pareto front learning in high-dimensional spaces is of high importance to many applications, e.g., in federated learning, it is important to have the number of objectives to be up to 10.", "theoretical_claims": "Yes, I checked. Eq. (14) defines the distance metric between the solution and the preference vector along the given direction. Although it is correct, the authors can explain this equation a bit more.", "experimental_designs_or_analyses": "The authors took extensive experiments on various applications to validate the effectiveness of the proposed solution. These experiments also show the fundamental role of the proposed approach in many applications. In addition to what is traditionally done in the experimental part, the authors also show a new recent application of the proposed approach to federated learning, which advances the methodologies of several promising problems in the recent papers published at other major venues such as NeurIPS, KDD and AAAI. In addition, to help show the reproducibility of the proposed solution, the authors can make their code available publicly or give more details on the design of the target neural network structure.", "relation_to_broader_scientific_literature": "This paper is related to two promising research topics, Pareto-front learning and federated learning, respectively. It is closely tied to the broader scientific literature (Okabe et al., 2004; Navon et al., 2020; Hoang et al., 2023; Cui et al., 2022; Tan et al., 2024; Chen et al., 2024). For example, Okabe et al. (2004) introduced VEDA, which laid the foundation for Voronoi-based methods in the field of multi-objective optimization (MOO). Navon et al. (2020) proposed the concept of Pareto-front learning (PFL) and Pareto HyperNetworks (PHNs) to address the MOO problems for machine learning tasks. Hoang et al. (2023) contributed to the understanding of multi-sample hypernetworks and hypervolume in PFL, however, their PHN-HVI scheme has limitations in covering the convex part of the Pareto front, which is significantly improved by this paper. Federated learning allows multiple data owners to collaboratively train machine learning models in a privacy-preserving way. Besides what have been done by these previous works, the authors of this paper also highlight a new application of PFL to federated learning where PFL is the foundation of the methodologies of several recent problems in federated learning, and PFL can be used to evaluate how important a data owner is to the other data owners. In many cases, such evaluation is the basis to optimize the collaboration relationships among data owners in multiple promising works at major venues in KDD’22, NeurIPS’24, and AAAI’24 (Cui et al., 2022; Tan et al., 2024; Chen et al., 2024).", "essential_references_not_discussed": "The references of this paper are sufficient.", "other_strengths_and_weaknesses": "Strengths:\n\tThe paper is well organized and easy to follow. \n\tThe overall framework is novel. The integration of Voronoi grid with genetic algorithms addresses sampling rays in high-dimensional spaces effectively and the novel loss function design balances Hypervolume optimization and Pareto front diversity.\n\tPareto front learning in high dimensional spaces is important. The research question of PFL has potential practical impact in real-world settings.\nWeaknesses:\n\tPlease refer to the part “Other Comments Or Suggestions” below", "other_comments_or_suggestions": "Somethings can be done to help non-expert readers better understand your work. Specifically, the authors can give more details about Algorithm 1, such as the number of parents involved in the tournament selection, the mutation rate, and how the mutation steps are handled if the offspring exceed the predefined parameter range.", "questions_for_authors": "Please refer to the above part ”Other Comments or Suggestions”", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: Strength:

1. The idea of applying the Pareto front (PF) to the FL is interesting and novel.

2. The paper is theoretically sound, which introduces the Voronoi paritition to help find solutions.

Weakness

1. Although the application is novel, the Pareto Front technique is directly applied with incremental modifications, which is limited in its novelty. 

2.  The motivation of the Voronoi paritition is not clear, which appears pursuing greater uniformity. 

3. The experimental improvement is small.
Decision: Accept (poster)