Gliding over the Pareto Front with Uniform Designs
Authors: Xiaoyuan Zhang
Genghui Li
Xi Lin
Yichi Zhang
Yifan Chen
Qingfu Zhang
Venue: NeurIPS 2024 poster (2024)
Keywords: uniform design; multiobjective optimization;  algorithmic fairness
Primary area: other

Abstract: Multiobjective optimization (MOO) plays a critical role in various real-world domains. A major challenge therein is generating $K$ uniform Pareto-optimal solutions to represent the entire Pareto front. To address this issue, this paper firstly introduces \emph{fill distance} to evaluate the $K$ design points, which provides a quantitative metric for the representativeness of the design. However, directly specifying the optimal design that minimizes the fill distance is nearly intractable due to the nested $\min-\max-\min$ optimization problem. To address this, we propose a surrogate ``max-packing'' design for the fill distance design, which is easier to optimize and leads to a rate-optimal design with a fill distance at most $4\times$ the minimum value.
    Extensive experiments on synthetic and real-world benchmarks demonstrate that our proposed paradigm efficiently produces high-quality, representative solutions and outperforms baseline methods.

Gliding over the Pareto Front with Uniform Designs
Xiaoyuan Zhang", Genghui Li " Xi Lin ", Yichi Zhang', Yifan Chen", Qingfu Zhang Department of Computer Science City University of Hong Kong;
College of Computer Science and Software Engineering. Shenzhen University Department of Statistics, Indiana University Bloomingtom d Departments of Mathematics and Computer Science, Hong Kong Baptist University
Abstract
Multiobjective optimization (MOO) plays a critical role in various real-world domains. A major challenge therein is generating K uniform Parcto-optimal solutions to approximate the entire Pareto front. To address this issue, this paper firstly introduces fill distance to evaluate the K design points, which provides a quantitative metric for the representativeness of the design. However, directly specifying the optimal design that minimizes the fill distance is nearly intractable due to the involved nested min max - min problem structure. To address this, we propose surrogate "max-packing" design for the fill distance design, which is easier to optimize and leads to a rate-optimal design with a fill distance at most 4x the minimum value. Extensive experiments synthetic and real- rerl-world benchmarks demonstrate that our proposed paradigm efficiently produces high- quality, representative solutions and outperforms baseline MOO methods.
Introduction
Multiobjective optimization (MOO) is widely used inform real- world decision-making across various fields, including materials science [20, 46, 5, 30), recommendation systems [29, 61,31,, and industrial design [44, 52, 50, 57J. An MOO problem (MOP) involves optimizing multiple conflicting objectives, which can be (informally) formulated as:
Covering
min f(x) - (f) fm (x)), #EX
(1)
where m is the number of objectives. Equation (1) vector optimization problem and it does not admit a total ordering, therefore, to compare solutions, the concept of Pareto optimality is introduced. A solution is called Pareto optimal if no other solution x' X dominates Domination occurs when fx(x') S f(x)) for all i 1, 1,.... m, with at least one strict inequality [37, 19]. In this paper, fxr denotes the Pareto objective a Pareto optimal solution.
Figure 1: Covering of Pareto Front (PF). Eight diverse Pareto objectives are used to cover the entire PF with a small covering radius.
The set of all Pareto optimal solutions is the Pareto set (PS), and their objectives form the Pareto front (PF).
Under mild conditions, a PS or PF forms a continuous (m- I)- manifold containing infinitely many solutions [25]. For general MOP. is intractable to precisely depict the entire PS or PF with a closed-form expression. Researchers thus turns to use a small number K of diverse Pareto optimal objectives to "represent" the entire PF. Several prior works have been focused on generating diverse
"Correspondigg to Qingfu Zhang, Contact: code integrated into the LibMOON library, available at httpp::///t //t:://iittt thub. com http///hanng22//11maee httts///lbbbb bmoon.
Jeityu.cdu.hk. The souree
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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solutions (see Section 2), but they lack formal definitions of representability and uniformity. In this paper, we define representability as the covering radius of size-K solution set that covers the entire PF. An illustrative example is provided in Figure 1, where eight uniformly distributed solutions cover the entire PF with a small radius.
In this paper, we first introduce the fill distance (FD) as the minimal covering radius of the Pareto objectives, which measures how well set of discrete solutions represents the true PF. A design with smaller covering radius is considered better representation of the PF. Next, we demonstrate that optimizing FD is challenging due to its nested min max min structure (Equation (4)). To overcome this, we maximize the minimal (max- min) pairwise distances between Pareto objectives, which bounds the minimal covering radius up to constant. Finally, we propose a bi-level optimization framework with neural network to efficiently solve this max min problem.
Our method, UMOD (Uniform the decomposition- baased MOO paradigm [58]. To show its effectiveness, we conduct comparative evaluations against methods on complex multiobjective problems with numerous local optimas and on fairness classification problems with thousands of decision variables. The contribution of this paper can be summarized as:
optimization based on Decomposition) is within
1. We introduce the fill distance as a metric for a set of Pareto solutions in MOO. We prove that the optimal fill distance design serves upper bound for the optimal Inverted Generational Distance (IGD) design, and that the max max-packing packing design is rate-optimal with respect to the fill distance. Therefore, the max-packing design provides effective surrogate for the optimal fill distance design.
2. We present a practical algorithm for identifying the maximum- packunngn-panggggg design of Pareto objectives, and formulate it as bi-level optimization problem. To expedite the optimization process, we introduce a neural network to avoid the frequent solving of the inner-loop optimization prob- lem. Additionally, we analyze the optimization bounds of the neural network-based approach in comparison to solving the original optimization problem directly.
3. Finally, we evaluate our approach against leading MOO methods, including evolutionary algorithms and gradient- based algorithms, on both synthetic and real-world problems. UMOD outperforms these methods in terms of both uniformity and efficiency, based on commonly used metrics in MOO.
Notations. In this paper, () (+) represents the Euclidean distance between vectors (0)) and
(h) with bold letters for vectors. Superscripts denote vectors, and subscripts (e.g., y1) indicate elements of a vector. A PF denoted by T. The objective space is y = { f(x),, x E )) Am is m
the m-D preference simplex; A.. {y ii=1 Vi y 0, E [m]] where [m] {1, (1...., m.).
W denotes a set.
2 Related works
In this section, review three lines of works to generate uniform diverse Pareto objectives. We focus our discussions on uniform/diverse Pareto objectives rather than Pareto solutions because our goal is to produce uniform Pareto solutions in the objective space ())), not the decision space (2').
2.1 Methods to generate diverse Pareto objectives
Various MOO methods can effectively generate diverse Pareto objectives, for both gradient-based and iooluttonn-copptattin (EC)-based frameworks. In the line of gradient-based methods, Pareto MultiTask Learning (PMTL) [32] generates Pareto objectives which are constrained in specific regions (sectors); MOO with Stein Variational Gradient Descent (MOO-SVGD) models objective vectors as particles, updating them through repulsive forces with kernel function to maximize their separation; Exact Pareto Optimization (EPO) [36] aligns solutions with user-specific preference vectors, fostering diverse distribution of Pareto objectives by specifying diverse preferences. For multiobjective evolutionary algorithms (MOEAs), NSGA2 [14] introduces the crowding distance and Pareto rank to achieve a diverse distribution of Pareto objectives; MOEA/D [58] and its variants generate diverse Pareto objectives by leveraging the positional relationship between preference vectors and Pareto objectives; Hypervolume-bart methods (e.g., SMS-MOEA [6]) maximize the set of solutions with the largest hypervolume both to enhance convergence and diversity. A distinction of the proposed

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UMOD method with the previously mentioned methods is that, for general MOPs, the distribution of the achieved Pareto objectives remains unknown, whereas the objective distribution of the proposed method has desirable properties.
2.2 Subset selection for multiobjectivv optimization
Another approach to generating a size-K K diverse Pareto set is subset selection. This method first generates large number of Pareto solutions, then selects K solutions to maximize hypervolume minimize the IGD indicator [23, 53, 9, 45]. Solving the subset selection problem, discrete optimization problem, is generally inefficient compared with continuous optimization problems. Recently, some approaches employ greedy algorithms [9,
(3, 33, 28] to obtain approximate solutions, with naive greedy methods typically providing a (1 - 1/e) guarantee. In contrast, our method addresses a continuous optimization problem on the PF, using gradient-based techniques to solve the established optimization problem to improve both accuracy and efficiency.
2.3 Preference adjustment methods in the decomposition- based MOO paradigm
Since the proposed method can also be classified under the preference adjustment category, we discuss the relationship between the proposed UMOD and preference/weigh" adjustment methods. The study of preference adjustment methods start from MOEA/D-AWE [40]. where its strategy is to remove the preference corresponding to the most crowded objective and add a preference corresponding to the most sparse one. Subsequently, several preference adjustment methods have been introduced [35], including DEA-GNG [34] and MOEA/D-SOM [22], which utilize neural gas networks to guide the selection of preference vectors. W-MOEA/D [21]. tw-MOEA/D [38], pał-MOEA/D [48], and MOEA/D-AWG [55] use mathematical models to shape the non non-dominated solutions and adjust preference vectors. The proposed method differs from other preference adjustment methods in two ways: (1) it models the PF with neural network for better accuracy and efficiency, and (2) it offers rigorous theoretical analysis for selecting preference vectors yielding uniformity and representativennss..
3 Pareto solutions with uniform designs
3.1 FD as upper bound of IGD
We first define fill distance (FD) [1]] of set Y (Y [y..)..... with the inverted generational distance (IGD) indicator [49] of set Y, famous metric in MOO. Formally, FD and IGD are defined as follows:
..... y (x)) and establish its relationship
Definition 1 (FD & IGD).
| min p(y, y ddy, eY
FD(Y)= = max min p(y. v²) max dist(y, Y), IGD(Y) - WET y'E?
(2)
DET
where p(., :) represents the Euclidean distance. The term min p(y. yy) represents the nearest dis- tance from point y the PF to the reference set Y. Therefore, FD( = maxyeT miny' sY p(y, y") denotes the covering radius Y, i.c., the largest radius within which at least one solution in covers the entire PF. However, IGD(Y), which represents the average distance from point on the PF to the set Y. lacks the clear geometric interpretation that fill distance offers. For MOO, the goal is to minimize set of Pareto objectives, i.e., Y C T, by optimizing either the FD IGD indicator: minger FD(Y) or minger IGD to reach a diverse distribution. Let the optimal sets be yFD and IIGD respectively. The following theorem compares FD and IGD.
Theorem (FD as an upper bound of IGD).
IGD(YIGD) < IGD(PFD) < FD(YFD)
The first inequality follows because VIGD minimizes IGD, and the second holds because the average distance (GD(YFD)) is no greater than the maximum distance (FD(FFD)). Theorem 2 shows that yFD, the optimal FD configuration, sets an upper bound for the IGD value. Since the optimal IGD configuration does not similarly bound FD, we focus on FD in this paper.
In this paper. "preference vector" and "weight vector" used interchangeably.
3

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3.2 Max-packing design as a surrogate of FD design
The minimization of FD involves solving the following nested min max min problem:
dDD min max min p(y, y"). TCT yET y'EY
A small dDD suggests that the optimal configuration YFD effectively covers the PF with a low covering radius. However, this triply nested structure is challenging optimize [54, 39], so propose using a max max-packing design surrogate for solving Equation (4).
dPack
max 8= max TCT YCT
min ISiCiSK
ply ,y 2))))
(5)
where ó represents the separation distance between two vectors. The optimal design YPack which solves the optimization problem (Equation (5)), is known as the max max-pacee tax-paccing design [8]. In this paper, we show that PPack effectively optimizes FD when the decision space is a PF, as d is bounded by Paack up to constant factor, independent of size K in Theorem 3.
Theorem (Surrogate for minimal FD). Consider a connected, compact" 3 PFT. The minimal fill distance dFD between T and size-K design pPPckppFDD will then be bounded as:
1 Pack s FD S. FDD(YPack) S &ack
1
(6)
Furthermore, the fill distance between T and the optimal design PPck induced by PPack is upper bounded by dPack, which is guaranteed to be upper bounded by 4dFD pPack is thus considered quality, rate-optimal representative for the whole PFT.
The second inequality dDD < Pacck is proved by Auffray al. [3], Pronzato [39], and we provide a tighter lower bound dPack /4, utilizing the topological property of a PF. The complete proof of Theorem 3 is left in Appendix A.I. Furthermore, under additional strict inequality condition dPack < pack K+1 on the PF, the max- max-packing design Waack serves as a dPack -overicn of T, which is established by Theorem 4. The subscript "K "K" specifically denotes the max max-packing distance d Pack for a size-K K design, similarly used for Fa to represent a size-K design.
K
Theorem 4. Consider a connected, compact PF (T), with the property dRack dK+b the max-
packing design Pack covers with radius of at most d'ackk
This theorem suggests that yPaak can represent T well since the maximal distance between any vector y E T and PPack is bounded by d'uckk
3.3 Characterizations of max-packing design
This section discusses key properties of the max max- maxx-pacinng design on a PF. For bi-objective problems, yPack shows favorable properties when Paac C T, as formalized in Theorem 5, with the proof in Appendix A.2.
Theorem 5 (Characterization of PPcck for biobjective problems). Let YPack (1)),..
((K) be
sorted by the first component, such that yi < ... S vi (K), For a compact, connected T, Pack is characterized as follows:
1. Equal spacing: (1)(1), y (2)) -..) (K-( y (K)), ). for K 23
2. Endpoint alignment: (1) and (K) are two endpoints (1) ( ((2)) of T, i.e,, y = =(()) TTT M2). for K 2 2.
(1)
linfyeT S1, U2] and y =p() = (supyeT Vı, ynf
Firstly, PPack including both the starting and ending points, spans the maximum range among all designs, which is desirable property. Secondly, equal pairwise distances between Pareto objectives yields an intuitive interpretation of uniformity. Maximizing hypervolume ensures the "equal spacing" property only for bi-objective linear PFs [4][Theorem 4], while the max-packing design only requires the PF to be compact and connected.
Remark.
A connected, compact set is also called rectifiable set.

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Besides this non non-asynnnnn asymptotical bi-objective results, we examine the asymptotic properties of Paakk with Theorem
Theorem 6 (Asymptotic uniformity [8]eThroe.m. 2.1)). As the number of set size K oc, the set sequence { weakly converge to a uniform distribution over a compact, connected T. Specifically, for any subset C. T with measure-zero boundary, the proportion of points in K lying within B converges to the proportion T occupied by B:
#(Y n B)
Vol(B)
lim K
= P
#(VPock) Vol(T)
where # denotes the number of points in a set, and "Vol" denotes the volume of a set.
Theorem 6 shows that as the solution set size K grows, "Ka approaches a uniform distribution.
Specifically, random variable Y Pack 4 UTi meaning Pack the categorical distribution where each y E PPack has probability 1/ converges in distribution to Uni((7).
4 Efficient optimization of a size- K uniform set
The original max-packing problem (Equation (5)) maximizes the minimal pairwise distance among Pareto objectives and can be reformulated as the following constrained optimization problem on the PF:
max
min ISijjsK p(y (0) ,001)
s.t. y y
( ET.
(8)
To constrain () yy() pairs as Pareto objectives, we solve decision variables of () as the optimal solution of the following modified Tehebycheff (mTche) aggregation function (Equation (9)),
y = h(A) = arg min y'Ey
y
22
π 0
R"
2
where z is a reference point (2; S ys: Vy E V. E [m)). This substitution is equivalent when the optimal solution of Equation (9) is unique, for any Pareto objective y. there exists a preference Am such that the optimal value Equation (9) matches y (explained Appendix A.4).
Substituting Equation (9) into Equation (8) yields the following bi-level optimization problems:
d Pack
max
min
((
00)
d Pack
min ISKISK ((h(O()),,
I0 ((R) ISiSSK"
max @1) ((.,
(M) arg min eyy
y
Vi (0(R))
, i [m]]
s.t.
((4)
0,
m-1
(10)
Exact Pareto optimal solutions can also be obtained using PMGDA [59], which achieves faster convergence with a suitable control parameter a. The function x()) converts a "preference angle" from an angle space [0, _ m-1 into a preference vector. The conversion relationship is detailed in Appendix C.3. We use ((() decision variables for easier optimization since A(0) is constrained in box. In the right equation, the Pareto objective is denoted as y h()) f((A())) Various bi- level optimization methods [47, 60] can be used to solve the problem (Equation (10)). For efficiency consideration, we use a gradient-based approach. Define 5 = min p(g ( ())) where (i",j") is the optimal pair from arg min(s.f) (1) ,1))). After some basic algebraic calculations, a6
can be
00
calculated by the following two equations:
a6
Oh(O(+))
h(0(i))) - h(OG"))
06
ah(06'))
h(O(() (((()))
00i))
p(h(O())) h(('))))))))))
200")
200")
(((() h(A())
B(nxm)
(mm1))
C(axm)
A(mx1)
For k + 1. ग is zero. While computing the gradient vector A straightforward, B and are challenging because h00 is a black-box function representing optimal values. Finite- difference estimation of B and C is impractical, requiring solving Equation (9) at least X m times.
5

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Instead, we use a neural model h trained on historical data (0. ( to approximate h. enabling efficient estimation via ah in Appendix A.3.
as We analyze the neural model's error in Theorem 7, with full details
Theorem 7 (Optimization error Enn introduced by using a network). Let (0) be an approximator of h()) such that || h(00)l S c. for every & [0. -]mm1 as commonly assumed in bi-level optimization, e.g.. [18]. Eq. (10). Can is the difference between the maximum of the minimal distances calculated using the approximate funttioo hig and the true function h. Then, can is bounded by 2e:
((ho( (b((1))
dPack S 2c.
Cun
min ISiSSSK
Remark. The error e, defined as (1)((0) h( 0l) S is both influenced by the covering radius R of the estimated solutions ho( ..... h (b(K)) and the fitting error Cie. where
éfit = max (0(k)) h(o(k))]l- For any 0 the error satisfies || (0) - h(d)ll <
L 1--00)11 ee ent S L. eft. where L is the Lipschitz constant of function (h,,( h)))), (( is the nearest solution to & among the estimated solutions, and R is the covering radius, which can be further reduced by adding more training pairs. For overparameterized networks, Efit can be considered as very small.
Practical algorithms. Due the space limit, the pseudo-codes of UMOD are provided in Algorithm 1 and Algorithm 2 in Appendix C.1. Initially, K diverse preferences are generated by the Das-Dennis algorithm [12]. Then either a decomposition-
multiobjective evolutionary algorithm gradient-based MOO with mTccee aggregation function is employed for producing preference angle and Pareto objective pairs (0, y). Evolutionary algorithms are preferred for problems with multiple local optimas, while gradient-based MOO methods are preferred for multiobjective machine learning (MOML) problems. Given (1, y) pairs, a PF model h (0) is fitted by optimizing the mean square estimation error. Finally, preference angles are updated to maximize the minimal pairwise distances of Pareto objectives. These two steps are repeated alternatively until convergence.
Experiments
The experiments compare UMOD with other methods on two types of MOPs: (I) those with multiple local optimas which can be solved by MOEAs efficiently, and (2) multiobjective fairness classification neural networks as decision variables. MOEAs and multiobjective fairesss problems are executed with 31/5 random seeds, separately. We employ seven indicators to assess performance of different
algorithms: (1) Hypervolume (1) [24], (2) IGD (L) [26], (3) Sparsity (4) [56], (4) Spacing (L) [43] (5) Uniformity (f), (6) Smooth Uniformity (1), and (7) Fill distance (L). Indicators 1-2 focus on convergence and diversity for multi-objective optimization (MOO), while indicators 3-7 evaluate solution uniformity. See Appendix B. for more detailed expression for these indicators.
5.1 Comparison with MOEAs
We demonstrate the effectiveness of our proposed method across diverse set of MOEA benchmark problems, in- cluding the ZDT [16], DTLZ [15]', and real-world testing problems [50]. Real-world testing problems include: four- bar truss design (RE21), reinforced concrete beam design (RE22), disc brake design (RE33), rocket injector design (RE37), car side design (RE41), and conceptual ma- rine design (RE42). Notably, RE41 and RF42 are complex four-objective problem having a large objective spaces. The prefix "RE" denotes this problem is real-world one.
UNCD
UPI0O Mlssin
ff
UMOD MOEAD on RE37
(b) Visualization on RF37
Figure 2: (a): UMOD solutions are more uniform. (b): A PF model can be trained with small number of solutions.
Baseline MOEAS include: MOEA/D [58], a decomposition- based approach; (2) MOEA/D-AWA [40],
*ZDT5 a discrete optimization problem and commonly excluded MOEA studies. $DTLZ7 7 is excluded because our theoretical results only apply to compact and connected PFs, which DTLZ 7 does not satisfy. Results DTLZ 5-6 are demonstrated in Appendix B.6.

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12
12
Sa
Lffone
Seamtnn Oreartg
Se Libann
Siverig
Caverny
63
63
60
36
34
62
022
-A2
-02
-026300
0250000
(a) LMPFE
(b) DEAGNG
(c) Subset selection
(d) SMS EMS-EMOA
1.
udona Deverhs
3.0
Sokzzonn Covertg
1.0
Covering
13
udors Cavertss
14
1O 8.P. TAA
025280 23 21
PUASIO 3-0
0L013 8.75
(c) NSGA3
(f) MOEA/D
MOEA/D-AWA (g)
(h) UMOD
Figure 3: Result comparison by different methods on ZDTI.
Table 1: Partial results for biobjective problems (full results are Table 7).
Indicntto
DFA-GING [14] LMPEF
Sulwet [5.45]
NSGA3 [77
SMS-FMOA [6] MOEA/D [58]
MOFA/D-AWA [40] UMOD
HV
1.3 (0.00) (6)
1.03 (0.00)
1.02 (0.00)
1.03 (0.00) (5)
1.04 (0.00) (1)
1.03 (0.00) (4)
1,05 (0.00) (3)
1.04 (0.00) (0)
(0.29)
(0.40)
5.28
5.42 (0.02)
5.19
Spacing
(1.74) (7)
3.51 (3)
5:00
1.91
(0.01)
3.69 (0.04)
2DT1
Spansty
4.00 (0.19) (7)
4:55 (0.12) (4) 2.59 (0.18) (0)
4.00 (0.00) (5)
4.48 (0.01) (2)
4.61 (0.00) (6)
4,51 (0.01) (3)
4:39 (0.00) (1)
Uniferm
133 (0.11) (6)
1.67 (0.32) (2) 038 (0.10) (7)
1,53 (0.00)
1,85 (0,03) (1)
1.51 (0.00) (4)
1,51 (0.00)
2,07 (0,01) (0)
SUnifor
(2.13)
(7.09)
0.49
1.00 (0.0)) (6)
1:23 (0.29)
(7)
1.55 (0.00)
(.16 (1)
(0.00) (4)
1.43 (0.000 (3)
1.04
(5)
1:23 (0)01)
11:24
1.24 (0.00) (6)
IGD
(0.28) (5)
4.61 (0.13) 5.15 (0.01) (6)
4,44 (0.00)
4.25 (0.02) (1)
5:40 (0.02) (?)
4.33 (0.34) (2)
4.12 (0.00)
Spacing
6.33 (3.16) (6)
3.62 (0.53) (4) 1.38 (0.00) (1)
5.71 (0.00)
3.19 (0.37) (2)
10.49 (0.04) (T)
3.47 (0:44)
0.12 (0.05)
RE21
Sparsaty
3.10 (6)
2.90
1.74
3.02
2.82
2,84 (0.01)
2.70
Uniform
(0.00)
1.26
Uniifomm
0.03 (0.17) (5)
0.12 (0.08) (3)
-0.17 (0.000 (7)
0:08 (0.00) (4)
0.20 (0.01) (1)
.0.17 (0.00) (6)
0.13 (.13300.1)) (2)
0.31 (0.00) (0)
Fill Disance
1.45 (0.20) (4)
1.21 (0.12) (3) 2.62 (0.01) (7)
1.47 (0.00)
1,09 (0,04) (1)
2.11 (0.01) (6)
1,15 (0,02) (2)
0,53 (0,00) (0)
IIV
5.14
4.14
5.57
3,86
0.57
3.86
3,57
1.29
1GD
4.86
2/43
2.29
4.43
Spacing
Rank Spunity
5.9
5.71
1.43
3.71
2.71
4.86
3.43
0.56
Uniferm
3.86
6.43
1.71
4.29
3.57
0.14
SUnifoem
3,57
6.86
3.14
3,14
0.14
1.
Souftins Dovirtt
3.0
Seariene Oweetig
63
64
044
64
00
UMOD
MOEAVD
-2 035 1a ol
(a) UMOD RE21
MOEA/D RE21
UMOD and MOEA/D
(d) Minimal distances
other objectives
Figure 4: Results on RE21 and DTL.Z2.

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LMPFE
Sutiett
UMOD
PF
PC
PF
(a) DEAGNG
(b) LMPFE
(c) Subset selection
(d) UMOD
Figure 5: Results on RE41 by different methods (full results are in Figure 8).
DEA-GNG
LMPFE
(a) DEAGNG
(b) LMPFE on RE41
(c) Subset selection
(d) UMOD
Figure 6: Results RE42 by different methods (full results are in Figure 9).
integrating adaptive weight adjustment; (3) NSGA3 [13, 27]. generating diverse Pareto objectives
through crowding distance; (4) SMS-EMOA [6], maximizing hypervolume for diverse solutions; (5) LMPFE [51], estimating the PF using multiple local models; (6) Subset selection [9, 45], choosing
solution set by hypervolume maximization". and (7) DEA-GNG [34], a preference adjustment method based on growing neural gas network. Methods (1)-(4) are classical MOEA methods implemented by Pymoo [7], while methods (5)-(7) are more recent methods. Full name of these methods are provided
in Table 4.
Table 2: Partial results on three-objective problems (full results are in Table 8).
|Idicator
IDEA- GNG [3]] LMPFE [51]
|Subset [15]
|SSGAJ [13]
SMS EMOA [6] MOEAD [5S]
|MDEDD AWA [40] UMOD
IIV
1.01 (0.01)
1.05 (0.00) (6)
1.08 (0.00)
1.06 (0.00) (5)
1.08 (0.00) (0)
1.06 (0.00) (3)
[.06 (0.00) (4)
1.07 (0.00) (2)
IGD
(0.52) (2)
12.46 (0.09) (3)
15.14 (0.00) (6)
12:54 (0.00) (3)
15.45 (0.25) (7)
12.55 (0.00)
12:55 (0.00) (4)
12.19 (0.84) (0)
Spacing
5.29 (1.72). (2)
261 (0.39)
8.97 (0.00)
5:45 (0.00) (3)
7.15 (0)51)
5:45 (0.01)
5.45 (0.01)
0.52 (0.21) (0)
Spursity
(0.13)
2.42
14)
(0.00)
Unitoom
1.58 (0.19) (5)
2.45 (0.13) (1)
0.96 (0.00) (?)
2.43 (0.00) (4)
1.51 (0.15) (6)
2.43 (0.00) (3)
2.43 (0.00) (2)
3.17 (0.09) (0)
SUniform
0,36 (0.23)
0.97 (0.04) ())
0.04 (0.00)
0.95 (0.00) (4)
0.18 (0.07)
0.95 (0.00) (2)
0.95 (0.00) (3)
1.15 (0.02) (0)
Fill Distance
(0.60)
2.09
3.54
2.39
2.39
(0.00)
(0.05)
HV
6.8
4.6
1.8
14.4
0.2
3.6
4.2
2.4
Spacing
4.4
1.6
0.6
Rank
Sparsity
1.4
3.6
1.6
4.6
4.8
5,6
4.6
1.5
Unicomm
5.8
2.8
34
3.4
0.8
SUniform
4.4
1.2
Fill Distance
6.6
422
52
4.2
0.4
We present the results for biobjective problems in Figure 3 and Table 17. By directly minimizing maximal pairwise distances, the uniform indicator (which corresponds to maximal pairwise distances, see Appendix B.I, metric 5) optimized effectively and ranks best among all methods. The fill distance, surrogate for maximal pairwise distance up to constant, also performs best among all methods. This indicates that solutions found by UMOD cover the true PF with the minimal covering radius among all methods. Figure 3 further confirms that the covering radius of UMOD is significantly smaller than other methods. We also observe that the IGD indicator (Appendix B. metric 2), representing the mean Euclidean distance between the true PF and the found size-K K solution set, is significantly improved. The significant improvement over IGD, well-established indicator of uniformity and convergence of Pareto solutions, suggests that our method finds high-
'Code: dttps://githur com/HHssaoLaaSUSCC/AAA...
For a better illustration, the unit of IGD, Spacing. Sparsity, Uniform, SUniform, and Fill distance are scaled by 0.0110001..0000.01100111 and 0.1.
8

---

Table 3: Statistical results on fairesss classification problems. Results are averaged on five random seeds. Results are presented in the the format of "(mean)(std))".
Indicator
[UMOD
Agg [37] Aga 1. [37]
Agg PBI [58]
[EPO [36]
|MGDA
|HVVGrdd [17]
6vair(t)
0.1049 (0.0000)
10.0151 (0.0000)
00.0006 (0.0000)
0.0232 (0.0000)
0.0166 (0.0000)
0.0201 (0. (0,0000)
[0.0421 (000000)
6sCnir(1))
0,0088 (0.0000)
-0.0849 (0.0020)
-0,1482 (0.0000)
-0.0764 (0.0000)
-0.0838 (0.0000)
-0.0738 (0.0000)
-0.0510 (0.0000)
Adult
HV(1)
Span(t>
0.0773 (0.0000)
0.6800 (0.0000)
0.0615 (0.0000)
0.6769 (0.0000)
0.6422 (0.0000)
0.0012 (0.0000)
0.6679 (0.0000)
0.0735 (0.0000)
0.0594 (0.0000)
0.6755 (0.0000)
0.0688 (0.0000)
0.6764 (0.0000)
0.6865 (0.9020)
0.0766 (0.0000)
Spacing(
0.8677 (0.0979)
2.9454 (0.0194)
0.0199 (0.0000)
2.3149 (0.0105)
2.9316 (0.0055)
5.6512 (0.0058)
1.9228 (0.0146)
Spansity(1)
1. 1.3914 (0.0057)
0.3067 (0.00044
0.0002 (0.0000)
0.3013 (0.0001)
0.3124 (0.0002)
0.8369 (0.0003)
0.4888 (0.0002)
(vsie(f)
0038011 (0.0027)
00.011 (0.0000)
00.0014 (0.0000)
0.0203 (0.0000)
0.00191
(0.0000)
00:0176 (0.0000)
[0.2199 (0.9020)
6sciir((1)
0.3148 (0.0014)
0000299 (0.0000)
.0.1126 (0.0000)
000332 (0.0000)
-0.0358 (0.0000)
-0.0360 (0.0000)
0.1165 (0.0000)
HV(1)
0.7262 (0.0012)
0.7394 (0.0000)
0.7256 (0.0000)
0.7359 (0.0000)
0.7448 (0.0000)
0.7428 (0.0000)
0.76225 (0.0000)
Compass
Span(1>
0.2155 (0.0006)
0.1950 (0.0000)
0.1783 (0.0000)
0.1954 (0.0000)
0.1979 (0.0000)
0.1986 (0.0000)
0.2089 (0.0000)
Spacing(]]
4.4137 (9.16
25.4158 (0.1 208)
27. 1222 (0.0185)
22.8444 (0.2152)
25.7459 (0.2029)
26.4816 (0. (548)
7.6151 (0.0198)
Sparsity(L)
19.7485 (1.5139)
12,5492 (0.0864)
12.4902 (2. 1463)
10.8758 (0.0985)
13,1004 (0.1647)
13.8356 (0.0960)
10.0604 (0.0233)
quality Pareto solutions and also implics an inherent theoretical connection between IGD and minimal pairwise distance maximization, warranting further investigation.
We would like to mention another advantage of UMOD it can handle PFs of different scales, as demonstrated in Figure 4(a)/(b). Unlike using fixed preference vectors, which can result in non-uniform Pareto objectives, UMOD ensures uniformity in the objective space, remaining the uniformity of the achieved distribution unaffected by the scale of PF.
For three-objective problems, DTLZZ1 owns a simplex-like PF, making DTL.ZI the only problem where uniform preferences result in uniform Pareto objectives. For DTL.22 problem with a sphere-like PF, using uniform preferences on the simplex fails to produce uniform Pareto objectives by MOEA/D. As shown in Figure 4(c), objectives solved by MOEA/D around the center of the PF are sparse, while those around the margin are dense. In contrast, UMOD produces uniform Pareto objectives on the PF. The minimal distances from one Pareto objective to the rest of the solutions, sorted by index, are shown in Figure 4(d), indicating that only UMOD achieves the maximal minimal pairwise distance. The results for the real-world RE37 problem are shown in Figure 2(a), indicating MOEA/D produces inefficient duplicated solutions on the boundary of the PF. In contrast, UMOD effectively reduces duplicated solutions by maximizing the minimal pairwise distance on the PF, as duplicated solutions have a minimal pairwise distance of zero. A visualization of the learned PF is shown in Figure 2(b), indicating that a PF model can be learned efficiently by using only small number of solutions. Most indicators related with uniformity, such as IGD, uniform distance, and FD outperforms other methods significantly, indicating UMOD finds much more uniform and representative solutions. The HV indicator of UMOD is the 3-rd best and comparable to SMS-EMOA, a method directly optimizes the HV indicator.
Finally, we discuss the results for four objeettee problems, shown in Figures 8 and 9 and Table 9. UMOD outperforms DEAGNG and LMPFE by discovering broader PF (Figure 5), as it directly maximizes the minimal pairwise distance on the PF. Although subset selection performs similarly to UMOD on the RE41 problem, it relies on an inefficient two-phase optimization to select subset from larger solution set, making it less efficient for four-ohjective problems. On RE41, UMOD significantly outperforms other methods in IGD, finding more representative solutions. On RE42, UMOD ranks highly (best or second best) in uniformity indicators like spacing, sparsity, and smooth uniformity. Due a challenging PF region, IGD indicators for UMOD, NSGA3, and SMS-EMOA are similar, with these three methods outperforming the others.
5.2 Results on fairness classification
We compare our method against other gradient-based MOO methods on multiobjective fairness tasks involving the Adult [2] and Compass [1] datasets. The decision variables are neural network parameters optimized for classification accuracy and fairness. The first objective is binary cross- entropy classification loss, and the second is a hyperbolic tangent relaxation of the difference of equality of opportunity (DEO) loss [4]] 6]. Data and network architecture details are provided in Table 6 in Appendix B.3.
9

---

Results with five solutions are illustrated in Figure 7 compared with PMGDA [59], Hypervolume- gradient method (HVGrad), EPO [36]. modified Tchebycheff aggregation function method (Agg- mTcche) [58]. For the Adult dataset, all methods run for 30 epochs, and for the Compass dataset, 20 epochs. A detailed description of these methods is Appendix C.4. real-world problems, the true PF range is often unknown, making it difficult for preference-baset methods to select uniform preference vectors. these vectors exclude PF endpoints, parts the PF can not be recovered. In contrast, HV-Grad and the proposed UMOD method do not rely on preferences and can automatically identify a large PF. Figure validates Theorem 5, showing that UMOD identifies both endpoints of the true PF, while HVGrad's endpoints cannot be determined.
Numerical results are presented in Table 3. In real-world problems. objectives often vary in scale, making uniform preference vectors non-equivalent to uniform objective vectors. UMOD, however, is designed to generate uni- form objective vectors independent of scale, as evidenced by its superior performance in uniformity and soft uniformity indicators. Ad- ditionally, UMOD's ' spacing indicator, mea- suring neighborhood distance deviation, is sig- nificantly lower than other methods (except
PAGDA
615
DPC
414
(a) Adult
(b) Compass
Figure Result on fairness classification.
Agg-LS, which produces duplicate solutions in the Adult dataset, resulting in the lowest spacing). UMOD's Span indicators outperform other methods a lot, demonstrating its ability to recover a more complete PF, desirable feature. Many methods show similar HV indicators, but their solution configurations vary, suggesting HV optimization is a coarse measure of uniformity. UMOD's signifi- cantly better uniformity performance indicates that directly optimizing uniformity, in UMOD, is promising approach for generating diverse Pareto solutions.
Conclusions and further works
Conclusions. In this paper, we have proposed a new understanding of longstanding problem in MOO, generating K uniform and representative Pareto objectives, through searching for max- packing design on the PF. We provide rigorous analysis of the resulting objective design, and in particular, we show this design will asymptotically converge to the uniform measure over Pareto front. With this new paradigm, we also empirically demonstrate how the space-filling design we obtain can benefit downstream performance with both synthetic and real-world MOO tasks. Overall, believe that we pave a new way for crate-joptimally configuring the Pareto objectives.
Future works include applying UMOD to large-scale real-world multiobjective problems, such as material design, vaceine design, and recommendation systems. The broader impacts of this work discussed in Appendix D.1.
Acknowledge
GH Li offers guidance on evolutionary computation, while YF Chen proves Theorem 3. The work described in this paper was supported by the Research Grants Council of the Hong Kong Special Administrative Region, China [GRF Project No. CityU 11215622).
10

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Supplementary Material
Table of Contents
A Complete proofs of theoretical results
16
A.1 Upper and lower bounds for space filling design
16
A.2 Configuration of dPack for bi-objective problems
16
A.3 Theoretical results for optimization bounds
A.4 Proof the "equivalent conversion" argument
B Experiment details
19
B.1 Metrics
19
B.2 Full name of multiobjective methods
21
B.3 Detailed hyperparameters and licences
21
B.4 Visualization for four objective problems
.o..............................
22
B.5 numerical results
22
Results on DTLZS and DTLZ6
25
C Method details
26
C.I Practical algorithms
26
C.2 Problem formulations
26
C,3 Conversion between a preference and a preference angle
27
C.4 Baseline methods used faimesss classification problem
28
Duplicated solutions issues caused by mTcche aggregation function
28
D Miscellanies
29
D.1 Broader impacts
29
D.2 Limitations
29
The appendix comprises four sections:
1. Appendix A: Detailed proof's for the main paper's conclusions.
2. Appendix B: Additional experimental details.
Appendix C: Method details omitted from the main paper.
4. Appendix D: Discussion on the broader impact of our method.
15

---

A Complete proofs of theoretical results
This section provides complete proofs for the theoretical results. We first provide the properties for max-packing and space-filling designs Appendices A. and A.2. Lastly, in Appendix A.3, prove for the bi-level optimization bound by using neural network as an approximation for the inner loop optimization problem.
A.1 Upper and lower bounds for space filling design
In the following content, we proves for Theorem 3 in the main paper, i.e.,
7 Pack < dDD S <FD(YPack) S dack
Proof. Following the derivation to attain Equation (3) in Pronzato [39]. we can prove dFD I dack as well the claim that the fill distance between T and the optimal design yPack induced by dfPuck upper bounded by Pack
Next, we prove dFD 2 EdPack by contradiction. Consider YFD is the design that induces dDD, we have that each point in Pack must be within dFD-ball centered at point in VFD, and the condition Pack requires that specific ball will only contain a single ( E PPakk otherwise YPack will not be dsPkk -packin.
dFD <
Since dDD d.ffuckk for all & [K] the corresponding dFl-ball completely contained in the larger (dPack (dPark/2)-ball centered at (h)), However, we note T is connected, and thus there exists certain y 'T outside all the (dPack (dPock/2)-balkk /2)-ball;; this certain point will not be covered by all the dFD- -bal centered at points in WFD well. Finally, the existence of the certain point contradicts the claim that yFD is ddD -covering.
Lastly, we prove for Theorem 4 in the main paper by a contradiction.
Proof. Since K+1
dPack, K is the maximal packing number under distance dPack.
Assume PPack is not dfk dPack -ooverco.. Then there exists (K+1) such that ( +1), y()) < d daack for all [K]. This contradicts the assumption, WPack U (K++))) forms PPack covering. implying packing number of K + 1 when K was maximal. Thus, WPack is Paak coverrig.
A.2 Configuration of dack for bi-objective problems
In the following, prove Theorem 5 in the main paper. Before that, prove for Lemma 8 to describe property of (2) (2)), the distance between y and y when (1) ()()))))))) TT.
Lemma 8. For biobjective problem function is strictly decreasing with respect to the first element (1h (1) of (2) and strictly increasing with the first element (y) (2), of (2)
Proof. We consider a new vector (1) E T such that ( < (. Since 7() E T. (2) and ÿ())
cannot dominate each other, implying _(1) > v2 The distance between this new solution (()) and
(2) is:
16

---

v(ol y2 (02") )2
V Voi 61 vi )2 (02 (1) + 62 1/2 12
(61. > 00)
- V (oi" - v/ )2 + (v) v1 (2)
+ (y5 ()
(2) )² + +
(1) 8/2
3/2
2 V (vf" - vi yz (w2") - u2 12 C
(C>0)
ply (1) (0))
(11)
The previous equations show that (1) , (2) ) is strictly decreasing with respect to the first element
of (1)), Similarly, considering a new vector g(2² E T, 5/ > (2) (1/2 < 1/2 (22),, using the same
calculation method used in Equation (11), it is proved that p(y g(2)) > ply ( ())) This
indicates that p(y ()) is strictly increasing with respect to the first element of (²)
With this established Lemma, are now geared up for the complete proof.
Proof. The proof consists of two parts.
Part 1 proves that the neighboring distances (1) , (f+ are equal for all i [K 1]. Part 2 proves that ( (()) and (k)) 22 p(1) and (2) are the two endpoints of PF.
Mly (K y ( by contradiction. We denote d(0) the distance between y and y ((+t)) and i' and j' are two indices such that
Part 1.
We prove p(y ((2) =
(dt)) dli't1,,.... d(6)) d .'...d''--)
(12)
d1 > d09)
Without loss of generality, assume 1' < j.. We now aim to derive a contradiction under the condition given by Equation (12). The approach to iteratively decrease the first element of ((++1) by a small margin E > O, yielding g('+1), By Lemma 8, this adjustment ensures that () 2 g((+1) < ((y y (1++)) while p( (++1), , (()))) > (ft() ( (+++)). Specifically, for
each & such that 1' + +1sks k 2. (t) is updated to g(k) according to the following rules:
M , (1) - e = pl g(++)),
p( (((tt)) = (() , j(ktt)). ((yb'-1), yb6) ,,0') + p(jll' -1) أ(ا').
I' ++ssksj -2
(13)
When E is sufficiently small, jr = arg min (p((() (1'+))) p("''=1) "
g()')) jJ' meaning the minimal index before and after adjustment remains unchanged. However, the value of the adjusted distance p(ğ('), ) z +)) is increased from p pg'',,," y +1)) by showing the original design is not a max maxx-packing design, leading to a contradiction. If conditions
{ (di') d(''t)) d(5)) (((+1) (.....---))
d0'-1) (,.0____-'-
(14)
< d0")
hold, similar argument as above can be used to derive contradiction. Since the selection of i' and j arbitrary, this implies that for any interval hetween and j', both Equation (12) and Equation (1 (14) cannot hold simultaneously. This leads to the following equality, which aligns with our intended design goal:
p(y (2) =... = ply KK (11) y (K)).
(15)
Remark. The key component of our proof is the validity of Lemma 8, which applies to two-dim PF only. This leads to a contradiction with both Equation (12) and Equation (14), forming the core of our argument.
17

---

Part 2. We prove y()) = p()) by contradiction. The proof for y
pl follows
similarly. Assuming y * pl we replace y² with p(), forming a new configuration
(p()) y y (K) Based on Part 1, where we showed equal neighboring distances between
vectors, the condition
=(y p y22) > ( y0 =... = (K-) (K())
holds.
Next, we replace y( 2 K 1, ensuring
y(k-1) with T(2),... ......((kk))
g((-1) on the PF, such that 8/ = v1² - -c()), for
p(g .g((tt)) > p(y (()))) iE [K - 1].
By Lemma 8, moving (1)) to 7()) results in p(j(1) (()) > ply y Iteratively shifting
(2) ...... y (K-)) 1) ensures 51 Vi
(2) and on, until 01 K- < y (k-1), completing the process.
This leads contradiction, proving that y (()) and (K) 2
A.3 Theoretical results for optimization bounds
In this part, we prove for Theorem 7, which bounds the optimization error caused by neural network in the bi-level optimization problem (Equation (10)).
Proof. Consider the function p(:,-), -), which measures the distance between two vectors. Given our assumption, we derive the error between distances computed under and h For any two points (0, y () in the image of h, the error in their distances compared to h& can be bounded as follows:
( y()) p(h h (0()))
|1 - 100 - |1 - 12(06))11
S (1) h2(((()) (0())) - (
(16)
S 1y ()) (1) ||
(0))) y +
2c.
This follows from the triangle inequality and the assumption (0) h())) e.
Given this pairwise bound, the overall configuration of points such that the minimization of distances among points under hs will either match or exceed the minimization under h within the bounds of 2e:
p(h (1)()) ( = ISKGSK
min
min isicsK ( 2c.
(17)
Considering the maximization of these minimum distances across configurations 0(1) (11),.... ((K),
we have:
min 0m 0 (00) ISKJSK
((ha(0(0), ),
max 000 o.... (K0) IsKJEK ((y(0,
min
(18)
2c 2c.
Thus, the error in the optimal maximal minimal distance under the model transformation can be bounded by 2c, completing the proof.
A.4 Proof of the "equivalent conversion" argument
This argument is actually direct corollary the following two lemmas.
Lemma (Adapted from [10], Theorem 3.1), solution æ weakly Pareto optimal iff there exists weight vector X such that x is (one of) optimal solution of the modified Tchebycheff function.
18

---

DEAGNG-PI
LMPFE
MOEAD
NSGAS
SMS
Subset
UMOD
RAMI
DEAGNG-PS
LMPHE
MUEAD
NSGAS
SMS
Subset
UMOD
DEAGNG.P4
LMPPE
MOEAD
NSGA3
Sulset
UMOD
Figure 8: Results RE41.
Lemma 10 (Modified from [37], Theorem 2.6.2). If an aggregation function is decreasing w.r.t. when f((x) S fi(x'), vi [m] and at least index fz(x) < fz(2'), then one of the optimal solution x² of (x(((=)) is weakly Pareto optimal solution for the original MOP. In addition, if the optimality is unique, x² is Pareto optimal.
vector f(x (L.e., (x(((x)) S 9x(((='))
Based on the first Lemma, we know that for any weakly Pareto objective, there a corresponding preference vector that solving the modified Tchebycheff function recover this vector. Furthermore, since we assume the uniqueness of the optimality the modified Tchebycheff function, thus according to the second lemma, solving the modified Tehebycheff function only yields Pareto optimal solutions. Combining these two arguments, we achieve that for any Pareto optimal objective, there exists a preference vector such that solving the corresponding modified Tchebycheff function yield this Pareto optimal vector.
B Experiment details
This section has four parts. In Appendix B.I, explain the metrics used in the experiments in details. In Appendix B.3, we list the necessary hyperparameters and license. In Appendix B.4, we visualize the results on four-objective problems by projection. Lastly, Appendix B.5 list for all numerical results for all experiments.
B.1 Metrics
To evaluate the uniformity and quality of these solutions, various performance indicators, detailed and mathematically expressed below.
1. Hypervolume (HV) (1) [24] both assesses convergence to PF and solution diversity. Low HV values suggest poor convergence to the PF, while comparisons are significant when HVs are substantially high. The hypervolume indicator measures the dominated volume by at least one
19

---

DEAGINGPP
LMPFE
MOEAD
NSGAA
SMS
Sabset
UMOD
DEAGNG-P2
LMPPE
MOEAD
NSGAA
SMS
Sahset
UMOD
SIZE
DEAGNGGP3
LMPFE
MOEAD
NSGAA
SMS
Subset
UMOD
Figure 9: Results on RE42.
objective belongs to the set Y with reference point
HV,- Vol((u E Y. y 3 y 1 r)).
2. IGD [26] indicator of set A with reference set Ž is defined as:
IGD(A)
d2
where d, represents the Euclidean distance from 24 to the nearest distance in the set of A.
3. Sparsity (4) [56] is measure calculated from the squared distances among solution vectors that are sorted according to their non-dominance levels [14]. The mathematical definition is given by:
N-1 Sparsity = m
0 ((111) 2
j=1
where ((t are the objective vectors arranged in non-dominated sorting order from the (2) ,.."",,...""""......... ((M)). Here, represents the number of objectives, and N is the number of solu- tions. lower Sparsity value indicates a more uniformly distributed set of solutions along the Pareto front. Specifically, for Pareto front with two-dimensional linear shape, the sparsity indicator reaches its minimum when the objectives are spaced equidistantly.
4. Spacing (L) [43]: This metric assesses solution distribution uniformity by calculating the standard deviation of d(t) the minimal distance between solution (6) and its nearest neighbor, where d( min (6) (f)). A lower spacing indicator implies evenly spaced solutions, reflecting uniform distribution. spacing indicator of zero indicates that all Pareto objectives have equally minimal neighborhood distances.
20

---

Table 4: Full name table, which has three parts. The first part is related with evolutionary algorithms, the second part is related with gradient-basee methods, the third part is related to indicators, while the last part related to multiobjective optimization concepts.
Short Name
Full name
DEA-GNG
Decomposition based Evolutionary Algorithm guided by Growing Neural Gas Evolutionary algorithm with Local Model based Pareto Front Estimation Nondominated Sorting Genetic Algorithm 3
LMPFE
NSGA3
SMS-EMOA
S Metric Selection based Evolutionary Multiobjective Optimization Algorithm MultiOhjectiiv Evolutionary Algorithm based on Decomposition
MOEA/D
MOEA/D-AWA
MOEA/D with Adaptive Weight Adjustment
UMOD
Uniform Multiobjective Optimization based on Decomposition
MOO-SVGD
Muetibjeective Optimization Stein Variational Gradient Descent
EPO
Exact Pareto Optimization
PMGDA
Preference based Multiple Gradient Descent Algorithm
Agg-LS
Aggregation function based on Linear Scalarization
Agg-PBI
Aggregation function based on Penalty Based Intersection
Agg-Tche
Aggregation function based Tehebycheff Scalarization
IGD
Inverted General Distance
HV
HyperVoluus Volume
MOO
MultiObjective Optimization
MOP
Multiobjective Optimization Problem
Uniformity (f) and Smooth Uniformity (t) indicators, as introduced by [42], evaluate the distribution of solutions. The Uniformity indicator, dunit is defined as the minimum distance between any two solutions:
Uuiif
(0) ,0))) IsKGSs
min
On the other hand, the Smooth Uniformity indicator, &unt. incorporates logarithmic exponential function to average distances among solutions. It introduces a sensitivity parameter 1) (set to 20 in this study) to highlight the overall distribution of distances:
Uuii
2 nK(K 1)
log Σ ISiCjSK
exp(1)
B.2 Full name of multiobjective methods
To avoid confusion, we provide the full names of the bascline MOO methods Table For SMS-MOEA (MOEA using S-Metric Selection), the term "S-metric" refers to the "hypervolume" metric.
B.3
Detailed hyperparameters and licences
Table 5 lists the hyperparameters for implementing MOPs solved by evolutionary algorithms. The system used features an Intel Core 17-10700 CPU and a NVIDIA RTX 3080 GPU.
Our method is implemented in the MOEA/D framework using Pymoo [7], without modifying the MOEA/D hyperparameters. The main difference is the use of a PF model (a fully-connected neural network) map preference angles to Pareto objectives and update preference vectors. The hyperparameters for the PF model are listed in Table 5.
For fairness classification problems, we use an additional fully connected network to classify input features into corresponding classes. The parameters of this network serve as the decision variables (x) for multiobjective problem. Details of the faimess classification problems are shown in Table 6.
21

---

Table Faimnsss classification problem details and network architectures. Act. is the short name for
activation.
Dataset
Features
Architecture
Act. function
# Params
Samples
Sensitivity
Adult
88
88-60-25-1
ReLU
6891
34188 4319
Sex
Compass
20
20-60-25-1
ReLU
2811
Sex
Table 5: Hyper-paranee used in UMOD-MOEA
Hyper parameters
Values
parameters
Values
Crosover
SBX (Simulated Binary Crossover)
(Polynomial mutation)
SBX mating threshold
0.5
SBX offsprings
PM mutation probability
0.9
PFL optimizer
SGD
PFL network
(m-1): 1)- 128-1128--2 128-128
PFL activation function
ReLU
PFI Learnnng rate
le-3
PFI. training epoch
1000
Number preferences (m
Number of preferences
21
Number of preferences (m ()
35
Prefereace initialization
Das-Deunis [12]
Number neighborhood MOEA/D
Probability mating
Number of experiments (MOEA)
31
Number of experiments (MOML)
Number liiness value obj)
40,000
Number litness value (3 obj)
126,000
Number of fitness value ohj)
350,000
(License) To close this subsection, would like to mention that the license used for Adult follows Creative Commons Attribution 4.0 International (CC BY 4.0) license and Compas is supported by Database Contents License (DbCL) v1.0 license.
B.4 Visualization for four objective problems
This section presents the visualization results for four-objective problems (see Figures and 9). Due to the difficulty of visualizing four-dimensional space, we project the Pareto objectives into 3-D spaces: (f1, f2, fa), (f1, f2, fa), (f1, fa, f4), and (f2, fs, f4), labeled P-1 to P-4 in Figures 8 and 9,
Despite losing some information the projections, meaningful conclusions can still be drawn from these figures:
DEAGNG and LMPFE can find partial parts the true Pareto front. For the P1, P2, and P4 projections, DEAGNG typically finds only small portion in the upper right of the 3-D Pareto front, while IMPFE misses a small part this region. contrast, the proposed method captures a more extensive span of the Pareto front, covering the largest area.
For four-objective problem, MOEA/D finds many duplicate Pareto objectives on the PF boundary by using fixed preference vectors. This highlights the importance of finding optimal preference vectors achieve a more uniform PF.
The proposed UMOD method finds more uniform Pareto objectives on the Pareto front compared to NSGA3, SMS-EM EMS-EMOA, and the Subset selection method. This aligns with Table 9 in the main paper, showing UMOD has a much lower IGD indicator than these three methods.
B.5 Full numerical results
We report the full numerical results for bi-objective, tri-objective, and four-objective problems Tables 9, respectively. Each experiment was conducted with 31 random seeds. For each data point, X (C)), X is the mean value, (Y) is the standard deviation, and (C") is the rank among all eight methods.
The hypervolume, Uniform, and Smooth Uniform (SUniform) are preferred larger, while IGD, spacing, sparsity, and fill distances preferred smaller. The optimal result averaged across all seeds marked in bold. In the last row of each table, we calculate the mean rank each indicator, with the highest one in bold.
The tables show that, with a small solution budget, the uniformity indicators IGD, Spacing, Uni- form, SUniform, and FD achieve the best results, outperforming previous methods significantly.
22

---

SMS-EMOA achieves the highest HV value among all methods, but HV is only rough mea- sure of uniformity, validating that optimizing hypervolume alone does not ensure the best uniform distribution.
These tables also validate the effectiveness of maximizing pairwise distances, proper surrogate for fill distance. This suggests that in practice, maximal packing distance can more tightly bound the minimal fill distance. An interesting finding is that the IGD indicator, an important measure in MOO serving as the average covering radius of the size- K optimized set, is optimized by maximizing the pairwise distance. This interesting empirical finding is worthy of further investigation.
Table 7: Full results for biobjective problems, based on 31 random seeds, include the standard deviation and rankings across all methods. The ranking values in the last row are averaged across all problems.
Indicattor
DEE.-GNG
LMPFFE
Subset
ISSGAA
|SMS-EMOA
MOEADD
MOEA/D-AWA| CAWAALMON LMOD
HV
1.02 (0.00) (6)
L03 (0.00) (2)
1,02 (0.00) (7)
1.00 (0.00) (5)
1.04 (0.00) (1)
1.03 (0.00)
1,03 (0.00) (3)
1.04 (0.00) (0)
IGD
5.68 (0.29)
5.42 (0.21)
5.21 (0.60)
5:28 (0.00)
5.22 (0.07) (2)
5.30 (0.00)
3.42(0.02) (5)
5.19 (0,00)
Spacing
(1.74) (7)
(2)
3.51 (0.47) (3)
5:00
191 (0.46) (1)
5.92 (0.01) (6)
(0.04) (4)
0.12
ZDTT
Spariity
(0.19) (7)
4.55 (0.12) (6)
2.59 (0.18) (0)
4.00 (0.00) (5)
4.48 (0.01) (2)
4.61 (0.00) (6)
4.51 (0.01)
4.39 (0:00) 11)
Uniform
1.33 (0.11) (6)
L67 (0.32) (2)
0.89 (0.10) (7)
1.53 (0.00) (2)
1.85 (0.02) (1)
1.51 (8)
1.51 (0.00) (5)
2.07 (0.01) (0)
SUnifoom
(6)
0.09
0.49
0.74
(0.00)
0.55 (0.00) (3)
0.77 (0.00)
Distance
(.60 (6)
(0.09) (2)
2,00 (0.18)
1.55 (0.00)
(0.05)
(0.00)
(0.03) (3)
1.04 (0.01)
IIV
(0.00)
(0:07)
0..63 (0.00)
00.71
0.77 (0.00)
(0.00)
(0.00) (2)
0.77
IGD
5:43 (1) (4)
(6.80 (7)
6.69 (0.00 (6)
5.31 (0.00) (2)
14) (5)
5.31 (0.00)
5.29 (0.00) (1)
5.23 (0.01) (9)
Spacing
3:92 (1.99) (6)
2.11 (0.88) (1)
3.022 (0.000 (5)
3.01 (0.00) (4)
4.87 (0.53) (7)
3.01 (0.01)
2.30 (0.03) (2)
0.25 (0.23) (0)
2DT2
Spuriity
4.66 (0.09) (5)
8.87 (5.33) (7)
2.26 (0.00) (0)
4.54 (0.00)
4.69 (0.06) (6)
4.54 (0)
4,52 (0.00) (2)
4.45 (0.00)
Uniform
(0:27)
1.25 (0.64) (6)
0.85
1.63
1.71 (0.11)
(0.00)
(0.02) (1)
2,05
SUnifoom
0.57 (0.10) (5)
0.16 (0.68) (5)
0.00 (0.00) (7)
0.68 (0.00) (2)
0.63 (0.02) (4)
0.68 (0.00)
0.71 (0.00) (1)
0.78 (0.00) (0)
Fill Distance
1.36 (4.1)))))
2.82 (1.88) (7)
2.39 (0.00) (6)
1.24 (0.00) (3)
1.63 (0.09) (5)
1.24 (0.00)
1,23 (0.00) (1)
1.07 (0.03) (0)
HV
0.91 (0.00) (0)
0.900 (0.01) (2)
0.90 (4)
0..89.0.000 (7)
[0.91 (0.02) (1)
0.90 (0.00)33
0.89 (0.02) (S)
(0.89x0.000) (6)
IFD
(0.55)))5) (1)
38.78 (8.31 (2)
38.58 (0.00) (0)
90:45 (0.03) (5)
50.23 (2 (2.25) (4)
19.09 (0.00) (3)
39.94 (2.08) (7)
99.83 (2.10)
Spacing
6.60 (i)
12.38 (0.00)
5.12 (2.21)
(0.00) (2)
(0.96) (6)
3:57 (1
ZDT3
Spursity
(0.300.6)) (5)
3.83 (0. (6)
7.27 (0.00 000 (7)
3,80 (0:21)
3.59 (0 (1)
(0.00) (2)
3.70 (0. 549 (3)
3.45 (0.40) (9)
Unifoem
0.39 (2)
1.42 (0.26) (0)
0.00 (0.000
(7)
0.69 (0 54) (4)
0.75 (0.07) (3)
0.66 (0.00)
0.28
(0.20) (6)
1.20 (0. 19) (1)
SUniform
0.06 (0.28) (3)
0.39 (0.10) (0)
0.91 (0.00) (7)
0.04 (0.33)
-.11 (0.09) (5)
0.09 (0.00) (2)
-0.32 (0.13) (6)
0.3 (.304)
FilI
(0.02) (4)
(0.00)
8.87 (0.000
8.58
9.22 (0.71)
88 (0.00)
9.23
9.22.(0.71) (0.7
HV
(0.550 (7)
1.03 (0.01)(5)
1.02 (0.00)
1.003
1.04 (0.00)
(0.00)
1.05 (0.00) (5)
1.04
KGD
38.29 (19.75)))
5.65 (0.19)
5.49 (0.017 (5)
5.28 (0.00)
5.19 (0003) (0)
5.31 (0.01) (3)
5.45 (0.05) (4)
5.21 (0:01) (3)
Spacing
(2.972 (5)
4.69 (1.93) (4)
2.72 (0.22) (2)
5:01 (0.00) (6)
1.60 (0:58) (1)
5.93 (0.01)
3.66 (0.05) (3)
0.22 (0.86) (0)
ZDT4
Sparsity
(1.2 (9)
4.71 (0.16) (7)
2,52 (0.00) (1)
4.60 (0.00) (5)
4.47 (0.01) (3)
4.62 (0.00) (6)
4.52 (0.02) (4)
(0.01)
Uniform
118 (0.6()))))))
(7)
1,06 (0.00)
1.53
(0.02) (4)
2,05 (0.01)
Stnnffem
0:45 (0.22) (5)
0.10 (0.01) (6)
0.49 (0.000 (3)
0.75 (0.02) (1)
0:49 (0.00)
0.53 (0.00) (7)
0.77 (0.00)
Fill Distance
(3.59) (7)
1.33 (0.09) (2)
2.03 (0.02) (6)
1.54 (0.00) (5)
(0.03) (1)
1.52 (0.02) (4)
1.45 (0.01) (3)
1.06 (0.01) (0)
HV
(0:00) (6)
0.66 (0.00) (5)
0..44 (0.00) (7)
0..06 (0.00) (2)
[0.6 (0.00) (1)
0.66 (0.00) (4)
0.66 (0.00) (5)
0.66 (0.00) (0)
IGD
4.81 (0.60)
4.25 (0.09) (3)
15.28 (0.00) 173
6.19 (1)
4.20 (0.02) (2)
4.32 (0.00)
4.37 (0.00) (5)
4.15 (0.09) (0)
Specing
5.04
(0.99)
2.12
1.94
(0.00)
(6)
0.32
Spanity
(:24 (7)
2.95 (0.06) (6)
1,83 (0.00)
2,89 (0.000
2.88 (0.02) (2)
2.91 (0.00) (4)
2:95 (0.00) (5)
2.86 (0:000 131
Unifoem
0.88 (0.21) (6)
1.23 (0. 19)19) (4)
0.79 (0.00) (7)
1.32 (0.00) (2)
1.48 (0.05) (1)
1.23 (0.00)
0.97 (0.01) (5)
1.67 (0.00) (0)
SUnifomm
0.06 (.066 (6)
0.26 (0:06) (4)
40.29 (0.00) (7)
0.31 (0.00)
0.33 (0.02) (1)
0.28 (0.00)
0.22 (0.00) (5)
0.35 (0.00)
Hill Distance
(0.30)
105 (0.08) (5)
3.89 (0.00)
0.54
(0.00)
(0.00) (4)
0.51 (0.01)
HV
1.24
1.23 (0.01) (7)
1.24 (0.00)
1.24 (0.00) (2)
1.24 (0.00) (1)
1.24 (0.00) (6)
1.24 (0.00) (3)
1.24 (0.00)
(0.28) (5)
(0. (4)
5.15 (0.01)
4.44
(0.02) (1)
(0.02) (7)
(0.04) (2)
Spacing
(..36) (6)
3.62 (0.93) (4)
1.58 (0.00) (1)
5.71 (0.01) (5)
3.19 (0.37) (2)
10.49 (0.04) (7)
3.67 (0.41) (3)
0.12 (0.05) (0)
RE21
Spuriity
(.10 (6)
2.90 (0.10) (5)
1.74 (0.00) (0)
3:02 (0.00) (5)
2.82 (0.02) (2)
3.87 (0.02)
2.84 (0.01) (3)
2.70 (0.00) (1)
Uniform
0:95
1,36 (0.00)
1.26
(0.05) (3)
1.62
SUniform
(0.77)))0))))
0.12 (0.00) (3)
00.17 (0000) (7)
0.05 (0.00)
0.20 (0.01) (1)
-0.17 (0.000
(0.01) (2)
0.31 (0.00)
Fill Distance
1.45 (0.20) (4)
1.21 (0.12) (3)
2.62 (0.01) (7)
1.47 (0.00) (5)
1.09 (0.04) (1)
2.11 (0.01) (6)
1.15 (0.02) (2)
0.83 (0.00) (0)
IIV
1.17 (7.000))0 (7)
L.88 (0.00) (5)
(.18:00.00) (4)
(0.00) (5)
1.18 (0.00) (0)
1.18 (0.00) (3)
1.18 (0.00) (2)
1.18 (0.00) (1)
IGD
4.63 (0.40) (5)
4.93 (0.25) (7)
4.48 (0.00) (4)
4.35 (0.00) (1)
4.42 (0.04) (2)
4.92 (0.00) (6)
4.47 (0.02) (3)
4.25 (0.022 (0)
Spocing
2.78 (0.95) (2)
3.11 (1.00) (5)
6.57 (0.00)
2,90 (0.00)
3.08 (0.12)
6.78 (0.01) (7)
2.25 (0.64) (1)
0.34 (0.02)
Spuniity
(0.34) (5)
(0.22)
2.64 (0.00)
2.77
(0.03) (4)
2.70
Uniform
1.08 (0.16) (2)
0.96 (0.01) (5)
0.59 (0.00) (7)
1.05 (0.00) (3)
1.03 (0.01) (4)
0.83 (0.00) (6)
1.09 (0.16) (1)
1.56 (0.01) (0)
Snnioomm
0.08 (0.01) (4)
0.02 (0:05) (5)
0.12 (0.00) (7)
(0.900..00 (2)
(0.02) (L)
00.06 (0.00) (6)
0.14 (0.03) (3)
0.30 (0.01) (0)
Fill Distance
1.39 (0.32) (6)
1.54 (0.14)
1,33 (0.00)
1.06 (0.0000.0)
1.04 (0.06)
1.33 (0.00)
1.29 (0.00) (3)
0.90 (0.01)
HV
5:14
4.14
5.57
3.86
@.57
1.86
3,57
1.29
2.43
Spocing
5.71
2.71
3,86
3,86
5.14
3.57
0.14
Rank
Spunity
5:20
5:71
1.43
8.71
2.71
4.86
3.45
0.566
Uniform
3.86
6.43
1.71
4.29
3.57
0.14
SUniform
6.86
3.14
0.14
Distance
3.57
5.57
3.43
3.29
0.36
23

---

Table 8: Full numerical and ranking results on three-objective problems.
JIndicator
DEA-GNNG
JLMPFF
|Subsr
NSSGA3
|SMS-FMDA
MOFAD
|MOFA/D-AAA
JUMO
Hv
1.51 (0.11
1. (8.00)
1.69 (0.00) (6)
1,09 (0.00)
1.69 (0.00)
1.69 (0.00)
(0,00)
1.69 (0,00000)))
(4.62 (7)
(0.10) (4)
4.94 (0.00) (6)
(0.00) (0)
4,86 (0.05)
4.86 (0:01) (2)
4,95 (0,09) (5)
4.87 (0.00) (2)
Spacing
1.12-0.87 (7)
1.14 10)100100000 (3)
1.94 (0.00) (6)
0.01 (0.00) (0)
1.49 (0.35) (4)
0.02 (0.00 ())
(.77(1.75) (5)
0.09 (0.02) (2)
DTLZI
Sparsity
03.00 (0.22) (0)
0.48 (0.02) (3)
0.36 (0.00) (1)
0.74 (0.00) (6)
0.41 (0.02) (2)
0.75 (0.00) (7)
0.73 (0.01) (4)
0.73 (0.00) (5)
Unition
(0.03)
(0.051
(0.00) (6)
0.86
1.41 (0.00)
1.29 (0.01)
SUiaform
(0.300
0.94 (3)
0.99 (0.00)
-0.90000.000 (2)
(0:95
(5)
.0.00 (0.00)
.0.95 (0.05)
-0.90 (0.00) (0)
Fill Distance
4.59 (0.64) (7)
1.00 (0.13) (6)
1:01 (0.00) (4)
0.77 (0.00) (0)
0.96 (0.05) (3)
0.77 (0.00) (1)
1.01 (0.28)
0.75 (0.00) (2)
HV
1.01 (0.01) (7)
1.05 (0.00) (6)
1:08 (0.00) (3)
1.06 (0.00) (5)
1.08 (0.00) (0)
1.06 (0.00) (3)
1.006 (4)
1.00
1.077 (0.00) (2)
IGD
(0.52 (2)
12:46 (0.09) (1)
15:14 (0.00) (6)
12.54 (0.00) (3)
15.45 (0.25)
12.55 (0.00) (5)
12.55 (0.00) (4)
12.19 (0.044 (0)
Spacing
5.29:11772)C
(0.39) (3)
(0.00)
7.15
5.45
(0.5010001155
0.52 (0.21)
DTLZ2
Spursity
(0.13) (0)
(0.13) (2)
2.42 (0.001 (3)
2.43 (0.00) (4)
2.75 (0:14)
2.43 (0:00)
(0.300 (2)
1.61 (0.32) (1)
Unitoom
1.58 (0.19) (5)
2.45 (0.13) (1)
0.96 (0.00) (7)
2.43 (0.00) (4)
1.51 (0:15) (6)
2.43 (0.00) (3)
2.4000 (2)
3.17 (0.09) (0)
Stiniforma
1.36 (0.22) (5)
0.97 (0.04) (3)
4.04 (0.00) (7)
0.55 (0.00) (4)
0.18 (0.07)
0:95 (0.00) (2)
[195 (0.00) (3)
1.18 (0.02) (0)
Fill Distunce
(0.600
(0.30) (4)
(0,00) (7)
2.59
3.42
2.59
2.37 (0.06)
IV
(0.200 (7)
1.02 (0.02) (6)
1:08 (0.00) (3)
1.06 (0.00) (4)
1.08 (0.00)
1.06 (0.00) (2)
100000..00 (3)
1.06 (0.00) (5)
(16.9(1
(4)
(0.00 (5)
(0.01)(1)
(6)
12.56 (0.01)
12.57 (0.01)
12.31 (0.09) (0)
Spacing
(3.25 (6)
#22.58 (557.38) (7)
8.98 (0.00) (5)
5.46 (0.01) (3)
3.49 (0.50) (4)
5.45 (0.01) (2)
5.34 (0.09) (1)
1.93 (1.43) (0)
DTL23
Spursity
5.86 (5.81) (6)
5433.68 (8:00.995) (7)
2.42 (0.00) (3)
2.43 (0.01) (4)
2.70 (0.17) (5)
2:43 (0.00) (3)
2.43:(0.01) (2)
1.70 (0.09) (9)
Unitoom
0.25:0022)
1.20 (5)
(6)
2.44
1.48
2.44
(0.000
2.72 (0.54)
SUnaform
-1.27 (0.68)
(0.57) (5)
0.04 (0.00)
(0.00)
0.17 (0.09)
0.95 (0:00)
0.96 (0.91) (3)
(0.12)
Fill Distunce
7.83 (2.95) (7)
(0.55) (6)
3.54 (0.00) (5)
2.59 (0.00) (1)
3.33 (0:23) (4)
2.59 (0.00) (3)
2.59 (0.00) (2)
2.39 (0.10) (0)
IIV
0.96 (7)
(100600001) (6)
1.08 (0.00) ())
1.06 (0.00) (4)
1.08 (0.00)
1.06 (0.00) (3)
(0.00) (5)
1.07 (0 (0.00) (2)
IGD
(15.30) (?)
12:78 (0.16) (4)
15:14 (0.00) [5]
12.54 (0.00) (1)
15.48 (0.30) (6)
12.55 (0.00) (2)
(2.50 (3)
12.19 (0.07) (0)
Spacing
(0.81) (3)
(0.34) (1)
8:97 (0.00) (7)
5.45 (0. (5)
7.67 (0.60)
5:45 (0.00)
(.36 (2)
0.53 (0.54) (0)
DTTZZ4
Spursity
(0.22) (0)
(0.18) (7)
(0.00) (3)
(0.00)
2.81
(0.00)
(0.05)
1.77
Unitoom
1.26 (0.64) (6)
2.55 (0.09) (1)
0.96 (0.00) (7)
2.43 (0.00) (2)
1.38 (0.16) (5)
2.43 (0.00) (4)
2.4300.000 (3)
3.07 (0.15) (0)
SUniform
0.06 (0.76) (6)
(0.06) (1)
0.01 (0.00)
0.95 (0.00) (4)
0.15 (0.08) (5)
0.95 (0.00) (3)
(.97 (2)
1.18 (0.02) (9)
Fill Distance
(..21
2.83 (0.24) (4)
(0.00) (6)
2.39 (0.00) (1)
3:41 (0.23) (5)
2.59 (0.00) (3)
2.59 (2)
2.41 (0.14) (0)
HV
(0.0)) (6)
(0.0)) (5)
1.10 (0.00) (0)
1.01 (0.01) (5)
1.07 (0.00) (1)
0.98 (0.00) (7)
(0.02) (4)
1.07 (0.01) (2)
(4.20
12.24 52) (2)
12.17 00.00) (1)
(0.27)
4.89 (1.90) (3)
16.54 (0:05)
(1.35
(0.20)
Spucing
7.47 (2.16) (4)
(1.5 (7)
5.74 (0.00) (3)
11.92 (1.2 (1.21)))))
8.30 (0:400447 (6)
8.22(0..55) (5)
5.23 (1 (I)
RF37
Spirsity
1.05 (0.50) (1)
(.53.) (4)
0.86 (0.01) (0)
1.74 (0.16) (5)
1.45 (0) 10) (3)
3.08 (0.05) (7)
2.47
(0.40) (6)
1.39 (0.07) (2)
Unifomm
0.23:00330 (4)
1.17 (0.56) (8)
0.59 (0.00 (0) (3)
0.01 (0 (6)
0.99 (0.09)
0.00 (0.00)
(.04 (5)
0.94 (0.53) (2)
SUniform
(0.08
(0.33) (0)
6666
-0.94
-0.37 (0.04) (7)
(0.01)
(0
(0 (1)
Fill Distance
(1.46) (7)
(0.93) (1)
(0.00) (4)
3.44 (0.34) (2)
3.73 (0.18)
4.82 (0:04) (6)
(0.05) (5)
2.93 (0.32) (0)
HV
6.8
4.6
1.8
4.4
4.2
2.4
ICD
4.6
1.8
4.6
1.4
0u6
Spacing
4.4
2.8
5.6
3.6
3.4
3.6
[US
Rank
Sparsity
1.4
4.6
Unifomm
5.8
58
3.4
3,4
SUiiiform
5.8
3.4
4.4
3
9.2
0.2
Fill Distance
6.6
4.2
5.2
4.2
3.2
3.2
0.4
Table 9: Numerical results on four-objective problems.
Indicatur
DDEA GNG
LMPPPE
|Subeet
NSGA3
SMS EMOA
MODEA/
MOEED AWA| UMOD
HV
0.63 (0.16) (7)
1.07 (0.00)
1.14 (0.00) (0)
1.08 (2)
1.00 (0.00) (5)
1000 (6)
1.02 (0.00000))) (4)
[.11 (0.00)
Spand Spacing
IGD
41,21 (12.66
17.56 (2)
15.15 (1)
19.47 (0.76)
(1.771.4) (4)
24.95 (0.26) (6)
(0.67)
14.36 (0.34)
4.75 (0.99) ())
4.11 (0.74) (0)
7.45 (0.07)(2)
10.81 (0.36) (7)
9.57 (0.04) (5)
9.2 (0.00) (4)
9.86 (1. 66. (6)
8.16 (0.8900.9)
RE41
Spunsity
0.40 (0.11) (0)
1 1.06 (0.09) (4)
0.75 (1)
0.99 (0.03)
1.28
(5)
2.34 (0.06) (7)
1.03 (0.00) (6)
0.77 (0.01)
Unifomm
0.07 (0.07)
2.04 (0.01)
1.10 (0.02)
0.01 (5)
0.16 (3)
0.00000000
0.00 10.00)000
0.54
SUniform
(5)
0.24 (0.02)
(1.01
1.31 ())
2.00 (0.02)
1.88 (0.05) (6)
(2)
Fill Distence
11.41 (0.12) (7)
5.74 (5)
3.39 (0.00) (0)
5.27 (3)
4.28 (0.52) (2)
5.71 (0.07) (4)
5.88 (0.02) (6)
4.15 (0.02)
(0.02) (3)
(0.0000.01000 (2)
[06] (1.
0.52 (5)
(0.02) (4)
(0.440000)) (7)
00551 (0.04) (6)
I.62 (0.01) (0)
ICD
(8.16 (6)
23.75 (3.02) (3)
25.33 (0.09) 21.01 (2.74) (0)
22.33 (1)
50.31 (0.70) (7)
26.16 (0.62)
(3.75(1.05) (2)
Spacing
10.80 (5)
8.51 (9.05) (3)
12.28 (0.06) (7)
11.81 (1.8)(10.46)) (6)
8:31 (1.18) (2)
6.52 (0.47) (0)
9:48 (1.32)
8.01 (0.64)())
RF42
Spursity
(7.42)
2.43
(0.01)
1.18 (1)
3.45 (0
(0.04)
Uniform
(0.02 (4)
(0.34) (0)
0.00 (0.00) (6) 0.03(0.03)
0.05 (0.05) (2)
0.00 (0.00) (7)
0.00 (0.00) (5)
0.07 (0.05)
SUniform
-1.47 (0.12) (4)
.0.61 (0.11) (0)
-2.53 (0.00 (7) -1.45 (.A5:00.12) (3)
-1.31 (0 (2)
-2.2 (0.04)
-2.12 16: (5)
-1.25 (0. (4)
MastiD
29.05 (3)
35.21 (5.92) (5)
35.59 (0:01) 28.04 (7.95) (0)
(1.72 (2)
54.41 (3)
34.75 (2.23)
37.36
24

---

(a) UMOD
NSGA3 Figure 10: Comparison of using different methods on DTL.75.
AWA
(c) MOEA/D
(c) SMS-EMOA
(a) UMOD
(b) AWA
(c) MOEA/D
(d) NSGA3
(e) SMS-EMOA
Figure 11: Comparison using different methods on DTLZ6.
B.6 Results on DTLZS and DTLZ6
The PFs of DTLZ5 and DTLZ6 identical, which degenerate 1-dimensional hyper-curve within the three-objective space. The visualization results are shown Figures 10 and 11 and numerical results shown in Table 10.
Table 10: Numerical results on DTLZ5 and DTLZ6 problems.
Spacing
Sparsity
HV
Uniform
Smooth Uniform
IGD
FD
UMOD
0.0073
0.0131
0.6995
0.0867
-0.0634
0.0294
0.085
MOEAD
0.0871
0.0645
0.6352
0
-0.1978
0.143
0.3002
AWA
0.0909
0.0312
0.6697
0
-0.1802
0,0712
0.1697
SMS-MOEA
0.0454
0,0147
0.7035
0.0783
-0,0753
0.0307
0. 1099
NSGA3
0.0599
0.0289
0.6623
0.0005
-0.1417
0,0683
0.1802
UMOD
0.0125
0.0128
0.7011
0.0738
-0.0618
0.0285
0.0731
MOEAD
0.0843
0.0599
0.6352
0
-0.2032
0.143
0.3002
AWA
0.0908
0.0312
0.6697
-0.1694
0.0712
0.1697
SMS-MOEA
0.0426
0,0143
0.7036
0.072
-0.0752
0.0305
0.1099
NSGA3
0.0524
0.0415
0.6623
-0.1357
0.0931
0.2972
Table 10 highlights that UMOD significantly outperforms other methods in ensuring evenly dis- tributed solutions. In the decomposition-bee based framework, for such degenerated problems, different preferences may correspond the same Pareto objective. The reason behind generating duplicate solutions is explained in Appendix C.5. And therefore, MOEA/D and MOEA/D-AV tend to produce duplicate solutions. NSGA3 also found to produce duplicate solutions easily. SMS-MOEA avoids duplicate solutions by maximizing the hypervolume. SMS-MOEA surpasses UMOD in hypervolume, but UMOD considerably outperforms SMS-MOEA IGD and FD, which are of interests this paper.
25

---

C Method details
C.1 Practical algorithms
In this section, we present practical algorithms to solve the maximal packing problem in the Pareto front (Equation (10)). We start by generating initial uniform distribution of preference vectors, Then, we use either multiobjective evolutionary algorithms (MOEAs) or gradient- haade MOO to solve for the preference angle and Pareto objective pairs. MOEAs suitable for problems with local optimas, while gradient-basee MOO is efficient for neural network problems with millions of decision variables. Next, we fit Pareto front model learn the expression of h and re-determine the preference angles by maximizing the pairwise distances. These two steps are repeated alternately.
Algorithm 1 Uniform Multiobjective Optimization (UMOD)
1: Input: Initial K uniform preferences {A(1), A(K) by Das-Dennis method [12]. Initial solutions (ert),...,.x(k))))
2: for 1 to N do
3: Run MOEA/D with mTche aggregation function or gradient-based MOO using
(ar(1)...., (k()) as initial solutions under preferences (X(1).... A(K)).
4: Train model he to predict Pareto objectives by the preference angles using mean square
estimation with objlc-obeective pairs (0(), yy)))
5: Update (0(1,... ..... (()) by Algorithm 2.
6: Recalculate preference vectors, (2) A (K)
7: Update the initial solutions (ax(1), ..... kk)) by MOEA/D or gradient-based m'Tche using the
last generation of solutions as a warm start.
8: end for
Algorithm 2 Recalculate Preference Angles (ALG {
1: Input: The initial configuration (g(1)....,0(k)) and hg.
2: for i = 1 to Nopt do
3: Calculate the indexes for the minimal pairwise objectives: min
(i",j") arg
(0(00) (0)). ISKjSK
4: Update the positions of (0)))) ",(')))
(ali") ((()t)
clip
)(
)AG.0 O
pld*) -- clip
r
Ag.u.
200")
where A h(((01)))
1 (o(')) T
(0h
5: end for
6: Output: The updated preference angles
C.2 Problem formulations
For completeness, ZDTI, ZDT2, and DTLZI problems described as follows.
26

---

ZDTI.
(x) I1,
f2(x) - g(x) h(fi())),
9(x) 1+ n- 1=2
9
Σ Si
(19)
((f((z). =1 v f
0 x 11,
i [n].
The PF of ZDTI is f22 = = S
ZDT2.
fi(x) 21, [2(2) = g(x) (((I)))
O(0) = 11 9
Σ Is.
(20)
2
((f(()).(()) = f(())/9()),
0 x4 $1,
i E [m].
The PF of ZDT2 is 12 = 1 A11.
DTLZ1.
f1(x) 21 + g(x)),
f2(2) = 21 (1 =1) - g(x)),
2
fa(Ø) = (1
(21)
n
g(x) 100((n 2) + Σ (r. - 0.5) Σ co0 0.5))),
0 24 ie [n].
The PF of DTLZI is 0.54g (3-dim simplex).
C,3 Conversion between preference and preference angle
The preference vector À and preference angles i are easily inter-convertible via the following equations. This one-to-one, differentiable mapping allows conversion between I and A. While belongs the m-D simplex, & lies within the box constraint [0, 2 11m- For optimization purposes, 9 is more manageable because it can be easily projected onto [0, 2 31n=1 whereas projecting À onto the m-D simplex is more complex.
01 arg cos( /x1).
Vx2
Aa sin? (0;; (U2))
1/2 arg
sin 11
As sin (01) sin sinn?(pp (02) cos (1z).
Vλg
0/3 arg cos
sin 1) sin 02
(22)
Am II sin (0.).
Am-1
arg COS
[1:-1
2
sin
27

---

C.4 Baseline methods used in fairness classification problem
This subsection describes baseline gradient methods for fairness classification problems (Section 5.2). We introduce three aggregation functions:
1. Agg-LS (Linear Scalarization):
9x (z) = Σ Af((z)).
2. Agg-Tche (Tchebycheff):
9X
Tebe
(a)
max (Ax(J((a)) 2;)}, iE(m
where z is a reference point (e.g., ideal point) which dominates the entire PF, i.e., I y, for all y E T. For the modified Tchebycheff (Agg-mTTche) function, the modification involves replacing X; in the equation above.
with x,
Agg-PBI (Penalty-Basee Intersection):
(x) = d1 pdl (z X
PBI 0
+ ( (= d,Aill:
|
where # is the same reference point as introduced. For gradient-bbsed multiobjective optimization methods, solution is updated as æ t- x -) (a) where 1) is small positive learning rate, and the upperscript agg denotes PBI, LS,, or Tche. We also would like briefly introduce the other three gradient based method, PMGDA (Preference- base Multiple Gradient Descent Algorithm) [59], EPO (Exact Pareto optimization) [36], and HVGrad (Gradient-based HV maximization method) [17]. PMGDA and EPO employ gradient-based techniques to precisely identify Pareto solutions at the intersection points where the Pareto objectives converge with the PF. On the other hand, HVGrad leverages gradient ascent on the hypervolume to maximize the hypervolume metrie, thereby optimizing the overall dominance of the solution set.
C,5 Duplicated solutions issues caused by the m'Tche aggregation function
fz
Pareto front
(
y'
(
Figure 12: Duplicated solutions generated by ATch. Different preference vectors (1) and (2) correspond to the same optimal objective vector y².
In this section, we discuss when Tchebycheff aggregation methods yield duplicated Pareto objectives. Appendix C.5 illustrates this, showing preference vector (1) intersecting the at the optimal solution y", where
v
y Am
(23)
X
If the preference vector does intersect the Pareto front (2))),, y is the Pareto front endpoint with the highest Tehebycheff value. In Appendix C.5, y" is the optimal value, so preferences ( and A (2) correspond to the duplicated Pareto objective y
28

---

D Miscellanies
D.1 Broader impacts
By optimizing multiple conflicting objectives, these algorithms enhance decision -ariiiinnmmminng in fields such healtheare, product design, and trustworthy machine learning. product design, multiobjec- tive optimization balances trade-off's between capacity and cost, facilitating the effective release of products that represent the entire Pareto front. In trustworthy machine learning, the UMOD method designs a series of classifiers that balance fairness and accuracy, improving our understanding of different Pareto-optimal classifiers.
UMOD is a foundational algorithm, and its broader impact depends on its downstream applications. We believe UMOD itself does not have direct negative social impact.
D.2 Limitations
While we have illustrated UMOD's success, we acknowledge some limitations. First, UMOD does not address disconnected Pareto fronts such DTLZ7, Theorems and assume connected Pareto front to ensure uniform distribution. Second, we have not considered problems with discrete, binary decision variables, constrained MOPs, our approach requires continuous Pareto front. Adapting UMOD to these complex MOO problems is our next goal. Finally, UMOD requires a neural model to estimate the Pareto front shape. Although training and preference updating with neural networks are efficient, this adds extra operations compared to traditional multiohjective algorithms, which brings inconvenience.
29

---

NeurIPS Paper Checklist
1. Claims
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We explicitly enumerate contributions in the end of the introduction part.
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Justification: The proposed method is fundamental algorithm and is not directly related to human subjects.
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Reviews:
[{"review_id": "sMutAf8NEj", "reviewer": "Reviewer_AxUm", "summary": "This study addresses the problem of generating K uniform Pareto-optimal solutions for multi-objective optimization problems. It introduces a new metric, fill distance, to evaluate the uniformity of the solution set on the Pareto front. This metric is then used as the objective for evolutionary optimization, where a surrogate for fill distance is first proposed, and the problem is then solved using a neural network-based bi-level optimization method.", "questions": "1. How sensitive is the algorithm to the initialization method for K preferences?  \n2. What are the settings for the algorithms in the experiments? For instance, what are the maximum fitness evaluations and the number of independent runs for each algorithm?  \n3. Were statistical tests performed to compare the algorithm's performance?", "limitations": "It is recommended to provide a sensitivity analysis of the algorithm concerning different values of K.", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "1. A new metric is proposed to evaluate the uniformity of the solution set on the Pareto front.  \n2. Some theoretical analysis is provided for the surrogate of fill distance.", "weaknesses": "1. While there are many metrics for evaluating the uniformity of the solution set on the Pareto front, they are not discussed, and the proposed fill distance is not compared with their ability to assess uniformity.  \n2. There is no discussion on how the proposed fill distance ensures that it can (at least to some extent) accurately evaluate the uniformity of the solution set on the Pareto front.  \n3. The standard benchmark set only considers ZDT and DTLZ problems, omitting more complex problem instances.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate your valuable reviews and hope our response addresses your concerns. \n\n**W1. Fill distance (FD) is not discussed and compared with other metrics for achieving and access uniformity.**\n\nWe beg to argue some discussion has been provided.\n\n1. **Hypervolume (HV)**: As noted in lines 124-125, HV is a coarse uniformity metric. It results in equally spaced solutions **only for a linear Pareto front**, while for a non-linear PF, the configuration of K solutions is generally unknown.\n2. **Sparsity**. The sparsity indicator considers the sum of squared distances between neighboring Pareto objectives in the **non-dominated sorted order**. However, summing in the non-dominated sorted order does not have a directly meaning for (more than) three-objective problems. \n3. **IGD**. Difference between FD and IGD please refer to general response 1. \n\nMoreover, directly using FD and IGD as indicators for a uniform configuration is challenging because they assume the true PF is known. In practice, we optimize a surrogate max-min distance for FD. The table below highlights the differences between optimizing the FD surrogate and optimizing HV to guide preference adjustment in UMOD. Most uniformity indicators show that optimizing the FD surrogate yields better results.\n\n**Table 1. Comparison between using FD and HV in UMOD.** \n| Problem | Method | Spacing $\\downarrow$ | Sparsity $\\downarrow$ | HV $\\uparrow$ | Uniform $\\uparrow$| Soft Uniform $\\uparrow$| IGD $\\downarrow$| FD $\\downarrow$|\n|--|--|--|--|--|--|--|--|--|\n|   ZDT1  | FD | **0.001** | **0.027** | 1.054  | **0.162** | **0.017**| 0.040| **0.082** |\n| | HV     | 0.018 | 0.027 | **1.054** | 0.148 | 0.014 | **0.040**| 0.091|\n|   ZDT2  | FD| **0.001** | **0.027** | 0.725 | **0.162** | **0.018** | **0.041** |**0.081**|\n| | HV     | 0.039 | 0.028| **0.727** | 0.145 | 0.008 | 0.046 | 0.136 |\n|   RE21  | FD | **0.001** | **0.016** | 1.252 | **0.125** | **-0.020**  | **0.032**|**0.064**|\n| | HV     | 0.026 | 0.017| **1.253** | 0.099|-0.028|0.033| 0.090|\n|   RE22  | FD| **0.002** | **0.017** |1.190| **0.125** | **-0.019**  | **0.033** | **0.066** |\n|  | HV     | 0.010| 0.017| **1.192** | 0.107|-0.020| 0.033| 0.072|\n\n \n**W2. How does FD evaluate uniformity?**\n\n1. Uniformity can be measured by the discrepancy between the empirical distribution of representative points and the uniform distribution over $\\mathcal{T}$ (Fang et al. 2000). FD can be rewritten exactly as the Wasserstein(W)-$\\infty$ distance between a discrete distribution (induced from the input designs) and uniform distribution $U(\\mathcal{T})$. **Optimizing the FD is equivalent to minimizing the W-$\\infty$ distance from a design to $ U(\\mathcal{T}) $**. Moreover, Minimizing FD is also called to produce space-filling designs (Pronzato et al, 2016), which is a kind of uniform distribution. \n\n2. For the proposed rate-optimal design approaching minimal FD, Thm. 3/4 demonstrate that the design will asymptotically converge to $ U(\\mathcal{T}) $ and yield an equal spacing design, representing uniformity (see Fig. 1, Line 123).\n\n**W3. More complex problem instances besides ZDT and DTLZ in standard benchmark?**\n\nBesides ZDT and DTLZ, we experimented with real-world multiobjective test suites (**RE** problems, Tanabe et al., 2020) and complex multiobjective machine learning problems. The reason we consider more RE problems is that Tanbabe el al. claimed that \"they (current synthetic problems) are likely to include unrealistic properties which may lead to overestimation/underestimation\". Additionally, we conduct new results on synthetic **UF** problems (Li et al., 2008) and **Fi** problems (Lin et al., 2022) (cf one-page pdf). \n\n**Q1. Sensitivity to the preferences initialization?**\nWe do the following ablation studies (cf one page pdf). \n\n- **Initial preferences span from $[0, 1]$ to $[0.5, 0.5]$**. UMOD then expands this to the full preference space, finding Pareto solutions up to $[0.9, 0.1]$.\n- **Non-uniform preferences (truncated Gaussian)**. UMOD still achieves uniform Pareto objectives, demonstrating robustness to non-uniform initialization.\n\n**Q2. What are the algorithm settings used in the experiments, such as number of fitness evaluations and independent runs?**\n\nThe experiment involves **31** independent runs for MOEA and **5** for multiobjective machine learning tasks. The maximum fitness evaluations are **32,000** for bi-objective, **126,000** for three-objective, and **350,000** for four-objective optimization (except for DTLZ5,6 since they are degenerated). Detailed hyper-parameters are in **Table 5**.\n\n**Q3. Statistical tests?**\n\nAlgorithms are ran for **31** times independently to obtain robust experimental results. We conducted the **Wilcoxon rank-sum test** at a significance level of 5%. The p value (%) against others are:\n\n**Table 2. p-value of Wilcoxon rank**\n|  | SMS  | NSGA3 | MOEA/D | AWA |\n|--|--|--|--|--|\n| IGD      | 1.76 | 3.92  | 0.34   | 0.01 |\n| Spacing  | 0.02 | 0.01  | 0.06   | 0.01 |\n| Sparsity | 0.2  | 0.01  | 0.01   | 0.07 |\n| Uniform  | 0.02 | 0.01  | 0.01   | 0.01 |\n| SUniform |0.07| 0.07  | 0.07   | 0.07 |\n| MaxGD    |1.15| 3.38| 2.9 | 0.67 |\nwhich indicates UMOD is significant. \n\n**L1. Sensitivity analysis concerning different values of $K$.**\n\nWe investigate **K=8, 12, 20, and 30** (cf attached PDF), showing UMOD can handle **a wide range of K values**.\n\n-----------\n\nReference\n\nUniform Design: Theory and applications. Fang et al. Technometrics. 2000.\n\nMinimax and maximin space-filling designs: some properties and methods for construction. Pronzato et al. Journal de la Société Française de Statistique. 2017.\n\nAn Easy-to-use Real-world Multi-objective Optimization Problem Suite. Tanabe et al. Applied Soft Computing. 2020.\n\nMultiobjective optimization problems with complicated Pareto sets. Li et al. TEVC. 2008. \n\nPareto Set Learning for Expensive Multi-Objective Optimization. Lin et al. NeurIPS 2022.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XC6GLwyhVG", "reviewer": "Reviewer_5qjZ", "summary": "In this paper, the authors focus on finding K uniform Pareto-optimal points that are capable of representing the entire Pareto front and propose a new metric, namely fill distance to quantify the effectiveness of these K points. To minimize the fill distance easily, the authors adopted a surrogate model, called 'max-min'. Furthermore, the authors design an effective multi-objective algorithm, namely UMOD.", "questions": "1. I know the situation of solutions reflected by different metrics (such as the convergence of IGD and the uniformity of Fill Distance), but is there a consistent relationship between the classic IGD indicator and the Fill Distance indicator? (Calculated results listed in Tables 1 and 2 seem to confirm this viewpoint)\n2. Is it necessary to have the Fill Distance indicator? What impact will it have on measuring the performance of different algorithms? What adverse effects will the lack of this indicator have?\n3. I am curious whether it is reasonable to generate K uniformly distributed points from PF based on Euclidean distance and use them as representative solutions for PF. Especially for irregularly distributed PF situations. In practical problems, optimization problems often contain several complex constraint conditions, leading to scattered arrangements when the solution space is mapped to the target space, resulting in many irregular distributions of PF.", "limitations": "Authors can summarize the drawbacks or shortcomings of existing indicators and delve into the necessity of the Fill Distance metric, as well as the impact it will have on evaluating the fairness of different algorithm performance.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The paper introduces a novel way of generating K uniform Pareto objectives, which is quite original. The paper has a good quality. The authors provided a rigorous proof process for the rationality of using the surrogate model, and the experimental results verified the effectiveness of the designed UMOD. The presentation of the paper is clear. The contribution of the paper is thus significant.", "weaknesses": "Section 2.1 and 2.2： What are the drawbacks of the existing methods for generating diverse solutions? You need to have a simple summary. The motivation of your design is not very clear without saying the drawbacks of existing methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for pointing out key passages that could be complemented with the necessary background for a broader audience.\n\n**W1. What are the drawbacks of the existing methods for generating diverse solutions? You need to have a simple summary.** \\\nFor indicators, please refer our response to Reviewer AxUm in W1 and W2. We will include the response as a simple summary in Appendix B1. \n\n**Q1. Is there a consistent relationship between the classic IGD indicator and the Fill Distance indicator?**\n\nThe detailed relationship between IGD and FD can be found in our general response 1. Thank you for the insightful question. We will add the discussion to the next revision.\n\n**Q2. Is it necessary to have the Fill Distance indicator? What impact will it have on measuring the performance of different algorithms? What adverse effects will the lack of this indicator have?**\n\nThe related discussion can be found in our general response 1. We argue FD is a better metric than its $L_1$ counterpart IGD. FD has better theoretical properties than IGD, especially in ensuring each point in the Pareto front can essentially be well represented (while IGD may dismiss some points).\n\nSpecifically for your **first** sub-question, we believe the FD indicator is necessary. Briefly:\n  - The FD indicator represents the entire Pareto front. Optimizing it provides the minimal covering radius, making it a space-filling design .\n  - FD is robust to the distribution of Pareto solutions approximating the whole front.\n  - We establish a rate-optimal design between FD and maximizing the minimal pairwise distance, which is easier to optimize, and provide a detailed solution distribution for this design.\n\nFor your **second** sub-question, since the FD indicator measures uniformity, a large FD indicator means the solutions generated by the algorithm are generally more uniform.\n\nFor your **last** sub-question, the main weakness is that to measure the level of uniformity, the FD indicator assumes the Pareto front is connected, which may not be suitable for disconnected problems like DTLZ7.\n\n\n**Q3. I am curious whether it is reasonable to generate K uniformly distributed points from PF based on Euclidean distance and use them as representative solutions for PF.Especially for irregularly distributed PF situations. In practical problems, optimization problems often contain several complex constraint conditions, leading to scattered arrangements.**\n\nWe use Euclidean distance because when $K$ is large enough it will well approximate geodesic distance on manifolds; in most experiments the strategy works well. However, due to the theoretical limitation (Theorem 3/4 assumes compactness and connectivity), UMOD does struggle with a disconnected Pareto front. For irregular, disconnected Pareto fronts, a potential solution is to generate uniformly distributed Pareto solutions on each segment separately. We will investigate this approach in the future work. \n\n**L1. Authors can summarize the drawbacks or shortcomings of existing indicators and delve into the necessity of the Fill Distance metric, as well as the impact it will have on evaluating the fairness of different algorithm performance.**\n\nThanks for your advice. To address your concern, we conduct comparisons with  the hypervolume indicator, the sparsity indicator, and the IGD indicator. \n - Hypervolume is a coarse uniformity metric as HV maximization brings uncontrollable configurations of Pareto solutions. As remarked in Line 124-125, only a linear Pareto front results in equally spaced solutions when maximizing hypervolume. \n - The sparsity indicator considers the sum of squared distances between neighboring Pareto objectives. However in problems with more than two objectives, this indicator lacks a clear interpretation.\n - For IGD, please refer to our general response 1. In summary, it is a weaker metric than FD.\n\nFor the impact of FD on evaluation, we argue FD is a better metric than its $L_1$ counterpart IGD. FD has better theoretical properties than IGD, especially in ensuring each point in the Pareto front can essentially be well represented (while IGD may dismiss some points).\n\nSpecifically,\n - The FD indicator represents the entire Pareto front. Optimizing it provides the minimal covering radius, making it a space-filling design (Joseph et al, 2016).\n - FD is robust to the distribution of Pareto solutions approximating the whole front.\n - We establish a rate-optimal design between FD and maximizing the minimal pairwise distance, which is easier to optimize, and provide a detailed solution distribution for this design. \n\nWe will incorporate the discussion into the next revision.\n\n-----------------------------------------------------\n\nReference\n\nSpace-Filling Designs for Computer Experiments: A Review. Joseph et al. 2016.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DCBTi8rPel", "reviewer": "Reviewer_JBGb", "summary": "The paper presents a new approach to effectively represent the entire Pareto front in multi-objective optimization problems. The authors propose using fill distance as a new metric for uniformity in MOO, addressing the challenge of quantifying the representativeness of design points.", "questions": "- The paper mentions the use of neural networks for approximating the Pareto front. Could the authors elaborate on the choice of the neural network architecture and how it impacts the optimization process?\n- Are there any additional limitations or specific considerations for higher-dimensional Pareto fronts?\n- While the paper demonstrates the method on several benchmark problems, what are the scalability considerations when applying UMOD to larger, more complex problems, especially those with a high number of objectives or decision variables?\n- It would be interesting to know how the algorithm performs on DTLZ5-7,  where the assumptions are not  satisfied.", "limitations": "The authors have acknowledged the complexity of the optimization landscape introduced by the neural network. They could further discuss how their method handles cases where the Pareto front is not well-approximated by the neural network or where the front is highly irregular.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper presents a novel approach to MOO by introducing the concept of fill distance and a surrogate objective function.\n- The UMOD algorithm provides bounds on the optimization error. Besides, it demonstrates that as the size of the solution set K increases, the obtained Pareto set will asymptotically converge to a uniform distribution over the Pareto front, which ensures the effectiveness of the algorithm\n- The paper makes comprehensive empirical evaluations.", "weaknesses": "- The proposed method relies on a neural network to approximate the Pareto front. The performance of the method could be sensitive to the quality of the neural network model, its training data, and hyperparameters.\n- While the paper claims that UMOD is efficient, there is limited discussion on the method to scale to larger problems.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank for the detailed feedbacks from the reviewer. We hope our following response can address your concerns more or less. \n\nSince W1 and Q1 are related. We combine the answers here.\n\n**W1. The proposed method relies on a neural network to approximate the Pareto front. The performance of the method could be sensitive to the quality of the neural network model, its training data, and hyperparameters.\\\nQ1. Could the authors elaborate on the choice of the neural network architecture and how it impacts the optimization process?**\n\nNeural networks in this paper serve as a universal approximator of the continuous mapping function from preference angles to Pareto objectives. Although it is a feature of NN that it will suffer from the sensitivity to the architectures, training data, and hyperparameters, this feature will allow substantial improvement of the NN performance through careful investigation of the problem structure, which is widely accepted in modern machine learning.\n\nOn the other hand, the experiments show that indeed the performance of NN is empirically robust. Here, we construct 6 NN architectures and exam the performance sensitivity. \n\n| Model | Architecture                                                                      |\n|-------|-----------------------------------------------------------------------------------|\n| $M_1$ | $(m-1) - 256 - 256 - 256 - m$ |\n| $M_2$ | $(m-1) -128 - 128 - 128 - m$ |\n| $M_3$ | $(m-1)  - 64 - 64 - 64 - m$    |\n| $M_4$ | $(m-1) - 256 - 256 - m$                    |\n| $M_5$ | $(m-1) - 128 - 128 - m$                    |\n| $M_6$ | $(m-1) - 64 - 64 - m$                      |\n\n**Table 1. Model  architecture. Non-linear activation functions are ReLU.**\n\nThe table shows that network structure minimally impacts the final results. Due to space constraints, experiments on other problems are not reported here but will be included in the revised paper.\n\n| Architecture | Spacing$\\downarrow$ | Sparsity$\\downarrow$ | HV$\\uparrow$    | IGD$\\downarrow$   | FD$\\downarrow$     |\n|--------------|---------|----------|-------|-------|--------|\n| M1           | 0.0030  | 0.0445   | **0.7120**| 0.0522| 0.1051 |\n| M2           | 0.0015  | **0.0445**   | 0.7119| **0.0522**| 0.1045 |\n| M3           | **0.0011**  | 0.0445   | 0.7119| 0.0523| 0.1054 |\n| M4           | 0.0016  | 0.0445   | 0.7119| 0.0523| 0.1052 |\n| M5           | 0.0013  | 0.0445   | 0.7119| 0.0523| **0.1043** |\n| M6           | 0.0054  | 0.0445   | 0.7116| 0.0522| 0.1078 |\n\n**Table 2. Results on various models for the ZDT1 problem.**\n\nThen, since W2 and Q3 are related. We answer then together. \n\n**W2. While the paper claims that UMOD is efficient, there is limited discussion on the method to scale to larger problems.\\\nQ3. What are the scalability considerations when applying UMOD to larger, more complex problems, especially those with a high number of objectives or decision variables.**\n\nIn Section 4.2 (Pages 8-9), we extend UMOD to solve **large-scale** machine learning problems, specifically fairness classification problems with around **2891/6811** neural network parameters as **decision variables**. \n\nThese experiments demonstrate that UMOD can handle very **large-scale** problems efficiently. During optimization, we only need to fit a model where the input is the preference angle and the output is the Pareto objectives, with dimensions $m-1$ and $m$, respectively, independent of the number of decision variables $n$.\nAdditionally, generating new preference angles has complexity related to $m$ rather than $n$. Considering $n \\geq m$, those two operations are efficient. Apart from these operations, UMOD functions similarly to standard MOEA/D, contributing to its efficiency.\n\nAlso, in Section 4.1, we have also conducted experiments on **four-objective** problems, namely RE41 and RE42. Our visual results (Full results see Figure 8 and Figure 9) demonstrates that UMOD performs baseline methods a lot in finding uniform K Pareto solutions on the entire PF. \n\n**Q2. Are there any additional limitations or specific considerations for higher-dimensional Pareto fronts.**\n\nFor high-dimensional Pareto fronts, we donot have very specific designs and we note three key differences: (1) More samples are needed to train the Pareto front model, so the number $K$ of candidate solutions should be larger; (2) Training iterations of the Pareto front model are increased; (3) Preference angle adaptation iterations are also increased.\n\n**Q4. It would be interesting to know how the algorithm performs on DTLZ5-7, where the assumptions are not satisfied.**\n\nFor discussion on DTLZ5-7 problems, please refer to our general response 2.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HvvcafZLsl", "reviewer": "Reviewer_N5xS", "summary": "This paper attempts to address a challenging problem in the field of MOO: how to generate a uniform set of solutions on the PF. The authors try to directly characterize uniformity and propose fill distance to quantitatively measure it. Given that fill distance is difficult to optimize directly, the paper further introduces a surrogate problem and theoretically analyzes the relationship between the surrogate problem and the original problem. Finally, the paper presents an MOA, named UMOD, based on MOEA/D. UMOD models the PF through an MLP and adaptively adjusts perference weights by optimizing the surrogate problem. Empirical studies demonstrate the effectiveness of UMOD.", "questions": "1. P1, Line 30. The dimension of PF can be considered as $m-1$, provided that PF is not a degenerate one. However, the dimension of PS has no relation to $m$.\n\n2. P1, Line 36. It is “crowding distance”, not “crowded distance”. Moreover, NSGA-II uses crowding distance, whereas NSGA-III does not. This mistake also appears in P2, Line 80.\n\n3. P3, Line 88. HV is to be maximized, while IGD is to be minimized.\n\n4.  P5, Line 163. $\\partial \\delta/\\partial \\theta^{(j^*)}$ is $-CA$ or $C(-A)$ instead of $C-A$.\n5.  P6, Footnote. I agree that DTLZ7 is not connected. But why do DTLZ5-6 not meet the requirements? DTLZ5-6 are both compact and connected.\n6.  P9, Line 279. Regarding these baselines as \"SOTA\" might be controversial. Of course, I don't think that a novel method must be able to outperform all past ones, but the baselines selected here are much lower than the SOTAs I've seen in recent years (with HV>3 on Adult and Compass). I think using these algorithms as baselines is ok, but please do not call them \"SOTA\".\n7.  P17, Line 586. The authors might have confused IGD and IGD+, which are two different indicators. In the caption (Line 586), the authors write IGD, on the left side of the equation they write IGD+, and on the right side of the equation it is IGD again.\n8.  P25, Fig. 10. The caption should be placed below the image.\n9.  The method proposed in this paper can be classified as a weight adaptation mechanism. Weight adaptation is a popular research direction in MOO, and there is a substantial amount of related work that shares some similarities with the focus of this paper. Therefore, I suggest that the authors include these related works in the literature review.\n10. Some experimental settings are missing, e.g., the reference point for HV and the reference set for IGD. These settings are important for reproducibility.", "limitations": "The discussion on limitations is presented in Section D.2 of the supplementary material. The authors' discussion of the limitations is somewhat perfunctory. Many limitations are mentioned in the main text, such as the inability to solve constrained problems and the inapplicability to disconnected PFs; however, they are overlooked in the limitations section. I suggest that the authors reconsider and summarize the limitations of the proposed method, as this will help readers gain a more comprehensive understanding of the contributions of this work.", "rating": 6, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "This paper tackles a longstanding challenge in MOA: producing uniformly distributed solutions on the PF. Generally, this paper is well written and easy to follow. \n\nI am really impressed by the theoretical analysis of uniformity. In the fields of MOO and evolutionary algorithms, past theoretical research has mainly focused on convergence, while theoretical analysis of distribution has long lacked progress. This paper introduces new mathematical tools for the analysis of diversity, making significant theoretical contributions to improving the uniformity of solutions. Previous MOAs typically used some mechanisms based on intuition to maintain distribution, such as crowding distance. These mechanisms lack theoretical basis and often do not perform well in practical applications. This paper is among the first to propose an MOA with theoretical guarantees for diversity, paving a new path for future research in MOO.", "weaknesses": "Although the theoretical part is impressive, I am not fully convinced that UMOD is a novel and outstanding MOA.\n\n**One of my major concerns is about the fill distance.** I acknowledge that the theoretical analysis of fill distance is very helpful. However, fill distance does not seem like a novel concept in MOO; actually, it can be regarded as a variant of IGD. The proposed fill distance is $\\max_{y \\in T} \\min_{y^\\prime\\in\\mathbb{Y}\\_K} \\rho(y,y^\\prime),$ and IGD is $\\operatorname{mean}\\_{y \\in T} \\min\\_{y^\\prime\\in\\mathbb{Y}\\_K} \\rho(y,y^\\prime)$. The difference is on that the fill distance uses $\\max$ whereas IGD uses $\\operatorname{mean}$. Therefore, minimizing the fill distance is generally equivalent to minimizing IGD, on which there have been many researches in the field of MOO. Moreover, a significant advantage of IGD over fill distance is that it can reflect the overall distribution of almost all non-dominated solutions, whereas fill distance only focuses on the maximal distance. This means that almost all the solutions may contribute to IGD, but only one solution contributes to fill distance. This directly leads to a drawback of the proposed UMOD algorithm: it can only update the nearest pair of weight vectors each time, while the gradients for the others are zero. This may limit the efficiency of the UMOD algorithm. This issue will be further discussed below.\n\n**Another concern is on efficiency.** Although the authors claim that efficiency is one of the main highlights of this paper (L139), I did not find results regarding efficiency, such as resource consumption, convergence curves, etc. Some critical experimental settings are also missing; for example, it does not seem to mention how many function evaluations UMOD and these baselines used. Without this key data, I cannot be convinced that the proposed algorithm is efficient. On the contrary, for the following reasons, I think UMOD might be much more expensive than the baselines.\n\nThe first reason is that the authors does not seem to take efficiency into account when designing the optimization objective. As mentioned above, the original problem (Eq. 2) and the surrogate problem (Eq. 3) proposed are both min/max optimization problems. Therefore, in each iteration, only the nearest pair of weights will be updated, while the others, which may also be disorganized, remain unchanged. This may make the weight updates very slow. It's like moving the two closest points slightly apart each time, which may require many iterations for all the points to achieve a uniform distribution. Additionally, the theorems and optimization objectives have a strong assumption that $\\mathbb{Y}_K\\subset T$. This means that before the solution set converges well to the PF, the theoretical analysis does not hold, making the weight updates potentially meaningless.\n\nThe second reason is the use of neural networks. In some previous studies on expensive MOO, DL models are used because fitness evaluation often takes much more time than training a model. However, in the general MOPs discussed in this paper, to balance efficiency, it is rare for MOAs to use neural networks to simply maintain distribution. In UMOD, training the PF model requires thousands of epochs each time, and this model needs to be repeatedly trained many times. Therefore, the training process of the PF model is quite expensive. Furthermore, as mentioned earlier, this model is only effective when the solution set has already converged to the PF, which raises a question: if the solutions have already converged, why incur such a high cost to repeatedly train a neural network just to make the distribution of some solutions more uniform?\n\nBased one these reasons and the minor ones listed below, I do not recommend accepting this paper in the current form. Since some of these problems may be easy to fix, I will increase my rating if some of concerns are adequately addressed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the exhaustive feedback and, especially, for acknowledging that our work and contributions are significant to the community. Due to space limit, we will **directly take the advice on writing** and incorporate them into the next revision.\n\n**W1. (1) Minimizing the fill distance is generally equivalent to minimizing IGD. (2) Limited efficiency of the UMOD.** \n\nThe comment mainly includes two points: \n1. whether FD is new given IGD (c.f. our general response 1)\n2. whether our proposed algorithm is efficient (deferred to W2).\n\n**W2. (1) Efficiency concerns. (2) If the solutions have already converged, why just make some solutions more uniform?**\n\n**Firstly**, we aim to address your concern with **efficiency**. \n\n*Conceptually*, optimizing the max-min problem (Eq.7) is naturally efficient due to the following reasons:\n- This optimization problem is solved by **gradient ascent**, which is more efficient than the evolutionary computations (ECs).\n- The gradient calculation is **closed-form** and GPU-accelerated.\n- Although only two preferences are updated, they reflect global information, eliminating the need to calculate the rest of the gradients.\n- The PF model is **lightweight**, taking preference angle ($(m-1)$-D) as input and outputting objective ($m$-D). \n\nFor algorithm configurations, see our response to AxUm(Q2). In bi-objective problems with a maximum fitness value of 32,000 (4000*8, with 8 as the main population), preferences are updated every 1000 iterations, resulting in at most *three* updates or retrainings of the Pareto front model. Thus, the Pareto front model is not trained **repeatedly**.\n\n*Empirically*, we analyze time consumption using the bi-objective ZDT1 optimization as an example. Finding the best preference configuration typically takes **500** iterations (**1s**), while training the Pareto front model requires only **10** iterations (**75ms** with GPU acceleration, see the convergence curve provided in a one-page PDF). The total runtime for UMOD is **95.49s**, with preference optimization accounting for a negligible **3.14%** and **0.24%** of the total time.\n\nWe also would like to make two remarks. \n - The two nearest preferences are updated repeatedly (rather than one time) to find the optimal configuration until convergence (convergence curve cf one-page PDF). \n - The max-min optimization can be accelerated by optimizing the max-softmin objective.\n\n**Secondly**, we explain why **finding uniform Pareto objectives is necessary**. \n\nThe main purpose of this paper is to study generating **$K$ uniform Pareto solutions** that represent the entire Pareto front in a **single run**. This is meaningful because:\n\n- Real-world applications (e.g., product releasing) often require producing only $K$ Pareto products, making the study of topK uniform solutions beneficial.\n- Too many solutions can overwhelm users, especially for many-objective problems. A small number of representative solutions is helpful during the early stage.\n\nPrevious methods, such as \"subset selection\" (Qian et al., 2020), achieve a $1 - 1/e$ precision guarantee by selecting topK representative solutions. Table 7/8 shows UMOD outperforms subset selection. Additionally, UMOD is more efficient, requiring fewer main populations (**40** vs **10** for UMOD) and taking less time (**142.2s** vs **95.49s** for UMOD).\n\n**Q1. However, the dimension of PS has no relation to $m$.**\n\nThe effective dimension of the Pareto set (PS) manifold in the $n$-D Euclidean space is $m-1$ (see Ye et al., 2022, p.1, line 12). For biobjective optimization, PS is a 1-D curve embedded the $n$-D space.\n\n**Q5. Results on DTLZ5-7?**\n\nPlease refer to general response 2. \n\n**Q6. Regarding these baselines as \"SOTA\" might be controversial.** \n\nWe will use the term \"baseline\" rather than \"SOTA\" in the revised paper.\n\n**Q7. IGD and IGD+?**\n\nIt is a typo and has been fixed. \n\n**Q9. Literature review of weight adaptation methods.** \n\nWe will add a new subsection to discuss weight adaptation method. \nSpecifically, we will follow the survey (Ma et al, 2020) as well as other recent progress to review the existing 6 categories of weight adaptation methods.\n\nWe remark the main **differences** between our proposed UMOD and other weight adjustment methods are: \n1. the theoretical motivation to improve the uniformity of solutions. \n2. the first practical neural network implementation (UMOD) to fit the Pareto front model (with neglectable training time $<100$ms). \n\n**Q10. Some experimental settings are missing, e.g., the reference point for HV and the reference set for IGD.**\n\nAs shown in the already uploaded core source code, we simply follow the conventional settings (e.g., using the default `pareto_front()` function in pymoo to calculate the reference set for IGD). We will detail these settings in Appendix B.3.\n\n**L1. The authors' discussion of the limitations**\n\nWe intentionally skipped the already mentioned limitations in Appendix D.2. As per your advice, we plan to reorganize D.2 as follows:\n\nWhile we have illustrated UMOD's success, it is also crucial to understand the limitations thereof. \nFirst, UMOD theoretically does not address disconnected Pareto fronts such as DTLZ7, as we note Theorems 3 and 4 both assume a connected Pareto front to ensure uniform distribution. \nSecond, cases containing discrete decision variables or constrained MOPs are also dismissed, as a continuous Pareto front is required for UMOD. \nFinally, the neural network complicates the analysis of the optimization landscape, which is a common issue for using neural networks.\n\n---\nReference:\n1. Pareto Navigation Gradient Descent: a First-Order Algorithm for Optimization in Pareto Set. Ye et al. UAI. 2022.\n2. A Survey of Weight Vector Adjustment Methods for Decomposition-Based Multiobjective Evolutionary Algorithms. Ma et al. TEVC. 2020.\n3. Subset Selection by Pareto Optimization with Recombination. Qian et al. AAAI. 2020.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We sincerely appreciate all helpful feedback and comments from the reviewers. In this part, we first address some general comments raised by the reviewers. 

**Q1 (Asked by 5qjZ and N5xS). Relationship between the convergence of IGD and the uniformity of Fill Distance (FD). The advantages of two indicators?**

We first illustrate the relation between Inverted Generational Distance (IGD) and Fill Distance (FD); and then explain the reasons why we prefer FD over IGD. 

We (informally) recall the definitions:
$$\mathrm{IGD}(\mathbb{Y}) = \mathbb{E_{\mathbf{y} \sim U(\mathcal{T})}} \min_{\mathbf{y}' \in \mathbb{Y}} \rho(\mathbf{y}, \mathbf{y}'), \mathrm{FD}(\mathbb{Y}) = \max_{y \in \mathcal{T}} \min_{\mathbf{y}' \in \mathbb{Y}} \rho(\mathbf{y}, \mathbf{y}'),$$
where $\mathcal{T}$ denotes the Pareto front, and $\rho$ is the Euclidean distance.
Given a set $\mathbb{Y}$, we denote $d(\mathbf{y}; \mathbb{Y}) = \min_{\mathbf{y}' \in \mathbb{Y}} \rho(\mathbf{y}, \mathbf{y}')$ as the distance function between $\mathbf{y}$ and  $\mathbb{Y}$.

We can then rewrite $\mathrm{IGD}(\mathbb{Y}) = \\|d(\cdot,\mathbb{Y})\\|\_{1, U(\mathcal{T})}$, the $L_1$ norm of distance functional $d(\cdot,\mathbb{Y})$ under the measure $U(\mathcal{T})$, and $\mathrm{FD}(\mathbb{Y}) = \\|d(\cdot,\mathbb{Y})\\|\_{\infty, U(\mathcal{T})}$, the $L_\infty$ norm of the distance functional. By the classic property that $L_\infty$ norm dominates $L_1$ norm, we can immediately conclude that $\mathrm{IGD}(\mathbb{Y}) \leq \mathrm{FD}(\mathbb{Y})$ and thus FD is a stronger metric than IGD. More specifically, a design with tightly controlled FD must imply small IGD, while the reverse is not always true.

(Reviewer N5xS) With the observation above, it is thus improper to claim **minimizing FD is generally equivalent to minimizing IGD**.
FD is a stronger metric, and in many fields, the design of an algorithm to provably attain small $L_\infty$ norm is harder than $L_1$ norm.
In addition, we argue **all** solutions essentially contribute to FD. FD, as the $L_\infty$ norm of the distance functional, is naturally the uniform upper bound over **all** solutions, and we recall convergence in $L_\infty$ norm is equivalent to uniform convergence.
Indeed, any outlier point $y$ away from the design $\mathbb{Y}$ will be captured by FD, while not by IGD.

We finally summarize the reasons why we choose FD as our motivated objective function:
- The attractive theoretical properties mentioned above.
- FD can be geometrically understood as covering radius. 
- FD induces a tractable surrogate (Thm. 1). We note, both IGD and FD, cannot be directly optimized without knowing the true PF.
Unlike our proposed surrogate, the practical computation of IGD relies on the empirical average over the approximation to the Pareto front, while the quality of the approximation is usually unguaranteed.


**Q2 (Asked by JBGb, N5xS). Results on DTLZ 5-7?**

1. UMOD performs **exceptionally well** on DTLZ5/6, as shown in Figure 3 of the one-page PDF. This is because DTLZ5/6 has a degenerated Pareto front, meaning most preference vectors correspond to duplicate Pareto objectives. 

- This property makes calculating specific preference vectors corresponding to uniform Pareto objectives more effective (by encouraging to produce different solutions). 
- In contrast, using fixed preference vectors results in many duplicate Pareto solutions, which is inefficient. While MOEA/D-AWA reduces duplicated solutions through heuristic methods, UMOD significantly outperforms it. 
- Additionally, UMOD achieves a much smaller covering radius compared to NSGA3 and SMS-MOEA. Although SMS-MOEA is competitive with UMOD on some uniformity indicators, it take more time (around 2x) compared with UMOD. 

Visual results for DTLZ5 and DTLZ6 are presented in the **one-page PDF**.

**Table 1:** Result on DTLZ5 (15 solutions), max func. evaluation 90,000. 
| Method | Spacing$\downarrow$ | Sparsity$\downarrow$ | HV$\uparrow$    | IGD$\downarrow$   | FD$\downarrow$ | Runtime (min.)|
|--------|---------|----------|-------|-------|-------|-------|
| UMOD   | **0.0197** | **0.0131** | 0.6999 | **0.0291** | **0.076** |10.34|
| AWA    | 0.0495  | 0.0282   | 0.6783 | 0.0639 | 0.1513 |10.01|
| MOEAD  | 0.0871  | 0.0645   | 0.6352 | 0.1430  | 0.3002 | 9.85|
| NSGA3  | 0.0599  | 0.0289   | 0.6623 | 0.0683 | 0.1802 | 0.61|
| SMS    | 0.0454  | 0.0147   | **0.7035** | 0.0307 | 0.1099 | 16.99|

**Table 2:** Result on DTLZ6 (15 solutions), max func. evaluation 90,000.
| Method | Spacing   | Sparsity | HV     | IGD   | FD | Runtime (min.)|
|--------|-----------|----------|--------|-------|---------|-----|
| UMOD   | **0.0315**| **0.0136**| 0.7008 | 0.0505| **0.0941** | 11.56|
| AWA    | 0.0904    | 0.0313   | 0.6697 | 0.0715| 0.1697  | 10.34|  
| MOEAD  | 0.0843    | 0.0599   | 0.6352 | 0.143 | 0.3002  | 9.34|
| NSGA3  | 0.0524    | 0.0415   | 0.6623 | 0.0931| 0.2972  | 0.73|
| SMS    | 0.0426    | 0.0143   | **0.7036** | **0.0305** | 0.1099  | 21.23|


Further comparisons with more recent methods will be included in the revised paper.


2. For the current version of UMOD, it is not suitable to solve DTLZ7 since DTLZ7 has a disconnected Pareto front. Theorem 3 and Theorem 5 assume that the Pareto front is compact and connected to build a uniform distribution of Pareto objectives, but DTLZ7 is disconnected. To handle disconnected Pareto fronts, one approach is to run UMOD on each segment separately. We will explore this in future work.

Meta-review: The authors have made substantial improvements in response to the reviewers' concerns, effectively addressing several key issues raised.

One of the primary concerns involved the novelty and effectiveness of the proposed fill distance metric, specifically its distinction from existing metrics like IGD. The authors provided a thorough rebuttal, clarifying its strong theoretical properties for ensuring uniform coverage of the Pareto front. Additionally, the authors emphasized the practical benefits of their surrogate optimization approach, which has been shown to outperform traditional methods in various benchmarks.

Another significant improvement was in the detailed discussion of the algorithm's efficiency, particularly regarding its use of neural networks. The authors provided detailed analysis to demonstrate that the optimization process is competitive with other state-of-the-art methods. This addresses the reviewers' concerns and strengthens the contribution of this paper. The authors also responded well to queries regarding the scalability of UMOD.
Decision: Accept (poster)