Title: Gliding over the Pareto Front with Uniform Designs
Venue: NeurIPS 2024
Accepted: yes
Keywords: uniform design; multiobjective optimization;  algorithmic fairness
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

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 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 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 × the minimum value. Extensive experiments on synthetic and real-world benchmarks demonstrate that our proposed paradigm efficiently produces highquality, representative solutions and outperforms baseline MOO methods.

1 Introduction

Multiobjective optimization (MOO) is widely used to inform realworld decision-making across various fields, including materials science [20, 46, 5, 30], recommendation systems [29, 61, 31], and industrial design [44, 52, 50, 57]. An MOO problem (MOP) involves optimizing multiple conflicting objectives, which can be (informally) formulated as:

where

m

is the number of objectives.

Equation (1) is a vector optimization problem and it does not admit a total ordering, therefore,

to compare solutions, the concept of

A solution is called

Pareto optimality

Pareto optimal is introduced.

if no other solution dominates

it.

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′

∈ X

Domination occurs when

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i

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, with at least one strict inequality [37, 19].

denotes the Pareto objective

for all

In this of a Pareto optimal solution.

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 -1)-dim manifold containing infinitely many solutions [25]. For a general MOP, it 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

solutions (see Section 2), but they lack formal definitions of representability and uniformity. In this paper, we define representability as the covering radius of a sizeK 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 a set of discrete solutions represents the true PF. A design with a smaller covering radius is considered a 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 a constant. Finally, we propose a bi-level optimization framework with a neural network to efficiently solve this max -min problem.

Our method, UMOD (Uniform Multi-Objective optimization based on Decomposition) is within the decomposition-based 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:

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 as an upper bound for the optimal Inverted Generational Distance (IGD) design, and that the max-packing design is rate-optimal with respect to the fill distance. Therefore, the max-packing design provides an effective surrogate for the optimal fill distance design.

We present a practical algorithm for identifying the maximum-packing design of Pareto objectives, and formulate it as a bi-level optimization problem. To expedite the optimization process, we introduce a neural network to avoid the frequent solving of the inner-loop optimization problem. Additionally, we analyze the optimization bounds of the neural network-based approach in comparison to solving the original optimization problem directly.

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, ρ ( y ( a ) , y ( b ) ) represents the Euclidean distance between vectors y ( a ) and y ( b ) , with bold letters for vectors. Superscripts denote vectors, and subscripts (e.g., y i ) indicate elements of a vector. A PF is denoted by T . The objective space is Y = { y | f ( x ) , x ∈ X} . ∆ m is the m -D preference simplex; ∆ m = { y | ∑ m i =1 y i = 1 , y i ≥ 0 , i ∈ [ m ] } , where [ m ] = { 1 , . . . , m } . Y denotes a set.

2 Related works

In this section, we review three lines of works to generate uniform or 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 ( Y ), not the decision space ( X ).

2.1 Methods to generate diverse Pareto objectives

Various MOO methods can effectively generate diverse Pareto objectives, for both gradient-based and evolution-computation (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 a kernel function to maximize their separation; Exact Pareto Optimization (EPO) [36] aligns solutions with user-specific preference vectors, fostering a 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-based 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

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 multiobjective optimization

Another approach to generating a sizeK diverse Pareto set is subset selection. This method first generates a large number of Pareto solutions, then selects K solutions to maximize hypervolume or minimize the IGD indicator [23, 53, 9, 45]. Solving the subset selection problem, a discrete optimization problem, is generally inefficient compared with continuous optimization problems. Recently, some approaches employ greedy algorithms [9, 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/weight 2 adjustment methods. The study of preference adjustment methods start from MOEA/D-AWA [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], pa λ -MOEA/D [48], and MOEA/D-AWG [55] use mathematical models to shape the 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 a neural network for better accuracy and efficiency, and (2) it offers a rigorous theoretical analysis for selecting preference vectors yielding uniformity and representativeness.

3 Pareto solutions with uniform designs

3.1 FD as an upper bound of IGD

We first define fill distance (FD) [11] of a set Y ( Y = [ y (1) , . . . , y ( K ) ] ) and establish its relationship with the inverted generational distance (IGD) indicator [49] of a set Y , a famous metric in MOO. Formally, FD and IGD are defined as follows:

Definition 1 (FD & IGD) .

where ρ ( · , · ) represents the Euclidean distance. The term min y ′ ∈ Y ρ ( y , y ′ ) represents the nearest distance from a point y on the PF to the reference set Y . Therefore, FD( Y ) = max y ∈T min y ′ ∈ Y ρ ( y , y ′ ) denotes the covering radius of Y , i.e., the largest radius within which at least one solution in Y covers the entire PF. However, IGD( Y ) , which represents the average distance from a point on the PF to the set Y , lacks the clear geometric interpretation that fill distance offers. For MOO, the goal is to minimize a set of Pareto objectives, i.e., Y ⊂ T , by optimizing either the FD or IGD indicator: min Y ⊂T FD( Y ) or min Y ⊂T IGD( Y ) to reach a diverse distribution. Let the optimal sets be Y FD and Y IGD respectively. The following theorem compares FD and IGD.

Theorem 2 (FD as an upper bound of IGD) .

The first inequality follows because Y IGD minimizes IGD, and the second holds because the average distance ( IGD( Y FD ) ) is no greater than the maximum distance ( FD( Y FD ) ). Theorem 2 shows that Y FD , 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.

3.2 Max-packing design as a surrogate of FD design

The minimization of FD involves solving the following nested min -max -min problem:

A small d FD suggests that the optimal configuration Y FD effectively covers the PF with a low covering radius. However, this triply nested structure is challenging to optimize [54, 39], so we propose using a max-packing design as a surrogate for solving Equation (4).

where δ represents the separation distance between two vectors. The optimal design Y Pack , which solves the optimization problem (Equation (5)), is known as the max-packing design [8]. In this paper, we show that Y Pack effectively optimizes FD when the decision space is a PF , as d FD is bounded by Y Pack up to a constant factor, independent of size K in Theorem 3.

Theorem 3 (Surrogate for minimal FD) . Consider a connected, compact 3 PF T . The minimal fill distance d FD between T and a sizeK design Y Pack / Y FD will then be bounded as:

Furthermore, the fill distance between T and the optimal design d Pack induced by Y Pack is upper bounded by d Pack , which is guaranteed to be upper bounded by 4 d FD . Y Pack is thus considered a quality, rate-optimal representative for the whole PF T .

The second inequality d FD ≤ d Pack is proved by Auffray et al. [3], Pronzato [39], and we provide a tighter lower bound d Pack / 4 , utilizing the topological property of a PF. The complete proof of Theorem 3 is left in Appendix A.1. Furthermore, under an additional strict inequality condition d Pack K < d Pack K +1 on the PF, the max-packing design Y Pack K serves as a d Pack K -covering of T , which is established by Theorem 4. The subscript ' K ' specifically denotes the max-packing distance d Pack K for a sizeK design, similarly used for Y Pack K to represent a sizeK design.

Theorem 4. Consider a connected, compact PF ( T ), with the property d Pack K < d Pack K +1 , the maxpacking design Y Pack K covers T with radius of at most d Pack K .

This theorem suggests that Y Pack K can represent T well since the maximal distance between any vector y ∈ T and Y Pack K is bounded by d Pack K .

3.3 Characterizations of a max-packing design

This section discusses key properties of the max-packing design on a PF. For bi-objective problems, Y Pack shows favorable properties when Y Pack ⊂ T , as formalized in Theorem 5, with the proof in Appendix A.2.

Theorem 5 (Characterization of Y Pack for biobjective problems) . Let Y Pack = [ y (1) , . . . , y ( K ) ] be sorted by the first component, such that y (1) 1 ≤ . . . ≤ y ( K ) 1 . For a compact, connected T , Y Pack is characterized as follows:

Equal spacing: ρ ( y (1) , y (2) ) = . . . = ρ ( y ( K -1) , y ( K ) ) , for K ≥ 3 .

Endpoint alignment: y (1) and y ( K ) are two endpoints ( p (1) , p (2) ) of T , i.e., y (1) = p (1) = [inf y ∈T y 1 , sup y ∈T y 2 ] and y ( K ) = p (2) = [sup y ∈T y 1 , inf y ∈T y 2 ] , for K ≥ 2 .

Remark. Firstly, Y Pack , including both the starting and ending points, spans the maximum range among all designs, which is a 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 .

Besides this non-asymptotical bi-objective results, we examine the asymptotic properties of Y Pack with Theorem 6.

Theorem 6 (Asymptotic uniformity [8](Theorem. 2.1)) . As the number of set size K → ∞ , the set sequence { Y Pack K } weakly converge to a uniform distribution over a compact, connected T . Specifically, for any subset B ⊂ T with measure-zero boundary, the proportion of points in Y Pack K lying within B converges to the proportion of T occupied by B :

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, Y Pack K approaches a uniform distribution. Specifically, random variable Y Pack K d - → Unif( T ) , meaning Y Pack K , the categorical distribution where each y ∈ Y Pack K has probability 1 /K , converges in distribution to Unif( T ) .

4 Efficient optimization of a sizeK 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:

To constrain y ( i ) , y ( j ) pairs as Pareto objectives, we solve decision variables of y ( i ) 's as the optimal solution of the following modified Tchebycheff (mTche) aggregation function (Equation (9)),

where z is a reference point ( z i ≤ y i , ∀ y ∈ Y , ∀ i ∈ [ m ] ). This substitution is equivalent when the optimal solution of Equation (9) is unique , as for any Pareto objective y , there exists a preference λ ∈ ∆ m such that the optimal value of Equation (9) matches y (explained in Appendix A.4). Substituting Equation (9) into Equation (8) yields the following bi-level optimization problems:

Exact Pareto optimal solutions can also be obtained using PMGDA [59], which achieves faster convergence with a suitable control parameter σ . The function λ ( ϑ ) converts a 'preference angle' from an angle space [ 0 , π 2 ] m -1 into a preference vector. The conversion relationship is detailed in Appendix C.3. We use λ ( ϑ ) as decision variables for easier optimization since λ ( ϑ ) is constrained in a box. In the right equation, the Pareto objective is denoted as y = h ( ϑ ) = ˜ h ( λ ( ϑ )) . Various bilevel optimization methods [47, 60] can be used to solve the problem (Equation (10)). For efficiency consideration, we use a gradient-based approach. Define δ = min ( i,j ) ρ ( y ( i ) , y ( j ) ) , where ( i ∗ , j ∗ ) is the optimal pair from arg min ( i,j ) ρ ( y ( i ) , y ( j ) ) . After some basic algebraic calculations, ∂δ ∂ϑ can be calculated by the following two equations:

̸

For k = i ∗ , j ∗ , ∂δ ∂ϑ ( k ) is zero. While computing the gradient vector A is straightforward, B and C are challenging because h ( ϑ ) is a black-box function representing optimal values. Finitedifference estimation of B and C is impractical, requiring solving Equation (9) at least n × m times.

Instead, we use a neural model h ϕ trained on historical data ( ϑ, h ( ϑ )) to approximate h , enabling efficient estimation via ∂ h ϕ ∂ϑ . We analyze the neural model's error in Theorem 7, with full details in Appendix A.3.

Theorem 7 (Optimization error ϵ nn introduced by using a network) . Let h ϕ ( ϑ ) be an approximator of h ( ϑ ) such that || h ϕ ( ϑ ) -h ( ϑ ) || ≤ ϵ , for every ϑ ∈ [0 , π 2 ] m -1 , as commonly assumed in bi-level optimization, e.g., [18], Eq. (10). ϵ nn is the difference between the maximum of the minimal distances calculated using the approximate function h ϕ and the true function h . Then, ϵ nn is bounded by 2 ϵ :

Remark. The error ϵ , defined as || h ϕ ( ϑ ) -h ( ϑ ) || ≤ ϵ , is both influenced by the covering radius R of the estimated solutions h ϕ ( ϑ (1) ) , . . . , h ϕ ( ϑ ( K ) ) and the fitting error ϵ fit , where ϵ fit = max k ∈ [ K ] || h ϕ ( ϑ ( k ) ) -h ( ϑ ( k ) ) || . For any ϑ , the error satisfies || h ϕ ( ϑ ) -h ( ϑ ) || ≤ L || ϑ -ϑ ( i ′ ) || + ϵ fit ≤ L · R + ϵ fit , where L is the Lipschitz constant of function ( h ϕ ( ϑ ) -h ( ϑ )) , ϑ ( i ′ ) is the nearest solution to ϑ among the estimated solutions, and R is the covering radius, which can be further reduced by adding more training pairs. For overparameterized networks, ϵ fit can be considered as very small.

Practical algorithms. Due to 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-based multiobjective evolutionary algorithm or a gradient-based MOO with mTche aggregation function is employed for producing preference angle and Pareto objective pairs ( ϑ, 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 ( ϑ, y ) pairs, a PF model h ϕ ( ϑ ) 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.

5 Experiments

The experiments compare UMOD with other methods on two types of MOPs: (1) 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 fairness problems are executed with 31/5 random seeds, separately. We employ seven indicators to assess performance of different algorithms: (1) Hypervolume ( ↑ ) [24], (2) IGD ( ↓ ) [26], (3) Sparsity ( ↓ ) [56], (4) Spacing ( ↓ ) [43], (5) Uniformity ( ↑ ), (6) Smooth Uniformity ( ↑ ), and (7) Fill distance ( ↓ ). Indicators 1-2 focus on convergence and diversity for multi-objective optimization (MOO), while indicators 3-7 evaluate solution uniformity. See Appendix B.1 for more detailed expression for these indicators.

5.1 Comparison with MOEAs

We demonstrate the effectiveness of our proposed method across a diverse set of MOEA benchmark problems, including the ZDT 4 [16], DTLZ [15] 5 , and real-world testing problems [50]. Real-world testing problems include: fourbar truss design (RE21), reinforced concrete beam design (RE22), disc brake design (RE33), rocket injector design (RE37), car side impact design (RE41), and conceptual marine design (RE42). Notably, RE41 and RE42 are complex four-objective problem having a large objective spaces. The prefix 'RE' denotes this problem is a real-world one.

Baseline MOEAs include: MOEA/D [58], a decomposition-based approach; (2) MOEA/D-AWA [40],

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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 a solution set by hypervolume maximization 6 ; 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.

We present the results for biobjective problems in Figure 3 and Table 1 7 . By directly minimizing maximal pairwise distances, the uniform indicator (which corresponds to maximal pairwise distances, see Appendix B.1, metric 5) is optimized effectively and ranks best among all methods. The fill distance, a 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.1, metric 2), representing the mean Euclidean distance between the true PF and the found sizeK solution set, is significantly improved. The significant improvement over IGD, a well-established indicator of uniformity and convergence of Pareto solutions, suggests that our method finds high-

quality Pareto solutions and also implies an inherent theoretical connection between IGD and minimal pairwise distance maximization, warranting further investigation.

We would like to mention another advantage of UMOD is 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 a PF.

For three-objective problems, DTLZ1 owns a simplex-like PF, making DTLZ1 the only problem where uniform preferences result in uniform Pareto objectives. For DTLZ2 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 a 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-objective problems, shown in Figures 8 and 9 and Table 9. UMODoutperforms DEAGNG and LMPFE by discovering a 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 a subset from a larger solution set, making it less efficient for four-objective 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 to 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 crossentropy classification loss, and the second is a hyperbolic tangent relaxation of the difference of equality of opportunity (DEO) loss [41][Eq. 6]. Data and network architecture details are provided in Table 6 in Appendix B.3.

Results with five solutions are illustrated in Figure 7 compared with PMGDA [59], Hypervolumegradient method (HVGrad), EPO [36], modified Tchebycheff aggregation function method (AggmTche) [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 in Appendix C.4. In real-world problems, the true PF range is often unknown, making it difficult for preference-based methods to select uniform preference vectors. If these vectors exclude PF endpoints, parts of 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 7 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 uniform objective vectors independent of scale, as evidenced by its superior performance in uniformity and soft uniformity indicators. Additionally, UMOD's spacing indicator, measuring neighborhood distance deviation, is significantly lower than other methods (except

0.30

0.35

0.40

0.45

0.50

0.55

0.60

f

1

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

f

2

UMOD

Agg-mTche

PMGDA

HVGrad

EPO

(a) Adult

0.25

0.50

0.75

1.00

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PMGDA

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(b) Compass

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, a desirable feature. Many methods show similar HV indicators, but their solution configurations vary, suggesting HV optimization is a coarse measure of uniformity. UMOD's significantly better uniformity performance indicates that directly optimizing uniformity, as in UMOD, is a promising approach for generating diverse Pareto solutions.

6 Conclusions and further works

Conclusions. In this paper, we have proposed a new understanding of a longstanding problem in MOO, generating K uniform and representative Pareto objectives, through searching for a maxpacking 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, we believe that we pave a new way for (rate-)optimally configuring the Pareto objectives.

Future works include applying UMOD to large-scale real-world multiobjective problems, such as material design, vaccine design, and recommendation systems. The broader impacts of this work is discussed in Appendix D.1.

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Supplementary Material

Table of Contents

The appendix comprises four sections:

Appendix A: Detailed proofs for the main paper's conclusions.

Appendix B: Additional experimental details.

Appendix C: Method details omitted from the main paper.

Appendix D: Discussion on the broader impact of our method.

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 in Appendices A.1 and A.2. Lastly, in Appendix A.3, we 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.,

Proof. Following the derivation to attain Equation (3) in Pronzato [39], we can prove d FD ≤ d Pack as well as the claim that the fill distance between T and the optimal design Y Pack induced by d Pack is upper bounded by d Pack .

Next, we prove d FD ≥ 1 4 d Pack by contradiction. Consider Y FD is the design that induces d FD , we have that each point in Y Pack must be within a d FD -ball centered at a point in Y FD , and the condition d FD < 1 4 d Pack requires that a specific ball will only contain a single y ( k ) ∈ Y Pack otherwise Y Pack will not be a d Pack -packing.

Since d FD < 1 4 d Pack , for all k ∈ [ K ] the corresponding d FD -ball is completely contained in the larger ( d Pack / 2) -ball centered at y ( k ) . However, we note T is connected, and thus there exists a certain y ∈ T outside all the K ( d Pack / 2) -balls; this certain point will not be covered by all the d FD -ball centered at points in Y FD as well. Finally, the existence of the certain point contradicts the claim that Y FD is a d FD -covering.

Lastly, we prove for Theorem 4 in the main paper by a contradiction.

Proof. Since d Pack K +1 > d Pack K , K is the maximal packing number under distance d Pack K .

Assume Y Pack K is not a d Pack K -covering. Then there exists y ( K +1) such that ρ ( y ( K +1) , y ( i ) ) < d Pack K for all i ∈ [ K ] . This contradicts the assumption, as Y Pack K ∪ { y ( K +1) } forms a d Pack K -covering, implying a packing number of K +1 when K was maximal. Thus, Y Pack K is a d Pack K -covering.

A.2 Configuration of d Pack for bi-objective problems

In the following, we prove Theorem 5 in the main paper. Before that, we prove for Lemma 8 to describe a property of ρ ( y (1) , y (2) ) , the distance between y (1) and y (2) when y (1) , y (2) ∈ T .

Lemma 8. For biobjective problem function g is strictly decreasing with respect to the first element ( y (1) 1 ) of y (1) and strictly increasing with the first element ( y (2) 1 ) of y (2) .

Proof. We consider a new vector ˜ y (1) ∈ T such that ˜ y (1) 1 < y (1) 1 . Since ˜ y (1) ∈ T , y (1) and ˜ y (1) cannot dominate each other, implying ˜ y (1) 2 > y (1) 2 . The distance between this new solution ˜ y (1) and y (2) is:

The previous equations show that ρ ( y (1) , y (2) ) is strictly decreasing with respect to the first element of y (1) . Similarly, considering a new vector ˜ y (2) ∈ T , ˜ y (2) 1 > y (2) 1 ( ˜ y (2) 2 < y (2) 2 ), using the same calculation method used in Equation (11), it is proved that ρ ( y (1) , ˜ y (2) ) > ρ ( y (1) , y (2) ) . This indicates that ρ ( y (1) , y (2) ) is strictly increasing with respect to the first element of y (2) .

With this established Lemma, we are now geared up for the complete proof.

Proof. The proof consists of two parts.

Part 1 proves that the neighboring distances ρ ( y ( i ) , y ( i +1) ) are equal for all i ∈ [ K -1] .

Part 2 proves that y (1) = p (1) and y ( K ) = p (2) . p (1) and p (2) are the two endpoints of a PF.

Part 1. We prove ρ ( y (1) , y (2) ) = . . . = ρ ( y ( K -1) , y ( K ) ) by contradiction. We denote d ( i ) the distance between y ( i ) and y ( i +1) and i ′ and j ′ are two indices such that

Without loss of generality, assume i ′ < j ′ . We now aim to derive a contradiction under the condition given by Equation (12). The approach is to iteratively decrease the first element of y ( i ′ +1) by a small margin ε > 0 , yielding ˜ y ( i ′ +1) . By Lemma 8, this adjustment ensures that ρ ( y ( i ′ ) , ˜ y ( i ′ +1) ) < ρ ( y ( i ′ ) , y ( i ′ +1) ) while ρ (˜ y ( i +1) , y ( i +2) ) > ρ ( y ( i +1) , y ( i +2) ) . Specifically, for each k such that i ′ +1 ≤ k ≤ j ′ -2 , y ( k ) is updated to ˜ y ( k ) according to the following rules:

When ε is sufficiently small, ˜ j ′ = arg min { ρ (˜ y ( i ′ ) , ˜ y ( i ′ +1) ) , . . . , ρ (˜ y ( j ′ -1) , ˜ y ( j ′ ) ) } = j ′ , meaning the minimal index before and after adjustment remains unchanged. However, the value of the adjusted distance ρ (˜ y ( j ′ ) , ˜ y ( j ′ +1) ) is increased from ρ ( y ( j ′ ) , y ( j ′ +1) ) by ε , showing the original design is not a max-packing design, leading to a contradiction. If conditions

hold, a similar argument as above can be used to derive a contradiction. Since the selection of i ′ and j ′ is arbitrary, this implies that for any interval between i ′ and j ′ , both Equation (12) and Equation (14) cannot hold simultaneously. This leads to the following equality, which aligns with our intended design goal:

Remark. The key component of our proof is the validity of Lemma 8, which applies to a two-dim PF only. This leads to a contradiction with both Equation (12) and Equation (14), forming the core of our argument.

̸

Part 2. We prove y (1) = p (1) by contradiction. The proof for y ( K ) = p (2) follows similarly. Assuming y (1) = p (1) , we replace y (1) with p (1) , forming a new configuration [ ˜ y (1) ( p (1) ) , y (2) , . . . , y ( K ) ] . Based on Part 1, where we showed equal neighboring distances between vectors, the condition

holds.

Next, we replace y (2) , . . . , y ( K -1) with ˜ y (2) , . . . , ˜ y ( K -1) on the PF, such that ˜ y ( i ) 1 = y ( i ) 1 -ϵ ( i ) , for 2 ≤ i ≤ K -1 , ensuring

By Lemma 8, moving y (1) to ˜ y (1) results in ρ (˜ y (1) , y (2) ) > ρ ( y (1) , y (2) ) . Iteratively shifting y (2) , . . . , y ( K -1) ensures ˜ y (2) 1 < y (2) 1 and so on, until ˜ y ( K -1) 1 < y ( K -1) 1 , completing the process.

This leads to a contradiction, proving that y (1) = p (1) and y ( K ) = p (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 ρ ( · , · ) , which measures the distance between two vectors. Given our assumption, we derive the error between distances computed under h and h ϕ . For any two points y ( i ) , y ( j ) in the image of h , the error in their distances compared to h ϕ can be bounded as follows:

This follows from the triangle inequality and the assumption | h ϕ ( ϑ ) -h ( ϑ ) | ≤ ϵ .

Given this pairwise bound, the overall configuration of points is such that the minimization of distances among points under h ϕ will either match or exceed the minimization under h within the bounds of 2 ϵ :

Considering the maximization of these minimum distances across all configurations ϑ (1) , . . . , ϑ ( K ) , we have:

Thus, the error in the optimal maximal minimal distance under the model transformation can be bounded by 2 ϵ , completing the proof.

A.4 Proof of the 'equivalent conversion' argument

This argument is actually a direct corollary the following two lemmas.

Lemma 9 (Adapted from [10], Theorem 3.1) . A solution x is weakly Pareto optimal iff there exists a weight vector λ such that x is (one of) an optimal solution of the modified Tchebycheff function.

MOEAD

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DEAGNG-P1

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Lemma 10 (Modified from [37], Theorem 2.6.2) . If an aggregation function is decreasing w.r.t. vector f ( x ) (i.e., g λ ( f ( x )) ≤ g λ ( f ( x ′ )) when f i ( x ) ≤ f i ( x ′ ) , ∀ i ∈ [ m ] and at least one index j f j ( x ) < f j ( x ′ ) , then one of the optimal solution x ∗ of g λ ( f ( x )) is a weakly Pareto optimal solution for the original MOP. In addition, if the optimality is unique, x ∗ is Pareto optimal .

Based on the first Lemma, we know that for any weakly Pareto objective, there is a corresponding preference vector that solving the modified Tchebycheff function can recover this vector. Furthermore, since we assume the uniqueness of the optimality of the modified Tchebycheff function, thus according to the second lemma, solving the modified Tchebycheff 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.1, we 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, we use various performance indicators, detailed and mathematically expressed below.

1. Hypervolume (HV) ( ↑ ) [24] both assesses convergence to a 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

DEAGNG-P1

LMPFE

MOEAD

NSGA3

SMS

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SMS-EMOA

DEAGNG-P2

LMPFE

MOEAD

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LMPFE

MOEAD

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objective belongs to the set Y with a reference point r .

IGD [26] indicator of a set A with a reference set Z is defined as:

where d i represents the Euclidean distance from z i to the nearest distance in the set of A .

Sparsity ( ↓ ) [56] is a 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:

where ˜ y ( i ) are the objective vectors arranged in a non-dominated sorting order from the set { y (1) , . . . , y ( N ) } . Here, m represents the number of objectives, and N is the number of solutions. A lower Sparsity value indicates a more uniformly distributed set of solutions along the Pareto front. Specifically, for a Pareto front with a two-dimensional linear shape, the sparsity indicator reaches its minimum when the objectives are spaced equidistantly.

̸

Spacing ( ↓ ) [43]: This metric assesses solution distribution uniformity by calculating the standard deviation of ˜ d ( i ) , the minimal distance between a solution y ( i ) and its nearest neighbor, where d ( i ) = min j ∈ [ m ] ,j = i ρ ( y ( i ) , y ( j ) ) . A lower spacing indicator implies evenly spaced solutions, reflecting uniform distribution. A spacing indicator of zero indicates that all Pareto objectives have equally minimal neighborhood distances.

5. Uniformity ( ↑ ) and Smooth Uniformity ( ↑ ) indicators, as introduced by [42], evaluate the distribution of solutions. The Uniformity indicator, δ Unif , is defined as the minimum distance between any two solutions:

Onthe other hand, the Smooth Uniformity indicator, ˜ δ Unif , incorporates a logarithmic sum exponential function to average distances among solutions. It introduces a sensitivity parameter η (set to 20 in this study) to highlight the overall distribution of distances:

B.2 Full name of multiobjective methods

To avoid confusion, we provide the full names of the baseline MOO methods in Table 4. 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 i7-10700 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) to 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 a multiobjective problem. Details of the fairness classification problems are shown in Table 6.

(License) To close this subsection, we 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 8 and 9). Due to the difficulty of visualizing four-dimensional space, we project the Pareto objectives into 3-D spaces: ( f 1 , f 2 , f 3 ) , ( f 1 , f 2 , f 4 ) , ( f 1 , f 3 , f 4 ) , and ( f 2 , f 3 , f 4 ) , labeled P-1 to P-4 in Figures 8 and 9.

Despite losing some information in the projections, meaningful conclusions can still be drawn from these figures:

1. DEAGNG and LMPFE can find partial parts of the true Pareto front. For the P1, P2, and P4 projections, DEAGNG typically finds only a small portion in the upper right of the 3-D Pareto front, while LMPFE misses a small part of this region. In contrast, the proposed method captures a more extensive span of the Pareto front, covering the largest area.

2. For a 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 to achieve a more uniform PF.

3. The proposed UMOD method finds more uniform Pareto objectives on the Pareto front compared to NSGA3, SMS-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 in Tables 7 to 9, respectively. Each experiment was conducted with 31 random seeds. For each data point, X ( Y )( 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 are preferred smaller. The optimal result averaged across all seeds is marked in bold. In the last row of each table, we calculate the mean rank of each indicator, with the highest one in bold.

The tables show that, with a small solution budget, the uniformity indicators IGD, Spacing, Uniform, SUniform, and FD achieve the best results, outperforming previous methods significantly.

SMS-EMOA achieves the highest HV value among all methods, but HV is only a rough measure of uniformity, validating that optimizing hypervolume alone does not ensure the best uniform distribution.

These tables also validate the effectiveness of maximizing pairwise distances, a 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 sizeK optimized set, is optimized by maximizing the pairwise distance. This interesting empirical finding is worthy of further investigation.

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B.6 Results on DTLZ5 and DTLZ6

The PFs of DTLZ5 and DTLZ6 are identical, which are a degenerate 1-dimensional hyper-curve within the three-objective space. The visualization results are shown in Figures 10 and 11 and numerical results are shown in Table 10.

Table 10 highlights that UMOD significantly outperforms other methods in ensuring evenly distributed solutions. In the decomposition-based framework, for such degenerated problems, different preferences may correspond to the same Pareto objective. The reason behind generating duplicate solutions is explained in Appendix C.5. And therefore, MOEA/D and MOEA/D-AWA tend to produce duplicate solutions. NSGA3 is 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 in IGD and FD, which are of interests in this paper.

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 an initial uniform distribution of preference vectors. Then, we use either multiobjective evolutionary algorithms (MOEAs) or gradient-based MOO to solve for the preference angle and Pareto objective pairs. MOEAs are suitable for problems with local optimas, while gradient-based MOO is efficient for neural network problems with millions of decision variables. Next, we fit a Pareto front model to learn the expression of h ϕ 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 { λ (1) , . . . , λ ( K ) } by Das-Dennis method [12]. Initial solutions { x (1) , . . . , x ( K ) } .

2: for n = 1 to N do

3: Run MOEA/D with mTche aggregation function or gradient-based MOO using { x (1) , . . . , x ( K ) } as initial solutions under preferences { λ (1) , . . . , λ ( K ) } .

4: Train a model h ϕ to predict Pareto objectives by the preference angles using mean square estimation with angle-objective pairs ( ϑ ( i ) , y ( i ) ) .

5: Update ( ϑ (1) , . . . , ϑ ( K ) ) by Algorithm 2.

6: Recalculate preference vectors, λ (1) , . . . , λ ( K ) .

7: Update the initial solutions { x (1) , . . . , x ( K ) } by MOEA/D or gradient-based mTche using the last generation of solutions as a warm start.

8: end for

Algorithm 2 Recalculate Preference Angles (ALG_Update)

1: Input: The initial configuration { ϑ (1) , . . . , ϑ ( K ) } and h ϕ .

2: for i = 1 to N opt do

3: Calculate the indexes for the minimal pairwise objectives:

4: Update the positions of ( ϑ ( i ∗ ) , ϑ ( j ∗ ) ) :

5: end for

6: Output: The updated preference angles { ϑ (1) , . . . , ϑ ( K ) } .

C.2 Problem formulations

For completeness, ZDT1, ZDT2, and DTLZ1 problems are described as follows.

The PF of ZDT1 is f 2 = 1 - √ f 1 , 0 ≤ f 1 ≤ 1 .

ZDT2.

The PF of ZDT2 is f 2 = 1 -f 2 1 , 0 ≤ f 1 ≤ 1 .

DTLZ1.

The PF of DTLZ1 is 0 . 5∆ 3 (3-dim simplex).

C.3 Conversion between a preference and a preference angle

The preference vector λ and preference angles ϑ are easily inter-convertible via the following equations. This one-to-one, differentiable mapping allows conversion between ϑ and λ . While λ belongs to the m -D simplex, ϑ lies within the box constraint [0 , π 2 ] m -1 . For optimization purposes, ϑ is more manageable because it can be easily projected onto [0 , π 2 ] m -1 , whereas projecting λ onto the m -D simplex is more complex.

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:

Agg-LS (Linear Scalarization):

Agg-Tche (Tchebycheff):

where z is a reference point (e.g., ideal point) which dominates the entire PF, i.e., z ⪯ y , for all y ∈ T . For the modified Tchebycheff ( Agg-mTche ) function, the modification involves replacing λ i with 1 λ i in the equation above.

Agg-PBI (Penalty-Based Intersection):

where z is the same reference point as introduced. For gradient-based multiobjective optimization methods, solution x is updated as x ← x -η ∂g λ ( x ) ∂ x , where η is a small positive learning rate, and the upperscript agg denotes PBI, LS, or Tche. We also would like to briefly introduce the other three gradient based method, PMGDA (Preference-based 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 metric, thereby optimizing the overall dominance of the solution set.

C.5 Duplicated solutions issues caused by the mTche aggregation function

𝒇𝟏

𝒇𝟐

O

Pareto front

𝒚 ∗

𝝀 (𝟏)

𝝀 (𝟐)

In this section, we discuss when Tchebycheff aggregation methods yield duplicated Pareto objectives. Appendix C.5 illustrates this, showing a preference vector λ (1) intersecting the PF at the optimal solution y ∗ , where

If the preference vector does not intersect the Pareto front ( λ (2) ), y ∗ is the Pareto front endpoint with the highest Tchebycheff value. In Appendix C.5, y ∗ is the optimal value, so preferences λ (1) and λ (2) correspond to the duplicated Pareto objective y ∗ .

D Miscellanies

D.1 Broader impacts

By optimizing multiple conflicting objectives, these algorithms enhance decision-making in fields such as healthcare, product design, and trustworthy machine learning. In product design, multiobjective optimization balances trade-offs 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.

UMODis a foundational algorithm, and its broader impact depends on its downstream applications. We believe UMOD itself does not have a 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 as DTLZ7, as Theorems 3 and 4 assume a connected Pareto front to ensure uniform distribution. Second, we have not considered problems with discrete, binary decision variables, or constrained MOPs, as our approach requires a 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 multiobjective algorithms, which brings inconvenience.

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes] .

Justification: We explicitly enumerate our contributions in the end of the introduction part.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: We list the limitations in the last section (Section 6).

3. Theory Assumptions and Proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: The assumptions are included in theorems and the full proofs are provided in Appendices A.1 to A.3 respectively.

4. Experimental Result Reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: We provide experiment details in Appendix B.3.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes] .

Justification: The model implementation and the code are attached as supplementary materi- als. Source codes have been open to the public.

6. Experimental Setting/Details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes] .

Justification: The details experiment settings are provided in Appendix B.3.

7. Experiment Statistical Significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes] .

Justification: All numerical results are averaged on 31 random seeds.

8. Experiments Compute Resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes] .

Justification: Details are provided in Section 5 (Experiment settings).

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Answer: [Yes] .

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Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes] .

Social impact is discussed in Appendix D.1.

11. Safeguards

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Answer: [Yes] .

Justification: Licenses are provided in Appendix B.3.

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Justification: The proposed method is a fundamental algorithm and is not directly related to human subjects.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\nThis 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.\n\nWEAKNESSES:\nAlthough 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.\n\nQUESTIONS:\n1. 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.\n\nLIMITATIONS:\nThe 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.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\n- 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.\n\nWEAKNESSES:\n- 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.\n\nQUESTIONS:\n- 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.\n\nLIMITATIONS:\nThe 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.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-10"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn 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.\n\nSTRENGTHS:\nThe 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.\n\nWEAKNESSES:\nSection 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.\n\nQUESTIONS:\n1. 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.\n\nLIMITATIONS:\nAuthors 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.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-12"}

comments: 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.

STRENGTHS:
1. A new metric is proposed to evaluate the uniformity of the solution set on the Pareto front.  
2. 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.  
2. 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.  
3. The standard benchmark set only considers ZDT and DTLZ problems, omitting more complex problem instances.

QUESTIONS:
1. How sensitive is the algorithm to the initialization method for K preferences?  
2. 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?  
3. 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.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 2
DATE: 2024-07-14