Gliding over the Pareto Front with Uniform Designs

Authors: xiaoyuan  zhang
genghui  li
xi  lin
yichi  zhang
yifan  chen
qingfu  zhang
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-13 03:50:13

Abstract: Multiobjective optimization (MOO) plays a critical role in various real-world domains. A major challenge therein is generating $K$ uniform Pareto-optimal solutions that represent the entire Pareto front. To address the very challenge, this study first introduces \emph{fill distance} to evaluate the $K$ design points, which provides a quantitative metric for the representativeness of the design. However, the direct specification of the optimal design that minimizes fill distance is almost intractable considering the nested $\min-\max-\min$ optimization problem. We further propose a surrogate to the fill distance, which is easier to optimize and induce a rate-optimal design whose fill distance proves at most $4\times$ the minimum one. Rigorous derivation also shows that asymptotically this induced design will converge to the uniform measure over the Pareto front. Extensive experiments on synthetic and real-world benchmarks demonstrate that our proposed paradigm efficiently produces high-quality, representative solutions and outperforms baseline methods.

Main paper: 
 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 ).Various MOO methods effectively generate diverse Pareto objectives, for both gradient-based and evolution-computation (EC)-based frameworks. In the line of gradient-based methods, Pareto Mul-tiTask Learning (PMTL) [33] produces 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 to maximize their separation; Exact Pareto Optimization (EPO) [37] aligns solutions with user-specific preference vectors, fostering a diverse distribution of Pareto objectives by utilizing varied preferences. For multiobjective evolutionary algorithms (MOEAs), NSGA2 [14] introduces the crowding distance and Pareto rank to achieve a diverse distribution of Pareto objectives; NSGA3 [13, 27] introduces a new diversity preservation technique based on reference points; MOEA/D [59] 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 greatest hypervolume to enhance diversity. A key 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, important distribution characteristics of its achieved Pareto objectives can be determined by UMOD.Another approach to generating a size-K diverse Pareto set is subset selection. This method first generates numerous objectives on the PF, then selects K solutions to maximize hypervolume or minimize IGD [23, 53, 9, 46] . Subset selection, a discrete optimization problem, is generally inefficient to solve compared with continuous optimization problems. Recently, some approaches employ greedy algorithms [9, 34, 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.Since the proposed method can be classified under the preference adjustment category, we also discuss the relationship between the proposed UMOD and preference/weight2 adjustment methods. Preference adjustment stems from MOEA/D-AWA [41] , 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 [36] , including DEA-GNG [35] and MOEA/D-SOM [22] , which utilize neural gas networks to guide the selection of preference vectors. Other approaches, such as W-MOEA/D [21] , tw-MOEA/D [39] , paλ-MOEA/D [49] , and MOEA/D-AWG [56] , use mathematical models to shape the non-dominated solutions and adjust preference vectors, achieving a diverse distribution of Pareto-optimal objectives. The proposed method differs from other preference adjustment methods in two key ways: (1) it models the PF using a neural network, offering both accuracy and efficiency, and (2) it provides rigorous theoretical analysis for selecting preference vectors that ensure uniformity and representativeness.3 Pareto solutions with uniform designsIGD(A) = 1 |Z| |Z| i=1 d i ,where d i represents the euclidean distance from z i to the nearest distance in the set of A.3. Sparsity (↓) [57] 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:Sparsity = 1 N -1 m j=1 N -1 i=1 ỹ(i) j - ỹ(i+1) j 2 ,where ỹ(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.4. Spacing (↓) [44] : 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],j̸ =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. δ Unif = min 1≤i<j≤K ρ(y (i) , y (j) ).Conversely, the Soft Uniformity indicator, δUnif , 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:δUnif = - 2 ηK(K -1) log 1≤i<j≤K exp(η • ρ(y (i) , y (j) )). 
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 generalized distance (IGD) indicator [54] of a set Y, a famous metric in MOO. FD and IGD are defined as follows:EQUATIONwhere ρ(•, •) represents the Euclidean distance between two vectors. 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 is an upper bound of IGD).EQUATIONThe first inequality holds by definition since Y IGD minimizes IGD, while the second inequality holds because, for a fixed configuration Y FD , the average distance (IGD(Y FD )) is always less than or equal to the maximum distance (FD(Y FD )). Theorem 2 establishes that the optimal configuration of FD (Y FD ) provides an upper bound for the IGD value. However, to the best of our knowledge, the optimal configuration of IGD does not offer a similar bound for FD. This distinction is one of the reasons we focus on discussions on FD in this paper.The minimization of FD involves solving the following nested min -max -min problem:EQUATIONA small d FD indicates that the optimal configuration Y FD can well cover the entire PF with a relatively low covering radius. However this triply nested structure is known difficult to be optimized [55, 40] . Thus, we plan to seek a surrogate max-packing problem,EQUATIONwhere δ represents the separation distance between two vectors. The optimal design Y Pack , which solves the optimization problem (Equation ( 5)), is known as the best-packing design [8] . In this paper, we show in Theorem 3 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. Theorem 3 (Surrogate for minimal FD). Consider a connected, compact3 PF T . The minimal fill distance d FD between T and a size-K design Y Pack /Y FD will then be bounded as: EQUATIONThis section discusses key properties of the max-packing design on a PF. For bi-objective problems, Y Pack exhibits favorable properties when Y Pack ⊂ T , as formalized in Theorem 5. The proof is provided in Appendix A.2.Theorem 5 (Characterization of Y Pack for biobjective problems). Let Y Pack = [y (1) , . . . , y (K) ] be sorted by increasing first component, such that y(1) 1 ≤ . . . ≤ y (K)1 . For a compact, connected T , Y Pack is characterized as follows: 1. Equal spacing: ρ(y (1) , y (2) ) = . . . = ρ(y (K-1) , y (K) ).2. 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 ].Remark. According to Theorem 5, first, Y Pack , including both the starting and ending points, spans the maximum range among all configurations, which is desirable. Second, equal pairwise distances between Pareto objectives yields an intuitive interpretation of uniformity. Maximizing hypervolume ensures the "equal spacing" property for bi-objective linear PFs [4] [Theorem 4], while our design only requires the PF to be compact and connected.Besides this bi-objective results, we examine the asymptotic properties of Y Pack by Theorem 6. Theorem 6 (Asymptotic uniformity [8] (Theorem. 2.1)). As 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:EQUATIONwhere # denotes the number of points in a set, and "Vol" denotes the volume of a set.  
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:EQUATIONTo 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)),EQUATION)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. This argument is proved in Appendix A.4.Example 7. Specific formulations of h(λ) on famous MOPs include:1. ZDT1:y 1 = kλ 1 , y 2 = k(1 -λ 1 ), where k = (2 -λ 1 --3λ 2 1 + 4λ 1 )/2(λ 1 -1) 2 .2. ZDT2:y 1 = kλ 1 , y 2 = k(1 -λ 1 ), where k = (λ 1 -1 + 5λ 2 1 -2λ 1 + 1)/2λ 2 1 . 3. DTLZ1: y i = 0.5λ i , for i = 1, 2, 3.The formulations and PFs of the three problems are provided in Appendix C.2. These examples indicate that uniform Pareto objectives are only achievable when the true PF is affine to simplexes (e.g., on DTLZ1 problem), where h(λ) is also affine. In most other cases, the non-linearity of h(λ) prevents uniform Pareto objectives. Substituting Equation (9) into Equation (8) yields the following bi-level optimization problems: (1) ,...,ϑ ϑ ϑ (K) min 1≤i<j≤K ρ(y (i) , y (j) )         d Pack = max ϑ ϑ ϑEQUATIONThe function λ(ϑ ϑ 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 ϑ ϑ) as decision variables for easier optimization since λ(ϑ ϑ ϑ) is constrained in a box. In the right equation, the Pareto objective is denoted as y = h(ϑ ϑ ϑ) = h(λ(ϑ ϑ ϑ)). Various bi-level optimization methods [48, 62] can be used to solve problem (Equation ( 10)). For efficiency consideration, we propose to 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 simple algebraic calculations, ∂δ ∂ϑ ϑ ϑ can be calculated by the following two equations:∂δ ∂ϑ ϑ ϑ (i * ) = ∂h(ϑ ϑ ϑ (i * ) ) ∂ϑ ϑ ϑ (i * ) B(n×m) h(ϑ ϑ ϑ (i * ) ) -h(ϑ ϑ ϑ (j * ) ) ρ(h(ϑ ϑ ϑ (i * ) ), h(ϑ ϑ ϑ (j * ) )) A(1×m) ⊤ , ∂δ ∂ϑ ϑ ϑ (j * ) = - ∂h(ϑ ϑ ϑ (j * ) ) ∂ϑ ϑ ϑ (j * ) C(n×m) h(ϑ ϑ ϑ (i * ) )h(ϑ ϑ ϑ (j * ) ) ρ(h(ϑ ϑ ϑ (i * ) ), h(ϑ ϑ ϑ (j * ) )) A(m×1) ⊤ ,The remaining terms ∂δ ∂ϑ ϑ ϑ (k) , k ̸ = i * , j * are zero vectors. Calculating the gradient vector A is straightforward by using basic algebra calculations. However, computing the gradient matrices B or C is difficult because h(ϑ ϑ ϑ) is a black-box function, representing the optimal values of an optimization problem. Estimating matrices B or C using finite difference is impractical, as it requires solving Equation (9) for at least n × m times, which is highly time-consuming. To efficiently estimate ∂h ∂ϑ ϑ ϑ , we propose using a neural model h ϕ to approximate h based on historical data (ϑ ϑ ϑ, h(ϑ ϑ ϑ)), allowing us to estimate ∂h ∂ϑ ϑ ϑ via ∂h ϕ ∂ϑ ϑ ϑ . Since a neural network is introduced, we analyze its induced error in Theorem 8, with the full proof in Appendix A.3. Theorem 8 (Optimization error ϵ nn introduced by using a network). Let h ϕ (ϑ ϑ ϑ) be an approximator of h(ϑ ϑ ϑ) such that ||h ϕ (ϑ ϑ ϑ) -h(ϑ ϑ ϑ)|| ≤ ϵ, for every ϑ ϑ 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 2ϵ:ϵ nn = max ϑ ϑ ϑ (1) ,...,ϑ ϑ ϑ (K) min 1≤i<j≤K ρ(h ϕ (ϑ ϑ ϑ (i) ), h ϕ (ϑ ϑ ϑ (j) )) -d Pack ≤ 2ϵ.Remark. The error ϵ, defined as ||h ϕ (ϑ ϑ ϑ) -h(ϑ ϑ ϑ)|| ≤ ϵ, is influenced by the covering radius R of the estimated solutions h ϕ (ϑ ϑ ϑ (1) ), . . . , h ϕ (ϑ ϑ ϑ (K) ) and the fitting error ϵ fit , whereϵ fit = max k∈[K] ||h ϕ (ϑ ϑ ϑ (k) ) -h(ϑ ϑ ϑ (k) )||.For any ϑ ϑ ϑ, the error satisfies||h ϕ (ϑ ϑ ϑ) -h(ϑ ϑ ϑ)|| ≤ L||ϑ ϑ ϑ -ϑ ϑ ϑ (i ′ ) || + ϵ fit ≤ L • R + ϵ fit , where L is the Lipschitz constant of function (h ϕ (ϑ ϑ ϑ) -h(ϑ ϑ ϑ)), ϑ ϑ ϑ (i ′ )is the nearest solution to ϑ ϑ 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 negligible.Practical algorithms. Due to the space limit, the pseudo-codes for UMOD are provided in Algorithm 1 and Algorithm 2 in Appendix C.1. Initially, we generate K approximately uniform preferences using the Das-Dennis method [12] . Then either a multiobjective evolutionary algorithm based on decomposition (MOEA/D) [59] or a gradient-based MOO with the mTche aggregation function is employed for producing preference angle and Pareto objective pairs (ϑ ϑ ϑ, y). MOEA/D is suitable for problems with many local optima, while gradient-based MOO is more efficient for multiobjective multitask learning (MOMTL) problems. Based on (ϑ ϑ ϑ, y) pairs, a PF model h ϕ (ϑ ϑ 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 steps are repeated for N iterations until convergence. 
experiments: 
 The experiments compare UMOD with other methods on two types of MOPs: (1) those with many 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 (↓) [57] , (4) Spacing (↓) [44] ,(5) Uniformity (↑), ( 6) Soft 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. We demonstrate the effectiveness of our proposed method across a diverse set of MOEA benchmark problems, including the ZDT series4 [16] , and DTLZ problems 1-6 [15] 5 , as well as various real-world problem instances [50] . The real-world cases include: four-bar 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 problems that involve extensive fourdimensional objective spaces. The prefix "RE" denotes real-world scenarios.Table 1 : Partial results for biobjective problems (full results are in Table 7 ).Indicator DEA-GNG [35] LMPFE [51] Subset [9, 46] NSGA3 [13, 27] Multiobjective testing problems often have many local Pareto optimal solutions, leading to comparisons primarily with MOEAs. These include: (1) MOEA/D [59] , a decomposition-based approach;(2) MOEA/D-AWA [41] , integrating adaptive weight adjustment; (3) NSGA3 [13, 27] , generating 9 ). 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, 46] , choosing a solution set by hypervolume maximization 6 ; and (7) DEA-GNG [35] , 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 minimal covering radius, which satisfies our main purpose. 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 size-K 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 highquality 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 varying scales, as demonstrated in Figure 4 Finally, we discuss the results for four-objective problems which are reported in Figures 8 and 9 and Table 9 . A key advantage of UMOD over DEAGNG and LMPFE is its ability to discover a broader PF (Figure 5 ) since it directly maximizing the minimal pairwise distance on the PF. Although subset selection performs similarly to UMOD on the RE41 problem, it relies on a inefficient two-phase optimization problem in the manner of electing a subset of problems from a much larger solution set. Generating a large number of solutions for a four-objective problem is considered inefficient. On RE41, UMOD significantly outperforms other methods in IGD, indicating UMOD find more representative solutions. On RE42, UMOD achieves high rankings (the best or second best) in uniformity indicators such as spacing, sparsity, uniform, and soft uniform. To close the discussion, we would like to mention that due to a challenging PF region that all methods find difficult to detect, the IGD indicators for UMOD, NSGA3, and SMS-EMOA are close and these three methods outperform the other methods a lot.In this subsection, we compare our methods with baseline methods on multiobjective fairness classification problems, specifically Adult [2] and Compass [1] . The decision variable is the neural network parameter used to classify both accuracy and fairness. The first objective, data classification accuracy, uses binary cross-entropy loss, while the second objective employs a hyperbolic tangent relaxation of the difference of equality of opportunity (DEO) loss [42][Eq. 6] . Details of the data and network architecture are provided in Indicator EPO [37] PMGDA [60] Agg-LS [38] Agg-PBI [59] Agg-Tche [38] HVGrad [17] UMOD Adult HV 1.17 (0.00) (3) 1.13 (0.00) (4) 1.20 (0.00) (0) This experiment validates our method's applicability to large-scale machine learning tasks and its robustness against stochastic gradients. Strong baselines include Exact Pareto Optimization (EPO) [37] , Preference-based MGDA (PMGDA) [60] , Hypervolume-gradient (HVGrad) [17] , and aggregation function-based methods like linear scalarization (LS), Penalty-based Intersection (PBI), and weighted Tchebycheff [38] . For the aggregation functions expressions, please refer to Appendix C.4. These gradient-based methods are supported by a recent MOO library called LibMOON [61] . Full name of these methods are provided in Table 4 . The visualization of different methods on the Adult problem is shown in Figure 7 , and the numerical results are in Table 3 . UMOD methods consistently find uniform Pareto objectives across the entire PF, recovering a larger Pareto set compared to other methods. The uniformity indicators outperform baseline methods. For the Adult problem, the Tchebycheff aggregation method is a strong competitor because the PF scales of the two objectives are similar. On the Compass problem, where objectives vary in scale, UMOD finds more uniform solutions compared to the Tchebycheff aggregation function, which tends to produce sparse and duplicated solutions at the lower right of the PF. The EPO method exhibits numerical instability issues in finding marginal solutions, leading to an incomplete PF and lower uniformity indicator, as reported in [31] . In both tasks, HV-Grad, EPO, and PBI aggregation methods identify only a small portion of the PF. Consequently, the spacing indicators of HV-Grad and EPO are better than UMOD since these methods find a narrower PF. 
conclusions and further works: 
 Conclusions. In this paper, we have proposed a new understanding of a longstanding problem in MOO, generating K uniform Pareto objectives, through introducing fill distance as a metric for the representativeness of the design. Moreover, we suggest an easy-to-optimize surrogate to the fill distance to specify the rate-optimal space-filling design. 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. 
: 
 Multiobjective optimization (MOO) is frequently utilized to guide the real-world decision-makings, such as in the field of material sciences [20, 47, 5, 30] , recommendation systems [29, 63, 32] , and industrial design [45, 52, 50, 58] . An MOO problem (MOP) involves multiple conflicting objectives, which can be informally formulated as:EQUATIONwhere m is the number of objectives. Noted that Equation (1) is a vector optimization problem and it does not admit a total ordering. To address that, the concept of Pareto optimality is therefore introduced. A solution is called Pareto optimal if no other x ′ ∈ X can dominate them. Domination occurs if f i (x ′ ) ≤ f i (x) for all i in 1 to m, with at least one strict inequality [38, 19] . The image f (x) of a Pareto optimal solution is called a Pareto objective in this paper. The set of all such optimal solutions is the Pareto set (PS), and their objectives constitute 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. There is number of prior works focused on generating diverse solutions (see Section 2), however they lacked formal definitions of representability and uniformity.In this paper, we define representability as the covering radius of a size-K solution covering the entire PF. An illustrative example is provided in Figure 1 , where eight uniformly distributed covers the entire PF with a small radius. In addition to representability, we describe uniformity through the configuration of non-asymptotic Pareto objectives and asymptotic uniformity when the number of K tends to infinity.In this paper, we first introduce the fill distance (FD) as the minimal covering radius of the Pareto objectives, measuring how well discrete solutions represent the true PF. A configuration with a small covering radius is considered a good representation of the PF. Secondly, we find that optimizing FD is challenging due to its nested min -max -min structure (Equation ( 4)). To address this, we maximize the minimal (max -min) pairwise distances of Pareto objectives that bounds the minimal covering radius up to a constant. Finally, we design a bi-level optimization framework based on neural network approximation to efficiently solve this max -min problem.Our method, called UMOD (Uniform Multi-Objective optimization based on Decomposition), is an important extension of the decomposition-based MOO paradigm [59] . We conduct comparative evaluations against methods on complex multiobjective problems with numerous local optimas and on fairness classification problems with thousands of decision variables. Empirical results demonstrate the effectiveness of our proposal. The contribution of this paper can be summarized as:1. We introduce the fill distance for a set of Pareto objectives as a new uniformity metric in multiobjective optimization, establishing a framework for measuring uniformity in MOO. Additionally, we reveal a surrogate objective function: maximizing the minimal pairwise distances of a size-K solution set on the PF, yielding a design with a fill distance that bounds the minimal fill distance up to a constant, regardless of the number (K) of solutions. 2. We propose constructing the maximizing minimal distance design as a bi-level optimization problem and using neural networks to enhance efficiency. We provide optimization error bounds for this problem and present a practical solution algorithm, UMOD. 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 surpasses these methods in terms of uniformity and efficiency on popular metrics used in multiobjective optimization.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 distinguish vectors, and subscripts (e.g., y  
 
 Appendix: 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 designIn the following content, we proves for Theorem 3 in the main paper, i.e., In the following, we prove Theorem 5 in the main paper. Before that, we prove for Lemma 9 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 9. For biobjective problem function g is strictly decreasing with respect to the first element (y1 ) of y (1) and strictly increasing with the first element (y1 ) of y (2) .Proof. We consider a new vector ỹ(1) ∈ T such that ỹ(1)EQUATION1 . Since ỹ(1) ∈ T , y (1) and ỹ(1) cannot dominate each other, implying ỹ(1)EQUATION2 . The distance between this new solution ỹ(1) and y (2) is:EQUATION1 -y(2)EQUATION1 -y1 ) + (yEQUATION2EQUATION2 -y2 )EQUATION1 -y1 ) 2 + (yEQUATION2) 2 + C (C > 0)> ρ(y (1) , y (2) ).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 ỹ(2) ∈ T , ỹ(2)EQUATION(2)2 ), using the same calculation method used in Equation (11) , it is proved that ρ(y (1) , ỹ(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 thatEQUATIONWithout 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 ỹ(i ′ +1) . By Lemma 9, this adjustment ensures that ρ(y (i ′ ) , ỹ(i ′ +1) ) < ρ(y (i ′ ) , y (i ′ +1) ) while ρ( ỹ(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 ỹ(k) according to the following rules:EQUATIONWhen ε is sufficiently small, j′ = arg min{ρ( ỹ(i ′ ) , ỹ(i ′ +1) ), . . . , ρ( ỹ(j ′ -1) , ỹ(j ′ ) )} = j ′ , meaning the minimal index before and after adjustment remains unchanged. However, the value of the adjusted distance ρ( ỹ(j ′ ) , ỹ(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 conditionsEQUATIONhold, 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: ρ(y (1) , y (2) ) = . . . = ρ(y (K-1) , y (K) ).Remark. The key component of our proof is the validity of Lemma 9, 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 [ ỹ(1) (p (1) ), y (2) , . . . , y (K) ]. Based on Part 1, where we showed equal neighboring distances between vectors, the condition ρ( ỹ(1) , y (2) ) > ρ(y (2) , y (3) ) = . . . = ρ(y (K-1) , y (K) )holds.Next, we replace y (2) , . . . , y (K-1) with ỹ(2) , . . . , ỹ(K-1) on the PF, such that ỹ(i)1 = y (i) 1 -ϵ (i) , for 2 ≤ i ≤ K -1, ensuring ρ( ỹ(i) , ỹ(i+1) ) > ρ(y (i) , y (i+1) ), i ∈ [K -1].By Lemma 9, moving y (1) to ỹ(1) results in ρ( ỹ(1) , y (2) ) > ρ(y (1) , y (2) ). Iteratively shifting y (2) , . . . , y (K-1) ensures ỹ(2)1 < y (2)1 and so on, until ỹ(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) .In this part, we prove for Theorem 8, 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:EQUATION)This follows from the triangle inequality and the assumption |h ϕ (ϑ ϑ ϑ) -h(ϑ ϑ 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 2ϵ:EQUATION)Considering the maximization of these minimum distances across all configurations ϑ ϑ ϑ (1) , . . . , ϑ ϑ ϑ (K) , we have:EQUATION)Thus, the error in the optimal maximal minimal distance under the model transformation can be bounded by 2ϵ, completing the proof.This argument is actually a direct corollary the following two lemmas. Lemma 10 (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. Lemma 11 (Modified from [38] , 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.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.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 HV r = Vol({y|∃y ′ ∈ Y, y ′ ⪯ y ⪯ r}). [26] indicator of a set A with a reference set Z is defined as:To avoid confusion, we provide the full names of baseline MOO methods in Table 4 . For SMS-MOEA (MOEA using S-Metric Selection), "S-metric' is another name for "hypervolume".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 . 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.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 Soft 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 size-K optimized set, is optimized by maximizing the pairwise distance. This interesting empirical finding is worthy of further investigation. 10 . 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 many local optima, 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 iterated until convergence.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) }.Train a model h ϕ to predict Pareto objectives by the preference angles using mean square estimation with angle-objective pairs (ϑ ϑ ϑ (i) , y (i) ).Update (ϑ ϑ ϑ (1) , . . . , ϑ ϑ ϑ (K) ) by Algorithm 2.Recalculate preference vectors, λ (1) , . . . , λ (K) .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 ϑ ϑ (1) , . . . , ϑ ϑ ϑ (K) } and h ϕ . 2: for i = 1 to N opt do 3:Calculate the indexes for the minimal pairwise objectives:(i * , j * ) = arg min 1≤i<j≤K h ϕ (ϑ ϑ ϑ (i) , ϑ ϑ ϑ (j) ).Update the positions of (ϑ ϑ ϑ (i * ) , ϑ ϑ ϑ (j * ) ):               ϑ ϑ ϑ (i * ) ← clip   ϑ ϑ ϑ (i * ) + η ∂h ϕ (ϑ ϑ ϑ (i * ) ) ∂ϑ ϑ ϑ (i * ) )A ϕ , 0, π 2   ϑ ϑ ϑ (j * ) ← clip   ϑ ϑ ϑ (j * ) -η ∂h ϕ (ϑ ϑ ϑ (j * ) ) ∂ϑ ϑ ϑ (j * ) A ϕ , 0, π 2  , where A ϕ = h ϕ (ϑ ϑ ϑ (i * ) )-h ϕ (ϑ ϑ ϑ (j * ) ) ρ(h ϕ (ϑ ϑ ϑ (i * ) ),h ϕ (ϑ ϑ ϑ (j * ) )) ⊤ . 5: end for 6: Output: The updated preference angles {ϑ ϑ ϑ (1) , . . . , ϑ ϑ ϑ (K) }.For completeness, ZDT1, ZDT2, and DTLZ1 problems are described as follows.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. UMOD is a foundational algorithm, and its broader impact depends on its downstream applications. We believe UMOD itself does not have a direct negative social impact.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.In this paper, a preference vector or a weight vector denotes the same meaning.A connected, compact set is also called a rectifiable set.ZDT5 is a discrete optimization problem and is commonly excluded in MOEA studies.DTLZ 7 is excluded because our theoretical results only apply to compact and connected PFs, which DTLZ 7 does not satisfy. Results on DTLZ 5-6 are demonstrated in Appendix B.6.Code: https://github.com/HisaoLabSUSTC/GAHSS.For a better illustration, the unit of IGD, Spacing, Sparsity, Uniform, SUniform, and Fill distance are scaled by 0.01,0.01,0.01,0.1,0.1, and 0.1. 


Decision: 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.

Reviews: review: paper_summary: paper_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.
strength_weakness: strength_weakness: 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: 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: limitations: It is recommended to provide a sensitivity analysis of the algorithm concerning different values of K.
paper_summary: paper_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.
strength_weakness: strength_weakness: 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.
questions: 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)
2. 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?
3. 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: 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.
paper_summary: paper_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.
strength_weakness: strength_weakness: - 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.
- While the paper claims that UMOD is efficient, there is limited discussion on the method to scale to larger problems.
questions: 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?
- Are there any additional limitations or specific considerations for higher-dimensional Pareto fronts?
- 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?
- It would be interesting to know how the algorithm performs on DTLZ5-7,  where the assumptions are not  satisfied.
limitations: 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.
paper_summary: paper_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.
strength_weakness: strength_weakness: Although the theoretical part is impressive, I am not fully convinced that UMOD is a novel and outstanding MOA.

**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.

**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.

The 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.

The 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?

Based 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.
questions: 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 $m$.

2. 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.

3. P3, Line 88. HV is to be maximized, while IGD is to be minimized.

4.  P5, Line 163. $\partial \delta/\partial \theta^{(j^*)}$ is $-CA$ or $C(-A)$ instead of $C-A$.
5.  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.
6.  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".
7.  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.
8.  P25, Fig. 10. The caption should be placed below the image.
9.  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.
10. 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: 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.

Comments: title: Final conclusion
Further experiment results and discussions (Sensitivity of different $K$, FE, independent runs)
FE for many-solution problems
Numerical results for larger K
Kindly ask for feed-backs
Thanks for your feedback
Results on DTLZ5 and 6
Source Code of UMOD
About Source Code of UMOD
Results on DTLZ5 and 6
We choose a small population because it is hard!
Results comparison with different FE on three objective problems.
Detailed Settings of the Comparison Experiments between LSSA and UMOD
Comparison between UMOD and LSSA
Analysis of different FEs
Kindly requiring for further comments
Thanks for your score increasing
Author's response
comment: Dear reviewer, 

Thanks a lot for pointing necessary references (e.g., LSSA), scoring raising, and your consist support for our work. We want to make a final response to your concerns, and we guarantee we will include these issues in out paper, since you have spent so much previous time and professional suggestions.

Here is our final response to your comments:

- **Comprehensive empirical study**: We will add extensive empirical studies, including all ablation and sensitivity analyses, beyond what was presented during the rebuttal.

- **Larger population and FE issues**: We will include results for both small and large populations in the main paper and discuss the impact of different FEs in MOEAs to appeal to a broader audience.

- **Complex PF**: Handling complex Pareto fronts (PF) is critical in the MOEA community. UMOD is well-suited for degenerated PFs (e.g., DTLZ5/6) in the decomposition framework, as it helps find preferences corresponding to representative solutions. For disconnected PFs, the challenges include (1) improving PF model estimation with techniques like self-organized mapping networks, and (2) optimizing the max-min distance, which is more difficult.

- **Limitations**: We will discuss limitations in detail, including the use of more FEs and the ability to handle complex PFs.
Dear Reviewer,

We have new results that directly address your concerns regarding to the setting of different $K$'s, which we summarize below for your convenience.

> **Question: Sensitivity analysis for different values of $K$?**

Thank you for raising this important issue. We agree that analyzing different values of $K$ is crucial. In our main paper, we focused on a more challenging scenario using small $K$ values ($K$=8 for bi-objective and $K$=21 for three-objective problems) because:

- Approximating the Pareto front with a small number of solutions is more difficult, while larger values of $K$ make this easier.
- For multiobjective evolutionary algorithms (MOEAs), handling smaller $K$ values is more challenging due to the complexities in the MOEA/D framework. However, we inventively do not want to delve into technical details in the main paper, as UMOD is a general framework applicable to both evolutionary and gradient-based MOO.
- Fitting a neural network with fewer solutions is challenging, but we demonstrated that even with small $K$, the Pareto front model fits well, highlighting the benefits of using neural networks in this context.
- Theoretically, as per Theorem 4 in the main paper, maximizing the minimum pairwise distances ensures that as $K$ increases, the distribution converges asymptotically to a uniform distribution over the Pareto front.

Extending UMOD to a larger number of solutions is therefore easier. We present two new results:

- The one-page PDF shows promising visual results for different values of $K$.
- In Table 6 of our response to N5xS, we demonstrate that using larger $K$ values (K=50 for bi-objective and K=91 for three-objective problems) yields even better results, outperforming classical methods in most uniformity indicators. For example, the filling distance (covering radius) is only half that of MOEA/D, indicating a more precise coverage of the Pareto front.

We plan to include these results and comparisons with advanced methods like LSSA (Wang et al. 2023) in the revised paper.


**Question: Hyperparameters (FE, Seed Number) Used in MOEA?**

In addition to the previously mentioned hyperparameters, the study explores the impact of varying FE (Function Evaluations). Detailed results can be found in Table 5 under "Results Comparison with Different FE on Three-Objective Problems." The sections "FE for Many-Solution Problems" and "Analysis of Different FEs" provide further insights, along with our response to N5xS.

------

Reference:

- Wang et al., "Enhancing Diversity by Local Subset Selection in Evolutionary Multiobjective Optimization," IEEE TEC, 2023.
Thank you for your response. With your patient explanation and additional results, I think now I have a more comprehensive understanding of your work. I increased my rating to 6, and I think this paper can be weakly accepted. I do not require further clarification.

Moreover, I strongly suggest the authors include a more comprehensive empirical study in the camera-ready version (if accepted). Currently, most of the presented results use a very small population size. I acknowledge that it is usually a difficult task to perform well with a small population size, but I think a good MOA should be compatible with a wide range of population sizes. To my knowledge, most related work may adopt a population size of around 100 for 3-obj problems. Additionally, this work tends to adopt a very large FE. UMOD performs well with a large FE because it can make use of extensive evaluations to refine the distributions, thus achieving a better benchmark performance, but MOEA/D cannot. I think a wide range of audiences may be interested in the performance of UMOD with a larger population size and a smaller FE. 

Another limitation of this work is it does not touch on the real complex PFs. The PF of the adopted problems is very simple. Although the reviewers are interested in the results of DTLZ7, the authors refused to present them. This work does not provide new insights into how to deal with complicated PFs, which I think is a very important topic with high practical value. The authors said the theories do not hold for a disconnected PF, so the proposed method is not applicable. I suspect the real limitation may lie in the PF learning. The PF learning mechanism in this paper does not seem to be able to learn a disconnected PF. The current model cannot model the boundary of the PF; thus if the weight vectors go outside the boundary, uniformity can no longer be guaranteed. I recommend an interesting test suite called IMOP, which contains many extremely complicated PFs. The authors could adopt these problems for future research.

Last, I would ask the authors to include a more comprehensive summary of the limitations of UMOD in the camera-ready version. I also suggest the authors double-check the experimental details such as the reference set for IGD. PyMOO is not a well-verified library, so it may have a lot of bugs.
Dear Reviewer N5xS,

For bi-objective problems, the FE count is 75,000 (50 * 1500 generations), similar to problems with fewer sub-problems. For three-objective problems, the FE count is 273,000 (91 * 3000 generations). As mentioned in our response under "Analysis of Different Iterations," three methods could significantly reduce FE, allowing for further reductions.

We also wish to clarify our paper's motivation:

- We aim to propose an (the first) optimizable indicator for MOO that, when optimized, ensures the algorithm returns uniformly distributed Pareto objectives. Optimizing such indicator is elegant and easy to understood (just like optimizing the HV), while methods like LSSA needs some complex mechanisms. Optimizing this indicator yields uniform solutions for both evolutionary and gradient-based problems. Our experiments confirm we achieved this goal. 
- Our goal for uniform Pareto objectives is to cover the entire Pareto front, significantly outperforming in terms of FD (covering radius). The main paper focuses on using a small population to cover the whole PF (a harder case). In the rebuttal, we demonstrate that UMOD also performs well with a larger population (K).
- While there is room for technical and engineering improvements to reduce FE, our primary focus is on the original idea, theoretical results, and designing a practical algorithm. We will actively leverage techniques to reduce FE.
- For the machine learning problems addressed in this paper, function evaluation is relatively inexpensive, making FE less critical in gradient-based methods. UMOD is the first method to find uniform Pareto objectives with low computation.

If you find our rebuttal helpful, we kindly ask you to consider raising your score for our paper. Thank you for your encouraging and constructive feedback. 

Best regards, 5680 authors
Thank you for your results. Could you report the FE budget of this experiment?
**Table 6. Results for large solution numbers (K).**

| Problem         | Method   | Spacing    | Sparsity   | HV         | Uniform    | Soft uniform | IGD | FD |
|-----------------|----------|------------|------------|------------|------------|--------------|------------|------------|
| ZDT1 (sols=50)  | UMOD     | **0.0018** | **0.0009** | **1.0973** | **0.0270** | **-0.2381**  | **0.0072** | **0.0156** |
|                 | MOEAD    | 0.0129     | 0.0010     | 1.0964     | 0.0215     | -0.2458      | 0.0078     | 0.0347     |
|                 | AWA      | 0.0130     | 0.0011     | 1.0961     | 0.0215     | -0.2431      | 0.0079     | 0.0347     |
|                 | SMS-MOEA | 0.0052     | 0.0009     | 1.0972     | 0.0195     | -0.2390      | 0.0075     | 0.0173     |
|                 | NSGA3    | 0.0168     | 0.0012     | 1.0962     | 0.0217     | -0.2456      | 0.0075     | 0.0361     |
| ZDT2 (sols=50)  | UMOD     | **0.0016** | 0.0009     | 0.7639     | **0.0239** | **-0.2374**  | **0.0074** | **0.0162** |
|                 | MOEAD    | 0.0049     | **0.0009** | 0.7632     | 0.0198     | -0.2379      | 0.0075     | 0.0167     |
|                 | AWA      | 0.0049     | 0.0009     | 0.7632     | 0.0202     | -0.2380      | 0.0075     | 0.0167     |
|                 | SMS-MOEA | 0.0098     | 0.0010     | **0.7642** | 0.0207     | -0.2405      | 0.0090     | 0.0404     |
|                 | NSGA3    | 0.0048     | 0.0009     | 0.7636     | 0.0206     | -0.2379      | 0.0075     | 0.0172     |
| DTLZ1 (sols=91) | UMOD     | 0.0009     | 0.0006     | 1.7016     | 0.0553     | -0.2839      | 0.0203     | 0.0341     |
|                 | MOEAD    | **0.0003** | **0.0007** | 1.7002     | **0.0581** | **-0.2843**  | **0.0203** | **0.0328** |
|                 | AWA      | 0.0086     | 0.0007     | 1.7002     | 0.0000     | -0.2837      | 0.0205     | 0.0531     |
|                 | SMS-MOEA | 0.0092     | 0.0044     | 1.6930     | 0.1064     | -0.0926      | 0.0491     | 0.1013     |
|                 | NSGA3    | 0.0005     | 0.0006     | **1.7017** | 0.0566     | -0.2842      | 0.0203     | 0.0333     |
| DTLZ2 (sols=91) | UMOD     | **0.0054** | **0.0013** | 1.1446     | **0.1276** | **-0.1607**  | 0.0542 | **0.1023** |
|                 | MOEAD    | 0.0277     | 0.0013     | 1.1406     | 0.0896     | -0.1688      | **0.0535**     | 0.1131     |
|                 | AWA      | 0.0255     | 0.0013     | 1.1414     | 0.0759     | -0.1678      | 0.0537     | 0.1130     |
|                 | SMS-MOEA | 0.0335     | 0.0018     | **1.1510** | 0.0443     | -0.1967      | 0.0797     | 0.1928     |
|                 | NSGA3    | 0.0273     | 0.0013     | 1.1419     | 0.0903     | -0.1683      | 0.0536     | 0.1134     |

From Table 6, we can find that, UMOD outperforms previous MOEA methods on ZDT1, ZDT2, and DTLZ2 problems, particularly in covering radius and evenly spread indicators. Since DTLZ1 is the only problem that uniform preferences induces uniform Pareto objectives, MOEA/D with uniform preferences perform the best on DTLZ1. Despite not directly optimizing IGD, UMOD significantly excels on IGD indicators, especially for ZDT1 and ZDT2. We will incorporate these results in our paper. 

If you have any further concerns, please let us know.
Dear Reviewer AxUm,

We sincerely thank you for your time and effort in reviewing our work, especially given your busy schedule. As the discussion phase wraps up, we kindly ask for your feedback on our responses. We hope to confirm that we've addressed your concerns and are open to any further questions or discussions.

If our answers meet your expectations, we would appreciate it if you could consider adjusting your score and confidence. We look forward to any further dialogue.

Thank you for your consideration.

Best regards, Paper 5680 Authors
We sincerely appreciate your positive feedback and are pleased that we were able to address your concerns.
Thank you for your detailed response. My concerns have been addressed. I have decided to maintain my original positive score.
After correcting the IGD calculation, the revised DTLZ5 and DTLZ6 results are as follows.

**Table 6. Results on DTLZ5.**
|          | Spacing    | Sparsity   | HV         | Uniform    | Soft uniform | IGD        | FD         |
|----------|------------|------------|------------|------------|--------------|------------|------------|
| UMOD     | **0.0073** | **0.0131** | 0.6995     | **0.0867** | **-0.0634**  | **0.0294** | **0.0850** |
| MOEAD    | 0.0871     | 0.0645     | 0.6352     | 0.0000     | -0.1978      | 0.1430     | 0.3002     |
| AWA      | 0.0909     | 0.0312     | 0.6697     | 0.0000     | -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     |

**Table 7. Results on DTLZ6.**

|          | Spacing    | Sparsity   | HV         | Uniform    | Soft uniform | IGD        | FD         |
|----------|------------|------------|------------|------------|--------------|------------|------------|
| 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          | -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          | -0.1357      | 0.0931     | 0.2972     |


Table 6/7 highlights that UMOD significantly outperforms other methods in ensuring evenly distributed solutions. While NSGA3, MOEA/D, and MOEA/D-AWA produce duplicate solutions, SMS-MOEA is the only method with a comparable HV to UMOD. However, UMOD surpasses SMS-MOEA considerably in IGD and FD.
When calculating the Pareto front (PF) for DTLZ5 and DTLZ6, we encountered an intriguing bug. After thorough investigation, we discovered that results may vary on different PCs due to a minor issue in pymoo (version 0.6.1.1). When using `Remote.get_instance().load("pymoo", "pf", "dtlz5-3d.pf")` to calculate the PF, the file is loaded from `A\\envs\\B\\Lib\\site-packages\\pymoo\\data\\pymoo\\pf\\dtlz5-3d.pf`, where 'A' is the anaconda installation directory and 'B' is the environment name.

Interestingly, the `dtlz5-3d.pf` file is 387KB, while `dtlz6-3d.pf` is only 32KB, indicating that the PF for DTLZ6 is incomplete. To resolve this issue, we used the `dtlz5-3d.pf` file for both DTLZ5 and DTLZ6 calculations.

The minor bug in pymoo has been fixed. We will report the DTLZ5/6 results and the performance of UMOD with large populations soon.
When we submitted the code, there were still some privacy issues in the project that we hadn't had time to address. Therefore, we mainly provided the main file to explain the algorithm's running process.

For UMOD solving multitask learning problems, recently, there is a open-sourcing implement of UMOD in the libmoon framework. Code is available at [here](https://github.com/xzhang2523/libmoon/blob/main/libmoon/tester/run_mtl_clean_pref.py) by setting `solver` as `uniform`.
I was trying to run the code uploaded by the authors, but I noticed that they only uploaded the `umod.py` file without other critical libraries and files like `xy_util` and `mo_util`, so the code cannot run actually. Could you please provide a full program?
Thank you for your response. I have checked the code of PyMOO. I carefully read `dtlz.py` at https://github.com/anyoptimization/pymoo/blob/main/pymoo/problems/many/dtlz.py. I think the setting of `n_partitions` does not affect the results on DTLZ5 and DTLZ6 because `ref_dirs = UniformReferenceDirectionFactory` is only used in DTLZ1-4. Moreover, I am also very confused as to why running on a new computer could result in the wrong result.
You can view our visual results on the one-page summary, where results on DTLZ5 and 6 are accurate. DTLZ5 and DTLZ6 were run on a new computer. However, there's a minor bug in the IGD function of the pymoo library (line 256 in dtlz.py), where the default partition is set to 15. This causes numerical instability in approximating the PF and IGD. We've adjusted it to `n_partitions=50`. This line of code has been changed to `ref_dirs = UniformReferenceDirectionFactory(3, n_partitions=50).do()    # Original is 15`.

We are aware of this issue, and all other results are correct (they are by our main PC). We will re-run the IGD values for DTLZ5 and DTLZ6.

New results will be updated soon.
We chose a small population to test whether a few solutions can approximate the entire Pareto front (PF). With a larger population, fitting the network is easier. **Our key point is that even with fewer solutions, UMOD with a neural model produces uniform Pareto objectives**.

We use mean in Table 2.
**Table 5 Results comparison with different FE on three objective problems.**
| Problem | Iterations | Method | Spacing   | Sparsity  | HV        | Uniform   | Soft Uniform | IGD       | MaxGD     |
|---------|------------|--------|-----------|-----------|-----------|-----------|--------------|-----------|-----------|
| DTLZ1   | 63,000     | UMOD   | 0.0079    | **0.0067**| **1.6926**| 0.1181    | -0.0903  | 0.0491| 0.0861    |
|         | 63,000     | MOEAD  | **0.0002**| 0.0075    | 1.6917    | **0.1408**| **-0.0899**      | **0.0488**    | **0.0750**|
|         | 84,000     | UMOD   | 0.0044    | **0.0070**| **1.6929**| 0.1274    | -0.0902  | 0.0489    | 0.0857    |
|         | 84,000     | MOEAD  | **0.0002**| 0.0075    | 1.6917    | **0.1408**| **-0.0899**      | **0.0488**| **0.0750**|
|         | 105,000    | UMOD   | 0.0031    | **0.0071**| **1.6929**| 0.1315    | **-0.0900**  | 0.0488    | 0.0843    |
|         | 105,000    | MOEAD  | **0.0002**| 0.0075    | 1.6914    | **0.1408**| -0.0900      | **0.0488**| **0.0750**|
|         | 126,000    | UMOD   | 0.0025    | **0.0072**| **1.6929**| 0.1331    | **-0.0900**  | 0.0488    | 0.0820    |
|         | 126,000    | MOEAD  | **0.0002**| 0.0074    | 1.6913    | **0.1404**| -0.0901      | **0.0487**| **0.0750**|
| DTLZ2   | 63,000     | UMOD   | **0.0151**| **0.0174**| **1.0637**| **0.2783**| **0.1155**   | **0.1223**| **0.2387**|
|         | 63,000     | MOEAD  | 0.0545    | 0.0243    | 1.0624    | 0.2433    | 0.0947       | 0.1263    | 0.2514    |
|         | 84,000     | UMOD   | **0.0044**| **0.0177**| **1.0625**| **0.3180**| **0.1178**   | **0.1218**| **0.2370**|
|         | 84,000     | MOEAD  | 0.0546    | 0.0243    | 1.0626    | 0.2433    | 0.0947       | 0.1263    | 0.2515    |
|         | 105,000    | UMOD   | **0.0070**| **0.0173**| **1.0661**| **0.3100**| **0.1178**   | **0.1222**| **0.2433**|
|         | 105,000    | MOEAD  | 0.0546    | 0.0243    | 1.0625    | 0.2433    | 0.0947       | 0.1263    | 0.2515    |
|         | 126,000    | UMOD   | **0.0042**| **0.0169**| **1.0730**| **0.3237**| **0.1204**   | **0.1250**| **0.2477**|
|         | 126,000    | MOEAD  | 0.0546    | 0.0243    | 1.0626    | 0.2433    | 0.0947       | 0.1263    | 0.2515    |
| DTLZ3   | 63,000     | UMOD   | **0.0150**| **0.0184**| 1.0531    | **0.2937**| **0.1177**   | **0.1237**| 0.2576|
|         | 63,000     | MOEAD  | 0.0545    | 0.0243    | **1.0564**| 0.2439    | 0.0958       | 0.1267    | **0.2524**    |
|         | 94,500     | UMOD   | **0.0064**| **0.0183**| 1.0586    | **0.3134**| **0.1208**   | **0.1236**| **0.2266**|
|         | 94,500     | MOEAD  | 0.0546    | 0.0244    | **1.0601**| 0.2436    | 0.0950       | 0.1264    | 0.2518    |
|         | 126,000    | UMOD   | **0.0025**| **0.0176**| **1.0610**| **0.3213**| **0.1177**   | 0.1219| **0.2351**|
|         | 126,000    | MOEAD  | 0.0546    | 0.0244    | 1.0602    | 0.2436    | 0.0950       | **0.1157**    | 0.2749    |
| DTLZ4   | 63,000     | UMOD   | **0.0142**| **0.0231**| **1.0711**| **0.3057**| **0.1194**   | 0.1252| **0.2672**|
|         | 63,000     | MOEAD  | 0.0546    | 0.0243    | 1.0626    | 0.2434    | 0.0947       | **0.1155**    | 0.2748    |
|         | 94,500     | UMOD   | **0.0045**| **0.0178**| **1.0645**| **0.3181**| **0.1195**   | 0.1233 | **0.2326**|
|         | 94,500     | MOEAD  | 0.0547    | 0.0243    | 1.0627    | 0.2433    | 0.0947       | **0.1155**    | 0.2748    |
|         | 126,000    | UMOD   | **0.0024**| **0.0170**| **1.0631**| **0.3226**| **0.1178**   | 0.1223| **0.2544**|
|         | 126,000    | MOEAD  | 0.0546    | 0.0243    | 1.0626    | 0.2433    | 0.0947       | **0.1155**    | 0.2748    |

In previous discussion, we have identified additional methods to further reduce FE and will incorporate them in the revised paper. Even with the original strategy, FE can be reduced to approximately 63,000 from 126,000 for 21 solutions. Table 5 highlights key insights:

- For the DTLZ1 problem, as shown in the main paper, uniform preferences yield uniform Pareto objectives. UMOD and MOEAD perform similarly, indicating that UMOD generates nearly uniform Pareto objectives.
- While further improvement beyond 63,000 FE is minimal, UMOD significantly outperforms MOEAD on DTLZ2 and DTLZ4.
Thank you for your response, and thank you for your effort in improving the empirical study. However, I am confused about some of the numbers you reported. The PF of DTLZ5 and DLTZ6 is the same. However, the IGD of UMOD on DTLZ5 is 0.63, and the IGD of UMOD on DTLZ6 is 3.23. Why do they differ so much on the same PF? Moreover, is the number in the table an average or median?

In addition, the population size used in your experiment is significantly smaller than that employed in other studies. In practical applications, a larger population is often necessary to adequately approximate an irregular or high-dimensional PF. I am curious about whether UMOD encountered difficulties when using a larger population size, such as 100 solutions on 3-objective problems.
Thank you for your consistent support and interest in our work.

The settings compared with LSSA are the same as in our main paper (Table 5, Appendix). We will include all additional settings discussed during the rebuttal.

**Table 3. Settings of the comparison between LSSA and UMOD.**
|                           | Bi-objective | Three-objective (DTLZ 1-4) | Degenerated three-objective (DTLZ 5-6) |
|---------------------------|--------------|-----------------------------|----------------------------------------|
| Number of solutions        | 8            | 21                          | 16                                     |
| Number of seeds            | 31           | 31                          | 31                                     |
| Func. Eval. (FE)           | 32,000       | 126,000                     | 96,000                                 |
| Number of decision variables | 30        | 30                          | 30                                     |

**Table 4. Software for LSSA and UMOD.**
|          | LSSA        | UMOD (for MOEA) |
|----------|-------------|-----------------|
| Software | PlatEMO 3.4 | Pymoo 0.6.1.1   |

The FE numbers and UMOD results follow those in our main paper. As discussed, the **FE budget can be significantly reduced**. For example, we reduced the FE for the bi-objective case from 32,000 to 12,000 **without affecting UMOD's performance**. Improved FE results for three-objective problems will be released soon.
Thank you for your response. The results are very impressive. Can you provide some details about the settings? For example, the number of solutions, the number of decision variables, the number of objective functions, and the FE budget.
**Table 2. Comparison with LSSA (Wang et al. 2022). IGD, spacing, and sparsity are scaled by 100, while uniformity, soft uniformity, and FD are scaled by 10 for a better illustration.**

|       |      | HV            | IGD            | Spacing       | Sparsity      | Uniform       | Soft uniform   |FD|
|-------|------|---------------|----------------|---------------|---------------|---------------|----------------|---------------|
| ZDT1  | UMOD | **1.04**      | **5.19**       | 0.12          | **4.39**      | 2.07          | 0.77           | **1.04**      |
|       | LSSA | 1.04          | 5.22           | **0.1**       | 4.45          | **2.09**      | **0.78**       | 1.04          |
| ZDT2  | UMOD | **0.71**      | 5.23           | **0.25**      | **4.45**      | **2.05**      | **0.78**       | 1.07          |
|       | LSSA | 0.71          | **5.19**       | 0.54          | 4.45          | 2.03          | 0.78           | **1.05**      |
| ZDT3  | UMOD | 0.89          | 39.8           | 3.57          | **3.45**      | **1.2**       | **0.3**        | 9.22          |
|       | LSSA | **0.91**      | **38.45**      | **3.27**      | 3.88          | 1.53          | 0.42           | **8.87**      |
| ZDT4  | UMOD | **1.04**      | **5.21**       | **0.22**      | **4.39**      | 2.05          | 0.77           | 1.06          |
|       | LSSA | 1.02          | 5.41           | 0.34          | 4.49          | **2.06**      | **0.79**       | **1.05**      |
| ZDT6  | UMOD | **0.66**          | **4.18**       | **0.12**      | **2.86**      | **1.67**      | **0.35**       | **0.81**      |
|       | LSSA | 0.66          | 4.54           | 3.04          | 3.07          | 1.01          | 0.17           | 1.2           |
| DTLZ1 | UMOD | **1.69**      | **4.87**       | **0.09**      | **0.73**      | 1.39          | -0.9           | **0.78**      |
|       | LSSA | 1.42          | 32.77          | 3.06          | 1.93          | **1.91**      | **0.47**       | 3.67          |
| DTLZ2 | UMOD | **1.07**      | **12.1**       | **0.52**      | **1.61**      | **3.17**      | 1.18           | 2.37          |
|       | LSSA | 1.07          | 12.23          | 2.2           | 2.17          | 2.96          | **1.21**       | **2.22**      |
| DTLZ3 | UMOD | **1.06**      | **12.3**       | **1.93**      | **1.7**       | 2.72          | 1.13           | **2.39**      |
|       | LSSA | 0             | 1345.87        | 281.52        | 923.37        | **6.72**      | **3.45**       | 136.54        |
| DTLZ4 | UMOD | **1.07**      | **12.1**       | **0.83**      | **1.77**      | **3.07**      | 1.18           | 2.41          |
|       | LSSA | 1.06          | 12.39          | 2.29          | 2.34          | 2.99          | **1.23**       | **2.32**      |
| DTLZ5 | UMOD | **0.72**      | **0.63**       | **0.7**       | 0.03          | 0             | -3.62          | **0.24**      |
|       | LSSA | 0.7           | 2.67           | 0.51          | **1.11**      | **0.96**      | **-0.72**      | 0.54          |
| DTLZ6 | UMOD | 0.69          | 3.23           | **1.06**      | 0.06          | 0             | -3.65          | **0.46**      |
|       | LSSA | **0.7**       | **2.68**       | 1.16          | **1.11**      | **0.87**      | **-0.73**      | 0.57          |

Thank you for mentioning LSSA as a strong baseline. The code is available at [GitHub](https://github.com/ilog-ecnu/LSSA) and uses PlatEMO version 3.4. Key observations from Table 1 are:

- UMOD outperforms LSSA on most problems, particularly for the IGD indicator, where UMOD excels in 8 out of 11 cases.
- Our reimplementation shows that LSSA struggles with a small population size, notably on ZDT5 and DTLZ3, which are one of the focus of this paper.
- LSSA does outperform UMOD on ZDT3 (disconnected case), and we plan to adapt UMOD for such scenarios.

Additionally, UMOD offers novel theoretical insights, including the relationship between IGD, fill distance, and max-min distance, along with error analysis. Beyond evolutionary problems, which are a part of this paper, UMOD also effectively addresses large-scale machine learning challenges using pure gradient-based methods.

------
Reference

[1] Enhancing Diversity by Local Subset Selection in Evolutionary Multiobjective Optimization. Wang et al. IEEE TEC. 2022.
Thank you for your effort in providing new results. However, the new results are not very convincing to me because the selected problems are too simple. For example, the PS is ZDT1 lies in $x_i=0$ for $i\ge 2$. All the decision variables are clipped to [0,1], so there is actually only one effective decision variable. Therefore, ZDT1 does not need much budget for searching and converging. I suggest the author use DTLZ1 instead of ZDT1 to provide a more convincing experiment setting. I am not asking for new results, and this is only my advice for improving this new experiment if the authors are considering including it in the paper.
**Table 1. Results under different function evaluations. (K=10).**
| Problem | Func. Eval. (FE) | Method | Spacing    | Sparsity   | HV         | Uniform    | Soft Uniform | IGD        | FD   |
|---------|------------|--------|------------|------------|------------|------------|--------------|------------|------------|
| ZDT1    | 15,000     | UMOD   | **0.0007** | **0.0266** | **1.0536** | **0.1616** | **0.0167**   | **0.0403** | **0.0809** |
|         | 15,000     | MOEAD  | 0.0506     | 0.0284     | 1.0503     | 0.1175     | -0.0047      | 0.0414     | 0.1353     |
|         | 20,000     | UMOD   | **0.0009** | **0.0266** | **1.0536** | **0.1613** | **0.0167**   | **0.0403** | **0.0810** |
|         | 20,000     | MOEAD  | 0.0503     | 0.0283     | 1.0505     | 0.1175     | -0.0047      | 0.0414     | 0.1353     |
|         | 25,000     | UMOD   | **0.0007** | **0.0266** | **1.0537** | **0.1620** | **0.0168**   | **0.0403** | **0.0813** |
|         | 25,000     | MOEAD  | 0.0509     | 0.0285     | 1.0503     | 0.1175     | -0.0047      | 0.0414     | 0.1353     |
|         | 40,000     | UMOD   | **0.0003** | **0.0266** | **1.0537** | **0.1624** | **0.0167**   | **0.0403** | **0.0816** |
|         | 40,000     | MOEAD  | 0.0504     | 0.0283     | 1.0504     | 0.1175     | -0.0047      | 0.0414     | 0.1353     |
| RE21    | 15,000     | UMOD   | **0.0015** | **0.0164** | **1.2513** | **0.1241** | **-0.0202**  | **0.0322** | **0.0661** |
|         | 15,000     | MOEAD  | 0.0845     | 0.0237     | 1.2447     | 0.0721     | -0.0580      | 0.0423     | 0.1792     |
|         | 20,000     | UMOD   | **0.0013** | **0.0164** | **1.2512** | **0.1250** | **-0.0202**  | **0.0322** | **0.0648** |
|         | 20,000     | MOEAD  | 0.0853     | 0.0239     | 1.2452     | 0.0721     | -0.0581      | 0.0425     | 0.1812     |
|         | 25,000     | UMOD   | **0.0007** | **0.0164** | **1.2513** | **0.1266** | **-0.0203**  | **0.0322** | **0.0643** |
|         | 25,000     | MOEAD  | 0.0853     | 0.0239     | 1.2452     | 0.0721     | -0.0581      | 0.0425     | 0.1812     |
|         | 40,000     | UMOD   | **0.0011** | **0.0164** | **1.2520** | **0.1256** | **-0.0202**  | **0.0321** | **0.0645** |
|         | 40,000     | MOEAD  | 0.0853     | 0.0239     | 1.2453     | 0.0721     | -0.0582      | 0.0425     | 0.1812     |
| RE22    | 15,000     | UMOD   | **0.0675** | **0.0200** | 1.1802     | **0.0050** | **-0.0800**  | **0.0406** | **0.0854** |
|         | 15,000     | MOEAD  | 0.0460     | 0.0194     | **1.1867** | 0.0601     | -0.0489      | 0.0385     | 0.1197     |
|         | 20,000     | UMOD   | **0.0676** | **0.0198** | 1.1809     | **0.0036** | **-0.0786**  | **0.0402** | **0.0866** |
|         | 20,000     | MOEAD  | 0.0460     | 0.0194     | **1.1867** | 0.0601     | -0.0489      | 0.0385     | 0.1197     |
|         | 25,000     | UMOD   | **0.0676** | **0.0198** | 1.1809     | **0.0036** | **-0.0786**  | **0.0402** | **0.0866** |
|         | 25,000     | MOEAD  | 0.0460     | 0.0194     | **1.1867** | 0.0602     | -0.0489      | 0.0385     | 0.1197     |
|         | 40,000     | UMOD   | **0.0677** | **0.0199** | 1.1822     | **0.0033** | **-0.0792**  | **0.0403** | **0.0875** |
|         | 40,000     | MOEAD  | 0.0460     | 0.0194     | **1.1868** | 0.0601     | -0.0489      | 0.0385     | 0.1197     |



UMOD is integrated within the Pymoo environment, which supports both evolutionary algorithms (popular in multi-objective optimization) and gradient-based ML problems. LSSA is found in the PlatEMO environment, with some differences in function evaluations (FE) between Pymoo and PlatEMO. For bi-objective problems, LSSA requires around 50,000 iterations (100 ( subproblem) x 500 (iterations) ).

Table 1 highlights two key points:
- Without subset selection, MOEA/D fails to produce uniform Pareto solutions even with large iterations, suggesting that **it is worthy to use extra FEs to find the topK uniform solutions**, which is the motivation of this paper. 
- **Reducing function evaluations** from 40,000 (our original setting) to 15,000 (K=10) is possible **without significantly impacting performance**. Results remain **robust when FE > 15,000**. 
(40,000 is for K=10, while 32,000 (previously mentioned) is for K=8.)

Further improvements  the number of FE could be made in three ways:
- 1. Using the last iteration as a warm start for generating uniform solutions, as MOEA/D provides good approximations.
- 2. Making the epoch for calculating the optimal preference configuration adaptive to save computation.
- 3. Using all populations to train the Pareto front rather than the main population (the method currently used).
Dear reviewer, 

We are more than willing to engage with you further to provide additional results to address any of your remaining concerns. Thanks for your effort on our work!

Best regards,  Paper 5680 Authors
Thanks for your score increasing. We are conducting more and more experiments to address your concerns and we will report them to you as soon as possible.
Thank you for your detailed response. I have increased my rating.
We deeply appreciate your prompt response. We hope that our following answers can address your concerns more or less. 

> **Q1.** Why the PS dimension is $m-1$? 

This means that the effective dimension of the Pareto set is $m-1$. This conclusion can be found in Theorem 2.2, Equation (6), on page 5, line 18 of Hillermeier et al. (2001). For completeness, we provide a rough analysis here. For a detailed proof, please refer to Hillermeier et al. (2001), page 5. 

Under a mild KKT condition (Equation (2), Hillermeier et al., 2001), the gradients of all solutions that belong to the Pareto set satisfy $n-m+1$ gradient constraints. Therefore, the dimension of the tangent space of the Pareto set is $n - (n-m+1) = m-1$. This indicates that the dimension of the Pareto set manifold is $m-1$.  

For example, the PS for a typical 2-objective optimization problem is a 1-D curve, and the PS for a typical 3-objective optimization problem is a 2-D surface. For the classic ZDT1, no matter how large the decision number $n$ is, its PS is always the 1-D line from [0,0,..,0] to [1,0,..,0].   

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Reference

Generalized Homotopy Approach to Multiobjective Optimization, Journal of Optimization Theory and Applications. 2001. C. Hillermeier. 


> **Q2.** About UMOD "require fewer main populations". I think a good algorithm should work well for adjustable population size, so I am confused about what "require fewer main populations" means.

The claim of "fewer populations" compares UMOD with subset selection in finding $K$ representative solutions. UMOD maintains a population size of $K$, whereas subset selection requires a larger initial pool of solutions to choose the final $K$. Specifically, we choose to use $4K$ solutions for subset selection. Therefore, we say UMOD uses fewer populations than subset selection.

UMOD does work well for a wide range of population sizes. Due to the space limit, we have reported that UMOD works for a large range of different populations size $K$, which is provided in the one-page PDF. If needed, we can report more experimental results. 


> **Q3.** Comparison between UMOD and subset-selection. 

UMOD offers two key advantages over energy-based subset selection:

 - **Single-Phase Algorithm**: Energy-based subset selection involves a "two-phase" process: (1) generating a large number of candidate solutions and (2) performing subset selection. This two-phase approach can be computationally expensive, especially when the candidate pool is large. Additionally, determining the appropriate size of the candidate pool to ensure representative solutions is uncertain. In contrast, UMOD operates as a single-phase algorithm, eliminating the need for an extensive candidate generation phase.

- **Efficient Optimization**: Energy-based subset selection is a discrete optimization problem, which can be computationally intensive. UMOD, on the other hand, employs a gradient-based method for selecting representative solutions, leveraging continuous optimization. Gradient-based methods are generally more efficient and faster compared to discrete optimization problems, leading to more timely and resource-effective solutions.

We argue that training a neural network is not a big trouble and should be widely adopted in multi-objective optimization due to its robustness (see response to Q1, JBGb) and quick training time (75ms).

We will soon compare it with LLSA (Wang et al., 2023). While we currently lack numerical results for LSSA, **Table 3 in the LSSA paper indicates that LSSA underperforms compared to SMS-MOEA on the ZDT4, DTLZ1, DTLZ5, and DTLZ6 problems. In contrast, UMOD consistently outperforms SMS-MOEA on those tasks. The results compared with SMS-MOEA serve as evidence that UMOD performs better than LSSA**.  

---
Reference

Enhancing Diversity by Local Subset Selection in Evolutionary Multiobjective Optimization. Wang et al. 2023. IEEE TEC. 

> **Q4** When the solutions are not well converged. In such situations, is the PF model meaningless?

**Firstly**, our results show that solutions easily converge to the PF, but uniformity of those solutions is a big issue. Therefore, we use converged Pareto solutions to train PF models in our experiment.
**Secondly**, according to our results, even if solutions are not fully converged, the fitted model remains meaningful. A rough Pareto front model still aids in finding more uniform solutions through solving the optimal preference configuration problem.

> **Q5.** The FE budget seems very high.

This question can be answered from two aspects. 
- UMOD is the first method, to our knowledge, to reliably find uniform Pareto objectives, as shown in Table 1 of our response to AxUm, where the spacing indicator is nearly zero. Increasing the FE budget is worthwhile for achieving uniform Pareto objectives.
- The FE budget can be largely improved. Previously, we do not conduct much effort to reduce it. We will provide new results for you soon.
Thank you for your response. Some of my concerns have been adequately addressed. However, I still have some concerns about the proposed UMOD algorithm.

1. This is a minor comment, but I have some reservations about your response. You provided a paper (Ye et al., 2022, p.1, line 12) to support your claim that "the dimension of PS is $m-1$". Actually, this paper did not explicitly prove this claim, so I think the statement in that paper may also be incorrect. The vector function may be a projection from $\mathbb R^n\to \mathbb R^m, (n>m)$. In this regard, I can not find any evidence of why the dimension of PS is $m-1$ without further assumptions, because there may be multiple $x$ resulting in the same $f(x)$.

2. About UMOD "require fewer main populations". I think a good algorithm should work well for adjustable population size, so I am confused about what "require fewer main populations" means.

3. About the limits of subset selection. I think "subset selection" is a problem, not an algorithm. There are many algorithms for solving subset selection. The $1-1/e$ bound is the guarantee of naive greedy algorithms. Actually, there are many subset selection methods that could achieve better results. Therefore, I do not agree with the authors claiming that the $1-1/e$ bound is a shortcoming of subset selection. There is a recent work [1] that also focuses on uniformity design but is based on subset selection. The results presented in [1] seem comparable with, or even better than, UMOD, regarding uniformity. Moreover, [1] seems more efficient because it does not require training a network. [1] can effectively deal with disconnected PFs like DTLZ7, whereas UMOD cannot. [1] optimizes for uniformity by minimizing an energy function, which is similar to UMOD which optimizes Eq. (3). Another representative work [2] focusing on weight adaptation also uses subset selection and archiving strategies to optimize for uniformity, and achieves very promising results in various PFs. I am not asking for new results, but to my knowledge and based on the results reported in this paper, UMOD does not seem to perform better than the real SOTA baselines regarding uniformity.

4. The assumptions $y\in T$ in the theory part is very ideal. Moreover, the PF model can not learn the PF when the solutions are not well converged. In such situations, is the PF model meaningless?

5. The FE budget seems very high. The maximum fitness evaluations are 32,000 for bi-objective, 126,000 for three-objective, and 350,000 for four-objective optimization. It makes me believe that UMOD needs much more budget compared with the baselines.

References

[1] Enhancing Diversity by Local Subset Selection in Evolutionary Multiobjective Optimization. IEEE TEC.

[2] What Weights Work for You? Adapting Weights for Any Pareto Front Shape in Decomposition-based Evolutionary Multi-Objective Optimisation. Evolutionary Computation.
We thank the reviewer for the encouraging response as well as the additional suggestion. 

We agree that examining the performance on multi-objective combinatorial optimization problems would be interesting. We note, however, considering the disconnected Pareto front thereof, the proposed method theoretically may be inappropriate. Nevertheless, we think that our method would be an interesting baseline; as we mentioned in the general response, we are more than willing to explore the potential recipe "run UMOD on each segment separately" in the future work.
The authors have provided detailed responses to my comments, including theoretical rigorous proof, and detailed textual supplements.

I think the revised paper is acceptable.

Additionally, I strongly encourage the authors to conduct similar research on multi-objective combinatorial optimization problems. In such problems, PF is discrete and discontinuous, and current practices generally use a set of non-dominated solutions from different algorithms to replace PF. Your research results may lead to many interesting discoveries in this field.