ParetoFlow: Guided Flows in Multi-Objective Optimization
Authors: Ye Yuan
Can Chen
Christopher Pal
Xue Liu
Venue: ICLR 2025 Poster (2025)
Keywords: Multi-objective optimization; flow matching; classifier guidance.
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: In offline multi-objective optimization, we introduce a ParetoFlow method, specifically designed to guide flow sampling to approximate the Pareto front.

Abstract: In offline multi-objective optimization (MOO), we leverage an offline dataset of designs and their associated labels to simultaneously minimize multiple objectives. This setting more closely mirrors complex real-world problems compared to single-objective optimization. Recent works mainly employ evolutionary algorithms and Bayesian optimization, with limited attention given to the generative modeling capabilities inherent in such data. In this study, we explore generative modeling in offline MOO through flow matching, noted for its effectiveness and efficiency. We introduce \textit{ParetoFlow}, specifically designed to guide flow sampling to approximate the Pareto front. Traditional predictor~(classifier) guidance is inadequate for this purpose because it models only a single objective. In response, we propose a \textit{multi-objective predictor guidance} module that 
assigns each sample a weight vector, representing a weighted distribution across multiple objective predictions. A local filtering scheme is introduced to address non-convex Pareto fronts. These weights uniformly cover the entire objective space, effectively directing sample generation towards the Pareto front. Since distributions with similar weights tend to generate similar samples, we introduce a \textit{neighboring evolution} module to foster knowledge sharing among neighboring distributions. This module generates offspring from these distributions, and selects the most promising one for the next iteration. Our method achieves state-of-the-art performance across various tasks. Our code is available.

Published as a conference paper at ICLR 2025
PARETOFLOW: GUIDED FLOWS IN MULTI-OBIECTIVE OPTIMIZATION
Ye Yuan' Can (Sam) Chen!.2ei Christopher Pal Xuue 2
McGill, MMILA Quebec Al Institute, "Polytechnique Montreal, *Canada CIFAR Al Chair
ABSTRACT
In offline multi-objective optimization (MOO), we leverage offline dataset of designs and their associated labels to simultaneously minimize multiple objec- tives. This setting more closely mirrors complex real-world problems compared to single-objective optimization. Recent works mainly employ evolutionary algo- rithms and Bayesian optimization, with limited attention given the generative modeling capabilitiess inherent in such data. In this study, we explore generative modeling in offline MOO through flow matching, noted for its effectiveness and efficiency. We introduce ParetoFlow, specifically designed to guide flow sam- pling to approximate the Pareto front. Traditional predictor (classifier) guidance is inadequate for this purpose because it models only single objective. In re- sponse, we propose multi-objeccive predictor guidance module that assigns each sample a weight vector, representing weighted distribution across multiple ob- jective predictions. A local filtering scheme is introduced to address non-convex Pareto fronts. These weights uniformly cover the entire objective space, effec- tively directing sample generation towards the Pareto front. Since distributions with similar weights tend to generate similar samples, introduce neighboring evolution module to foster knowledge sharing among neighboring distributions. This module generates offspring from these distributions, and selects the most promising one for the next iteration. Our method achieves state-of-the-ar perfor- mance across various tasks. Our code will be available here)
INTRODUCTION
Offline optimization Kim et al. 20225), a fundamental challenge science and engineering, involves minimizing a black-box function using solely offline dataset, with diverse applications ranging from molecule design Sarkisyan et al. (2016); Angermüller et al. (2020); Hu et al. (2025a) to neural architecture design Lu et al. (2023). Previous research primarily focuses single-objective opti- mization, aiming to optimize a single desired property Trabucco et al. (2022); however, this fails to capture the complexities of real-world challenges that often require balancing multiple conflicting objectives, such as designing a neural architecture that demands both high accuracy and minimal pa- rameter count Lu et al. (2023). In this study, we explore offline multi-objective optimization (MOO), leveraging an offline dataset of designs and their associated labels to simultaneously minimize mul- tiple objectives.
The pioneering work Xue et al. (2024) adapts evolutionary algorithms Deb et al. (002): Zhang & Li (2007) and Bayesian optimization Daulton et al., (2023); /Zhag Golovin 2200)) Qing et al. (2023) to handle the offline MOO setting. Besides, some studies design controllable generative models that manage multiple properties Wang et al.) (2024). However, these studies generally either focus on different settings, such as online optimization Gruver et al. (2023) and white-box optimization Yao ct al. 2024), or utilize less advanced generative models, such as VAEs Wang et al. (2022). None of these studies fully exploits the potential of advanced generative modeling in offline MOO.
"Equal tech contribution with random order: Can designs algorithm/drafis paper: Ye conducts experiments. Tech lead: chencan 421 8 gmail com or can, chen@rile 1a. quebec. I Equal senior contribution with random order.

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To bridge this gap, we employ a flow matching framework, renowned for its effectiveness and ef- ficiency over diffusion models Lipman et al. (2023); Le et al. (2023); Polyak et al. (2024) to in- vestigate generative modeling in offline MOO. We introduce the ParetoFlow method, specifically designed to guide flow sampling to approximate the Pareto front. The Pareto front is defined as the set of optimal objective values that are not dominated by any other points. As illustrated in Fig- ure Ja), the solid blue curve represents the Pareto front in a two-dimensional objective space for the conflicting objectives fi and f2.
Traditional predictor (classifier) guidance Dhariwal & Nichol (202)), focusing solely on a single objective, fails to adequately explore the entire Pareto front. As demonstrated in Figure [a), direct- ing sample generation from pure noise (circles) towards single objective, such as fi or f2. yields isolated Pareto samples (pentagrams) without capturing the full spectrum of optimal samples. To address this, we propose Module 1, termed multi-objective predictor guidance, which assigns each sample a weighted distribution. This distribution, characterized by weight vector across multiple objective predictions, guides sample generation towards its corresponding Pareto-optimal point. To navigate non-convex Pareto fronts, this module adopts a local filtering scheme, to filter out samples whose objective prediction vector deviates from the weight vector. These weight vectors uniformly cover the entire objective space, thereby effectively guiding sample generation towards the Pareto front. As shown in Figure [b), uniform weight vectors w represent five weighted distributions over Si and f2, ensuring that the generated samples (pentagrams) approximate the Pareto front.
(a)
(b)
(c)
Traditional Predictor Guidance Multi- Oittttvbeeevvee Predictor Guidance
Neighboring Evolution
1.0
100,000
1.0
11,0,000 0.01
1.0
Objective Function fz
0.8
0.8
S 34, 0.39)
0.8
x 10.7%, 0.29)
0.6
[1000 120
0.6
(000, 2.15
0.6
Knowledge
10.50 0:30 *
Sharing.
0.4
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0.6
0.8 1.0
Objective Function f
Objective Function f
Objective Function f.
Pareto Front
Pure Noise
* Generated Samples
Sampling Trajectory
Figure 1: Motivation of Module 1 in (b) and Module 2 in (c).
Distributions with similar weight vectors tend to generate similar samples. As shown in Figure Ic), the distributions with weights w² and 3h are neighboring and they generate similar sample along to introduce Module termed neighboring evolution,
the sampling trajectory. This motivates to foster knowledge sharing among these neighboring distributions. We propose to generate diverse offspring samples from neighboring distributions, and select the most promising one for the next it- eration. For instance, consider w² with w as its sole neighbor in Figure We generate offspring samples from both 22 and w3 and then select the most promising one ine-iieet by dashed circle-at as the next iteration state for the 2 distribution. A similar scheme applies to wa assum- ing w² as its neighbor. This module facilitates valuable knowledge sharing between neighboring distributions (w² and wi), enhancing the effectiveness of the overall sampling process.
To summarize, our contributions are three-fold:
We explore the use of generative modeling in offline MOO by introducing ParetoFlow, Curetfollo,specfffcally designed to effectively steer flow sampling to approximate the Pareto front.
We propose multi-objective predictor guidance module that assigns a uniform weighted distri- bution to each sample, ensuring comprehensive coverage of the objective space.
We establish neighboring evolution module to enhance knowledge sharing among distributions with close weight vectors, which improves the sampling effectiveness.
Classifier guidance, initially for classification, adapted to predictor guidance generalize to regression.
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2 PRELIMINARIES
2.1 OFFLINE MULTI-ORJECTIVE OPTIMIZATION
Offline multi-objective optimization (MOO) seeks to simultaneously minimize multiple objectives using offline dataset D of designs and their corresponding labels. Consider a design space denoted as X' C I where d represents the dimension of the design. In MOO, we aim to find solutions that achieve the best trade-offs among conflicting objectives. Formally, the multi-objective optimization problem is defined as:
Find x E X' such that there is no æ E X' with fx) < f(x)),
(1)
where f X' + R is a vector of m objective functions, and x denotes Pareto dominance. A solution x: said to Pareto dominate another solution x" (denoted as f(x) f(x")) if:
Vi E (I, m........ [x)) S f1(x") and 3j E (I,...,m} such that fạ < { (x").
(2)
In other words, no worse than x" in all objectives and strictly better in least one objective. A solution x" is Pareto optimal if there is other solution E X that Pareto dominates x", The set of Pareto optimal solutions constitutes the Pareto set (PS). The corresponding set of objective vectors, defined as {f(x))
PS}. is known as the Pareto front (PF).
The goal of MOO is to identify a set solutions that effectively approximates the PF, providing a comprchensive representation of the best possible trade-off's among the objectives.
2.2 FLOW MATCHING
Flow matching, an advanced generative modeling framework, excels in effectiveness and efficiency over diffusion models Lipman et al. (003)); Le et al. (2023); Polyak et al. (2024). At the core of this framework lies conditional probability path pr 21), [0, 1]. evolving from an initial distribution Po( σ1) = g(x)) to an approximate Dirac delta function P1 (x: x1) z 8(x x1). specific point æ from the distribution Pdata and is driven by the conditional vector field u, (x 21). A neural network, parameterized by 8, learns the marginal
This evolution is conditioned
vector field uax, I):
0(x t; 0) v(x,t) Emiopi las) [ue [x)]]
This modeled vector field, ((x, t: 6), functions as a neural Ordinary Differential Equation (ODE), guiding the transition from (xx) to (2))
Following (Pooladian et al. 2023), the process begins by drawing initial noise zo from g(vo). This noise is then linearly interpolated with the data point #11
æ / (1 æo + L. æ1, Tp ((30)
The derivation of the conditional vector field is straightforward: ue xx (z. x)/(1 t)-
Alternatively, this can be expressed as ue (w æ1) = æ - øo- Training this conditional flow matching model involves optimizing the following loss function:
(1m) (1, t;
(æ) - xx)ll
(5)
We can then use the learned vector field ((x, t; 0) to generate samples by solving the neural ODE,
3
METHOD
describe two modules of our PuretoFlow method: multi-objective predictor guidance in Sec-
tion 3.1 and neighboring evolution in Section 3.2 The full algorithm is detailed in Algorithm |
3.1 OBJETVOBE PREDICTOR GUIDANCE
In this section, first elucidate the concept of predictor guidance within the flow matching frame- work. Next, we detail the formulation of a weighted distribution driven by uniform weight vector. Finally, we introduce local filtering scheme designed to effectively manage non-convex PFs.
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Algorithm ParetoFlow: Guided Flows in Multi-Ohjective Optimization
Input: Offline dataset D, time step At, number of offspring O, number of neighbors K
1: Train objective predictors (f1 {:)) for properties on 'D in supervised manner.
2: Train the vector field ö(æ,, t; 0) using the flow matching loss from Eq. (5).
3: Generate uniform weight vectors {W")
Ji=1 using the Das-Dennis method.
4: Identify neighboring distributions for each w' using (1)
5: Initialize the Pareto-optimal set PS to retain high-quality samples.
6: Generate N initial noise { from a standard Gaussian distribution.
7: for to 1 do
8: > Example for a single distribution i*/
9: + This process is parallelized across N distributionss/
10:
Set the next iteration as = At.
M t Obetttettttttt ve Predictor Guidancex/
12:
Calculate the weighted distribution using Eq. (8).
13:
Compute the guided vector field (( l, y; 0) from Eq.
14:
Derive diverse samples for the in distribution using Eq. 10
Ne NNNeihhorrin whborring Evolut ionw/
16:
Form the neighboring offispring set X, based on ((t). Apply the local filtering scheme to filter X; to X Select the next-iteration state at, with the weighted objective using Eq. 12)
17:
18:
19:
If x is superior, update PS with i; (ar)).
end for
21:
Return PS
Predictor Guidance. Originally, classifier guidance proposed to direct sample generation to-
ward specific image categories Dhariwal & Nichol (2021). This concept has been adapted for regres- sion settings to guide molecule generation Lee al. (2023); Tian al. (2024); Chen et al. (2025).
In this paper, we term this technique predictor guidance for a generalization. Based on Lemma in Zheng et al. (2023). we derive predictor guidance in flow matching as:
1 1-t -VV
i(x,, f. y: 0) ((x t: 1;
Vaz logps(y t).
(6)
where (y t) represents the predicted property distribution. Further details can be found in Appendix A.1 Training the proxy at different time steps is resource resource-intensive Therefore, we approximate this by leveraging the relationship between 27 and ay;:
Pa (y xe, t) = Py a (x.,,
simplified to p (a:)). This guides the generation of æ; towards samples with the property y-
Weighted Distribution. The preceding discussion typically pertains to generating samples to satisfy single property y. whereas our framework designed to optimize multiple properties simultane- decompose the multi-
ously, denoted y = [xx) fm (æ)]. To manage this complexity, objective generation challenge into individual weighted objective generation subproblems. Specifi- cally, we define weight vector [W1, w2, Wm], where each wi >> 0 and Σ Wi 1. The weighted property prediction expressed as:
m Σ
fa (3) =
fe((z)(( (a:);: Bi)wi,
where f. predicts the 4th objective for 24, trained using only æ data, and the negative sign indicates minimization. We then formulate the weighted distribution as Lee et al. (2023)
Pa(y E (2x)) w) = (
(8)
where 7 is scaling factor and Z is the normalization constant. Integrating this into Eq, 6 leads to:
8(x,, y; 0) = (x1, f) + - fu
(9)
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This vector field drives sampling towards desired properties within the weighted distribution. Equa- tions (8) and ( are applied in Algorithm Lines 12 and 13, respectively.
Using the Das-Dennis approach Das & Dennis (999). which subdivides the objective space into equal partitions to generate uniform weight vectors, we produce N weights w. Each weight maps to sample, effectively covering the entire objective space. For the inh sample 2 at time step the Euler-Maruyama method Kloeden et al. 1992) applied to advance to the next time step At:
2 = æ + ((=), t, y: 0) + gv
(10)
where s = t + At indicates the next time step, = 0.1 denotes the noise factor, and e is a standard Gaussian noise term. This process is on Line 14 in Algorithm [ Unlike standard ODE sampling, this additional noise term enhances diversity and improves exploration of the design space.
Local Filtering. Using Eg. TOp, sampling reach any point the Pareto Front (PF) if it is con- vex. As shown in Figure a), a weight vector w = [0.5, 0.5] successfully guides sample generation to the /1 f2 Pareto-optimal point. In such convex case, set of uniform weight vectors fectively direct sample generation across the entire PF. However, with a non-convex PF as depicted in Tigure[[b), the same weight vector skews the sampling toward favoring single objective, either fi or f2, making it challenging to approach the fi = f2 Pareto-optimal point or its vicinity.
ef-
(a)
(b)
(c)
Convex PF
Non- Non-Convex PF Filter
Non-Convex PF Filter
1.0
1.0
1.0
Objective Function f2
0.8
(0,0, 1.34
0.8
10.50 50
0.8
0.6
0.6
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Objective Function fi
Objective Function f
Objective Function fi
Pareto Front
Pure Noise
* Generated Samples
Sampling Trajectory
Figure 2: Local filtering:
samples outside the hypercone are filtered out as shown in (c).
To overcome this, we confine the sampling space for each weighted distribution to a hypercone, characterized by apex angle 2, as depicted in Figure For any given sample a from Eq. 10) the angle as between the prediction vector ('(a()) [ ( (æt); 31), ... fm (â) (.t); 3m, and the weight vector wf is calculated. Samples where exceeds / 22 are filtered out in Figure c). Inspired by Wang et al. (2016). 4 is calculated as 2> Φi//m, /m, where dij is the angle from the Jeh closest weight to w³. This setup ensures that the sampled objective vectors align closely with the weight vector, enabling effective discovery of Pareto-optimal solutions at the hypercone boundaries and enhancing the diversity of the generated samples.
This local filtering scheme is also employed in the Neighboring Update outlined Section 3.2 and specifically applied at Line 17 of Algorithm
3.2 NEIGHBORING EVOLUTION
In the prior module, we discuss sampling from a single weighted distribution while overlooking the potential interactions between different distributions. In this section, we define neighboring distributions and introduce module to foster knowledge sharing among them.
Neighboring Distribution. Weighted distributions with similar weight vectors are likely to pro- duce similar samples, which could benefit from potential knowledge sharing. Since each weighted distribution is defined by unique weight vector, we define neighboring distributions based on the proximity of their weight vectors. For a distribution associated with w', its neighbors are identified the K distributions whose weight vectors have the smallest angular distances to wh:
Aii) = {i w E KNN(W" K, (w/))))))) )
(11)
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Here, KNN(W', K, (w0)(w)) denotes the set of the K nearest weight vectors to w;- By definition, distribution is also considered a neighbor of itself. It is outlined on Line in Algorithm !I
Neighboring Update. As mentioned, neighboring distributions generate similar samples, and aim to leverage this similarity to foster knowledge sharing. Since e introduces randomness in Eq.4 we obtain different next step states æt where each state can be viewed as offspring. We can generate O offspring for ě', denoted as 1 Given that there are K neighboring sam- ples for sample this results in set of K-O offspring X; = { j E (i),,, o E 21, 0}}. Line 16 in Algorithm I outlines this step. All candidates in this set are likely to satisfy the weighted distribution well, as they are guided by similar weighted distributions j E (N))..
1o=1-
We aim to update the current sample æ using the neighboring set X,, The local filtering scheme from the previous module filters out X; exclude objective predictions not aligned with w'. The remaining viable offspring are termed X Subsequently, the next iteration for æi is updated as:
arg max fw. (a) aR EX!
(12)
This determines the next state æi. for each of N samples and is detailed in Line 18 of Algorithm
Pareto-optimal Set Update. While directly selecting the final N samples from the flow generation effective, we also aim to retain all high-quality samples during generation. To achieve this, we maintain PS consisting of N samples, where the i-th sample the best for fw (-; (). The PS is initialized with non-dominated samples in the offiin dataset following Xue et al. (2024). Using Eq. (12.. we compare xy with the i-th sample in PS. If æ is superior, we update PS with Φ (x)); otherwise, we retain the existing sample. This step is specified at Line 19 in Algorithm I Finally, we apply non-dominant sorting to PS and select 256 candidates for evaluation.
4 EXPERIMENTS
we compare our ap- We conduct comprehensive experiments to evaluate our method. In Section 4.4 proach to several baselines to assess performance. In Section 4.5 we demonstrate the effectiveness of our proposed modules.
4.1 BENCIMARR OVERVIEW
We utilize the Off MO0- -ee0hBBBehhhh which summarizes and collects several established bench- marks Xue al. (022).. We explore five groups of tasks, each task with dataset D and ground- truth oracle for evaluation, which is not queried during training. For discrete inputs, we convert them to continuous logits as suggested by /Trabucoo et al. (2022); Xue et al. (2024).
Tasks. (1) Synthetic Function (Synthetic) Xue et al. (0044): This task encompasses several subtasks involving popular functions with 2-3 objectives, aiming to identify PS with 60,000 offline designs. We exclude the DTLZ2-6 tasks as recommended by the authors due to evaluation errors
(2) Multi-Objective Neural Architecture Search (MO-NAS) Dong & Yang 2020); Lu et al. (202)); Li et al. (202I): This task consists of multiple sub-tasks, searching for neural architecture that optimizes multiple metrics, such as prediction error, parameter count, and GPU latency.
(3) Multi-Objective Reinforcement Learning (MORL) Todorov et al. 2012;; (a) The MO-Swimmer sub-task involves finding a dimension-9, 734 control policy for a robot to maximize speed and energy efficiency; (b) The MO-Hopper sub-task aims to find dimension- 10, to optimize two objectives related to running and jumping.
10,144 184 control policy for robot
(4) Scientific Design (Sci-Design): (a) This Molecule Zhao al. 2021 sub-task aims to optimize two activities against biological targets GSK3,3 and JNK3 in a dimension-32 molecule latent space, using 49,001 offline points. (b) The Regex sub-task aims to optimize protein sequences to maximize the counts of three bigrams, using 42,048 offline points. (c) The ZINC sub-task aims to maximize the logP (the octanol-water partition coefficient) and QED (quantitative estimate of drug-likeness) of small molecule. (d) The RFP sub-task aims to maximize the accvest- surface area and the stability of RFP in protein sequence designs.

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(5) Real- World Applications (RE) Tanabe & Ishibuchi (2020: This category encompasses variety of practical optimization challenges. including four-bar truss and pressure vessel design. The MO- Portfolio task Fabozzi et al. (2088) is also included here, which focuses on optimizing expected returns and variance of returns in a 20-dimensional portfolio allocation space.
The original Off-MOO-Bernt also includes some combinatorial optimization tasks such MO- TSP, MO-CVRP, and MO-KP. While these could potentially be incorporated under a generative modeling framework Sun & Yang (2023), the decoding strategy required is rather complex. As this paper focuses on a general guided flow matching method, we have opted to exclude these tasks, given the sufficient variety of other tasks already available for our evaluation.
Evaluation. We follow the evaluation protocol in Xue et al. (022)). Each algorithm outputs 256 solutions for evaluation. Each task has a reference point, and we compute the hypervolume metric, which measures the volume between the proposed solutions and the reference point. A larger hy- pervolume indicates better solutions. We report the P percentile measure, employing 50 in this study. Specifically, we rank the solutions using nondominated sorting Deb et al. (2002), remove the top - P% of solutions, and then report the hypervolume of the remaining solutions.
100 and
4.2 BASELINE METHODS
Following Xue et (2024) we compare two primary groups of methods: DNN- and GP-based methods, along with some notable generative modeling methods.
DNN-Based Methods:
These methods utilize surrogate DNN models combined with evolutionary algorithms to optimize solutions. We assess three configurations: (1) End-to-End Model (F2R): Out-
puts m-dimensional objective vector for a design æ. enhanced by multi-task training techniques such as GradNorm Chen et al. (2018) and PeGrad Yu et al. (2020). (2) Multi-Head Model (MH): Uses multi-task learning train a single predictor, employing the same techniques as End-to- End. (3) Multiple Models (MM): Maintains m independent predictors, each using techniques like COMs CoMsTrabuccoo al. (2021). ROMA Yu al. (2021). IOM et al. (2022), ICT Yuan al. (2023), and Tri-mentoring Chen et al. (203a)) The default evolutionary algorithm is NSGA-II Deb et al. 2002). , with results cited from the original study. Additionally, we compare MOEA/D Zhang Li 2007 + MM due to its superior performance. We further expand our comparison to include more
ICTa
traditional approaches in Appendix A.2 A.2
Bayesian Optimization compute the acquisition function to select new de- signs, which are then evaluated using a predictor model. Techniques include: Hypervolume- based qNEHVI Daulton et al. (2021). Scalarization-base qParEGO Daulton et al. (2020), and Information cooommtion-thhorrtiic-hoe
GP-Based Methods:
theoretic-based JES Hvarfner et al. (2022). referrncce results from Xue et al. (2024).
We communicate with the authors and use the updated benchmark data for all MO-NAS tasks and the wollwwrld application tasks RE21, RE34, RE35, RE36, RE41, RE42, and RE61. For these tasks, rely the latest results provided by the authors, rather than those published in the paper.
Generative Modeling Methods: (1) PROUD Yao et al. (2024) enhances diversity by incorporating hand-designed penalties into diffusion sampling process. (2) LaMBO-2 Gruver et al. (2023) utilizes the acquisition function to guide diffusion sample generation. (3) CorvVAE Wang et (2022) em- ploys a VAE to decipher semantics and property correlations, adjusting weights in the latent space. (4) MOGFNs Jain et al. (2023) incorporates multiple objectives into the GFlowNet framework.
4.3 TRAINING DETAILS
Our objective is to derive 256 design samples. However, since the Das-Dennis method may not generate exactly 256 uniform weights, we generate slightly more, resulting in over 256 samples. We then use learned predictors for non-dominant sorting to select the top 256 samples. We set the number of neighboring distributions, K, to be m + 1, where m is the number of objective functions, and set the number of offspring, O, to be 5. The sensitivity of these hyperparameters is further examined in Appendix A.3 We follow the predictor training configurations outlined in Xue et al. (2024) and flow matching training protocols described in Tomezak (2022). Additional hyperparameter details are provided in Appendix A.4 the computational overhead is discussed in Appendix A.5
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Table I: Average rank of different methods on each type of task in Off-MOO-Benchh
Methods
Synthetic
MO-NAS
MORL
Sci-Design
RE
All Tasks
D-Best
16.82 6.28
14.42 = 4.11
15.00 = 4.00
13.75
6.91
18.06 = 3.93
16.02 + 5.13
5.13
E2E
10,91 i 8.20
6.05 1 3.32
12.50 1 1.50
9.75 1 4.97
9.69 1 5.65
8.73 5,88
E2E- + GradNorm
12.64 + 6.68
13.42 + 5.54
8.50 + 0.50
13.50
5.12
14.19 + 5.87
13.31 + 5.87
E2E + PcGrad
9.45 9.45 6.37
6.42 = 3.18
16.50 - 2.50
14.00
10.88 = 6.17
9.40 5.70
MH
11.35 7.19
5.26 2 3.93
12,00 = 4.00
1.250
10.00 = 5.67
8.87 6.00
MH + GradNorm
10.45 1: 6:21
16.42 4.84
18,00 2.00
1.44
17.00 1: 4.72
15.27 1: 5.64
MH + PeGrad
11.45 4.58
6.84 +
18.50
+.50+0.50 0.50
13.50
11.06 6.24
10.08 5.46
4
6.74
16.50
6.75 4.32
6.75
6.69 3.46
6.71 = 4.31
MM COMs
13.00
9.53
1.50
12.25 6.83
4.75
12.15 5.06
MM RoMA
13.27 1.553
5.75
10.00
12.00 2.455
5.14 +
10.27
MM IOM
3.78
5.37
6.50
10.75 1.92
6.73 3.88
MM ICT
14.45
8.33
9.50
12.50 7.12
11.75
6.54
11.12 5.77
MM Tri-Mentor
5.75
10.50 + 1.50
13.00 3.54
10.50 1: 5.82
10.27 5.65
MOEA/D MM
10.55 4.83
1.05
11.00 + 1.00
10.75 6.87
12.12 6.62
11.81 + 5.66
MOBO
10.91 + 4.42
3.82
17.00 + 0.00
8.25 + 6.61
11.00 + 5.79
12.37 + 5.32
MOBO-
13,36 + 3.98
16.63
3.77
21.00 + 0.00
12.75
8.04
17.69 = 4.55
16.13 + 4.91
MOBO-JES
17.27 + 3.11
22.00
0.00
21.00 + 0.00
18.75
5.63
13.62 = 5.19
18.13 + 5.00
PROUD
8.55 - 6.33
14.53 4.43
2.50
0.50
6.25 + 3.49
5.75 = 5.02
9.46 = 6.39
LaMBO-2
10.18 6.55
14.37 1: 4.66
3.00 1 1.00
5.00 1 1.22
5.00 - 4.72
9.44 6.49
CorrVAR
11.73 + 6.14
17.74 2.95
4.50 + 0.50
8.00 + 4.18
9.56 + 6.00
12.69 + 6.35
MOGEN
10.55 + 6.04
15.95 = 3.98
3.50 - 1.50
5,50 + 4.50
5.88 = 4.97
10.42 + 6.63
ParctoFlow (ours)
4,00 = 3.88
3.47 = 4.26
1.00 = 0.00
2.75 = 1.48
2.44 = 3.45
3.12 = 3.77
4.4 RESULTS AND ANALYSIS
Table displays the average ranks of the 100th percentile results for all methods across various task types. Detailed hypervolume results for both the 100th and 50th percentiles are reported in Appendix AA.6 and Appendix A.7 respectively. Two separator lines distinguish: (1) DNN-based methods from GP-based methods, and (2) GP-based methods from generative modeling methods. D(best) denotes the best solution set the offline set, characterized by the largest HV value. The last column summarizes the average rank of each method across all tasks. In each task, the best and second-best ranks are highlighted in bold and underlined, respectively. We provide visualization results for C-10/MOPI and MO-Hopper, and a case study on C-10/MOPS, in Appendix A.8
We make the following observations: (1) As shown in Table and Figure 3 our method ParetoFlow consistently achieves the highest ranks across all tasks, underscoring its effectiveness. (2) Both DNN-based and generative modeling-basee -baiin methods frequently outperform D(best), illustrating the strength of predictor and generative modeling. (3) GP-based methods often underperform D D(best). We hypothesize this is because these methods, typically used in online optimization to select subse- quent samples, are less effective this offline context. (4) Within the generative modeling category, ParetoFlow surpasses other methods, including diffusion-based methods like PROUD and LaMBO- 2. the VAE-based method Cor,VAE, and the GFlowNet-based method MOGEN, highlighting the superiority of our ParetoFlow method. (5) MO-NAS and Sci-Design tasks are predominantly dis- crete, with MO-NAS having a higher dimensionality. Generative modeling methods show reduced effectiveness MO-NAS relative Sci-Design. This performance gap may stem from the diffi- culty in modeling high-dimensional discrete data.
4.5 ABLATION STUDIES
Table 2: Ablation Study on ParetoFlow.
Methods
7DT2
C-10/OP1
MO-Hopper
Zine
RF23
Equal
6.15 0.23
4.64 0.03
5.58 + 0.38
4.14 + 0.14
4.75 0.00
First
5.58 0.38
4.59 0.02
5.25 0.23
4.00 0.14
4.89
w/o local
5.78 0.15
4.65 = 0.03
5.44 0.21
4.36 0.04
6.44
w/o neighbor
6.43 = 0.01
4.64 0.00
5.62 0.16
4.40
6.08 0.20
w/o PS
6.45 0.52
4.49 1 0.00
5.00 1 0.02
4.40 0.00
5.28 1 0.21
ParetoFlow
6.79 0.16
4.77 + 0.00
5.69 + 0.03
4.49 0.06
6.32 + 0.46
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We use PlretoFlow as the baseline to assess the impact of removing specific modules, with results detailed in Table [ We conduct these ablation studies on representative subtasks: ZDT2 for Syn- thetic, C-10/MOPI for MO-NAS, MO-Hopper for MORL, Zinc for Sci-Design. and RE23 for RE.
Mujti-Objectivee Predictor Guidance: This module employs uniform weights for batch samples. In our ablation study, we explore: (1) Equal: Equal weight assigned to every sample across all ob- jectives. (2) First: Weight applied solely to the first objective. Both variants underperform compared to the full ParetoFlow, demonstrating the advantage of our uniform weight scheme. Equal generally outperforms First, suggesting that focusing on single objective can bias sample generation.
Additionally, evaluate the impact of excluding the local filtering scheme (w/o local) deter- mine its importance. The performance drop observed without this scheme underscores its effec- tiveness in managing non-convex Pareto Fronts. Additionally, we measure pairwise diversity using Nj Σin Ej-i+1 d(yr::)). Yi), where d denotes the Euclidean distance. This metric is applied to samples from both PoretoFlow and PoretoFlow w/o local. For ParetoFlow w/o local, diversity decreases from 5.144 2.080 in ZDT2, from 0.832 0.827 in C-10/MOP1, from 0.897 to 0.181 in MO-Hopper, from 0.721 to 0.495 in Zinc, and from 0.991 to 0.814 in RE23, This indicate that the local filtering scheme enhances performance by improving the diversity of the solution set. We further compare local filtering performance on convex and non-convex tasks in Appendix A.9 We also include in Appendix 10 detailed comparisons between flow matching and diffusion models, as well as between the Dennns method and another weight generation strategy.
We omit this module (w/o neighbor) to observe the effects on sample gen- eration, focusing exclusively on direct offspring without leveraging neighboring samples. Removing this module leads to performance decrease as detailed in Table 2 demonstrating the effectiveness of neighboring information. Besides, we found that employing the neighboring module significantly improves the selection of the next step's offssring. In the sampling process, the majority of offspring selected from the neighborhood outperform those from their own distribution: 67.33% for ZDT2, 73.67%for C-10/MOP1, 58.33% for MO-Hopper, 81.33% for Zinc, and 61.98% for RE23, high- lighting the pivotal role of this module in the sampling process. Besides, we observe that only 12% of the points in C- C-10/MOP1 and 1% in MO-Hopper are duplicates. The higher duplication rate in C- C-10/MOP1 is primarily due to the decoding of continuous logits back to the same discrete values. This observation underscores the effectiveness of ParetoFlow.
Neighboring Evolution:
Lastly, we examine the performance of our method without the Pareto Set (PS) update, relying only on the final samples produced through the sampling process. The observed performance degradation confirms the critical role of the PS update, indicating that final samples alone are insufficient.
RELATED WORK
Offline Multi-Objective Optimization. The primary focus of MOO research is the online setting. which involves interactive queries to a black-box function for optimizing multiple objectives si- multancously Diang et al. (2023); Park et al. (2023): Gruver et al. (2023). However, offline MOO presents a more realistic setting, as online querying can be costly or risky Xue et al.] (2024); Kim et al. (2025). In this context, two traditional methods adapted with trained predictor as the oracle: Evolutionary algorithms employ a population-base -aased search strategy that includes iterative parent selection, reproduction, and survivor selection Deb et al. (2002); Zhang & Li (2007). Altern- tively, Bayesian optimization leverages the learned predictor model to identify promising candidates through an acquisition function, with sampled queries advancing each iteration Daulton al. (2023);
Zhang & Golovin (2000); Golovin((020),
Q et al. (2023). Additionally, several predictor training techniques such as COMs Trabucco et al. (2021), ROMA Yu al.) (202I), NEMO Fu & Levine (02T). ICT Yuan et al. (2023), Mentooig Chen et al. (2023a), GradNorm Chen et al. (0188), and PeGrad Yu et al.
(020) are adopted to enhance training efficacy.
Our ParetoFlow method is inspired by the seminal evolutionary algorithms MOEAD Zhang & Li (2007) and LWS Wang et al. (2016), which use a weighted sum approach Ma et al. 2020) to guide populations and facilitate mutation among neighbors. The generation concept in these algorithms corresponds to the time step concept in our method. The primary distinction of our method is its gen- erative modeling aspect: we train an advanced flow matching model on the entire dataset, enabling the exploration of data generative properties. This capability allows our sampling process to access the sample space that traditional evolutionary algorithms are unlikely to reach. We further explore
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the relationship between evolutionary algorithms and flow models in our ParetoFlow framework in Appendix A.12
Guided Generative Modeling, Several studies have developed generative models to produce sam- ples meeting multiple desired properties. For instance: |Wang et al. (202I) integrates structure- property relations into conditional transformer for biased generative process. Wang et al. (2022)
employs a VAE model to recover semantics and property correlations, modeling weights in the la-
tent space. TTaaasvska et al. (2022) applies multiple gradient descent on trained EBMs generate
new samples, although training EBMs for each property can be complex. Han et al. (2023) explores distinet setting aimed at generating modules that fulfill specific conditions. Zhu et al. (2023)
uses GFlowNet as the acquisition function and Jain et al. (2023) integrates multiple objectives into GetowNet. Yao et al. (2024) introduces diversity through hand-designed diversity penalties instead of uniform weight vectors, focusing on white-box setting. Gruver et al. (2023) investigates online multi-ohjective optimization within diffusion framework, using the acquisition
guide sample generation. Kong et al. (024) applies multi-objective guidance under diffusion framework but only uses equal weights for all properties, failing to capture the Pareto Front. Chen al. (2024);
Yuan et al. 2024) also explore guided diffusion models; however, their focus is limited single- objective optimization. Instead of focusing on optimization techniques, Hu et al. (2025b) investi- gates auto-regressive diffusion models for molecule design, using the protein pocket condition. These studies vary in setting and approach, often using generative models that are either less advanced or challenging to train. Unlike these efforts, our work combines the advanced genera- tive model of flow matching with evolutionary priors in traditional algorithms, an intersection never explored in the existing literature.
directional
CONCLUSION
In this work, we apply flow matching to offline multi-objective optimization, introducing ParetoFlow. Our multi-objective predictor guidance module employs a uniform weight vector for each sample generation, guiding samples to approximate the Pareto-front. Additionally, our neigh- boring evolution module enhance knowledge sharing between neighboring distributions. Extensive experiments across various benchmarks confirm the effectiveness of our approach. We discuss ethics statement and limitations in Appendix A.13
7 ACKNOWLEDGEMENTS
This research was partially funded by the Fonds de recherche du Québec Nature et technologies. We also gratefully acknowledge CIFAR for its support through the AI Chairs program.
We thank Mattie Tesfaldet and Alexander Tong from Mila, along with Chin- Wei Huang from Mi- crosoft Research, for their insightful discussions on score-based models. We further appreciate Jiarui Lu from Mila for his valuable suggestions regarding the presentation of this paper.
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1.015
1.015
1.005
1.000
0.995
0.995
0.900
0.990
0.980
0.980
0.975
0.975
16 -iigpa
C-100Mor:
setaareeeeernneenn Rank
Number neighbors
Number offspringg
Figure 3: Sticks and triangles are rank medians and means.
Figure 4: Sensitivity the number of neighbors K.
Figure 5: Sensitivity to the mumber of offspring O.
A APPENDIX
A.1 PREDICTOR GUIDANCE IN FLOW MATCHING
From Lemmal I in Zheng et al. (2023), the guided vector field is derived as:
Uar,,, 0) a, + by Va, p(æ;( ())
(13)
where dee and b, (dio a.oi) de a
With az and or t, we simplify Eq. (13):
1 1-t Vz, (, xt
1 log lv)
(14)
The oog-robbability function is expressed as:
logp(x; y) log (o: + log p Te; t) logp(y)
(15)
where pe (xx) represents the data distribution learned by the flow matching model and p x denotes the predicted property distribution.
Substituting these expressions leads to:
i(x,, 1, -tt + 1 1-1 1-' Vn, logro(x:) + 1 1- A-'V.,
1
Vm, log logpaly x..()
(16)
= 0(x1, t: 0) + - -'Va, log x.,
t
A.2 EXTENDED COMPARISONS
We have expanded analysis include widely recognized methods such NSGD-III Deb & Jain 2013) and MINS-EMOAAA Beume et al. (2007), applied to the same five tasks highlighted in our ablation studies. Our findings in Table demonstrate that ParetoFFlow consistently outperforms these traditional approaches, reinforcing the effectiveness and robustness of our method.
Table 3: Comparison of NSGD-III and SMS-EMOA
Methods
ZDT2
C-10/MOP1
MO-Hopper
Zine
RE23
NSGD-III + MM
5.74 0.05
4.71 0.01
5.31 0.13
4.15 + 0.06
4.96 = 0.04
SMS-EMOA + MM
6.23 0.09
4.73 + 0.00
5.45 - 0.23
4.33
+ 0.09
5.67 = 0.08
PlretoFlow (ours)
6.79 + 0.16
4.77 0.00
5.69 0.03
4.49
0.06
6.32 0.46
A.3 HYPERPARAMETER SENSITIVITY
This section examines the sensitivity of our method to various hyperparameters- naaaaaaammeeeeeeaaaael..... the num- ber of neighbors (K), the number offspring (O), the number of sampling steps (f), the scaling factor (7) in (8. the noise factor (g) in Eq. 10 across tasks: the continuous MO-Hopper
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Hopper
1.000
1.015
C-10MOP3
1.000
0.950
1,000
0.925
0.900
3..90
0.875
0.985
3.g
Samsing step
0.850
Value ofalinng factory
0.900
Value nolse
Figure 6: Hypervolume as a function of t.
Figure 7: Sensitivity to the scaling factor of 7.
Figure 8: Sensitivity the noise factor
and the discrete C-10/MOP1. Hypervolume metrics are normalized by dividing by the default hy- perparameter result facilitate comparative analysis, unless specified otherwise.
Number of Neighbors (K): Tested values include 1, 2, 3, 4, and 5, with K = 3 as the default. As shown in Figure performance remains stable with changes K. While performance generally increases with K, indicating more neighbors provide more useful information, it stops the increase K 4, likely limited by predictor accuracy and redundancy at higher neighbor counts.
Number of Offspring (O): We vary the number of offspring, testing values of 3, 4, 5, 6, and 7, with 0 as the default. As illustrated in Figure [S] performance stable across different O values. Performance tends to increase with larger O as more offspring provide additional options for subsequent iterations; however, this benefit is offset by increased computational costs.
Number of Sampling Steps ()): We analyze the impact of the number of sampling steps our method's effectiveness. The normalized hypervolume the Pareto set (PS) is plotted as a function of time step in Figure 0 We observe a general increase in hypervolume with increasing Addi- tionally, the robustness of our method to changes is further examined in the Appendix A.II
Scaling Factor (7): The effect of varying is investigated with values 0, 1, 2, 3, and 4, and y 2 as the standard setting. As indicated in Figure / performance remains stable across changes in y, and improves from y 0 to 7 2, demonstrating the effectiveness of increasing predictor guidance.
Noise Factor (g): The effect of varying is investigated with values 0.025, 0.05, 0.1, 0.2, and 0.4, and g 0.1 as the default. As shown in Figure performance is quite robust to changes in g.
A.4 TRAINING DETAILS
We adopt the predictor training configurations from Xue al. (2024). utilizing multiple model setup. Each predictor consists of a 3-layer MLP with ReLU activations, featuring hidden layer size of 2048. These models are trained over 200 epochs with a batch size of 128, using the Adam optimizer Kingma & Ba (215)) at an initial learning rate of 1 x 10-3 3, and decay rate of 0.98 per epoch. Flow matching training follows protocols from Tomezak (2022), employing a 4-layer MLP with SeLU activations and hidden layer size of 512. The model undergoes 1000 training epochs with early stopping after 20 epochs of no improvement, with batch size of 128 and the Adam optimizer.
The approximation a (ax.) proves inaccurate when is near 0, leading to an unreliable predictor. Consequently, set 7 2 only if exceeds predefined threshold; otherwise, determine this threshold by evaluating the reconstruction loss between & (æx) and Φ As illustrated in Figure 9 for the tasks C-10/MOP1 and MO-Hopper, the reconstruction loss remains below 0.2
0. We
when exceeds 0.8. Therefore, we establish the threshold at 0.8.
We employ the simplest setup of Multiple Models (MM) within PlretoFlow. As detailed in Table we experiment with the IOM setup for comparison. The results indicate that the IOM setup performs similarly or slightly worse than the MM setup.
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Loss
1.020
3.0
Hopper
2.8
G-10/MPP1
1.015
2.6
MD-hopper
2.4
Hypervolume ratio
1.010
2.0
1.005
0 1.8
1.6
1.000
1.44
0 9.2
0.995
0.8
0.990
8 0.6
0.4
0.985
0.980
0.0
0.1
0.2
0.3
0.4
0.5 Timet
0.6
0.7
0.8
0.9
1.0
250
500
1000
2000
4000
Number of sampling steps
Figure 9: Reconstruction loss as function of the time step t.
Sensitivity to the number of sampling steps
Figure 10:
Table 4: Comparison of ParetoFlow with Vanilla Multiple Models and with IOM Multiple Models.
Methods
ZDT2
C-10/MOP
MO-Hopper
Zinc
RE23
ParetoFlow (default)
6.79 +: 0.16
4.77 + 0.00
5.69 + 0.03
4.49 + 0.06
6.32 + 0.46
ParetoFlow + 10M
6.32 + 0.22
4.75 4 0.01
5.97 + 0.13
4.45 : 0.04
6.15 1 0.14
A.5 COMPUTATIONAL COST
All experiments are conducted on workstation with an Intel 19-12900K CPU and an NVIDIA RTX3090 GPU. The computational cost of method consists of three components: predictor training, flow model training, and design sampling. Detailed task information and time costs are summarized in Table (in minutes). For discrete tasks, report the dimension of the converted logits. Our method is efficient, completing most tasks within 10 minutes.
We also benchmark some baseline methods in Table 6 Considering the minimal overhead and significant performance gains of our method compared to baselines, we believe it provides a practical solution for researchers and practitioners seeking effective results without sacrificing speed.
Table 5: Time cost of ParctoFlow.
Components
ZDT2
C-10/MOP1
MO-Hopper
Zinc
RE23
Design dimension
10184
378
Number of offline points
60000
12084
4500
48000
60000
Number of objectives
2
2
Predictor training (min)
3.99
0.76
1.03
3.21
3.94
Flow model training (min)
2.36
1.28
2.51
6.56
1.58
Design sampling (min)
0.55
0.25
0.64
0.42
0.63
Overall time cost (min)
6.90
2.29
4.18
10.19
6.15
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Published as a conference paper at ICLR 2025
Table 6: Time cost of baselines.
Tasks
C-10/MOP1
MO-Hopper
F2E
1.20
1.17
MH
1.24
1.17
MM
1.70
1.80
MOEA/D MM
1.50
1.36
MOBO
0.12
33.68
PROUD
2.23
4.05
LaMBO-2
2.54
4.22
CAr
1.81
2.89
MOGEN
5.73
10.62
ParetoFlow (ours)
2.29
4.18
To provide more detailed insights, have conducted thorough analysis across a diverse set of MO-NAS tasks from the NAS-Bench series, including C-10/MOP1, MOP2, MOP5, MOP6, and MOP7, as detailed in Table 7 Our findings indicate that despite variations in the number of objec- tives and design dimensions, the computational cost our method remains consistent. Notably, for tasks with higher dimensional objectives such as MO-Swimmer and MO-Hopper, computational efficiency is comparable to tasks with lower dimensions. This consistency underscores our method's scalability across range of computational demands. Additionally, method consistently achieves strong performance, further attesting to its robustness.
Table 7: Task- Cask-Specific Details
Tasks
C- 1-1OMOPPI
C-10/MP22
C-10MOPS
C-I0MOPS
C-10MOPP
MO-Hopper
MO-Swimmer
Raw Input Dimension
Input Dimension (Logits)
31
31
24
24
21
NJA
N/A
Number Objectives
Number of Offline Samples
12084
26316
9875
9375
9376
4500
8571
Predictor Training
Flow Model Training (min)
1.28
5.92
1.20
5.18
1.19
2.51
2.53
Design Sampling (min)
Overall Time Cost (min)
2.29
8.83
3.06
7.42
4.08
4.18
4.97
Rank of ParetaFlow
A.6 100TH PERCENTILE RESULTS
present the 100th percentile results with 256 solutions, demonstrating that our method, ParetoFlow, performs well across different tasks. For each task, algorithms within standard deviation of having the highest performance bolded.
As shown in Tables 8
10 and 13
Table 8: Hypervolume results for synthetic functions.
30:43
8:22
3.57
0.06
1:51
15:11
101
4.70
5:05
5.45
1.76
00.00
0100
EZE PeCindl
10.85 2
10.02-20.0
4.5 0.00
2.37 1.25
4.1.2200.1 0.17
00.19 0.00
6.54 0.03
5.20:20000 2.00
5.65 103-200
5.1.5
2/04 2-22
10:2 0.35
10:3--100
-.15
11:00
0.00
MIT Patird
10:: 0.00
10.19 mm
4:35
2.55
mm
11:0
m:00
1-111
211 1.1
3M
10.60 0.00
10.78:20.00
4.50 1100
2.37
6.28 0.000
40.54 0.00
1000
1.35 2:000
5.80 0.00
6.34 5120
4.01 2:00
0.06
J1.75
2:07
4
BoMA
100: 0.00
301
2.51
1.300
11.00
1.77
MMN 17M
1843 0.80
10.74 cups
4:38 11:0
2.35 3.74
3.77 6m
00.00
0.00
1.21 c.01
3.01 8m
1.3 2:19
130 0.0
100
10:75
0:33 0.05
86
11:0
B/G
3.11
3:31 1.111
MOEAJD MM
1066 6.0600000
10.90 2.12
4.7750000
0.31 362
4.01 22
50:000000
4.44 0.64
1:21-40::0
1.38 0.16
4.7 0L15
4.20 0m
XO0BO
10.20:0.000
DUMt 2
1010
1012 370
106 10.7
0:29
mm 6.70
9-35 cups
3.7
NIA
00.00
L.
3/M
mm 110
1:74 1::00
3:33111331
1111 mm
200 1001
131 131
1:21 BTM
1:3 101
1.01 110
10 m.00
6:22
4.7%
Suos
1m0
MIGI
4.03
1.11 11.35
GurVAE
MOGEN
1001 0.00
0.13 4 0.02
4.7 2.75m
5.88 3.05
400 mm
46 0.0
5.37 0.08
110 8.06
L30-4 861
1,5 0.00
Pack Go05
10:55 005528..0 8.00
1000 0000
0.000
3:25
1.0 2002
10.96 6.00
1:30 0.00
0.70 0:16
5.85 0.00
5.15 0008
102 0004
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Published as a conference paper at ICLR 2025
Table 9: Hypervolume results for MO-NAS (Part 1).
Methods
C-IMDDPI
C-INMOP2
C-UUMOPS
C-IIMMDP
C-UUMOPS
C-IUMOPS
C-10MOP7
C-JMMPPS
C-INMMPS
Dilea)
4.72
10.42
9.21
18.02
00.73
105.56
299.67
4.35
9.64
E2E
4.70
10:45 2 0.31
10.15
0.00
21.47
2: 2:042
43.17
1:0.02
103.52 0.a5
491.5 0.38
6.62 22 2.0.28
10.34 0.04
Cerm
6.5---00 0.00
1101
0.000
19:05
0.38
0.01
1.59
157.55
594 +-0.31
PeCirdd
670-+-00 0.00
10.48 0.02
10.16-1
0.90
21:35
0.37
0.01
106.14
184.60
0.17
0.03
0.00
0.00
GradNom
1:21
10:45
0.21
102.25
2.69
0.00
10:44
0.00
401.23
0.00
MM COMS
10:44
20.91
0.00
MM RoMA
4.77 0.00
10.05 0.00
10.16-11 0..00
21.67
0.225
05.06 0.30
103-75 + 0.57
483.16 0.00
0.35 0..00
0.76
MM ROM
6.75-1 0- 0..01
10.05 + 0.00
13.07 0:00
21.50
0.55
09.20
0.00
106.57 0.05
400-77
0.00
5.66 0.000
10.35 0.00
MIM KCT
4.74 2: 0,01
10:40 11.00
0.06 0,000
20.00 8: 11:00
29.17
0,00
106.29
4: 0.01
201 2: 0.00
4.12 03000
B,22 0.00
MM + To Menter
4.77 0.00
10.00 0.50
10.13 11:00
21.58 6- 0.08
43.05
01:05-2:10 11:00
106.41 0.01
201. 88
0100
4.13 1:1002
5.84 0.21
MOEAID MM
6.74 2.0.00
0.67 0.0.0
0.80 03.07
21.90 22 0:31
48.84 0:00
103. 89 = 0.18
102.23
2.69
6.19 2.020
9.62 0.02
MOBO
6.74 0.000
10.43 D.51
8.58 0.00
20.85
=2 0.02
44.89
1: 0.01
33 D.07
483.07 4.01
6:12 0.000
K.77 0.00
MOBO &PuEGO
6.63 0.00
10:44 2: 0.00
8.94 0.00
20.01
2: D.06
0.00 37.21
94.72 4 6.91
300.55 0.15
4.00 0.300
8.36 0.00
MORO-IFS
NA
N/A
W/A
N/A
N/A
N/A
FROUD
171300001
10:35 1020
K00
15:11 1:38
0:12
16035 11:75
LMBO-2
0.27
100.30 1.82
47651
13.64
0.00
10:43
17.53 0.72
47.00 0:30
4G4.27
0.36
17.91
10:0
4:900
Table 10: Hypervolume results for MO-NAS (Part 2).
Mebedd
IN-IKMMPPI
IN-IR/MOP
IN-IK/MUR
IN-KMMMP
IN-IR/MOP'
IN-RIMMP
IN-IUMIOT
IN-R/MMUS
IN-IKMOPP
4.50 0001-..11
1.56 01.12
8-55 0.00
1.51 4-0.5
1.72 11,09
9:91 0-11
1.37 0.19
0:51
19.51 1.05)
8:18 18:4
122R GroNNom
6.11 0.76
1:47 (1:0)
S:1 0.01
1.663.0.3
0.99 (0.01
0:54 0.19
11121:1 0100
6:61:60 66.15
17:58 1..55
5:6 6-062
10.05 0.00
1.10 19:5
0.81 0//7
3:51 0113
0:52 6.06
19:53
6:17
2:00
10.10=000 0:07
20:30
GaadNora
4.45
2:35
5.01
2:366
5,00
MB PoGual
4.07 0 0:30
4.33:2 0.01
10:00 0.01
4.38 0.05
6 0.022
0.56 0.13
3.04 2: 0.00
0.57 = 0.06
10.25 2 0.00
0.10 061
MM
4.50 0 0.30
3.54
10.05 0.00
4.09 0.00
4.02 0.00
0.50-20.2 0.24
4.14 0113
0 1.00-: 0.00
10.02 0.00
0.10-000 0.00
COOS
20.07
MDC 1OM
3 11.00
0,00
BUU2 0,00
6.45 9312
3.53 8:00
5186 0:33
4 0.11
0400 6.06
23:31 2: 0.10
9.16 062
KI5E ICT
6.06 0.30
4:35 (1.0
0.00 0.00
6.68 9322
9.00 0.06
5.99 9.00
3:18 1.12
9:30 0.29
19.05 2 2.175
0.00 066
Menar
4.52
0.91
4:25 01113
9.07 0.00
MOORO
4.9 5:31
4:5:2:01
6305
1 0-05
3:30 1.0
6:5 1076
3:5
1837 2:00
MORO MORD--PPWEED PWWFID
3.50 000-4000
4.95 0.01
23 6.14
1.16 1.00
3.04 0.00
0.52 - 0.04
1.55 6.05
6.67
6.1.2
16:33 4
8 0.04
MOBO-JES JES
NW
N0A
NAA
N/A
NAA
NAA
375
NAA
N/A
5,12 0,65
1:1:2
5350
4.0
5:20 02202133
1:5
SOL 5004
1803 2
L375
L
0.07
CarVAA
0.56 0.00
415
0:27
5.95
3.00
8:16 0.08
3,45 0/0
0.82
1833
1 0.10
9.17 0.33
MOGIN
4.57 0.01
421 1 0110
0:51 0.06
5.06 1.19
0.18 1.06
0:22 0,00
149 0.1
6:80 1.00
15.00 0.00
10/00 0.08
PaCEEFO louns)
4.05 0.01
0.33
10.05 0.04
4.03 6.00
4.703:0.10
10.05 1.005:00000
2.0
0.70 0.10
20.59 0.00
0.30 0000
Table 11: Hypervolume results for MORL.
Methods
MO-Hopper
MO-Swimmer
D(best)
4.21
2.85
E2F
4.76 0.25
2.77 + 0.03
E2E GraiNNom
5.02 1 0.04
2.90 1 0.07
E2F + PeGrnd
4.60 + 0.227
2.49 + 0.05
MH
4.57 0.22
+
MH + GradNorm
3.78 4.005
2.69 + 0.24
MH PeGrad
0.61
+
MM
4.58 4.58+0..19
2.60 + 0.15
MM COMs
0.17
2.71
MM + RoMA
0.23
2.78 + 0.20
MM + 10M
0.49
MM + ICT
1.67 0.12
3.11 + 0.08
MM + Tri-Mentor
0.11
2.82
MOEA/D + MM
4.75 - 0.28
2.86 + 0.19
MOBO
MOBO-jPPurEGG
NIA
NIA
MOBO-JES
N/A
PROUD
5.65 + 0.00
3.43 + 0.04
LaMBO-2
5.66 0.083
3.41 0.12
ConYAA
5.61
3.38 - 0.08
MOGEN
5.54 0.00
3.43
ParetoFlww (ours)
5.69 0.03
3.50 = 0.07
21

---

Published as a conference paper at ICLR 2025
Table 12: Hypervolume results for scientific design.
Methods
Molecule
Regex
RFP
ZINC
Dibest)
2.82
4.01
E2E
2.30 0.48
2,80 + 0.00
3.50
0.04
4.17 #: 0.00
GradNorm
1.10 0.03
4.11 0.30
E2E + PeGrad
1.54 0.53
2.80 + 0.00
3.84.1...0
1: 0.05
4.16 :: 0.08
MH
2.08 0.00
2,80 + 0.00
3.15 0.00
4.16+0.0
MH GradNorm
1.62
4.08
0.32
MH + PeGrad
1.22 1-22++0.. 0.100
2,80 + 0.00
4.19++0.22
4.12 0.02
MM
2.78 2: 0.00
2,80 0.00
4.40 0.02
4,40
4.16
MM COMs
2.30 0.00
2.21 + 0.17
4.14
4.12 + 0.05
MM RoMA
1.65 0.00
2,80 + 0.00
0.29
4.16
MM
MM + ICT
1.37 0.17
2,80 + 0.00
4.41
4.10 0.07
MM + Tri Mentor
2.03 2.03::0.00
2,80 +. 0.00
4.12
4.06 0.01
MOEA/D MM
1.47 0.09
3.96
+ 0.15
4.52 0.05
MOBO
0:008
5.12 I 0.17
3.74
0.00
4.26 = 0.00
MOBO-gParEROO
3.33
MOBO-JES
2.10
N/A
N/A
NIA
PROUD
0.45
3,26 = 0.00
1.225
137 0.03
LaMBO-2
2.18
0.22
4.3
CorVAE
1.71
3.26
+ 0.00
4.19
4300
MOGEN
1.76 0.07
3,26 - 0.00
0.02
4.36 1.02
PuretoFlow (ours)
2.91 0.11
3,96
0.00
4.49 0.06
Table 13: Hypervolume results for RE.
20.3
233
2311
121.0:0
18100
18:0
138
158:
430
$00.1:00
1841-40 641
1532 431
49 15.
300
10440.0
124.00
1487 120
153
-4.35
13.00
300 38.480
150
01 11-1121
1020400
1191000
11:00
100
112
18:00 10
155 115
28:12
10:
A.7 50TH PERCENTILE RESULTS
We evaluate the 50th percentile performance of 256 solutions. As shown in Table 14 our method achieves the highest overall ranking based the 50th percentile results. For each task, algorithms with performance within one standard deviation of the best are bolded. Detailed hypervolume
results presented in Tables 15
16
17
18
19 and 20
22

---

Published as a conference paper at ICLR 2025
Table 14: Average rank of 50th percentile results on each type of task in Off-MOO-Bench.
Methods
Synthetic
MO-NAS
MORL
Sci-Design
RE
All Tasks
D(best)
13.18 + 4.86
9.26 = 4.93
5.00 + 3.00
6,00 = 2.92
14.06 + 4.74
11.15 + 5.48
E2E
12.64 1 8.04
5.63 1 3.33
14.50 1 3.50
13.50 1 3.50
8.75 1 5.95
9.02 1 6.30
E2E + GradNorm
14,36 + 6.65
15.42 + 4.97
9.50 + 0.50
11.75 + 1.30
12.88
+ 6.28
13.90 + 5.74
E2E + PcGrud
11.64 6.76 6.76
6,58 + 4.11
16.50 = 1.50
10.00 + 4.85
10.25
+ 6.78
MH
9.73
1.84::4.88
8.50 1 5.50
10.00
4.64
11.25
1: 6.08
8.75 6.35
MH + GradNorm
12.45:7.33
17.26 4.73
16.00 4.00
16.75
+ 3.70
16.75 + 5.31
16.00 - 5.78
MH PeGrad
11.09:44.5
8.05
10.00
16.00
4 3.24
12,00 +: 5.39
10.60 5.03
MM
4.91
6.32
15.00 2.00
13.75
3.11
6.75 - 4.58
7.06 + 4.95
MM+ + COMs
12.91
10.32 5
13.50 1.50
11.00
3.61
14.06 1: 4.31
12.19 4.84
MM + RoMA
13.82 6.58
5.53 4.25
11.50 4.50
11.25
3.56
11.88 + 5.18
9.90 6.12
MM+ IOM
3.19
6.16
12.00 3.00
11.75
3.56
8.81 +: 4.88
7.63 4.58
MM + ICT
14.64
10.58 4.43
11.00 + 6.00
13.75 + 3.83
12.31 + 6.65
12.23 + 5.49
MM + Tri-Mentor
10.91 5.60
11.68 = 4.38
12.00 = 0.00
13,00 + 4.85
9.69 2 6.06
11.02 + 5.27
MOEA/D + MM
5.400
10.79 1 6.39
13.00 1 7.00
10.25 1 6.76
9.50 1 5.99
10.00 1 6:20
MOBO
12.73 1: 5.69
15.74 1.77
18.50 = 0.50
11.00 1 6.04
14.88 6.45
14.58 1: 5.18
MOBO-jPaFFGO
12.91 + 4.01
17.37 3.42
21.00 + 0.00
10.25 + 6.30
18.94 + 3.65
16.50 + 4.84
MOBO-JES
17.00 + 3.52
21.32 = 2.90
21.00 = 0.00
21.50 + 0.87
16.12 + 4.44
18.81 + 4.22
PROUD
9.09 + 4.76
14.05 = 3.41
4.50 = 2.50
3.75 = 1.92
6.62 = 5.58
9.56 = 5.73
LaMBO-2
9.73 5.12
13.37 1: 4.29
4.00 + 0.00
4.50 1: 1.80
7.69 - 6.03
9.81 1 5.76
VAL
11.91 + 5.25
17.42 3.76
6,50 + 1.50
5.75 + 2.86
11.19 +: 5.50
13.02 + 5.92
MOGEN
9.55 5.35
15.47
+ 4.11
4.50 + 1.50
4.00
+ 3.08
8.44 + 5.07
10.75 + 6.01
PlretoFlowoour>
5.82 1 3.76
6.63 1 3.73
1.00
0.00
2.50
0.87
3.12 1 3.76
4.85 3.97
Table 15: Hypervolume results for synthetic functions.
Mehob
DTT.ZI
DTL.TT
OreeTo:
VLMOPI
VI.MPP2
VLMOPS
ZDTTL
2DT2
ZDT3
ZDT4
ZDT6
10:0 6.00
6:37 c-0.7
4:35
0/10 13003
115 1.11
95
2:10 1.50
3:11 1-0.00
3-16
3.11 6.3/3
1.87 4.
E GradNian
10.65 = 0.00
542 2,15
2.52 11.0
8000:::000 2/00
2407 2:1 1.21
22.31
11.17
6.70 1.00
4.01 2:0.1
5.27 = 11.1
5,00 2 5002
2,05 2.22
0.10
0.00
5:00
4:6
2.10
10.81 cume
4:50 1:1--115
0:55
115
96.33
0.00
1:97
4.30 0.00
MIT CootVoor
10:0 0.00
p.5 c:33
11:00
0100 9.00
105 6m
19.50 05
1:3 1:333
411 1.00
1:51 0.00
10 0.00
10:3: m.
4:38 8::0
1:47 1:47
3300 1:31
05.23 1.56
8:10 11:0
5:26 8.0
3.09
300 11155
MM COM-
10:00 0.04
6.0
2.552:035 0.31
0000 0:00
Los 22
00.2
8.52 0.10
4.55 0-11
3.44 0.07
4.3 005
XM4 RaMA
10.0 2
10.01 = 2.16
2:00 10.00
3.00
1:0020000
01.08 1..%%
4.90 0.01
1-02--00.0
2:57 005
1
MAS ITT
0.00
6:11 c.15
310
Bu16 5.55
03.50
2.56
0:32
3.77 105
SI0 Ta-Maaku
1001 0.00
0.75 6.00
8:50 10:0
9.73 0.07
1.00 2:00
91.0 0.88
4.58 RUG
1.53 3.83--00.01
3.45
8 0:22
2.00
MOEADD
0.00
10.90
0.00
0.31
4.01
4.44 0.04
1:20+-0.00
0.00
4.57
0.00
X00BO
MCBO-PPaBBO
10:00-200 0.00
0.65 0.14
3.10=000 0.09
0.00 9:00--33.0
1.00 0.10
06.82
2A:2==000 0.06
6.11 0E6
4.85 2 2.005
4.0 8.11
4.31 D67
2 0.06
10:06 0.30
6:25 C/10
3:36 8000
1.39 10301
SAT IND4
1.0.87
4.72 0:10
3,77
1,87 11.00
LaMB3O-3
1030 0.066
6.30
4.77
2.78
440 1.00
01.17
1:00
1:6.80
0.34
4.24
0.55 1UN
CorViAL
10.6 2 0.30
5.43 C
4.76 11:0
2 0.53
20100 1.35
1.11 I00
1:x2 c.1.3
4.7 0:20
1.36 2u82
1:36 1IM
9:01
0.06
C.13
4.81 0.00
0.000
0.78
Table 16: Hypervolume results for MO-NAS (Part 1).
Mihhake
C-NMMPP
C-HUMIF
C-16MPP3
C-IIMIP
C-I0MOPS
C-RMMOI
C-RMMPP
C-IMMPS
C-IMMIP
Piboon)
4.72
10.02
03.79
105.55
300.57
4.38
9:61
0.15
0.05
18:17 0.11-
102.55+
498
1.11
-1.00
9.85 0.03
CinalNoem
0.01
0.08
0.77
50.25 7.18
(76.1
8.30
3.56
11.1
8:34 0.66
10:42 0.00
45:11
104 0.05
0.00
9.55
0.04
101.411 2.05
MB PeGrad
4.71
10.19
0.11
20.20
101.07 1.71
2.45
9.75 0.00
10.07 0.00
0.00
19.96
0.00
100.04 0.40
481.21
0.000
0.00
COM
10:35 0.000
0.00
11:00
45.41
100.22 0.07
406.90
11:0
9.57 0.00
MM 10M
1.00-22 2 0.02
10:45:t:00 (0.05:00..00 0.00
9.00
0.00
20.00
0.07
15.05 8- 0.00
162 0.78
151.43
0.00
4.54 0.00
9.80 0.00
MM RoMA
4.73 0.00
10.42:2:0.00
10.01 0.00
20.06
0.05
47.64 2 0.00
161
1.65
472.22
0.00
4.16
0.00
9.02 0.04
MM ICT
1G5 =: 0.04
10.15:2::.00
5.48 D.60
18.55
0.00
45.80 2: 0.00
164.22 à 0.31
47L.66
0.00
4.26 = 0.00
9.16 0.00
MM + Ti--irttr
4.09 8 0.30
10.02-4 0::00-000:0)
0.445 0.00
19.77
0.211
38 90
0.00)
1020T 8 0.
485.00 0T 6.15
0.24 002
8.22 0.30
MORAIT- MM
4.73 + 0.05
0.00
9.53
0.00
20.24
0.54
17 56 0.06
101 33 C104
470.63
3:36
4.12
0.0
S + 0.66
MTDRR
1.50 7 0770
10 0.00
8:30 000
15:73
0:16
10 0..07
T00AS 7.81
182.01
01:33
1.11 1000
7:00 0.0
M0B0-
4.53 = 0.07
6:30 +
0.19 8.45
0.44 19.62
37.17 + 0.00
01.58 5.95
357. 57 8.91
4.15 = 0.06
8.25 0.16
MOBO-IES
NIA
NiA
NIA
N/
N/A
NIA
N/A
NIA
N/
PROUD
10000
2 0.00
8.50 A.10
17.96
0.50
Ja0N 25 2.82
IMEDY 2.10
406.12
0:32
4.00 = 1.10
5.03 O.AY
LMMB0-
1.66
10:01
0.00
8:37
0.10
02
033
45 7.23
100 18 2.00
654
1.35
0:00
000
774 0.51
DorVAR
4.55
10:39-
0.03
8.12
0.1
10.75
0.99
11
90.71 0.91
158.3
7.19
3.87
0.10
611 0.11
MOGEN
1.00 0.05
0,00
8.20
0.14
16.92 0.07
15
100
112.90
6.79
3,00
011
8.25 0.22
Panes Flw (eurs)
4.74
10.47 0.01
9:32
0.1
37
105.70
5.14
4.57
000
5006
23

---

Published as a conference paper at ICLR 2025
Table 17: Hypervolume results for MO-NAS (Part 2).
Mrrhish
IN-KMMPPI
PN-IKMPPS
IN-TRMOPS
IN-TRINIPP
IN-RRMMOPS
IN-KIMON
IN-RAMMPT
IN-IKMMOPS
INIKMMOM
Nasterchol---rr
ESB
4:5
0:5-2:0:12
1.50=:0.00
0::7
6.4=0.11
228 CralNien
100 000-2000
4.30:200 0.81
1:3 2.122
610 11.05
5.94 1: 0139
1.08
5:49 119-2-0.1
0.2.1
10.55 2.035
5:41
EJE IXGma
1.45 0.03
1.35 0.11
9:88 6.0,2
6.15 1.12
5:01 0.01
1.45 0.06
3,66 0662
9.20 0.05
10.31 2.030
9383 0.41
MH
4.50
4.45-210 -::-0.10
5.0.8
4.57 0.00
0.06
0.1-
0.00
30.00 1:013
Petina
1:41
4.50 0.00
L.3 6-0.5
0.13
19.31 +.0.5
XBM
1.52 2 0.00
4.44 0.82
9.32 0.00
4.45 0.00
4.22:::000
4.00 0.15
9.45 0,02
10.08 2.0.3
2 0005
XISC CORds
4.37 0.01
4.21 0.05
7.84 2.04
4.23 EUB
6:83 0.02
0.81 0.32
5.70 0..4
3-40-2000 0.94
10.01 2.0.3
5/01 0.49
RodGA
0.00
0.0.4
MM - RET
1.10
4:30 0.00
8.30 BUG
0.00
0.33
18.71 0.88
5-30 0.11
MM Th-Menter
1.37 0,05
4:38 0181
9:73 00
8.11 0.00
1.28 I.60
8.00
0.98
3.97 0.62
9:15 0.17
11.81 1.20
6.75 0,83
MOBAD BVD MM
4.35
118-2000 0.00
3.46 0,07
0,67 0.30
4.27 0LB4
6.87
0.73
0.23
5.67 0.00
7.60=2000 0.14
18:93
+-0.66
8.81 0,07
MOBO
103 0001
2.05
1538
MOBOJIES
NIA
NGA
N6A
NiA
NAA
NIA
NA
NIA
N6A
5-06 0.15
PROCUD
0110
011:00011
2001 1130
310 1T0
7.53 0.35
16.11 2:
8:70
LAMBO-3
1.38 0.00
4:19 0.02
0.0
0.10
8.64 9.84
5.75
0.446
16.41 1:33
0.40
137
1.030
Table 18: Hypervolume results for MORL.
Methods
MO-Hopper
MO-Swimmer
(hhest)
4.21
2.85
E2E
3.68 0.00
2.04 + 0.10
B2E
GradNorm
3.91
E2E
PoGrad
3.72 = 0.01
1.90 + 0.05
MH
MH + GradNorm
3.67 - 0.00
1.98 0.12
PeGrad
MM
3.76
1.91 0.02
MM
MM RoMA
4.00
1.95 1 0.06
MM
MM + ICT
1.701.0001
2.38 1 0.11
MM Tii-Mentor
3,82
MOEA/D MM
4.75
0.28
0 1 0.19
MOBO
0.00
MOBO-JPuFEG
NIA
NIA
PROUD
4.84
2.32 0.24
LaMBO-2
4.77
2.41 0.21
ConVAE
4.76 0.01
2.23 1: 0.20
MOGEN
1.75 0.03
2.35 = 0.21
ParetoFlow
5.56 0.001
2.93 0.09
Table 19: Hypervolume results for scientific design.
Methods
Molecule
Rogex
REP
ZINC
D(best)
2.26
3.05
3.75
1,06
E2E
1.07 0.07
2.05 + 0.00
3.64
0.05
3.95 + 0.04
E2B GradNNrm
1.07 0.07
-
3.00
3.92 0.00
F2F + PeGrad
2.12 + 0.04
2.05 + 0.00
3.70 + 0.05
3.89 0.06
2.05
3.74
3.%
MH + GradNorm
1.00 0.00
2.05 0.00
3.69 0.01
3.82
MH + PeGrad
1.00 0.00
2.05 0.00
3.68 0.02
0.00
MM
0.09
0.00
3.00
3.81
MM COMs
0.14
2,38 0.33
0.00
3.86 0.02
MM RoMA
0.00
2.05 0.00
0.09
0.02
MM + IOM
1.02 002:000 0.01
2.05 0.00
3.76 0.03
3.91 0.02
MM + ICT
1.02 0.02
2.05 0.00
3.67 0.000
3 0.07
MM Tri-Mentor
0.17
3.00
MOEA/D + MM
1.47 0.09
2,99 + 0.00
3.62
4.52 = 0.05
MOBO
1.02 0.02
3.42 + 0.25
3.70
0.01
3.90 0.01
MOBO-pPurEGO
1.96 0.12
3.17 + 0.11
3.33 0.00
4.00 0.03
MOBO-EE JES
1.00 0.00
NA
N/A
N/A
PROUD
1.67 0.166
3.26
0.00
4.1.
4.26 0.22
LaMBO-2
1.67 0.166
3.26
0.00
4.09 0.118
4.17 0.28
CorVAR
1.58 0.02
3.26
0.00
4.07 0.07
4.09 0.18
MOGEN
ParctoFlow (ours)
1.09
0992200 0.09
3.26 = 0.00
1.00
4.43 0.04 0.04
24

---

0.8 Published as a conference paper at ICLR 2025
Table 20: Hypervolume results for RE.
530
110
2112
420
18:1 5800
500 124
0.391 3.58
141-180
221
530 6:1
13-0005
18:03 6,80
213-48-5
18:54437
121
441
201 610
-21-20
386
281-00
553
13
10-13
1414
152
115 4:2
11 1146-944
153
1101-12
111
613
41
611
11
A.8 VISUALIZATIONS AND CASE STUDY DETAILS
provide C-10/MOP1 and MO-Hopper visualization results in Figure 11 parisons between offine samples and samples generated by ParetoFlow, clearly demonstrating the superior quality of the latter.
This features com-
Illustration C-10/MOP1
Illustration on MO-Hopper
0.9
Offline Samples
Offline Samples Samples
Gen. Samples
N.
Objective Function
Objective Function
0.7
0.4
0.2
0.5
00
0.0
0.0
0.2
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Objective Function f
Figure 11: Illustrations of ParetoFlow on two tasks C-10/MOPI and MO-Hopper.
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Figure 12: C-10/MOPS case study: (1) samples prioritizing prediction error and model complexity; (2) samples focusing on prediction error and hardware efficiency; (3) samples emphasizing predic- tion error, model complexity, and hardware efficiency.
We have conducted a detailed case study on C- C-10/MOP5, focusing on optimizing prediction error, model complexity, and hardware efficiency, outlined in Lu et al. 2003). The case study analyzes
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three sets of solutions as detailed in Figure where the weight vector for unconsidered objectives is set to zero. We discuss the frequency of operators used in these sets, noting that the 3x3 convolution is consistently preferred across three sets for its effectiveness in reducing prediction error, while the 3x3 average pooling is less favored. Additionally, there a notable shift from the use of 1x1 convolutions to 'none' operators in moving from the first the second set, suggesting trade-off for better hardware efficiency. This analysis provides insights into the structural preferences and performance trade-off's in the generated architectures.
A.9 EFFECTIVENESS OF LOCAL FILTERING
To further verify the effectiveness of local filtering, conduct experiments the convex-PF task ZDTI and the nonconvex-PF task ZDT2. When we remove the local filtering, the performance of ZDTI nearly does change: from 4.30 + 0.02 to 4.29 + 0.04. In contrast, the performance on ZDT2 drops obviously: from 6.79 = 0.16 to 5. 5.78 + 0.15. This demonstrates the effectiveness of our local filtering scheme in handling nonconvex PFs and verifies its underlying motivation.
A. FURTHER ABLATIONS
To substantiate the advantages of flow matching over diffusion models, we replace flow matching in our ParetoFlow framework with diffusion model SSnng et al. (202T) and conduct comparisons on two tasks: MO-Hopper and C- C-10/MOPI. The results Table consistently demonstrate the superior performance of flow matching in our context.
Table 21: Comparison between flow matching and diffusion models
Methods
C-10/MOP1
MO-Hopper
ParetoFlow w/ Diffusion
4.65 + 0.05
5.54 + 0.07
ParetoFlow (ours)
4.77 + 0.00
5.69 = 0.03
We further compare our Das-Deniss with another weight generation strategies. The Das-Dennis method is widely used in multi-objective optimization studies due to its simplicity and ease of use, as it does not require optimization. However, a limitation of this method is that does not allow for specifying exact number of weights. On the other hand, the Riesz s-Energy method Hardin & Saff (205)) allows for precise control over the number of weights generated. However, this approach involves optimization process, making it more complex implement. We conduct experiments on C-10/MOP1 and MO-Hopper using both strategies, and find that their performance quite close,
as shown in Table 22
Table 22: Comparison between Das-Dennis and Riesz s-Energy
Methods
C-10/MOP1
MO-Hopper
ParetoFlow w/ Riesz s-Energy
4.77 + 0.00
5.55 + 0.10
ParetoFlow (ours)
4.77 + 0.00
5.69 + 0.03
A.11 SENSITIVITY TO THE NUMBER OF SAMPLING STEPS
As shown in Figure 10 our method is robust to changes in the number of sampling steps T.
A.12 RELATIONSHIP BETWEEN EVOLUTIONARY ALGORITHMS AND FLOW MATCHING
We further discuss the relationship between evolutionary algorithms(EA) and flow matching. In our flow sampling process, each intermediate noisy sample æ; be mapped to clean sample (æ,)) This mapping bridges flow matching and EA, where flow matching handles æ, and EA operates on ær Specifically, in our algorithm, the weighted predictor fw (ă, (ax:) to select promising #;
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for the next iteration. This predictor selection, originally part of EA and applied to æı, is integrated into flow matching through its application to 24. This relationship demonstrates the integration of EA and flow matching.
To illustrate the efficacy of integrating EA with flow matching, consider the performance of each component in isolation. Removing flow matching from framework leaves solely with EA. Table I demonstrates that the EA method NSGA-II, performs worse than our integrated approach, highlighting the value added by flow matching. Conversely, excluding EA results system where flow matching operates on uniformly weighted objectives for guided sampling, but lacks the crucial selection and local filtering processes. Our experiments on tasks like MO-Hopper and C- C-10/MOPI show that this configuration leads to inferior hypervolume (HV) results, as depicted in Table 23 further validating the significance of combining both strategies:
Table 23: ParetoFlow w/o EA
Methods
C-10/MOP1
MO-Hopper
ParctoFlow w/o EA
4.53 = 0.00
5.06 + 0,00
ParetoFlow (ours)
4.77 = 0.00
5.69 0.03
This comparative analysis clearly supports the effectiveness our integrated method, demonstrating that each component contributes significantly to the overall performance.
A.13 ETHICS STATEMENT AND LIMITATIONS
Ethics Statement. Our method, ParetoFlow, holds promise for accelerating advancements in new materials, biomedical developments, and robotic technologies by simultancously optimizing mul- tiple desired properties. Such advancements could drive significant progress in these fields. How- ever, like any powerful tool, ParetoFlow also carries risks of misuse. A potential concern is the application of this technology in designing systems devices for malevolent purposes. For exam- ple, inappropriately used, the optimization capabilities could aid in developing more effective and energy-efficient robotic weaponry. It is, therefore, imperative establish robust safeguards and strict regulations to control the application of such technologies, especially in critical sectors.
Limitation. While method shows considerable promise, its effectiveness heavily relies the accuracy of the underlying predictive models. In highly complex applications such as protein se- quence design Ferruz intricately linked, simple predictive models may fall short in capturing these complexities, leading to suboptimal performance. Consequently, speciffe strategies may be essential for accurately modeling such complex scenarios. For instance, employing advanced protein models Lin et al. (2023); CChnn et al. (2023c) could enhance the modeling of protein sequences. Future research should consider integrating domain-specific insights into the predictor modeling, thus improving the method's 's ability to handle complex challenges more effectively.
(2022); Chen et al. (2023b 2022). where amino acid interactions are
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Reviews:
[{"review_id": "sYEl3UD2g6", "reviewer": "Reviewer_YcyJ", "summary": "This paper presents *ParetoFlow*, an innovative approach aimed at offline multi-objective optimization (MOO) that leverages the flow-matching generative modeling framework. The authors introduce two key modules: a *multi-objective predictor guidance module* that uses uniform weight vectors to direct sample generation toward approximating the Pareto front and a *neighboring evolution module* for sharing knowledge between distributions with similar objectives. These techniques are designed to improve coverage of both convex and non-convex Pareto fronts and enhance sample diversity.", "questions": "1. How does ParetoFlow manage potential overfitting to specific tasks or datasets, given that it optimizes multiple objectives using a learned generative model?\n\n2. Can the authors elaborate on the choice of the Das-Dennis method for generating uniform weights? Are there scenarios where alternative weight distribution methods could be more effective?\n\nI am open to revising my score upwards if the authors can satisfactorily address these concerns during the discussion phase.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The integration of flow matching in offline MOO is a fresh perspective compared to the typical use of evolutionary algorithms or Bayesian optimization. This method captures the potential of generative models for addressing complex, multi-faceted optimization challenges.\n\n- The proposed module that fosters information exchange among adjacent distributions is a standout feature. This knowledge-sharing mechanism supports enhanced exploration and refinement of solutions.\n\n- This method was evaluated across diverse benchmarks, demonstrating state-of-the-art performance and outperforming existing MOO methods such as PROUD, LaMBO-2, and CorrVAE, especially in generating high-quality Pareto-optimal solutions.\n\n- The provided code and training details are clear.", "weaknesses": "- You only cited [1] to state that Flow matching outperforms Diffusion models, which I find insufficient. This is especially true in the context of multi-objective optimization, where specific problems require specialized analysis.\n\n- Although the results on benchmarks with high-dimensional objectives, such as MO-NAS, are mentioned, there could be more detailed insights into how well the method scales with increased complexity in terms of computational cost and solution quality.\n\n- The comparisons focus more on generative modeling approaches. A broader comparison that includes more traditional, non-generative MOO techniques could strengthen the context of ParetoFlow's advantages and limitations.\n\n[1] Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oo0K6V9n9Y", "reviewer": "Reviewer_4YDk", "summary": "This paper proposes ParetoFlow, a novel method for offline multi-objective optimization (MOO) using flow matching. It introduces a multi-objective predictor guidance module and a neighboring evolution module to improve the efficiency and effectiveness of flow sampling in approximating the Pareto front. The paper presents extensive experimental results on various benchmarks, demonstrating ParetoFlow’s superior performance compared to existing MOO methods.", "questions": "1. Can the authors offer a case study for the generated architectures in terms of structure and performance on MO-NAS?\n2. How to guarantee the validity of the generated samples? For instance, an architecture with the valid computational flow in the NAS task.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The paper is well-written and easy to follow.\n2. The proposed modules, multi-objective predictor guidance and neighboring evolution, provide a well-motivated approach to guiding flow sampling and promoting knowledge sharing among neighboring distributions.\n3. The experimental results of ParetoFlow on various benchmarks are promising.", "weaknesses": "1. The paper mentions the computational cost of ParetoFlow but does not provide a detailed comparison with other generative modeling methods.\n2. Providing some visualization results for the generated samples would be beneficial.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UN3EEpoyLr", "reviewer": "Reviewer_Cej3", "summary": "This paper propose a new MOO method.", "questions": "see advantages and weakness.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. **The title is engaging and draws interest.**\n\n2. **Overall, the paper is well-written and easy to follow.**\n\n3. **Section 2.2 is clearly explained. However, could you elaborate on how using a normalizing flow would alter or improve the results?**", "weaknesses": "1. **Insufficient Mathematical Rigor**: The paper lacks adequate mathematical depth and rigor. Incorporating more detailed mathematical formulations, proofs, and theoretical underpinnings would strengthen the credibility and scholarly value of the work.\n\n2. **Incorrect Formalization in Line 105**: The statement on line 105 is inaccurate. A Multi-Objective Optimization (MOO) problem cannot be \"formally\" formulated merely as a vector optimization problem. Such a formulation is more of an informal notational convenience rather than a formal definition. Clarification and correct formalism are needed here.\n\n3. **Unclear Purpose of Equation (9)**: The role and application of Equation (9) are not clearly explained. It is difficult to understand its significance and how it fits into the overall argument of the paper. Providing a thorough explanation or context for this equation would enhance comprehension.\n\n4. **Unconvincing Integration of EA and Flow Matching**: The combination of Evolutionary Algorithms (EA) and flow matching presented in the paper does not seem convincing. Additional justification, evidence, or examples demonstrating the effectiveness of this integration would help in substantiating this approach.\n\n5. **Inappropriateness of Section 4.6 in Main Text**: Section 4.6 may not be suitable for inclusion in the main body of the paper. It might be more appropriate to place this section in an appendix or supplementary material to maintain the focus and flow of the main content.\n\n6. **Unconvincing Experimental Results Using Hypervolume Metric**: The experiments conducted are not entirely convincing, particularly due to the exclusive use of the hypervolume (HV) metric. If alternative methods such as SMS-MOEA were employed, it is questionable whether the proposed method would perform as well as, or better than, SMS-MOEA. Including comparisons with such methods and using a variety of performance metrics could provide a more comprehensive evaluation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1d8lYti3Yx", "reviewer": "Reviewer_FugU", "summary": "This paper applies flow matching to offline multi-objective optimization. Authors propose a multi-objective predictor guidance module which generates weigts for each objective using Das-Dennis approach to guide the sample generation towards the Pareto front, and neighboring evolution module is proposed to enhance knowledge sharing between the generation of different weights. Experimental results show the effectiveness of the proposed method.", "questions": "1. In Section 3.1 authors consider the case of non convex PF. I wonder whether such cases show up in the experiemnts? If so, does the designed filtering successfully process such cases?\n\n2. In the Neighboring Update, the next state is the best one in the K nearest weight vectors, will it make a large ratio of vectors generate the same solution and lead to early convergence?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "1. The concept of ParetoFlow is innovative and creative.\n\n2. The experimental results show that the proposed method has promising performance on the tested problems and surpass the baselines.", "weaknesses": "1. The proposed method relies on the accuracy of the objective predictors, which might be a bottleneck on problems with complex landscapes and limited offline dataset.\n\n2. It seems that the required weight vectors will grow exponentially (i.e., a problem with 10 objectives neesds at least 1024 vectors to ensure each objective has at least 2 weights). It may be time consuming to generate and sample using these weights, while less weights may not fill the PF.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)