Pareto-Conditioned Diffusion Models for Offline Multi-Objective Optimization
Authors: Jatan Shrestha
Santeri Heiskanen
Kari Hepola
Severi Rissanen
Pekka Jääskeläinen
Joni Pajarinen
Venue: ICLR 2026 Oral (2026)
Keywords: Multi-Objective Optimization
Conditional Diffusion Models
Primary area: generative models
TL;DR: We propose Pareto-Conditioned Diffusion (PCD), a novel framework for Offline Multi-Objective Optimization

Abstract: Multi-objective optimization (MOO) arises in many real-world applications where trade-offs between competing objectives must be carefully balanced. In the offline setting, where only a static dataset is available, the main challenge is generalizing beyond observed data. We introduce Pareto-Conditioned Diffusion (PCD), a novel framework that formulates offline MOO as a conditional sampling problem. By conditioning directly on desired trade-offs, PCD avoids the need for explicit surrogate models. To effectively explore the Pareto front, PCD employs a reweighting strategy that focuses on high-performing samples and a reference-direction mechanism to guide sampling towards novel, promising regions beyond the training data. Experiments on standard offline MOO benchmarks show that PCD achieves highly competitive performance and, importantly, demonstrates greater consistency across diverse tasks than existing offline MOO approaches.

Published as a conference paper at ICLR 2026
PARETO-CONDITIONED DIFFUSION MODELS FOR OFFLINE MULTI-OBIECTIVE OPTIMIZATION
Jatan Shrestha* Aalto University
Santeri Heiskanen"
Kari Hepola Tampere University
Aalto University
Severi Rissanen
Pekka Jääskeläinen Tampere University
Joni Pajarinen Aalto University
Aalto University
ABSTRACT
Multi-objective optimization (MOO) arises in many real-world applications where trade-offs between competing objectives must be carefully balanced. In the offline setting, where only a static dataset is available, the main challenge is generalizing beyond observed data. We introduce pureto-Conditioned Diffusion (PCD), novel framework that formulates offline MOO as a conditional sampling problem. By conditioning directly on desired trade-offs, PCD avoids the need for explicit surro- gate models. To effectively explore the Pareto front, PCD employs a reweighting strategy that focuses on per--performin samples and a reference-directio mech- anism to guide sampling towards novel, promising regions beyond the training data. Experiments on standard offline MOO benchmarks show that PCD achieves highly competitive performance and, importantly, demonstrates greater consis- tency across diverse tasks than existing offline MOO approaches.
INTRODUCTION
Multi-objective optimization (MOO) tackles real-world problems involving conflicting objectives (Miettinen 1998 Deb 200T). Such scenarios arise in domains from scientific discovery engi- neering, where performance must be balanced against cost and energy. In many of these domains, evaluating candidate designs is prohibitively expensive, time-consuming, even risky. For instance, in biological sequence design, each experiment or simulation incurs substantial cost (Angermueller al. 2020). In such settings, online optimization methods, which rely on repeatedly querying a back--bx box objective function, are often impractical (Trabucco et al. (002). This challenge has moti- vated the development of offline MOO, which instead relies on a pre-collected dataset of previously evaluated designs, without access to the true objective during optimization (Xue et al. 2024).
black-
The offline setting presents unique and significant challenges. Unlike online methods, algorithms cannot query the true objective function during training and are restricted to static dataset (Tra-) bucco et al. 2022). This constraint introduces a fundamental tradeoff as the algorithm must remain faithful to the data avoid unreliable predictions, yet also explore beyond the observed designs to identify superior solutions (Trabucco et al. 2022). Moreover, offline MOO requires finding a set of solutions that balances multiple objectives, rather than single optimum (Xue et al. 2024). This leads to a particularly difficult form of generalization: algorithms must be creative enough to find solutions that improve upon the training data, while being conservative enough stay within the reliable boundaries of the training data manifold (Trabucco et al.) 2022 Xue et al. 2024).
The most dominant approach offline MOO is model-based optimization (Kim et al. 2026). typ- ically using a surrogate model (Trabucco et al. [2021] Yu et al. 2021 Yuan et al. 2023b Qi et al.) 2022 Chen al. 2023 Daulton al. 2020; 2021). The surrogate trained on the static dataset to approximate the black- box objective and then used with multi-objective evolutionary algorithms (MOEAs) (Deb) 2001 Deb et al. (2022) to search for promising solutions. More recent approaches use generative models to directly synthesize solutions (Kim et al. 2023). For instance, ParetoFlow
Equal contribution. Correspondence to: jatan, sheeannshreetthaaaallt... eanteri. hei..sseeeaneeeaat....tt
Project website: http:///ttesgooglll..om//////ee/

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Offline Training
Sampling
Static Multi-Objective Dataset
D
Noise
Condition
Reweighted
D
Target Objectives
Pareto-Conditioned Diffusion Model
Train
Sample
Pareto-Conditioned Diffusion Model
Designs
Figure Overview of the PCD framework, which reframes offline MOO a conditional sampling problem. Training: A conditional diffusion model is trained on static dataset, using novel reweighting strategy emphasize high-quality solutions near the Pareto front. Sampling: At inference, the model directly generates novel designs conditioned on target objectives. This end-to- end approach sidesteps the need for the explicit surrogate models and separate optimizers required by prior methods.
Yuan et al. 2025) employs flow matching-basse generative models guided by a multi-objective predictor to generate samples along the Pareto front. However, these methods still rely on surrogate predictors to guide the generation process, inheriting the same potential risks of exploiting inaccu- racies in the surrogate model (Trabucco al. 202I).
Contribution. In this work, we propose Tareto-Conditions Diffusion (PCD), a novel framework that reframes offline MOO as a conditional sampling problem, enabling the generation of high- target trade-off's (see Fig. D Unlike prior methods (Xue) ct al. 2024; Yuan et al. 2025) that rely on multi-stage pipelines surrogate predictors or separate optimization algorithms, PCD offers more direct, end-to-end framework. It unifies the generation of candidate solutions and the modeling of the Pareto front into single conditional generative model, simplifying the overall optimization process without requiring explicit objective function approximation.
quality solutions conditioned directly
To enable this framework, introduce two key algorithmic contributions: (1) multi-objective reweighting strategy that biases the model to focus on high-performing regions of the data distri- bution, and (2) reference-direction mechanism generate diverse and high-quality conditioning points that enable generalization beyond the observed dataset. We provide extensive empirical val- idation on a wide range of standard offline MOO benchmarks. These experiments demonstrate that PCD achieves highly competitive performance and, more importantly, exhibits significantly greater consistency across wide variety tasks than existing offline MOO approaches. The effectiveness of our proposed reweighting and conditioning mechanisms is confirmed through rigorous ablation studies.
RELATED WORK
Offline Multi-Objective Optimization. Prior work in offline MOO has primarily focused on model-based approaches (Xue et al.) 2024 Kim et al.] 2026). These methods train surrogate model on a static dataset to approximate the black-box objective function. Surrogates can take the form of Deep Neural Networks (DNNs) (Trauucco et al. 2021 Yu et al. 2021 Qi et al. 2022 Yuan jet al., 2023b Chen et al. 2023). which scale well to high-dimensional search spaces, or Gaussian Processes (GPs) (Daulton al. 2020 202T), which provide uncertainty estimates useful for explo-
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ration. Once the surrogate is trained, separate search algorithm, most commonly MOEAs such as NSGA-II (Deb) 2001 Deb et al. 2002). is used to optimize the learned proxy to find promising so- lutions (Xue al. 2024). The performance of this multi-stage pipeline, however, is fundamentally bottlenecked by the accuracy of the surrogate model (Trabucco et al. 2021).
Generative Modeling Methods. More recent approaches use generative models to directly syn- thesize solutions (Kim al. 2023). The primary advantage these methods their ability to gen- erate diverse designs without relying on a separate, explicit search algorithm (Kim et al. (026)) For instance, ParetoFlow (Yuan et al. 2025) and PGD-MOO (Annadani et al. (025) employ generative models guided by multi-objective predictor. Similarly, LaMBO (Stanton et al. 2022) integrates
Bayesian optimization to steer its diffusion model, while MOGFNs (Jain et al. 2023) utilize gen-
erative flow networks often steered by scalarization functions. However, the effectiveness of these
methods is still dependent on surrogate predictor or a scalarization function to guide the generator Annadani al. 2025).
Yuan et al.
2025
In contrast, our method, PCD, removes this dependency on intermediate guidance mechanisms like explicit surrogate models scalarization. PCD frames offiin MOO conditional sampling problem, generating high-quality solutions by conditioning the diffusion model directly on target objectives. doing so, PCD extends prior work on single-objective conditional generation, such as Model Inversion Networks (MINs) (Kumar & Levine 2020) and Denoising Diffusion Optimization Models (DDOM) (Krishnamoorthy et al., (023), the more complex multi-objective setting, Our work is also distinct from reward-directe diffusion models (Yuan et al.) 2023a), which focus on maximizing single reward signal rather than balancing a set conflicting objectives.
3 BACKGROUND
Offiine Multi-Objective Optimization. Let f : - R" be veltur-valued function that maps solution from the solution space X' to the m-diimensional objective space R". A multi-objective optimization (MOO) problem can be formally defined as
min (xx = [fi fm))) #EA
(1)
where E X' C CR' is solution vector of dimension d, and f(x) consists of m objective functions that must be minimized simultaneously (Miettinen) 1998 Deb (001). Since there is generally no single solution xx that minimizes all objectives simultaneously, solutions compared using Pareto dominance.
Definition 1 (Pareto Dominance), An objective vector f(x) is said to Pareto-dominate another vector f(x'), denoted f(x) fxx), if Vi: f((x) f((x') with at least one strict inequality.
Definition 2 (Pareto Optimality). A solution x. X called Pareto-optimal if no other solution E X dominat- it. The set of all such solutions is the Pareto-optimal set (PS) and their corresponding set of objective vectors, xx*) x E PS the Pareto front (PF).
The goal of MOO is to approximate the PF with a diverse and well-distributed set solutions. When evaluating the objective function is expensive, as in drug discovery or architecture search, offline MOO uses a pre-collected dataset D = { {xx ys) ) with y, = f(x,), without direct access to f during training (Xuc et al.) 2024. Kim et al. 2026). Offline methods rely solely on this static dataset D, though a limited budget Q of new queries to if may be allowed during evaluation. Solution quality typically measured by the Hypervolume (HV) indicator, which is calculated relative to a reference point derived from the best-known solutions when the true PF is unknown (Zitzler &
Thiele 1998
Zitzler et al. 2007).
Conditional Diffusion Models. Diffusion models are a powerful class of generative models that produce high-quality and diverse samples by iteratively denoising corrupted data (Ho et al.] 2020 Song et al. 2021b). Building on the Elucidated Diffusion Model (EDM) formulation (Karras et al, 2022). , we extend it to the conditional setting. The clean data is drawn from joint distribution (x. y ~ Paasa (x, y), and the conditional smoothed distribution is defined as p(æ o) ), obtained by adding Gaussian noise of standard deviation a to The evolution of noisy conditional sample
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p(æ o: as the noise level decreases is described by the ordinary differential equation (ODE):
dz/do - (Do(æ y. - 2//0.
(2)
The denoiser network Da is trained to minimize the expected L2 denoising error:
arg min (x Paa P 2o0 Pa + n: y.o) - x:
Here, Purin (0) specifies the noise distribution used during training and X(o) is a weighting factor between noise levels. Sampling starts from pure noise 2o p(æ: y; mmux) and follows the ODE down to = 0. yielding a sample from the conditional data distribution.
To steer generation towards desired conditioning target y. we employ classifier-free guidance (CFG) (Ho & Salimans] 2022). This modifies the ODE by combining conditional denoiser Dø(#; y, 0) with an unconditional denoiser Dp(æ; o), where the unconditional denoiser is imple- mented using the same network by randomly dropping conditioning during training. The modified ODE is:
dz/do= = - (>>D( y. + (11-11 Do o) - x)/a,
(4)
where y controls the guidance strength. Setting y > amplifies the influence of the conditioning target, guiding the samples to regions consistent with y.
4 METHOD
Building on the conditional diffusion framework, PCD reframes offline MOO as a conditional gen- eration problem, where represents candidate solutions and the conditioning vector y corresponds to the target objectives. The diffusion model learns to directly model the conditional distribution p(a: o) of solutions æ given objectives y. In the following sections, we describe the two key components that enable this approach: (1) a reweighting strategy to emphasize high-quality designs near the Pareto front, and (2) reference-direction mechanism to steer sampling toward preferred trade-offs and generalize beyond the observed dataset.
Training via Multi- Oujeti-Oeeeetive Reweighting. As stated, the goal of offline MOO is to discover new solutions that outperform those in the offline dataset DD To achieve this, the model must gen- eralize accurately near well-performing samples, while modeling of low-performing regions less important. A natural way do this is place higher emphasis on high-performance points by reweighting the dataset (Kumar & Levine 2020 Krishnamoorthy et al., 2023). Simply pruning
poorly performing points, in (Xue al.) (002)). ), can increase the variance of the training objective
(Kumar& & Levine 2020 Krishnamortthy et al. 2023).
A key challenge when reweighting the offline dataset is how to rank the samples. As noted earlier, there is no total ordering between f(x) and f(x'), so simple strategies, such as measuring the distance to the best point in the dataset, are not feasible. Instead, points can be ranked based on the number of points that dominate them. Specifically, one can compute the dominance number (Zhao ct al. 2022) :
o(x) = Σ [ff()) - f(x'),
(5)
EEED
where I is the indicator function. Intuitively, a lower dominance number indicates that a point performs better relative other points in the offline dataset.
To reweight the dataset, the objective space Th is first partitioned into grid of NB bins of equal width. Points in a bin are then weighted to emphasize bins that 1) contain more points, and 2) are werl-performing on average. We implement this using the following multi-objective reweighting Krishnamoorthy et al., 2023):
scheme, similar to (Kumar & Levine 2020
Σ B. 1 O(Tb,)
B B. K
w/ =
exp(
(6)
where [B|| is the size of the i-th bin, Db, is the j-th sample of i-th bin and K, T are hyperparameters controlling smoothing over bin sizes and the temperature of the weighting distribution. The first
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91
91
91
Figure 2: Overview of the conditioning points generation procedure: a) The objective space is partitioned via direction vectors, and points are ranked based non-dominated sorting. b) Each direction vector is paired with the point closest to it in perpendicular distance (black arrow). The rest of the points are paired the vector with the least amount of assigned points (gray arrow). c) A diverse set of conditioning points is generated by extrapolating the assigned points along the direction vectors and adding Gaussian noise.
term emphasizes bins with more points, while the second assigns higher weight to bins containing regh-performance points. Therefore, Equation 6 achieves both desired properties. Appendix A.2 discusses practical implications of reweighting in more detail.
To put it all together, Equation is applied to Equation resulting in the reweighted denoising L2 objective:
= arg min (( PaPans Prepiiss 2) X(o) [1D + t x
(7)
where wyy) the reweighting factor from Equation 6 and A(a) is the noise-level-dependen weight- ing term.
Generating a diverse set of conditioning points. A crucial question in conditional sampling is how to select the target conditioning points, which we denote as ÿ. While simple tactics exist for single-objective optimization (Kumar Levine (020), they are ineffective for MOO where the goal is to approximate the entire Pareto front. We therefore propose a two-stage procedure to generate a set of Q conditioning points ý. designed to be both high-quality and diverse. The procedure is described briefly below, with full pseudocode and explanation in Appendix A..
To collect the set of representative points, we propose to use a procedure motivated by the survival method in NSGA-III (Deb & Jain 2014). Firstly, the objective space is partitioned by generating L direction vectors w; via Riesz s-E --Eergy method (Hardin Safi 2005). Secondly, the points in the offline dataset are ranked via non-dominated sorting, resulting in a list of fronts (Fi, F2, ... FM)) where each front consists assign each point direction vector w, Initially, each direction vector gets assigned point that is closest to in perpendicular distance. Then, the rest of points in front F; are iterated over one-by- one and assigned to direction vector with the least amount of points the moment. This procedure is continued until Q (point, direction vector) pairs have been created.
set of points with the same rank. Lastly, we iterate over the fronts to
After collecting the set of representative points, the points are extrapolated towards the assigned direction vector to improve their quality over the points in the offline dataset. However, in practice the number of solutions Q is often larger than the number of direction vectors L. Therefore multiple points get extrapolated towards the same direction, which decreases the diversity of the conditioning points. To mitigate this, the final conditioning points are obtained by adding zero-men cero-mean Gaussian noise to the extrapolated points, akin to the Gaussian mutation operator used in evolutionary algo- rithms. This reweighted objective, paired with reference direction sampling, forms the core of the proposed PCD.
Sampling with Classifier-Free Guidance. For each conditioning point á generated by our reference-directio mechanism, sample corresponding solution by applying Classifier-Free Guidance (CFG) (Ho & Salimans| 2022). As formally defined in Section CFG steers the denoising process by creating weighted combination of the model's general and target-specific predictions.
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Algorithm PCD Training and Sampling
Input: Dataset ((r,, y:)] hyperparameters for reweighting (Np, K, and sampling
(Q, 7, mmnn max)
Output: Candidate set X with X = Q
Phase 1: Training
1:
Initialize model parameters
2: Train denoiser D3 on D using the reweighted objective (Eq.
Phase 2: Diffusion Sampling
3: Generate conditioning points Y= (ū:) using Algorithm
4: X < 0
5: for each 9 E Y do
6: Sample x: (O, (0 mas
Start sampling from pure noise
7: for from max to min over S steps do
8:
(a; Ú, 0) + 7Da(a; ú, + (1 Do(=)) 9: Update CDE-STEP(R, Do
Apply Classifier-Free Guidance b Perform step of denoising
10: end for
11: X X {ω}
12: end for
13:
return X
This well-established technique, widely used for high-fidelity image generation (Saharia et al. 2022 Rombach et al. 2022), strongly enforces adherence to the conditioning signal. In our setting, this biases the generated sample æ to be highly faithful to the target trade-off y. yielding a final set of solutions that accurately covers the desired region of the Pareto front. The complete training and sampling procedure is summarized Algorithm
5 EXPERIMENTAL VALIDATION
We conducted extensive experiments to validate the performance of the proposed method against variety baselines from the literature. In Section [.1 we give an overview of the used benchmark, while Section 5.2 introduces the baselines. Section 5.3 discusses the main results, and Section 5.4 validates the performance of the proposed components via ablation studies.
5.1 OVERVIEW
Our proposed approach is evaluated using the comprehensive Offline MOO benchmark (Xue al.] 2024). We consider five diverse task categories: I) synthetic, 2) multi cujtti-objective reinforcement learn- ing (MORL). 3) real-world applications (RE), 4) scientific design, and 5) multi-objective neural ar- chitecture search (MONAS), each of which is described in more detail below. For benchmark tasks that involve discrete values, we convert them into continuous logits, following the standard practice in prior work (Trabucco et al. 2022 Xue et al. 2024).
Tasks, The synthetic task set consists of 11 commonly used multi-objective test functions, includ- ing DTLZI, _ OmniTest, VI.MOP1-3 and ZDT1-4, 14, ZDT6. These are all cantinuous-valuee functions with datasets of 60,000 points each, featuring two or three objectives with varying Pareto front shapes.
The MORL task set contains two continuous high-dimensional problems, MO-Hopper and MO- Swimmer, each with two objectives, where goal is to generate optimal parameters for neural network control policy. The offline datasets are composed of policy parameters collected during PGMORL training (Xu et al. 2020).
The Real-World Applications (RE) task set consists of engineering design problems, such pres- sure vessel and rocket injector design (Tanabe & Ishibuchi 2020). These tasks have 4-7 dimen-
'Tasks DTI.72-DT.776 are excluded due evaluation errors in the original henchmark: https:// gi github.com amda -boo/offiin-moo//// sa/es/18

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sional search spaces and 2-6 objectives. Unlike the previous task sets, several RE tasks involve mixed-integer search spaces.
The Scientific Design task set includes one molecule design problem and three discrete protein design problems. The molecule task is optimized in a 32-dimensional latent space, while the protein tasks involve generating sequences of 32 to 200 tokens to optimize two conflicting objectives.
The MONAS task set aims to optimize the accuracy, latency, and parameter count of a neural net- work. With up to 34 decision variables, this is the only purely categorical task set in our evaluation, posing a unique challenge for continuous generative models.
Excluded Tasks, While the original benchmark suite by (Xue et al. 2024) included combinatorial tasks, we exclude them here. Adapting diffusion models to combinatorial domains requires substan- tial modifications to the denoising process and decoding strategy (Sun & Yang 2023] Sanokowski et al. 2024), which is beyond current scope. Following prior work (Yuan et al. 2025), we focus on continuous, discrete and categorical tasks, which we believe sufficiently illustrate the strengths and limitations of our proposed method.
Setup. The quality of the generated solution set is measured using the Hypervolume (HV) indi- cator, following the original benchmark (Xue et al., 2024). Formally, HV measures the volume of the objective space that is weakly dominated by solution set (Zitzler & Thiele 1998). HV is comprehensive metric that condenses the quality of a set into a single number while simultaneously measuring its 1) uniformity, 2) spread, and 3) convergence (Zintgraf et al. (015). making it ideal for comparing methods. Following prior work (Trabucco et al. 2022] Xue et al. 2024). we eval- uate performance on different subsets of the generated solutions. Specifically, we first generate set of Q = 256 candidate solutions. We then report the HV on the top P percentile of this set for P E {100, 75, 50). For P < 100, the lowest-performing (100- P1% of solutions fore HV calculation. All results are reported as the mean and standard deviation over random seeds unless specified otherwise. Detailed hyperparameters for our method PCD and training procedure are provided in Appendix A.3
removed be-
5.2 BASELINES
Our proposed method is compared against a comprehensive set of baselines, primarily drawn from the Off-MOO-Ben benchmark (Xue et al.] 2024). The majority of these methods are surrogate- based: they first train a model on the offline data to approximate the objective functions, and then this learned proxy guide search algorithm. These baselines primarily differ in the type of surrogate model used, for which we consider both DNN- based and GP-based approaches.
DNN-based surrogates categorized into three subclasses based on their architecture. End-to-end (E2E) models use a single DNN to output an m-dimenniooal vector representing all objectives. In contrast, Multiple Models (MM) maintain separate regressor for each of the m objectives. As each model is trained on single objective, this approach allows for pairing with powerful offline single-objective optimization techniques such as COMs (Trabucco al. 2021). ICT (Yuan et al., 2023b). 1OM (Qi et al. 2022), and Tri-Mentoring (TM) (Chen et al. 2023). Finally, inspired by multi-task learning. Multi-Hese (MH) models use shared feature encoder with separate output heads for each objective, enabling efficient information sharing. MH models are often paired with training techniques like PCGrad (PC) (Yu et al., 2020) and GradNorm (GN) (Chen et al. 2018) to improve generalization.
For GP-based surrogates, we only consider the prominent NEHVV algorithm (Daulton et al. 2020)
Following the findings of (Xue al. 2024),, omit other GP models for two reasons: 1) training them on large offline datasets is computationally expensive, and 2) DNN-based models have been shown to outperform them most offline MOO tasks. Regardless of the surrogate type (DNN or
GP), the final solution set is generated using the NSGA-II search algorithm (Deb et al. 2002) with budget of Q 256 evaluations on the true oracle.
Finally, we include generative model baselines. ParetoFlow (Yuan et al. 2025) is the only other generative model specifically developed for offline MOO that we are aware of and is our primary
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Table 1: Average rank (1) of PCD and baseline methods across five task categories. Ranks are calculated based on the 100th percentile HV. Bold and underlined rows indicate the best and runner up methods respectively. PCD achieves the best overall average rank, demonstrating its strong and consistent performance.
Method
Synthetic
MORL
RE
Scientific
MONAS
Avg. rank
D(best)
5.45:0.11
1.70:10.27
2.60:10.07
9.35+1.14
11.53:10.06
7.43
MOBO
8.69+-030
14.60+00.42
10.00-+0.33
6.75++0.47
8.11++0.0
8.81+0.34
E2E + GN
7.33::.55
5.70+22.14
7.06:10.32
5.35+1.38
9.33+0.53
7.82=10.40
E2E + PC
5.93--0.25
3.50++1.22
6.224-0.33
4.30-t1.32
6.60-40.40
6.01--0.29
E2B
6.16110.0
9.70:22.08
6.061t0.30
4.20+11.0
5.13::.22
0.71
MH GN
8.82+-0.53
8.90--2.16
8.14
5.05++2.14
12.57-10.40
9.84-10033
MH PC
8.87=0.45
10.90=#1.08
6.74
6.15+
7.46
7.68=
MH
6.181:053
8.00=1.41
6.141:.29
5.80+0.89
5.88:40.4
6.10
MM + COMS
8.02-10.47
3.60+11229
6.54
3.85+-0.68
7.22+40.43
6.80+-0.13
MM + ICT
6.73=:0.6
9.10
5.44:0.32
5.05=t0.74
8.42#00.0
0.08
MM IOM
5.16410.1
12.70+10.91
5.76.10.52
4.40+1.15
5.77-10.50
5.80+-020
MM + TM
6,55+0.82
7.90+22116
5.78:0.22
5,90
7.87:4.39
6.91
MM
6.07+40.0
9.50+10.79
5.94
6.55++0.93
4.97-00.46
5.80+0.21
ParetoFlow
2.44
8.50
1.74
9.05++0.27
11.19:00.52
6.74
PCD (ours)
3.38-10.20
5.50--3.30
1.51--0.13
4.05++0.33
7.54
0.30
comparisor While LaMBO (Stanton et al. 2022) is related diffusion-based approach, it was used generate some of the benchmark datasets themselves. Including LaMBO baseline could therefore introduce bias, so it is excluded. We also include simple but strong baseline, D(best), which represents the performance the non-dominated set of solutions present in the original offline dataset.
5.3 RESULTS
Table summarizes our primary experimental findings, showing the average rank each method across the five task categories based the 100th percentile HV. The table also presents the total average rank aggregated over all tasks. For detailed per-task HV scores at the P E {100, 75, 50) percentiles, please see Appendix A.II A.12 and A.13 respectiievl.l.l.
The results in Table [ clearly demonstrate the effectiveness of our approach. PCD achieves the best overall rank, highlighting its consistent high performance across diverse set of problems. This consistency is particularly noteworthy, as these results were obtained using single, fixed set of hyperparameters, underscoring the method's robustness the offline setting where such tuning is inherently difficult. Even in the MORL and MONAS task sets, where its performance is weaker relative to other categories, PCD still outperforms the other generative baseline, ParetoFlow (uaan) et al.) 2025). by considerable margin.
The relatively weaker performance on MORL tasks can be attributed the extremely high- dimensional search space. In fact, this benchmark proved challenging for all evaluated methods. Notably, and contrast to some prior reports (Yuan et al.) (225), we found that no method could produce solution set superior to the non-dominated points already present in the offline datase This finding highlights the immense difficulty of the MORL tasks and underscores the need for further research in this area.
Similarly, the performance on MONAS is impacted by its purely categorical search space, which presents challenge for our continuous diffusion model. While diffusion models have achieved ex- cellent results on discrete and graph-based data, doing so typically requires specialized architectural
PPGD-MOO Annadani al. 2025 is omitted hecause the results in the original study were limited
synthetic tasks, and we were unable to obtain results on our broader benchmarks using the official code. We hypothesize that the difference in our results and those of prior work may he due to updated datasets released by (Xue et al. 2024
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Table 2: Average 100th percentile HV (1) on 5 representative tasks with different data process- ing and sampling strategies. Bold and underlined rows indicate the best and runner up methods respectively. The results validate the effectiveness of both our proposed reweighting and reference- direction mechanisms.
Variant
ZDT2
MO-Swimmer-v2
RE34
Regex
C10/MOP2
Ideal + N/A
0.59
1.76=#021
9.19:10.26
5.60
10,46:0.001
D(best) N/A
7.54
3.58+00006
9.32++0.01
4.84
10.44:10.02
Ref. Dir. + N/A
7.89:10.19
3.53+0.11
10.11+00033
5.55+0..25
10.47=00.01
Ref. Dir. + Pruning
5.64
3.63+0.07
10.16+0.003
4.2011064
10.55=00.08
PCD
0.25
3.69:
10.17:10.04
4.8010.87
10.59=1.04
changes not employed in this work (An al. 2024 Niu et al.
2020 Jo al. 2022). Integrating
these urchitectural advances into the PCD framework is promising direction for future work.
5.4 ABLATION STUDIES
Ablations on PCD components. We conduct an ablation study to validate the two primary com- ponents of PCD introduced in Section the multi-objective reweighting scheme and the reference- direction mechanism for generating conditioning points. We compare three data processing tech- niques: 1) no processing (N/A), 2) simple pruning of low-quality points as (Xue et al. 2024), and 3) our proposed data reweighting. Additionally, we compare three strategies for generating condi- tioning points: our proposed set of points in the offline dataset (Dbbst)). and simpler bascline (Ideal) that randomly samples non-dominated points from the dataset and extrapolates them toward the ideal objective vector, a strategy common in offline single-objective optimization. Note that the full PCD method combines our proposed reweighting with the reference-direction mechanism.
(Ref. Dir.) mechanism, conditioning on the best
The results in Table SableZconfirm the effectiveness of both of our proposed components. First, using the reference-directe mechanism to select conditioning points consistently improves or maintains per- formance compared to both the simplistic "Ideal" and "D(best)" strategies. The benefit is especially pronounced on the MO-Swimmer task, where our approach nearly doubles the HV compared to the "Ideal" strategy, highlighting the critical importance of generating high-quality and diverse condi- tioning points. Second, the table showcases that our proposed reweighting strategy consistently and significantly outperforms the simple pruning approach from (Xue et al., 2024). Interestingly, for the ZDT2 and Regex tasks, any form of data processing appears to harm model performance. We hy- pothesize this is due to the quality of the underlying datasets these specific cases; when a dataset is less skewed (i.e., most points discard valuable training signal for minimal gain. As explained further in Appendix [A. these tasks appear to be the exception, as we observe no similar performance degradation in other benchmarks.
already high-quality), aggressive reweighting pruning can
Effect of guidance scale. Finally, we ablate the effect of the classifier-free guidance scale, 7. As discussed in Section 3 y balances the influence of the conditional and unconditional components of the denoiser. Since our goal is to generate solutions superior to those in the offline dataset, we hypothesized that a stronger guidance scale (7 1) would improve performance by foreing the model to adhere more closely to our novel conditioning points. Figure 3 plots the resulting HV. normalized by the performance of our default scale, 7 2.5 across five different tasks.
The results in Figure 3 show that increasing the guidance scale yields surprisingly limited perfor- mance gains, with performance saturating or even slightly decreasing for 7 > 2.5. We attribute this effect to two factors: 1) The dataset is already reweighted to prioritize high-performing samples. This means the diffusion model is inherently biased toward generating better solutions, even with weak conditioning. 2) The conditioning points near the data manifold. Our reference-direction mechanism generates novel targets by extrapolating from the best existing data. While these targets new, they remain close to the learned distribution. Consequently, the unconditional component of the model is already adept at generating similar high-quality solutions, reducing the marginal benefit of stronger conditional guidance.

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Published as a conference paper at ICLR 2026
MO 20-mmimmer-22
ZINC
RE22
VLMOP2
IN- - 1K/OP?7
Hypervohums ratio
1.04
1.00
0.96
0.92
2.0
2.5
5.0
8.0
Guidance scale y
Figure 3: Effect of the guidance scale y on the normalized HV ratio across five representative tasks. Performance saturates quickly, showing limited benefit from strong guidance.
6 SUMMARY AND LIMITATIONS
This paper introduced Pareto-Conditional Diffusion (PCD), novel framework that reframes offline multi-objective optimization as a conditional sampling task. By learning to generate high-quality solutions conditioned directly on desired trade-offs, PCD elegantly sidesteps the need for explicit surrogate models or hand-crafted scalarization schemes. To make this approach effective, we in- troduced two key components: multi-objective reweighting scheme bias the model toward the Pareto front, and an NSGA-IIl-insppe points from the offline data. Our extensive experiments demonstrated that PCD achieves compet- itive performance, driven by its remarkable consistency and robustness across wide variety of benchmarks using single set of hyperparameters.
mechanism to generate diverse set of novel conditioning
Despite its strong performance, PCD has several limitations. On extremely high-dimensional con- tinuous tasks such as MORL (~10,000 dimensions), performance is limited by applying an MLP denoiser directly to the parameter space. Future work could extend PCD by using Latent Diffusion Model (Rombach et al. 2022) to operate in latent space or by adopting a Transformer- based denoiser dinensiiens neural network parameters (Wang et al. 2024 Peebles et al. 2022). For discrete optimization tasks with categorical search spaces such as MONAS, the PCD) framework
(Peebles & Xie| (2023). both of which have demonstrated success in generating
be readily extended using
continuous-space diffusion methods for categorical data (Dieleman et al. 2022 which preserve the applicability of standard classifier-free guidance.
Alternatively, one could employ fully discrete diffusion models (Austin et al. 2021): while standard
guidance is not applicable in this setting, recent work has developed effective extensions for discrete guidance (Nisonoff et al.] 2025 Schiff et al., 2025). Finally, PCD can be naturally extended to combinatorial optimization problem ems (e.g., TSP. CVRP) by handling permutation constraints within the sampling loop (Sun & Yang 2023). While outside the scope of this study, this could be achieved by combining constrained diffusion frameworks (Sun Yang. 2023] (annoooski al., (2024) with the aforementioned discrete guidance techniques, effectively transferring our key contributions to the combinatorial setting.
ACKNOWLEDGEMENTS
This work was supported by the Research Council of Finland (Decisions Nos. 353198, 353 353199, 357301, 339730, 362408, and 334600). Santeri Heiskanen was supported by the Ministry of Ed- ucation and Culture's 's Doctoral Education Pilot in Finland (Decision No. VN/3137/2024-KKS Finnish Doctoral Program Network Artificial Intelligence, AI-DOC). We acknowledge the com- putational resources provided by the Aalto Science-IT project.
ETHICS STATEMENT
Our work, Pareto-Connn Conditioned Diffusion (PCD), is a general- for real-world applications in engineering and scientific design. However, we recognize its poten-
general-purpose purpose optimization method intended
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Published as a conference paper at ICLR 2026
tial for misuse, the method could be applied harmful outcomes. We emphasize the need for oversight and safeguards to mitigate potential risks. Because PCD operates in an offine setting, may also inherit or amplify biases from training data, underscoring the importance of careful dataset curation.
REPRODUCIBILITY STATEMENT
To ensure reprodueibility, we provide supplementary materials with submission, including:
Source Code: A complete implementation of proposed method, Pareto- Coneitoondd Diffusion (PCD).
Environment and Data: A README file with instructions to set up the environment, along with links to all benchmark datasets and baseline implementations.
Scripts and Configurations: Scripts to reproduce the results presented in the paper, in- cluding the exact training configurations, hyperprrameter,,,rr and random seeds.
Our implementation is available at https//gghhub..com/aaaaa/////// 12/PCD
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A APPENDIX
A.1 RELATION BETWEEN DATASET QUALITY AND REWEIGHTING
As shown our main ablation study (Section 5.4), our data reweighting scheme significantly im- proves performance tasks like C10/MOP2 and MO-Hopper, yet detrimental to performance on ZDT2. We hypothesized that this discrepancy stems from differences in the quality of the underlying offline datasets. To investigate this, we analyze the distribution of normalized dominance numbers for each task, shown in Figure 4 A narrow distribution suggests that most samples are of similar, relatively high quality. In contrast, a wide, right-skewed distribution indicates a large variance in sample quality, with a long tail of low-performing data points.
ZDT2
C10/MOP2
MO-Hopper-v2
10
Densily
1.5
0.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
Normalized dominance number
Figure 4: Normalized dominance number distributions reveal dataset quality differences. The narrow distributions of C10/MOP2 and MO-Hopper indicate consistently high-quality datasets, whereas the long-tailed distribution of ZDT2 signals high variance in sample quality.
Figure clearly supports hypothesis. The C10/MOP2 and MO-Hopper tasks exhibit narrow distributions, concentrated near zero. ZDT2, however, shows a much wider, long-tailed distribution, confirming it contains more varied set of samples. While dataset with high variance in quality like ZDT2 may seem like an ideal candidate for reweighting, our results show the opposite can be true. When the sample quality is highly varied, our reweighting scheme assigns tiny, near-zero weight to the vast majority of points in the long tail. This can effectively remove significant portion of the training data, which, despite being lower-quality, may still contain valuable learning signal about the data manifold. This raises natural follow-up question: could this issue be mitigated by tuning the reweighting hyperparameters?
A.2 REWEIGHING SENSITIVITY ANALYSIS
Based on our analysis in Section A.1 we hypothesized that the performance degradation on high- variance datasets like ZDT2 could be mitigated by adjusting the reweighting hyperparameters. Specifically, increasing the temperature parameter 7 in Equation 6 should make the weighting dis- tribution more uniform, effectively "softening" the reweighting and preserving more of the learning signal from lower-quality samples. To validate this, we perform sensitivity analysis on 7 for the same three tasks: ZDT2, C10/MOP2, and MO-Hopper.
Figure plots the normalized hypervolume performance of our default value, = 0.05. As predicted, the effect of T is highly task-dependent. For ZDT2, the task with the highest sample variance, increasing the temperature yields substantial performance improvement of up to 15%. In contrast, for the low-variance C10/MOP2 dataset, per- formance is almost entirely insensitive shows a modest improvement.
function of т. The results are normalized by the
The MO-Hopper task, which has moderate variance,
This outcome confirms intuition: the higher the variance in dataset quality, the higher temper- ature 7 is required to achieve optimal performance. Although we explored systematic ways to tune (e.g., estimating from the standard deviation CDF of the dominance number distribution),
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1.15
ZDT2
C10/MOP2
1.10
MO-hopper-v2
Hyporvolung
1.00
0.05
1.00
2.50
5.00
7.00
Reweighing temperature
Figure 5: Effect the reweighting temperature r on normalized performance. For the high- variance ZDT2 dataset, increasing significantly improves performance. The effect is minimal on the lar-nariance datasets.
found the default value (r = 0.05) to be robust across the majority of tasks. However, for outlier datasets with high variance like ZDT2, we recommend a simple visual heuristic: if the distribution of dominance numbers exhibits "long tail," increasing effectively "softens" the reweighting to preserve more training data. We note that the other reweighting hyperparameter, K, which controls smoothing over bin sizes, was observed to have negligible impact performance in preliminary experiments, and thus was not included in this analysis.
A.3 TRAINING DETAILS
We build upon the open-source EDM implementation (Karras et al., 2022) introduced in SynthER (Lu et al.] (202), adapting it to our specific training setup and dataset. Following SynthER, our denoising network adopts the MLP architecture with skip connections from the previous layer as proposed in Tolstikhin et al. (202T), with the diffusion noise level encoded using a Random Fourier Feature (RFF) embedding (Rahimi & Recht 2007). The network has base width 512 and depth of 4, and is trained with batch size of 512 for up to 12k steps with early stopping.
2022) during training with probability 0.25 and use the same denoiser configuration for all tasks. We maintain an exponential moving average (EMA) of the model parameters (Karras et al. 2022) and use the EMA weights for sampling to improve sta- bility (Karras et al.) 2024). Sampling is performed with the EDM stochastic differential equation (SDE) sampler using default hyperparameters, increasing the number diffusion timesteps to 1024 to enhance sample quality. The default hyperparameters for all components of our PCD frame- work- inckuiinl the denoiser network, the EDM sampler, our multi-objective reweighting scheme, and our conditioning point generation mechanism- are summarized in Table 3
We apply CFG dropout (Ho & Salimans,
Table 3: Default hyperparameters for PCD framework including Residual MLP Denoiser for reweighted train- ing (Ieft) and EDM Sampling (right). If multiple values were used, bolded value indicates default value.
Residual MLP Denoiser
EDM
Parameter
Value(s)
Parameter
Value(s)
No. of Bins K
30
No. of Diffusion Steps
1024
10
0.002
0.05
Omxx
80
No. of Layers
4
Sama
80
Width
512
Smin
0.05
Batch Size
512
Smm:
50
RFF Dim
16
Sucse
1.003
Activation
ReLU
Query Budget Q
256
Optimizer
Adam W
No. of Reference Directions
Learning Rate
3 10-4
CFG Dropout Proh.
0.25
learning Rate Schedule
Cosine Annealing
CFG Guidance Scale
{1.0), 2.0, 2.5, 5.0, 8.0)
Denoiser Training Steps
12,000 (early stopping)
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A.4 COMPUTATIONAL COST
A key component of our reweighting strategy is the computation of the dominance number o(æ) (Eq. (5) for each sample in the offline dataset. Naively, this calculation scales quadratically with the dataset size, mm) However, we emphasize that this is a one-time preprocessing step performed prior to training. For standard offline MOO benchmarks (up N 60, 000), this calculation is inexpensive in practice when utilizing optimized non-dominated sorting algorithms (e.g., from the pygmo library (Biscani & Izzo, 2020)). Empirically, computing the dominance numbers for our largest dataset (RE34) takes less than 20 seconds, as shown in Table While the quadratic cost did not pose issues for the benchmarks considered in this work, may become a limitation for extremely large datasets, since the worst-case complexity remains quadratic.
To examine inference costs, we report the wall-clock time for sampling, averaged over three seeds, across four representative tasks for PCD and the mredictor-guides ParetoFlow baseline (Yuan al. 2025), All experiments for this study were conducted on a workstation with an Intel i5-12600K CPU, 64GB of memory, and an NVIDIA GeForce RTX3080 GPU. Table hiigllgghs the key find- ing: PCD sampling generates a batch of 256 solutions faster than ParetoFlow on most tasks. Al- though PCD employs CFG (Ho & Salimans (022), which requires two forward passes per step, it avoids the computationally expensive and memory-intensivy backpropagation through the network required by the baseline's predictor-guidant (Dhariwal & Nichol 2021
Table 4: Computational cost comparison. We report the one-time dataset reweighting cost (sec- onds) and sampling times (seconds) averaged over 3 runs. PCD demonstrates faster sampling speeds compared to the generative baseline ParetoFlow across four representative tasks, while reweighting adds negligible overhead.
Method
RE34
MO-Hopper-v2
Molecule
C10/MOP2
Dataset Info
No. of Samples
60,000
4,500
49,001
26,316
No. of Objectives
3
2
3
No. of Dimensions
5
10,184
32
31
Reweighting Time (sec)
0..37
0.090110.000
I 11.8
0.94+
Sampling Time (sec)
PCD ParetoFlow
0.16 6.08
3.22
1.80
2.02
17.20+0..1
36.91
5.81
A.5
DETERMINISTIC SAMPLER VS. STOCHASTIC SAMPLER
Recent work has established a strong theoretical connection between diffusion models (Ho et al.
2020 Song al. 2021b) and flow matching (Lipman et al., 2023 Liu al., 2023), showing that they can be unified under the framework of continuous-time generative models (Gao et al.) [025] Albergo et 2025). In particular, when using Gaussian probability flow paths, the sampling pro- cess for both frameworks is governed by an ODE that maps a Gaussian noise distribution to the data distribution. Consequently, using diffusion model with deterministic ODE sampler is mathe- matically equivalent to the sampling process of standard Gaussian flow matching (Gao et al.) 2025). The primary practical differences lie in the parameterization of the vector field, the noise schedule, and the weighting of the training objective (Gao et al. (025). These design choices have been ex- tensively studied within the diffusion context, most notably in the EDM paper (Karras et al. 2022)
This work uses the parameterization derived in EDM.
To validate our choice of sampler, we compare our default stochastic EDM (an SDE-like sampler) against its deterministic counterpart (ODE). The results in Table 3 demonstrate that the stochas- tie sampler outperforms the deterministic one on most tasks. We hypothesize that the randomness inherent to the stochastic EDM solver aids exploration, which is particularly beneficial for discov- ering diverse, high-quality solutions in offline MOO. The notable exception is the discrete Regex task, where the deterministic sampler performs better. We hypothesize that this is due the final
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decoding step: mapping continuous logits to categorical tokens is sensitive to noise, and the deter- ministic trajectory the ODE sampler leads to more stable token sequences (Song et al. (0211). Consequently, adopt the stochastic sampler the default configuration, Regex the only exception.
Table 5: Average 100th percentile HV (1) on different tasks comparing EDM with stochastic and deterministic samplers. Bold and underlined rows indicate the best and runner up methods respectively, The default stochastic sampler shows superior performance on most tasks.
Method
C10/MOP2
MO-Hopper-v2
RE34
Regex
ZDT2
Stochastic PCD
10.59=0.04
5,95=0.18
10.17=4.044
4.80++0.87
6.25:+0.06
Deterministic PCD
10.54+0.03
4.47=0.00
9.47+00228
0.56+00.8
5.00++-0.04
Stochastic PCD (w/o reweighting)
10.47=0.01
6.01=0.58
10.11:0.03
5.55
0.8910.19
Deterministic PCD (w/o reweighting)
10.49=0.06
4.47=0.00
9.2440.26
5.68:10.31
4.9840003
A.6 FURTHER ABLATIONS ON SAMPLING HYPERPARAMETERS
To further analyze the components of the proposed method, we perform multiple ablations on the key design choices related to generating conditioning points. Specifically, we examine the effect of: 1) varying the number of selected base points (Table(6); 2) increasing the noise scale and extrapolation distance applied to the generated conditioning points (Figure 66;; and 3) changing the method used for generating direction vectors (Table
A.6.1 NUMBER OF CONDITIONING POINTS
We begin by examining the number of conditioning points selected at the second stage of condi- tioning point generation. Recall that if the number of selected points less than the number of generated solutions Q (as is often the case), multiple conditioning points will be generated from the same selected point. Therefore, the more points are selected, the fewer new points are generated for each selected point. As seen from Table 6 increasing the number of conditioning points tends to improve performance, with the exception of Regex, where using J = 4 points performs the best. This indicates that choosing sufficient number of points from the existing dataset is required for achieving reasonable performance.
Table 6: Average 100th percentile Hypervolume (f) on 5 different tasks using varying numbers of conditioning points J. Bold and underlined rows indicate the best and runner up methods respec- tively. Increasing the number of selected points generally yields higher performance, with 32 achieving the best results on the majority tasks.
No. cond. points
C10/MPP2
MO-Hopper-v2
RE34
Regex
7DT2
J=4
10.3110.12
4.471:0.0
6.56.10.83
5.26
4.87:10.08
J=8
10.37+0.08
4.47+00.0
7.19
4.69
0.92+00.55
J=16
10.41
4.47:00.0
8.33-10.59
4.69
4.97-00.33
J=32
10.59
5.95
10.17=0..0
4.80+0.87
6.25+00.6
J=64
0.04
4.47+00.00
0..22
4.81+
4.004
A.6.2
NOISE SCALE AND EXTRAPOLATION DISTANCE
Figure 6 iillssrates the effect of increasing the noise scale and extrapolation distance, the two pri- mary parameters controlling the generation novel conditioning points. Intuitively, increasing the noise scale produces a more diverse set of conditioning points, while increasing the extrapolation distance pushes the targets further from the original data distribution. As shown in Figure(6) increas- ing the noise scale has a limited yet positive impact performance without significant downsides, likely due to the increased diversity of the generated targets. Conversely, increasing the extrapola- tion distance offers negligible gains and significantly increases run-to-rum variance MO-Hopper. As discussed in Section 5.3] MORL tasks proved challenging for current approaches; consequently, pushing conditioning targets further from the training distribution increases model uncertainty. For
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C10/MOP2
MO MO-Hopperw2
RE23
Hyporvolume rabi
1.12
1.08
1.04
1.00
0.05
0.50
0.70
1.00
0.1
0.2
0.3
0.4
0.5
Noise scale 5
Extrapolation distance
Figure 6: Impact of noise scale and extrapolation distance conditioning point generation. Left: Increasing the noise scale has a limited, yet positive impact on performance. Right: Increasing the extrapolation distance leads to diminishing returns.
the remaining tasks, we hypothesize that the limited gains stem from the model's inability to gener- alize effectively to regions farther away from the training distribution.
A.6.3 RIESZ S- S-ENERG Vs.
Generating conditioning points requires set of reference direction vectors. Table 7 compares our default method, Riesz s-Energy (Hardin & Saff, 2005), against the classical Das-Dennis method (Das & Dennis 1998). While the Riesz s-Energy method allows one to precisely set the number of generated direction vectors, it requires method is simpler and requires no optimization. However, it does not allow for explicit control over the number of generated direction vectors. As shown in Table 7 while Das-Dennis slightly outper- forms Riesz s-Energy on most tasks, the performance difference is marginal. This demonstrates that PCD robust the underlying strategy used for generating direction vectors.
optimization procedure. In contrast, the Das- Das-Dennis
Table 7: Average 100th percentile HV (f) comparing Riesz s-Energy (default) and Das-Dennis methods for generating direction vectors. Bold rows indicate the best method. The simpler Das- Dennis method yields slightly better results, though the difference is minimal, indicating robustness.
Method
C10/MOP2
MO-Hopper-v2
RE34
Regex
ZDT2
Riesz s- e-Enrgyy
10.59=0.04 10.58=0.04
5.95 6.16:10.30
10.17=0.04
4.80
6.25
Das-Dennis
10.23:10.12
5.21
0.13
A.6.4 NUMBER DENOISING STEPS
Finally, we perform an ablation to study the effect of increasing the number of denoising steps. While increasing the sampling budget typically improves generation quality in standard diffusion literature (Nichol & Dhariwal 2021: Salimans & Ho 2022). Figure 7 shows that this is not the case here. With the exception of Regex, the proposed method achieves consistent performance regardless of the number of denoising steps. We attribute this to the fact that we are generating points relatively close to the training distribution, allowing the model to produce high-quality samples even with fewer steps. The deviation in Regex aligns with our discussion in Section A.5) discrete data requires a decoding step, which sensitive to approximation errors. Although the MONAS task (C10/MOP2) is also discrete, it is far simpler, as each of its decision variables takes only 3 values. In contrast, Regex variables have up to 72 options, making the decoding process far more susceptible to the small errors caused by fewer sampling steps.
A.7 CASE STUDY 1: THE EFFECTIVENESS OF PARETO-CONDITINING
To provide qualitative understanding of our Pareto-con conettoondiiionnng mechanism, we present a case study contrasting tasks: MO-Hopper, challenging benchmark where method struggles,
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C10/MOP2
MO-Hopperrv2
RE34
Regex
ZDT2
Hypervolume ratio
0.9
32 64
128
256
512
1024
Number denoising steps
Figure 7: Hypervolume ratio as function of denoising steps. Performance remains consistent across increasing numbers of steps for most tasks, demonstrating that PCD generates high-qualij solutions efficiently even with fewer sampling steps.
and RE21, task where PCD significantly outperforms all baselines. These cases correspond to the quantitative results presented in Table and Table T2 respectively.
MO MO-Hopper-v2
RE21
D(best)
Outcome You
Cond. Point
1.5
2.5
0.0
0.0
-1.5
-2.5
-2
1.5
Figure 8: Qualitative analysis of Pareto-condittoning effectiveness. Left (MO-Hopper): On this high-dimensional task, the model struggles to generate solutions that improve upon the first ob- jective. Right (RE21): On this lower-dimensional task, the conditioning is highly effective: the generated points (ymm) closely align with and even outperform their conditioning targets (ý).
Figure 8 visualizes the relationship between the best solutions from the original dataset D(best), our generated conditioning points ÿ. and the objective values of the final generated solutions Your MO-Hopper, the model's struggle is evident. Despite being conditioned on targets ý with better per- formance on the first objective y1, generated points your fail to achieve significant improvements, likely due to the difficulty of generalizing in the task's nigh-dimensional search space. In stark con- trast, the conditioning mechanism is highly effective on RE21. Here, the generated points your con- sistently match that when the underlying data distribution well-captured by the model, PCD can successfully generalize to superior regions of the Pareto front.
Pareto-dominate their respective conditioning targets ÿ. This demonstrates
To further study the capabilities of PCD, we examine its ability to generate points within the training distribution. More specifically, partition the training dataset into fronts (Fi,..., FM) using
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MO-Hopperrv2
RE21
Cond. point y
3.0
Outcome Yout
1.5
1.5
0.0
0.0
-1.5
-1.5
-2
-1.5
0.0
1.5
91
91
Figure 9: Analysis of the model's ability to reconstruct the original data distribution. Colors represent distinct non-dominated fronts (F)) extracted from the training set. The model is condi- tioned on the objective vectors of the original data (circles, j); successful reconstruction is achieved the generated solutions (crosses, Your) overlap with these circles. Left (MO-Hopper): In this high-dimensional task, the model is unable to capture the original data distribution it was trained on. Right (RE21): this simpler task, the model faithfully reconstructs the original conditioning points.
dominated sorting. Then, we condition the model on the objective vectors from these fronts F; to validate whether it can reconstruct the data distribution it was trained on. Figure[jsowcasses a sharp contrast: while the simpler, lower-dimensional RE21 task, PCD captures the underlying distribu- tion almost perfectly, it fails to do so in the more complex, high-dimensional MO-Hopper task. We highlight that while the positive results on RE21 demonstrate the potential of the proposed approach, the MO-H HO-Hopper results indicate clear need for further development enable generalization to very high-dimensional data distributions.
A.8 CASE STUDY 2: SCALABILITY TO MANY-OBJECTIV PROBLEMS
While our main experiments demonstrate PCD's strong performance on standard tasks, of these benchmarks are limited to 2 or 3 objectives. To investigate PCD's scalability to many-objective problems, conduct ablation study on the synthetic DTL.ZI and DTLZ7 functions, varying the number of objectives m {3, 4, 5, 6). To facilitate this experiment, we collect offline datasets following the protocol described by Xue et al.] (202)). Specifically. we used the NSGA-III (Deb) & Jain 2014) algorithm to collect the dataset of N 60, 000 samples per task, aggregated across multiple random seeds.
repoots the relative improvement in Hypervolume (HV) achieved by PCD over the best solu- tion set available in the offline dataset (D(best)). By analyzing the relative improvement rather than raw HV values, we normalize for dataset quality and objective scaling, allowing to isolate PCD's performance trend as dimensionality increases. As shown in Table PCD achieves consistent im- provements over the baseline regardless of the number of objectives. This highlights the scalability of our reweighting and conditioning mechanisms, demonstrating that PCD effectively generalizes even in high-dimensional objective spaces.
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Table Scalability analysis on many-objective DTLZ tasks. We report the relative Hypervolume improvement (1) of PCD over the best offline dataset solutions (D(best)). PCD achieves consistent performance improvements regardless of the number of objectives (m).
No. objectives
Task
4
DTLZI
1.29:00007
1.25 1.17=0.01
1.84:0.16
1.37:0.08 1.21=00004
DTLZ7
1.33++002
1.19+0.02
A.9 DETAILED LOOK AT CONDITIONING POINT GENERATION
Section introduced the procedure to generate a set of diverse conditioning points ÿ by utilizing a procedure similar to the survival selection from NSGA-III (Deb & Jain 2014), which is detailed in Algorithm 2 We begin by generating I. direction vectors via Riesz s-Energy method (Hardin & Safi) 2005), which partition the objective space R Next, goal is select diverse subset of well- performing points from the objective space and assign these points to one of the direction vectors. To do so, an empty S is initialized hold pairs of points and direction vectors (y/, W:). Then, set of fronts ./f...., FM) is obtained via non-dominated sorting. The procedure continues by iterating over each front F. During each iteration, we identify the point with the shortest perpendicular distance to each direction vector and assign it to that vector (lines 8-13 in Alg. 2 Finally, the remaining points in F; (which have no direction vector assigned them), are iterated over one-by- one and assigned to the direction vector that currently has the fewest assigned points (lines 14-22 in Alg.
After obtaining the set of (point, direction vector) pairs S, the points extrapolated towards the corresponding direction vectors. Finally, Gaussian noise added to the points to ensure further diversity. The procedure described in Algorithm 2 follows very closely to the survival procedure of NSGA-III with one key difference: in NSGA-III the points in the current population are assigned to the closest direction vector, and only if there is need for more points, the points get assigned to the direction vectors with the least number of points. On the contrary, in our procedure, only points from front get automatically assigned the direction vector closest them. The rest of the points in the current front are assigned based on the number of points each direction vector has currently. This is crucial for ensuring that the conditioning points have an adequate coverage of the objective space in cases where the offline dataset itself has poor coverage of the objective space.
A. USE OF LARGE LANGUAGE MODELS
Finally, during the preparation of this manuscript, large language models (LLMss) were used assist in writing and literature discovery, but all content was reviewed and verified by the authors, who are solely responsible for the scientific claims of this work.
A.11 100TH PERCENTILE RESULTS
Tables
10
12
13
15 and 16 showcase the per task 100th percentile hypervolume (f) In each task, the best performing model is bolded, while the runner up
across all considered tasks.
underlined.
A.12 75TH PERCENTILE RESULTS
Tables
and 24 showcase the per task 75th percentile hypervolume (1) across all considered tasks. In each task. the best performing model is bolded, while the runner up
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Algorithm 2 Generating Diverse Conditioning Points
Input: Dataset objectives Y (y)) Number of reference vectors Query budget Q
Output: Set of Q conditioning points Y
e.g., via Riesz s-Energy method D Rank data points into fronts 2: (f.....,PM) + NonDominatedSort Initialize the of selected (point, vector) pairs
1: W
3: S+0
4: p pt Initialize niche counts for each w E W to 2ero
5: it1
6: while |S|
< do do P: +
7:
Initialize points be considered from current front Find point y E F; with minimum perpendicular distance to w
8:
for each reference vector w W do
9:
10:
Add (y: w) to S
11:
Add y to temporary set P.
12:
p[w] p|w) +1
Update niche counts for each reference vector
13:
end for
14:
for y F2 P: do
15:
Find reference vector Wmin with the minimum niche count p[Wmin]
5 5 SU {wy, Wmin]]
Add the selected pair our main set } Increment the niche count Remove the point from consideration
17:
p|u p|w wiu t p[Wainn +1
P. P. {
19:
if S Q then
20:
break
Exit if we have collected enough points
21: end if
22: end for
23: i 1-i+1
Move to the next front
24:
end while
25:
Yextrapolated Excrappolateoonnt(((S
Extrapolate each point along its vector
Y Y exttrapolated + NN 02 T I)
D Add noise for diversity
27: return Y
Table 9: Average 100th percentile hypervolume for synthetic tasks (part I)
Method
ZDT1
ZDT2
ZDT3
ZDT4
ZDT6
D(best)
4.17
4.68
5.15
5.46
4.61
E2E + GN
2.81
3.17+0.00
3.75+-0.00
4.87+L0.05
3.82+L0.03
E2E + PC
2.81=0.00
3.17+0.00
3.75+0.00
4.84+0000
3.78+0.08
E2E
2.81+0.00
3.17+0.00
3.76+-0.00
4.82+0.02
3.83+0.01
MOBO
2.38+0.01
2.76+0.00
3.13+0.05
4.93
3.76+0000
MH GN
2.8110.00
3.1740.00
3.7610.00
4.83
3.7410.04
MH PC
2.4110.27
2.68+0.01
3.6810.16
4.83+0.04
3.7610.04
MH
2.8110.00
3.1710.00
3.7610.00
4.87
3.8310.01
MM + COMs
2.81+0.00
3.1740.00
3.75+0.00
4.84
3.82+0.04
MM + ICT
2.81+0.00
3.1740..0
3.76+0.00
4.86+0.04
3.83+0.02
MM + IOM
2.81+0.00
3.1740.00
3.73+0.05
4.85+0.03
3.84+0.01
MM + TM
2.81+0.00
3.17400.0
3.73+0.05
4.80
3.84+0.00
MM
2.81
3.1710.00
3.7610.00
4.8510006
3.00
ParetoFlow
4.1510004
5.7110.21
5.2210.11
4.8610.08
4.51
PCD (ours)
4.47+00.2
6.25+0006
5.361-0.36
5.23+0006
4.56+00.55
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Table 10: Average 100th percentile hypervolume for synthetic tasks (part 2)
Method
DTLZI
DTLZ7
OMNITEST
VLMOPI
VLMOP2
VLMOP3
D(best)
10.60
8.56
4.53
0.41
1.78
45.65
E2E GN
10.63110.00
8.95110.00
4.5510.14
0.29=0.00
2.8010.16
41.93.11.13
E2E PC
10.63110.00
8.9510000
4.6910.06
0.29-0.00
3.3410.01
42.45.10.00
E2E
10.61+0.01
8.95=0.00
4.69++0.55
0.29=0.00
3.36++0.00
42.45+00.00
MOBO
10.40+00.06
3.96=0.10
4.76+0.01
0.29=0.00
2.08=0.16
38.99+0.14
MH GN
10.6311.001
8.9510.00
4.61
0.08
3.15.0.07
30.1416.91
MH PC
10.63110.00
8.9510.00
4.48.10.22
0.20
2.98.10.09
41.85.00.11
MH
10.63110.00
8.9510.00
4.61
0.29-0.00
3.32.10.11
42.45.10.00
MM + COMs
10.62:00.00
8.95=0.00
4.54+0.15
0.23=0.13
1.68+0.04
42.44++0.02
MM + ICT
10.63=00.00
8.95=0.00
4.56+10.08
0.29=0.00
3.35=0001
42.43+00011
MM + IOM
10.6310.00
8.9510.00
4.72.10.03
0.29-0.00
3.32.10.33
42.4510.001
MM TM
10.63
8.9510.01
4.5910.10
0.29
3.35
42.41110.01
MM
10.63:10.00
8.95-10.00
4.71
0.29 0.00
3.35.10.01
42.45:10.00
ParetoFlow
11.15#0.44
10.54=0444
4.78+00.00
N/A
4.22:10.02
45.92
PCD (ours)
13.63:10.78
11.35=0.18
4.76:10.01
0.09=0.01
2.93+0.28
45.91
Table 11: Average 100th percentile hypervolume for scientific design tasks
Method
Molecule
RFP
Regex
Zine
D(best)
2.91
7.11=0.00
3.96:10.00
4.52=0.00
E2E + GN
2.77+-0.07
7.88-0.6
4.30
E2E + PC
2.91+00.00
7.22-0.02
4.74+-0.0
4.73-0.04
E2E
2.9110.00
8.11=0.49
3.98+0.00
4.71=0.07
MOBO
1.58:10.02
7.21=0.00
6.76:10.28
4.60=0.00
MH GN
2.91100000
8.10
4.1910.20
4.62-0.06
MH + PC
2.91
7.41=0.49
4.52+0.45
4.65=0.03
MH
2.91#0.00
7.21=0.02
3.98-10.0
4.66=0.00
MM COMs
2.91+0.000
7.66-0.61
4.61+-0.30
4.72-
MM ICT
2.91+0.00
7.87-0.61
4.30+H0.35
0.0T
MM IOM
2.91
7.66=0.62
4.83+0.00
4.69=0.05
MM + TM
2.87+0.06
7.83=0.65
4.83+0000
4.64=0.01
MM
2.91+0000
7.21=0.03
3.87-10224
4.66=0.00
ParetoFlow
2.06+
7.07=0.09
5.11+0.39
4.40=0.03
PCD (ours)
4.07+00.1
7.24
4.80-10.87
4.70-0.04
Table 12: Average 100th percentile hypervolume for RE tasks (part 1)
Method
RE21
RE22
RE23
RE24
RE25
RE31
RE32
D(best)
4.10
4.78
4.75
4.60
4.79
10.60
10.56
E2E + GN
2.60++0.00
3.18+0.00
0.00++0.00
1.66+0.00
3.81
5.06-0.73
4.96++0.13
F2E PC
2.60++0.00
1.91+1.774
0.00+0.00
1.66+0.00
3.29+0.71
6.54-0.13
7.07+0032
E2E
2.60++0.00
3.18++0.00
0.00++0.00
1.66+00.00
3.93++0.07
6.62-0.00
7.14+t0.20
MOBO
1.98:10.02
0.00+10.00
0.00+10.00
0.00+10.00
0.00+10.00
0.00=0.00
0.00:10.00
MH GN
2.58++0.03
1.91+1.74
0.00++0.00
0.00++0.00
3.19:0.61
5.97-0.36
4.37+10.3
MH PC
2.60++0.00
1.91+1.774
0.00++0.00
0.00+0.000
0.00++0.00
6.42-0.46
5.54+33.16
MH
2.60++0.00
3.18+0.00
0.00+0.00
1.66+0.00
3.41+
6.62-0.00
7,28+00.0
MM + COMs
2.59-10.00
1.64+1160
0.00+00.00
1.66+0.00
3.98++0.00
5.67-1.56
7.28++0.00
MM + ICT
2.60+10.00
3.18:10.00
0.00:10.00
1.66:00.00
3.98:10.00
6.54-0..8
7.28
MM + 1OM
2.60:10.00
3.18:10.00
0.00110.00
1.66100000
3.93
6.62-00.0
7.28:10.00
MM + TM
2.60-10.00
3.18:10.00
0.0011.00
1. 1.6600000
3.2740.64
6.62-0000
7.28
MM
2.60+0.000
3.18++000
0.00++0.00
1.66+0.00
3.77+0.08
6.62-0.00
7.28+0.00
ParetoFlow
4.09-10.06
4.78++0.07
4.80-E0.02
5.04
4.96:t0.15
10.56:10.14
11.03=0.38
PCD (ours)
4.35+10.00
4.89:10.11
4.84:10.0
188800002
5.06:10.03
11.35:+0.06
19.94=5.75
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Table 13: Average 100th percentile hypervolume for RE tasks (part 2)
Method
RE33
RE34
RE35
RE36
RE37
RE41
RE42
RE661
Dibest)
10.56
9.30
10.08
7.61
5.57
18.27
14.52
97.49
E2F + GN
0.00+0.00
6.31+00.00
0.00+-0.00
0.00+0000
3.34+00.00
15.13+0.000
0.00++0.00
45.11+00.00
E2E PC
0.0010.00
6.31110.00
0.0010.00
0.00.10.00
3.34.00.00
15.12.10.00
0.00.10.00
45,11
E2F
0.00++0.00
6.31+00.0
0.00+0000
0.00++000
3.34+-0.00
15.1240..00
0.00++0.00
45.11++0.00
MOBO
0.00=0.00
4.02++0.04
0.00+0000
0.00=0.00
2,80++0.00
13.03++0.03
0.00=00.00
0.00=0.00
MH GN
0.00++0.00
5.89++038
0.00+00.00
0.00+0000
3.334-0.00
14.99+0.12
0.00++0.0
45.11++0.00
MH PC
0.00=0.00
6.31+0000
0.00+0000
0.00=0.00
3.34+0.00
15.12=+0.00
0.00=00.00
45.11=0.000
MH
0.00=00000
6.31=0.00
0.00=10.00
0.00=00.00
3,34+0.00
15.12=0000
0.00=00.00
45.11=00000
MM+ COMs
0.00=0.00
6.31=0001
0.00+0000
0.00=0.00
3.33++0.00
15.10=0.044
0.00=00.00
45.01=0.000
MM + ICT
0.00=0.00
6.31=0.00
0.0010.000
0.00=00.00
3,34+0.00
15.13=0.00
0.00=00.00
45.11=0.000
I0M
0.00:10.00
6.31110.00
0.0010.00
0.0010..0
3.34110.00
15.13:10.00
0.00-10.00
45,11110.00
MM TM
0.00
6.11--0.01
0.00
0.00:0000
3.34:-0.00
15.12:40.00
0.00
45,11
0.0010..00
6.31110.00
0.0010.000
0.00.10.00
3.34110.00
15.12:10.00
0.00110.00
45,1110.000
ParetoFlow
10.84:0.224
11.69 1,33
11.12 0.43
0.17
6.51 0.58
18,20-10.41
54.44
112.08 11.62
PCD (ours)
10.97=0.00
10.17=0004
10.87=0.04
8.16=0011
5.97+0.01
18.14:0.02
15.49:40.03
NIA
Table 14: Average 100th percentile hypervolume for MONAS tasks (part 1)
Method
C10MOPI
C10MOP2
CIdMOP!
CIMMOP4
CIOMOPS
CI0MOP6
C10MOP7
CI0MOPS
C10/MPP9
D(best)
4.47
10.47
9:21
18.62
45.11
103.78
422.29
4,70
10.79
10:48=0.07
10.05-0.02
20.84-0.20
102.42:4.75
445.77+20.M6
4.614008
10.72-0.16
E2E PC
4.4740.01
10.52=0.07
10.18=0.01
21.64=0.13
49.22::0.09
105. 005.99::0.49
492.21 1.55
4.6520.08
10.46:20.29
EZR
4,47+00.00
18:55-0:.5
10.19-0.01
21. 21.70-0.13
48.87=0.05
106.17+0.36
494.92-1.3 1,70
4.77+0.03
10.64+0.15
MOBO
4.46++0011
10.31
9.57=00.0
20.80-0.01
49.01
106.54-00.04
304 00407700
4.85++002
11.05=0.05
10.45
9.53=20.26
19.50=1.33
48.47:50.41
82.40=4.76
4.4110.16
10.02-00.09
MH PC
4.46++0.00
18.53-00.04
10.19-0.82
21.67-0.08
49.13+00.12
136-03-00.99
492. 27 2.51
4.54+0.06
10.68+0.22
MH
4.47++0.01
10.47=0.03
10.17-00.02
21 21.26-0.45
49.18=00.07
105 88-00.64
490.43-
4.72+00077
11,10=0,27
MM COMs
4.69:0.01
10.49:00.33
9.952:0.06
20.84=0.33
47.86:20.75
12:0026
484.34:3.79
4.90:10.07
11.0
MM
4.46+0.07
10.52-00.5
10.00-0.07
20.85-0.19
48.98-001
106.08-00.35
4.6240.10
10:34+0.18
MM e JOM
4.46++0.01
10.42=00.05
10.12=0.04
21.71-0.04
49. 10.20=00.07
106. 28-028
494.33-
4.89+0.06
11.50=0.05
MM TM
4,4820001
10.46:20.09
10.14=00.0
21. 01.36-0..66
49.06-20.15 14
17:047200.17
491 12: 1686
4.5750.07
10:22::0.35 34
MM
4.46++0001
10.49-00.06
10.20-0.01 0.01
21,77
48.96-40-17
33--33-
400
4.82++0.05
10.99-0.26
ParetaPlaw
4.48+0.02
10.45=0.001
9.48==0.15
19.50=0.39
48
19:
435.00-+20.16
4.65+0008
10.79-#0.08
PCD (ours)
4.50:10.01
10.99:0.04
9.75
20.43:00.12
41.88.00.66
105 105.82:10.7
NIA
4.7910.02
10.99:10.19
Table 15: Average 100th percentile hypervolume for MONAS tasks (part
Method
IN-1K/MOP1
IN-IKIMUP2
IN-I//MPP3
IN-1K/MOP4
IN-IKIMUPS
IN-1K/MPP6
IN-IKMMP
IN-1KMMPS
IN-IK/MPP
2Nbest)
4.47
4.53
10.06
4.15
4.30
9.15
3.94
9.28
18,07
E2E - GN
4.63:0.04
4.16=0.08
0.93:007
4.44:0055
4.46:00.7
9:71
4:40:0009
9. 9.17::0.1 10:00.10
1884:1..0 30
E2E + PC
4.7010.04
4.07=0.10
102110033
4.50:20.10
4.531:0.9
9.8810.14
4.38:10.12
9.582.0.13
19.521:.22
E2E
4.201005
4.05-0.09
102510033
4.47.10.08
1.6210.02
100310.18
4.52:10.10
9. 9.721.0.0
2011210.13
MOBO
4.345085
4.27-0.84
9:970004
4.43-50110
4.52=00.5
9.546:111
3.94=0.05
9.20+ 10
19.75-00199
MII + GN
4.47:10.16
4.06-0.14
7.43:115 19
4.26:10.1 18
4,68.:0.11 0
9.41:0433
4.31:0.11
7,17
14.48 1.25
PC
4.641:0022
4,16=00007
1000710100
4.40:20.05
4.59200.0
9.7855125
4.45:00.02
9.502:0.1
19.28 10
MI
4.65100.5
3.81-0.22
10.1711005
4.58110.4
4.61100.2
10.0710110
4.5910.04
9 9,50.0..0
19.09 10.18
MM COMs
4.64:0.07
4.34-0.02
10.19:001
4.29-00222
4.51=0.11
4.15=0.08
9,73-01:01
2011 X0177000
1.09
4.42.10.7
4,57.00.10
9.7610.18
4:40:10.09
TE 10T
4.67:0002
4.09-0.17
10.2210022
4.00:10.45
4.55:20.12
9.561:0006
4.40-20133
0.
TM
4.6111003
4.1120012
4.18.10.03
4.57:0.03
9.7310.15
4.40.10.32
0.3910.16
19.2210177
MM
469=0005
4.18-0.5
4.59-0.09
0.64-0.85
10.34=0.17
456-007
9.65=0000
ParetoFlow
4.46110.1
0.35
9.660008
4.2811007
4.45
9.3100.19
3.81
9.30..0.09
10.5
PCD (ouns)
4.63.10.5
4.43.0.02
102200003
4.36.10.07
4.56.0..2
9.76101111
4.22.00.55
9.38.00.03
188311031
A.13
50TH PERCENTILE RESULTS
Tables
26 27
28
30 and 32 showcase the per task 50th percentile hypervolume (1)
across all considered tasks, In each task, the best performing model is bolded, while the runner up
is underlined.
26

---

Published as a conference paper at ICLR 2026
Table 16: Average 100th percentile hypervolume for MORL tasks
Method
MO-Hopper-v2
MO-Swimmer-v2
D(best)
6.33
3.70
E2E GN
5.94:10.33
3.65++0.02
E2E + PC
6.22+t0.15
3.66+10.00
E2E
5.98:E0.34
0.68
MOBO
4.75:10.00
0.86+
MH GN
5.73+:0.26
3.49
MH + PC
4.99:t0.34
3.42
MH
5.64+00.54
3.20
MM + COMS
6.14+0.15
3.68
MM + ICT
5.67++0.26
3.12+1.111
MM + IOM
4.95+:0.4
1.95
MM + TM
5.93+0.30
3.55=
MM
5.99+0.119
1.61
ParetoFlow
5.92
3.50
PCD (ours)
5.95+0..18
3.69
Table 17: Average 75th percentile hypervolume for synthetic tasks (part 1)
Method
ZDT1
ZDT2
ZDT3
ZDT4
ZDT6
D(best)
4.17
4.68
5.15
5.46
4.61
E2E + GN
2.60
2.77+0.00
3.17+0.00
4.59+0.15
3.79+00.5
E2E + PC
2.81+0.00
3.17400.00
3.1610.20
4.54
3.78+00.0
E2E
2.8110.00
3.1710000
3.4810.13
4.4310.23
3.82
MOBO
2.36.10.11
2.7440.01
3.01+0.05
4.60:10.06
3.70
MH GN
2.7810.02
3.1710.00
3.5410.00
4.4110.11
3.73
MH PC
2.26
2.68
3.27+0.22
4.55+0.15
3.76+L0.4
MH
2.81+0.00
3.1740.00
3.52+0.03
4.55+00.9
3.76+0000
MM + COMs
2.7910.02
3.1740.00
3.5410.00
4.4910.10
3.80
MM + ICT
2.8010.00
3.1740.00
3.5410.00
4.48
3.80
MM + IOM
2.80
3.1710.00
3.0610.09
4.5310.06
3.8210.04
MM + TM
2.78
3.1740.00
3.5410.00
4.47+0.20
3.82+0.04
MM
2.00
3.1710.00
3.5310.02
4.5710.18
3.76
ParetoFlow
4.11
5.45+0025
5.21+0.12
4.77+0.11
4.40
PCD (ours)
0.15
5.84
4.0910.16
4.91
4.31
27

---

Published as a conference paper at ICLR 2026
Table 18: Average 75th percentile hypervolume for synthetic tasks (part 2)
Method
DTLZI
DTL.77
OMNITEST
VIMMPP
VI.MPP2
VI.MP33
D(best)
10.60+0.00
8.5611.00
4.53:0.000
0.41=0.00
1.7810.00
45.65
E2E GN
10.63:10.00
8.8611.00
3.7810.16
0.16=0.01
2.1710.21
39.8711.22
E2E PC
10.6110.00
8.8610.00
4.6710.07
0.13=0.04
3.3410.01
42.4411.01
E2E
10.28+0.22
8.86++000
4.67+00.8
0.14=0.03
3.35+00001
42.45++0.00
MOBO
10.07++0.09
3.89=0.14
4.61+0.01
0.16-0.01
1.73+0.03
38.98+0.14 14
MH GN
10.6110.03
8.8510.01
3.4410.32
0.06-0.03
2.5110.09
29.5715.85
MH PC
10.6310.00
8.8610.00
4.3111024
0.16-0.06
2.8210021
38.20112.94
MH
10.6210.000
8.8610.00
4.30.0..0
0.20=0.05
3.3110.01
42.44.10.00
MM + COMs
10.30+0.04
8.86+0.00
4.25+0.32
0.18-0.12
1.65+0.02
1..72+t.499
MM + ICT
10.55=00.07
8.86+0.00
4.44+0.17
0.11=0.02
3.34+0.01
42.39++0.02
MM + IOM
10.6310.00
8.8610.00
4.72.10.03
0.29-0.00
3.3110.04
42.44.10.00
MM TM
10.6110.02
8.8510.01
4.1810..3
0.04
3.3510.01
42.34.10.06
MM
10.6110.02
8.8610.00
4.70.10.0
0.09
3,3410.01
42.45.1.001
ParetoFlow
10.83+0.36
10.32=0.43
4.78+0.00
N/A
4.21=00.33
45,90-10.02
PCD (ours)
13.59:1.78
11.26=0.15
4.34
0.01=0.01
2.12:00.4
45.81+10.04
Table 19; Average 75th percentile hypervolume for scientific design tasks
Method
Molecule
RFP
Regex
Zinc
D(best)
2.91=0000
7.11=0.00
3.96+0.00
4.52=0.00
E2E + GN
1.55++0.08
7.14=0.03
3.16:10.8
4.52=0.05
E2E + PC
2.47++0.07
7.15-0.03
4.15-10.77
4.54
E2E
2.41+00.1
7.17=0.02
2.99+-0.00
4.57=0.04
MOBO
1.18:0.66
7.21=0.01
5.35:10.22
4.31=0.05
MH GN
2.36=10.07
7.15=0.03
3.33-1046
4.52=0.07
MH + PC
1.73+1.004
7.11=0.04
4.21+0.41
4.54=0.05
MH
2.41=0001
7.15=0.04
2.99-1.00
4.56=0.05
MM + COMs
2.53+0.03
7.14=0.04
4.52+0.45
4.59=0.05
MM ICT
2.507
7.18
3.53+1081
4.60-0.05
MM + IOM
0.00+-0.00
7.18-0.11
4.46-00.2
0.03
MM TM
1.8340.45
7.13-0.05
3.73+1.00
4.53=0.06
MM
2.50
7.16=0.03
2.9910000
4.60-0.04
ParetoFlow
1.75+00.2
7.04
5.03+10.47
4.40=0.03
PCD (ours)
1.8410.09
7.19=0.02
4.1410025
4.50
Table 20: Average 75th percentile hypervolume for RE tasks (part 1)
Method
RE21
RE22
RE23
RE24
RE25
RE31
RE32
D(best)
4.10
4.78
4.75
4.60
4.79
10.60
10.56
E2E + GN
2.57+0.00
0.00:10.00
0.00:10.00
0.00:10.00
0.00=10.00
0.00=0.00
0.00:10.00
E2E PC
2.57+0.00
0.00:10.00
0.00:10.00
0.00:10.00
0.00=10.00
1.00=2.24
2.30++2.20
F2E
2.57++0.00
0.00++0.00
0.00++0.00
0.00++0.00
0.00++0.00
2.49-2.58
3.15+22.05
MOBO
1,94++0.01
0.00++0.00
0.00++0.00
0.00+0.00
0.00+0.00
0.00-0.00
0.00+00.0
MH GN
2,40++0.32
0.00++000
0.00++000
0.00++0.00
0.00+00.00
2.30-3.19
0,99+11.35
MH PC
2.57+40.00
0.00+10.00
0.00+10000
0.00:10.00
0.00+1.000
2.41=3.32
0.61
MH
2.57+0.00
0.00+0000
0.00+0.00
0.00=0.00
0.00=0.00
5.32-0.94
3.12422.10
MM COMS
2.56+00.01
0.00+0.00
0.00+0.00
0.00+0.00
0.00+0.00
0.00=0.00
1.41++1.25
MM ICT
2.57+0.00
0.00+0.00
0.00+0.00
0.00+0.00
0.00+0.00
5.00-1143
1.32+1.81
MM 1OM
2.57+00.0
0.00+0.00
0.00+0.00
0.00++.000
0.00+0.00
3.17=2.06
1.13+1.57
MM +
2.57+00.0
0.00+0000
0.00+0.00
0.00+10.00
0.00+0.00
5.72-0012
1.13
MM
2.57100.0
0.00:10.00
0.00:10.00
0.00+10.00
0.0010.00
2.32-3.18
2.76:10.1
ParetoFlow
4.0511005
4.72:10.08
4.75
4.51100.44
4.81
10.31110008
11.00
PCD (ours)
4.28-10.01
4.74
4.83-10.00
4.86-10.01
4.86-10.01
10.7441.13
10.67=0.03
28

---

Published as a conference paper at ICLR 2026
Table 21: Average 75th percentile hypervolume for RE tasks (part 2)
Method
RE33
RE34
RES5
RE36
RE37
RE41
RE42
RE61
D(best)
10.56
9.30
10.08
7.61
5.57
18.27
14.52
97.49
0.0010.00
6.29-1.01
0.00.10.00
0.00.10.00
3.33.10.00
14.76.10.16
0.00110.00
45.0910.01
E2E + PC
0.00+-0.00
6.28-
0.00++0.00
0,00++0.00
3.33-+0.00
14.84+-0.11
0.00+40000
45.09-00.00
0.00+00000
6.27=0.02
0,00+0.00
0,00++0000
3.35=00.0
14.81+0.10
0.00+00.00
45, 15.08=0002
MOBO
0.00=00000
3.80=0.10
0.00::0.00
0.00+00.00
2.79=00.00
12.99:0..06
0.001:0.00
N/A
MH GN
0.001:0.00
5.36
0.00110.00
0,00.10.00
3.28:10.04
14.23-10.71
0.00110.00
45 0.00
MH + PC
0.00=0000
6.27=0.02
0.00::0.00
0.00+00.00
3.33=00.00
14.78::0.14
0.001:0.00
45 0.001
MH
0.001:0.00
6,28
0.00110.00
0,00110.00
3.33:10.00
14.81
0.00110.00
45. 0.00
MM +
0.00+-0.00
6.29--0.01
0.00++0.00
0.00+-0.00
3.33++0.01
14.84+-0.18
0.00+-0.00
45. 0.00
MM ICT
0.00+00.00
6.27=0.02
0.00+0.00
0,00+0000
3.33=40.00
14.70=0.12
0.00+00000
45 0.00
+0M
0.00=00000
6.29=0.02
0.00+:0.00
0.00+0000
3.33=00.00
14.85+0.12
0.00+:.00
45 09002
MM+ + TM
0.00110.00
6.28
0.00:10.00
0,00110.00
3.33:10.00
14.7810.009
0.00100.00
45. 1.00
0.00::0.00
6.29 0.011
0.00
0.00
3.30
0.001
45
PuretoFlow
10 10.84-0.24
1..75+1.11
11.12+0.43
8.01--0111
6.19=00.1
17.88++0.49
35.94=7.81
111.35-1133 11.74
PCD (aurs)
10.86-0.07
9.96-0.04
10.82-40.01
7.924-0.04
5.96++0.02
18.05++0.00
15.42-0.044
N/A
Table 22: Average 75th percentile hypervolume for MONAS tasks (part 1)
Method
CI8MOPI
CINMOP3
C16MOP3
CI6МOP
CIUMOPS
CI6MOPS
CIOMOPT
CHUMOPS
C16/MРР9
Plbest)
422.29
4,70
E2E GN
4.4710.02
10.45
9.9220.15
20.65-0.19
48.35.10.48
92.88_00.7
130.13:11.84
4.3710.10
10.14:0.14
E2E + PC
1.M100.01
10.47:0.02
10.11=0.01
21.29-0.21
49.01-0.11
101.93.0.53
191. 1.25
4.5010.11
10.1910.27
EZB
4.4520001
10.46200.22
10.12-0001
21.33-0.25
48.7620.04
104.96-22.03
493.24 2,336
4.5610.10
10.24:0.18
MOBO
4.4410.01
10.43
9.41200.6
20.55-0.15
48.62.10.09
105.83-0.05
291945
4.7410.04
10.54.10.27
Mil+ GN
4.45100.0]
10.4110.01
9,37.10.3
19.30=1.25
42.5923.888
75.36_4.92
391.79.12177
4.1810.17
9.59-0.24
MH PC
4,44:20.01
10.46:20.02
10.11=0.04
21.20-0011
48.73:20.19
103,54:2.116
485.73:22448
4.36::0.06
10.38:10.22
MH
4.45110.00
10.45
10,11
21.02-0.40
49.01
105.10.10.57
485.29
4.4910.06
10.6811..27
MM COMs
4.4510.02
10.1510.02
9.7810007
20.53_0.12
103.2910.91
17919113.11
4,7710006
10.81-0077
MM + ICT
4.4410.01
10.45
0.87==0.08
20.59-0.30
1..25:
106. 10:10.11
488. 3.222.
4.4510.10
9.78-0.18
MM + 10M
4.45110.11
10.41
10.05-0.06
21.46-0.21
48.85.00.36
105. 74.057
493.0911.77
4.7910.12
11.33-0.12
MM + TM
4.4510002
10.12:00.85
10.01=0.11
21.02-0.18
48.59:10.49
105. 29.0.0
181.7917.11
4.4610.09
9.86=0.51
MM
4.4440.02
10.45
10.14=0.03
21.24=0.28
48.75:10.02
104.6::0.87
492.55 1.95
4.6610009
10.65
ParetoFlow
4.64100.4
10.44.0.01
9.35
19.4
41.4422.00
102. 102.22
424.23
4.4810.12
10.58
PCD (ours)
4.4320001
10.43:0.033
9.72:.0.09
20.25-00119
40.63:10.44
101.4911.10
NIA
4.6710:02
10.66:10.13
Table 23: Average 75th percentile hypervolume for MONAS tasks (part 2)
Method
IN-IR/MOPI
IN-IKMMOP2
IN-IK/MOP3
IN-IK/MOP4
IN-IKIMOPS
IN-IK/MOPG
IN-IK/MOP7
IN-IK/MOPH
IN-IK/MOP)
2Nbest)
4.47
4.53
10.06
4,15
4.30
9.15
3.94
9.28
18,07
E2E + GN
4.52#0.09
4.07=0.10
9.57+0001
4.3400.8
4,20=0.10
9.20++1.3
421=0.11
9.09+0.16
1878+0031
E2G PC
4.68:0.04
3.92-0.17
10 1910022
4.3710.09
4.421.0.07
96310.16
4.20:0.15
9.6510.11
19.3310.2 20
EZE
3.79=00.1
102440022
4.40:20.06
4.53-0.00
9.77
4.47:00.11
9.67200000
195441011
MOBO
4.311000
4,21-0.03
4.35100.6
0.44
9.5310013
3.79
9.25
19.65-101
4.39:0.16
3.87-0.22
7.00-1
4.18-00.21
4.33-00.21
9.30-+014 43
411-00112
7,05=1.2 1.73
14.45
PC
4.36.10.06
9.5410.24
4.63:00022
3.66=11.26
4.
4.52:00.34
0.54:0.06
9.53:10307
COMS
4.61:10.07
4.16-00.99
10.1710011
4.16:10.25
4.42:0.12
141.1001
4.3110.10
9,7020.34
ICT
467+000
3.91-0.21
1002-+008
431+0.05
4,47=0.00
9.1+0014
421-007
MM 10M
4.6410..3
3.25.0.25
10.201003
4.0610.46
4.43.0.08
0.3110003
4.25
9.7910..3
20.25 0.12
MM TM
4.5710.02
0.25
100210005
1.3610033
4.50:0.04
0.55:0113
4.2110.34
9.37
19.32100188
MM
4.67110.44
3.99-0.21
10271:0.6
4.5119.06
0.56:00.4
9.9114133
4.50:00.8
9.50200.09
198811028
ParctoFlow
4.37+0.07
4. 4.3400.6
9.38+0.10
4.6-00.88
4.4100.55
9.20--0277
3.63-0.12
9. 9.26+0.009
1886-+052
PCD (eurs)
4.49+0.01
4.240..55
10.88+002
4.20-0033
4.40+0.01
9.16+0077
3.11-0114
9.33+0.06
18.7
29

---

Published as a conference paper at ICLR 2026
Table 24: Average 75th percentile hypervolume for MORL tasks
Method
MO-Hopper-v2
MO-Swimmer-v2
D(best)
6.33
3.70
E2E GN
5.57:t0.04
3.60=+0.03
E2E + PC
5.62++0.01
3.65++0.01
E2E
5.27::0.48
0.89
MOBO
4.75:t0.00
0.86+10.00
MH GN
5.20=t0.42
1.08
MH + PC
4.75
1.04=:0.0
MH
5.07+00.44
2.46++1.42
MM + COMS
5.63+00022
3.64+00.0
MM + ICT
4.75++0.00
2.12+11322
MM + IOM
4.75
1,18:0036
MM + TM
4.75+00.00
1.11
MM
5.58+
1.00
ParetoFlow
5.59
2.93
PCD (ours)
5.07
2.82
Table 25: Average 50th percentile hypervolume for synthetic tasks (part 1)
Method
ZDT1
ZDT2
ZDT3
ZDT4
ZDT6
D(best)
4.17
4.68
5.15
5.46
4.61
E2E + GN
2.04
2.76+0.01
3.16+0000
4.49+0.13
3.74+0002
E2E + PC
2.7410000
3.17400.0
3.05+0.01
4.47+00.1
3.73+0.05
E2E
2.7410.00
3.1710000
3.1110.05
4.36.10.3
3.7610000
MOBO
2.35+0.02
2.7140.03
2.9240.04
4.43
3.70
MH GN
2.64.0.17
3.1710.00
3.1910.01
4.37.0.15
3.73
MH PC
2.25
2.68
3.1440.19
4.49+0.13
3.74+0.01
MH
2.74+0.00
3.1740.00
3.15+0.05
4.4940.10
3.76
MM + COMs
2.7410.00
3.1740.00
3.0710.00
4.42.0.12
3.70
MM + ICT
2.74+0.00
3.1740.00
3.13+0.03
4.11
3.74
MM + IOM
2.7410.00
3.1710.00
2.9710.01
4.4810.07
3.76100000
MM + TM
2.60
3.1740.00
3.20+0.01
4.44
3.71
MM
2.70
3.1710.00
3.1010.01
4.5011.18
3.76
ParetoFlow
4.10+0007
5.1240.10
5.02+0.18
4.68+00.13
4.33+00.3
PCD (ours)
4.02-10.07
5.5110010
3.3710.19
4.72L0..10
0.00
30

---

Published as a conference paper at ICLR 2026
Table 26: Average 50th percentile hypervolume for synthetic tasks (part 2)
Method
DTLZI
DTL.77
OMNITEST
VIMOPP
VI.MOP2
VI.MP33
D(best)
10,60
8.56
4.53
0.41
1.78
45.65
E2E GN
10.31
8.77+10.0
3.37110.14
0.00=0.00
1.9910.20
37.46.10.92
E2E PC
10.30:10.09
8.7710.00
4.2410.03
0.00=0.00
3.2710.07
40.1511.118
E2E
9.86++0211
8.77+00.0
4.31+0.21
0.00-0.00
3.33+0.02
40.27++0.00
MOBO
9.81+0.09
3.81=0.17
4.25+0.04
0.00=0.00
1.64+0.02
38.98+0.14 14
MH GN
10.1011..1
8.7811000
3,2611021
0.04-0.02
2.02.10.55
29.1915.40
MH PC
10.31110005
8.7710.00
4.0111021
0.09-0.07
2.5410021
35,96:11.79
MH
10.2110.05
8.77.00.0
4.04.10.41
0.00
3.1510.02
40.66110.7
MM + COMs
10.06+00008
8.78+0.01
4.01++0.3
0.06-0.13
1.55+00033
19.03+1.441
MM + ICT
10.22:40.05
8.77=0.00
4.12:40.3
0.00=0.00
3.34+0.01
39.29+00300
MM + IOM
10.3310..05
8.77.0000
4.2910..3
0.19-0.05
3,2510.05
40.2310.006
MM TM
10.24.10.04
8.7710.01
3,93.0.46
0.00
3.3410.01
38.88
MM
10.17110.08
8.77.10.00
4.2810022
0.00
3.3210.011
40.6910.95
ParetoFlow
10.43+0044
9.44=0448
4.78+00.00
N/A
4.21=00.33
45.45+0.41
PCD (ours)
12.11:0.551
10.61=0.07
3,70+1.17
0.00=0.00
1.80:10.03
45.68:10.03
Table 27: Average 50th percentile hypervolume for scientific design tasks
Method
Molecule
RFP
Regex
Zinc
D(best)
2.91=0.00 1.24
7.11=0.00
3.96+0.00
4.52=0.00
E2E + GN
6.96=0.13
2.9911000
4.42=0.02
E2E + PC
1.51+-0.92
7.03-0.12
2.99+0.00
4.39=0.09
E2E
2.26++0.07
7.10=0.03
2.99+0000
4.40=0.04
MOBO
0.00+:0.00
7.13=0.00
5.0511066
4.03=0.13
MH GN
1.23::069
6.97=0.05
3.16:1038
4.34=0.09
MH + PC
0.45+11001
7.05=0.01
3.76+1.47
4.33=0.12
MH
1.83+1.02
7.00=0.21
2.99+1.00
4.42=0.02
MM + COMs
2.31
7.07=0.04
3.4410.64
4.43=0.02
MM ICT
2.30
7.05
3.16-10.8
4.34
MM + IOM
0.00+-0.00
7.08
2.99+0.00
4.39-0.03
MM TM
1.1910.66
7.08=0.04
3.3610.82
4.35=0.07
MM
2.30+10.00
7.11=0.04
2.9910.00
4.45-0.03
ParetoFlow
1.26+
6.87=0.12
4.72+10.54
4.15=0.25
PCD (ours)
1.65+0.005
7.14
4.1410.25
4.32
Table 28: Average 50th percentile hypervolume for RE tasks (part 1)
Method
RE21
RE22
RE23
RE24
RE25
RE31
RE32
D(best)
4.10
4.78
4.75
4.60
4.79
10.60
10.56
E2E + GN
2.541:0.0
0.00:10.00
0.00:10.00
0.00:10.00
0.00=10.00
0.00=0.00
0.00:10.00
E2E PC
2.54+00.0
0.00:10.00
0.00:10.00
0.00:10.00
0.00=10.00
0.00=0.00
0.49+1.11
F2E
2.54+10.00
0.00++0.00
0.00++0.00
0.00++0.00
0.00++0.00
3.49-3.18
0.68+11.53
MOBO
1,90+-0.02
0.00++0.00
0.00++0.00
0.00+0.00
0.00+0000
0.00=0.00
0.00+0.00
MH GN
2.33-+0.29
0.00++0000
0.00++.00
0.00++0.00
0.00+0.00
1.16=2.60
0.00++0.00
MH PC
2.54-10.00
0.00+10.00
0.00+00000
0.00:10.00
0.00+10.00
0.00=0000
0.00:10.00
MH
2.54+00.00
0.00+0000
0.00+0.00
0.00=0.00
0.00=0.00
0.48=1.08
0.61
MM COMS
2.32+00.22
0.00+0.00
0.00++000
0.00+0.00
0.00+0.00
0.00=0.00
0.20+0.46
MM ICT
2.54++0.00
0.00+0.00
0.00+0.00
0.00+0.00
0.00+0.00
2,18-2.33
0.63+1.44
MM 1OM
2.54+00.00
0.00+0.00
0.00+0.00
0.00++.000
0.00+0.00
2,17=2.16
0.00++0.00
MM +
2.54+10.0
0.00+0.00
0.00+0.00
0.00:10.00
0.00+0.00
2.32-3.18
0.00+10.00
MM
2.5410000
0.0010.00
0.0010.00
0.00+10.00
0.0010.00
2.01=2.81
1.37
ParetoFlow
3.9810.08
4.65
4.6610.00
4.43-10.03
4.81
10.22:10.55
10.87=0.39
PCD (ours)
4.22-10.01
4.4910.08
4.82-10.00
4.85-10.01
4.84110.00
10.53-10.02
10.64=00.00
31

---

Published as a conference paper at ICLR 2026
Table 29: Average 50th percentile hypervolume for RE tasks (part 2)
Method
RE33
RE34
RE35
RE36
RE37
RE41
RE42
RE61
D(best)
10.56
9.30
10.08
7.61
5.57
18.27
14.52
97.49
E2E GN
0.00:1:.00
6.15:10.16
0.001:0.00
0.00:1:0000
3.32::0.01
14.47:10.05
0.00:10.00
45.05:1:0.00
E2E + PC
0,00+00000
6.224:0:01
0.00+:0.00
0.00:10.00
3.31=0.00
14.44:10.16
0.00+:0.00
45.062:0.08
E2E
0.00::0.00
6.21:1:000
0.001:0.00
0.00:10.00
3.31:00.00
14.49:1.12
0.00:10.00
45.07:10.03
MOBO
0.00++0.00
3.56-10.02
0.00
0.00--0.00
2.77-0.00
12.89--0.03
0.00++0.00
N/A
MH GN
0.00+00.00
4.56:t:0.4
0.00++0.00
0.00+00.00
3.11=0009
13.92:40073
0.00++0.00
44.72-+0.38 0.35
0.00
6.22.1..01
0.00
0.001:0.00
3.00=00.0
14.52:10.18
0.00:+0000
45.06-1:0.03
MH
0.00++0.00
6.21++0.00
0.00++0000
0.00++0.00
3.31=0.00
14.55++0.18
0.00++0.00
45.09++0.01
MM + COMs
0.00+0000
6,2244002
0.00++000
0.00+00000
3.22=00055
14.40:00115
0.00=+0.00
45.10::0.00
MM ICT
0.00++0.00
6.20++0.06
0.00++0.00
0.00++0.00
3.31-00.0
14.47+-0.10
0.00++0.00
45.06-+0.03
MM + JOM
0.00+0000
6.22++0.02
0.00+0000
0.00+0.000
3.31=0.00
14.43+0.09
0.00+00.00
45,0 45.07=+0.03
MM+ IM
0.00::0.00
6.061:029
0.00+:0000
0.00:0000
3.31=0.00
14.45:1.24
0.00+00.00
45.0910.02
MM
0.00:1:000
6.21
0.00100000
0.00110.00
3.31=0000
14.57:10.04
0.00:10.00
45 05.06.10.02
ParetoFlow
10,58=0.19
9.641.0.79
00.85=0.37
7.411:.24
5.89=0.51
17.7740.50
20.08::4.86
106.15=6.82
PCD (ours)
10.80=0.08
9.47:0009
10.57=0.03
7.71::0.03
5.83=0.02
17 17.98:0.03
15. 19:0077
N/A
Table 30: Average 50th percentile hypervolume for MONAS tasks (part 1)
Method
C10MOP1
C1OMOP2
CIOMOP3
CINMOP
CIMMOPS
CI6MOP6
C10MOP7
C10MMOP
CI6/MPP9
PPbest)
4.47
10.47
9:21
18.62
45.1
103.78
422.29
4,70
10.79
EZR. + GN
4.43+0001
10.41-0009
9. 9.25++0.14
19.20
47.16-00.77
89 89.27-1.06 1.06
41839-55.31
4.13+0.26
9.59-0.24
E2E, PC
4.41+0002
10.43=50.02
10.01=0.04
20.06-0.24
48 48.40=00.22
103.72:0024
3.000
4.3750.07
9.87=0.33
EZE
4,41:00.03
10.40=0.36
10.01-0000
20.47 0.25
48.43=00.24
10381=22.47
9300
4.43:0.55
9.96-0.14
MOBO
4.42+0001
10.40=00.02
9.27+0001
19.01-0.2 0.45
48.06-90.46
105.42-0004
2H1. 58--4-9-
4.62+0.13
10.00+0.28
MH GN
4,40:20.06
10.28:20.2
0.96:0047
17.71=1.21
39.43:22.66 56
71 71,98-5.95 5.98
365 1771:19 19.94
4.02::0.15
9.24-046
MH
4,41++0.03
10.30-00.18
20.19-0.22
47.83-00.40
98.46-5.65
10.03-0.32
4,42+00.1
10.43-00.01
0.00
19.98-0.35
48.89=0.02
2.44
476
4.39+:0.04
10.36+0.37
MM + COMS
4.43:00111
10:44:0001
9.70:20.08
19.38-0.11
44.39:12.16
10:115:1::47
461 28 7.22
4.65::0.04
10.55:0.11
MM + ICT
4,42+0.01
10:43-00.11
9.69==0.08
18.63- 1.03
45.65-+1.25 26
10404-0-49
477.05-33001
4.35+-0.09
9.47-0.11
MM JOM
4.40+0005
10.28-0.22
9.91+0.10
20.12 20.12-0.25
47.59-00.78
104.61+0.11
482.85++4.11
4.71+00.15
11.22-4.13
MM + TM
4.4210001
10.31
9.92:00133
19.66:0.45
47.72:81.00 01
103,33:1 104
162.91111.81
4.351:0.77
9.57=045
MM
4.38++0.02
10:44-00.1
10.02-0.155
10.11-0.38
48.5K-H0. 12
102.62-0.95
485. 56-339944
4.53+00.14
10.33-0.48
ParcioFlow
4.37100.66
10. 10.000
9.0010.42
18.26.0078
39.0010.60
100.52:2.07
18..12111177
4,3010.08
10.37:0.100
PCD (ours)
4.4110.01
10.26:00.3
9.3120.11
18.40_00.8
39.61.0.400
101.5711.19
NIA
4.6010.01
10.3420.13
Table 31: Average 50th percentile hypervolume for MONAS tasks (part 2)
Method
IN IK/MOPI
IN IK/MP?
IN IK/MPPS
IN IN-IKMMP
IN IK/MPPS
IN JK/MPPS
IN IK/MPP
IN IK/MPPS
IN IN-IXMMPP9
Dibest)
4,47
4.33
10.05
4,15
4.30
9.15
3.94
9.28
18.07
E2G + GN
4.41.10.16
3.77.0.30
9.4100.33
4.10.10388
3.42.0.14
86210.96
3.71
8.5910.26
17.6810.99
EZE
4.66::0.14
3.84:0.15
101755002
4.18:0018
4.3-0006
9.3680111
400:00177
9.342:0.13
190550034
46610004
166-0.10
10.00000
434-0006
0.45-0.01
0.51+0007
438-00.9
0.90-D.11
19.50+041
1.6510.13
MIl
4.2510.21
8.910..66
6.72.1..6
461=0001
397=91116
9946:118
424:0.05
4.39::0.05
9.286:120
422=00088
9.15:00.16
461-0002
3.42-0.18
11:000
442++002
0.46::0.5
9
4.45-0.08
9.45=00.77
19.30
COMs
4.50+0.9
13:000
3.95+0.27
4.35-0015
746+0.80
3.91-0.05
0. 0.60-0.00
19,00-LL
MM ICT
4.64110.3
3.79
9.2.00088
4.23.10.06
1.3
9.201.0113
4.15.00.6
8.55
1005
MM 10M
4.59:50.06
3.76-1133
10.174003
1.93:10.55
0.
9.06-5111
4.18:=0.06
9.74-11066
20.10-0.21
MM TM
4.52:00.3
0.22-..3
0.
4.20-002
0.42
0:23--0100
0.2--.335
9. 16:60.22
1887:0.35
VM
465-+0.03
1.55-0.44
10
4.43-0.05
4.58-0.55
9,70--022
4.4-0.09
9. 0.52-0.07
19
ParetaFlow
4,29110.55
0.25
9.440005
4.10.10.18
4,2610.16
9.1110024
3311
8.851
18.4410.50
PCD (ours)
4.41-10.02
0.06
1010010.100
4.11-10.33
4,2710.022
8.8210.02
3.30.10.15
8. 8.96210.02
17.6310771
32

---

Published as a conference paper at ICLR 2026
Table 32: Average 50th percentile hypervolume for MORL tasks
Method
MO-Hopper-v2
HO-Swimmer-v2
D(best)
6.33
3.70
E2E + GN
5.06:L0.44
1.02:1.17
E2E+ PC
0.84
3.61
E2E
4.92:t0.39
0.87+0.01
MOBO
4.75+t0.00
0.86+#.00
MH GN
4.90-L0.35
0.97
MH + PC
4.75+00.00
0.95++0.1
MH
4.91
0.97
MM + COMs
5.32+0.34
3.50+0.26
MM + ICT
4.75+00.00
0.96++0.10
MM + IOM
4.75+00.00
1.00
MM + TM
4.75
0.93
MM
4.75-L0.00
0.92+1.10
ParetoFlow
5.38
2.66
PCD (ours)
4.83:10.02
2.40
33


Reviews:
[{"review_id": "Bo3E2FrA9H", "reviewer": "Reviewer_mkio", "summary": "This paper focuses on offline multi-objective optimization (MOO) and addresses the core challenge of generalizing beyond observed data in static datasets. The motivation stems from existing offline MOO methods, which either rely on error-prone surrogate models or multi-stage pipelines (e.g., ParetoFlow) and struggle to eliminate dependencies on intermediate guidance mechanisms.\n\nThe proposed core method, **Pareto-Conditioned Diffusion (PCD)**, reframes offline MOO as a conditional sampling problem using diffusion models. It integrates two key components: a multi-objective reweighting strategy (prioritizing samples near the Pareto front) and an NSGA-III-inspired reference-direction mechanism (generating diverse conditioning points).\n\nExperiments on five benchmark categories (synthetic, MORL, real-world, scientific design, MONAS) show PCD achieves competitive HV scores and outperforms baselines (e.g., ParetoFlow) in cross-task consistency with a single set of untuned hyperparameters. Ablation studies validate its reweighting and reference-direction components.", "questions": "See weaknesses", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. PCD eliminates the need for explicit surrogate models or scalarization schemes (common in existing methods like MOBO or ParetoFlow), simplifying the optimization pipeline and reducing risks of exploiting surrogate inaccuracies.\n2. The multi-objective reweighting strategy (Equation 6) balances dense bins and high-performance samples, while the reference-direction mechanism (inspired by NSGA-III) ensures diverse, high-quality conditioning points. This design is sound and well-motivated.\n3. PCD maintains consistency across tasks with a single set of hyperparameters, and ablation studies confirm the soundness of its mechanism.", "weaknesses": "1. PCD underperforms on some challenging tasks, including MORL (high-dimensional) and MONAS (purely categorical). I expect more discussion and guidance for future research. For example, are there targeted modifications (e.g., dimensionality reduction, specialized denoiser architectures) that might address this?\n2. The paper explicitly excludes combinatorial tasks (citing needs for specialized denoising) but provides no technical discussion of how PCD could be extended to this subset of MOO.\n3. The reference-direction mechanism involves extrapolating points along direction vectors and adding Gaussian noise. Please provide more intuition for key choices (e.g., extrapolation distance, noise variance).\n4. The sensitivity analysis (A.2) shows τ must be adjusted for datasets with high quality variance (e.g., ZDT2). How can practitioners **systematically estimate dataset variance** (e.g., via dominance number distribution) and select it without extensive tuning?", "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": "ZgElk1F2za", "reviewer": "Reviewer_pgCD", "summary": "The paper proposes Pareto‑Conditioned Diffusion (PCD), an end‑to‑end framework for offline multi‑objective optimization (MOO) that removes explicit surrogate predictors. PCD trains a conditional diffusion model to sample designs x conditioned directly on target objective trade‑offs y. Two key components make this work in the offline regime: (1) a multi‑objective reweighting of the static dataset that emphasizes bins with high‑quality, non‑dominated samples (using dominance numbers and binning) (2) a reference‑direction mechanism to generate a diverse set of conditioning points by assigning offline points to uniformly spread directions and extrapolating slightly beyond the dataset. The model is then sampled with classifier‑free guidance. On Off‑MOO‑Bench, PCD reports competitive HV.", "questions": "See above", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Framing offline MOO as conditional sampling is elegant: a single model generates candidate sets conditioned on desired trade‑offs, sidestepping surrogate‑then‑optimizer pipelines.\n2. Method components are well‑motivated:\n- The reweighting combines dominance‑based quality with bin density, to emphasize promising regions without discarding too much signal\n- The reference‑direction procedure provides diverse, high‑quality conditioning targets for sampling\n3. The paper evaluates across different task families and runs targeted ablations that isolate each module’s effect; Table 2 shows the reference‑direction mechanism and reweighting both help on several tasks.\n4. The authors highlight contradictions with prior works or expectations and give plausible hypotheses. This transparency is commendable.", "weaknesses": "1. Computation of dominance numbers. Computing the dominance number o(x) for reweighting is quadratic in dataset size if done naively (Eq. 5). The paper would benefit from complexity notes or empirically evaluating the time cost for sampling with the proposed PCD method.\n2. Benchmarks are mostly m≤3 objectives. It’s unclear how the reference‑direction generation and conditioning scale when m grows (e.g., direction coverage, sampling stability, and the number of conditions L required). A small synthetic study with m>3 would help.", "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": "lKHVLaqZgm", "reviewer": "Reviewer_QMRg", "summary": "This paper proposes Pareto-Conditioned Diffusion models (PCD) for offline multi-objective optimization (MOO), which aims to obtain a group of high-performing (i.e., Pareto optimal) final solutions across multiple objectives using only a fixed offline dataset $ \\mathcal{D} = \\\\{ (\\boldsymbol{x}_i, \\boldsymbol{y}_i) \\\\} _{i=1}^N$, without any online iterative evalution. While prior works mainly focus on modeling $p(\\boldsymbol{y}|\\boldsymbol{x})$ and obtaining the inputs that maximize the model’s output, PCD models $p(\\boldsymbol{x}|\\boldsymbol{y})$ instead via conditioned diffusion models, extending the practice of [1] to the MOO scenarios. Specifically, unlike directly parameterizing the distribution, the authors train the models via binning the objective space $\\mathbb{R}^m$, quantified by the dominance number, and assigning weights to each bin, calculated by considering both the number of points inside the bin and performance based on the dominance number. After training, they generate a diverse set of condition $\\boldsymbol{y}$ and sample for final solutions via classifier-free guidance. Experimental results demonstrate the superiority of PCD over a wide range of surrogate-based methods.", "questions": "- Beyond the sampling procedure delivered in Fig. 5,  can PCD recover the dataset distribution by conditioning on different objective space, like the Fig. 4 in [1]? For example, I suggest you examine the modeling performance of PCD by conditioning on different Pareto front layers in the offline dataset (or the test data provided by [6]) and validating if the model can recover the ground-truth points.\n- How does the hypervolume change as the sampling timesteps promote?\n\nI can understand that adding experiments on all task from [6] is quite expensive during rebuttal. I would be pleased that you conduct additional experiments on some representative tasks.\n\n## References\n\n[1] Diffusion models for black box optimization. ICML 2023.\n\n[2] ParetoFlow: Guided Flows in Multi-Objective Optimization. ICLR’25. \n\n[3] Preference-Guided Diffusion for Multi-Objective Offline Optimization. NeurIPS’25. \n\n[4] A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II. IEEE TEvC, 2002. \n\n[5] MOEA/D: A multiobjective evolutionary algorithm based on decomposition. IEEE TEvC, 2007.\n\n[6] Offline multi-objective optimization. ICML 2024.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 2, "presentation": 4, "contribution": 3, "strengths": "- This paper is well-written and easy to follow.\n- The idea of extending forward-surrogate-free (i.e., $\\boldsymbol{y} \\to \\boldsymbol{x}$ instead of $\\boldsymbol{x} \\to \\boldsymbol{y}$) methods to offline MOO is novel. It is still unknown that which of the modelings is better, and the discussion in this paper brings new insights into this field.\n- Compared to recent generative-based methods [2-3], PCD does not rely on forward surrogate or classifier, which may introduce compounding error in both the surrogate and the generative model. Instead, PCD employs classifier-free guidance to sample final solution, which is simpler and shows flexible inference control in diffusion model community.\n\nOverall, I think this paper is of high quality. Once my concerns listed below are addressed, I am willing to adjust my ratings.", "weaknesses": "- The multi-objective reweighing scheme is somehow heuristic, mainly regarding the choice of dominance number as ranking metric.\n    - From my perspective, the dominance number is just a quantification of the Pareto dominance. However, in MOO, there are still other metrics to further comprehensively examine the goodness of the solutions, e.g.,  the Pareto dominance together with crowding distance from NSGA-II [4] (also used in [3]) and the scalarization mechanism from MOEA/D [5]. I suggest comparing some of them.\n- Typos:\n    - Not correctly using ``\\citep{}``, in line 53, 292, 761.\n    - line 324: “Baselines.” → “Baseline”", "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": "AvSnFmHDYb", "reviewer": "Reviewer_MLZS", "summary": "The paper proposes Pareto-Conditioned Diffusion (PCD) for offline multi-objective optimization (MOO). Instead of learning explicit surrogates and optimizing them with MOEAs, PCD trains a conditional diffusion model over designs conditioned on objective vectors. Two key ingredients: (i) a multi-objective reweighting scheme that emphasizes bins with high-quality nondominated solutions; and (ii) a reference-direction conditioning set (NSGA-III-inspired with Riesz s-energy vectors + extrapolation + Gaussian jitter) to generate diverse target trade-offs.", "questions": "See Weakness.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The dominance-aware bin reweighting and NSGA-III-style conditioning looks intuitive and principled.", "weaknesses": "1. > Unlike prior methods (Xue et al., 2024; Yuan et al., 2025) that rely on complex, multi-stage pipelines of surrogate predictors\nor separate optimization algorithms, PCD offers a more direct, end-to-end framework.\n\nIn my view, ParetoFlow is classifier guidance while this PCD is classifier-free guidance. I do not think the former is more complicated. Although paretoflow requires two stages: predictor training and generative model training, these are established pipelines for that and thus they should be simple instead of complex.\n\n2. The propose conditioning search looks interesting. Have u compared the uniform vectors from Das-Dennis?\n\n3. It looks weird in MORL that no methods can beat against the offline best? so what is the point of all methods?\n\n4. Have u tried flow models instead of diffusion models? \n\n5. I doubt whether we really need diffusion models for some tasks mentioned here. Diffusion models can be very useful it the data has meaningful patterns to learn like image, text, protein. In this context, some designs are pure math numbers. Do they have meaningful patterns?", "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 (Oral)