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Improving Pareto Front Learning via Multi-Sample Hypernetworks

Long P. Hoang, Dung D. Le, Tran Anh Tuan, Tran Ngoc Thang

arXiv:2212.01130v7cs.LG

TL;DR

Existing Pareto-front learning methods ignore relationships among solutions, limiting the quality of the learned front. PHN-HVI generates multiple preference-conditioned solutions and maximizes their hypervolume; reported comparisons show full baseline-front coverage in one setting and a uniform toy-example distribution, though COSMOS does not converge.

  • Problem

    Existing PFL methods ignore relationships among Pareto solutions during optimization, a gap the paper connects to the quality of the learned front.

    Source: §1 · p. 1

  • Method

    PHN-HVI uses a hypernetwork to generate multiple solutions from trade-off preferences and maximizes the hypervolume defined by those solutions.

    Source: §1 · p. 1, §3 · p. 2, §6 · p. 7

  • Results

    PHN-HVI's front covers baselines entirely in Quadrant I and is described as uniformly distributed in the toy example, while PHN-HVI does not converge on COSMOS.

    Source: Image Classification · p. 6, Toy Examples · p. 6, Text Classification and Regression · p. 6

  • Takeaways & Limitations

    A variety of solutions representing different trade-off preferences offers a global view of the current Pareto front.

    Source: §1 · p. 1

    Hypernetwork-based PFL cannot approximate disconnected Pareto fronts, leaving effective PFL without a hypernetwork as an open question.

    Source: §6 · p. 7

Abstract

from arXiv · show

Pareto Front Learning (PFL) was recently introduced as an effective approach to obtain a mapping function from a given trade-off vector to a solution on the Pareto front, which solves the multi-objective optimization (MOO) problem. Due to the inherent trade-off between conflicting objectives, PFL offers a flexible approach in many scenarios in which the decision makers can not specify the preference of one Pareto solution over another, and must switch between them depending on the situation. However, existing PFL methods ignore the relationship between the solutions during the optimization process, which hinders the quality of the obtained front. To overcome this issue, we propose a novel PFL framework namely PHN-HVI, which employs a hypernetwork to generate multiple solutions from a set of diverse trade-off preferences and enhance the quality of the Pareto front by maximizing the Hypervolume indicator defined by these solutions. The experimental results on several MOO machine learning tasks show that the proposed framework significantly outperforms the baselines in producing the trade-off Pareto front.

1 Introduction

Multi-objective optimization requires choices among conflicting objectives, but many methods require preferences in advance or separate models for Pareto-front points. The paper frames Pareto Front Learning as a one-shot hypernetwork mapping from preferences to Pareto solutions, then proposes multi-sample hypervolume guidance.

1 Introduction

  • Many MOO methods require advance trade-offs or a separate model per front point, while existing PFL methods use random preferences and ignore relationships among solutions.These limitations matter when users must choose or switch trade-offs on the fly.Source: §1 · p. 1
  • PHN-HVI generates multiple Pareto solutions from diverse preference vectors and maximizes their defined hypervolume to guide front learning.The framework builds on Pareto Front Learning with hypernetworks and evaluates the approach against baselines.Source: §1 · p. 1

2 Preliminary

  • PFL with a hypernetwork is formulated as one-shot approximation of the entire Pareto front, mapping a preference vector to a Pareto-optimal solution.The formulation sets the basis for the proposed framework.Source: §2 · p. 2

3 Multi-Sample Hypernetwork

The method generates several target networks from sampled preferences and uses their joint objective losses to guide Pareto-front learning. It combines hypervolume optimization with preference sampling strategies, and also describes a Stein variational alternative whose investigation is deferred.

3 Multi-Sample Hypernetwork

  • For each sampled preference r_i, the hypernetwork h generates a target network with parameters θ_i.The method samples p preference vectors for the multi-sample hypernetwork.Source: §3 · p. 2

HV Indicator Hypernetwork

  • PHN-HVI maximizes the generated solutions' hypervolume, whose maximum implies each θ_i is Pareto optimal; cosine similarity spreads the front and aligns solutions with preferences.The hypervolume gradient approach is described as scalable for domination-ranked fronts.Source: HV Indicator Hypernetwork · pp. 2–3
  • Only hypernetwork parameter φ is optimized; at inference, one preference vector r yields its corresponding Pareto local optimum θ_r.The update direction combines the hypervolume indicator with the cosine-similarity term.Source: HV Indicator Hypernetwork · p. 3

Partitioning Algorithm

  • Partition sampling seeks even preference coverage via Delaunay subregions, but is used only for two objectives because partitions grow exponentially; higher-dimensional cases sample from Dir(α).Delaunay triangulation constructs the partition, while the algorithm switches to Dirichlet sampling for three or more objectives.Source: Partitioning Algorithm · p. 3

Stein Variational Hypernetwork

  • The Stein variational alternative generates p target networks and uses a minimum-norm vector from the loss-gradient convex hull; multi-sample profiling remains future work.The authors reserve investigation of this multi-sample extension for future work.Source: Stein Variational Hypernetwork · pp. 3–4

4 Related Work

Related work spans classical multi-objective optimization, multi-task learning, and hypernetwork-based Pareto-front learning. The paper distinguishes prior methods by their trade-off handling, front coverage, and computational constraints.

4 Related Work

  • Classical MOO approaches include evolutionary methods such as VEGA and NSGA-III, scalarization, monotonic optimization, and common-descent methods.The cited scalarization methods require convex-function conditions to approximate the entire front, while common-descent methods cannot take decision-maker preferences.Source: §4 · p. 4
  • Multi-task methods address task balancing or gradient conflicts, but balanced-solution approaches do not simulate trade-offs; several preference-based and finite-front methods also face rising computational cost.The passage notes that computational cost can become impractical as preference vectors and objective functions increase.Source: §4 · p. 4
  • Prior hypernetwork PFL methods combine hypernetworks with scalarization or EPO, or concatenate preferences with data; their inherited drawbacks or unclear preference-data interaction remain concerns.PHN-LS and PHN-EPO retain features and drawbacks of their optimization techniques, while efficient interaction between least-squares loss and cosine similarity is not assured in all situations.Source: §4 · p. 4

5 Experiments

Experiments compare PHN-HVI with existing methods on toy problems and multi-task learning datasets. Results vary by setting: PHN-HVI produces well-distributed fronts in several cases, while the regression analysis finds diminishing gains from adding rays.

5 Experiments

  • The experiments compare PHN-HVI with PHN-LS, PHN-EPO, and COSMOS, evaluating methods with shared, well-spread preference vectors and selecting multi-task models by validation hypervolume.Toy examples use 200 evenly distributed test preferences; multi-task experiments select the model with the highest validation-set HV.Source: §5 · p. 4

Toy Examples

  • In Problem 3, PHN-LS and PHN-HVI can profile the disconnected Pareto front by removing dominated solutions, whereas PHN-EPO produces only part of the front.In Problem 4, PHN-HVI produces a fairly uniform distribution from 231 uniform 3D rays; PHN-LS fails, while COSMOS and PHN-EPO are widespread but somewhat chaotic.Source: Toy Examples · p. 6

Image Classification

  • On Multi-Fashion+MNIST, PHN-HVI’s Pareto front outperforms and covers the baselines, while PHN-LS solutions are not widely dispersed.The comparison evaluates 25 uniformly distributed preference vectors.Source: Image Classification · p. 6

Text Classification and Regression

  • On the Drug Review dataset, PHN-HVI has greater hypervolume than previous techniques and permits the Pareto front to move deeper; COSMOS does not converge.The reported COSMOS issue is attributed to mapping a preference vector to a very high-dimensional embedding feature.Source: Text Classification and Regression · p. 6

6 Conclusion and Future Work

The conclusion describes PHN-HVI as using multiple trade-off preferences followed by hypervolume maximization, and notes that hypernetwork-based PFL struggles with disconnected Pareto fronts.

6 Conclusion and Future Work

  • PHN-HVI uses a variety of trade-off vectors simultaneously, followed by hypervolume maximization.Source: §6 · p. 7
  • The conclusion’s figures examine loss-space partitioning and the impact of hyperparameter λ.Source: Figure 7 · p. 7, Figure 8 · p. 7
  • Hypernetwork-based PFL cannot approximate disconnected Pareto fronts well, prompting the question of whether PFL can work effectively without a hypernetwork.Source: §6 · p. 7

A Experimental Details

The experimental details specify a two-hidden-layer hypernetwork and task-specific training settings for image, text, and regression experiments.

A Experimental Details

  • The hypernetwork is an MLP with two hidden layers, linear heads, hidden dimension 100, and dropout at its last layer.Its parameters are selected based on references to the original papers.Source: A Experimental Details · pp. 7–8
  • Image classification uses Adam, batch size 256, up to 400 epochs, and PHN-HVI learning rate 5e−4; PHN-LS and PHN-EPO use α = 0.2.PHN-HVI uses α = (0.5, 0.5) and dropout 0.1; early stopping and learning-rate reduction are also specified.Source: A Experimental Details · p. 8
  • Text classification and regression use Adam, batch size 256, up to 400 epochs, and learning rate 1e−3; COSMOS does not converge on this task.PHN-HVI uses α = (0.5, 0.5), and all methods use dropout 0.1.Source: A Experimental Details · p. 8
  • Multi-output regression runs up to 1000 epochs with batch size 64/512 for Jura/SARCOS and normalizes outputs using dataset-specific scales.PHN-HVI uses α = (1/7, . . . , 1/7); no dropout is used for these methods.Source: A Experimental Details · p. 8

B Warm up for Multi-Sample Hypernetwork

The multi-sample hypernetwork receives a one-epoch warm-up to help its quality and extra-criterion functions interact.

B Warm up for Multi-Sample Hypernetwork

C Additional results of Partition

Additional results examine partition sampling across datasets, reporting both a favorable case and a potential drawback.

C Additional results of Partition

D Incompatibility of LS and Cosine Similarity

In Toy Problem 3, COSMOS does not converge because the Linear Scalarization solution is very different from a preference ray. PHN-HVI’s evenly spread points stay close to preference rays, allowing the HV function and cosine similarity to interact effectively.

D Incompatibility of LS and Cosine Similarity

E Training cost

Training requires running and storing p target networks per iteration, though their forward computations can be parallelized and adding rays eventually brings little improvement. At test time, inference takes the same time as the baselines.

E Training cost

  • Adding rays eventually no longer significantly improves results, while the forward computations of p target networks can run in parallel.The approach still requires running and storing p target networks at each iteration.Source: E Training cost · p. 8
  • The method advances results over other methods without extra trainable parameters, despite its more expensive training cost.Source: E Training cost · pp. 8–9
  • At testing, Multi-Sample Hypernetwork and the baselines have identical inference time because each preference ray provides its corresponding target network.Source: E Training cost · p. 9

F Selecting the HV reference point

The HV reference point should be fixed across iterations and equal on every coordinate when the true Pareto front is unknown and no objective is prioritized. A point above randomly initialized losses is often sufficient; early losses above the reference point can be temporarily rescaled.

F Selecting the HV reference point

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