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Simple Extensions of Single-Objective Acquisition Functions and Hedge Strategies for Multi-Objective Bayesian Optimization

Haris Moazam Sheikh

arXiv:2609.31940v1cs.LGmath.OCstat.ML

TL;DR

MOBO must handle trade-offs among objectives when preferences are unknown, and existing methods often use specialized formulations. This paper extends single-objective acquisitions and hedge strategies with Pareto-aware hypervolume selection; its methods match or outperform more complex state-of-the-art approaches across benchmarks.

  • Problem

    In non-preferential MOBO, preferences among competing objectives are often unknown, so methods must promote coverage across Pareto trade-offs.

    Source: §1 · p. 1

  • Method

    The framework uses single-objective acquisitions to generate candidates and hypervolume-based selection, and extends hedge strategies to adaptively choose acquisitions using performance feedback.

    Source: §1 · p. 2, §4.1 · p. 5, §4.2 · p. 6

  • Results

    Across the benchmark suite, the proposed methods match or outperform more complex state-of-the-art MOBO approaches, and no current baseline dominates uniformly.

    Source: §6 · p. 8, §7 · p. 9

  • Takeaways & Limitations

    A lightweight Pareto-aware extension of classical single-objective acquisitions can yield consistently strong performance across multi-objective problems.

    Source: §6 · p. 9

    qHAX-α relies on an inner multi-objective optimization step, which generally uses evolutionary or population-based algorithms that do not naturally use efficient gradient-based optimization.

    Source: §6 · p. 9

Abstract

from arXiv · show

Multi-objective Bayesian optimization (MOBO) is commonly approached through specialized acquisition functions or scalarization schemes designed to explicitly account for trade-offs among non-preferential objectives. In this work, we show that such complexity might be unnecessary. We propose a framework that extends standard single-objective acquisition functions directly to the multi-objective setting through a hypervolume-based transformation. We further extend hedge strategies for acquisition functions, which are typically used only in single-objective optimization, to the multi-objective regime. Our approach requires minimal modification to existing Bayesian optimization pipelines and avoids the need for bespoke multi-objective formulations. We demonstrate how a broad class of commonly used single-objective acquisition functions and hedge strategies can be adapted in a principled manner to handle multiple objectives, while preserving their intuitive interpretation and computational efficiency. Empirically, we evaluate the proposed methods across a range of synthetic and real-world multi-objective benchmarks. Despite their simplicity, our extensions consistently match or outperform more complex state-of-the-art MOBO methods in terms of optimization performance and sample efficiency. These results suggest that effective multi-objective Bayesian optimization can be achieved by reusing and carefully extending well-established single-objective acquisition strategies, offering a simpler and more flexible alternative to existing approaches.

1 Introduction

In non-preferential multi-objective problems, preferences may be unknown and the aim is to approximate Pareto trade-offs. This work reuses single-objective acquisitions and extends hedge strategies through Pareto-aware, hypervolume-based selection.

1 Introduction

  • The framework uses standard single-objective acquisitions to propose candidates, then applies Pareto-aware hypervolume ranking to select evaluations.This approach is presented as preserving classical acquisitions’ interpretability while enabling non-preferential MOBO.Source: §1 · p. 2
  • Because no single acquisition performs best across all problem classes, hedge strategies use historical performance to choose among acquisitions.In MOBO, the method must also account for objective scaling, geometry, and Pareto-front coverage.Source: §1 · p. 2
  • MO-Hedge adaptively selects among acquisition functions using hypervolume-based performance feedback.The paper addresses the relative lack of hedge strategies for MOBO.Source: §1 · p. 2
  • Across multi-objective benchmarks, the proposed methods consistently match or outperform more complex state-of-the-art MOBO approaches.The paper presents these methods as requiring minimal pipeline disruption.Source: §1 · p. 2
  • The paper concludes that extending established single-objective strategies can offer a simpler, more flexible alternative to specialized multi-objective acquisition design.The empirical evaluation spans multi-objective benchmarks.Source: §1 · p. 2

2 Preliminaries

The preliminaries define Bayesian optimization for expensive black-box objectives and describe multi-objective optimization through Pareto-optimal solutions and set-based quality indicators.

2 Preliminaries

  • For non-preferential objectives, the goal is to identify non-dominated solutions rather than a single optimum.A solution Pareto-dominates another when it is no worse on every objective and strictly better on at least one.Source: §2 · p. 2
  • The paper assumes maximization by default unless stated otherwise.Source: §2 · p. 2
  • Hypervolume and inverted generational distance are set-based indicators used to assess finite approximations of the Pareto front.IGD averages distances from a discrete representation of the true front to each approximation’s nearest neighbor; its reference front is often a high-quality approximation.Source: §2 · p. 3
  • Bayesian optimization sequentially selects evaluations of an expensive black-box objective using a Bayesian prior and acquisition function.The acquisition balances exploration and exploitation through the posterior mean and variance.Source: §2 · p. 3

3 Related Work

Related work covers scalarization, hypervolume-based acquisitions, and information-theoretic methods for multi-objective Bayesian optimization. These approaches differ in how they represent trade-offs and in the computational machinery they require.

3 Related Work

  • Scalarization reduces vector objectives to a scalar, often using randomized weights, but performance depends on the scalarization and objective scaling.Collapsing objectives can produce uneven coverage of complex Pareto geometries, particularly in higher dimensions.Source: §3 · pp. 3–4
  • Hypervolume-based methods select evaluations for their contribution to dominated hypervolume, while EHVI targets expected hypervolume gains.qEHVI and qNEHVI extend EHVI to parallel batches and noisy observations, respectively, but may require decompositions, Monte Carlo estimation, or integration over dominance regions.Source: §3 · p. 4
  • HVKG selects evaluations by maximizing expected future hypervolume under a one-step Bayes-optimal policy.Its nested expectations and future Pareto-structure optimizations can be costly as the finite Pareto approximation grows.Source: §3 · p. 4
  • Information-theoretic methods choose evaluations to reduce uncertainty about Pareto-optimal inputs or objectives.PESMO targets Pareto-optimal inputs, PFES the Pareto front, and JES both; approximations such as truncated Gaussian sampling or Monte Carlo integration are commonly used for intractable entropy computations.Source: §3 · p. 4

4 Methodology

The methodology adapts single-objective acquisition functions through Pareto-diverse candidate generation and hypervolume ranking, and extends hedge strategies to select acquisition functions using hypervolume-based performance feedback.

4.1 Hypervolume-Ranked Acquisition Extension

  • qHAX-α forms Pareto-diverse candidates from standard single-objective acquisitions, then selects evaluations by posterior-mean hypervolume ranking.The ranking promotes Pareto-front convergence and coverage without nested expectations or look-ahead optimization.Source: §4.1 · pp. 4–5
  • For parallel batches, qHAX-α greedily re-ranks candidates after each selection without refitting surrogate models.If candidates are exhausted before the requested batch is complete, it uses a uniformly sampled random input as a fallback.Source: §4.1 · p. 5
  • Hypervolume's monotonicity and submodularity give greedy selection a (1 −1/e)-approximation to the optimal single-step gain within the candidate set.This is a myopic, per-iteration guarantee.Source: §4.1 · p. 5

4.2 Multi-Objective Hedge Strategy

  • MO-Hedge selects among acquisition functions using probabilities proportional to historical performance measured in objective space, rather than committing to one acquisition in advance.Unlike qHAX-α's within-iteration candidate ranking, MO-Hedge adapts which acquisition to trust over time.Source: §4.2 · p. 6
  • Each portfolio acquisition independently proposes candidates, and its candidate history is evaluated retrospectively under the current surrogate posterior.The independent candidate-generation step is embarrassingly parallel.Source: §4.2 · p. 6
  • MO-Hedge rewards acquisitions by hypervolume gains from posterior-mean predictions of their historical candidates, then updates selection probabilities with exponentiated weights.The strategy uses a Pareto-aware reward signal while retaining adaptive acquisition selection.Source: §4.2 · p. 6
  • The recalibrated-history gains used in the algorithm violate the conditions for the standard Hedge regret bound, although initial tests favored this variant over the bounded alternative.Freezing past rewards would make the standard bound apply under the same assumptions.Source: §4.2 · p. 7

5 Benchmarking Details

The benchmarking compares qHAX and MO-Hedge variants with established MOBO baselines on synthetic and engineering problems, using Pareto-front quality indicators across serial and batch settings.

5 Benchmarking Details

  • The comparison includes qHAX-UCB and qHAX-EI, two MO-Hedge portfolios, and established hypervolume, scalarization, lookahead, and entropy-based baselines.MO-Hedge 1 includes a random sampler; MO-Hedge 2 uses four strong acquisition strategies.Source: §5 · p. 7
  • Evaluation covers nine benchmarks: seven synthetic functions and two engineering design problems, across different input dimensions and objective counts.The engineering problems are Welded Beam and Vehicle Crash Safety.Source: §5 · p. 7
  • Each strategy receives 30 shared initial points and 75 additional evaluations, with independent objective noise and 20 runs in serial (Q = 1) and batch (Q = 3) settings.Noise standard deviation is set to 1% of the absolute mean objective value over the initial design.Source: §5 · p. 7
  • Performance is measured by hypervolume and inverted generational distance gaps relative to an approximate NSGA-II reference front; Figure 1 shows batch hypervolume gaps.Its solid lines show means and shaded regions show ±1 standard deviation over 20 runs; lower gaps are better.Source: §5 · p. 8, Figure 1 · p. 8

6 Discussion

Across benchmarks, qHAX-α performs competitively with specialized MOBO baselines and sometimes outperforms them; qHAX-UCB is notably strong without a bespoke multi-objective acquisition function. MO-Hedge is motivated by benchmark-dependent acquisition performance and shows robust results, while qHAX-α has low, stable runtime and an inner-optimization limitation.

6 Discussion

  • Across the benchmark suite, qHAX-α is competitive with specialized state-of-the-art MOBO baselines and sometimes outperforms them, while qHAX-UCB achieves the best or near-best final hypervolume gap on several tasks.Log-normalized IGD shows similar trends; scalarization-based and noisy hypervolume baselines remain competitive on specific functions but rarely outperform qHAX-α variants overall.Source: §6 · p. 8
  • No individual MOBO acquisition strategy performs well uniformly across benchmarks, motivating adaptive selection rather than committing to one strategy in advance.Source: §6 · p. 9
  • MO-Hedge 1 reliably tracks its strongest constituent across benchmarks, while both MO-Hedge variants show robust performance across the benchmarking suite.MO-Hedge 1 uses a deliberately mixed-strength portfolio; MO-Hedge 2 uses a stronger portfolio and also performs consistently well.Source: §6 · p. 9
  • qHAX-α exhibits low and stable runtime across benchmarks; Table 1 reports mean runtime per single optimization run over 20 runs in batch mode Q = 3.Source: §6 · p. 9, Table 1 · p. 9
  • The qHAX-α framework relies on an inner multiobjective optimization step to generate Pareto-diverse candidates, typically using evolutionary or population-based algorithms such as NSGA-II.These algorithms do not naturally use efficient gradient-based optimization, though hybrid evolutionary search and local gradient-based refinement can alleviate the limitation.Source: §6 · p. 9

7 Conclusion

The paper concludes that Pareto-diverse candidate generation and hypervolume-based selection extend single-objective acquisition functions into MOBO, while MO-Hedge adaptively selects among acquisition functions. Its background describes how classical acquisition functions balance exploration and exploitation and how hedge strategies use performance feedback to select among them.

7 Conclusion

  • qHAX-α extends standard single-objective acquisition functions to MOBO through Pareto-diverse candidate generation and hypervolume-based selection, and the paper reports competitive or superior benchmark performance at lower, more predictable computational cost.MO-Hedge adaptively selects acquisition functions based on historical Pareto-front contributions and performs consistently across problem classes.Source: §7 · p. 9

A Classical Single-Objective Acquisition Functions And Hedge Strategies

  • Single-objective acquisition functions use a surrogate posterior’s predicted mean and uncertainty to choose evaluations, balancing exploitation of promising regions against exploration of uncertain ones.Source: A Classical Single-Objective… · p. 12

A.1 Single-Objective Acquisition Functions

  • Single-objective BO optimizes an acquisition utility to choose the next evaluation point; EI incorporates improvement magnitude, while UCB uses an optimistic strategy whose exploration is controlled by β_t.Under suitable assumptions and an appropriate β_t schedule, GP-UCB has a sublinear cumulative regret guarantee.Source: §A.1 · p. 12

A.2 SO Hedge Strategies for Acquisition Selection

  • Portfolio-based hedge strategies adaptively select among acquisition functions because their relative performance can vary with objective characteristics; GP-Hedge treats acquisitions as arms and updates rewards after evaluations.The stated regret guarantee concerns acquisition selection relative to the best fixed acquisition in hindsight, not direct optimization of the unknown objective.Source: §A.2 · p. 12

B.1 Benchmark Test Function Details

The appendix specifies a nine-problem MOBO benchmark suite spanning synthetic and engineering design tasks, with varying input dimensions and numbers of objectives. The functions cover varied Pareto-front geometries and distance-function structure.

Benchmark suite

  • The suite contains seven synthetic benchmarks and two engineering design problems, with input dimensions and objective counts summarized in Table 2.The suite includes ZDT2, ZDT4, ZDT6, DTLZ1–3, Branin–Currin, unconstrained Welded Beam, and Vehicle Crash Safety.Source: §B.1 · p. 13, Table 2 · p. 15

ZDT benchmarks

  • ZDT2 tests convergence to a smooth, non-convex Pareto front, ZDT4 has a highly multimodal distance function, and ZDT6 has a non-uniform front and nonlinear first objective.ZDT2 and ZDT4 attain their Pareto-optimal sets when the remaining decision variables are zero.Source: §B.1 · p. 13

DTLZ benchmarks

  • DTLZ1 combines a linear Pareto front with a multimodal distance function, DTLZ2 has a spherical front and unimodal distance function, and DTLZ3 combines DTLZ2’s mapping with DTLZ1’s multimodality.The stated Pareto-optimal conditions set the distance variables to 0.5.Source: §B.1 · p. 14

Engineering and other benchmarks

  • Branin–Currin combines two established single-objective benchmarks, while the unconstrained Welded Beam task is a bi-objective structural design problem.The Branin–Currin implementation clips x2 away from zero for numerical stability.Source: §B.1 · p. 15
  • Vehicle Crash Safety is a three-objective engineering design problem based on surrogate models for automobile crash worthiness.It has five design variables, each bounded between 1 and 3.Source: §B.1 · p. 15

B.2 Algorithm Implementation in BoTorch

The implementation uses BoTorch surrogates and candidate-optimization routines for proposed and baseline strategies, with shared comparison settings. The appendix also reports computational resources and introduces additional results.

B.2 Algorithm Implementation in BoTorch

  • Proposed qHAX-α and MO-Hedge and the baseline strategies are implemented in BoTorch, using ModelListGP and SumMarginalLogLikelihood for model-based methods.The proposed methods use pymoo’s NSGA-II for Pareto-front generation in acquisition space.Source: §B.2 · pp. 15–16
  • The baselines include scalarization, noisy hypervolume, lookahead, and entropy-search strategies, with qHVKG using a one-step expected-hypervolume objective and qMOJES using sampled Pareto sets.qHVKG uses 10 fantasies and 10 Pareto points; qMOJES uses 20 Pareto-set samples and 40 Pareto points per sample.Source: §B.2 · p. 16
  • Baseline candidate generation uses optimize_acqf over bounded domains, with normalized inputs and common optimization settings unless otherwise stated.The default settings are 10 restarts, 500 raw samples, batch limit 5, and 100 iterations.Source: §B.2 · p. 16

B.3 Computational Resource Details

C.1 Batch Mode (Q = 3) IGD Results

In batch optimization, qHAX-UCB and hedge-based strategies provide strong, stable IGD performance across benchmarks. The results also indicate that no single acquisition strategy is uniformly best.

C.1 Batch Mode (Q = 3) IGD Results

  • qHAX-UCB consistently performs strongly with relatively low variance, while qHAX-EI is competitive but more problem-dependent.IGD measures distance from the approximation to a reference Pareto front; lower values indicate more accurate, uniformly covered approximations.Source: §C.1 · p. 17
  • Hedge strategies maintain competitive, stable IGD across benchmarks and avoid stagnation seen in some baselines, while no single acquisition is uniformly optimal.The qHAX-α framework provides robust Pareto-front coverage alongside strong hypervolume performance.Source: §C.1 · p. 17, Figure 2 · p. 17

C.2 Sequential Mode (Q = 1) Benchmarking Results

In sequential optimization, qHAX-UCB generally reduces hypervolume and IGD gaps quickly with low variance, while MO-Hedge tracks strong constituent strategies. qHAX-α also has stable runtime relative to more specialized baselines.

C.2 Sequential Mode (Q = 1) Benchmarking Results

  • Across most problems, qHAX-UCB reduces hypervolume and IGD gaps fastest while maintaining low variance; baselines show mixed behavior and none dominates across tasks.The sequential results report hypervolume and IGD trajectories across benchmarks; lower gaps are better.Source: §C.2 · p. 18, Figure 3 · p. 19, Figure 4 · p. 19
  • Both MO-Hedge variants perform consistently well, matching or tracking their strongest constituent acquisition on each problem.The results characterize hedge-based selection as supporting robust, efficient performance on black-box objectives with unknown structure.Source: §C.2 · p. 18
  • qHAX-α has stable, predictable runtime, while specialized MOBO baselines have substantially higher and more variable computational costs.Its runtime is largely insensitive to problem structure, objective count, or batch size; Pareto-front generation is the dominant cost.Source: §C.2 · p. 18, Table 3 · p. 18
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