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Probability Distribution of Hypervolume Improvement in Bi-objective Bayesian Optimization
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Derives the exact probability distribution of hypervolume improvement for bi-objective GP posteriors via cell partitioning, and proposes ε-PoHVI, an acquisition function based on HVI's CDF, which outperforms ε-PoI and EHVI on high-uncertainty WOSGZ problems.
Problem
Hypervolume improvement is a random variable under GP posteriors, but existing multi-objective acquisition functions (EHVI, ε-PoI) only use its mean or ignore its distribution, which is unreliable when HVI dispersion is large.
Approach
Partition the objective space into cells defined by the current Pareto approximation, express HVI within each cell as z1*z2 plus a constant (products of truncated Gaussians), derive its exact PDF/CDF, and define ε-PoHVI as 1 minus the HVI CDF at ε; also generalize HVI to assign negative values to dominated points.
Key results
- On the WOSGZ problems, ε-PoHVI variants substantially outperform ε-PoI and EHVI, while EHVI leads on ZDT and RE problems.
- Across 14 test problems, ε-PoHVI-scaling wins significantly in 25 pairwise comparisons and loses in 8 (Wilcoxon rank-sum), versus 11/20 for ε-PoI.
- The exact HVI CDF is roughly one order of magnitude faster than Monte Carlo (10^4 samples) at comparable numerical accuracy.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ε-PoHVI-scaling instead of ε-PoI | DGEMO Bayesian optimization · 14 selected test problems | hypervolume convergence: 9/3/2 | Helped |
| Use ε-PoHVI-scaling instead of EHVI | DGEMO Bayesian optimization · 14 selected test problems | hypervolume convergence: 8/1/5 | Helped |
| Use ε-PoHVI-scaling instead of ε-PoHVI-smoothing | DGEMO Bayesian optimization · 14 selected test problems | hypervolume convergence: 8/5/1 | Helped |
| Use ε-PoHVI (either control scheme) instead of ε-PoI | DGEMO Bayesian optimization · WOSGZ problems | hypervolume convergence | Helped |
| Use ε-PoHVI instead of ε-PoI | DGEMO Bayesian optimization · ZDT1, ZDT2, and ZDT4 | hypervolume convergence | Mixed |
| Use ε-PoHVI-scaling instead of ε-PoHVI-smoothing | DGEMO Bayesian optimization · All test problems except ZDT3 | hypervolume convergence | Helped |
| Use ε-PoHVI instead of EHVI | DGEMO Bayesian optimization · ZDT and RE problems | hypervolume convergence | Hurt |
Surprising findings
- EHVI leads on the ZDT and RE problems, so ε-PoHVI's advantage is confined to the high-uncertainty WOSGZ problems.
- ε-PoI also suffers less from high HVI dispersion than EHVI despite not using HVI quantiles.
Limitations
- Exact distribution derived only for bi-objective problems with independent GPs per objective.
- Computational cost of epsilon-PoHVI is quadratic in the Pareto front size, higher than epsilon-PoI and EHVI.
- Empirical evaluation limited to synthetic test problems and one real-world problem.
Experimental setup
Datasets: ZDT1-4 ZDT6 WOSGZ1-8 four bar truss design (RE)
Benchmarks: ZDT1 ZDT2 ZDT3 ZDT4 ZDT6 WOSGZ1-8 RE
Metrics: hypervolume Wilcoxon rank-sum test win/tie/loss counts CPU time CDF accuracy vs Monte Carlo
Baselines: epsilon-PoI EHVI ParEGO Joint Entropy Search
Builds on: DGEMO epsilon-PoI expected hypervolume improvement ParEGO
Closest papers by meaning
- Expected Hypervolume Improvement Is a Particular Hypervolume Improvement (AAAI 2025 · similarity 0.92)
- Simple Extensions of Single-Objective Acquisition Functions and Hedge Strategies for Multi-Objective Bayesian Optimization (NeurIPS 2026 · similarity 0.87)
- BOtied: Multi-objective Bayesian optimization with tied multivariate ranks (ICML 2024 · similarity 0.87)
- Pareto Front-Diverse Batch Multi-Objective Bayesian Optimization (AAAI 2024 · similarity 0.85)
- Optimal Scalarizations for Sublinear Hypervolume Regret (NeurIPS 2024 · similarity 0.85)
- Posterior Sampling-Based Bayesian Optimization with Tighter Bayesian Regret Bounds (ICML 2024 · similarity 0.84)
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