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MOBO-OSD: Batch Multi-Objective Bayesian Optimization via Orthogonal Search Directions
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Proposes MOBO-OSD, a batch multi-objective Bayesian optimization method that solves GP-surrogate subproblems along orthogonal search directions over an approximated convex hull of individual minima, augmented with Pareto front estimation and Kriging-believer batch selection.
Problem
Sample-efficient multi-objective Bayesian optimization of expensive black-box functions, requiring Pareto front diversity, good hypervolume, scalability to many objectives, and batch evaluation.
Approach
Approximates the convex hull of individual minima (CHIM) from observed data using boundary points, places well-distributed orthogonal search directions on it (Riesz s-energy), solves NBI-style subproblems on GP posterior means with confidence-bound constraints, densifies candidates via first-order Pareto front estimation, and selects batches by hypervolume improvement with Kriging believer and exp
Key results
- On the Car Side Design ablation without PFE, HV rises from 145.12 (n_beta=20) to 160.28 (n_beta=500), while the default with PFE reaches 177.48, indicating PFE provides efficiency gains beyond denser weight vectors.
- MOBO-OSD consistently outperforms baselines (qParEGO, USeMO, DGEMO, PDBO, JES, qEHVI, NSGA-II, NBI) across five synthetic and four real-world problems in sequential (batch size 1) and batch settings (sizes 4, 8, 10).
- qEHVI and DGEMO are the strongest baselines after MOBO-OSD in batch settings, with HV close to MOBO-OSD.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase nβ from 20 to 500 without PFE (ablation) | MOBO-OSD · DTLZ2-M2 | HV: 0.4041 -> 0.4164 | Helped |
| Increase nβ from 20 to 500 without PFE (ablation) | MOBO-OSD · VLMOP2 | HV: 0.2713 -> 0.3159 | Helped |
| Increase nβ from 20 to 500 without PFE (ablation) | MOBO-OSD · Car Side Design | HV: 145.1195 -> 160.2797 | Helped |
| Add PFE (default nβ=20) instead of removing PFE at nβ=20 (ablation) | MOBO-OSD · DTLZ2-M2 | HV: 0.4041 -> 0.4217 | Helped |
| Add PFE (default nβ=20) instead of removing PFE at nβ=20 (ablation) | MOBO-OSD · VLMOP2 | HV: 0.2713 -> 0.3383 | Helped |
| Add PFE (default nβ=20) instead of removing PFE at nβ=20 (ablation) | MOBO-OSD · Car Side Design | HV: 145.1195 -> 177.4782 | Helped |
| Vary nβ across the tested values (ablation) | MOBO-OSD · Nine benchmark problems | Hypervolume difference | No clear effect |
Surprising findings
- Without the PFE component, simply increasing the number of weight vectors to 500 closes much of the gap to the default method, suggesting PFE is an efficiency aid rather than essential.
Limitations
- Only handles noiseless observations.
- Subproblem formulation relies on posterior mean and GP-based constraints, which may be limited under noise.
Experimental setup
Datasets: DTLZ2 ZDT1 VLMOP2 Speed Reducer Car Side Design Marine Design Water Planning
Benchmarks: DTLZ2 ZDT1 VLMOP2 Speed Reducer Car Side Design Marine Design Water Planning
Metrics: hypervolume logarithmic hypervolume difference
Baselines: qParEGO USeMO DGEMO PDBO JES qEHVI NSGA-II NBI
Builds on: Normal Boundary Intersection Pareto Front Estimation Kriging Believer ParEGO
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