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Trading Off Quality and Uncertainty Through Multi-Objective Optimisation in Batch Bayesian Optimisation
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Proposes POEE, a batch Bayesian optimisation method that treats posterior mean (exploitation) and posterior uncertainty (exploration) as two objectives and selects a batch from a dynamically updated Pareto front using NSGA-II and TOPSIS.
Problem
Batch Bayesian optimisation of expensive black-box functions needs batches that balance exploitation and exploration, but existing multi-acquisition-function Pareto methods often pick similar solutions.
Approach
Treat quality (posterior mean) and uncertainty as two objectives, select the most exploitative point first, then iteratively pick trade-off points via TOPSIS from a Pareto front that is recomputed after each selection because the GP posterior variance changes with the pending selected points.
Key results
- On 14 problems with q=5, POEE has the best mean regret on most problems and wins statistically against RS (14/0/0), TS (13/1/0), and GIBBON (13/0/1) per Table 2.
- POEE achieves the best mean regret on Push4, and is statistically equivalent to the best method KB on Push8.
- Under observational noise (sigma=0.1), POEE statistically outperforms most peers, notably LP (12/2/0), PLAyBOOK (13/1/0), and AEGiS.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a dynamically updated quality-uncertainty Pareto front when selecting batch solutions (ablation) | POEE · 14 synthetic and practical problems | regret | Helped |
| Select the most exploitative solution first (ablation) | POEE · 14 synthetic and practical problems | regret | Helped |
| Use TOPSIS to select subsequent trade-off solutions (ablation) | POEE · 14 synthetic and practical problems | regret | Helped |
| Use POEE instead of the compared batch optimization methods | Batch Bayesian optimization · 14 synthetic and practical problems (q=5) | regret | Helped |
Surprising findings
- Weights favouring exploration slightly outperform exploitation-leaning weights in the sensitivity analysis (0.2/0.8 marginally better than 0.8/0.2), and the mid weight 0.4/0.6 is best overall.
- MACE's performance declines as batch size grows, plausibly due to selection of more similar solutions within a batch.
Limitations
- Scalability to higher dimensions may be limited due to standard GP model; PCA or trust regions suggested as mitigation.
- NSGA-II is not re-run after each uncertainty update, so the dynamic Pareto front uses a fixed candidate pool.
- Fixed weight (0.4, 0.6) may not suit all problems; the paper notes different functions prefer different weights.
Experimental setup
Datasets: WangFreitas Branin BraninForrester Eggholder GoldsteinPrice SixHumpCamel Hartmann6 Ackley Griewank GSobol Push4 Push8
Benchmarks: WangFreitas BraninForrester Branin Eggholder GoldsteinPrice SixHumpCamel Hartmann6 Ackley (2D, 10D) Griewank (2D, 10D) GSobol (10D) Push4 Push8
Metrics: regret Wilcoxon signed-rank test win/equal/loss counts
Baselines: Random search qEI Local Penalisation PLAyBOOK Kriging Believer Thompson Sampling epsilon-shotgun AEGiS GIBBON MACE Gupta TuRBO
Builds on: NSGA-II TOPSIS MACE epsilon-shotgun AEGiS
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