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Bayesian Optimization with Preference Exploration using a Monotonic Neural Network Ensemble
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Proposes BOPE-MoNNE, which models the decision maker's utility with a monotonic neural network ensemble trained on pairwise comparisons and selects comparisons with an independent EUBO variant (IEUBO), outperforming GP-based and other BOPE baselines on six test problems.
Problem
Bayesian optimization with preference exploration: finding the solution maximizing an unknown, monotonic decision-maker utility over expensive multi-objective outputs, learned from noisy pairwise comparisons.
Approach
Replace the GP utility surrogate with a monotonic neural network ensemble (positive weights via exponential transform) trained with a pairwise hinge loss, with output normalization to handle scale invariance, and use an independence-assumption EUBO (IEUBO) to choose comparisons among observed outputs.
Key results
- BOPE-MoNNE achieves the best performance among BOPE methods across the six test problems, followed by BOPE-BMNN and BOPE-GP, with regret below 0.001 in examples.
- MoNNE attains the highest uncertainty quality (e.g., 0.924 on DTLZ2 vs 0.622 for GP), and the highest accuracy on most problems (e.g., 0.96 on ZDT1 vs 0.92 for GP).
- Ablation shows that removing the monotonicity constraint or the ensemble degrades performance, and even without monotonicity BOPE-MoNNE still outperforms BOPE-GP.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use BOPE-MoNNE instead of BOPE-GP as the utility surrogate in BOPE | BOPE · six test problems | simple regret | Helped |
| Use MoNNE instead of GP for utility uncertainty estimates | utility surrogate · DTLZ2 | uncertainty quality: 0.622 (\pm 0.01) -> 0.924 (\pm 0.014) | Helped |
| Use MoNNE instead of GP for utility uncertainty estimates | utility surrogate · VLMOP3 | uncertainty quality: 0.625 (\pm 0.01) -> 0.915 (\pm 0.022) | Helped |
| Use MoNNE instead of GP for utility uncertainty estimates | utility surrogate · ZDT1 | uncertainty quality: 0.674 (\pm 0.012) -> 0.951 (\pm 0.025) | Helped |
| Use MoNNE instead of GP for utility uncertainty estimates | utility surrogate · VehicleSafety | uncertainty quality: 0.63 (\pm 0.016) -> 0.863 (\pm 0.045) | Helped |
| Use MoNNE instead of GP for utility uncertainty estimates | utility surrogate · OSY | uncertainty quality: 0.64 (\pm 0.015) -> 0.878 (\pm 0.022) | Helped |
| Use MoNNE instead of GP for utility uncertainty estimates | utility surrogate · CarCabDesign | uncertainty quality: 0.658 (\pm 0.008) -> 0.951 (\pm 0.08) | Helped |
| Add a monotonicity constraint (ablation) | BOPE-MoNNE · various test problems | simple regret | Helped |
| Add ensemble learning (ablation) | BOPE-MoNNE · various test problems | simple regret | Helped |
| Replace the ensemble with a single model for acquisition sampling (ablation) | BOPE-MoNNE · various test problems | simple regret | Hurt |
| Use IEUBO instead of EUBO for preference queries (ablation) | MoNNE-based BOPE · utility-noise setting | simple regret | Helped |
| Use a linear utility surrogate instead of MoNNE | BOPE · linear utility function | simple regret | Helped |
Surprising findings
- HMC-trained Bayesian neural networks perform poorly on pairwise comparison data due to scale invariance and independent priors, requiring thousands of comparisons to fit a simple function.
- BOPE-Linear slightly outperforms BOPE-MoNNE when the true utility is linear, but MoNNE is better on the other problems.
Limitations
- Experiments use simulated decision makers with synthetic utility functions and output problems rather than real DMs.
- Monotonic network training with pairwise data is limited to small training sets and shallow architectures, and the optimal depth varies across problems.
- Restricting comparisons to observed outputs and the IEUBO independence assumption are heuristic choices without theoretical guarantees.
Experimental setup
Datasets: DTLZ2 VLMOP3 ZDT1 OSY VehicleSafety CarCabDesign
Benchmarks: DTLZ2+Linear VLMOP3+Exponential ZDT1+LinearExponential OSY+Quadratic VehicleSafety+KumaraswamyCDF CarCabDesign+CobbDouglas
Metrics: simple regret uncertainty quality (modified cross-entropy) prediction accuracy
Baselines: Random qNParEGO qNEHVI PBO BOPE-Linear BOPE-GP BOPE-BMNN Known Utility
Builds on: BOPE EUBO qNEIUU PBO Composite BO
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