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Batched Energy-Entropy acquisition for Bayesian Optimization
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Proposes BEEBO, a statistical-physics-inspired batched acquisition function for GP-based Bayesian optimization that combines a closed-form energy (softmax-weighted expected objective) and information gain with a temperature trading off exploration and exploitation.
Problem
Batched Bayesian optimization with large batches, where common batch acquisition functions (q-EI, q-UCB) require intractable high-dimensional integrals or MC sampling, leaving explore-exploit control poor at large batch sizes; also needs handling of heteroskedastic noise.
Approach
Defines a free-energy-like acquisition, energy (expected softmax-weighted batch objective, closed-form via Taylor approximation) minus temperature times information gain, where the information gain for a GP has closed form via log-determinants of posterior covariances, so the batch can be optimized jointly by gradient ascent without sampling.
Key results
- meanBEEBO achieves the highest mean normalized best observed value at kappa=1.0 (sqrt kappa) with 0.828 vs 0.801 for q-UCB across 33 test experiments with Q=100.
- maxBEEBO yields a mean relative batch instantaneous regret of 0.256 vs 0.872 for q-UCB at the lowest explore rate, showing q-UCB's final exploitative batch is only slightly better than random.
- Under heteroskedastic noise on 3-optimum Branin, BEEBO preferentially optimizes the low-noise optimum while q-UCB remains risk-neutral.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use mean energy instead of softmax-weighted energy (ablation) | BEEBO · 33 test problems, Q=100, lowest explore rate | normalized best objective (mean): 0.811 -> 0.795 | Hurt |
| Use mean energy instead of softmax-weighted energy (ablation) | BEEBO · 33 test problems, Q=100, medium explore rate | normalized best objective (mean): 0.790 -> 0.828 | Helped |
| Use mean energy instead of softmax-weighted energy (ablation) | BEEBO · 33 test problems, Q=100, highest explore rate | normalized best objective (mean): 0.744 -> 0.788 | Helped |
| Use mean energy instead of softmax-weighted energy (ablation) | BEEBO · 33 test problems, Q=100, medium explore rate | relative batch instantaneous regret: 0.265 -> 0.257 | Helped |
| Use mean energy instead of softmax-weighted energy (ablation) | BEEBO · 33 test problems, Q=100, highest explore rate | relative batch instantaneous regret: 0.292 -> 0.261 | Hurt |
| Replace q-UCB with meanBEEBO | Bayesian optimization · 33 test problems, Q=100, lowest explore rate | normalized best objective (mean): 0.758 -> 0.795 | Helped |
| Replace q-UCB with meanBEEBO | Bayesian optimization · 33 test problems, Q=100, medium explore rate | normalized best objective (mean): 0.801 -> 0.828 | Helped |
| Replace q-UCB with meanBEEBO | Bayesian optimization · 33 test problems, Q=100, highest explore rate | normalized best objective (mean): 0.678 -> 0.788 | Helped |
| Use BEEBO's information gain to account for input-dependent observation noise | BEEBO · Heteroskedastic Branin | mean distance to optimum 1 | Helped |
Surprising findings
- q-UCB's relative batch instantaneous regret is only slightly better than a random batch in many cases even with explore term set to zero, attributed to MC-based acquisition optimization difficulties at large Q.
- Kriging Believer with LogEI can perform very competitively for large batches, especially on robot control problems.
Limitations
- Numerical instability of the softmax expectation approximation for beta above about 5.
- Under unknown heteroskedastic noise, the surrogate uncertainty on the noise is not used, requiring a fully Bayesian treatment.
- Cubic scaling in the number of observations N for GP inference limits large-N use.
Experimental setup
Datasets: Ackley Levy Rastrigin Rosenbrock Styblinski-Tang Shekel Hartmann Cosine Powell Embedded Hartmann 6 Branin
Benchmarks: Ackley Levy Rastrigin Rosenbrock Styblinski-Tang Shekel Hartmann Cosine Powell Embedded Hartmann 6 Branin with heteroskedastic noise
Metrics: normalized best observed objective value relative batch instantaneous regret mean distance to optima
Baselines: q-UCB Kriging Believer LogEI
Builds on: q-UCB GIBBON UCB LogEI BoTorch
Closest papers by meaning
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