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A Unified Framework for Entropy Search and Expected Improvement in Bayesian Optimization
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Shows EI is a special case of a variational lower bound on Max-value Entropy Search (MES) and proposes VES-Gamma, a Gamma-variational acquisition function that generalizes EI and is evaluated on synthetic, GP-sample and real-world BO benchmarks.
Problem
Expected Improvement and information-theoretic acquisition functions are usually viewed as fundamentally distinct, and MES approximations use rigid variational forms for the posterior of the maximum value y*.
Approach
Views MES through a Barber-Agakov variational lower bound (ESLBO) on p(y*|D, y_x); with an exponential variational family this bound reduces to EI, and a Gamma family generalizes it to VES-Gamma with auto-determined shape and rate parameters.
Key results
- VES-Exp and EI pass the two-sample Kolmogorov-Smirnov test at 92.6%-99.8% of iterations across six benchmarks.
- VES-Gamma outperforms EI and MES on 100-dimensional GP samples with length scales 0.05, 0.1 and 0.25, and on the 388-dimensional SVM benchmark.
- VES-Gamma is much slower: about 10.9 s per BO iteration versus 1.6 s for EI and 1.1 s for MES.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use VES-Gamma instead of EI | Bayesian optimization · Branin | optimization performance | Helped |
| Use VES-Gamma instead of EI | Bayesian optimization · Levy | optimization performance | No clear effect |
| Use VES-Gamma instead of EI | Bayesian optimization · Hartmann | optimization performance | Helped |
| Use VES-Gamma instead of EI | Bayesian optimization · Griewank | optimization performance | No clear effect |
| Use VES-Gamma instead of EI | Bayesian optimization on 100-dimensional GP prior samples · GP prior, ℓ=0.05, 0.1, 0.25 | optimization performance | Helped |
| Use VES-Gamma instead of EI | Bayesian optimization on 100-dimensional GP prior samples · GP prior, ℓ=0.5 | optimization performance | No clear effect |
| Use VES-Gamma instead of EI | Bayesian optimization · SVM | optimization performance | Helped |
| Use VES-Gamma instead of EI | Bayesian optimization · Mopta08, Lasso-DNA | optimization performance | No clear effect |
| Use VES-Gamma instead of EI | Bayesian optimization · Rover | optimization performance | No clear effect |
| Use VES-Gamma instead of MES | Bayesian optimization · Rover | optimization performance | Hurt |
| Use VES-Gamma instead of MES | Bayesian optimization · SVM | optimization performance | Helped |
| Use VES-Gamma instead of EI | Bayesian optimization · Ackley | optimization performance | Helped |
Surprising findings
- On Rover, MES achieves the best results while VES-Gamma performs comparably to EI.
- Under the noiseless setting, the exponential-family VES variant converges in one effective step to the same maximizer as EI.
Limitations
- VES-Gamma is computationally expensive due to inner Monte Carlo iterations and timed out on 1000-dimensional Lasso-Hard after 48 hours.
- Theory is restricted to noiseless observations; extension to noisy settings is open.
- Interpretation of the ESLBO terms, including the 'anti-EI' term, remains unclear.
Experimental setup
Datasets: Branin Levy Hartmann Griewank Ackley Michalewicz Rover Mopta08 Lasso-DNA Lasso-High Lasso-Hard SVM
Benchmarks: Branin Levy Hartmann Griewank Ackley Michalewicz Rover Mopta08 Lasso-DNA SVM GP prior samples
Metrics: simple regret (log) best function value per iteration Kolmogorov-Smirnov test pass rate runtime per BO iteration
Baselines: EI LogEI MES
Builds on: Max-value Entropy Search Expected Improvement Barber-Agakov bound LogEI BoTorch
Closest papers by meaning
- Batched Energy-Entropy acquisition for Bayesian Optimization (NeurIPS 2024 · similarity 0.89)
- FunBO: Discovering Acquisition Functions for Bayesian Optimization with FunSearch (ICML 2025 · similarity 0.89)
- BONSAI: Bayesian Optimization with Natural Simplicity and Interpretability (NeurIPS 2026 · similarity 0.88)
- Cost-aware Bayesian Optimization via the Pandora's Box Gittins Index (NeurIPS 2024 · similarity 0.88)
- A General Framework for User-Guided Bayesian Optimization (ICLR 2024 · similarity 0.88)
- Approximation-Aware Bayesian Optimization (NeurIPS 2024 · similarity 0.88)
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