We Still Don't Understand High-Dimensional Bayesian Optimization
Bibliographic record
Abstract
Bayesian optimization is often limited not only by the complexity of the surrogate model but also by the difficulty of optimizing the acquisition function. We study this difficulty for Gaussian-process surrogates with the spherically projected linear kernel of Doumont et al. That kernel is exactly Bayesian linear regression on a finite feature map whose nonconstant features lie on a sphere, so on the unconstrained spherical design space the posterior mean is a linear function of the spherical coordinate and the posterior variance is a quadratic function. Our main result is that maximizing Expected Improvement (EI) or Log Expected Improvement (LogEI) for this surrogate reduces exactly to a single-variable search over a mean–variance Pareto frontier, where each frontier point is the solution of a classical trust-region subproblem. The acquisition optimization is therefore one-dimensional and independent of the ambient dimension, which enters only through the linear algebra used to evaluate the frontier. This gives an interpretable explore–exploit dial and separates one source of high-dimensional robustness (a tame acquisition geometry) from the statistical expressiveness of the surrogate. We also explain why this one-dimensional picture is exact for unconstrained spherical optimization but is broken by the box constraints present in most real Bayesian-optimization tasks.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.007 | 0.029 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.008 |
| Scholarly communication | 0.004 | 0.018 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.005 | 0.016 |
| Insufficient payload (model declined to judge) | 0.008 | 0.005 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".