From High Dimensions to One: The Exact Acquisition Frontier for Spherical Linear Bayesian Optimization

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.