Optimal Bayesian Optimization with Stereographic Functions

In Bayesian optimization, the reward-maximizing policy solves a Bellman equation whose state is an entire posterior distribution, and that recursion is intractable; practice substitutes acquisition functions that look a single query ahead. We give a setting in which the optimal policy can instead be written down. The unknown function is linear in the stereographic projection of its input into ${\mathbb{R}}^n$, which makes it a smooth function of the $n-1$ input coordinates with one peak and one valley; evaluations are priced, and probing one known direction costs more than probing any other; and the learner is scored on a single final query, against a point drawn at random from those it could afford. The terminal reward turns out to be a quadratic form in the posterior mean whose weight matrix annihilates the expensive direction, so information about that direction has no value. The Bellman equation then admits a closed-form solution, and the optimal query schedule is non-adaptive: it can be computed in full before any data is seen. Choosing that schedule reduces to water-filling over the eigenvalues of the prior, which we solve in $O(n^3)$ time, and the resulting design puts no weight on the expensive direction, so every query it issues costs only the base price. Experiments check the closed-form value function against Monte Carlo estimates and measure how much exact optimality is worth relative to structured baselines.