Prequential E-Values for Selected-GP Near-Optimality Certificates

When optimizing an expensive black-box function sequentially, as in hyperparameter optimization, we may want to stop once the best evaluated value is certified within $\varepsilon$ of the global optimum. Such a certificate needs two ingredients: a lower confidence bound for the selected value and an upper confidence envelope over unevaluated points, typically supplied by a Gaussian process (GP). GP-UCB-style stopping rules are valid when the kernel and constants defining this envelope are fixed before the run, but the practical temptation is to tune the envelope from the same adaptive evaluations and then certify as if it had been fixed. We use prequential e-values to make this selection auditable: starting from a predeclared set of fully specified GP/RKHS envelopes, each candidate is tested by its own one-step-ahead e-process, contradicted candidates are deleted, and certification uses the largest upper bound among the survivors. If at least one declared envelope is valid for the objective, the resulting stopping rule is valid at arbitrary stopping times. On a 512-seed noisy RBF stress sweep, it roughly halves false-certification risk at comparable power versus fit-then-certify. Against random fixed GP precommitment on smooth $d=3,4$ objectives, each additional false certificate is accompanied by 3.0 and 13.5 additional correct certificates, respectively.