CONTRAST-LEVEL POSTERIOR REGRET CERTIFICATES FOR QUADRATIC COMPOSITES WITH UNCERTAIN GP HYPERPARAMETERS

Learning Gaussian-process hyperparameters changes uncertainty in nonlinear com- posite objectives. We study posterior regret certificates for finite-design squared- norm losses over continuous hyperparameter families. A common posterior cou- pling yields event-transfer bounds from signed-quadratic contrast coefficients without assuming sub-Gaussian composite tails. Under a correctly specified model, predictable observations and valid retained-family coverage, the bounds control joint false certification at a fixed terminal time. Structured two-design posteriors separate intrinsic linear ambiguity from the logarithmic penalty of a specified optimized concentration formula. An explicit event-specific remainder then gives a numerically validated finite separation: at the same posterior and parameter box, our certificate is below a fixed risk threshold while an optimally tilted, sign- conditioned information bound remains inconclusive. Informative observations reduce disagreement across a fixed family, with a sharp boundary rate, eventual almost-sure narrowing and a failure case for an unobserved relevant direction. Direct robust probability remains tighter in the finite example, and the information bound wins in a high-information case. The results characterize uncertainty and certificate conservatism; they do not establish faster Bayesian optimization