Robust Bayesian Optimization via Tempered Posteriors

Bayesian optimization (BO) iteratively fits a Gaussian process (GP) surrogate to accumulated evaluations and selects new queries via an acquisition function. Under local misspecification, this feedback loop can produce overconfidence precisely in the region guiding subsequent decisions. We develop a tempered GP-based BO framework that raises the likelihood to a power α\in(0,1]. For a generalized family of improvement acquisitions indexed by g, including probability of improvement (PI, g=0) and expected improvement (EI, g=1), we derive finite-time cumulative regret bounds with adaptively learned kernel hyperparameters. The analysis shows that tempering reduces the noise-driven confidence and information-gain contributions to regret, while a deterministic RKHS term prevents arbitrarily aggressive tempering from being uniformly beneficial. It also clarifies the role of the acquisition function: positive-order g-EI rules preserve the usual information-gain regret behavior, whereas zero-jitter PI is more exploitative and admits a weaker worst-case guarantee. Motivated by our theoretic findings, we propose a prequential procedure for selecting α online: it decreases α when realized prediction errors exceed model-implied uncertainty and returns α toward one as calibration improves. Empirical results demonstrate that tempering provides a practical yet theoretically grounded tool for stabilizing BO surrogates under localized sampling.