[Paper Review] Robust Bayesian Optimization via Tempered Posteriors

Jiguang Li, Hengrui Luo|arXiv (Cornell University)|Jan 11, 2026

Advanced Bandit Algorithms Research0 citations

TL;DR

The paper introduces tempered (alpha- tempered) posterior updates in Bayesian optimization to curb surrogate overconfidence, derives regret bounds for tempered GP surrogates with generalized improvement acquisitions, and proposes an online alpha-tuning scheme.

ABSTRACT

Bayesian optimization (BO) iteratively fits a Gaussian process (GP) surrogate to accumulated evaluations and selects new queries via an acquisition function such as expected improvement (EI). In practice, BO often concentrates evaluations near the current incumbent, causing the surrogate to become overconfident and to understate predictive uncertainty in the region guiding subsequent decisions. We develop a robust GP-based BO via tempered posterior updates, which downweight the likelihood by a power $α\in (0,1]$ to mitigate overconfidence under local misspecification. We establish cumulative regret bounds for tempered BO under a family of generalized improvement rules, including EI, and show that tempering yields strictly sharper worst-case regret guarantees than the standard posterior $(α=1)$, with the most favorable guarantees occurring near the classical EI choice. 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.

Motivation & Objective

• Motivate robustness of BO surrogates under local misspecification and overconfidence.
• Embed likelihood tempering into Bayesian linear and GP surrogates within BO to stabilize updates.
• Derive cumulative regret bounds for tempered GP surrogates under generalized improvement rules (including EI).
• Propose a tuning-friendly online procedure to select the tempering parameter alpha in BO.
• Demonstrate practical performance benefits of tempering on benchmark functions through experiments.

Proposed method

• Formulate tempered posterior updates by raising the likelihood to a power alpha in (0,1].
• Develop a GP surrogate with tempered posterior updates and derive predictive mean and variance under alpha (mu_{t,alpha}, sigma_{t,alpha}).
• Compute a closed-form for the generalized EI (g-EI) acquisition under tempered GP posteriors (including a rescaling nu_t and the tau_g(v) function).
• Establish cumulative regret bounds for tempered BO that depend on alpha and the g parameter, with explicit gamma terms and determinant growth bounds.
• Propose a prequential, tuning-light schedule to adapt alpha online based on calibration and realized prediction errors.
• Provide analytical comparison to standard alpha=1 theory, showing tempered posteriors can yield sharper worst-case guarantees in nonparametric GP settings.

Experimental results

Research questions

• RQ1 Does tempering the likelihood in BO improve robustness to local misspecification of the surrogate?
• RQ2 How do tempered posteriors interact with generalized improvement acquisitions like PI and EI in terms of regret?
• RQ3 Can we derive explicit regret bounds for tempered GP-based BO that depend on alpha and g, and how do these compare to the standard posterior?
• RQ4 Is there a practical online method to adapt alpha during BO without heavy tuning?
• RQ5 Do tempering strategies offer empirical performance gains on benchmark BO problems?

Key findings

• Tempering the surrogate via alpha in (0,1] downweights local misspecification and can prevent overconfident posteriors in BO.
• For the tempered GP, a closed-form expression for the g-EI acquisition is derived, enabling analysis of the tempered BO algorithm.
• The regret bounds show that, for fixed g, tempering yields strictly sharper worst-case guarantees than the standard alpha=1, with strongest gains near EI settings.
• In the linear surrogate baseline, tempered updates do not improve leading-order EI regret under correct specification, highlighting the value of tempering mainly in nonlinear GP regimes.
• A prequential online schedule is proposed to adapt alpha, decreasing alpha when prediction errors exceed model uncertainty and returning to 1 as calibration improves.
• Empirical results indicate tempering stabilizes BO surrogates and improves robustness under localized sampling.