Robust Bayesian Optimization via Tempered Posteriors

TL;DR

This work introduces robust Bayesian optimization by tempering GP surrogates with a power parameter $\alpha\in(0,1]$, addressing overconfidence under local misspecification. It derives regret bounds for tempered surrogates across the generalized EI family $g$, showing that tempering yields strictly sharper guarantees than the standard posterior and is particularly beneficial when the surrogate is prone to local overconfidence. A tunable online tempering schedule based on information matching adapts $\alpha$ during optimization, converging to 1 in well-specified settings and to a smaller value under misspecification. Empirical results on benchmarks and a materials-optimization dataset demonstrate tempered BO can stabilize exploration and improve performance for more exploitative acquisition regimes, with careful behavior in highly exploratory settings.

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.

Figures (8)

• Figure 1: Schematic procedure of BO with possible tempered posterior. The tempered posterior part is indicated by blue boxes, setting $\alpha_t\equiv1$ in these blue boxes reduces to the regular BO.
• Figure 2: Tempered probability of improvement in one dimension. Each row shows iteration $t\in\{5,10,15\}$. Left panels plot the true function \ref{['eq:blackbox_fun']} as a dashed curve and, the posterior mean functions of the surrogates in solid curves for each $\alpha\in\{0.1,0.5,1.0\}$. Bigger dots are the current candidates, smaller dots are the selected historical candidates. The table reports the best observed value of $y$ so far for each $\alpha$. Right panels plot the probability of improvement curves with their maximizers marked. The surrogate uses a Matèrn kernel with $\nu=2$ and a white noise term, observations have standard deviation $0.05$, five shared random initial points are used at iteration 0.
• Figure 3: Effects of Tempering on Selected benchmarks: iterations v.s. log-regret
• Figure 4: Effects of Tempering across Acquisition Functions: Iterations v.s. Best Observed SQUID Voltage
• Figure 5: Normalized average regret $D_t(\alpha,g)$ for fixed kernel parameters, with $\alpha = 0.55$ and $g \in \{0.0,0.5,1.0,1.5,2.0\}$.

Theorems & Definitions (35)

• Theorem 3.1: EI regret bound (fixed $\alpha$)
• Lemma 3.2: Determinant growth
• Corollary 3.3
• Proposition 4.1
• Theorem 4.2
• Theorem 4.3
• Remark 4.4
• Proposition 5.1: Limits of the Adaptive Schedule
• proof
• Lemma D.1: Coordinate-wise monotonicity of EI