Stopping Bayesian Optimization with Probabilistic Regret Bounds

TL;DR

The paper tackles stopping Bayesian optimization with probabilistic guarantees by introducing a probabilistic regret bound (PRB) criterion under Gaussian process models. It couples scalable GP-based sampling with robust Monte Carlo testing to decide when a solution is within $\epsilon$ of the optimum with probability at least $1-\delta$, and proves almost-sure termination to yield an $(\epsilon,\delta)$-optimal solution under mild assumptions. The authors provide practical techniques for simulating stopping events and for adaptive, confidence-based stopping decisions (PRB_MC), plus theoretical convergence results. Empirically, PRB demonstrates competitive stopping behavior, often reducing the number of evaluations, while highlighting sensitivity to model calibration and the importance of uncertainty quantification. Overall, PRB offers a principled pathway to model-based stopping in BO with concrete guarantees and practical guidelines for deployment.

Abstract

Bayesian optimization is a popular framework for efficiently tackling black-box search problems. As a rule, these algorithms operate by iteratively choosing what to evaluate next until some predefined budget has been exhausted. We investigate replacing this de facto stopping rule with criteria based on the probability that a point satisfies a given set of conditions. We focus on the prototypical example of an $(ε, δ)$-criterion: stop when a solution has been found whose value is within $ε> 0$ of the optimum with probability at least $1 - δ$ under the model. For Gaussian process priors, we show that Bayesian optimization satisfies this criterion under mild technical assumptions. Further, we give a practical algorithm for evaluating Monte Carlo stopping rules in a manner that is both sample efficient and robust to estimation error. These findings are accompanied by empirical results which demonstrate the strengths and weaknesses of the proposed approach.

Figures (6)

• Figure 1: Overview of PRB stopping behavior when $f: [0, 1]^2 \to \mathbb{R}$ is drawn from a model with noise variance $\gamma^2 = 10^{-4}$. Regret bounds $\epsilon > 0$ dictate how close $f(\v{x})$ must be to the optimum $f^{*}$ for $\v{x} \in \c{X}$ to be satisfactory. Tolerances $\delta > 0$ upper bound the chance of returning an unsatisfactory point. Left: Percent of runs that stopped before time $T=128$. Middle: Percent of stopped runs that returned $\epsilon$-optimal points. Right: Median number of trials performed by stopped runs.
• Figure 2: Left: Posterior mean and two standard deviations of $f$ (blue) given eight noisy observations (black dots). The goal is to find a point $\v{x} \in \c{X}$ whose true function (black) value is within $\epsilon > 0$ of the optimum $f^*$ (orange star). Middle: Draws of $f_t \sim \c{GP}(\mu_t, k_t)$ and $f^{*}_t$ (orange stars). Right: Estimators for $\Psi_t$. Ground truth (dashed black) was established using location-scale sampling on a dense grid. The joint-sampling strategy from \ref{['sec:method_sampling']} is shown in blue. Competing methods analytically integrated out $f_t(\v{x}) \mid f_t^{*}$ by approximating it with: $f_t(\v{x})$, $f_t(\v{x}) \mid f_t(\v{x}) \le f_t^*$, or $f_t(\v{x}) \mid f_t(\v{x}) \le f_t^* \land f_t(\v{x}_t^*) = f_t^*$ where $f_t^*$ and $\v{x}_t^* \in \mathop{\mathrm{\arg\max}}\limits_{\v{x} \in \c{X}} f_t(\v{x})$ were jointly sampled.
• Figure 3: Left: Median number of draws used by \ref{['alg:prb_mc']} to decide if the expectation of a Bernoulli random variable $Z \sim \operatorname{Bern}(p)$ exceeds $\lambda = 10^{-5}$ (chosen arbitrarily). Middle: Empirical CDFs of $\Psi_t^n$ when optimizing draws from known priors $\c{GP}(0, k)$ in two, four, and six dimensions with noise variance $\gamma^2 = 10^{-6}$ (solid, $\circ$) or $\gamma^2 = 10^{-2}$ (dashed, $\times$). PRB parameters were set to $\epsilon=0.1$ and $\delta_{\mathrm{mod}}=\delta_{\mathrm{est}}=2.5\%$. Right: Runtimes for \ref{['alg:prb_mc']} using the generative strategy from \ref{['sec:method_sampling']} and a (wall) time limit of roughly one thousand seconds.
• Figure 4: Median number of samples (show in $\log_{10}$) used by \ref{['alg:prb_mc']} to decide if the expectation of $Z \sim \operatorname{Bernoulli}(p)$ exceeds $\lambda = 2.5\%$ using different types of intervals with nominal coverage probability $1 - \delta_{\mathrm{est}}$. The number of samples drawn is seen decrease in both $\delta_{\mathrm{est}}$ and $|*|{p - \lambda}$.
• Figure : BO with Monte Carlo PRB

Theorems & Definitions (19)

• Proposition 0
• Proposition 0
• proof
• Definition 1
• Proposition 2
• proof
• Proposition 3
• proof
• Lemma 3
• proof