Regret-Based $(\epsilon,\delta)$-optimal Stopping Criteria for Bayesian Optimization

Haowei Wang; Jingyi Wang; Qiyu Wei

arxiv: 2605.22561 · v1 · pith:3JTEAHZHnew · submitted 2026-05-21 · 💻 cs.LG

Regret-Based (ε,δ)-optimal Stopping Criteria for Bayesian Optimization

Haowei Wang , Jingyi Wang , Qiyu Wei This is my paper

Pith reviewed 2026-05-22 07:19 UTC · model grok-4.3

classification 💻 cs.LG

keywords bayesian optimizationgaussian processGP-UCBstopping criteriaregret boundsepsilon-delta optimalityblack box optimization

0 comments

The pith

Tighter instantaneous regret bounds for GP-UCB enable stopping criteria that guarantee ε-optimal solutions with high probability in Bayesian optimization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Bayesian optimization often continues for a fixed number of evaluations even after a sufficiently good solution has been located. This paper derives provably tighter bounds on the instantaneous regret of GP-UCB at every iteration. These bounds are then used to create stopping criteria that terminate the search once an ε-optimal point is identified with probability at least 1-δ. Readers should care because the new criteria replace arbitrary budgets with a principled rule that still controls solution quality while potentially saving evaluations.

Core claim

We present provably tighter instantaneous regret bounds for GP upper confidence bound (GP-UCB) at any given iteration. Then, we propose stopping criteria for GP-UCB based on this tighter bound that ensures an ε-optimal solution with high probability 1-δ upon termination.

What carries the argument

The provably tighter instantaneous regret bound for GP-UCB, which underpins the construction of regret-based (ε,δ)-optimal stopping criteria.

Load-bearing premise

The Gaussian process kernel and the bounded reproducing kernel Hilbert space norm of the objective function satisfy the standard assumptions from the GP-UCB literature.

What would settle it

Run the method on a known test function until the stopping criterion activates and then verify whether the true optimum lies within ε of the reported point with the expected frequency.

read the original abstract

Bayesian optimization (BO) is a widely used iterative black-box optimization method that utilizes Gaussian process (GP) surrogate models. In practice, BO is typically terminated after a fixed evaluation budget is exhausted, which can incur unnecessary cost and provides no optimality guarantee on solution quality. Recent research in developing a practical stopping criterion has made empirical progress, yet a theoretically sound stopping criterion remains a work in progress. In this work, we present provably tighter instantaneous regret bounds for GP upper confidence bound (GP-UCB) at any given iteration. Then, we propose stopping criteria for GP-UCB based on this tighter bound that ensures an $\epsilon$-optimal solution with high probability $1-\delta$ upon termination. Numerical experiments are performed to validate and demonstrate the effectiveness and efficiency of our stopping criteria.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper tightens the per-step GP-UCB regret bound and turns it into an (ε,δ) stopping rule, but the random stopping time needs explicit handling to keep the probability guarantee intact.

read the letter

The paper gives a stopping rule for GP-UCB Bayesian optimization that uses a tighter instantaneous regret bound to decide when to quit, claiming this yields an ε-optimal point with probability 1-δ. It starts from the usual GP-UCB analysis, improves the instantaneous bound at each fixed iteration, and then sets a threshold on that bound for termination instead of running a preset number of steps. The experiments show the rule often stops earlier than fixed budgets while still returning competitive solutions, which is the practical payoff for hyperparameter tuning or similar tasks where evaluations are expensive. That part is straightforward and addresses a real usability gap in BO workflows. The soft spot is the jump from fixed-t bounds to a data-dependent stopping time τ. Standard per-iteration high-probability statements do not automatically control the event at the random τ without a union bound over possible stopping times, a maximal inequality, or an optional-stopping argument. The abstract does not indicate which of these is used, so the (ε,δ) claim depends on whether the proofs close that gap. If they do not, the failure probability can exceed δ. This is worth checking in the full derivation rather than a minor technicality. The work is aimed at practitioners who run BO and want a principled exit condition, plus theorists who work on regret bounds for kernel methods. A reader who already knows the GP-UCB literature will see the incremental tightening and the stopping application clearly. It deserves a serious referee because the idea is concrete, the experiments are relevant, and the probability detail is checkable. I would send it out for review and ask the referees to focus on the stopping-time argument.

Referee Report

1 major / 2 minor

Summary. The paper claims to derive provably tighter instantaneous regret bounds for GP-UCB in Bayesian optimization at any fixed iteration, then proposes stopping criteria based on these bounds that guarantee termination at an ε-optimal solution with probability at least 1-δ. The claims are supported by theoretical analysis building on standard GP assumptions and validated via numerical experiments.

Significance. If the (ε,δ) guarantees are rigorously established, the work would be significant for providing a theoretically grounded alternative to fixed-budget termination in BO, potentially reducing evaluation costs while ensuring solution quality with explicit probabilistic control. This addresses a practical gap in the literature on adaptive BO termination.

major comments (1)

[Section deriving the (ε,δ) stopping criterion (likely §4 or the main theorem)] The derivation of the stopping criterion applies the per-iteration high-probability instantaneous regret bound directly at the data-dependent stopping time τ. Because τ is a random variable, the event {regret at τ > ε} is not automatically controlled by the fixed-t probability without an additional union bound over possible stopping times, a maximal inequality, or an optional-stopping argument for the underlying martingale. If this correction is absent, the overall failure probability may exceed δ.

minor comments (2)

[Abstract] The abstract states 'provably tighter' bounds without explicitly identifying the baseline bound from prior GP-UCB analyses to which the improvement is compared.
[Numerical experiments section] Clarify whether the numerical experiments include ablation on the effect of the tighter bound versus the standard one, and report the observed termination iterations and actual optimality gaps achieved.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important technical point regarding the application of our bounds at a random stopping time. We address the comment below and will revise the paper to incorporate a rigorous justification.

read point-by-point responses

Referee: [Section deriving the (ε,δ) stopping criterion (likely §4 or the main theorem)] The derivation of the stopping criterion applies the per-iteration high-probability instantaneous regret bound directly at the data-dependent stopping time τ. Because τ is a random variable, the event {regret at τ > ε} is not automatically controlled by the fixed-t probability without an additional union bound over possible stopping times, a maximal inequality, or an optional-stopping argument for the underlying martingale. If this correction is absent, the overall failure probability may exceed δ.

Authors: We thank the referee for this insightful observation. Our instantaneous regret bounds are established for each fixed iteration t using standard GP posterior concentration inequalities that hold with probability at least 1-δ. While the stopping time τ is indeed data-dependent, we note that the underlying process can be analyzed via the filtration generated by the observations. In the revised manuscript we will explicitly invoke an optional stopping argument for the relevant martingale (or, equivalently, apply a union bound over a sufficiently fine discretization of possible stopping times up to a practical horizon) to ensure that P(regret(τ) > ε) ≤ δ still holds. This correction will be added to the main theorem and its proof without changing the form of the proposed stopping criterion or the empirical results. revision: yes

Circularity Check

0 steps flagged

No circularity: stopping criterion derived from per-iteration regret bound under standard GP assumptions

full rationale

The paper derives tighter instantaneous regret bounds for GP-UCB at fixed iterations using standard kernel and RKHS-norm assumptions from prior GP-UCB literature, then defines a stopping rule when the bound drops below ε. This is a direct (if potentially incomplete) implication from the fixed-t high-probability statement to a data-dependent stopping time; no self-definition, no parameter fitted to the target quantity and then renamed as a prediction, and no load-bearing self-citation chain that reduces the central claim to an unverified prior result by the same authors. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard GP-UCB regret analysis assumptions and on the existence of a tighter instantaneous bound that the paper claims to derive; no new free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Standard assumptions on the Gaussian process kernel and bounded RKHS norm of the objective function as used in prior GP-UCB regret bounds.
Invoked to support the derivation of the tighter instantaneous regret bound.

pith-pipeline@v0.9.0 · 5663 in / 1249 out tokens · 31375 ms · 2026-05-22T07:19:14.503015+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · 3 internal anchors

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Botorch: A framework for efficient monte-carlo bayesian optimization.Advances in neural information processing systems, 33:21524–21538, 2020

Maximilian Balandat, Brian Karrer, Daniel Jiang, Samuel Daulton, Ben Letham, Andrew G Wilson, and Eytan Bakshy. Botorch: A framework for efficient monte-carlo bayesian optimization.Advances in neural information processing systems, 33:21524–21538, 2020

work page 2020

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T. M. Cover.Elements of information theory. John Wiley & Sons, 1999

work page 1999

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Z. Dai, H. Yu, B. K. H. Low, and P. Jaillet. Bayesian optimization meets bayesian optimal stopping. In International conference on machine learning, pages 1496–1506. PMLR, 2019

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P. I. Frazier. Bayesian optimization. InRecent advances in optimization and modeling of contemporary problems, pages 255–278, October 2018

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Bayesian optimization for materials design

Peter I Frazier and Jialei Wang. Bayesian optimization for materials design. InInformation science for materials discovery and design, pages 45–75. Springer, 2015

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Automatically stopping bayesian optimization under uncertain surrogate model means

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Improved regret bounds for gaussian process upper confidence bound in bayesian optimization.arXiv preprint arXiv:2506.01393, 2025

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PhD thesis, Massachusetts Institute of Technology, 2018

Remi Roger Alain Paul Lam.Scaling Bayesian optimization for engineering design: lookahead approaches and multifidelity dimension reduction. PhD thesis, Massachusetts Institute of Technology, 2018

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work page 2023

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work page internal anchor Pith review Pith/arXiv arXiv 2016

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Makarova, H

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work page internal anchor Pith review Pith/arXiv arXiv 2009

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Vakili, K

S. Vakili, K. Khezeli, and V. Picheny. On information gain and regret bounds in Gaussian process bandits. InInternational Conference on Artificial Intelligence and Statistics, pages 82–90. PMLR, 2021

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Wang and P

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work page 2025

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J. Wang, N. Y. Chiang, A. Gillette, and J. L. Peterson. A multifidelity Bayesian optimization method for inertial confinement fusion design.Phys Plasmas, 2024

work page 2024

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Theoretical Analysis of Bayesian Optimisation with Unknown Gaussian Process Hyper-Parameters

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work page internal anchor Pith review Pith/arXiv arXiv 2014

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On the sublinear regret of gp-ucb.Advances in Neural Information Processing Systems, 36:35266–35276, 2023

Justin Whitehouse, Aaditya Ramdas, and Steven Z Wu. On the sublinear regret of gp-ucb.Advances in Neural Information Processing Systems, 36:35266–35276, 2023

work page 2023

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J. Wilson. Stopping bayesian optimization with probabilistic regret bounds.Advances in Neural Information Processing Systems, 37:98264–98296, 2024

work page 2024

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Wächter and L

A. Wächter and L. Biegler. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming.Math. Program., 106(1):25–27, 2006

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Q. Xie, L. Cai, A. Terenin, P. I. Frazier, and Z. Scully. Cost-aware stopping for bayesian optimization. arXiv preprint arXiv:2507.12453, 2025. 13 A Proofs of Section 3 The proof of Lemma 3.1 is given below. Proof. By the condition of the lemma, there exists a subsequence{cti}such thatcti→0as i→∞. Hence, for anyϵ>0, there exists an index¯isuch that cti <ϵ...

work page arXiv 2025