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Gaussian Process Upper Confidence Bound Achieves Nearly-Optimal Regret in Noise-Free Gaussian Process Bandits

Shogo Iwazaki

arXiv:2502.19006v2cs.LG

TL;DR

Noise-free GP-bandit theory had suggested that GP-UCB was suboptimal despite its empirical success relative to nearly optimal non-adaptive methods. This paper derives refined posterior-variance bounds for GP-UCB and obtains nearly optimal noise-free regret, including constant cumulative regret for the squared exponential and suitable Matérn kernels. The results rely on information-gain analysis and an RKHS-bounded objective, with stated scope limitations for some kernel regimes and eigenfunction assumptions.

  • Problem

    Prior theory indicated suboptimal noise-free GP-UCB regret, despite empirical success and nearly optimal competitors using non-adaptive sampling.

  • Method

    The paper bridges noisy-regime information-gain analysis to noise-free settings through algorithm-independent upper bounds on observed GP posterior standard deviations.

  • Results

    GP-UCB attains nearly optimal regret, including constant O(1) cumulative regret for the squared exponential kernel and Matérn kernels with d > ν.

  • Takeaways & Limitations

    The posterior-standard-deviation bounds may extend nearly optimal noise-free analyses beyond GP-UCB to other confidence-bound algorithms.

  • Takeaways & Limitations

    The stated results rely on an information-gain bound using a uniform eigenfunction-boundedness assumption whose validity is questioned on general compact domains.

Abstract

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We study the noise-free Gaussian Process (GP) bandits problem, in which the learner seeks to minimize regret through noise-free observations of the black-box objective function lying on the known reproducing kernel Hilbert space (RKHS). Gaussian process upper confidence bound (GP-UCB) is the well-known GP-bandits algorithm whose query points are adaptively chosen based on the GP-based upper confidence bound score. Although several existing works have reported the practical success of GP-UCB, the current theoretical results indicate its suboptimal performance. However, GP-UCB tends to perform well empirically compared with other nearly optimal noise-free algorithms that rely on a non-adaptive sampling scheme of query points. This paper resolves this gap between theoretical and empirical performance by showing the nearly optimal regret upper bound of noise-free GP-UCB. Specifically, our analysis shows the first constant cumulative regret in the noise-free settings for the squared exponential kernel and Matérn kernel with some degree of smoothness.

1 Introduction

The paper addresses the gap between GP-UCB’s empirically strong performance and prior theory suggesting suboptimal noise-free regret. It proves nearly optimal guarantees, including constant cumulative regret for the squared exponential kernel and sufficiently smooth Matérn kernels.

  • Motivation: Existing nearly optimal algorithms [Iwazaki and Takeno, 2025a, Salgia et al., 2024] use non-adaptive sampling, whereas GP-UCB selects queries adaptively.Examples include uniform sampling [Salgia et al., 2024] and maximum variance reduction [Iwazaki and Takeno, 2025a].
  • Contributions: The analysis matches both cumulative-regret lower bounds and simple-regret lower bounds [Bull, 2011] up to polylogarithmic factors for Matérn kernels.
  • Empirical motivation: Figure 1 compares GP-UCB with REDS [Salgia et al., 2024] and PE [Iwazaki and Takeno, 2025a] using average cumulative regret over 3000 runs across three kernels.The settings use d = 2, with Matérn smoothness ν = 5/2 and ν = 3/2 in two panels.
  • Contributions: The key technical contribution is an algorithm-independent upper bound on observed posterior standard deviations that bridges noisy-regime information-gain analysis to noise-free settings.The bounds may extend nearly optimal noise-free guarantees to other confidence-bound algorithms beyond GP-UCB.

2 Preliminaries

The paper formulates noise-free GP bandits over compact domains, assumes the objective lies in a bounded-norm RKHS, and uses GP posterior quantities to define GP-UCB. Kernel choice controls function regularity and information-gain complexity.

  • Noise-free GP bandit problem: In the noise-free GP bandit problem, the learner queries points, observes exact function values, and is evaluated by cumulative or simple regret.The objective is a pointwise-evaluable black-box function on a compact subset of R^d.
  • Gaussian process model: The GP posterior is computed from queried inputs and evaluations, while fully correlated inputs are removed from the effective set E(X_t).For commonly used squared exponential and Matérn kernels, E(X_t) equals the query sequence when there are no duplicate inputs.
  • Kernel function and information gain: The squared exponential and Matérn kernels encode different smoothness structures, and their information gain determines kernel-dependent regret complexity.The Matérn smoothness parameter is ν, while the squared exponential and Matérn information-gain rates differ as functions of dimension and sample size.
  • GP-UCB: GP-UCB selects x_t by maximizing posterior mean plus an exploration term β^1/2σ, then updates the posterior after observing f(x_t).The algorithm starts with an empty query set and uses the RKHS norm bound B to initialize β^1/2.
  • Regularity assumption: The analysis assumes f belongs to the RKHS of a known positive-definite kernel, with k(x,x) ≤ 1 and ||f||_k ≤ B < ∞.This assumption supports confidence bounds for the objective.

3 Refined Regret Upper Bound for Noise-Free GP-UCB

The paper derives nearly optimal cumulative and simple regret bounds for noise-free GP-UCB by refining posterior-standard-deviation analysis through information gain. The resulting bounds match known lower bounds up to polylogarithmic factors and yield constant cumulative regret for squared exponential and sufficiently smooth Matérn kernels.

  • 3 Refined Regret Upper Bound for Noise-Free GP-UCB: GP-UCB achieves cumulative regret matching the lower bound up to polylogarithmic factors, including constant O(1) regret for squared exponential and Matérn kernels when d>ν.The implied constants depend on problem and kernel parameters, including B, d, ℓ, ν, and the domain diameter.
  • 3 Refined Regret Upper Bound for Noise-Free GP-UCB: The refined analysis also matches the simple-regret lower bound up to polylogarithmic factors for the Matérn kernel.
  • 3 Refined Regret Upper Bound for Noise-Free GP-UCB: These posterior-deviation bounds directly tighten GP-UCB regret through the standard UCB selection rule and noise-free confidence bound.The cumulative and simple regret inequalities depend respectively on the sum and minimum of observed posterior standard deviations.
  • 3 Refined Regret Upper Bound for Noise-Free GP-UCB: The main technical contribution is an algorithm-independent upper bound on cumulative and minimum posterior standard deviations, obtained by bridging information gain analysis from noisy to noise-free GP models.The proof compares noise-free posterior deviations with noisy-model deviations, uses variance monotonicity, and applies an elliptical potential count bound.
  • 3 Refined Regret Upper Bound for Noise-Free GP-UCB: Lemma 3 holds for any input sequence, unlike prior algorithm-specific bounds, and therefore addresses whether the conjectured rates extend beyond non-adaptive sampling.The conjecture was previously validated only for specific non-adaptive algorithms [Li and Scarlett, 2024, Iwazaki and Takeno, 2025a, Salgia et al., 2024].
  • 3 Refined Regret Upper Bound for Noise-Free GP-UCB: The proof strategy may extend the analysis of confidence-bound algorithms beyond GP-UCB and to other kernels when suitable information-gain bounds are available.The paper specifically discusses noisy-setting algorithms and possible neural tangent kernel extensions, while noting that Lemma 3 itself is limited to squared exponential and Matérn kernels.

4 Discussion

The discussion identifies unresolved questions about squared-exponential simple-regret rates, Bayesian constant regret, and GP-UCB’s empirical simple-regret performance.

  • Lower bound for squared exponential kernel: The optimal simple regret rate under the squared exponential kernel remains unknown, and the authors conjecture that further improvement may be possible.Their conjecture is motivated by a mismatch between known information-gain upper and lower bounds.
  • Constant cumulative regret in Bayesian setting: Existing analysis gives logarithmic Bayesian regret in SE and Matérn settings with d < ν, while the frequentist result guarantees constant O(1) regret.The paper identifies designing a Bayesian algorithm with constant regret or proving a matching lower bound as an open direction.
  • GP-UCB in simple regret minimization: GP-UCB has nearly optimal worst-case simple regret under the Matérn kernel, but its empirical performance appears inferior to EI in simple-regret minimization.The source leaves open whether this gap reflects constants, logarithmic factors, or a deeper difference between worst-case analysis and empirical behavior.

5 Conclusion

The paper establishes nearly optimal regret for noise-free GP-UCB through a new posterior-standard-deviation bound tailored to the noise-free setting. The bound may also support extending noisy-setting confidence-bound analyses to noise-free algorithms.

  • 5 Conclusion: A tight posterior-standard-deviation upper bound tailored to noise-free GP bandits is the key theoretical component behind GP-UCB’s nearly optimal regret.The bound is formalized as Lemma 3.
  • 5 Conclusion: Lemma 3 may allow existing noisy GP-bandit results to be translated into noise-free variants by replacing their noisy posterior-deviation bound.The paper presents this as a potential broader application beyond GP-UCB.

A.1 Proof of Theorems 1 and 2

The proof derives cumulative and simple regret bounds from a deterministic noise-free confidence bound and the GP-UCB selection rule. It then combines these inequalities with the preceding equations to obtain the theorems.

  • A.1 Proof of Theorems 1 and 2: The proof begins by stating a deterministic confidence bound that applies to any input sequence in the noise-free setting.The bound is attributed to prior results including [Kanagawa et al., 2018], [Lyu et al., 2019], and [Vakili et al., 2021a].
  • A.1 Proof of Theorems 1 and 2: The cumulative-regret derivation uses the deterministic confidence bound together with the UCB-selection rule for each queried point.These steps establish the intermediate inequalities for R_T.
  • A.1 Proof of Theorems 1 and 2: The simple-regret derivation similarly applies the confidence bound and the definition of the selected recommendation before combining the resulting inequalities with Eqs. (11)–(13).The proof concludes by combining these intermediate bounds with the earlier equations.

A.2 Proof of Lemma 3

The proof of Lemma 3 obtains posterior-standard-deviation bounds from information-gain estimates for squared exponential and Matérn kernels. It then controls the resulting expressions using monotonicity, integration, and Gamma-function identities.

  • A.2 Proof of Lemma 3: The proof uses Lemmas 4 and 5 together with the posterior-variance bound σ(x_t; X_{t−1}) ≤ k(x_t, x_t) ≤ 1 to establish the intermediate equations.Lemma 8 supplies a detailed version of the posterior-standard-deviation upper bound.
  • A.2 Proof of Lemma 3: Lemma 8 assumes a compact domain, bounded kernel diagonals, and information-gain upper bounds for squared exponential and Matérn kernels.The constants may depend on d, ℓ, ν, and the domain diameter, but not on T.
  • A.2 Proof of Lemma 3: The resulting posterior-standard-deviation bound depends on fixed constants from the information-gain estimates, which yields the conclusions of Lemma 3 for fixed d, ℓ, and ν.The proof relates these constants to the implied information-gain bounds.

B.1 Experimental settings for Figure 1

The experiments define RKHS objective functions on a 50 × 50 grid, use matched kernels with fixed parameters, and evaluate the setup across 3000 seeds. Figure 2 separately compares GP-UCB and EI under two Matérn smoothness settings over 100 seeds.

  • Objective function: The objective function is a 50-term kernel expansion with independently sampled coefficients and locations.The coefficients are sampled uniformly from [−1, 1], while locations are sampled uniformly from [0, 1]^2.
  • Kernel and domain: The input domain is a uniformly aligned 50 × 50 grid on [0, 1]^2, with the same kernel used for data generation and the GP model.The lengthscale is fixed at ℓ = 0.25, and the confidence width is set to the exact RKHS norm.
  • Experimental protocol: Experiments use 3000 different seeds under the described setting.
  • Figure 2 comparison: Figure 2 compares GP-UCB and EI for simple regret under Matérn kernels with ν = 3/2 and ν = 5/2 across 100 seeds.

B.2 Comparison between EI and GP-UCB

Under the same experimental setting, GP-UCB’s simple-regret performance is consistently worse empirically than EI, despite GP-UCB’s nearly optimal worst-case regret.

  • Comparison between EI and GP-UCB: GP-UCB’s empirical simple-regret performance is consistently worse than EI’s under the same experimental setting.The comparison is reported using the results shown in Figure B.2.
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