Title: Improved Regret Bounds for Gaussian Process Upper Confidence Bound in Bayesian Optimization
Venue: NeurIPS 2025
Accepted: yes
Keywords: Gaussian process bandits

regret analysis

Bayesian optimization
Abstract: This paper addresses the Bayesian optimization problem (also referred to as the Bayesian setting of the Gaussian process bandit), where the learner seeks to minimize the regret under a function drawn from a known Gaussian process (GP). 
Under a Mat\'ern kernel with some extent of smoothness, we show that the Gaussian process upper confidence bound (GP-UCB) algorithm achieves $\tilde{O}(\sqrt{T})$ cumulative regret with high probability. Furthermore, our analysis yields $O(\sqrt{T \ln^2 T})$ regret under a squared exponential kernel. These results fill the gap between the existing regret upper bound of GP-UCB and the current best upper bound provided by Scarlett [2018]. The key idea in our proof is to capture the concentration behavior of the input sequence realized by GP-UCB, enabling us to handle GP's information gain in a refined manner.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the theoretical performance of the GP-UCB algorithm in Bayesian Optimization, which presents a refined analysis showing that the existing GP-UCB achieves\n$\\tilde{O}(\\sqrt{T})$ regret for Matérn kernels and $O(\\sqrt{T \\ln^{4} T})$ regret for the squared exponential kernel under certain conditions. The core contribution lies in analyzing the actual input sequence generated by GP-UCB rather than relying solely on worst-case information gain bounds, and thus improves the analytical performance of GP-UCB.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n1. This paper establishes new high-probability regret bounds for the standard GP-UCB algorithm, which improve over the classic bounds in Srinivas et al. [1] and match or approach state-of-the-art [2].\n\n2. One advantage is that the algorithm analyzed in the paper does not require prior knowledge of regularity constants around the maximizer compared to state-of-the-art [2], making the result more practically relevant.\n\n3.  The presentation is very clear and easy to follow. The core proof  idea of capturing the concentration behavior of the input sequence by GP-UCB is well illustrated.\n\n\n**Weaknesses**\n\n1. While the paper provides a refined regret analysis, the algorithm itself seems to remain the standard GP-UCB without modification. As such, the contribution is best viewed as an analytical extension rather than a methodological advance. It illustrates that an existing algorithm can achieve near-optimal performance under more careful analysis, but does not propose new algorithmic ideas or frameworks.\n\n2. Aside from the regret decomposition built on the core insight of leveraging the concentration behavior of the input sequence under GP-UCB, the overall proof strategy appears to rely heavily on established techniques from prior work. While well-executed, this may limit the broader technical novelty of the paper.\n\n3. The improved bound for the Matérn kernel holds only under specific smoothness conditions (e.g., $2v + d \\leq v^2$, shown in Theorem3), which may not hold in many practical settings, especially in high dimensions.\n\n4. While the paper is theoretical, a small empirical experiment could help illustrate how the actual information gain or input concentration behaves in practice, supporting the core intuition beyond the single example in Figure 1.\n\n[1] Srinivas, Niranjan, et al. \"Gaussian process optimization in the bandit setting: No regret and experimental design.\" ICML (2010).\n[2] Jonathan Scarlett. \"Tight regret bounds for Bayesian optimization in one dimension.\" ICML (2018).\n\nQUESTIONS:\nCould the authors provide more precise characterizations or examples where the realized information gain $I(X_T)$ deviates substantially from $\\gamma_T$​? Are there quantifiable metrics for how close GP-UCB is to the ideal case in practice?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents an improved regret bound for the famous GP-UCB algorithm under Bayesian setting (i.e. Gaussian-Process Bandits). While standard GP-UCB uses the worst-case maximum information gain (MIG), corresponding to the scenario where the algorithm always queries the maximum variance, the paper observes that this worst-case bound will be very loose in the later stage of optimisation, where the algorithm queries point around the optimum. Authors split the total regret of the algorithm into two parts and show that the regret of this \"later\" stage asymptotically dominates the regret of the \"early\" stage. Then, due to the fact that the radius of the region in which the algorithm queries points in the \"later\" stage is smaller than that of full domain, this results in substantial reduction in MIG. Importantly, the resulting bound for SE kernel does not depend on dimensionality (previous bound had exponential dependance) and for the Matern kernel, the dependance of the bound on the dimensionality is significantly improved.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n+ The bound seems monumental, as it significantly improves the dependance on dimension of the problem. Previously it was believed that exponential scaling of the bound with dimensionality for more complex kernels is unavoidable due to the curse of dimensionality. \n+ Although GP-UCB was known for years, it seems like no-one was able to significantly improve on the initial regret bound derived by (Srinivas et al, 2010). The fact that for this relatively simple algorithm one can derive much better regret guarantees than previously thought, is definitely very important to the community.\n+ As far as I understand, the paper also presents the first non-worst case analysis of the MIG, which seems like a very important step for the community that was essentially relying on the worst-case analysis of (Srinivas et al, 2010) for the last 15 years.\n+ I believe due to the novelty of the approach, the work has the potential to inspire a number of follow-up works, rethinking the regret bounds in BO in general\n\nWeaknesses:\n- For Lemmas 1 and 2, authors just state that these properties are existing and known, do not provide any proof for them and just cite other paper. The problem, however, is that none of the cited papers actually proved these properties (to the best of my knowledge), they merely used them as assumptions. As such it would appear, Lemmas 1 and 2 are Assumptions that have to be made for the presented analysis to hold, rather than being results that follow from previous assumptions. I thus believe that these Lemmas should be changed into Assumptions.\n- In Corollary 8, page 23, just above equation 123, authors write $\\gamma_T(\\lambda^2)$. I believe in the adopted notation, $\\gamma_T(\\cdot)$ took a set of points as an argument, but $\\lambda$ is a number, so not sure what is meant by that.\n- (minor) I believe the formula for Matern kernel on page 3 is wrong, I believe the distance between points should be raised to the power of $\\nu$\n\nQUESTIONS:\n- The authors present a very detailed analysis regarding the MIG in the reduced search space. I wonder is it really necessary to go through it and instead think about reducing the search space as increasing the length scale. Clearly if the radius of input space shrinks from $1$ to $\\eta < 1$, then we could define a GP on the original space with radius $1$ with a longer length scale $\\hat{l} = \\frac{l}{\\eta} $ and then map the points from the reduced space to the original space as $\\hat{x} = \\frac{x}{\\eta}$. We can easily check that for a stationary kernel $k_{\\hat{l}}(\\hat{x},\\hat{x}^\\prime) = k(\\frac{\\hat{x}}{\\hat{l}},\\frac{\\hat{x}^\\prime}{\\hat{l}}) = k(\\frac{x}{l},\\frac{x^\\prime}{l}) = k_{l}(x,x^\\prime)$ and thus $\\max_{x:||x||\\le \\eta} \\sigma_{l}(x)  = \\max_{\\hat{x}:||\\hat{x}||\\le 1} \\sigma_{\\hat{l}}(\\hat{x})$. We could then just plug the MIG bounds with explicit dependence on lengthscale [1,2] and obtain the results that $\\gamma_T(\\{x: ||x||\\le \\eta\\}) = \\eta^d \\gamma_T(\\{x: ||x||\\le 1\\})$, which I believe is much stronger than $\\frac{1}{\\log^d(\\frac{1}{\\eta})}$ that authors derive. Is the analysis above correct for the problem setting authors consider? If so, would it make your final bound even stronger?\n\n- Is it correct to think that the proposed result effectively says BO can \"escape\" the curse of dimensionality? This seems to contradict the common knowledge. Could authors explain whether this is what their result implies and if so, why it seems to contradict the empirical experience.\n\n- In theorems authors state they treat dimensionality $d$ as constant. However, since the derived results seems monumental it terms of reduced dimensionality dependence, I would like to see the full order scaling with $d$ of the bound. I assume, for large enough $T$ this scaling will be dominated by the stated bound, but it would still give more context if authors were able to provide an explicit bound.\n\n[1] Berkenkamp, Felix, Angela P. Schoellig, and Andreas Krause. \"No-regret Bayesian optimization with unknown hyperparameters.\" Journal of Machine Learning Research 20.50 (2019): 1-24.\n\n[2] Ziomek, Juliusz, Masaki Adachi, and Michael A. Osborne. \"Bayesian Optimisation with Unknown Hyperparameters: Regret Bounds Logarithmically Closer to Optimal.\" The Thirty-eighth Annual Conference on Neural Information Processing Systems (2024).\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-06-29"}

comments: SUMMARY:
The paper studies the standard Bayesian optimization problem under the assumptions that the objective function is sampled from a Gaussian Process prior and the noise is Gaussian. It focuses on the standard GP-UCB algorithm and provides an improved regret bound under certain assumptions that hold for the Matérn family of kernels. Specifically, the paper removes the dependence on the maximum information gain $\gamma_T$—which can grow polynomially with $T$—and replaces it with logarithmic terms, leading to tighter theoretical guarantees.

STRENGTHS AND WEAKNESSES:
The paper fills a significant gap in the literature by deriving regret bounds for GP-UCB that match the known lower bound. I consider this a substantial contribution to the field.

A limitation is the assumption on the smoothness parameter of the kernel in Theorem 3, which the authors themselves acknowledge. However, the reason for this restriction is not clearly discussed, and it is difficult to understand the necessity of this assumption from either the main text or the appendix.

The main parts of proof are well explained in Section 3.2.

A very minor comment: the quotation marks around favorable on line 191 should be corrected.

QUESTIONS:
1. Can the authors explain why the assumptions on the smoothness parameter of the kernel are necessary? Specifically, where does the proof break down if these assumptions are not satisfied?

2. Can the authors elaborate on why the proof technique used in Section 3.2 does not extend to GP-TS, in contrast to GP-UCB?

3. Can the authors comment on the noise-free setting? How would the results change if the noise variance were zero?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 4
ORIGINALITY: 3
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper improves the regret bound for the GP-UCP algorithm.\n\nThe algorithm applies to the following framework. There is a domain ${\\cal X}$ (restricted to $[0,1]^d$ in the paper) and a zero-mean Gaussian random field on it (sorry, I prefer a field to a process in this context). For every $x\\in{\\cal X}$ there is a fandom value $f(x)$ so that all joint distributions of $(f(x_1), f(x_2),\\ldots,f(x_n))$ are Gaussian, the means $Ef(x)$ are 0, and the covariances $\\mathrm{cov}(x_1,x_2)$ are given by a known kernel function ${\\cal K}(x_1,x_2)$. The learner can request a value at $x\\in {\\cal X}$ and get $f(x)+\\varepsilon$ corrupted by i.i.d.\\ Gaussian noise $\\varepsilon=\\varepsilon(x)$. Given a set of pairs $(x_i,f(x_i)+\\varepsilon)$, the conditional mean $\\mu(x)$ of $f(x)$ at an arbitrary $x$ is then given by the well-known kernel ridge regression formula $Y'(\\lambda I +K)^{-1}k(x)$ . \n\nHere the goal of the learner is to choose $x_t$ sequentially aiming to hit near $x^* \\in\\arg\\max f $ as soon is possible. The learner's performance is measured by the cumulative regret $\\sum_t (f(x^*) - f(x_t))$.\n\nA natural approach here is to choose $x_t$ maximising $\\mu$ as worked out on the basis of the previously observed values at $x_1,x_2,\\ldots,x_{t-1}$. However, this suffers from the usual optimism under uncertainty faults: the value $\\mu(x)$ may be high due to insufficient exploration. So instead of $\\mu(x)$ Srinivas et al (2010) suggest optimising an upper confidence bound $\\mu+\\beta$. \n\nThis paper tunes the algorithm to improve the regret bounds of Srinivas et al (2010) for the rbf and Matern kernels. The resulting bounds get within the logarithmic factor to the known lower bounds. \n\nThe authors try to explain the intuition behind the improvement through information gain on a sequence.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper improves on state-of-the-art results, which are quite fundamental to machine learning. It does not quite match the lower bounds, but is really close. As a downside, I would mention that the analysis only applies to specific kernels.\n\nQUESTIONS:\nNone\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-05"}