Improved Regret Bounds for Gaussian Process Upper Confidence Bound in Bayesian Optimization

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 Matérn kernel with a certain degree of smoothness, we show that the Gaussian process upper confidence bound (GP-UCB) algorithm achieves 𝑂 ( √ 𝑇) cumulative regret with high probability. Furthermore, our analysis yields 𝑂 ( √︁ 𝑇 ln 2 𝑇) regret under a squared exponential kernel. These results fill the gap between the existing regret upper bound for GP-UCB and the best-known bound provided by Scarlett [46]. The key idea in our proof is to capture the concentration behavior of the input sequence realized by GP-UCB, enabling a more refined analysis of the GP's information gain.

𝑇 ln 𝑇) cumulative regret. Then, the natural question is whether there is further room for improvement in the existing regret upper bound of GP-UCB. This paper provides an affirmative answer to this question by showing that GP-UCB achieves 𝑂 ( √ 𝑇) regret with high probability.

Contribution. We summarize our contributions as follows.

• We show that the GP-UCB proposed by Srinivas et al. [51] achieves 𝑂 ( √ 𝑇) regret with high probability under a Matérn kernel with a certain degree of smoothness (precise condition is provided in Theorem 3). Here, 𝑂 (•) is the order notation that hides polylogarithmic dependence. This result is comparable to state-of-the-art 𝑂 ( √ 𝑇 ln 𝑇) regret provided by Scarlett [46] up to a polylogarithmic factor and strictly improves upon the existing 𝑂 (𝑇 𝜈+𝑑 2𝜈+𝑑 ) upper bound of GP-UCB [51,58]. Here, 𝑑 and 𝜈 denote the dimension of the input domain and smoothness parameter, respectively.

39th Conference on Neural Information Processing Systems (NeurIPS 2025).

The existing theory of GP-UCB under the Bayesian setting utilizes the regularity conditions of the realized sample path of GP. We summarize the existing known properties of the GP sample path in the following lemmas. Lemma 1 (Lipchitz condition of sample path, e.g., [51]). Suppose 𝑘 = 𝑘 SE or 𝑘 = 𝑘 Matérn with 𝜈 > 2. Assume Assumption 1. Then, there exist the constants 𝑎, 𝑏 > 0 such that

Lemma 2 (Sample path condition for the global maximizer, e.g., [13,14,46]). Suppose 𝑘 = 𝑘 SE or 𝑘 = 𝑘 Matérn with 𝜈 > 2. Assume Assumption 1. Then, for any 𝛿 GP ∈ (0, 1), there exist the strictly positive constants 𝑐 gap , 𝑐 sup , 𝑐 quad , 𝜌 quad > 0 such that the following statements simultaneously hold with probability at least 1 -𝛿 GP :

1. The function 𝑓 has a unique maximizer x * ∈ X such that 𝑓 (x * ) > 𝑓 ( x * ) + 𝑐 gap holds for any local maximizer x * ∈ X of 𝑓 .

√ 𝑇), while it is strictly smaller than 𝑂 (𝑇 𝜈+𝑑 2𝜈+𝑑 ) of the original GP-UCB's analysis.