Title: Convergence Rates of Constrained Expected Improvement
Venue: NeurIPS 2025
Accepted: yes
Keywords: Bayesian optimization

constrained Bayesian optimization

constrained expected improvement

simple regret

convergence analysis
Abstract: Constrained Bayesian optimization (CBO) methods have seen significant success in black-box optimization with constraints.  One of the most commonly used CBO methods is the constrained expected improvement (CEI) algorithm. CEI is a natural extension of expected improvement (EI) when constraints are incorporated. However, the theoretical convergence rate of CEI has not been established.   In this work, we study the convergence rate of CEI by analyzing its simple regret upper bound. First, we show that when the objective function $f$ and constraint function $c$ are assumed to each lie in a  reproducing kernel Hilbert space (RKHS),   CEI achieves the convergence rates of $\mathcal{O} \left(t^{-\frac{1}{2}}\log^{\frac{d+1}{2}}(t) \right) \ \text{and }\ \mathcal{O}\left(t^{\frac{-\nu}{2\nu+d}} \log^{\frac{\nu}{2\nu+d}}(t)\right)$ for the commonly used squared exponential and Matérn kernels, respectively. Second, we show that when $f$ is assumed to be sampled from Gaussian processes (GPs), CEI achieves similar convergence rates with a high probability. Numerical experiments are performed to validate the theoretical analysis.

Convergence Rates of Constrained Expected Improvement

Abstract

Constrained Bayesian optimization (CBO) methods have seen significant success in black-box optimization with constraints. One of the most commonly used CBO methods is the constrained expected improvement (CEI) algorithm. CEI is a natural extension of expected improvement (EI) when constraints are incorporated. However, the theoretical convergence rate of CEI has not been established. In this work, we study the convergence rate of CEI by analyzing its simple regret upper bound. First, we show that when the objective function f and constraint function c are assumed to each lie in a reproducing kernel Hilbert space (RKHS), CEI achieves the convergence rates of O ( t -1 2 log d +1 2 ( t ) ) and O ( t -ν 2 ν + d log ν 2 ν + d ( t ) ) for the commonly used squared exponential and Matérn kernels ( ν > 1 2 ), respectively. Second, we show that when f is assumed to be sampled from Gaussian processes (GPs), CEI achieves similar convergence rates with a high probability. Numerical experiments are performed to validate the theoretical analysis.

1 Introduction

Bayesian optimization (BO) is an efficient method for optimizing expensive black-box functions without derivatives. It leverages probabilistic surrogate models, most commonly Gaussian processes (GPs), to balance exploration and exploitation in the search for optimal solutions [Frazier, 2018]. BO has found widespread success in diverse fields such as structural design [Mathern et al., 2021], machine learning hyperparameter tuning [Wu et al., 2019], robotics [Calandra et al., 2016], fusion design [Wang et al., 2024], etc.

While traditional BO is typically applied to unconstrained settings, many real-world problems involve black-box constraints that must be satisfied. This has motivated growing interest in constrained

Bayesian optimization (CBO), where surrogate models are also constructed for constraint functions [Bernardo et al., 2011] that are complex and expensive to evaluate, making CBO especially valuable in applications like engineering design [Song et al., 2024] and automated machine learning [Ungredda and Branke, 2024]. One of the very key difference between unconstrained and constrained optimization is that the feasible region for constrained optimization problem consists of the search space where all constraints must be satisfied. A general form of the constrained BO problem is:

where f : R d → R is the objective function, and c : R d → R m are the constraint functions. Both are defined on a compact input space C ⊂ R d . The objective and the constraint functions are both expensive black-box functions, that can only be evaluated through expensive physical or computer experiments. Throughout this paper, we consider the noise-free setting for both the objective and the constraints, i.e. , the function evaluations are deterministic and the true function values can be observed (see Remark 3.14 for discussion on the noisy case). In addition, a single constraint is considered, i.e. , m = 1 , for simplicity of presentation. We note that our analysis can be easily extended to multiple constraints (see Remark 3.13 for details).

Broadly, CBO methods can be categorized into implicit and explicit approaches [Amini et al., 2025]. Implicit methods modify standard acquisition functions to incorporate constraints via merit functions or feasibility weights. Explicit methods estimate the feasible region directly and restrict search to this region. Among these, the constrained expected improvement (CEI) [Schonlau et al., 1998, Gelbart et al., 2014, Gardner et al., 2014] stands out as one of the most basic and widely adopted methods. CEI is a natural extension of the well-known expected improvement (EI) function [Jones et al., 1998], where the acquisition function is computed as the product of EI and the probability of feasibility. Thanks to this simple and interpretable formulation, CEI has been successfully applied across domains, and it remains one of the default choices in many constrained BO software packages [Balandat et al., 2020].

Despite its empirical popularity, the theoretical understanding of CEI lags behind. In contrast, unconstrained EI has been more extensively studied. Under a frequentist assumption where the objective f lies in a reproducing kernel Hilbert space (RKHS), Bull [2011] established the convergence rate of EI by deriving the simple regret upper bound. Other works explored the density of sampled sequences [Vazquez and Bect, 2010] or connections between EI and optimal computing budget allocation [Ryzhov, 2016]. However, convergence rates ( i.e. , simple regret upper bound) for CEI have not been rigorously established-neither under frequentist nor under Bayesian settings. Here, Bayesian setting means the objective f is a function sampled from a GP.

Introducing constraints into EI significantly complicates the theoretical analysis. Unlike in the unconstrained case, the algorithm may need to explore infeasible regions to gain information on the constraint boundary. Furthermore, CEI's acquisition function is inherently more complex and nonconvex, posing challenges for analysis. On the other hand, the presence of constraints in CEI leads to changes in the sampling procedure. As a result, the key challenge to study the convergence rate of CEI lies in analyzing the exploration (searching for feasible regions) and exploitation (optimizing within feasible areas) since the feasibility threshold is unknown in the input space.

In this paper, we provide the first theoretical convergence rates for CEI, focusing on simple regret upper bounds under both the frequentist and Bayesian settings. Our convergence rates provide practitioners theoretical assurance for the practical deployment of CEI. We explain the technical challenges and how we address them in Section 3. Our contributions are summarized as follows:

Under the frequentist setting, we derive simple regret upper bounds of O ( t -1 2 log d +1 2 ( t ) ) for the squared exponential (SE) kernel and O ( t -ν 2 ν + d log ν 2 ν + d ( t ) ) for Matérn kernels ( ν > 1 2 ). These bounds are improved upon the direct extension of Bull [2011] to the constrained case for SE kernel with d ≥ 3 and Matérn kernels with d ≥ 3 , ν ≥ d d -2 . (see Theorem 3.7).

Under the Bayesian setting for the objective, we achieve similar simple regret upper bounds with high probabilities. These bounds are established based on the newly derived bounds (see Theorem 3.11) on the difference between the improvement function and its corresponding EI in the Bayesian setting.

This paper is organized as follows. In Section 2, we describe the basics and preliminaries of BO, including the CEI algorithm. In Section 3, the simple regret upper bounds of CEI are established in both settings. Numerical experiments to validate the theoretical results are given in Section 4. Conclusions are made in Section 5. All proof details are presented in the appendix.

2 Background

CBO mainly consists of two components: the GP surrogates for the black-box objective function f and constraint function c , and the constrained acquisition function as the sequential sampling rule guiding for the global optimum.

2.1 Gaussian process models for f and c

Without losing generality, let the mean function for the objective GP model prior be 0 and the covariance function (kernel) be k f ( x , x ′ ) : R n × R n → R . At sample point x t ∈ C , we denote the objective function value as f ( x t ) and the observed constraint function value is c ( x t ) . Given t sample points, denote x 1: t = [ x 1 , . . . , x t ] and f 1: t = [ f ( x 1 ) , . . . , f ( x t )] . Moreover, denote the t × t covariance matrix K f t = [ k f ( x 1 , x 1 ) , . . . , k f ( x 1 , x t ); . . . ; k f ( x t , x 1 ) , . . . , k f ( x t , x t )] . The posterior distribution of f ( x ) | x 1: t , f 1: t ∼ N ( µ f t ( x ) , ( σ f t ( x )) 2 ) can then be inferred using Bayes' rule as follows

where k f t ( x ) = [ k f ( x 1 , x ) , . . . , k f ( x t , x )] T . Similarly, denote the kernel for c as k c : R n × R n → R and the covariance matrix K c t = [ k c ( x 1 , x 1 ) , . . . , k c ( x 1 , x t ); . . . ; k c ( x t , x 1 ) , . . . , k c ( x t , x t )] . The posterior distribution for c is

where k c t ( x ) = [ k c ( x 1 , x ) , . . . , k c ( x t , x )] T , and µ c t ( x ) and ( σ c t ) 2 ( x ) are the posterior mean and variance for c , respectively. Here we use the subscripts f , c and superscripts f , c to distinguish between GPs for f and c . Choices of the kernels k f and k c include the SE and Matérn kernels, which are among the most popular kernels for GP and BO. Their definitions are as follows.

where l > 0 is the length hyper-parameters, r = ∥ x -x ′ ∥ 2 , ν > 0 is the smoothness parameter of the Matérn kernel, and B ν is the modified Bessel function of the second kind.

2.2 Constrained Expected Improvement

Acquisition functions are critical to the performances of BO algorithms. In the unconstrained setting, one of the most widely adopted acquisition functions is EI [Jones et al., 1998]. Given t samples, the improvement function of f used in EI is defined as

where f + t = min i =1 ,...,t f ( x i ) . The expectation of (3) conditioned on existing samples is EI, which has a closed form Brochu et al. [2010]:

where z f t ( x ) = f + t -µ f t ( x ) σ f t ( x ) . The functions ϕ and Φ are the probability density function (PDF) and the cumulative distribution function (CDF) of the standard normal distribution, respectively. The t +1 th sample using EI is chosen by

Taking into account the constraint, the constrained improvement function in CEI [Gardner et al., 2014] is defined as

where ∆ c t ∈ { 0 , 1 } is the feasibility indicator function where ∆ c t ( x ) = 1 if c ( x ) ≤ 0 and ∆ c t ( x ) = 0 otherwise. The incumbent f + t in CEI is augmented to be the best feasible observation. CEI assumes that f and c are conditionally independent [Gardner et al., 2014]. Taking the conditional expectation of (6), the CEI function is

where P t ( · ) is the probability of feasibility (POF) function for c ( x ) ≤ 0 . CEI chooses the next sample via

The CEI algorithm is given in Algorithm 1.

CEI can be extended to multiple constraints assuming conditional independence among the constraints [Gardner et al., 2014]. Our derived convergence rates can also be readily extended to multiple constraints, as we explain in Remark 3.13.

3 Convergence rates of CEI

We present our main results of convergence rates for CEI by establishing the simple regret upper bounds. Denote the optimal solution to the constrained optimization problem (1) as x ∗ . In the unconstrained case, the simple regret of EI is defined as f + t -f ( x ∗ ) [Bull, 2011]. In the constrained case, we use the current best feasible observation and compare it to the optimal solution f ( x ∗ ) , since one could have an infeasible sample point with smaller objective than f ( x ∗ ) . Given that f + t is already defined as the best feasible observation till iteration t in CEI, we continue to use

as the simple regret for CEI. In our analysis, we make the same underlying assumption as CEI that f + t exists. In the following, we first establish the convergence rate under the frequentist assumptions in Section 3.1, including an improved version of the rate under frequentist assumptions in Section 3.1.1. Then, we establish the convergence rate under Bayesian objective assumptions in Section 3.2.

3.1 Simple regret upper bound under frequentist assumptions

In this section, we present the simple regret upper bound for CEI under the frequentist setting. Moreover, by adopting the information theory-based bounds and techniques in the noise-free cumulative regret bound of upper confidence bound (UCB) [Lyu et al., 2019], we can derive an improved upper bound in some cases compared to Bull [2011]. The definition of RKHS is given below.

Definition 3.1. Let k be a positive definite kernel k : X × X → R with respect to a finite Borel measure supported on X . A Hilbert space H k of functions on X with an inner product ⟨· , ·⟩ H k is

called a RKHS with kernel k if k ( · , x ) ∈ H k for all x ∈ X , and ⟨ f, k ( · , x ) ⟩ H k = f ( x ) for all x ∈ X , f ∈ H k . The induced RKHS norm ∥ f ∥ H k = √ ⟨ f, f ⟩ H k measures the smoothness of f with respect to k .

In this section, we assume the following assumptions on the functions f and c .

Assumption 3.2. The functions f and c lie in the RKHS, denoted as H f k ( C ) and H c k ( C ) associated with their respective bounded kernel k f and k c , with the norm ∥·∥ H f k and ∥·∥ H c k . The kernels satisfy k f ( x , x ′ ) ≤ 1 , k c ( x , x ′ ) ≤ 1 , k f ( x , x ) = 1 , and k c ( x , x ) = 1 , for ∀ x , x ′ ∈ C . The RKHS norms are bounded above by constants B f and B c , respectively, i.e. , ∥ f ∥ H f k ≤ B f , ∥ c ∥ H c k ≤ B c . Moreover, the bound constraints set C is compact.

Technical Challenges under Frequentist Assumptions. The main challenge in establishing a simple regret upper bound for CEI is how to incorporate the constraint c ( x ) ≤ 0 and the probability of feasibility function P t ( x ) into the analysis. Existing regret bounds analysis on CBO methods often focus on UCB-type methods [Lu and Paulson, 2022, Zhou and Ji, 2022], for which the acquisition functions do not have the multiplicative structure between the objective and the constraint.

Under Assumption 3.2, both f and c are bounded on C by their RKHS norm bounds, as stated in Lemma B.1. The simple regret upper bound is given in the following theorem.

Theorem 3.3. Under Assumption 3.2, the CEI algorithm leads to the simple regret upper bound of

for some t k ∈ [ t 2 -1 , t ] , and c τB = τ ( B f ) τ ( -B f ) .

Sketch of Proof for Theorem 3.3. We start by noticing that the sum of the difference between consecutive best feasible observations is bounded, i.e. , ∑ T t =1 f + t -1 -f + t ≤ 2 B f . Then, we adopt a technique in Bull [2011] to find t k such that f + t k -f + t k +1 ≤ 2 B f k , where k ≤ t k ≤ 2 k and 2 k ≤ t ≤ 2( k +1) . Next, using the monotonicity of f + t , r t is bounded by r t k . Using the inequality between I f t and EI f t in Lemma B.4, we can bound r t k by the EI on objective: EI f t k ( x ∗ ) we transform EI f t k ( x ∗ ) into EI f t k ( x t k +1 ) by inserting the term P t k ( x ∗ ) , taking advantage of the multiplicative structure of CEI. The upper bound of r t then consists of the term 1 ∗ EI f ( x t +1 )

From the confidence interval | f ( x ) -µ f t ( x ) | (Lemma B.2) and the fact that f + t k -f + t k +1 ≤ f k , we can bound EI f t k ( x t k +1 ) . The constraint term 1 P t k ( x ∗ ) remains to be bounded. We use the confidence interval on | c ( x ) -µ c t ( x ) | in Lemma B.2 at x ∗ and the fact that x ∗ is a feasible solution to obtain a lower bound for P t k ( x ∗ ) . This concludes the proof.

. Then, P t k ( x ) t k k . 2 B

Remark 3.4 (Constraint in the simple regret upper bound) . The terms derived from the constraint function in (10) is 1 Φ( -B c ) , which emerges from the probability of feasibility function and µ c t ( x ) and σ c t ( x ) of the GP model of c ( x ) . Thanks to the multiplicative structure between the objective and constraint in I C t (6) and EI c t (7), the simple regret upper bound maintains a similar form.

It is clear from (10) that the convergence of r t relies on the posterior standard deviation σ f t ( x t +1 ) . Since t k increases with t , as σ f t ( x t +1 ) → 0 , so does σ f t k ( x t k +1 ) . In the noise-free setting, the posterior variance can be bounded via the maximum distance between sample points and a given point. To obtain the rate of simple regret bound, we use Assumptions (1)-(4) in Bull [2011] and focus on squared exponential (SE) and Matérn kernels. Recall that the smoothness parameter of the Matérn kernel is ν > 0 . Both the SE and Matérn kernels satisfy Assumptions (1)-(4) in Bull [2011], with SE kernel obtained as ν →∞ . Further, define

where α = 1 2 if ν ∈ N , and α = 0 otherwise. Then, for SE and Matérn kernels, σ f t k ( x t k +1 ) can be bounded with the following lemma.

Lemma 3.5 (Bull [2011]) . For the SE kernel, there exists constant C ′ > 0 so that given ∀ t ∈ N ,

holds for at most k times, for ∀ k ∈ N , k ≤ t and i = 1 , . . . , t -1 . For Matérn kernels,

holds at most k times.

In the constrained setting, we are able to obtain the same rates as those in the unconstrained case [Bull, 2011] using Lemma 3.5.

Corollary 3.6. Under Assumption 3.2, the CEI algorithm leads to the convergence rates of

for SE and Matérn kernels, respectively, where η is from (11) .

Corollary 3.6 shows that the CEI algorithm is guaranteed to find the best feasible point asymptotically with the rates elaborated in (14). Also, we point out that the choice of kernels and their parameters affect the convergence rates. Since the SE kernel can be viewed as a Matérn kernel with ν →∞ , its convergence rate is better than Matérn kernels with ν ≤ 1 . However, due to the limitations of the kernel analysis in Bull [2011] (see Remark 3.8), for ν ≥ 1 , SE and Matérn kernels have similar convergence rates in Corollary 3.6. As we present in the following section, improved rates for both kernels can be obtained in some cases.

3.1.1 Improved simple regret upper bound under frequentist assumptions

Next, we apply maximum information gain and the corresponding information theory to obtain improved simple regret upper bounds.

Theorem 3.7. Under Assumption 3.2, the CEI algorithm leads to the improved convergence rates of

Sketch of Proof for Theorem 3.7. The proof follows similar steps to that of Theorem 3.3 but further bounds σ f t k ( x t k +1 ) using γ f t . To do so, we first recognize that the bound using γ f t (Lemma A.3) is established in the noisy case where the posterior standard deviation has a different form as in (18). Using Lemma A.4, we can establish that the noise-free posterior standard deviation also satisfies ∑ t -1 i =0 σ f i ( x i +1 ) ≤ √ C γ tγ f t . Then, from Lemma A.5, we can find a small enough σ f i ( x i +1 ) . Specifically, choose k = [ t/ 3] , where [ x ] denotes the largest integer smaller than x . Thus, we have 3 k ≤ t ≤ 3( k + 1) . Then, there exists k ≤ t k ≤ 3 k such that f + t k -f + t k +1 ≤ 2 B f k and σ f t k ( x t k +1 ) ≤ √ tγ f t k . The rest of the proof follows from that of Theorem 3.7.

Remark 3.8 (Improved rate of convergence) . As mentioned above, the rates in Corollary 3.6 are the same as the known convergence rates for EI in Bull [2011]. Meanwhile, the rates in Theorem 3.7 is an improvement over those of Bull [2011] for SE kernel with d ≥ 3 and Matérn kernels with d ≥ 3 , ν ≥ d d -2 . To achieve this, we applied techniques from regret bound analysis on noise-free UCB[Lyu et al., 2019] that allows us to use maximum information gain to bound the sum of σ f t ( x t +1 ) . Then, we use our techniques in the proof of Theorem 3.3 to bound an individual σ f t k ( x t k +1 ) . In Bull [2011], the σ f t k ( x t k +1 ) is bounded by the Taylor expansion of the kernel functions. Therefore, the rates of decrease are limited to quadratic terms for both SE and Matérn kernels, since their Taylor expansions around 0 for ∥ x -x ′ ∥ 2 are quadratic at best. On the other hand, maximum information gain can lead to tight bounds on γ f t that take advantages of the spectral properties of the kernels [Vakili et al., 2021, Iwazaki, 2025]. Hence, using γ f t to bound σ f t k ( x t k +1 ) can produce a faster rate. As the open question raised in Vakili [2022] gets answered, further improvement of the convergence rates is possible, e.g. , using techniques from Iwazaki [2025].

3.2 Simple regret upper bound under Bayesian objective assumption

In this section, we present the simple regret upper bound for CEI under the Bayesian objective assumptions. We again use the maximum information gain to derive the simple regret upper bound.

Assumption 3.9. The bound constraint set C ⊂ [0 , r ] d is compact and convex. The objective function f is sampled from GP (0 , k f ( x , x ′ )) . Further, the objective function f is assumed to be Lipschitz continuous (of 1-norm) with Lipschitz constant L f with probability ≥ 1 -da f e L 2 f /b 2 f for some constants a f > 0 and b f > 0 . The kernels satisfy k f ( x , x ′ ) ≤ 1 and k f ( x , x ) = 1 . The constraint function c remains in the RKHS of k c , similarly to the frequentist setting.

In the remaining of this section we will work under Assumption 3.9.

Technical Challenges under Bayesian Assumptions. In addition to the challenges in the frequentist setting, the bounds on EI in the Bayesian setting are not available in current literature, to the best of our knowledge. Starting from the confidence interval on | f ( x ) -µ f t ( x ) | , we derive the bounds on | I f t ( x ) -EI f t ( x ) | with high probability, an important step towards the bound on r t . Noticeably, under the Bayesian setting, the bounds are satisfied with a given probability, e.g. , 1 -δ , where δ ∈ (0 , 1) .

The simple regret upper bound is given in the following theorem.

Theorem 3.10. Let β = 2log(6 c α /δ ) and β t = 2log(3 π t /δ ) , where c α = 1+2 π 2 π and π t = π 2 t 2 6 . Under Assumption 3.9, the CEI algorithm leads to the simple regret upper bound

for some t k ∈ [ t 2 -1 , t ] , c τ ( β ) = τ ( β 1 / 2 ) τ ( -β 1 / 2 ) , and constant M f > 0 with probability ≥ 1 -δ .

The constant M f is from Lemma C.1. The convergence rate is given in the next theorem.

Theorem 3.11. Let β = 2log(6 c α /δ ) and β t = 2log(3 π t /δ ) , where c α = 1+2 π 2 π and π t = π 2 t 2 6 . Under Assumption 3.9, the CEI algorithm leads to the convergence rates of

for SE and Matérn kernels, respectively, with probability ≥ 1 -δ .

Sketch of Proof for Theorem 3.10. Recall that f and c are assumed conditionally independent in CEI. We start from the bound on the confidence interval for f : | f ( x ) -µ f t ( x ) | ≤ β 1 / 2 σ f t ( x ) , with probability ≥ 1 -δ , where β = 2log(1 /δ ) , as in Lemma C.2. The confidence interval of c remains the same as in the frequentist setting. These are well-known results [Srinivas et al., 2009]. Then, we derive the subsequent bounds | I f t ( x ) -EI f t ( x ) | ≤ √ βσ f t ( x ) , where β = max { 1 . 44 , 2 log( c α /δ ) } and c α = 1+2 π 2 π with probability ≥ 1 -δ (Lemma C.5). Then, we prove the relationship in Lemma C.6 √

that I f t ( x ) ≤ τ ( β ) τ ( - √ β ) EI f t ( x ) with probability ≥ 1 -δ . We can now follow the general analysis framework in Section 3.1 and Theorem 3.3 to obtain the simple regret upper bound under Bayesian objective assumptions, while choosing t k with a more defined criterion.

Remark 3.12 (Comparison to the frequentist setting) . Comparing Theorem 3.7 to Theorem 3.11, the convergence rates in the frequentist and Bayesian settings are the same except for a log 1 / 2 ( t ) term. This is partially because simple regret focuses on the best feasible solution f + t and thus many of the parameters in Theorem 3.3 and 3.10 do not depend on t .

Remark 3.13 (Multiple constraints) . As mentioned in Section 1, our results can be readily applied to CEI with multiple constraints for both frequentist and Bayesian settings. Consider m constraints c i ( x ) ≤ 0 , i = 1 , . . . , m . Assuming conditional independence of the constraints, the CEI function is EI C t ( x ) = Π i m =1 P i t ( x ) EI f t ( x ) = Π i m =1 Φ ( -µ c i t ( x ) σ c i t ( x ) ) EI f t ( x ) , where P i t is the probability of feasibility function of constraint c i ( x ) ≤ 0 , and µ c i t ( x ) and σ c i t ( x ) are the posterior mean and standard deviation for c i , respectively. By making the assumption that each constraint function lies

in its corresponding RKHS of the kernel k c i , we have | c i ( x ) -µ c i t ( x ) | ≤ B c i σ c i t ( x ) , where B c i is the upper bound of RKHS norm associated with kernel k c i and function c i . We can then apply the analysis framework in this paper to obtain an upper bound similar to that of Theorem 3.3, where the term 1 Φ( -B c ) is replaced with Π i m =1 1 Φ( -B c i ) . We note that in the Bayesian objective setting, to ensure probability 1 -δ , the parameter β needs to increase with the number of constraints as well, e.g. , β = 2log(( m +5) c α /δ ) .

Remark 3.14 (Extension to the noisy setting) . Extending our analysis to the noisy setting is non-trivial, and we discuss the associated challenges for noisy objective and constraint functions separately. A noisy constraint function introduces additional complications in defining feasibility. If only noisy observations of the constraint values are available, the notion of a feasible sample and the definition of f + t becomes ambiguous. As a result, major modifications to the CEI algorithm are required to appropriately handle the uncertainty introduced by noise.

For the noisy objective function, CEI can be adapted similarly to the noisy EI formulation by treating the best feasible noisy observation as the incumbent. However, to the best of our knowledge, a theoretical guarantee on the simple regret bound for the noisy unconstrained setting remains unavailable. Recent work by Wang et al. [2025] provides a framework for deriving noisy simple regret bounds based on the best observed value, r s t = y + t -f ( x t ) , which can be extended to CEI. Specifically, by defining r s t as the simple regret for CEI with y + t denoting the best feasible noisy observation, a similar proof strategy as in Theorem 3.10 yields an analogous upper bound. In the Bayesian setting with i.i.d. Gaussian noise on the objective and noise-free constraint observations, the convergence rate of the upper bound on r s t can be obtained. However, we note that given the noise, r s t is possibly negative.

Remark 3.15 (Infeasible initial sample) . It is well known that CEI requires initial feasible sample [Gardner et al., 2014]. That is, f + t exists from the initial samples so that the CEI calculation can proceed. Methods proposed to address this issue typically employ separate strategies when no feasible samples are available and revert to the standard CEI formulation once feasibility is established [Lin et al., 2024, Letham et al., 2019]. In addition, introducing a tolerance parameter in the constraint can further mitigate this problem by allowing near-feasible points when the degree of violation is small. Remark 3.16 (Tolerance in constraints) . In gradient-based optimization methods, a tolerance for constraint violation is often used to improve the performance and flexibility of algorithms Wächter and Biegler [2006], Nocedal and Wright [2006]. Motivated by this, we introduce a tolerance parameter λ ≥ 0 , where a point x is considered feasible if c ( x ) ≤ λ and infeasible otherwise. The corresponding

CEI with tolerance is defined as EI C t ( x , λ ) = P t ( x , λ ) EI f t ( x ) = Φ ( λ -µ c t ( x ) σ c t ( x ) ) EI f t ( x ) . Clearly, the standard CEI formulation is recovered when λ = 0 .

The simple regret bound is affected by λ and should lead to 1 Φ( λ -B c ) in place of 1 Φ( -B c ) . In fact, we can replace 1 Φ( λ -B c ) with 1 / Φ ( λ σ c t k ( x ∗ ) -B c ) , which is time-varying. One can follow the proof of Theorem 3.3 to obtain this term, which emerges from the confidence interval of c at t k , x ∗ and c ( x ∗ ) ≤ 0 .

As the sample iteration increases, the inclusion of σ c t k ( x ∗ ) is important in balancing -B c that can lead to a large simple regret upper bound. We explain the intuition below. As t → ∞ , t k → ∞ and k → ∞ . We know σ f t ( x t +1 ) → 0 , and hence σ f t k ( x t k +1 ) → 0 and r t → 0 . That is, the simple regret upper bound of CEI with λ > 0 converges. Thus, x t approaches at least one of the optimal solutions. Suppose without losing generality, x t → x ∗ . Then, by definition σ c t k ( x ∗ ) → 0 . Consequently, we should have λ σ c t k ( x ∗ ) →∞ for λ > 0 . Then, we have Φ ( λ σ c t k ( x ∗ ) ) → 1 . Therefore, 1 / Φ( λ σ c t k ( x ∗ ) -B c ) → 1 and B c does not affect the simple regret upper bound asymptotically. We note that the convergence rate of CEI with tolerance remains similar since it is dominated by the maximum information gain of f .

4 Numerical experiments

Although this paper is primarily theoretical, we conduct numerical experiments to support the theoretical results. We apply the CEI algorithm to eight synthetic problems that are randomly generated from RKHS of kernels and GP priors, and five benchmark problems commonly used in the

CBO literature. These numerical experiments are not intended to demonstrate superior performance over the state-of-the-art CBO algorithms. Instead, they serve as empirical evidence for the theoretical analysis presented in this work. All experiments are conducted on M1 (16GB memory) 1 .

4.1 Synthetic problems

In this section, we study objective and constraint functions drawn from reproducing kernel Hilbert spaces (RKHSs) as well as from Gaussian process priors with Matérn ( ν = 2 . 5 ) and squared exponential (SE) kernels, across input dimensions d ∈ 2 , 4 . The domain is the hypercube [0 , 1] d . For RKHS cases (the frequentist setting), the functions are generated with a similar approach to Chowdhury and Gopalan [2017]. Specifically, both objective f ( x ) and constraint functions c ( x ) are generated by sampling from the RKHS associated with a chosen kernel (Matérn/SE kernels with a length scale of 0 . 2 ). Each function is constructed as a weighted sum of kernel evaluations at 100 randomly selected basis points, with weights drawn from a standard normal distribution. Formally, the function takes the form f ( x ) = ∑ n i =1 α i k ( x, X i ) , where k is the kernel, X i are basis points, and α i are random coefficients; c ( x ) is generated similarly. For the GP cases (the Bayesian setting), the functions are generated with an approach similar to Srinivas et al. [2009]. Specifically, we uniformly choose 1000 points in the design space and sample randomly from a multivariate Gaussian distribution defined by the GP prior with the chosen kernel.

For each synthetic problem, we conducted 100 independent trials. The number of initial design is set to 10 d , and 50 optimization iterations were performed for all cases. We plotted the log-log curve of simple regret against the number of iterations in Figure 1. In all cases, we consistently observed sublinear convergence patterns, which align well with our theoretical guarantees.

d

= 2

d

= 4

(c) RKHS, SE,

d

= 2

(d) RKHS, SE,

d

= 4

d

= 2

d

= 4

(g) GP, SE,

d

= 2

(h) GP, SE,

d

= 4

log(simple regret) vs log(iteration)

log(simple regret) vs log(iteration)

log(simple regret) vs log(iteration)

log(simple regret) vs log(iteration)

01

10°

101

10-1

(aless 6ol) taubau a(dw!s

(aje2s 6o1) lau6au a(dws

(ajes 6ol) taufau aldwls

(ajeos fol) aufau aldws

10

102

101

10-1

10

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10

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Iteration t (log scale)

Iteration t (log scale)

Iteration t (log scale)

Iteration t (log scale)

log(simple regret) vs log(iteration)

log(simple regret) vs log(iteration)

5 × 10-1

log(simple regret) vs log(iteration)

log(simple regret) vs log(iteration)

5 × 10-1

7 × 101

6 × 10-1

(ajes fol) taufau alduwis

(ajes o) afa a)dwls

6 × 10-1

Simple regret (log scale)

(ajess ol) aufau alduls

5 × 10

10-1

5 × 10-1

101 。

4 × 101

5 × 102

4 × 10-1

5 × 10-7 ↓

10

20

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Iteration t (log scale)

Iteration t (log scale)

Iteration t (log scale)

Iteration t (log scale)

4.2 Test problems

Next, we evaluate simple regret of CEI on five commonly used test problems in the literature of CBO. Specifically, Problem 1 tests performance in a small feasible region, which was previously studied in Gardner et al. [2014], Ariafar et al. [2019]. Problem 2 includes multiple constraints and local minimums, which has been used in Gramacy et al. [2016], Hernández-Lobato et al. [2015]. Problem 3 is a four-dimensional problem, previously studied in Picheny et al. [2016], Ariafar et al. [2019].

Problem 4 is the six-dimensional Hartmann problem, previously tested in Letham et al. [2019]. Problem 5 is the Rosenbrock function, where the global minimum lies in a narrow region. The mathematical formulations of the five functions are presented in Appendix D. For two-dimensional problems, we also include the contour plots of the objective and constraint functions in Appendix D. The SE kernel is used for the GP modeling (similar performance is observed for the Matérn kernel) and the hyper-parameters are estimated by a standard maximum likelihood method.

For each test problem, we conducted 100 independent trials with different random initial designs. When CEI fails to identify a feasible sample, we adopt the same heuristic strategy as in Letham et al. [2019]. The numerical results are summarized in Figure 2. The solid line represents the median of the simple regret, and the dotted lines represent the 25th percentile and 75th percentile of the simple regret, respectively. From the figures, we observe that CEI consistently reduces simple regret, aligning with the asymptotic convergence theories established in this paper. Accross all problems, the simple regret converges to 0 quickly. The 25th percentile and 75th percentile results demonstrate the good statistical properties of CEI.

8

simple regret

6

4

CEI: 75th

CE1: 25th

CEl: median

2

5

10

15

20

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30

optimization iterations

feasible objective

m

2

......

CEI: 75th

CEl: 25th

CEl: median

1

0

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optimization iterations

10

8

simple regret

CEI: 75th

CEl:25th

4

CEl:median

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optimization iterations

2.5

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0.0

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optimization iterations

3.5

3.0

regret

2.5

simple

2.0

1.5

IEEERSS

CEI: 25th

CEl: median

CEI:75th

1.0

0.5

0.0

0

10

20

30

40

50

optimization iterations

5 Conclusions

In this paper, we studied the simple regret upper bounds of the CEI algorithm, one of the most widely adopted CBO methods. Under both frequentist setting and Bayesian objective assumptions, we establish for the first time the convergence rates for CEI. Our results provide theoretical support and validation for the empirical success of CEI.

References

S. Amini, I. Vannieuwenhuyse, and A. Morales-Hernández. Constrained bayesian optimization: A review. IEEE Access , 13:1581-1593, 2025. doi: 10.1109/ACCESS.2024.3522876.

S. Ariafar, J. Coll-Font, D. Brooks, and J. Dy. ADMMBO: Bayesian optimization with unknown constraints using ADMM. Journal of Machine Learning Research , 20(123):1-26, 2019.

M. Balandat, B. Karrer, D. R. Jiang, S. Daulton, B. Letham, A. G. Wilson, and E. Bakshy. BoTorch: A Framework for Efficient Monte-Carlo Bayesian Optimization. In Advances in Neural Information Processing Systems 33 , 2020. URL http://arxiv.org/abs/1910.06403 .

J. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. Smith, and M. West. Optimization under unknown constraints. Bayesian Statistics , 9:229, October 2011.

E. Brochu, V. M. Cora, and N. De Freitas. A tutorial on Bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning. arXiv preprint arXiv:1012.2599 , December 2010.

A. D Bull. Convergence rates of efficient global optimization algorithms. Journal of Machine Learning Research , 12(10), 2011.

R. Calandra, A. Seyfarth, J. Peters, and M. P. Deisenroth. Bayesian optimization for learning gaits under uncertainty. Annals of Mathematics and Artificial Intelligence , 76:5-23, February 2016.

S. R. Chowdhury and A. Gopalan. On kernelized multi-armed bandits. In International Conference on Machine Learning , pages 844-853. PMLR, 2017.

P. I. Frazier. Bayesian optimization. In Recent advances in optimization and modeling of contemporary problems , pages 255-278, October 2018.

J. R. Gardner, M. J. Kusner, Z. Xu, K. Q. Weinberger, and J. P. Cunningham. Bayesian optimization with inequality constraints. In Proceedings of the 31st International Conference on International Conference on Machine Learning - Volume 32 , ICML'14, pages II-937-II-945. JMLR.org, 2014.

M. A. Gelbart, J. Snoek, and R. P. Adams. Bayesian optimization with unknown constraints. arXiv preprint arXiv:1403.5607 , 2014.

R. B. Gramacy, G. A. Gray, S. L. Digabel, H. K. lee, P. R. Ranjan, G. Wells, and S. M. Wild. Modeling an augmented lagrangian for blackbox constrained optimization. technometrics , 58 (1):1-11, 2016.

J. M. Hernández-Lobato, M. Gelbart, M. Hoffman, R. Adams, and Z. Ghahramani. Predictive entropy search for Bayesian optimization with unknown constraints. In International conference on machine learning , pages 1699-1707. PMLR, 2015.

S. Iwazaki. Gaussian process upper confidence bound achieves nearly-optimal regret in noisefree gaussian process bandits. arXiv preprint arXiv:2502.19006 , 2025.

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

D. R. Jones, M. Schonlau, and W. J. Welch. Efficient global optimization of expensive black-box functions. Journal of Global optimization , 13:455-492, 1998.

B. Letham, B. Karrer, G. Ottoni, and E. Bakshy. Constrained Bayesian optimization with noisy experiments. Bayesian Anal. , 14(2), 2019.

Q. Lin, J. Hu, Q. Zhou, L. Shu, and A. Zhang. A multi-fidelity bayesian optimization approach for constrained multi-objective optimization problems. Journal of Mechanical Design , 146(7): 071702, 2024.

C. Lu and J. A Paulson. No-regret bayesian optimization with unknown equality and inequality constraints using exact penalty functions. IFAC-PapersOnLine , 55(7):895-902, 2022.

Y. Lyu, Y. Yuan, and I. W. Tsang. Efficient batch black-box optimization with deterministic regret bounds. arXiv preprint arXiv:1905.10041 , 2019.

A. Mathern, O. S. Steinholtz, A. Sjöberg, et al. Multi-objective constrained Bayesian optimization for structural design. Structural and Multidisciplinary Optimization , 63:689-701, February 2021.

J. Nocedal and S. Wright. Numerical Optimization . Springer-Verlag, New York, 2006.

V. Picheny, R. B. Gramacy, S. Wild, and S. Le Digabel. Bayesian optimization under mixed constraints with a slack-variable augmented lagrangian. Advances in neural information processing systems , 29, 2016.

I. O. Ryzhov. On the convergence rates of expected improvement methods. Operations Research , 64(6):1515-1528, 2016.

M. Schonlau, W. J. Welch, and D. R. Jones. Global versus local search in constrained optimization of computer models. Lecture notes-monograph series , pages 11-25, 1998.

J. Song, Y. Cui, P. Wei, M. A. Valdebenito, and W. Zhang. Constrained bayesian optimization algorithms for estimating design points in structural reliability analysis. Reliability Engineering System Safety , 241:109613, 2024.

N. Srinivas, A. Krause, S. M. Kakade, and M. Seeger. Gaussian process optimization in the bandit setting: No regret and experimental design. arXiv preprint arXiv:0912.3995 , 2009.

J. Ungredda and J. Branke. Bayesian optimisation for constrained problems. ACM Trans. Model. Comput. Simul. , 34(2), April 2024.

S. Vakili. Open problem: Regret bounds for noise-free kernel-based bandits. In Conference on Learning Theory , pages 5624-5629. PMLR, 2022.

S. Vakili, K. Khezeli, and V. Picheny. On information gain and regret bounds in gaussian process bandits. In International Conference on Artificial Intelligence and Statistics , pages 82-90. PMLR, 2021.

E. Vazquez and J. Bect. Convergence properties of the expected improvement algorithm with fixed mean and covariance functions. Journal of Statistical Planning and inference , 140(11): 3088-3095, 2010.

J. Wang, N. Chiang, A. Gillette, and J. L. Peterson. A multifidelity bayesian optimization method for inertial confinement fusion design. Physics of Plasmas , 31(3), 2024.

J. Wang, H. Wang, C. G. Petra, and N. Y. Chiang. On the convergence of noisy bayesian optimization with expected improvement. arXiv preprint arXiv:2501.09262 , 2025.

J. Wu, X. Chen, H. Zhang, L. Xiong, H. Lei, and S. Deng. Hyperparameter optimization for machine learning models based on bayesian optimization. Journal of Electronic Science and Technology , 17(1):26-40, 2019.

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.

X. Zhou and B. Ji. On kernelized multi-armed bandits with constraints. Advances in neural information processing systems , 35:14-26, 2022.

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A Background information and preliminary results

A.1 Information gain

To obtain state-of-the-art simple regret upper bound, we utilize the information theory results that are well-established in previous literature [27, 8, 30]. Using f as the example, let A ⊂ C denote a set of sampling points. Assume that the observations are noisy at sample points with y A = f ( x A ) + ϵ A at x ∈ A , where ϵ A ∼ N (0 , σ 2 ) denotes the independent and identically distributed Gaussian noises. The maximum information gain is defined as follows.

Definition A.1. Given x A and y A , the mutual information between f and y A is I ( y A ; f A ) = H ( y A ) -H ( y A | f A ) , where H denotes the entropy. The maximum information gain γ f T after T samples is γ f T = max A ⊂ C, | A | = T I ( y A ; f A ) .

The rate of increase for γ f t depends on the property of the kernel. For common kernels such as the SE kernel and the Matérn kernel, γ f t has been studied in literature and the state-of-the-art rates of γ f t are summarized in Lemma A.2 [30, 15].

Lemma A.2. For GP with t samples, the SE kernel has γ t = O (log d +1 ( t )) , and the Matérn kernel with smoothness parameter ν > 0 . 5 has γ t = O ( t d 2 ν + d (log 2 ν 2 ν + d ( t ))) .

The maximum information gain γ c t for the constraint function can be defined similarly.

While γ f t is defined in the noisy case, we can readily apply it to the noise-free case and bound σ f t ( x t +1 ) using techniques similar to that in [20]. To do so, we note that given the Gaussian observation noise ϵ t ∼ N (0 , σ 2 ) in the GP model, the posterior prediction for f becomes

Similarly, we can define the posterior predictions for c with noise in the GP model as ˜ µ f t ( x ) and ˜ σ f t ( x ) . The sum of posterior variance for GP generated by (2) satisfy the next lemma, based on information theory [27] .

Lemma A.3. The sum of GP posterior variances given t samples satisfy

where C γ = 2 log (1+ σ -2 ) and γ f t and γ c t are the maximum information gains for f and c , respectively.

We have the following lemma for ˜ σ f t ( x ) and σ f t ( x ) .

Lemma A.4. The noise-free (GP) posterior standard deviation satisfies σ f t ( x ) < ˜ σ f t ( x ) for ∀ σ > 0 .

Proof. We first note that all the eigenvalues of K f t is smaller than those of K f t + σ 2 I , since K f t is symmetric and positive definite. Thus,

for ∀ σ > 0 and k f t ( x ) . Therefore, by their definitions (2) and (18), the proof is complete.

The posterior standard deviation under assumptions (1)-(4) in [6] in the frequentist and noise-free setting is given in Lemma 3.5, whose proof is given below.

Proof. By Lemma 7 in [6], there exists C ′ > 0 so that

at most k times, for ∀ k ∈ N , k ≤ t , and i = 1 , . . . , t -1 . Therefore, for the SE kernel,

at most k times. For Matérn kernels, we have (21) at most k times.

Using maximum information gain, a tighter bound on the posterior standard deviation can be obtained in the next lemma.

Lemma A.5. Given ∀ t ∈ N and i = 1 , 2 , . . . , t -1 , for SE kernel, there exists constant C ′ > 0 so that

holds for at most k times, for ∀ k ≤ t . Similarly, for Matérn kernels with ν > 0 . 5 ,

holds at most k times.

Proof. From Lemma A.4 and Lemma A.3, we can write that

where we use without losing generality C γ ≤ 1 . Therefore, for any k ∈ N and k ≤ t ,

at most k times. Therefore, for SE kernel, by Lemma A.2, there exists C ′ > 0 such that

at most k times. For Matérn kernels with ν > 0 . 5 ,

at most k times.

Next, we state some basic properties of ϕ , Φ and τ as a lemma.

Lemma A.6. The PDF and CDF of standard normal distribution satisfy 0 < ϕ ( x ) ≤ ϕ (0) < 0 . 4 , Φ( x ) ∈ (0 , 1) , for any x ∈ R . Given a random variable ξ sampled from the standard normal distribution: ξ ∼ N (0 , 1) , we have P { ξ > c | c > 0 } ≤ 1 2 e -c 2 / 2 . Similarly, for c < 0 , P { ξ < c | c < 0 } ≤ 1 2 e -c 2 / 2 . The function τ ( · ) is monotonically increasing.

The last statement in Lemma A.6 is a well-known result ( e.g. , see proof of Lemma 5.1 in [27]).

The next lemma proves basic properties for EI f t .

Lemma A.7. For ∀ x ∈ C , EI f t ( x ) ≥ 0 and EI f t ( x ) ≥ f + t -µ f t ( x ) . Moreover,

Proof. From the definition of I f t and EI f t , EI f t ( x ) ≥ 0 and EI f t ( x ) ≥ y + t -µ f t ( x ) follow immediately. By (4),

If z f t ( x ) < 0 , or equivalently f + t -µ f t ( x ) < 0 , (30) leads to EI f t ( x ) σ f t ( x ) < ϕ ( z f t ( x )) . If z f t ( x ) ≥ 0 , we can write EI f t ( x ) σ f t ( x ) < z f t ( x ) + ϕ ( z f t ( x )) . The left inequality in (29) is an immediate result of EI f t ( x ) ≥ f + t -µ f t ( x ) .

B Proofs for simple regret upper bound under frequentist assumptions

We state the boundedness result as a Lemma for easy reference.

Lemma B.1. Under Assumption 3.2, | f ( x ) | ≤ B f and | c ( x ) | ≤ B c for all x ∈ C .

Proof. By Assumption 3.2, we can write

The following Lemma is a well-established result [8].

Lemma B.2. At any given x ∈ C and t ∈ N , the confidence intervals satisfy

Next, we present a lemma on the relationship between I f t and EI f t , previously seen in [6].

Similarly, for ∀ x ∈ C and t ≥ 1 .

Proof. If f + t -f ( x ) ≤ 0 , then I f t ( x ) = 0 . Since EI f t ( x ) ≥ 0 , (35) is trivial. If f + t -f ( x ) > 0 , by Lemma B.2,

From the monotonicity of τ , we have

From Lemma B.3,

Applying (39) to (38) leads to

Proof of Theorem 3.3 is given next.

and therefore,

Lemma B.4. The improvement function I f t ( x ) and EI f t ( x ) satisfy

Since f + i -f + i +1 ≥ 0 , f + i -f + i +1 ≥ 2 B f k at most k times for any k ∈ N . Further, f ( x t ) ≥ f + t for ∀ t ∈ N . Choose k = [ t/ 2] , where [ x ] is the largest integer smaller than x so that 2 k ≤ t ≤ 2( k +1) . Then, there exists k ≤ t k ≤ 2 k so that f + t k -f + t k +1 < 2 B f k and f + t k +1 -f ( x t k +1 ) ≤ 0 . From Lemma B.4,

where the last inequality uses ϕ ( · ) < 0 . 4 . From Lemma B.2,

Using (44) in (43), we have

Next, we consider the function P t k at x ∗ . Using the fact that c ( x ∗ ) ≤ 0 , we have by Lemma B.2, µ c t k ( x ∗ ) ≤ B c σ c t k ( x ∗ ) + c ( x ∗ ) ≤ B c σ c t k ( x ∗ ) . (46)

Thus,

From the monotonicity of Φ , we have

Applying (48) to (43), we have

As t →∞ , t k →∞ and k →∞ . Further, if σ f t ( x t +1 ) → 0 , σ f t k ( x t k +1 ) → 0 and r t → 0 .

Proof of Corollary 3.6 is presented next.

Proof. We consider the convergence rate for r t under additional assumptions for the kernel. From Lemma 3.5, for both SE and Matérn kernels, σ f i ( x i +1 ) ≥ C ′ k -min { ν, 1 } d log η ( k ) at most k times for any k ∈ N and i = 1 , . . . , t .

Choose k = [ t/ 3] so that 3 k ≤ t ≤ 3( k +1) . Following the proof of Theorem 3.3, there exists k ≤ t k ≤ 3 k where f + t k -f + t k +1 < 2 B f k , f ( x t k +1 ) ≥ f + t k +1 , and σ f t k ( x t k +1 ) < C ′ k -min { ν, 1 } d log η ( k ) . Similar to (49), we can obtain

The convergence rates follow.

B.1 Proofs for improved simple regret upper bound under frequentist assumptions

Proof of Theorem 3.7 is given below.

Proof. From the proof of Lemma A.5, we know σ f i ( x i +1 ) ≥ √ tγ f t k at most k times for any k ≤ t and i = 1 , . . . , t .

Choose k = [ t/ 3] so that 3 k ≤ t ≤ 3( k + 1) . Following the proof of Theorem 3.3, there exists k ≤ t k ≤ 3 k where f + t k -f + t k +1 < 2 B f k , f ( x t k +1 ) ≥ f + t k +1 , and σ f t k ( x t k +1 ) < 3 √ tγ f t t -3 . Similar to (50), we can obtain

The convergence rates of the simple regret upper bound follow from Lemma A.5.

We provide the sample complexity of Theorem 3.7 below.

Corollary B.5. Under Assumption 3.2, the CEI algorithm achieves a ϵ sample complexity of

Proof. Using Theorem 3.7, to achieve simple regret of most ϵ for SE kernel, set

Solving asymptotically for t gives the sample complexity

Similarly, for Matérn kernels, we have which completes the proof.

C Proofs for simple regret upper bound under Bayesian objective assumption

We state the boundedness of f and c as a Lemma for easy reference.

Lemma C.1. Under Assumption 3.9, there exists M f > 0 such that | f ( x ) | ≤ M f with probability ≥ 1 -δ/ 3 . The constraint function is bounded by its RKHS norm bound | c ( x ) | ≤ B c .

We recall a well-known result on confidence interval of | f ( x ) -µ f t ( x ) | under Assumption 3.9 [27]. Lemma C.2. Given δ ∈ (0 , 1) , let β = 2log(1 /δ ) . For any given x ∈ C and t ∈ N ,

holds with probability ≥ 1 -δ .

Proof. We prove the inequalities for f . Under Assumption 3.9, f ( x ) ∼ N ( µ t ( x ) , σ 2 t ( x )) . By Lemma A.6,

Similarly,

Thus,

Let e -β 2 = δ and (54) is proven.

holds with probability ≥ 1 -δ .

The next lemma address I f t under the Bayesian assumption.

Lemma C.4. Under Assumption 3.9, the probability distribution of I f t satisfies

Proof. Under Assumption 3.9, at a given t , f ( x ) ∼ N ( µ f t ( x ) , σ f t ( x )) . Since I f t ( x ) ≥ 0 for all x , (59) follows immediately if a < 0 . For a ≥ 0 ,

Using basic properties of the standard normal CDF,

Next, we present the relationship between I f t ( x ) and EI f t ( x ) .

Lemma C.5. Given δ ∈ (0 , 1) , let β = max { 1 . 44 , 2 log( c α /δ ) } , where constant c α = 1+2 π 2 π . Under Assumption 3.9, at given x ∈ C and t ∈ N ,

Proof. Given a scalar w > 1 , we consider the probabilities

Consider the first probability in (61). From Lemma A.7, EI t ( x ) ≥ 0 for ∀ x and t . Therefore, σ t ( x ) w + EI t ( x ) > 0 . From Lemma A.7, Lemma C.4, and the monotonicity of Φ , we have

where the last inequality is from Lemma A.6.

For the second probability in (61), we further distinguish between two cases. First, consider -σ f t ( x ) w + EI f t ( x ) < 0 . From Lemma C.4,

Second, consider the premise -σ f t ( x ) w + EI f t ( x ) ≥ 0 . By Lemma C.4, we have

To proceed, we show that f + t -µ f t ( x ) ≥ 0 . Suppose on the contrary, f + t -µ f t ( x ) < 0 and thus z f t ( x ) < 0 . From Lemma A.7,

which contradicts the premise of this case. Thus, we have f + t -µ f t ( x ) ≥ 0 (and z f t ( x ) ≥ 0 ). From the definition (4), since Φ ∈ (0 , 1) ,

In addition, by the premise of this case and Lemma A.7,

Given that w > 1 and ϕ (0) ≥ ϕ ( z f t ( x )) , we have

As z t ( x ) ≥ 0 increases, ϕ ( z t ( x )) > 0 decreases. Thus, we have

Denote c 1 ( w ) = w -ϕ (0) w -ϕ (0)+ ϕ ( w -ϕ (0)) . Applying (69) to (67), we obtain

Applying (70) and (66) to (64), we obtain

Notice that ϕ ( c 1 ( w ) w ) < ϕ ( c 1 ( w )) < ϕ ( c 1 ( w )) w due to w > 1 . By the definition of Φ and mean value theorem,

where c 2 ( w ) = [1 -ϕ ( c 1 ( w ))] 2 + [ c 1 ( w )] 2 . The last inequality in (72) again uses Lemma A.6. Notice that c 2 ( w ) increases with w and for w ≥ 1 . 2 , c 2 ( w ) > 1 . Thus, e -1 2 w 2 > e -1 2 c 2 ( w ) w 2 for w ≥ 1 . 2 , which simplifies (72) to

where c π 1 = 1+ π 2 π . Therefore, by (71) and (73), if w ≥ 1 . 2 ,

Combining (74) with (62) and (63), we have

where c α = 1+2 π 2 π for w ≥ 1 . 2 . The probability in (75) monotonically decreases with w . Let δ = c α e -1 2 w 2 . Then, taking the logarithm of δ leads to log( 1+2 π 2 πδ ) = 1 2 w 2 . Let β = max { w 2 , 1 . 2 2 } , and the proof is complete.

The relationship between I f t ( x ) and EI f t ( x ) under the GP prior assumption is given in the following lemma.

Lemma C.6. Given δ ∈ (0 , 1) , let β = 2log(2 c α /δ ) , where c α = 1+2 π 2 π . At given x ∈ C and t ∈ N ,

holds with probability ≥ 1 -δ

Proof. From Lemma C.2, with probability ≥ 1 -δ , (54) stands. If f + t -f ( x ) ≤ 0 , then I t ( x ) = 0 . Since EI t ( x ) ≥ 0 , (76) is trivial. If f + t -f ( x ) > 0 , by Lemma C.2,

with probability greater than ≥ 1 -δ/ 2 . From the monotonicity of τ , we have

and therefore,

with probability greater than 1 -δ/ 2 . Using δ/ 2 in Lemma C.5,

with probability ≥ 1 -δ/ 2 . Applying (80) to (79) with union bound leads to

with probability greater than 1 -δ .

We can now prove Theorem 3.10.

with probability ≥ 1 -δ/ 3 . Next, consider f + i -µ f i ( x i +1 ) . Recall that β t = 2log(3 π t /δ ) . From Lemma C.3 Lemma A.4 and Lemma A.3, we have

with probability ≥ 1 -2 δ/ 3 via union bound. Given that max { f + i -µ f i ( x i +1 ) , 0 } ≥ 0 , max { f + i -µ f i ( x i +1 ) , 0 } ≥ 2 M f k + β 1 / 2 t k √ C γ tγ f t at most k times for any k ∈ N with probability ≥ 1 -2 δ/ 3 . Choose k = [ t/ 2] , where [ x ] is the largest integer smaller than x so that 2 k ≤ t ≤ 2( k + 1) . Then, choose the first index t k where k ≤ t k ≤ 2 k so that max { f + t k -µ f t k ( x t k +1 ) , 0 } < 2 M f k + β 1 / 2 t k √ C γ tγ f t . As discussed above, such a t k exists with probability ≥ 1 -2 δ/ 3 . We note that maximum information gain and its upper bound does not depend on the optimization path. Importantly, the choice of t k does not depend on random information after iteration t k .

From Lemma C.6 and β = 2log(6 c α /δ ) , with probability ≥ 1 -δ/ 3 ,

where the last inequality uses ϕ ( · ) < 0 . 4 . From the choice of t k , (84) leads to

with probability ≥ 1 -δ . Next, we consider the probability function P t k at x ∗ and x t k +1 . Using the fact that c ( x ∗ ) ≤ 0 , we have by Lemma C.2 at x ∗ and t k ,

Thus, we can write

From the monotonicity of Φ , we have

Using (88), the P t k functions have

Applying (89) to (85), we have

with probability ≥ 1 -δ .

The proof of Theorem 3.11 is next.

Proof. From Lemma A.5, σ f i ( x i +1 ) ≥ √ γ f t t k , where i = 0 , . . . , t -1 , at most k times for any k ∈ N and k ≤ t . Choose k = [ t/ 3] which leads to 3 k ≤ t ≤ 3( k +1) . Let t k be the first index in [ k, 3 k ] so that σ f t k ( x t k +1 ) ≤ √ tγ f t k and max { f + t k -µ f t k ( x t k +1 ) , 0 } < 2 M f k + β 1 / 2 t k √ C γ tγ f t , which exists with probablity ≥ 1 -2 δ/ 3 . Notice that β 1 / 2 t = O (log 1 / 2 ( t )) . Following the proof for (90), we have

with probability ≥ 1 -δ . Using Lemma A.2, the proof is complete.

D Test problems

The mathematical formulations of the testing problems in Section 4.2 are given in this section. The objective, constraint functions and the optimal f of Problem 1 is given below.

The objective, constraint functions and the optimal f of Problem 2 is given below.

The objective, constraint functions and the optimal f of Problem 3 is given below.

The objective, constraint functions and the optimal f of Problem 4 is given below.

b

1.00

0.75

5

5

0.50

4

0.25

E

X

0.00

2

0.25

2

0.50

1

0.75

0

0

1

2

3

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5

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0

1

2

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1.00

X1

X1

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2.00

d.0

0.50

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0.25

0.8

1.50

1.0

0.00

0.6

1.25

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0.25

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1.00

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X

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1.00

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0.00

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X1

X1

X1

10/

107

14

106

106

12

105

105

10

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8

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103

6

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101

100

100

2

0

2

4

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101

-2

0

2

4

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101

X1

X1

The objective, constraint functions and the optimal f of Problem 5 is given below.

The contour plots of Problem 1, 2, and 5 are given in Figure 3, 4, and 5.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers the constrained Bayesian optimization framework where one desires to minimize the function $f(x)$ is subject to the constraint $c(x) \\leq 0$, where both are black box functions. The expected improvement algorithm, based on a Gaussian process model, is a celebrated algorithm in the unconstrained setting. The constrained expected improvement (CEI) algorithm extends this by including a term in the expected improvement that accounts for the feasibility $c(x) \\leq 0$. This paper presents, to my knowledge, first convergence guarantees or regret bounds for the CEI algorithm. Pacifically, in both a frequency setting where are the functions are assumed to come from an RKHS, and also an amazing setting where they are assumes sampled from the Gaussen process, the authors prove appropriate convergence rates of the regret, which extends the results in the unconstrained case. The authors also specialize these results to the commonly used squared exponential and Matérn kernels, and present some numerics to show convergence of the regret.\n\nSTRENGTHS AND WEAKNESSES:\nAssessment:\nIn my view this is a moderately high quality submission in its current state. The results are technically sound. However, there are perhaps a few gaps in the results that make the paper a bit less mature than ideal (please see my detailed comments/questions below) in its current form. These might be able to be addressed in the rebuttal period. Although I did not read the appendix in detail, the claims appear well supported by extensive proofs. The paper is also written and organized OK; it clearly articulates the main ideas and results but the grammar and style could be improved somewhat throughout.\n\nStrengths:\n1. The results of this paper are significant, interesting, and have potential for impact. I like the topic, as I think Bayesian optimization is becoming more and more popular and important, both in theory and practice. Convergence rates for important Bayesian optimization algorithms, like CEI, is therefore fundamental and important.\n2. It is very good that this paper provides precise convergence rates for the regret, that all of the constants are explicitly spelled out, and that the arguments are outlined. I quite appreciate these level of transparency in the results.\n3. I am not the most well versed in the closely related recent literature. However, to the best of my knowledge, there are no overlapping works in the literature and this work appears to present original new insights, which I think will be valuable.\n\nWeaknesses:\nTo me the weaknesses concern two aspects of the bounds. These weaknesses are the main reasons for my current values of the Quality and Overall Score below, both of which could possibly be improved in the rebuttal. I list them below in my view on order of importance.\n\n1. I understand your rationale for defining regret as you do, but I wonder if by doing so you are being ``overly strict\" in the requirements of feasibility, leading to perhaps overly conservative bounds. What I mean is that, while the CEI expected improvement criteria is somewhat symmetric in how it multiples the constraint and objective criteria, your analysis is not. Instead, you require to only consider those points $x_t$ that actually are feasible with $c(x_t)$. As a result, many points may be thrown away. Furthermore, I think this manifests in the bound in the term $1/\\Phi(-B_c)$, which only gets worse as a product of multiple terms with multiple constraints (Remark 3.15). Is it possible to have a more \"parameterized\" version of a bound that might allow small violations in constraints $c(x_t) \\leq \\epsilon$ at the benefit of rejecting less points? I can imagine scenarios where decision-makers might be OK with this. And you can also recover your original bound by setting $\\epsilon = 0$. I am very curious to know your thoughts and if this approach is feasible, as I think it would improve the bounds a lot.\n2. To my understanding, your bounds implicitly involve a well-specified assumption where the kernel in Assumption 3.3/3.10 is the same as the kernel in the GP model in the algorithm; is this correct? If so, at the very least I think this must be acknowledged up front as a limitation of the current analysis. It is also of course interesting to see if such assumptions can be relaxed, but that might be beyond the scope of this current submission.\n3. I furthermore have several other questions below, which I don't inherently view as weaknesses per se, but I would appreciate to see addressed.\n\nQUESTIONS:\n1. Please see my question/comment on Weakness 1 above.\n2. Please see my question/comment on Weakness 2 above.\n3. Starting at the beginning of Section 2.1, there are a few instances of subscript $k_f$. Is this a typo? Should it be $k^f$?\n4. Why do you refer to the regret as \"simple\"?\n5. Theorem 3.7 seems more like a Corollary of Theorem 3.4 or is my understanding wrong? I guess Thm 3.4 is a generic result for all kernels, Thm 3.7 is its specialization to SE and Matérn kernels, and Thm 3.8 is a strengthening that uses the special structure of SE and Matérn kernels. \n6. Line 217:  Should Theorem 3.8 be Theorem 3.4?\n7. In Section 3.2, is the probability $\\delta$ over both sampling of $f$ and $c$? Is this probability affected or does it somehow affect the results more when you have more than 1 constraint?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper establishes the first theoretical convergence rates for the Constrained Expected Improvement (CEI) algorithm, a popular and widely used method for constrained Bayesian optimization (CBO), under both the frequentist and Bayesian settings. Despite its empirical success, the convergence rate of CEI had not been previously established. \n\nThe authors highlight that these results are not direct extensions from the unconstrained Expected Improvement (EI) analysis due to the complexities introduced by the constraints and the multiplicative nature of the CEI acquisition function. The paper also provides numerical experiments on both synthetic and benchmark problems to empirically validate the theoretical convergence results.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. This work is the first to establish the theoretical convergence rate for the Constrained Expected Improvement (CEI) algorithm, a widely used method for which such an analysis was previously missing. \n2. The paper provides convergence rates under two different standard settings: a frequentist setting, where the functions are assumed to be in a Reproducing Kernel Hilbert Space (RKHS), and a Bayesian setting, where the functions are assumed to be sampled from Gaussian Processes (GPs).\n3. The authors explicitly identify and address the technical challenges unique to analyzing CEI, such as the multiplicative structure of the acquisition function and the need to balance exploration for feasibility with exploitation for optimality.\n\nWeaknesses:\n1. The analysis is conducted for a noise-free setting, meaning the function evaluations are deterministic. This may not hold in many real-world black-box optimization problems where observations are noisy.\n2. The CEI algorithm and the corresponding analysis assume that the objective and constraint functions are conditionally independent. For the extension to multiple constraints, it is also assumed that the constraints are conditionally independent of each other.\n\nQUESTIONS:\n1.  The analysis is performed in a noise-free setting. What new technical challenges would arise when extending this analysis to account for noisy observations of the objective and constraint functions, and how would you anticipate the convergence rates would be affected?\n2. The CEI algorithm and the analysis assume that the objective and constraint functions are conditionally independent. How robust do you expect these convergence rates to be to violations of this assumption, which may occur in many practical engineering problems?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
This paper derives the convergence rate of the constrained expected improvement (CEI) algorithm, a widely used method in constrained Bayesian optimization. 
The analysis shows simple regret bounds for CEI on objective and constraint functions that lie in an RKHS and on those drawn from Gaussian processes.
The authors conducted experiments to confirm the results of the analysis.

STRENGTHS AND WEAKNESSES:
Strength:
This paper is the first work to derive the convergence rate of the CEI algorithm. In addition, the key points of the proof are summarized as proof sketches. These points increase the originality and clarity of this paper.

Weakness:
The numerical evaluation is not sufficient to check whether the derived convergence rate is reasonable.

QUESTIONS:
The analysis seems to assume that the number of constraint functions is one. However, I think there are several real-world problems that contain more than one constraint. Is it tedious work to extend the analysis result in such a situation?

I could not confirm how close the derived result was to the actual behavior of CIE from the result of the numerical experiment. Could you plot the derived convergence rate in Figures 1.a and 1.b (if possible)? In addition, please conduct experiments with different dimensions to confirm how it changes the convergence rate.

In Figures 2 and 3, I think the "infeasible" region is marked.

LIMITATIONS:
The analysis with more than 1 constraint function is desired in future work.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper provides a theoretical convergence analysis of the Constrained Expected Improvement (CEI) algorithm in Constrained Bayesian Optimization. It establishes simple regret bounds under both frequentist and Bayesian settings, showing improved convergence rates for squared exponential and Matérn kernels. Using information-theoretic tools, the analysis overcomes challenges from CEI’s multiplicative structure. Results are supported by experiments, offering theoretical justification for CEI’s practical success.\n\nSTRENGTHS AND WEAKNESSES:\nstrengths:\n1. This is the first work to derive simple regret upper bounds for the CEI algorithm under both frequentist and Bayesian settings.\n2. Under frequentist assumptions (RKHS model), they derive improved rates for SE kernel $\\mathcal{O}(t^{-1/2} \\log^{(d+1)/2}(t))$ and for Matérn kernel  $\\mathcal{O}(t^{-\\nu/(2\\nu+d)} \\log^{\\nu/(2\\nu+d)}(t))$. These rates improve over the direct constrained extension of Bull (2011), which achieved $\\mathcal{O}(t^{-1/d})$.\n3. The authors show that, with high probability, the same convergence rates hold when f and c are samples from GPs. This complements prior work like Srinivas et al. (2009) for unconstrained GP-UCB, extending it to CEI.\n4. The paper leverages maximum information gain bounds to obtain tighter regret bounds, following approaches in recent literature on UCB (e.g., Lyu et al., 2019; Vakili et al., 2021).\n\nweaknesses:\n1. The analysis requires the existence of an initial feasible sample, which is not always easy to guarantee in practice. Without it, the CEI acquisition function is undefined because the improvement is computed over the best feasible value.\n2. The convergence analysis is conducted only under a noise-free setting, where evaluations of both the objective and constraint functions are assumed to be exact.\n3. The analysis assumes a single constraint (or extends to multiple constraints under independence assumptions). However, real-world CBO often involves correlated or conflicting constraints, which are not addressed.\n4. There’s no non-asymptotic regret bound (e.g., PAC-style guarantees) to quantify performance in small-budget settings.\n\nQUESTIONS:\n1. How does the analysis handle the non-convexity and potential multimodality of the CEI acquisition function in practice? The regret bounds assume exact maximization of CEI at each iteration, but the acquisition function is non-convex and often only approximately optimized in practice. How sensitive are the theoretical guarantees to optimization error in the acquisition step?\n\n2.  How valid is the assumption of conditional independence between the objective and constraint GPs in real applications? The CEI analysis assumes conditional independence, yet in practice, objective and constraint functions may be correlated. How would the theoretical results change if this assumption were relaxed or violated?\n\n3.  Can the authors clarify how the feasibility probability is controlled in the regret analysis? The bound includes inverse terms like $1 / \\Phi(-B_c)$, which can be large or unstable if the feasibility boundary is close to zero. How robust is the bound when $c(x^*) \\approx 0$, i.e., the optimal point lies near the constraint boundary?\n\n4. Is there a path to extending the current analysis to noisy or approximate observation settings? Given that real-world evaluations are often noisy, can the authors discuss how their framework could be generalized to include noise, possibly following techniques used in noisy GP-UCB or Thompson Sampling literature?\n\n5. How does the multiplicative acquisition function structure affect the exploration-exploitation trade-off compared to additive or penalty-based methods? The CEI acquisition function is the product of EI and feasibility probability. How does this structure influence the theoretical exploration behavior, especially in high-dimensional or narrow-feasibility domains?\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-20"}