Optimistic Bayesian Optimization with Unknown Constraints
Authors: Quoc Phong Nguyen
Wan Theng Ruth Chew
Le Song
Bryan Kian Hsiang Low
Patrick Jaillet
Venue: ICLR 2024 poster (2024)
Keywords: Bayesian optimization
black box constraint
decoupled query
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: We introduce a constrained Bayesian optimization method with a theoretical guarantee for decoupled queries.

Abstract: Though some research efforts have been dedicated to constrained Bayesian optimization (BO), there remains a notable absence of a principled approach with a theoretical performance guarantee in the decoupled setting. Such a setting involves independent evaluations of the objective function and constraints at different inputs, and is hence a relaxation of the commonly-studied coupled setting where functions must be evaluated together. As a result, the decoupled setting requires an adaptive selection between evaluating either the objective function or a constraint, in addition to selecting an input (in the coupled setting). This paper presents a novel constrained BO algorithm with a provable performance guarantee that can address the above relaxed setting. Specifically, it considers the fundamental trade-off between exploration and exploitation in constrained BO, and, interestingly, affords a noteworthy connection to active learning. The performance of our proposed algorithms is also empirically evaluated using several synthetic and real-world optimization problems.

Published as a conference paper at ICLR 2024
OPTIMISTIC BAYESIAN OPTIMIZATION WITH UNKNOWN CONSTRAINTS
Quoe Phong Nguyen', Wan Theng Ruth Chew?, Le Song", Bryan Kian Hsiang Low² & Patrick Jaillet'
'LIDS and EECS, Massachusetts Institute of Technology, USA
SScooo of Computing, National University of Singapore, Singapore aphongmpe gmail i1. com, chew. ruth@u.. nus edu, 1e. song@u nus edu, lowkhê comp. nus. edu. sg, jai wwl.llemtttteeeee
ABSTRACT
Though some research efforts have been dedicated to constrained Bayesian opti- mization (BO), there remains a notable absence of a principled approach with a theoretical performance guarantee in the decoupled setting. Such setting involves independent evaluations of the objective function and constraints different inputs, and is hence a relaxation of the Nommonly-studier coupled setting where functions must evaluated together. As a result, the decoupled setting requires an adaptive selection between evaluating either the objective function or constraint, in ad- dition to selecting an input (in the coupled setting). This paper presents a novel constrained BO algorithm with provable performance guarantee that can address the above relaxed setting. Specifically, it considers the fundamental trade-off be- tween exploration and exploitation in constrained BO, and, interestingly, affords a noteworthy connection to active learning. The performance of our proposed algorithms is also empirically evaluated using several synthetic and real-world optimization problems.
INTRODUCTION
In real-world applications, we often encounter expensive-to-evaluate black-box objective functions that can only be assessed through simulations experimentation. For example, problems involve machine learning model (Wistuba al. [2018] Perrone et al..
optimizing the hyperparameters 2020), or choosing experiments in the fields of material and drug design (Schweidtmann et al. 201 8). To address these problems, Bayesian optimization (BO) has gained prominence as a widely adopted approach (Brochu al., 2010 Frazier 2018 Garnett 2022). It is iterative model-bases approach that employs a probabilistic model, c.g., a Gaussian process (GP), to estimate the unknown objective function. At each iteration, BO searches for the optimal solution by strategically selecting input query to evaluate the objective function, maintaining a balance between exploiting promising areas and exploring poorly-est poorly-estimated regions. BO encompasses many well-established techniques such as the probability of improvement (Kushner 1964), expected improvement (EI) (Mockus et al. 1978). Gaussian process upper confidence bound (GP-UCB) (Srinivas al. 2010), the anowledge-gradien based approach (Frazier et al., 2008), and information-theored approaches: entropy search (Hennig and Schuler 2012), predictive entropy search (PES) (eemández--obarl et al., 2014). and max-value entropy search (MES) (Wang and Jegelka 2017).
Beyond the black-box objective function, recent advancements in BO have focused on addressing the prevalent presence of black-box constraints. For example, there often exist prediction time constraints and class-wise performance constraints when tuning machine learning models (Hernández)
Lobato et al. 2016 Takeno et al. 2022). They are just costly to evaluate the objective function, Constrained BO has led to many BO extensions such as EIC (an El-based method) (Gardner et al.] 2014), knowledge gradient- based method (Chen et al. 2021), CMES-IBO (an IESS-bsed method) (Takeno et al., 2022). augmented Lagrangian approaches (Gramacy et al. 2016
Picheny
al. 2016), and upper trust bound (UTB) (a GP-UCBBbaase method) (Priem al. 2020

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However, existing approaches primarily concentrate on empirical performance and lack theoretical analysis to ensure consistent performance. Recently, theoretical studies for constrained BO have gained attention via the works of Lu and Paulson (2022) and Xu et al. (2023). While Lu and
Paulson 2022) perform the analysis by proposing a penalty- baseed regret, Xu et al. (2023) analyse the
cumulative regret due the objective function and the constraint violation separately.
Besides the above largely unexplored theoretical analysis, existing works often overlook the potential for evaluating the objective function and constraints independently different inputs, known as decoupled queries. Specifically, the above works, including the theoretical studies by Lu and Paulson (2022) and Xu al. (2023), require simultaneous evaluations of the objective function and constraints at an input query, known as coupled queries. The distinction between coupled and decoupled queries was first mentioned in the work of Gelbartt et al. (2014). They discuss a chicken-and-egg pathology which prevents extending myopic BO approach such as EIC to the decoupled setting. Later, Hernández-Lood Lobato et al. (2016) introduce a principled approach based on PES, namely PESC, to address constrained BO with decoupled queries. While PESC is functionally equivalent to lookhead approach, it leverages the symmetric property of mutual information to avoid performing the actual lookahead computation. Regrettably, its implementation is fairly complex, making less accessible to practitioners. Besides, lacks a theoretical performance guarantee. While ADMMBO (Ariafar et al., 2019) deals with decoupled queries, it deterministically alternates the evaluations of the objective function and constraints, without exploiting the benefits of an adaptive selection approach. Hence, the question of devising an approach that offers a theoretical performance guarantee, is adaptable to decoupled queries, and can be readily implemented by practitioners remains unanswered.
In this paper, we address this question by proposing simple algorithm with a theoretical performance guarantee, especially in the decoupled setting. Notably, our algorithm is myopic without expensive lookaheads. In Sec. 2] we introduce a regret that does not require any penalty parameter, unlike that in the work of Lu and Paulson (2022). Then, we discuss the
-etrration-xlliiiattinn trade-off in constrained BO in Sec. 3 Specifically, we introduce a new form of exploration, namely horizontal exploration, resulted from the presence of -lack-box constraints. It is to differentiate from the vertical exploration in unconstrained BO (Srinivas et al. 2010). Then, we design a unified approach that handles coupled and decoupled queries from this perspective. More importantly, our algorithms shown to be no-regret in Theorem 3.3 and App. B While the viewpoint of balancing exploration and exploitation is inherently grounded in BO, especially in the bandit setting like GP-UCB (Srinivas let al., 2010), Sec. 3.3 shows that the choice of the function to query can also be framed within the well-known uncertainty sampling paradigm in the active learning literature (Settles) 2009). In Sec. 3.4 we propose an estimator for approximating the optimal solution at each BO iteration with a theoretical performance guarantee. To empirically demonstrate the performance of our algorithms. we presents several experiments using both synthetic and real-world optimization problems in Sec.
2 CONSTRAINED BAYESIAN OPTIMIZATION AND REGRET DEFINITION
Let f be a real-valued black-box objective function and C be a finite set of real-valued black-box constraints. Let the compact subset X' Rdd be the input domain and d E N4 be the input dimension. We consider the following constrained optimization problem
max f(x) where the feasible region S é {x E X c(x) 2 A Vc E C} and X E Vc (I) XES
An equality constraint can be transformed into two inequality constraints. Let us denote the set of the objective function and constraints as F á [f) UC and denote a function in F as h.
To identify the optimal solution x & arg f(x), we employ BO which is an algorithm operating in a sequential manner. In the decoupled query setting, at iteration f, we gather a noisy observation of the function query he E F evaluated an input query x, E X'
(x.) hx(x:) + eh, (X:), en, (x.) ~ (.
(2)
For instance, at iteration t, the algorithm may decide to query the objective function (i.e., hr f) while it may query a constraint (i.e., hy = c for some c E C) at a different iteration t' * t. On the contrary, in the coupled query setting, at iteration functions h E F evaluated at the same input query x, E X. The coupled setting is less challenging since it does not require specifying the function query.
query for observations {y) (xx) heer of all
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Let Di,t denote the set of observed inputs of h until iteration t (including x,) and D, - UherDi,e Then, Do consists of initial observed inputs. Let yn (Dr) = yn (D,,)) & {yr (x),,, denote the set of observations from h at 21 BO effectively utilizes the acquired observations (Dn,t)))))) in the previous - iterations to formulate a strategy for determining the next input query x,, the next function query hus and an estimator, denoted x for approximating the optimal solution x". We will discuss the probabilistic model of E F given the acquired observations and a performance metric of BO, called the regret, the rest of this section. Then, we will elaborate on our strategy of choosing Xs, he, and x in the following sections.
Gaussian process. For each finite subset of (h(x))xE> follows multivariate Gaussian distribution. The GP is fully specified by its prior mean (x)) and its kernel (x, x') _ cov(h(x), h(x')). We employ the commonly-used squared exponential (SF) kernel. At iteration 4, given the observations yı (Dt-1) the previous
J, model h with Gaussian process (GP). It implies that every
iterations, the posterior distribution of h(x) follows Gaussian distribution with closed-form posterior mean and variance, denoted as Ph,--1 (x) and oi (x), respectively!
Regrets. To analyse the theoretical performance of constrained BO, we propose the following instantaneous regret including that of the objective function ry and the constraints Te.
r(x,) A max (x,)
where (x(x)) max(0, f(x*) f(x))
Vc C, re(x)) max(0, de c(xx))
Then, our goal is design BO algorithms that achieve sublinear cumulative regret
T r(x;) = 0.
lim T-0 T T
RT
lim
Σ
as it implies that x r(x,)= S Et= r(x)) approaches 0 as T approaches
Remark 2.1 (Instantaneous regret sum). Alternatively, the instantaneous regret can be defined as a sum of instantancous regrets the objective function and the constraints
sxx,)
A Σ rx())))
hEF
Let us consider the case of a single constraint = (co). For X + (x, =
and Tou (Xx) = 0 while Teg (x()= 1. then r(x;) r(x) while s(x:) = 1 s(x)) As result, s(x) is more effective than r(x) at measuring the suboptimality of solution. Nevertheless, a sublinear cumulative regret w.r.t. implies sublinear cumulative regret w.r.t. and vice versa since r(x,) < s(x)) < [F] r(x.). We revisit s sx,) in Sec. 3.4whhen discussing the estimator x
3 OPTIMISTIC BAYESIAN OPTIMIZATION WITH UNKNOWN CONSTRAINTS
To simplify the derivation, we consider the case of finite input domain X' and utilize the following Lemma 5.1 of Srinivas et al. (2010) with modification by applying the union bound for all functions
in F (Lu and Paulson 2022).
Lemma 3.1. Pick E [0, and set Br 2
/65). Then,
[h(x) - Ph,e- 1xx) s // 1x,)-1(x) Vx X W W
(6)
holds with probability 2 5. It suggests
wx.t-1(x) 1 (x) + 3. (x))-((x) and (xx-1(() 2 1xx) 3/ ( (7)
as the upper and lower confidence bounds of h(x) for all h E F, X. and > 1, respectively.
Please refer to Rasmussen and Williams 2006 for the closed-form expressions.
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3.1 INPUT QUERY
In the classic unconstrained GP-UCB work of Sriiivas et al. (2010) (i.e., C = 0). it balances between exploiting the current posterior belief by selecting those with high posterior mean ( and exploring inputs with highly uncertain evaluations of by selecting those with high posterior standard deviations (x). Specifically, GP-UCB (Srinivas et al. 2010) selects an input query
that maximizes an optimistic objective function evaluation (x
X
GP-UCB
arg max f (x) = arg max (11,5-1()) +
3/ (PPos.--(ii)
(8)
XEX
XEX
vertical exploration bonus
Let us call o the vertical exploration bonus as it encourages us to be optimistic about the unknown objective function evaluations It is shown as the blue region in Fig. I
In the presence of unknown constraints C, apart from the above optimistic objective function evalu- ation 2f,t-1 (x), we additionally consider an optimistic feasible region, denoted as Or, which may contain some infeasible inputs. It represents the exploration of the feasible region which we refer to as horizontal exploration as opposed to the vertical exploration in the optimistic objective function evaluation. Ideally, Or is superset of the feasible region S to ensure that the optimal solution x* remains in O, We consider the following optimistic feasible region
O, ف (x E 2 (x)+-(()) 2 de Y C}
which aligns with our goal that O C S with high probability since (e,--1xx) 2 c(x) with high probability (Lemma (.1).. D). By considering an optimistic feasible region, we avoid the subtle issue that the probability mass of the feasible region is 0 given the GP posterior beliefs of the constraints. This issue affects several existing approaches such as EIC, PESC, and CMES-IBO as discussed in App. A
Combining the vertical and horizontal explorations, we select the input query X that maximizes the optimistic objective function 11 restricted to the optimistic feasible region O,
X arg max uff.t1 (x).
(10)
X
We derive an upper confidence bound of (x,) similar to that GP-UCB, and an additional upper confidence bound of re(x;) which holds with probability 2 1 5 (see App. B
(x,) S 2/3//2 (x.) and re(x;) S 2////2 o.,t-1xx:)
(11)
If the queries are coupled, we can achieve a no-regret BO algorithm by simply obtaining observa-
tions (y) (xx) ] at iteration as elaborated in App. B This algorithm for the coupled setting is called UCB-C C to distinguish it from another algorithm, namely UCB-D, for the decoupled setting described in the next section.
While a condition resembling equation (9) is utilized in the work of Priem et al. (2020), they do not offer any theoretical analysis. Furthermore, they maximize a variant of the El criterion as opposed to the upper confidence bound ff in our approach. Besides, the optimization problem in equation (TO) can also be framed as the unconstrained penalized acquisition function in the work of Lu and Paulson 2022). It suffers from an extra penalty parameter requiring automated fine-tur fine-tuning, which is left as future work by Lu and Paulson| (2022). The algorithm most closely related to our UCB-C is the recent CONFIG algorithm (Xu et al.] (023), which is accompanied by theoretical bounds on the cumulative regret from the objective function and the constraints. However, decoupled queries remain unexplored in these studies. In the next section, we address this scenario by adaptively selecting the function query h while maintaining the theoretical performance guarantee.
In order for the choice of x, in equation (TO) to exist, O2 must be non-empty. This holds with probability 2 1 - ö according to Lemma 3. if the optimization problem is feasible. To avoid the subtle case of O, 9), we recommend setting the GP prior mean of constraint e to Ae in practice.
3.2 FUNCTION QUERY
To begin with, we provide insights into a rational function query selection strategy in the decoupled setting. Then, we will rigorously translate them into a concrete strategy.
'The term "vertical" refers to the output of f often plotted as the (vertical) y-axis, as opposed to the term "horizontal" which refers the input of f often plotted the (horizontal) J-axis, e.g., Fig. Tp.
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f(m)
w/a (()))())
c(x)
vertical exploration bonus
PLa (reduced by evaluating
Morr-1 (1))
Fa (xx
(x)))(()))
Ag MM
S1
24
St
74g
horizontal exploration (reduced evaluating c)
borizontal exploration
(a)
(b)
Figure 1: (a) Plot of the vertical and horizontal exploration bonuses in the space of the objective
function; and (b) Plot of the horizontal exploration bonus in the space of the constraint function.
Remark 3.2 (On rational function query selection strategy). In the decoupled setting, it can be inefficient by querying all functions in F at every iteration since the possibility of querying/evaluating functions in FF independently constraint at an input query that is likely to be feasible, which is illustrated in Fig- 2 in Sec. On the other hand, if there is significant risk of a constraint violation at x,, is likely that querying the "most-violated"" constraint eliminates x, from the feasible region (hence, from being x*), in which case querying the objective function at X is redundant. is illustrated in Figs. 2b-c in Sec.
left unexploited. On one hand,
unnecessary to evaluate any
From the regret analysis perspective, the decoupled setting poses a challenge in bounding the instantancous regret r(x)) at each iteration. It is because rxxe) depends all instantancous re- grets {rA (x)) while we do not evaluate all functions in F each iteration, unlike in the coupled setting. Therefore, it is necessary establish connection across (ry (xx((x)) (x)))
To motivate our choice of he, we partition the optimistic feasible region O₂ into a 1/2 1/-relaxed feasible confidence region St and an uncharted region Ut
O O U
U
(12)
horizontal exploration bonus
S, A (x E (,,11 A U4 Ve C) and u 4=
(13)
where is 2 a constraint-relaxation parameter (S, and Ur are illustrated in Fig. [p). Recall
that (x I c(x) holds with high probability (Lemma 3.1). so does S; C Sv, where S., {x X c(x) ) 2 > i Vc C) is ue-relaxation of S. Therefore, S, consists of feasible inputs w.r.t So, with high probability. Furthermore, any input x in the uncharted region Ud satisfies
Ee E C, (x) 2 de Ae 44 (x).
(14)
Hence, the uncharted region Ue consists inputs whose feasibilities w.r.t. S are unknown (be- cause (x) 2 d -x,)-1(x)) and whose risks of a constraint violation are sufficiently high (because A - (xx 2 142 for some e E C) as illustrated in Fig. Dp.
We interpret U4 as the horizontal exploration bonus. Its role in the optimistic feasible region O is analogous to the vertical exploration bonus 3! 1/22 j (x) in the optimistic objective function
evaluation u "f.--(()) (illustrated in Fig. T and equation (8) vs. equation (22). ). Let translate the intuitive cases in Remark 3.2 into the following concrete conditions.
Querying constraint when X, E U4. When x, E U₁ the risk of constraint violation at x; is sufficiently high (A, - -x,,-((x,)) (x,) 2 illustrated in Fig. [[b). From Remark 3.2 we query the "most-violated" constraint defined as the one with the highest risk of a constraint violation, i.e., arg max > -lc,--((;;). is noted that as decreases, the size of the uncharted region U, is non-decreasing (see Fig. (p)). Hence, we use v, to control the size U4, i.c., controlling the horizontal exploration bonus. However, assigning small value to is risky because we may excessively query a constraint, i.e., excessive horizontal exploration. Let us consider an extreme scenario: 144 = and there exists iteration such that constraint C is active xt, i.e., c(x;) Xe- In this case, X₂ - 4e,t-1 (x,) c(x,) - 1 2 0, so the algorithm will keep querying a constraint without querying the objective function.
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Algorithm UCB-D
Require: X, Do
1: Update GP posterior beliefs: {(HA, anoo)nEr
2: for + 1;t do
3: X; f arg max (xx1-1(x))
4:
a - arg max de- (x()
// most-vvillate constraint
5:
if Ae, - (xx) > 23,-' (x() then // El,
6:
he + ct
// query most-violated constraint
7:
else
// x E
8:
14 +
// query objective function
9:
end if
10:
Yh. (Dhu,r) ()) f yn, (Dhne-1) (yn, (x,)
Update GP posterior belief: phent Thee
12:
end for
Querying the objective function when E St. When St, is likely that is feasible solu- tion (relaxed by v4), illustrated Fig. [p. From Remark3.22 we prefer querying the objective function. However, assigning a large value to V4 is risky because may excessively query the objective func- tion, i.c. insufficient horizontal exploration. In particular, if 14 > max.ec maXxex² A (x). then U, = 0 and O = St. It means x, St and the algorithm queries the objective function.
To resolve the dilemma of setting 24 too large (excessively querying the objective function) or too small (excessively querying constraint), let vi to be "self-tuned" by tying its value with the vertical exploration bonus, i.e., setting Vt = 2,3/22 (x.)) The broad intuition that if 14 is too large, the algorithm repeats querying the objective function which reduces the vertical exploration
bonus 3//22 ff,-1(xx,) It, in turn, reduces V4 as 1 = 2/3//22 a f.t-1 fx:) On the contrary, it is too small only if 2,3 - is too small. From equation 11 it implies that r (x, is small, so it is justifiable to refrain from querying the objective function.
The resulting algorithm, called UCB-D,
for the decoupled setting is described in Algorithm [ In App.
prove the following Theorem 3.3
on the cumulative regret of Algorithm]] I
Theorem 3.3. The cumulative regret Rr of Algorithm is bounded by
Pr(R RT S FIT max WEF
CI 1
(15)
where C₁ & 8/log(1+07, and T adopted from the work of Srinivas et al. (2010) is the maximum information gain from observing T noisy evaluations of h.
from
Srinivas et al. (2010) show that Th is sublinear for some commonly used kernels including SE and Matérn kernels. Hence, Theorem 3.3 suggests that Algorithm [ resulls in a cumulative regret that
grows sublinearly when employing GPs with these kernels.
3.3 FUNCTION QUERY FROM ACTIVE LEARNING PERSPECTIVE
The choice the input query X, from GP-UCB exhibits an intriguing link to the concept of informa- tion gain found in the active learning literature, where one seeks the "most informative data point" or its approximate equivalent, the "most uncertain data point" as discussed in the work of Srinivas et al. (2010). Interestingly, one can view the choice of the function query h2 as an uncertainty sampling strategy as well (i.e., seeking the "most uncertain data point") (Settles) 2009).
Let us denote the upper confidence bounds of the instantaneous regrets w.r.t the objective function and constraint e in equation (IT) as
Ury, (x,) A 2/3 2/8,//011--1xx: (x,) 2 rx,)))
(16)
Ur (x:) A max(0, A - le,t-1(x,)) 2 re(x)).
(17)
While the nature of the uncertainty in active learning differs from that the instantancous regret in problem, interested in minimizing their values in both scenarios. Hence, employ the

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uncertainty sampling paradigm choose the function query by setting
h4 arg max (rx,)-1(Xx)) NEF
(18)
This resulting strategy, interestingly. coincides with the choice of h2 in Algorithm It is noted that the uncertainty sampling approach may not be effective when there exists uncertanty that cannot be reduced with observations, referred to as aleatoric uncertainty in the work Hüllermiier and Waegeman (202T). Fortunately, our instantaneous regrets are analogous to the epistemic uncertainty that can be reduced with observations. Specifically, the more observations we obtain, the better the objective function and constraints (hence, x*) are estimated. Thus, a smaller regret can be achieved. Remark 3.4. If the evaluation of h E F incurs cost ((h) > O, then we would like to choose the function query by maximizing cost-aware upper confidence bound of the instantaneous regret. Specifically, the function query is chosen as f2 = arg e (xxx,--(xx)//((0 the upper confidence bound of the instantaneous regret per unit cost (Swersky al. 2013
is interpreted as
3.4 ESTIMATOR OF THE OPTIMAL SOLUTION
Remark 2.1 states that (x) is more effective than r(x) in assessing the suboptimality of a solu- tion. Hence, would like use s(x) to propose an estimator x for approximating the optimal solution x". From equation [T), we obtain an upper confidence bound of sxx) at iteration !
sxx
(x)
S
Ura.t-1(x))
(19)
REF
WEF
where t is defined in equation 16 the lowest upper confidence bound of ((x)
and equation T7p.
We would like to select the input with as the estimator by considering all previous t iterations
& (c()))
(20)
where
arg min XEX
Σ 1
(x)
and
x(1) A arg min
Σ Ur
(21)
bEF
Then, App. D proves the following lemma.
Lemma 3.5. By picking the estimator in equation 20 it holds with probability 2 1 6 that
VI2 1, s(x;) S |F
F1
max Caine/l
(22)
heF
where Cn & 8/loo(11+0 (1+77 and " adopted from the work of Srinivas et al. 2010) is the maximum information gain from observing T noisy evaluations of h.
from
Hence, when Th, is sublinear (c.g., when the kernel is SE or Matérn (Srinivas et al. 2010)), the sum sāt) of instantancous regrets at the estimator approaches 0 as co.
4 EXPERIMENTS
This section validates the empirical performance of our algorithms (UCB-C in the coupled setting and UCB-D in the decoupled setting) by comparing with EIC (Gardner al.) 2014), ADMMBO (Ariafar et al., 2019), and the state-of-the-art CMES-IBO which significantly outperforms other existing approaches including EIC and PESC in the work of Takeno al. (2022). We did not conduct a comparison with PESC due to the challenge of maintaining a consistent initial configuration for PESC, as emphasized by Takeno et al.] (2022). Moreover, in the decoupled setting, PESC is not currently available in the primary branch of the Spearmint tool at /tttts:://gtitti This also highlights the complexity involved decoupled queries and its limited accessibility to practitioners. For these baselines, we select the estimator x arg max x maxxex
com/ implementing PESC for
HIPS/Spearmin
(x) such that Vc E C, Pr(c(x) 2 Xe) 2 'V0.95 as suggested by Takeno et al. (2022). This definition may be undefined the presence of an equality constraint, so we do not consider equality constraints in our experiments. The estimator in our algorithms is described in equation (2)). To illustrate both the instantaneous regrets of the objective function and constraints, we plot the average and standard error (over 10 repeated experiments) of the sum s sX;; (equation (5) of these regrets at the estimator against the number of queries [D,]_ The noise't 's standard deviation set a 0.01 Vh F. Additional details described in App. E
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10
Percantage funetion quaries
Co Ao h-
0.8
0.-
84
04
6A
02
0.2
0.0
20
02
D4
06
80
10
0.0
02
0.4
0.6
5.
10
02
02
04
06
0.0
1.
queries
(a) [S-A0]
(b) [S-AI]
(c) [S-A2]
(d) [S-A0]
CMES-1BO
ADMMBO
UCB-D
Percantage function quelees
2.0
2.8
30
1.a
0.6-
10-1
20
10t4
10ah
25
50
75
25
50
75
50 Number of
50
Number of quaries
Numbbr quaries
Number
Number of
(e) [S-A1]
(f) [S-A2]
(g) [S-A0]
(h) [S-A1]
(i) [S-A2]
Figure 2: Synthetic problems: (a-c) Plots of the upper confidence bound f (plotted as the gray heatmap), constraints (plotted as contour lines with arrows showing the side of the feasible region), the optimal x², the estimator x and input queries of UCB-D; (d-f) Plots of the percentage of the objective function (the dotted area) and constraints selected as h2 by UCB-D; (g-i) Plots of the instantaneous regret s(x)) against the number of queries.
4.1 SYNTIETIC PROBLEMS
The experiments are conducted on synthetic constrained optimization problems each labeled in the format 2f_Af=_of____ of_active_eeoeteeeeeeeee_ee__eee caatiee_ooosrrits______ }]: [S-A0], [S-A1], and [S-A2] with 0, 1, and active constraints, respectively (the formulations are described in App. E.T. In these problems, there are input dimensions so we can visualize the constraints and the input queries as shown in Figs. Figs.2p-c. p.c [S-A0]: The constraint is inactive at x* which is located distant from the boundary of the feasible region (Fig. 2). Thus, UCB-D does not require precise boundary estimations of S (i.e., of cu) to pinpoint x". This results in a sparse allocation of input queries around the boundary of S. Furthermore, around the estimator x a substantial number of queries are evaluated at the objective function (i.e., /,, f. as plotted by the yellow pluses) due to the high certainty that the input query is feasible, which aligns with Remark 3.2 Specifically, Fig. 2d shows that more than 70% of 60 input queries evaluated at the objective function f. plotted by the dotted area. Despite the distance between the estimator X; and the optimal x", the difference between f(x;) and f(x") is minimal because s( (X:) is small in Fig. 2 for UCB-D. [S-A 1]: At x", the constraint co is inactive, but unlike [S-A0), the constraint e is active (Fig. (p). Thus, it requires precise boundary estimations around x* to pinpoint x", but does not require precise boundary estimations of co. This results in sparse allocation of input queries around the boundary of co and a denser allocation of input queries around the boundary of C1, especially near X in Fig. 2p. Specifically, Fig. Z shows that only small number of the input queries are evaluated at the inactive co. [S-A2]: Both constraints co and cı are active at x" (Fig. 2b). Thus. UCB-D requires precise boundary estimations of both co and ei around x* to pinpoint x This results in a dense allocation of input queries around the boundaries of both co and C1, especially around x" in Fig. - Specifically, Fig. 2 shows that the input queries roughly allocated equally to the objective function and the 2 constraints. Despite the distance between x; and x", the difference between fx;) and f(x") ) is minimal because s(x;) is small in Fig. 2 for UCB-D. In Figs. 2b-c, we also observe that only minority of the function queries h, are located far away from the feasible region S. which aligns with Remark 3.2
cı
Regarding the instantaneous regret, Figs. 2-- show that our UCB-D converges faster than other algorithms. Hence, UCB-D is more efficien,, as explained by the above discussion. UCB-C performs competitively compared to the state-of-the-art CMES-IBE IBO as both are designed for the
8

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BO
ADMNBO
EIC
Weccettoge..o of faetiie
124
30
60
50
130
152
22
60
600
Number queetes
Number queries
Number querles
Number of queries
Mumber of queres
Number of queres
(a) [Gas]
(b) [CNN]
(c) [QChip]
(d) [Gas]
(e) [CNN]
(f) [QChip]
Figure 3: Real-world problems: Plots of (a-c) the instantaneous regret s s(x*) and (d-f) the percentage of function queries (by UCB-D) against the number of queries.
coupled setting. ADMMBO does not work well probably because it requires tuning the number of evaluations of f and at each BO iteration. In our experiments, ADMMBO evaluates f and e once at each BO iteration to be consistent with EIC, CMES-IBO, and UCB-C.
4.2 REAL-WORLD PROBLEMS
In this section, we introduce optimization problems utilizing real-world objective functions and constraints. These problems serve to assess the effectiveness of our algorithms in practice. We select a rorl-world problem optimizing a gas transmission compressor design, referred as [Gas], from Kumar et al. (2020). It consists of d = 4 input dimensions and has CJ constraint. The problem of tuning hyperparameters of a convolutional neural network (CNN), referred to as [CNN], is taken from the work of Takeno et al. 2022). In the [CNN] problem, a two-layer CNN is trained on a class- daaasiimbllneeed CIFARIO dataset. The goal is to maximize the overall accuracy across 10 classes subject to the constraint that the recall of each class is least 0.5, i.e., |C| = 10. There are d 5 hyperparameters to be optimized. The final experiment, referred to as [QChip], involves maximizing the coupling strength of a synthetic superconducting quantum chip (Yan al. 2018). While it is critical aspect for the chip's performance, coupling strength must be carefully controlled within the constraints of the coupling energy to prevent issues like noise, cross-talk between qubits, and poor gate fidelity (Kwon et al. 2021). In particular, we maximize the coupling strength subject to [C] [ constraints specifying the desirable range of the coupling energy, by adjusting d 11 geometric features that describe the physical dimensions and arrangement of quantum chip components. To create the groundtruth functions, we obtain dataset consisting of 393 data points. They are rigorously generated through extensive simulations using electrical simulation software (see App. E.3
Figs. BJ a-c) show that our UCB-D and UCB-C converge faster than other baseline methods. Therefore, when considering the same number of queries, our algorithms outperform other baseline methods in identifying superior designs for the above real-world problems. In Figs. rigs.3. 3 and 3J. we observe that the number of queries to the objective function dominates that to the constraints in the [Gas] and [QChip] experiments, hinting that the constraints are inactive at the optimal solution, aligning with the groundtruth. Fig. 3e shows that in the [CNN] experiment, UCB-D initially focuses on identifying a feasible input as it allocates few queries to the objective function at the start. It is noted that locating feasible input is more challenging in this experiment due to the large number of constraints.
CONCLUSION
In this paper, we propose novel constrained BO algorithm with provable performance guarantee that adaptively selects not only the input query but also the function query to account for the decoupled query. We formulate the algorithm from the standpoint of the fundamental exploration-exploiiaiie trade-off and, interestingly, cast the proposed algorithm under the uncertainty sampling paradigm in the active learning literature. As constrained BO solution requires only the confidence bounds of the function evaluations, we believe the approach can be applied to other BO problems such as BO of risk measures (Cakmak al. 2020 Nguyen al. 202Tb and meta-BO (Nguyen et al. 2023

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REPRODUCIBILITY STATEMENT
We have described detailed proofs for the theoretical results in App. B and D These proofs utilize assumption from the work Srinivas et al. (2010) elaborated Sec. Regarding the experimental results, we have included both the code and the datasets in the submission. We have also provided more detailed description of the experiment settings in App.
ACKNOWLEDGEMENT
This research is supported by Al Singapore, under grant DASG2-RP-2020-000
This research is part of the programme DesCartes and supported by the National Research Founda- tion, Prime Minister's 's Office, Singapore under its Campus for Research Excellence and Technological Enterprise (CREATE) programme.
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12


Reviews:
[{"review_id": "OwVVBgc2sx", "reviewer": "Reviewer_L67a", "summary": "The paper proposes an algorithm tackling BO with unknown constraints by (1) explicitly measuring the benefits of querying the objective and the benefits of querying the constraint(s), (2) maximizing the general benefits to achieve an efficient trade-off of learning the unknown constraint and optimizing the unknown objective. The theoretical analysis offers a convergence guarantee of the proposed CBO method, a significant advancement in the domain.", "questions": "1. Could the author include the line of work in constraint active search [2][3] in a discussion of related work? They are closely related to the CBO problem as it aims at searching feasible points efficiently within feasible regions defined by unknown constraints.\n\n2. There is a recent paper studying a similar CBO solution [4] to the proposed algorithm. It is unnecessary to include it in the paper due to its release publication time, but I encourage the author to explore it.\n\n**References**\n\n[2] Malkomes, G., Cheng, B., Lee, E. H., & Mccourt, M. (2021, July). Beyond the pareto efficient frontier: Constraint active search for multiobjective experimental design. In International Conference on Machine Learning (pp. 7423-7434). PMLR.\n\n[3] Komiyama, J., Malkomes, G., Cheng, B., & McCourt, M. (2022). Bridging Offline and Online Experimentation: Constraint Active Search for Deployed Performance Optimization. Transactions on Machine Learning Research.\n\n[4] Zhang, F., Zhu, Z., & Chen, Y. (2023). Constrained Bayesian Optimization with Adaptive Active Learning of Unknown Constraints. arXiv [Cs.LG]. Retrieved from http://arxiv.org/abs/2310.08751", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The key concepts and proposed algorithm mostly rely on the confidence interval, which bears good interpretability.\n\n2. The analysis extends GP-UCB results into the CBO setting.\n\n3. The figures are illustrative, and the paper is, in general, well-organized.", "weaknesses": "1. Though the author highlights the connection of the proposed method to active learning (AL), it lacks a discussion on the link to the existing AL methods. For example, the concepts, including the uncharted area and $\\nu_t$-relaxed feasible confidence region, resonate with the concepts in [1], and the analysis also bears connections.\n\n2. The definition of regret is unconventional and lacks sufficient discussion. Typically, in the CBO setting, the reward is only defined within the feasible region, as there is no reward incurred by querying the points that are infeasible. The regret here is defined on both the objective and constraints, which circumvent the problem of infinite instantaneous regret in cumulative regret analysis when querying points out of the feasible region.\n\n**References**\n[1] Gotovos, Alkis. \"Active learning for level set estimation.\" Master's thesis, Eidgenössische Technische Hochschule Zürich, Department of Computer Science,, 2013.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tjjxTyOr0W", "reviewer": "Reviewer_TVyc", "summary": "This paper studies the Bayesian optimization with unknown constraints. Different from previous work, this paper focuses on the decoupled setting where objective function and constraints are evaluated independently at different inputs. A new constrained BO algorithm is proposed, and empirical results show the effectiveness of the algorithm.", "questions": "1. How does the last equation hold in eq 8 by adding vertical exploration bonus?\n2. In last paragraph of Section 3.1, how does Lemma 3.1 imply that $O_t$ is non-empty with probability $>1-\\delta$?", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. This paper studies the constrained Bayesian optimization problem in the decoupled setting, which was a problem seldom studied before.\n2. The whole paper is well organized, and I like the illustration figures in Figure 1, which are helpful.\n3. Experiments on both synthetic and real-world problems are conducted to show the effectiveness of proposed algorithm.", "weaknesses": "1. In Introduction, the motivation of studying decoupled constrained BO (CBO) is unclear to me. What are the real-world applications of decoupled CBO? In which case should we evaluate objective function and constraints at different inputs? In that case, can we run several independent standard BOs to solve all problems separately? Or can we run multi-objective BO to solve it?\n2. I’m surprised to see definition of regret in eq (3) by combining objective function evaluation together with constraint violations. They may sit in totally different function ranges. Let F denote the range of objective function and let C denote the range of constraints. If C is much greater than F, then regret has little information about convergence. Also, the optimal point x* is defined w.r.t. objective function and $r_c$ is independent to x*. Why is $r_c$ is part of the regret?\n3. How do you solve optimization problems in Line 3 and 4 in Algorithm 1? Definition of $O_t$ seems like making solving Line 3 intractable.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dqGQ9drRbJ", "reviewer": "Reviewer_xiLP", "summary": "The paper addresses the challenge of constrained BO within a decoupled setting, where evaluations of the objective function and constraints occur independently at different inputs. The decoupled setting requires adaptive selection between evaluating the objective function or a constraint, alongside selecting an input.  Additionally, the paper empirically evaluates the performance of the proposed algorithms using both synthetic and real-world optimization problems.", "questions": "1. In plotting the UCB-C's regret against the number of queries (e.g., Figure 2g), as evaluations are conducted at both the objective function and constraints in each BO iteration, how are they plotted? Does the plot advance by jumping every #constraints + 1?\n2. Just curious, in cases where the query point is feasible, why don’t we evaluate the constraints together with the objective function? What are the advantages of strictly adhering to UCB-D?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "By allowing separate evaluations, the decoupled setting mirrors a more practical and realistic scenario, offering flexibility in the optimization process. This approach is more adaptable to various situations and potentially reduces computational expenses or time. Additionally, the proposed method is equipped with a theoretically proven performance guarantee.", "weaknesses": "n/a", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8QWQvLFGjS", "reviewer": "Reviewer_WGLv", "summary": "This paper proposes a constrained Bayesian optimization algorithm which aims at dealing with constrained black-box optimization under decoupled setting. The authors utilized confidence bound derived from Gaussian process to determine the function oracle to query, and derived the cumulative regret bound in terms of both maximization and summation computation way. Experiment on synthetic function and real-world application demonstrates the query-efficiency of the proposed UCB-D algorithm.", "questions": "1. As mentioned in weakness part, my major concern lies in the motivation of decoupling the function queries, since in many real-world applications, the objective and constraint values are simultaneously evaluated after one trial, and decoupling the function query seems not save the cost. Can you explain the motivation and potential application of decoupling the function queries?\n\n2. The result plot only shows the summation of regrets, which does not tell the found solution is feasible or not. Can you separately show the regret of  the objective function and constrant functions? \n\n3. In Figure 2 (h), the standard error of UCB-D is much larger than other baselines. Can you give some insights of why this happens?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The idea is clear and easy to follow.\n\n2. The paper is well-written, and the presentation from the active learning aspect helps better understanding the proposed algorithm.\n\n3. The experiment result well demonstrates the query efficiency of UCB-D.", "weaknesses": "1. The paper does not explicitly introduce the motivation of decoupling the function query, and seems that the chosen real-world benchmark does not has the property of decoupling the function queries.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)