Pareto-frontier Entropy Search with Variational Lower Bound Maximization
Authors: Masanori Ishikura
Masayuki Karasuyama
Venue: ICML 2025 poster (2025)
Keywords: Bayesian optimization
Multi-objective optimization
Mutual information
Primary area: probabilistic_methods->gaussian_processes

Abstract: This study considers multi-objective Bayesian optimization (MOBO) through the information gain of the Pareto-frontier. To calculate the information gain, a predictive distribution conditioned on the Pareto-frontier plays a key role, which is defined as a distribution truncated by the Pareto-frontier. However, it is usually impossible to obtain the entire Pareto-frontier in a continuous domain, and therefore, the complete truncation cannot be known. We consider an approximation of the truncated distribution by using a mixture distribution consisting of two possible approximate truncations obtainable from a subset of the Pareto-frontier, which we call over- and under-truncation. Since the optimal balance of the mixture is unknown beforehand, we propose optimizing the balancing coefficient through the variational lower bound maximization framework, by which the approximation error of the information gain can be minimized. Our empirical evaluation demonstrates the effectiveness of the proposed method particularly when the number of objective functions is large.

Pareto-frontier Entropy Search with Variational Lower Bound Maximization
Masanori Ishikura
Masayuki Karasuyama"
Abstract
(e.g., Knowles, 2006; Emmerich, 2005; Ponweiser et al., 2008; Belakaria et al., 2019; Suzuki et al., 2020; Qing et al., 2022; Tu et al., 2022).
This study considers multi-objective Bayesian optimization (MOBO) through the information gain of the pureto-fronticr To calculate the infor- mation gain, predictive distribution conditioned on the Pareto-frontier plays key role, which is defined as a distribution truncated by the Pareto- frontier. However, it is usually impossible to ob- tain the entire proett-fronterrii in a continuous do- main, and therefore, the complete truncation can- not be known. We consider an approximation of the truncated distribution by using a mixture distribution consisting of two possible approxi- mate truncations obtainable from subset of the Pareto-frontice which we call over- and under- truncation. Since the optimal balance of the mix- ture is unknown beforehand, we propose opti- mizing the balancing coefficient through the vari- ational lower bound maximization framework, by which the approximation error of the infor- mation gain can be minimized. Our empirical evaluation demonstrates the effectiveness of the proposed method particularly when the number of objective functions is large.
This study focuses on multi-objective BO (MOBO) based on the information gain of the Pareto-frontier. Since the op- timal solution of an MOO problem is not unique in general, the optimal values are represented tors, called the Pareto-frontier F Pareto-frontier entropy search (PFES) (Suzuki et al., 2020) considers the mutual in- formation between the preeto-rronter and candidate point as an acquisition function of MOBO. The effectiveness of the basic idea PFES has been repeatedly shown (Qing et al., 2022: Tu et al., 2022).
set of output vec-
In the cntommtiionthhooretti approaches considering the in- formation of the Pareto-frontier (Suzuki et al., 2020; Qing et al., 2022; Tu et al., 2022). the predictive distribution given the Pareto-frontier p(f(x) FF) plays key role in the information evaluation, where f(x) is a vector of objective function values at the input æ. In this distribu- tion, f() cannot be better than F because F should be the Pareto-frontier. Therefore, p(f(x) F²) becomes a truncated distribution (see Fig. 1(a) for which details will be discussed in Section 3.2). However, it is practically im- possible to obtain the entire F in continuous space, and we only obtain a finite size subset Fš C F (red stars in Fig. ((a).. This means that the exact truncation by F² shown Fig. 1(a) cannot be calculated. To avoid this is- sue, all the existing studies use approximations based on an overly truncated distribution created by F3 (Fig. I(d)). A drawback of this approach is that the effect of truncation by F on f(x) is always estimated stronger than the true truncation, because of which we call it over-truncation. which is often difficult for practical problems. On the other hand, Bayesian optimization (BO) based approaches to MOO, which use a probabilistic surrogate model (typi- cally, Gaussian process), have been widely studied recently
1. Introduction
Multi-objective optimization (MOO) of black-box func- tions is ubiquitous in a variety of fields such as materials science, engineering, drug design, and AutoML. Evolu- tionary algorithms have been classically studied for MOO, but they require a large number of function evaluations,
In this study, we introduce the variational lower bound maximization approach into the mutual information (MI) estimation of information--heorrtie MOBO, which is called Pareto-frontier Entropy search with Variational lower bound maximization (PFEV). In addition the conven- tional over-truncation, we also consider a conservative ap- proach, called under-truncation (shown as Fig. 1(C)). The under-truncation estimates the effect of truncation by F² weaker than the true truncation. Therefore, to balance two opposite approaches, we combine the over and the under-
Department of Computer Science, Nagoya Institute of Tech- nology, Aichi, Japan. Correspondence to: Masayuki Karasuyama kkarsuuyamae @ nitech.ac.jp>.
Proceedings of the 1nnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Parcio-fronticr Entropy Search with Variational Lower Bound Maximization
truncation in such way that mixture of them (Fig. (())) defines variational distribution of the MI approximation. We show that the mixture weight can be optimized through the variational lower bound maximization. This means that the optimal balance of the over and the under-truucattin can be determined so that the MI approximation error is minimized.
function is u/ Sa + E, where & NO o maise). The training dataset with observations is written as 'D == {æ,, y,)) where y,
We use in- dependent L GPs with a kernel function k(xx,'). Let k(()) == (A:(a,, a1 mi),..., -.-,,,K(,,,, ,(),,,)))) y' == (Vi,...,wh)) vh) and K be the matrix whose (i.j)-element is k(x,, The posterior p(1² D) of the 1-th GP (with O prior mean) written as (((r(z)).
where m((x) k((x) (K d hoose 1) y' and of(xr) = k(ær
Our contributions are summarized follows:
k(x) (K - noise 1) k(x). From independence, we have p(f= D) = (ieuu)p(E D)- For notational
PFEV is the first approach to continuous space MOBO that is based on a general lower bound of MI (exist-
ing work only shows lower bound under a restric- tive condition of two objective problems, for which we discuss in Section 4).
brevity, conditioning on D is omitted (e.g. p(f, DD) is written as p(fx)).
We newly introduce an under-truncation approxima- Further, we define variational distribution as mixture of distributions with the over and the
3. Pareto-frontier Entropy Search with Variational Lower Bound Maximization
tion for p=f | (F)).
We consider multi-objective Bayesian optimization (MOBO) based on mutual information MT(f.::F) between an objective function value fa and the Pareto- frontier F (Note that, throughout the paper, F is a random variable determined via the predictive distribution of fa). An intuition behind this criterion is to select x that provides the maximum information gain of the Pareto-frontier F The effectiveness of this approach is shown by (Suzuki et al., 2020), but it is known that accu- rate evaluation of MI(f : F*) is difficult. Our proposed method the first method introducing the variational lower bound maximization to evaluate Mi(fa:: F*). We call our proposed method frontir Entropy search with Variational lower bound maximization (PFEV).
and show how to optimize the mixture weight.
We also discuss properties and extensions of PFEV. For example, we show that our MI lower bound can be further lower bounded by PI (probability of improve- ment). We also discuss a Monte-Carlo approximation tailored to our lower bound. We further discuss sev- eral extended settings such parallel querying.
Through empirical evaluation on Gaussian process generated functions, benchmark functions, and ap- plication to machine learning hyper-parameter opti- mization, we demonstrate effectiveness of PFEV. We empirically observed that PFEV shows a particular difference from existing over-truncation based meth- ods when output dimension 2 3, in which the differ- ence of two truncation becomes more apparent.
3.1. Lower Bound of Mutual Information
A lower bound LB(x)) of MI(fa; F*) can be derived as
2. Multi-Objective Optimization and Gaussian Process Model
MI(f FiFT)
= = P(Ft) / p(f=1 ((falF"'oog p(f F²)
df df dF
We consider Bayesian optimization (BO) for a multi- objective optimization (MO0) problem of maximizing L 2 2 objective functions f X TR (1 1,...,),, L), where X' C Rd is input space. Let fa TT where fu. = (x)) The optimal solutions of MOO are characterized by the Pareto optimality. For given fa and far if f/ 2 ft. for VI E (1....,L) L} and there exists I that satisfies S/ > far, then "fa dominates fa.". written as fa far When fa is not dominated by any other fa', then a is Pareto-optimal. The Pareto- frontier F is a set of Pareto-optimal f that can be defined as F = (fa E Fx Sw o fa, Vf. E Fx). where Fx (f. RL Vo E X.).
p(fa)
= ((F)) -|D(F) [s p(f a |Ft) log (f F*)
df
p(fm)
+ DKL (p(Fa 1F)) (lf=) [F)) dF
2 E If JolS ((J=IF') F )F0) qf FF)
df
p(fa)
= -E Ss loog (f FT))
= LB(xr,
(1)
p(fa)
where DKL. is Kullback-Leible (KL) divergence, and qff. FF) is density function called a variational dis- tribution. A similar lower bound was first derived in the context of constrained BO (Takeno et al., 2022). This lower bound holds for any distribution qf F²) that satisfies
Each objective function is represented by the Gaussian pro- cess (GP) regression. The observation of the 1-th objective
2

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Parcio-frontier Entropy Search with Variational Lower Bound Maximization
the following support condition:
qf Fr)
supp(a(f== (F)) 2 supp(p(f= F))),
(2)
where supp is a set of non-zero points of given density, defined as supp(p(f] | F+)) = (f E R | p(f FF) * 0}. In addition to this condition, the lower bound LB(x) is derived by using the convention Olog0 = 0 (Cover & Thomas, 2006). We call the condition (2) variational distri- bution condition (VDC). Note that if VDC is not satisfied, LB(x) is not defined, because then, the probability mea- sure induced by p(f FF) is not absolutely continuous with respect to those of q(f... n. (F)).
A=
f
fx
(a)
(b)
qu (f 20
FF)
go F3))
go
q3 (f FF))
SL
3.2. Variational Lower Bound Maximization by Combining Over- and Taaeer-ruuntioon
A #
fz
f
ff
If q(f FF) = p(f x F*) for F almost every- where, the lower bound LB(x) is equal to Ml( However, the analytical representation of p(f= x F²) is not known. This stems from a common well-known dif- ficulty of inforoition-theoretic BO, and effectiveness of the truncated distribution based approximation has been repeatedly shown (c.g., Wang & Jegelka, 2017: Takeno
(c)
(d)
(c)
Figure (a) Predictive distribution truncated by F. (b) Ex-
amples of possible truncation given F3. (c) Under-truncation
based on F3. (d) tven-truncation based on Fs. (e) Mixture of (f a FF3 and (f F3).
et al., 2020; Perrone et al., 2019; Suzuki et al., 2020) not only for MOBO, but also for a variety contexts of BO problems (such as standard single-objective problems, con- straint problems, and multi-fidelity problems).
on the most conservative truncation shown in Fig. I(c). This truncation is defined by removing "the region that dominates F3" (any point that dominates F cannot ex- ist). The remaining region is written as A = R e -²_ 1 = f.3f' E F$}. The resulting truncated normal distribution
Define f 3 F* as that f is dominated by or equal least one element in F The truncation-base based approximation replaces the conditioning in p(fx FF) with fa 3 F by which we obtain qf
F²) p(f. fo F*):
(x)
if E A otherwise.
qu qf
0
2Z (x)
if fr E AF, otherwise,
a(f 1F))
where = (x) = plf E A ). We call this distribution PFTN-U (PFTN with under-truncation)) PFTN-U has a larger support supp(qu (fx FF)) 2 supp(p(f, FF)). and thus, VDC is satisfied. This truncation is conserva- tive in the sense that the region where the density function becomes zero is smaller than qf FF). by which it un- derestimates the effect of the condition fa - F* on f..
where AF' = If E R². If E F*} and 2F (x)) = p(f E A²) is the normalization constant. As shown in Fig. I(a), qff= F²) is a truncated nor- mal distribution in which only the region dominated by JF remains. Suzuki et al. (2020) call this distribution PFTN (Pareto-Frontier Truncated Normal distribution). PFTN is derived by the fact that if F is given, any Se dominat- ing F cannot exist. However, F cannot be obtained in practice in the continuous domain, and we only obtain lim- ited discrete points, such as those indicated by the red star points in Fig. ((a). A subset of F defined by these discrete points is written as F& C F (we discuss how calculate F3 in Section 3.3). In this study, consider combining two truncated distributions derived from Fss, instead of us- ing F that is not obtainable.
The second truncated distribution is based on the over- truncation shown in Fig (d). This is "the region dominated by F3",, defined as A S == {f E R I = 3. F3}. The resulting truncated normal distribution is
p ((f=)//25' (x)
if E A0, otherwise,
go go f FF)
where (x) = = E 430 In contrast to qu. 90 overly truncates the distribution in the sense that the region where the density function value becomes zero is larger than qffx F²), by which it overestimates the effect of
In Fig. 1(b), the orange dashed lines are examples of possi- ble truncation behind F3 and the red dot line is the true F+, which is unknown. The first truncated distribution is based
3

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Parcto-frontics Entropy Search with Variational Lower Bound Maximization
bound:
max LB(x, A) AE(0.1)
(5)
f2
This maximization implies the following property, which is well-known advantage of the variational lower bound maximization (Bishop & Bishop, 2023):
f
Figure 2. An example of truncation in three-dimensional output space. The resulting truncate regions have large difference.
Remark 3.1. From (1),
MI F*) -LB(x, X)
By- [DRL (p(fz) F²) 22( FF))]
the condition fx I F on fz. We call this distribu- is impor-
tion PFTN-O (PFTN with over-truncation). tant note that PFTN-O may not satisfy VDC (2) because supp(go(ff F3)) C supp(p(f.. FF)).
Therefore, (5) is equivalent to
min Ex- [DKL (p(f 1F)) 1x(xx [F))) xE(0.1]
As we have already discussed, to and qu over and under estimates the truncation by the true F*, Therefore, instead of using one of them, we consider the mixture defined as follows:
As result, (5) can be seen the minimization of difference between true p(f n. FF)) and gx-
Further, we have the following property:
F3) = (f F3) + (1--)go(ff, FS). d- + 1-1
Remark 3.2. (5) can be bounded from below (the proof is in Appendix A):
p(fa) if fa E
(a) (0)
max LB(x, Ey- p(f $ ATr) O. AE(0,1)
Ap(F Ap(fa)/Zi w)/Z( 3 (x)
fa Auio
(3)
otherwise,
p(f. & A can be seen as the probability of improve- ment (PI) from the region A .frmm which positivity (S) is also directly derived.
where AUO A A and A (0, is weight of the mixture. Because of X + O, we have supp(ax(fa FG)) 2 supp(p(fx, FF)), meaning that the mixture sat- isfies VDC. Figure ((e) shows an example of the mixture. The effect PFTN-U and PFTN-O can controlled by A. In the 2D illustration in Fig. the difference between the two truncations might appear small. However, as shown in Fig. 2, in three more dimensions, the two truncations are significantly different, obviously (further discussion is in Appendix L). Therefore, we conjecture that the balance of them can have a strong effect, particularly for problems with L 2 3.
While MI has trivial lower bound in general, this remark guarantees that (5) is larger lower bound than this trivial also see that if the candidate æ is
bound. Further, promising in sense of PI, (5) should have substantially larger value than 0.
As a result, the selection of æ is formulated as
max LB( A), 2,XE(0,11)
in which æ and À can be simultaneously optimized ((d+1)- dimensional maximization).
By substituting (x Fš) into LB(x), we define LB(x, A)
3.3. Computations
LB(x,
We employ the Monte-Caria (MC) estimation to calculate the expectation in LB(x,
(gu ( + N)go(fa FF3) }
-EF. log
p(fa.) EF -- [log[ Fb (x))((f E )
LB(x, A) ~
Σ (Fg.
log
Ft ( (x)](ff E A
K
Fi (x))(( A/
(4)
8''s + 7 (x) A
where [x)
T' (x)
(6)
(
(m))
and I is the indicator function. Importantly, the weight parameter À can be estimated by maximizing the lower
where F is of sampled pairs of (F3, f), for which sample pair is denoted as (Fz, f), and K |F] the

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Parcio-frontier Entropy Search with Variational Lower Bound Maximization
number of samples. Henceforth, variables with in- dicate sampled values. For Fs the same sampling strat- egy can be used as existing information-theeortie MOBO methods (Suzuki et al., 2020; Hernandez-Lobato et al., 2016), in which a sample path ji is approximately gen- crated by using random feature map (RFM) (Rahimi & Recht, 2008). We obtain F3 by solving the MOO maxi- mizing sampled (ax) for 7 = 1,....,.. L. This maximiza- tion can be performed by general MOO solvers such as the well-known NSGA-II (Deb et al., 2002). In the case of NSGA-II, [F'l can be specified, typically less than 100. can be easily performed by
which MAP (maximum a posteriori) becomes
OMAP (fAP(fz): + IIf E EAA
The detailed derivation is in Appendix D.I. OMAP can be seen as an average of p and T(f E AAO)). As a result, we obtain
IB(a, K
OMAP log 2x (xx (P
Σ
Computations of z and Z 53 the cell (hyper-rectangle) decomposition- -based approach as shown by (Suzuki et al., 2020) for which details are in Appendix B. This decomposition is only required once for each iteration of BO because it is common for all candidate x.
+ - 8 log) TI (x) =: LB(x, A). (8)
We empirically observe that the estimation variance can be improved by the prior. Instead, the bias to ş can occur, but because the number of samples is usually small (our de- fault setting is K 10 in the experiments, which is same as existing coommation-thooreti BO studies such as (Wang & Jegelka, 2017)), variance reduction has stronger ben- efit in practice. Further discussion on the accuracy of this estimator is in Appendices D.2 and D.3.
We also consider another way to numerically approximate LB( X) based on the following transformation:
LB(a, X)
=Ey' So [(ffa E A/²) log (x)+ A log log 13 (अ)
The procedure of PFEV is shown in Algorithm 1. Assume that we already have the posterior mean and variance of the GPs. Then, computations of (8) is O(K'L). where C is the number of hyper-rectangle cells in the decompo- sition. For the cell decomposition, hypervolume (QHV) algorithm (Russo & Francisco, 2014) as indicated by (Suzuki et al., 2020). Sampling of F re- quires ( for RFM with D basis functions and the cost of NSGA-II is also required. We empirically see that, for small I,, the computational cost of NSGA-II is dominant, and for large L, QHV becomes dominant, both of which are commonly required for several rnformation-theoreee MOBO (Suzuki et al., 2020; Qing et al., 2022; Tu et al., 2022). Compared with them, the practical cost of the lower bound calculation in CalcPFEV of Algorithm is often rel- atively small (see Appendix K.3 for details). As discussed in the end of Section 3.2, the acquisition function maxi- mization is formulated as (d + 1)-dimensional optimiza- tion (line 10 of Algorithm 1). On the other hand, it is also possible to optimize À for each given æ inner one- dimensional optimization. In the latter case, efficient com- putations can be performed by considering (8) (or (6)) is concave with respect to X. Further, we can show that the maximizer of À exists though the candidate À is defined as left open interval (0. 1]. Details of these discussions about the maximization problem of X are shown in Appendix C.
- Ex- (ols E Ab3 FF))logh ( (x)
+plsa E A0:0 F)) log (m)
employ the quick
Σ plaa E Ao¹1 (3 log 22 83 (z)
K
(Fgi)EF
+D(fa E Avio 1F) log vix (z))
Note that the second line is from Iif E A I(ff Am)o), and the third line is obtained by replac- ing EF with E,-E,, Since (6) can be rewrit- xx &'s + I(F= (x)+
ten as k E.tghee Ilfa E A60) ) A F' Ao)) log via (z). by comparing this re- re-written expres- sion with (7), we can interpret that (6) performs one-sample
approximations p(Sz E A fr) R I(Ja E A and p(f E A F) x Ilf E Agro)) We consider improving this approximations by introducing prior knowl- edge about a E A/3 f*) and p(f A F*))
Let 0 = p(Sa E
F.). We use approximation 0 = p(ff. A Ifa E A =
A
Z = (z) (z) = P as our prior knowledge. The approximation is replacement of the conditioning by F with the under-truncation f. E A Note that Zo0 $ (x) and L (x) are required even in the naive MC (6). P can be obtained without additional computations. A prior distribution having the mode at ş is introduced to estimate A. We use the beta distribution by
4. Related Work
For MOBO, a variety of approaches have been proposed, typically by extending single-objective acquisition function such as expected improvement and upper confidence bound
5

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Parcio-fronticr Entropy Search with Variational Lower Bound Maximization
Algorithm Pseudo-code of PFEV
bound of MI is Parallel Feasible Pareto Frontier Entropy
Function PFEV( Initial dataset Do )
Search ((PF) (Qing al., 2022), which also pointed out the problem of over-truncation. However, {PF) is still based only on over-truncation. A shifting parameter e E TR_ of the Pareto-frontier is heuristically introduced to mitigate over-truncation, for which theoretical justification is not clarified. Further, their criterion is not guaranteed as lower bound in general (only when L 2 with the assumption [F oc).
2: for = 0, ...... do
3: F F+ 0
4: for k = 1....... K do
5: Generate sample path f from the current poste-
rior of f
6: Obtain Fig by applying NSGA-II to f=
7:
Create cell-decomposition by QHV
8:
F FF.
9:
end for
5. Discussion on Extensions
10: (I A(+1) <- argmax,,, CALCPFEV(Z, F)
11: Evaluate f a and observe Mi+1
PFEV is a general framework so that we can drive exten- sions for the following four scenarios:
12: Di+1= + De U (1++1.3++1)
13: end for
14: Function (
Parallel querying: In parallel querying, we consider querying multiple points at one iteration, which is im- portant practical setting. Suppose that we consider Q > 1 and H, (fam..... fa for q S Q. The MI for all Q points MT(HQ:F)) represents the benefit of selecting Xg. but this results in a (Q x dd-dimensional optimization problem, which can be unstable. Instead, we follow the approach in (Takeno al., 2022), which is greedy selection based on conditional mutual information (CMI). When we select the (q + 1)-th after determining xx)), ,.... a.()), the MI can be decomposed into
(z), 15: Calculate ( (z), (x), and Ilfa e Aot) for
(Fs. f) E F
points selection. Let X2 = f x
16: return Approximate lower bound (8)
(e.g., Emmerich, 2005; Shah & Ghahramani, 2016; Zulu- aga et al.. 2016: Daulton et al., 2020; Ament et al., 2023). information-theoree
approaches (e.g.. Mernandeez-Lobtto et al., 2016; Belakaria et al., 2019) also have been extended from its counterpart of the single-objective BO (Hennig Hernández-Lobato et al., 2014; Wang & Jegelka, 2017). Among these, the most closely related method to the proposed method is PFES (Suzuki et al., 2020). PFES can be seen as a multi-objective extension of the max value based information-theoreti BO, called max-value entropy search (MES) (Wang& & Jegelka, 2017). Information -theoretic MOBO before PFES and other ap- proaches are reviewed in (Suzuki et al.. 2020). Here, we mainly focus on information-theoreete MOBO methods af- ter PFES. A more comprehensive related work including other information-theoreti methods and other criteria are discussed in Appendix I.
& Schuler, 2012:
F CMII mi F² HHg) where CMI(S 20 F H.y) = EH,[Mi[ [MI(f=: Hg)] is the MI conditioned on Hy. Since the first term does not depend on æ, only need optimize CMI to select æ. The calculation of CMI can be performed by adding sam- pled H. into the training dataset of the GPs, after which evaluation of the lower bound is almost same as the single querying. See Appendix E for details.
MI(H, U
Decoupled setting: In the decoupled setting, we can ob- serve only one selected objective function instead of query- ing all L objective functions simultaneously (Hernandez- Lobato et al., 2016). In PFEV, the criterion for the decou- pled setting can be defined as information gain from only one objective function fa. The lower bound of F+) can be derived by the same approach shown in Section 3. In this case, we need to ap- proximate p(fz F²) instead of p(f. F²). ) for which we can use the marginal distribution of qx (S F²). See Appendix F for detail.
The MI approximation with PFIN was first introduced by PFES. On the other hand, the MI approximation is based on a decomposition into difference of the entropy, which is classical approach in information-theoneti BO (Hernández-Lobre et al., 2014). In the context of con- strained BO, Takeno et al. (2022) revealed that the entropy difference based decomposition can cause a critical issue originated from a negative value of the MI approximation. The positivity of PFES has not been clarified. Further, PFES only uses over- trenccannntionn (PFTN-O). Therefore, ef- fect of the truncation by F= on fa is overly estimated.
which is the
After PFES, Joint Entropy Search (JES) (Tu et al., 2022) was proposed in which the joint entropy of the optimal x and F is considered. This approach is also only uses over-truncation. Another approach considering the lower
Joint entropy search: For PFEV, not only the informa- tion gain of the Pareto-frontier F but also the correspond- ing input x can be considered as MI ( X*,F*) where X = {a fa E F'). This can be seen as a JES extension of PFEV. When we derive the lower bound for
6

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Parcio-frontier Entropy Search with Variational Lower Bound Maximization
1.0
1.0
1.0
1.0
0.5
0.8
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02.5-
0a
0.
BM
0.4
0.0
20.6
0.5
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PRES
0.44
PRES
0.4
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Proposed
0.3
0.2
50
100
100
Iteratoo
Itrraion
Raratton
(a) (d, L) = (2,
(b) (d. L) = ()
(c) (d. L) (2,4)
(d) (d. L) = (2.5)
(e) (d, L) (2.6)
0.0
0.5
0.2
0.8
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6.7
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IES
ENi
PFZE5
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PRES
DMM
0.
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H225
203
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02
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0.1
Ito 00/2
Itoration
Boration
Mtoration
Roration
(f) (d, L) (3, 2)
gg) = (3,
(h) (d, L) = (3,4)
(i) (d, L) (3,
① (d. L) = (3,6)
Figure 3. Performance comparison in GP-derived synthetic functions (average and standard deviation of 10 runs).
MI(ff::X" by the same approach as Section 3, a vari- ational approximation for p( X", F*) is required. A ba-
MOBO-RS (Paria et al., 2020), and MESMO (Belakaria et al., 2019), which are mostly omitted in the main text for brevity of plots. We show results on GP-based synthetic functions, benchmark functions, and hyper-parameter opti- mization problems.
sic idea of JES (Tu et al., 2022) is simply add (X', FF)
into the training data of the GPs. In the case of PFEV, we can define the variational distribution qf X²FF*) by using the same mixture as (3). The only difference is that (X",F") is added to the training data of the GPs. Since we have not observed particular performance in improvement by this additional conditioning, we employ MI(fa. FF) as the default setting of PFEV. See Appendix G for detail.
Each evaluation run 10 times. As a performance metric, RHV (relative hyper-volume) was used. RHV is defined as the hyper-volume of the observed by the volume of the reference Pareto-frontice The ref- erence Pareto-fronticr was obtained by 10, 000 iterations of NSGA-II. All methods used GPs for f/ with kernel function k(x,, x²) = exp( | x' (2(k))( where CREF E R is a hyper-parament maximization was performed at every iteration to optimize GRBF. The number of samples of the optimal value or the Pareto-frontier in MESMO, PFES, {PF) and PFEV 10, each of which was performed by NSGA-II (1. 000 generations and population size 50). We used the DIRECT algorithm (Jones et al., 1993) for the acquisition function maximization. We selected 5 random æ as the initial obser- vations in D. Depending on problems, EHVI and JES were performed up to I, = 4 due to computational issues. Other settings are described in Appendix J.
frontier divided
Noisy observations: is also possible to derive the information gain obtained from noisy observation MI(y F²), where y fa + e with E ~ (O, - T). for brevity. In this case, instead of p(f= x F"), we need to consider p(ym F*). Through the relation p(y, FF) f I (S. F*)dfa. we can derive the lower bound based on the same approximation of p(f FF)) by using gx- See Appendix H for detail.
The marginal likelihood
though we mainly consider M
6. Experiments
We empirically verify the performance of PFEV by comparing mainly with EHVI (Emmerich, 2005), PFES
6.1. Synthetic Functions Generated by GPs
(Suzuki et al., 2020), {PF) (Qing et al., 2022), JES (Tu et al., 2022), and random search. In Appendix K. we added other methods such as ParEGO (Knowles, 2006),
Here, we consider synthetic functions from GPs as true ob- jective functions, i.c., f ~ GP k). in which GREE 0.1
7

---

Parcto-frontice Entropy Search with Variational Lower Bound Maximization
Pareco
MOBO-RS
MESMO
PFEES
PFZES
Proposed
0.8
0.9
16
13
1.
64
24
0.
GO
Reattve Wyperrve ummm
0.00
24
08
06
0.5
0.77
02
FHVI
0.8
FIM
(a) L = (b) L 3 (c) L = (d) L -=
(0) L 6
0.3
PF255
PFZES
Figure 4. Boxplots of RHV at 100-th iteration. Note that, here, methods omitted in Fig. (ParEGO, MOBO-RS, and MESMO) are also included, which are shown Appendix K.1.
Random
Random
Proposed
Propoved
50
100
30
75
100
Rerabio
Itaration
(a) Fonseca-Fleming
(h) Kursawe
was used in the kernel k. Since we require objective func- tions in a continuous domain, we used the RFM-based ap- proximation (the number RFM basis is D = 1000). The input dimensions are d (2, and the domain X is [0, 1je. The output dimensions L [2,3, 63.
1.0
0.9
0.66
0.7
0.5
Figure 3 shows the results. In all (a)-(), PFEV shows su- perior or comparable performance to existing methods. In these experiments, we empirically see that the performance of PFEV is often similar to PFES and {SF) for 2, and differences become clearer for L 2 3. This can also confirmed by Fig. 4, which shows boxplots of RHV at the 100-th iteration in all trials of Fig. 3. This result is consis- tent with our conjecture about the difference of two types of PFTN described in Section 3.2. In Appendix K.I, addi- tional results on different fREF and d = 4 are shown, in which similar tendency was confirmed.
EHVl
Rotetve
EHM JES PHES
0.5
JES
P955
P9255
Proposed
Propesed
0.4
25
50 Rocation
100
25
30
73
100
Iteration
(c) Viennet
(d) FES3
1.0
0.8
0.3
0.7
0.0
Setaave Hyper
0..5
6.2. Benchmark Functions
0.4
PYPS P12ES
P-225
We used benchmark problems called Fonseca-Fleming (d, L) 22, Kursawe (d, L) = (3, 2). Viennet (d, L) (2 and FES3 (d, L) = (3, problems (for details, see Appendix J). Further, we combine multiple prob- lems having the same input dimensions, i.e., created Fonseca+ Vensnnt+VVennet (d,
Random Proposed
Random Propesod
75
75
100
Rarabion
Itaration
(e) Fonseca + Viennet
(I) FES3 Kursawe
(2) and (d, = (3, 6). Since the input domain is shared (which is scaled to [0, 1 beforehand), only the output dimension increases compared with the original problems. The results are shown Fig. 5. Overall, the performance of proposed PFEV high among the compared methods. Here again, see that in problems with the output dimension 2 3, PFEV tends show its advantage. Additional results benchmark functions are also shown in Appendix K.2.
Figure 5. Performance comparison in benchmark functions (av- erage and standard deviation of 10 runs).
classification accuracies for each class, which means the output dimension also equals to the number of classes. Ex- periments were conducted on four datasets: Abalone and Waveform (both with 3 classes), and Pageblocks and Ges- turephase (both with 5 classes). For each dataset, we split the original data into training and test sets with an 8:2 ra- tio. Figure shows the results, where the vertical axis represents the hyper-volume of test accuracies across all classes, calculated using the reference point at O. Note that each objective function is [o, 1] (classification accuracy of each class), and therefore, the volume is also [0, 1] (if all classes achieve accuracy 1, the volume becomes 1). As observed in the figure, overall, PFEV shows sufficiently high performance among the compared methods.
6.3. Hyper- ppermpareetter Optimization of LightGBM
As an application example of hyper-parameter optimization problems, we consider optimizing class weights in multi- class classification problems using LightGBM (Ke et al., 2017) as the base model (this setting is from (Ozaki et al., 2024)). We focused solely on optimizing the class weight parameters, making the input dimension of BO equal to the number of classes. The objective functions are the test
8

---

Parcio-frontier Entropy Search with Variational Lower Bound Maximization
7. Discussion on Gradient-based Optimization of Acquisition Function
0.900
0.675
0.45
We employed DIRECT as optimizer of the acquisition function because it does not require 'initial points' unlike the gradient descent that requires the appropriate setting of the initial points (the number of initial points and loca- tions). Our purpose to focus more on differences of the acquisition functions and to reduce the other factors affect- ing the performance. On the other hand, evaluation using gradient-based approaches is also important future work because it is widely used in BO and should have high per- formance in general. We partially show comparison with a baseline using the gradient optimization (qLogNEHVI in BoTorch) in Appendix M.
ParEdO
0.825
FaréGo
0.40
EHVI
MO30-45 AS
0.800
EHMt
MORD-RS
0.35
HESSO
0.775
MESMO
0.750
PF2ES
0.725
PFZES
Random
Fandom
Proposed
0.700
Proposed
25
30
75
300
50 Itrriion
150
Taration
Abalone (4177 samples, features, classes)
(b) Waveform (5000 sam- ples. 40 features, 3 classes)
0.575
0.30
0.550
Note that the proposed acquisition function (8) mostly differentiable, except for the indicator T(J Age) in OMAP(fa). We consider that possible approaches are sim- ply ignoring this term in the gradient (regarding the gra- dient of the indicator as 0) using continuous approx- imation of the gradient. In the continuous approxima- tion, we replace I(f E Ag6) = p(f A where f NMfan a: pT) in which p > 0 is fixed smoothing pa- rameter. The right hand side of the approximation is dif- ferentiable with respect to x through the similar decom- position to (9) in Appendix B (note that fa is differential because it is generated from RFM). This can be interpreted as a counter-part of the standard CDF-based smoothing ap- proximation of an indicator function, extended the indi- cator of the Pareto dominated region. For the calculation, although the cell-based decomposition is required for A we can reuse the cells created in line 7 of Algorithm 1.
FaréGo MOBD R5
0.08
PorECO
0.525
MCCO RS
0.500
NESMO
0.06
MESNO
0.475
0.450
Randam Propesed
0.02
Random Proposed
0.425
50 Bereiion
100
25
50
75
100
Beration
(c) Pageblocks (5473 plcs, 10 features, classes)
(d) Gesturephase (9873 samples, 33 features, classes)
Figure 6. Results on hyper-parameter (elass-weights) optimiza- tion.
1.0
0.8
Relative Hypervo ume
Relative Hyperwo ume
0.7
0.8
0.6
0.5
0.6
0.4
Proposed[Bayes MAP!
ProposediBayes MAPI
8. Conclusions
Naiee MC)
NaieosedINaive MC)
0
25
50
75
100
0
25
50
100
We proposed a multi-objective Bayesian optimization ac- quisition function that based variational lower bound maximization. By combining two normal distributions, de- fined by under- and over-truncation of Pareto-frontier, we introduced a variational distribution as mixture of these two distributions based on which a lower bound of mutual in- formation can be constructed. Performance superiority was shown by GP-generated functions and benchmark func- tions. A current limitation includes theoretical guarantee of the MI approximation and convergence, which is impor- tant open problems for information possible future direction is to use a more complicated vari- ational distribution. We employ the simple one parameter (A) distribution because g should be estimated based on the K samples, which is usually quite small (10 in the exper- iments). However, make the lower bound tight, a more flexible distribution may be required.
Ieertion
Iteration
(a) (d, L)
6)
(b) (d, L) (3,6)
Figure 7. Comparison between two MC estimators (6) and (8) of PFEV using GP-derived synthetic functions.
6.4. Comparison of PFEV with Different MC Estimators
By using GP-derived synthetic functions, two estimators of PFEV (6) and (8) are compared. The same settings as in Section 6.1 were used. The results of (d, L) = (2, 6) and (d, L)- = (3.60 6) are shown in Fig. 7. We see that the MAP-based approximation (8) improves the performance compared with (6). We confirmed similar results on other Ge-derived functions, which is summarized in Fig. 21 in Appendix K.I.
BO. Another

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Parcio-frontier Entropy Search with Variational Lower Bound Maximization
Acknowledgement
tion. In Proceedings of the 40th International Confer- ence Machine Learning, volume 202 of Proceedings of Machine Learning Research, pp. 7167-7204, PMLR, 2023.
This work partially supported by JSPS KAKENHI (Grant Number 23K21696, 25K03182, 22H00300, and 23K17817), and Data Creation and Utllization Type Ma- terial Research and Development Project (Grant No. JP- MXPI122712807) of MEXT.
Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6(2): 182-197, 2002.
Impact Statement
This paper presents work whose goal advance the field of Bayesian optimization. There are many potential soci- etal consequences of our work, none which we feel must be specifically highlighted here.
Emmerich, M. T. Single-and multi multi-objective evolutionary design optimization assisted by gaussian random field metamodels. PhD thesis, Dortmund University, Ger- many, 2005.
Sánche, D. and Hemández-Lobato, D. Joint entropy search for multi-bbjective Bayesian optimiza- tion with constraints and multiple fidelities. In ESANN 2024 proceedings, European Symposium on Artificial Neural Networks, Computational Intelligence and Ma- chine Learning, 2024.
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Parclo-frontier Entropy Search with Variational Lower Bound Maximization
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Parclo-frontier Entropy Search with Variational Lower Bound Maximization
A. Proof of Remark 3.2
Next, we consider % 'that can be re-written as
The lower bound of (5) is derived by using the case of X =
2t =p f Av
1:
=1 - p (f $ A
max LB(x, A)
AE(1.1)
=1 - p p f = (fflfe
2 LBB2 x 1)
9 (f((x F&s) v(fx)
In the last equation, the region f & A is re-written by flipping the sign of F3 in such a way that the region is written as "dominated region". This enables us to use QHV, which decomposes "dominated region" into hyper- rectangles, for the computation of p(-f= 3 Iffife F3}) based on almost the same way as (9).
= Eo
log
= -EEr-
plSx F²) log 2 (2)
= - -Ey-
log 2 F (z)
log(1 - plf x $ (/²))) }}
= EEF.
C. Properties about Maximization of X
2 Es- pl & A ] 0
We first show the concavity. To simplify the notation, we unify (8) and (6)
B. Computations of Zoo and 2
LB A) =
1
Σ log
A
+ 1-1
{lfm)
K
Zu
Zo
(Pg.DEE
First, we consider the normalization constant of PFTN-O 2 = =ff E ) We decompose A into disjoint hyper-rectangles, denoted as A²² UC U...UC. Cc. where C, is a hyper-rectangle and C' is the number of hyper- rectangles. Algorithms decomposing a dominated region have been studied in the context of the Pareto nyper-volums computation. We use quick hypervolume (QHV) (Russo & Francisco, 2014), used also in PFES. Each hyper-rectangle C, is written as
()
) (1-E(fa))
+ log
Zu
where Zu (m), Zo % (a)), and {(f
0MAP (fa) when (8), and {(Fa) 1lf A0O when
(6). The second derivative is
LLB(x, A) Ox2
Σ
(Mizo - bv) + Lu
Zo - Zu
{(fa)
K
(Fg.)EFF
G = (et x x... x (4t uf].
x {(fa)) 50.
where &t and un are the smallest and largest values in the 1-th dimension of the i-th hyper-rectangle. From the in- dependence of the objective functions, 2/55 can be easily computed by
Thus, we see concavity.
Further, in the case of (8), we can show that the optimal X should exist for maxxe(o,11 LB( A) though À has left open interval as a domain. First, we can derive P E (0, 1) from its definition P = 2/3 -20(())) (x)
83 Since fn is the
20 :: = Σ
C
(
p(f Pl
Gaussian distribution, (x)( = =p(f E Ag3) E 10 and Z0 (z) < ( = plf E A E (0. Here, we used 0 + A c A. RR (neither A nor A
i=1 c.
C L
= EI
P(
is the empty set or the entire output space, and A C
C L
A holds as far the sampled Pareto frontier set Fz is
ΣI ==
(((..) +(a,)),
(9)
finite while the true Pareto frontier is continuous which is our problem setting). Therefore, we see MMPP E (0, 1). When À + O, the second term of (8) goes to 0 (from the definition of 7)x 5) On the other hand, the first term is finite even when x - 0. As a result, if X - 0, we see (8) goes to -00.
where a. = (C) - (o)))) and ã.ي = (uf (z))) and 1 is the cumulative distribution func- tion of the standard normal distribution.
13

---

Parcto-frontics Entropy Search with Variational Lower Bound Maximization
the number of 'success' and 8 is the number of 'fail' in Bernoulli trials. We can interpret that I(f E Age is sample from the Bermoulli distribution with the probability
1.0
0.8
Selected lambda
p(f E A |T) Therefore, we set m II(f x E A and L 1 - I(f E ) from which the mode of the posterior Beta(a + m,b E) can be derived as
0.6
0.4
1 + m
OMAP =
a a 1-1+me m - 1
rD + 1(f E AAO)
0
20
40
60
80
100
rp + r A/0
r(1 ()) (1 I(S A))))
Iteration
p + ++[l( E Ao
Figure 8. Transition of selected X for GP-derived synthetic func- tion (d- = 3, L = 4).
r+
In the main text, we employ r 1.
The lower bound estimation by MAP (8) can have an timation bias caused by the approximation p. though it is almost negligible when K is a typical setting (such as K = 10). This bias occurs because we only have one f= for each corresponding F3. Therefore, in each OMAP, the effect of the prior remains even when K is large. We can easily avoid this bias by setting r so that it decreases O when K increases, by which the estimator (8) converges to the usual MC estimator (6). In Appendix D.3, we show that, in practice, the MAP-based approach has an advantage for small K setting.
1.30
1.35
1.00
0.90
0:35
0.00
60
0.3
0.1
36
0.5
10
Figure 9. Density functions of the beta distribution when 0.8. For all r, the mode is 0.8.
D.2. Analyzing Variance
Figure shows the transition of during BO iterations (the average and standard deviation of 10 runs) for a GP-derived synthetic function (d 3, L 4). We see that À takes in- termediate values in (0, 1]. indicating that under- and over- truncated distributions are indeed mixed during BO itera- tions. As far as we have examined, no consistent increasing or decreasing tendency has been observed during iterations.
We re-write the estimator of the lower bound (8) as
1
Σ aFF))(())+ + +Fzg.
K
(Frr
where
a(F3)
=
log
Zy
(x)
2
3(x)
D. Combining Prior Knowledge in MC Estimator
log
b(Fg)
log
and
2 (x))
(x))
{(fa) E [0, 1] is 0MAP when (8) and is 1(f E A when (6). By using independence of the MC samples
D.1. Derivation of MAP
and law of total variance, the variance of this estimator is decomposed as follows:
We use the beta distribution for the prior 0 ~ Beta(a,b) b). The density function is
p(0) 8 11- 6-
V 3
Σ
(F
K
B(a,b)
K By.
V トー [a(F))(ffa) + b(F)
Using the approximation 0 s P. we set a - 1 = rp and b 1 = r(1 - p). where r 2 0 is a parameter. This sets the mode of p(0) as p. As shown in Fig. larger r has stronger peak at p. When r p0) becomes uniform in [0, 1].
V 31 (a(F3)((()) + b(Fs)
K
E+-
|a V [<( ]] Ex [b(Ft)
The posterior of the beta distribution with the Bernoulli distribution likelihood is Beta(a + m, b + f), where m is
+ V..
[a(F:) [((faa] + +bFF))
(10)
14

---

Parcio-frontier Entropy Search with Variational Lower Bound Maximization
In the first term of (10), only V [S(Fa)] changes de- pending on ç. From
Raym (-w MC Bays WA (r- 1) Mt
000
000
V ni (MAApfaa) v [ [Itf E A /4,
VC aill Eape ( V VIT0
RTC with Eaye MAP (
-00
WC
we see that MAP makes the variance of the first term 1/4. /4. The second term in (10) does not change depending on ç. In the case of the MAP estimate, the third term in (10) is
12'
Figure 10. MSE compared with the pseudo ground truth, created by Naïve MC with K = 10 The plot is the original scale, and the right plot the logio scale.
V
a 9) LE
[(fx)]
+ b(F3))
al ((Fg) =
+
+ff E {
+ +MFF3)
E. Parallel Querying
2
We here consider parallel querying in which Q queries should be selected every iteration. Let X₂ = {ee(t) ,..,),,.,,, ...,,,,()) and Hg = .....).....f...)) } for q S Q. Then, MI(HQ::)) can be a selection criterion for deter- mining Q points simultaneously. However, this leads to (Q x d)-dimensional optimization. Instead, we employ greedy strategy shown by (Takeno et al., 2022), in which the MI approximation can be reduced to a similar compu- tation to the case of single querying.
EAO F3 (Ft)
= -. al (F3))
P + p(f
+ +b(Fg)
2
If = pla E A033 1 Fr), this should be similar to V (o(Fs)olf& E A 1 +bF))) which is the variance in the case of {(fa) Ilfa 9 There- fore, under the assumption of) 인 p(f E A FT) the variance reduction is expected because of the variance reduction in the first term of (10).
Assume that we have already selected q < Q points Xy. for which observations are not obtained yet, and con- sider determining the (g + 1)-th point æ. In this case, MI(H, U Uf FF) should be maximized with respect to the additional æ. We see
D.3. Empirical Verification of MC Estimator of Lower Bound
MI(H, U Uf MI(Hg;;F) + CMI(f 2: FF (Hq),
Using two objective problem generated by GPs (d 1), we examine the accuracy of the MC estimator compared with the true lower bound. Hereafter, Naive MC indicates the calculation by (6), and MC with Bayes MAP indicates the calculation by (8). We calculate the lower bound at 100 grid points in x E [0. 1] with random training points. Re- garding the result by Naivee MC with K- 100 samples as pseudo ground-truth, compared with which we evaluate the estimation error. In addition to the fixed r setting, we here examine the setting = /10/ K, by which reduces with (K) (same the general convergence rate of the MC estimator) and r- = 1 when K- 10 in this setting.
where CMI(f | Hlg) =EH [MI(f fi Ft Hgg] is the MI conditioned on Hy. Since the first term does not depend on œ. we only need to consider the second term CMI(f HHy).
The lower bound of MI(fa:F* | Hg) is derived in the same way as the lower bound of Mf(f The only difference is that the GPs have additional train- ing data consisting of X, and Hy. Let Xx, (x) and V ().6()))) be X (x)) and 1)x F3 (æ) in which the GPs with additional q observations are used to calculate z F (x)) and 2 (x) as follows:
i.e., (4).
Figure 10 shows the results by mean squared error (MSE). MC with Bayes MAP shows r = 1 and r = / 110/K K, for which they have the same result when K- = 10. When the sample size K is small, MC with Bayes MAP has smaller errors for both the r settings compared with Naive MC. With the increase of K, all methods decrease MSE, but in the right log scale plot, the decrease of MC with Bayes MAP (r = 1) stagnates at around K- = 102-103. On the other hand, MC with Bayes MAP (r = /10/K) K) continues to decrease MSE because it can diminish the effect of the approximation in the prior.
(x) =
A
+
Z (x HWg))
%
3
(x Hlg)
X
(z)
z (x Hgg)
where z (x: Hg) = plf a E A Hly and z (xr
15

---

Parclo-frontier Entropy Search with Variational Lower Bound Maximization
H4) = p(f E Aoo Hg). Then, we can write
1.0
Bng M F Hyyl
0.9
Reletive Sypersolurg
Reledivs Hypersyolume
0..8
2 RH, R f 174
log(522(0))(11 E EA3)
0.7
0.7
+1 (T))( E A [
0.5
MOB0-RS
0.5
MC8O RS
EHИ
0.4
EHVI
= EF H. Ho
log (og(sta()) E 103)
0.3
JES
Proposed
JES
Proposed
25
50
75
50
75
150
Iteraton
Iteration
+1 (T)) E ool
(11)
(a) Q = 2
(b) Q = 3
Since the expectation joint expectation performed by using samples from the joint distribution of F". and Hy. Let Fg be a set of samples (Fg. H.g) from the joint distribution. Then, the MC approximation of the lower bound (11) is
be seen the
Figure 11. Results on parallel querying.
the MC approximation can be
Figure 11 shows the empirical evaluation for which the set- ting is same as in Section 6.1. Here, we set Q 2 and 3, and we used the (d. L) (3.3) GP derived functions. For comparison, ParEGO, EHVI, MOBO-RS and JES were used. ParEGO and EHVI consider the expected improve- ment when Q points are simultaneously selected, which is a well known general strategy (Shahriari et al., 2016). Un- like the single querying, the expectation is approximated by the MC estimation for which the number of samples was 100 and 10 for ParEGO and EHVI, respectively. For MOBO-RS, points selected by repeating Thompson sampling with different sample paths Q times (Kandasamy et al., 201 2018), in which the weights in the Tchebycheff (also known as Chebyshev) scalarization were also re-sampled. JES approximates the simultaneous information gain by Q points as indicated by (Tu et al., 2022). From the results, we can see that the proposed method shows sufficiently high performance in the parallel querying.
Σ He..
log O )
K
+ 1x.q t (x))(f E A
The sampling of Fg can be performed by almost the same procedure as the single querying. First, we generate the "entire function f" by using RFM. F3 is obtained by ap- plying NSGA-II to f. fa and H.g can be immediately ob- tained from RFM. Note that even when we perform the next (q + 2)-th point selection, we can reuse F3 and f. from which Hgt1 can also be immediately obtained.
The MAP-based approximation can also be applied to the parallel setting. The lower bound (1I) can be re-written as
I'ss (1(()) Ex SH, Es , I[f A log Xx
F. Decoupled Setting
For the decoupled setting. MI and its lower bound are
J's I(fa Abio) log 1x (0))
Ex Hy plf A/3 Fr, Hgg log J's (2. (x)
F's +plf ATTO F Hyg log 12 (x))
The prior approximation for p(5n E A Fr Hly) be- comes p(f. E A Fr, H,, R 2 H4)/Z (x Hly). Then, fMAP can be obtained by the same procedure shown in Section 3.3. As a result, we have an approxima- tion of the lower bound (11)
MT(FF: F*)
Σ
OMPP log Sx,
(x)
log 915 (F*)
DKL. (plSa (F') ll 9052 FF)
3 H.JEF
p(fa)
log 9(5t FF))
+ (1 0MAP) 1og
2 1y 1/
LDec (2,
p(fz)
16

---

Parcto-fronticr Entropy Search with Variational Lower Bound Maximization
We define qS/2 F*) as the marginal distribution of (ff= F)):
where Zer +((a..)) ((ger)) and C-' is the (L 1)- dimensional hyper-rectangle created by removing the 1-th dimension of C.- For p(fx A {.), we use the following relation:
Ul F²)
f axlfa FF)
= I Snv(S (F3) + 1 Fz)dfa'
pífa E A (t)
I plS ( E Jos
= 1 - =1-p(f -(
1 1
+ + p (z) E A DE
=((E) (z) / pLEE Mf D5
Since fz & A 5 in the last term can be written as "dom- inated region", shown in the end of Appendix B. We can calculate p(f= & A 83 f by the same dominated region decomposition as (13).
+ ( P( (f E Aso)))))
= ->(sa)(>> (z) A (Tm
+ (x) E A 12)),
(12)
Figure 12 shows the empirical evaluation for which the set- ting is same in Section 6.1. Here, we used the (d, L) (3,3) GP derived functions for three length scales. As a bascline, the hypervolume-based KG is used. Since KG is defined as the hyper-volume defined by the 'one-step ahead' posterior mean, it is easy to extend to the decou- pled setting as shown by (Daulton et al., 2023). We used BoTchh'
where fo the (L- 1) dimensional subvector of fa in which f/ is removed. Then, the lower bound and its MC approximation is obtained as
Dee (x,(,)) 2,
implemen- tation for the HVKG baseline, which is based on the 'one shot' optimization of KG in the decoupled setting. Due to the long computational time, we set the number of Pareto- optimal points in the one-step ahead posterior mean as 10 in HVKG. For fair comparison, our proposed method also set the size of the Pareto-optimal points in the sampled func- tion (i.e., the NSGA-II population size) 10. The number of samples (so called fantasy points) in HVKG is also set as 10. For the final evaluation of the performance, since the decoupled setting observes only one objective function I (xx)) ) in each iteration, the hyper-volume consisting of ob- served points is difficult to define. Instead, we employ an approach similar to so-called inference regret. At each it- eration f, we apply NSGA-II to the posterior mean p(xx), and obtain a set of the Pareto-optimal points X for (a()). We evaluate the hyper-volume defined by the ground-truth objective function values on X", and the plots are the max- imum values of the volumes identified until each iteration. From results, we see that the decoupled extension of PFEV has reasonable performance (Note that we only have about 1/3 observations compared with the same number of itera- tions of the coupled setting because only one of the objec- tive functions is observed). Due to the package differences (PFEV is based on GPy, while HVKG is BoToorc)), this comparison is not a completely fair comparison. However, it demonstrates that the proposed method can achieve com- (13) parable or superior performance to a standard package. A more fair and thorough comparison is our future work.
= Er log (x) E A (1/)
+ t + E A [2)
1
Σ x Ag6 (1)))
K
(FgT
+ 11 3 (x) E Avio {b))]
The probabilities pl E A (1) and plf x E Avio H can be analytically calculated. Let C; = (e), uf] x (C) (2 x ..... x (44, w be the cell partitions of the over- truncated region A08. Then, from the independence of fa. we see
531 Pia)) plaa E A
= | E Jos
C
Lus' e (') [ Jc.
p(S=")as='
c
Σ UU' E (e.. u(l) (T((or)) - ((ar))
Σ u E (C, w')) II
Zar,
141
17

---

Parcto-fronticr Entropy Search with Variational Lower Bound Maximization
Best Inference Relative Hyper-vokme
Biost Inferance Relative Hypervolume
0.70
Best Inference Relative Hyper-volume
0.8
0.50
0.7
0.40
0.50
0.35
0.45
Proposed
Proposed KG (BoTorch)
0.5
Proposed
0.40
KG (BoTorch)
KG (Bo Torch)
25
50
75
100
25
50
75
100
25
75
100
Iteration
Iteration
Iteration
(a) ERBL 0.05
(b) ÉRBL = 0.1
(c) ERBL = 0.25
Figure 12. Results on decoupled setting.
G. Joint Entropy Extension
The MAP-based approximation also possible based on transformation of (14):
Let X = {ar f E F*).
The lower bound is
Mi(fai20 X F²) F*) Ex. log {f x F*) p(fz)
Ex.,F pll E A X", F*) log o Hs)
In practice, we only obtain subset X2 S X², by which F3 = {f [fa]æex and His {(x, falff//eeeee are defined. In the variational distribution qff X" ;)), Hs can be seen as an additional training data of the GPs f Then, a natural extension of the variational distribution (3) is
+plfa E A |2 F² log log (x Hs)
Based on the same idea shown in Section 3.3, we approx- imate p(f E A X X²FF)) 2 (æ H (æ Hs). from which the MAP estimator can be defined.
(f Fs)
p(f Hss)
fa E A if ff E A
26 (m)
20 (x)2e))
Figure 13 shows the empirical evaluation for which the set- ting is same as in Section 6. Here, we used the (d, L) = (3, 4) GP derived functions for three length scales. We see that the original proposed method shows slightly better per- formance than the JES extension, though their behaviors are similar.
fff Hss)
20 (w)
otherwise.
where z ( Hs) p(fm E A Hs) and x / Hss) = p(f.. A Hs). Note that here simple addition
the conditioning on H.s is interpreted of the training data, and does not impose the conditions that "Hs is the optimal solutions", which is represented by the truncation (Because of this reason, we do not give to Hg). The resulting lower bound is
H. Noisy Observation
Let v/e = S& + E and yæ = ........ - where NO0 (00,0 noise)) The mutual information for noisy observa- tion is Ml(y.;F"), for which the lower bound can be de- rived
Ex F f- log S (x) Hs)I(f A
F' i1x (x Hs) E Ario (14)
MI(ya:FF
Ex ue log (11 Fr)
plya)
where 2-'
Hs)
and
(m)
% (m)
DRL. (p(U F
1/y0 (F))
Tx (x H.s)
The same MC approxima-
Ey vn log 9/y0 F
zy 3 Ha)H))
Ha)
tion (6) can be used evaluate (14).
plyx)
18

---

Pareto-frontier Entropy Search with Variational Lower Bound Maximization
1.0
0.6
0.9
Relative Hypervoluune
0.5-
Reletive Hypervolume
0.7
Reetivv Hygervolume
0.6
0.7
0.4
3..6
0.5
0.3
0.4
0.4
0.3
Proposed Proposed_ES
Proposed Proposed_ES
0.3
0.2
0.2
Proposed JES
25
50 Iteration
100
25
50
75
100
25
75
100
Iteration
Iteration
(a) ERBL 0.05
(b) ÉRBL = 0.1
(c) ERBL = 0.25
Figure 13. Comparison with JES extension of proposed method.
Since F and yn are independent if fa is given, define uy F²) by using (f F²) follows.
where E is set of L-dimension noise samples é from NO (10, Pooisn I), Vn fa. E is sample of V and Kg |E]. Note that, in this approximation, we transform EF v, to EF lEg. which is possible because of the in- dependence of the noise term. To evaluate this MC approx- imation, we need the conditional distribution p(f yz). written as
yyy F²)
- plUz (f F*)df
= -f plyz f (Zg (F3)
fz 12 ~ ((v((a)
+1 - (1- FF))dfz
-
p(va -f
( (x) E 1g3)
where
m1(x) = m<(x) + of(x)
+ + E dff
(1' - (x(x))
of(z) + o
= - Plu Plua. fa (x)dff
0
si(x) = =ff(x) of (x)) o noise
*
Plyx: f
Therefore, p(fa A 5s ju) can be calculated by using
the same procedure as 2 fs
plf E A shown in (9):
-p(ya) ) 5 (x)
p(fa ynd
SEEA
(
PSo vald
plfa E A F0 ())) =
Σ
/
p(fa
ddf
i=1 C:
= (5t E A 1y,)
Σ
p(fa Wi
+ ( A yz)
ΣI (Ba.)) - +((,))
Then, the MC approximation becomes
Ex. te log { FF (x)p(Sa E A ya)
where B,, = (E) - vi (x))) and B., = (ut
+ F3 (z)p(f Aro ym)}
For pifaa E A Uz), we can use a rela-
KK, Σ
Σ log f (x)p(ff E A Un)
tion p(f. E Ario V.) = p(f. E A
(ia) p(fa E
(F..NEFFEE BEE
A (m). The last term p(f, E A
(i)) can be cal-
culated by the same approach as Z = p(fa E Aa7)).
Fs
(x)p(( Avio {m)
described in the end of Appendix B.
19

---

Parcio-fronticr Entropy Search with Variational Lower Bound Maximization
Figure 14 shows the empirical evaluation for which the setting is same as in Section 6.1. Here, we used the (d, L) = (3,3) GP derived functions for three length scales. We added the independent noise (0, maise) with Tnoise = 0.1 to all the observations. The GPs have the same value of the noise parameter noise = 0.1. We compared with EHVI, MOBO-RS, and JES. EHVI and MOBO-RS can handle the noise just by incorporating it into the surro- gate GPs. JES (Tu et al., 2022) considers the information gain from noisy observations. PFEV shows similar results to its noisy observation counterpart. We empirically see that F*); sufficiently works even when the obser- vation contains a moderate level of noise. The investigation under stronger noise is our future work.
Lobato et al., 2016). PESMO defines an acquisition func- tion through the entropy of the set of Pareto-optimal lutions ø, which is based on complicated approximation by expectation propagation (EP) (Minka, 2001). On the other hand, Belakaria et al. (2019) proposed using the individual max-value entropy of each objective function, called max- entropy search for multi-objective opti- mization (MESMO). This largely simplifies the computa- tions, but obviously, information of the Pareto- frontter is lost. Another JES based approach has been recently pro- posed (Fernández-Sáncher & Hernández-Lobate 2024), which is in a more general formulation including multi- fidelity and constrained problems. Their computations are based on the Gaussian based entropy approximation, whose validity remains unclear.
I. Additional Discussion on Related Work
J. Detail of Experimental Settings
Although our main focus is on information-theeorticcc ap- proaches, here, other criteria are also reviewed. A clas- sical approach is the scalarization that transforms multi- ple objective functions into a scalar value, among which ParEGO (Knowles, 2006) is well-known method based on random scalarization. However, the information of the Pareto-frontier may be lost by the scalarization. The standard expected improvement (EI) has been extended based on measuring the improvement of the hyper- volume called EHVI (expected hyper-volume improvement) (Em- merich, 2005; Shah & Ghahramani, 2016). The GP up- per confidence bound (UCB) is another well-known gen- eral approach to BO (Srinivas et al., 2010). UCB based MOBO methods have been studied (Ponweiser et al., 2008; Zuluaga et al., 2013; 2016), but the setting of the con- fidence interval sometimes becomes practically difficult. SUR (Picheny, 2015) is based on the reduction of PI af- ter the querying, which is computationally quite expensive. A hypervolume-based multi-objective extension of knowl- edge gradient (KG) is considered by (Daulton et al., 2023). Naive computations of the hypervolume KG are compu- tationally intractable, and several approximation and ac- celeration strategies have been studied. For example, so- called one-shot strategy transforms the nested optimization into simultaneous optimization which makes computation much faster, but the dimension of the acquisition function optimization becomes high.
Bayesian optimization was implemented by a Python pack- age called GPy (GPy, since 2012). PFEV, PFES, and [PF) require C, that are hyper-rectangles decompos- ing the dominated region. To obtain C., we used quick hypervolume (QHV) (Russo & Francisco, 2014), which is efficient recursive algorithm originally proposed for the Pareto hyper-volumt calculation. In PFEV, we maximize X for each given æ by calculating LB(xr, >) for 11 grid points of X (10-3 0,1,0,1.1,, 0.2, ........ 1.0). In RFM used for sam- pling F (required in PFEV, PFES, MESMO, (PF) and JES), the number of bases D was 500. The GP hyper- parameter onoise is fixed as 10-4 In ParEGO, the coef- ficient parameter in the augmented Tehebycheff function was set 0.05 as shown in (Knowles, 2006). EHVI, the two reference points are required, shown as Uref and wyet in (Shah & Ghahramani, 2016). The worst point vector Dref is defined by subtracting 10-4 from the vector consisting of the minimum value of each dimension of y; in the training data. On the other hand, the ideal point vector Wrer is de- fined by adding to the vector consisting of the maximum value of each dimension of y,-
The definition of each benchmark function is as follows.
Fonseca-Fleming (Fonseca & Fleming, 1995)
We focus on the information-thheoreicc approach, which was first proposed for single-objective BO (Hennig & Schuler, 2012; Hernández-Lobato et al., 2014; Wang & Jegelka, 2017). Recently, (Cheng & Becker, 2024) pro- posed a different variational lower bound approach to singlc-objective BO, which is only for single-objective problems. A seminal work in the information-theone approach MOBO is the Predictive Entropy Search for Multi-Objective Optimization (PESMO) (Hernandez-
d
1
fi(x) = exp
Vd
i=1
d
27
f2(x)) = exp
I
+
Vā
subject to
- -4 x, S 4, E
20

---

Pareto-frontier Entropy Search with Variational Lower Bound Maximization
0.8
0.6
0.9
0.7
Reletive Hypervol volume
Relative Hyper-volume
Relative Hyper-ellmm
0.8
0.5
0.6
0.7
0.4
0.5
0.6
MOBO-RS
MOBO-RS
EHVI
0.4
EHVI
0.5
EHVI
0.3
JES
JES
JES
Proposed
Proposed
0.4
Proposed
Proposed_noist
Propesed_nos noisy
Proposed noisy
0.2
25
50
100
25
50
75
100
25
50
75
100
Iteration
Iteration
Iteration
(a) ERBL 0.05
(b) ÉRBL = 0.1
(c) ERBL = 0.25
Figure 14. Results with noisy observations in GP-derived synthetic functions (average and standard error of 10 runs)
Kursawe (Kursawe, 1990)
FES3 (Fieldsend et al., 2003)
2
f1(x)
- exp (0.22/03 +
exp (()
0.5
i=1
f((x) Σ 2,
3
f2(x))
Σ
||00 0.8
+ 55sin(;))]
fz(a) Σ
I4 sin(t - 1)? cos(i - 1)?
subject to
n; 5, 1 15i53.
d
Viennet (Vlennet et al., 1996)
fa(x) Σ
|x (0.25 cos(i 1) cos(2i
1=1
fi (I, = 0.5 (x² + x²) + sin (x² + y²²)
0.5)
f2(x,y)= (3r 2y + 4)²
(r y + 1)2 15
fa(x) Σ X1 sin [1000 ;) - 11
27
fa(x,0)= x² + 22
1.1exp (-(x² + 1²))
subject to 0 x S1, i (d].
subject to - -35 x, 3.
K. Additional Results of Empirical Evaluation
FES1 (Fieldsend et al., 2003)
(i/d)2 0.5
K.1. Additional Results on GP-derived Synthetic Functions
f1(x)) =
Σ
Fy exp
Additional results on GP-derived synthetic functions are shown in Fig. 15-17. The results are all combinations of CBBF {0.05, 0.1, 0.25). L E {2,3, 4, 5, 6). and d E (2, 4} (Note that the results in the main text Fig. 3 are also included). We here also show results of ParEGO, MOBO-RS, and MESMO, which are omitted in the main text. The same format of boxplots as Fig created from all the results are shown in Fig. 18. We also examined an put dimension wise boxplot in Fig. 19 which differences of all the methods gradually decrease with the increase of the input dimension. We further plot separately for each length scale setting of the true objective function in Fig 20. We see that, particularly for small length scale problems (which tends to be multi-modal functions), differences be- come small. We speculate that the differences were less
10mi
[2(2) = Σ
0t 0.5 cos
- 0.5
d
1=1
subject to 0 24 < 1,
i e [d].
FES2 (Fieldsend et al., 2003)
10mi
[1(z) =
I 0.5 cos
0.5
f2(x)
xi sin ((i 1) c)cos( 1)1u
fa(z)) Σ
Its 0.25 cos(i 1) cos(22- 2) 0.5
1m1
subject to 05 24 1, [d)
21

---

Parcio-frontier Entropy Search with Variational Lower Bound Maximization
apparent for challenging problems with high-dimensional and highly multi-modal functions.
Further, Fig. 21 is performance difference between Bayes MAP (8) and naive MC (6).
22

---

Parcto-fronticr Entropy Search with Variational Lower Bound Maximization
0.5
2.0
0.9
6.0
02
Nalatea hope-oo
0.7
0.7
NoSo-as
0.6
0.7
06
EMM
PereGo
0.8
JtS
0.5
0.5
MES/O
F5S
0.4
0.5
NESHO
0.3
PFZES
044
PFZES
0.4
RF255
SF2E5
FF265
Random
Randem
0.3
Random
Sandom
Random
Proposed
03
Propozed
Propesed
Procosed
Proposed
100
50
73
100
50
150
50
75
100
100
(a) (I, (RBF) = (2, 0.05) (b) (I, ENB)) (3, 0.05) (c) (I, LNB = (4, 0.05) (d) (L, CRBF) (5, 0.05) (e) (L, (RBF) = 0.05.
1.0
1.0
0.5
0.5
Hyperssluns
0.8
Perfico
KOSO
0.7
0.7-
MCB0
0.6
0.7
0.
ME5/0
0.5
C
0.5
PIZES
64
b.c
212E5
Random
Randam
Bandom
Sandom
0.5
Proposed
Proposed
Proposed
0.2
Proposed
Proposed
100
25
50
75
100
150
50
75
300
100
(1) (L, (RBF) = 02,0.1) (g) (L, (RRFR = 0.10 (h) (L, (L, (4,0.1) ( (RBF) = (5.0.)) ① (L, (RAF) = (6,0.1)
1.0
2.9
0.8
0.7
belanve
0.6
0.5
0.6
Proposed
PFIE5
0.5
Rondom
Propossdd
0.5
0.0
PF2ES
Proporsed
0.2
Ronddom
FF2E5
PrES
Proposed
25
50
1.0
300
(k) (L, (RBP) = (2,0.25) ()) (L, (RBF) = (3, 0.25) (m) (4, 04,0.25) (n) (L, (REF) (5, (0) (L, (RBF) = 0.,))
Figure 15. Results on Ge-derived synthetic function 2)
23

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Parcto-frontier Entropy Search with Variational Lower Bound Maximization
0.8
0.6
Retartur Hypee volumee
0.0
0.7
Por650
2.5-
Detetire tyyaeroau
0..5
Nosp
0.5
NDBO-It
0.6
0.4
Brid MESWO
PareCO
2..4
Pereso
NESMO
C4
033
JtS
JES
J5S
ITS
163
21ES
PIES
0.4
Random
PFZES
Random
PF2E5
RF255
Random
0.2
Sandom
FF2E5
0.2
Random
#F265
Proposed
Proposed
Propesed
Procosed
Proposed
0.2
100
25
50
75
100
50
150
50
100
50
100
(a) (I,, RBB) = (2, 0.05) (b) (I, ENB)) (3, 0.05 (c) (I, LNBr) = (4, 0.05) (d) (L, CRBF)
(5, 0.05) (e) (L, {RBF) = 0.00.
0.9
0.0
2.8
0.0
0.7
0.7
0.7
0.8
0.7
NOS0
0.6
6.5
0.6
Reative
JES
MESKO
0.4
Random
PIZES
Randam
0.3
Bandom
6.3
Sandom
21255
MES
0.3
0.2
Random
Proposed
0.3
Proposed
0.2
Proposed
Proposed
Proposed
100
25
50
75
100
150
50
75
100
50
100
(1) (L, (RBF) = 02,0 (g) (L, (RRFR = (3, 0.1) (h) (L, (EBFR = -4,,0) ( (L, (RBF) = (5.0.))
① (L, (RAF) = (6,0.1)
1.0
1.0
0.3
0.9
1483000 Seprratam
0.8
0.2
0.8
0.8
Pelanve Hyper
0.7
0.
0.5
055
0.5
54
PFIES
IF2ES
FF255
PF2E5
0.5
Rondom
0.3
Ronddom
Proposed
Propossed
324
Proposedd
0.2
Propoeed
0.2
Proposed
25
50
1.0
300
Itoraion
(k) (L, (RBP) = (2,0.25) ()) (L, (RBF) = (3, 0.25) (m) (L, (4.F))(444.022 (n) (L, (REE) = (5, (0) (L, (aBF) = 06,).2
Figure 16. Results on GP-derived synthetic function 3)
24

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Pareto-frontier Entropy Search with Variational Lower Bound Maximization
0.6
0.7
0.6
Rebittie wyper vo uime
0.5
0.5
0.6
No50-45
Nbeo-Rt
C4
0.3
BRWI
BMM
Pareo
Pereso
0.5
JTS
0.4
JES
0.3
F5
WESHO
0.4
MESSO
IES
PIES
0.4
Random
PFZES
PFZES
Randem
0.2
RF255
Random
Sandom
FF2E5
Random
PF265
Proposed
Proposed
Propesed
6.1
Procesed
Proposed
100
25
50
75
100
50
150
50
100
50
100
(a) (I,, (BB) = 0.05) (b) (I, ENB) (3, 0.05) (c) (I, LNBR) = (4, 0.05) (d) (I, CRBF)
(5, 0.05) (e) (L, {RBF) = 0.00.55
0.8
0.2
0.7
0.7
Reforve Myoemuol Luee
0.00
0.6
6.6
0.7
8.5
0.5
0.0
0.5
MCB0
0.5
JES
D.3
HES
0.4
Bandom
PIZES
Randam
3.2
R2255
Bandom
6.
Sandom
21255
P1PE5
Proposed
02
Propossd
Proposed
0.1
Proposed
Proposed
100
25
50
75
100
150
50
75
100
100
(1) (L, (RBF) = 02,0 (g) (L, (RRRR = (3. 0.1) (h) (L, (RBFR (4,0..1 (1) (L, [RBF) = (5.0.)) ( (L, (RAF) = (6,0.1)
1.0
0.9
0.0
2.0
0.8
0.7
6.7
0.5
0.7
0.6
0.6
0.0
petavve
0.5
2.c
0.4
0.4
PrES
0.5
PFIES
Rondor
0.3
PF2E5
0.3
Rondom
FF255
PF2E5
Proposed
Propossed
02
Proporedd
Propoeed
Proposed
25
50
1.00
300
(k) (L, (RBP) = (2,0.25) ()) (L, (RBF) = (3, 0.25) (m) (L, (LDF))(444.22 (n) (L, (REF) = (5, (0) (L, (aDF) = 06,).2
Figure 17. Results on Ge-derived synthetic function 1)
25

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Parcio-frontic Entropy Search with Variational Lower Bound Maximization
Paredo
RG
MESMO
FFEB
PF2E5
Praposed
88
24
66
24
06
06
64
24
24
94
Q4
02
22
22
02
02
(a) L
(b) L
(0)
L
(d)
L
(e)
L
Figure 18. Boxplots of RHV at 100-th iteration.
Par8oo
MOBO
MESMO
PFES
PFZES
Proposed
12
1.2
1.2
0.8
0.6
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.2
(a)
(b)
3
(c)
Figure 19. Boxplot of RHV 100-th iteration for each input di- mension.
Diff average RHV
0.0
ParEGO
MOBIORS
PFZES
Proposed
1.2
-0.1
1.0
1.0
-02
0.8
0.8
0.6
0.8
0.8
50
75 Heeraiion
100
0.4
0.4
0.4
0.2
0.2
Figure 21. Boxplot of difference of average RHV between Bayes MAP and naive MC (Bayes MAP minus naive MC). The differ- ences of the 10 runs average RHV are shown for BO iterations 50, 75, and 100. The GP-derived functions of d E {2,3, 4), L {2, 3, 4}, and bNBF E {0.05, 0.1, 0.253 were used (i.c., x points for each iteration). Since positive values indicate Bayes MAP is better than naive MC, we see that Bayes MAP shows slightly better performance across iterations.
(a)
2, <RBF
0.05 (b)
2, <BB 0.1 (c)
2, (RBF 0.25
1.2
1.2
1.2
1.0
1.0
0.8
0.8
0.2
0.2
(d)
ARBF
0.0G (e)
(f)
0.25
1.0
1.0
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
(g)
4. (RFF
0.00 (h) 4, <RBF
u.1 (i) à fRBF 0.26
Figure 20. Boxplot of RHV 100-th iteration for each input di- mension and length scale.
26

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Parcto-frontic" Entropy Search with Variational Lower Bound Maximization
0.9
0.0
0.6
Reative Hypervolume
0.0
Reggive Rypervoluuee
Roletvve hyper-volure
0.0
ParEGO NO30-RS
0,7
FaréGo MORO-RS
EHvi
EHVt
9.8
0.4
MESSYO
0.6
MESMO
MOBO-RS RS
EHVI
MOEO RS
EHVI
PFZES
0.5
PFZES
0.0
JES
JES
Random
Random
PFES
PFES
Proposed
0.4
Proposed
25
75
100
50
75
100
0.5
Proposed
Proposed
Iaaaiton
Iterationn
100
75
100
Iteration
Iteration
(a) Ponseca-Fleming
(h) Kursawe
(a) FES1
(b) FES2
1.0
0.9
0.7
Natativa taperookuna
0.6
0.7
MO30-45
MESMO
EAVI
0.5
MESMO
EIM
MGBO-HS
Relative sypervolums
0.6
0..7
6.6
JES
PIES
JES
PIES
EHVI
MOBO-RS
Relative
0..5
MOBO
PariGG
0.4
MUSMO
ВHVI
Proposed
Proposod
IES
MESMO
0.4
25
75
100
25
50
75
100
0.3
PFES
PF2ES
0.3-
PFZES
Propased
Proposed
(c) Viennet
(d) FES3
100
1.0
0.8
(c) FESI+ Kursawe
(d) FES2 Kursawe
Rabotie Hyporypuure youre
3.6
0.7
Figure 23, Performance comparison in additional benehmark functions (average and standard deviation of 10 runs).
0.6
0.6
PerfiO
Pertfo MOBO AS
HO30
044
from d 5 to 10. All experimental settings except for the input dimension identical to those in the main text. The results show that our method demonstrates sta- ble optimization performance the input dimension in- creases, similar to the cases presented in the main text. Ad- ditionally, we evaluated our method on the DTLZ bench- mark suite with (d, L) = (3,3), as shown Figure 27, On DTLZ2, DTLZ5, DTLZ6, and DTLZ7, our proposed method achieves higher RHV compared to other methods. For DTLZI and DTLZ3, all methods show similarly high RHV values (close to 1) with small differences between them. In the case of DTLZ4, EHVI and MESMO demon- strate better performance, while our method performs com- parably the remaining baselines.
P12E5
P12ES
Random
Random Procosed
Proposod
75
100
75
100
Iee Lerotion
Itoration
(e) Fonseca Viennet
(f) FES3 Kursawe
Figure 22. Performance comparison in benchmark functions (av- erage and standard deviation of 10 runs).
K.2. Additional Results on Benchmark Functions
Figure 22 presents an extended version of Fig. 5 from the main text, including additional baseline methods (ParEGO, MOBO-RS, and MESMO). Figure 23 shows additional results benchmark functions. FEST and FES2 are (d, L) = (3,2) and (d, L) (3,3), respectively. There- fore, Kurs+Kursawe and TES2+Kursawe are (d, L) (3, 4) and (d. L) = (3, 5), respectively. The experimen- tal setup is identical to that Section 6.1. These additional results further demonstrate our method's superior perfor- mance across various problem configurations.
To assess the scalability of the proposed method, we ex- tend our experiments to higher input dimensions. Fig- ures 24, 25, and 26 display the results for FESI, FES2, and FES3 benchmark functions with input dimensions ranging
27

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Parcto-frontier Entropy Search with Variational Lower Bound Maximization
0.70
0.0
0.65
Deletive Hoper-slomee
Polatve hypervelume
0.60
Relative Hypor
0.7
0.6
0.55
ParLGGO MOBO ВНИ
PartGO
PertiGO
30.50
Pari
0.6
EHVI
0.5
EHVI
Retelive
0.45
EHVI
MESMO
MESMO
MESMD
MESMO
0.5
PFES
0.40
PIZLS
P/ZLS
PIZES
0.35
PY 21S
Proposed
Proposed
Proposed
Proposed
0.4
25
50
75
100
50
75
100
75
100
50
100
Ieration
Iteration
Iteration
Iteration
(a)
5
(b) d
d
(h)
0.9
0.65
0.65
0.80
0.60
0.00
Relatvv hypercvllme
0.8
Reladiv Hypervoumee
0.75
0.55
0.53
0.70
0.50
0.50
0.7
ParECO MOBO-RS
ParEGo HOBO RS
0.45
RarEGO
0.45
FarEGO
ЕНИ
0.60
EHVI
Delabve
0.40
MOBO RS
EHVI
Reaaue
EHVI
MOBO-RE
FFES
PFES
0.35
PFES PFZES
PFES
PF2ES
PFZES
0.90
PF2ES
Proposed
Proposed
Proposed
0.25
25
50
75
100
25
50 Iteration
75
100
25
50
75
100
25
50 Iteration
75
100
Beration
Iteration
(c) 7=
(d) d
(c) d 7
(d) =
0.65
0.75
0.8
0.60
0.70
0.7
0.55
Palatwo Hysaravoluma
13.6
0.65
Weletee Hygee-
0.50
0.5
0.45
MUBO-HS
MOBO-RS
MOIrO-RS
MOB0 RS EHVI
EHMI
0.50
0.35
PFES
PFES
PFES
PFES
0.45
AF2ES
0.4
PF2E5
0.30
AF2ES
0.3
Proposed
Proposed
Proposed
Proposed
0.40
50 Iteration
25
25
75
50
75
100
Ieraaion
Iteration
Iterat
(c) d = 9
(f) d = 10
(c) d = 9
(f) d 10
Figure 24. Results on FESI.
Figure Results on FES2.
28

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Parcto-frontic" Entropy Search with Variational Lower Bound Maximization
0.7
0.6
Peladive Hypoer Hyporrvolene volumee
0.5
ParLGGO MOBO
PartGO EHVI
0.4
ВНИ
0.4
MESMO
MESMO
0.3
PIZLS
P/ZLS
Proposed
Proposed
25
50
75
100
50
75
100
Ineration
Iteration
(a)
5
(b) d=
0.65
0.55
Reladive Hyperseohuns
0.50
ParECo
ParEGo
0.40
MOBO-RS
MCBO-RS
РНИ
EHVI
0.30
FFES
PFES PFZES
PFZES
0.25
Proposed
0.2
Proposed
25
50
75
100
25
50
75
100
Ieeraion
Iteration
(c) 7
(d)
0.7
0.6
0.5
0.5
ParEGO
MOBO-HS
0.4
MOBO-R5
0.3
PFES
PFES PF2E5
AF2ES
0.2
Proposed
0.2
Proposed
25
Bleation
Ieeaaion
(c) d = 9
(f) d = 10
Figure 26. Results on FES3.
29

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Parcio-frontic Entropy Search with Variational Lower Bound Maximization
K.3. Computational Time
1.00
1.0
We here examine the computational time of PFEV. We used the GP-derived synthetic function (d 3) from Sec- tion 6.1. The training dataset randomly selected 50 points, and we evaluated the computational time (CPU time) required to calculate the acquisition function values for 100 points generated by the Latin hyper cube sampling. In PFEV, we evaluate time of calculating 11 grid points of À (10-3, 0.1, 0.2, ........ 1.0) for a given æ. The results are shown in Table 1, in which we evaluate ParEGO, PFES, EHVI and PFEV (Proposed).
0.90
Rolatve hyger-aduure
Relative Hygarualune
0.98
0.8
0.97
ParEGO MORD-RS
0.7
ParEGO MOBD-RS
0.96
MESMO
MESMD
0.95
FFES
PFES
PFZES
0.5
PFZES
0.94
Proposed
Proposed
25
75
100
0.4
50
75
100
Iteration
Iteration
(a) DTLZI
(b) DTLZ2
Obviously, ParEGO was quite fast because it applies the usual single-objective BO to the scalarized value. For L = 3 and 4, EHVI was also fast, but it becomes slow at = 5 and we stopped at = 6 because of the long computational time. The slow computation of EHVI at large L is widely known. PFEV and PFES were similar up to L = 4, but for L = 5 and L 6, PFEV was slower than PFES. This is because of the computation of QHV. For == 3 and L = 4, the computational time of NSGAlI was dominant in PFEV (Note that the same procedure is also performed in PFES). However, for = 5, QHV requires the similar cost as NSGAII, and for L = 6, the compu- tational cost of QHV became much larger than NSGAIL. PFEV requires QHV two times for each sampled Pareto- frontier (see Appendix B) while PFES requires QHV only once for each sampled Pareto-frontier. This was major rea- son of the difference of PFEV and PFES in = 6.
1.00
Rolativ hyparceluue
0.98
Rulative Hypervelume
1..6
0.96
PartGO
0.5
PartGO
MOBD
MOBD
0.04
EHVI
EMVI
MESMO
MESMO
PFES
PRES
0.92
PF2E5
PIEES
Proposed
0.3
Proposed
25
100
75
100
Iteration
Iteration
(c) DTLZ3
(d) DTLZ4
0.8
Delative Hi ypoer dorteme
Relativa Hi yooe volumee
0.7
0.6
0.74
PD
MOBO
0.5
PartGG MO8D
L. Difference of Over- and Under-Truncation
EHVI
EHVI
0.0
MESMO PFES
MESMO
PFBS
Here, discuss approximation error caused by replac- ing the dominated region by the entire Pareto-frontier AT with its counter-part created by Fs. i.e., A or A The difference actually can be small if S is sufficiently large, but practically intractable size of S is required even for rea- sonable size of L. To demonstrate this claim, we evaluate differences of hyper-volumes of A and A If the dif- ference of A and A are small, it suggests that the true F can be accurately approximated by F'g-
PFZE5
PIZSS
0.5
Proposed
Proposed
25
50
75
100
25
50
75
100
Iteration
Iteration
(c) DTL.75
(f) DTL.76
1.0
0.9
Relative Hyperookiee
0.8
0.7
RarEGO
In Fig. 28, we empirically investigate the ratio of the vol- ume of under- and over- truncations compared with the true volume. We assume that the Pareto-frontier the simplex (Ee(()) f = 1 and f 2 0), and approximation points F3 are sampled uniformly in the simplex as shown in Fig. 28 (a). The examples of the volume ratio is shown in Fig. 28 (b). In Fig. 28 (c), we see that, when L = 2, the difference between the truncations rapidly decreases, but for L > 3 shown in Fig. 28 (d) and (e), large differences remain even when the population size is 1,000 (Note that in this exam- plc, oven-truncation closer the true value than under- truncation, but in general, it depends on the shape of the
MCBO-RS
0.6
EHVI
0.5
PFES
PFZES
Propesed
0
25
50
75
100
Iteration
(g) DTLZ7
Figure 27. Results on DTLZ (d L =
30

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Parcto-fronticc Entropy Search with Variational Lower Bound Maximization
Table 1. CPU time evaluation (sec).
L = 3
L = 4
L = 5
== 6
ParEGO
0.95 + 0.08
1.15 + 0.06
1.38 + 0.07
1.62
PFES
21.43 = 0.20
29.18 + 0.39
45.23 = 1,41
127.64 = 24.44
EHVI
0.86 - 0.06
3.51 + 2.24
500.09 = 336.18
Proposed
Total
21.93 + 0,42
31.74 0.64
61.04 + 3.26
274,49
28.99
CALCPFEV NSGAII
0.59 = 0.05
0.13
5.04 0.49
24.76 + 3.71
19.59 0.37
25.66
31.36 + 0.27
37.26 + 0.22
RFM
0.59 0.00
0.80
1.00 0.00
1.19 + 0.00
QHV
0.42 0.03
2.72 + 0.26
22.41 + 2.65
209.81 25.68
true Pareto frontier which is unknown). The continuous Pareto-frontier is the (L dimenssioaa space, and we conjecture that the required number of points to keep the sufficient density approximation points F& increases ex- ponentially when increases.
About the difference between J and F3, increasing the number of MC samples cannot mitigate the essential prob- lem in principle. In our formulation, the over-truncation corresponds to X = O. Therefore, the over-truncation based lower bound is biased toward a smaller value, because the bound defined as the maximizer with respect to A. This discussion holds even the population (a, X) (i.e., be- fore introducing the sample approximation), and therefore, increasing the number of samples from the GPs does not solve this intrinsic bias.
M. Consideration Gradient-based Optimizer
Figure 29 shows results with NLogNEHVI (Daulton et al., 2020; 2021; Ament et al., 2023) for GP-derived func- tions (d = 3, L 3, 4, 5) and four benchmark func- tions. For gLogNEHVI, we used gLogNoisyExpected Hlyperolumellmrroveenntt of BoTorch. The reference point is defined by Vmin 0.1 - Vmiu), where Vmin and Ymax vectors consisting of the minimum and the maximum of each dimension in training y,, respec- tively. For qLogNEHVI, we used both the gradient descent (optimi ze accq function of BoTorch) and DIRECT for the GP-derived functions. We see that the gradient opti- mizer slightly improves the results, but DIRECT also has the similar performance. Overall, qLogNEHVI shows good performance, while PFEV also shows comparable perfor- mance. Although implementations tent (PFEV is GPy base and gLogNEHVI is BoTorch), we can see that performance in our results is not largely differ- ent from the well-known package.
not exactly consis-
31

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Parcto-frontier Entropy Search with Variational Lower Bound Maximization
f3
f2
f
ft
(a) Illustration of 3.
(b) The difference between two truncations is evaluated by the volume of the colored region defined in the L- dimensional unit cube [0, 1] When L = 2, the true volume is 1/2. In the plot (c), the areas of the orange and the blue regions are compared (each of which are divided by 1/2 (c), respectively).
The true Pareto-frontier is
assumed to be the simplex. The discretized points (red points) are sampled from uniform distribution on the simplex (Dirichlet distribu- tion).
1.15
Over truncation
Over truncation
Ratio VS true volume
1.10
Under truncation
Ratio true volume
2.0
Under truncation
1.05
1.5
1.00
0.95
1.0
0.5
0.85
200
400
600
800
1000
200
400
600
800
1000
Num. of Pareto optimal points
Num. of Pareto optimal points
(c) I =
(d) T. 3
8
Over truncation Under truncation
Ratio vs true volume
4
200
400
600
800
1000
Num. of Pareto optimal points
(e) L 4
Figure 28. Hyper-volume relative differences between over- and under- truncations. The volume defined in the unit cubc [0, 1 The vertical axis is the ratio compared with the true volume.
32

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Parcio-fronticy Entropy Search with Variational Lower Bound Maximization
0.9
0.8
Relative Hyper-volume
0.7
0.6
EHV
HESHO
PF2ES
0.4
@laghEHVil@radd RUOENEINVDDRECTI
Proposed
25
50
75
100
Iteration
(a) I.
0.9
0.8
Relative ilper-volume
0.7
0.6
ParEGO
0.5
MESHO
0.4
glogNEMW/1Grad)
0.3
şloyMPHvjDBFCT
0.2
Proposed
25
50
75
100
Iteration
(b) L
Relative Hyper-volume
0.8
0.6
MOBO RS HESHO
0.4
glogNEWW/lGndl
BLOONEHVIDRBECI
0.2
Random
25
50
75
100
Iteration
(c) L -= 5
Figure 29. Comparison with NLogNEHVI.
33


Reviews:
[{"review_id": "Zt9vzQzaDX", "reviewer": "Reviewer_SDSC", "summary": "The paper considers an information-theoretic acquisition function -- namely Pareto Frontier Entropy Search -- for multi-objective Bayesian Optimization. The authors propose a novel \"variational\" approximation to the predictive distribution given the Pareto frontier based on over and under truncation of the Pareto set. This results in an acquisition function that requires jointly optimizing over the (one-dimensional) variational parameter and the query location(s). The authors describe the implementation and computational details and discuss useful extensions. Finally, the paper demonstrates the empirical performance of the proposed approach in a number of benchmarks against common baselines.\n\n### Update after rebuttal.\nMy overall assessment of the paper remains the same. The authors' response to my review clarify some questions but it also highlights that relying on a gradient-free optimization method for the acquisition function optimization has its limits. I don't think this should make or break the paper if it's sufficiently discussed in an updated version (the additional results generated for the rebuttal should help with that); ideally providing a continuous relaxation of the indicator as discussed in the authors' response. I think this is a solid paper and \nthink it should be accepted, but would not revolt if the majority of other reviewers feels differently.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your constructive comments.\n\n1) HVKG in Appendix F: The linked figure ([link](https://anonymous.4open.science/r/AuthorResponse-0652/Fig20-Decoupled.pdf)) shows a comparison with BoTorch HVKG (called qHypervolumeKnowledgeGradient, which is based on the well-known 'one shot' optimization of KG) in the decoupled setting (Appendix F). Because of the long computational time, we set the number of Pareto optimal points in the one-step ahead posterior mean as 10 in HVKG. For fair comparison, the proposed method also set the size of the Pareto optimal points in the sampled function (i.e., the NSGA-II population size) as 10. The number of samplings (so called fantasy points) in HVKG is also set as $10$. Although there is the implementation difference (the proposed method is the GPy based implementation), we consider the result shows the performance of the proposed method is sufficient compared with an existing package.\n\n2) qEHVI: We performed evaluation with qLogNEHVI. See response #4 of reviewer QnYc.\n\n3) Higher input dimension: For FES1, 2 and 3 benchmark functions, results on the $10$ dimensional input are shown in Figure 19 of Appendix K.2. We further show the results on $d = 5, \\ldots, 9$ of the same benchmark functions [here](https://anonymous.4open.science/r/AuthorResponse-0652/FES.pdf), which were performed before the review becomes open. The results indicate that the proposed method stably performs optimization similar to the cases of the main text. Evaluating more higher input problems on realistic scenarios is one of our future work.\n\n4) About DIRECT: We employed DIRECT because it does not require `initial points' unlike the gradient descent that requires the appropriate setting of the initial points (the number of initial points and locations). Our purpose is to focus more on differences of the acquisition functions and to reduce the other factors effecting the performance. On the other hand, evaluation using gradient-based approaches is also important future work because it is widely used in BO. We partially show comparison with a baseline using the gradient optimization (qLogNEHVI) in response #4 of reviewer QnYc.\n\n5) About mixture: As the reviewer suggested, it is possible to use a mixture depending on the location (e.g., defining $\\lambda$ as a function of $\\boldsymbol x$, which can be estimated by the variational lower bound maximization). We employ the simple global mixture only with one parameter because it should be estimated based on the K samples, which is usually quite small ($10$ in the experiments). Introducing more complicated variational distribution is a possible future direction.\n\n6) Positivity of PFES: PFES calculates an approximate MI through a decomposition as a difference of the entropy. On the other hand, Takeno 2022 et al. show possible negativity of the naive application of MES to the constraint BO, which has a similar formulation to PFES (the entropy difference based MI decomposition and truncation of a multi-dimensional predictive distribution by a box shaped region). This suggests that PFES also might have a risk of having a negative value, but positivity of PFES has not been proven so far. \n\n7) Relation of the input dimension and performance: As the reviewer suggested, there is a tendency for the differences between methods to decrease as the input dimension increases. We confirmed this tendency by the boxplot ([link](https://anonymous.4open.science/r/AuthorResponse-0652/Fig24-25-boxplot_input_dim.pdf)), created by the same way as Figure 4. According to the figure of the first page, differences of all the methods gradually decrease (not only the proposed method). In the second page, we further plot separately for each length scale setting of the true objective function. We see that, particularly for small length scale problems (which tends to be a multi-modal function), differences become small. We speculate that the differences were less apparent for challenging problems with high-dimension and highly multi-modal functions.\n\nWe will also reflect other comments such as plotting and references.", "claims_and_evidence": "Yes, claims are clear and demonstrated based on convincing evidence.", "methods_and_evaluation_criteria": "Yes, the proposed method is sound and the evaluation is clear and comprehensive (modulo the discussion of higher-dimensional settings, see below)", "theoretical_claims": "Yes, I checked the main claims in the MT and spot-checked some of the details in the appendix.", "experimental_designs_or_analyses": "* Overall the experiments and conclusions drawn are solid. \n* The plots in the MT are all **extremely** hard to read. Please consider moving some of the technical details in the MT into the appendix to make room for more readable plots. You may also want to select a few representative results and relegate the rest of the empirical results to the appendix.\n* In a similar vein as above, the comparison to HVKG in Appendix F seems potentially problematic due to the simplifications in the implementation: \"we evaluate the posterior mean hyper-\nvolume after adding sampled fl(x) into the GPs only by using the pre-defined grid points in X (uniformly taken 5 points in each dimension is used\"). This feels like comparing an optimized implementation of PFEV against a simplified implementation of HVKG. A comparison against the implementation in [A] would be more meaningful.\n\n[A] S. Daulton, M. Balandat, and E. Bakshy. Hypervolume knowledge gradient: A lookahead approach for multi-objective Bayesian optimization with partial information. In  International Conference on\nMachine Learning, 2023.", "relation_to_broader_scientific_literature": "* The key contribution is a novel variational approximation of the predictive distribution given the Pareto frontier.\n* This formulation is quite elegant and effectively addresses an issue with the approximation that becomes more and more problematic as the number of outcomes increases, an issue that the paper identifies and explains nicely.", "essential_references_not_discussed": "AFAICT the paper discusses all of the essential references in this space as far as the methodology is concerned. But some of the comparisons with the EHVI-based methods do not use state-of-the-art implementations though as discussed in [A,B,C].", "other_strengths_and_weaknesses": "**Strengths**\n* The paper is well-written and clearly identifies the key contributions.\n    * For instance, I really liked the use of Figure 1, this helps a lot with visualizing the over- and under-truncation and makes it a lot easier to follow the paper.\n* The authors provide a comprehensive review of related work. \n* The empirical evaluation setup is solid, and the baselines considered are meaningful - however, some do not use state-of-the-art implementations (see other comments)\n* The performance of the proposed approach especially on problems with many outcomes is demonstrated clearly (albeit only for low input dimensions).\n* The paper has very comprehensive supplementary material that goes into detail on many relevant extensions / variants of the main contribution (parallel, decoupled, joint, noisy).\n\n**Weaknesses**\n* The primary weakness of the approach is that it uses gradient-free optimization of the acquisition function (in this case the DIRECT algorithm). These methods are known to perform very poorly in higher dimensions, which presumably is the reason why the largest evaluated input dimension is d=3 (d=4 in the supplementary material). This weakness / limitation of the approach is currently not discussed at all in the paper. \n    * I would like to see some ablations of PFEV on higher-dimensional inputs compared to approaches that use gradient-based optimization of the acquisition function such as qEHVI/qNEHVI from [B]/[C] (assuming that EHVI as evaluated in the paper also used DIRECT).\n\n[B] S. Daulton, M. Balandat, and E. Bakshy. Differentiable expected hypervolume improvement for parallel multi-objective bayesian optimization. In Advances in Neural Information Processing Systems, volume 33, 2020.\n\n[C] S. Daulton, M. Balandat, and E. Bakshy. Parallel bayesian optimization of multiple noisy objectives with expected hypervolume improvement. In Advances in Neural Information Processing Systems, volume 34, 2021.", "other_comments_or_suggestions": "* \"Note that if VDC is not satisfied, L(x) is not defined because of log 0\" <- \"not defined because of log 0\" is a rather odd statement; this not being defined is simply b/c p would not be absolutely continuous w.r.t. q.\n* \"number of samplings\" -> number of samples", "questions_for_authors": "* Your approach uses a \"global\" mixture distribution for the PF truncation. Is it possible / would it be advisable to do have this mixture depend on the location (in the outcome space)? I'm thinking of settings in which over-truncation is more accurate in one part and under-truncation is more accurate in another part of the outcome space.\n* Positivity of PFES has not been clarified - what does this mean? Is this about the formulation or about the approximation / algorithm?\n* The results in Fig 3. show a pattern where PFEV and MOBO-RS perform quite a bit better than PFES in d=2, but the difference is less for d=3. Is this b/c PFEV is more challenging to optimize as the dimension increases?", "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": "xo9SphevvN", "reviewer": "Reviewer_w3eW", "summary": "This paper introduces Pareto-frontier Entropy Search with Variational Lower Bound Maximization (PFEV), an acquisition function for multi-objective Bayesian optimization (MOBO). It addresses a key limitation in prior information-theoretic MOBO methods, which rely on over-truncation when approximating the predictive distribution hence leading to inaccurate approximations. Instead, PFEV models this distribution as a mixture of over- and under-truncated approximations and optimizes the mixture weight variationally to minimize approximation error. Empirical results show that PFEV outperforms existing methods, particularly in high-dimensional output spaces ($L ≥ 3$), with superior relative hyper-volume performance on both Gaussian process-simulated and benchmark functions.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your constructive comments.\n\n1) About benchmark function: The results on DTLZ is shown [here](https://anonymous.4open.science/r/AuthorResponse-0652/Fig26-DTLZ.pdf). Note that these were performed before the review opening. We see that, in DTLZ 2, 5, 6, and 7, the proposed method shows high RHV among compared methods. On the other hand, in DTLZ 1 and 3, all the methods show high RHV (close to 1), and differences among methods are small. In DTLZ4, EHVI and MESMO showed good performance, while the other methods were similar performance.\n\n2) Mixture approximation: We employ one parameter ($\\lambda$) model, because this parameter should be estimated by the small number of samples (i.e., $K$, for which we used 10). Using more complicated model is possible (e.g., defining $\\lambda$ as a function of $\\boldsymbol x$), but there is a risk of the over-fitting to the small samples. We think that investigating the performance of more complicated variational distribution is one of important directions.\n\n3) Hyper-volume decomposition: In current our experimental setting, the hyper-volume is exactly calculated by a divide-and-conquer based approach (QHV). Therefore, even if the output dimension increases, the accuracy of the decomposition is maintained. On the other hand, the cell-based decomposition can take longer time for a high dimensional output space (e.g., $L \\geq 7$). Then, introducing approximate decompositions is a possible approach. The cell decomposition is performed for calculating cumulative distribution function (CDF) of the predictive distribution. We plan to consider a pruning strategy that stops the divide-and-conquer procedure if the density $p(\\boldsymbol f(\\boldsymbol x))$ is sufficiently small in the (intermediately) decomposed region.\n\n4) Transition of $\\lambda$: The linked figure ([link](https://anonymous.4open.science/r/AuthorResponse-0652/Fig27-lambda-transition.pdf)) shows the transition of $\\lambda$ during BO iterations (the $10$ runs average) for a GP-derived synthetic function ($d = 3, L = 4$). We see that $\\lambda$ takes intermediate values in $(0,1]$, indicating that under- and over- truncated distributions are indeed mixed during BO iterations. As far as we examined so far, any increasing or decreasing tendency have not been observed during iterations.\n\n5) Line14-16: The line 14-16 in Algorithm 1 is the definition of the function `CalcPFEV', which calculates the acquisition function value. This function is used in line 10 of Algorithm 1.\n\n6) $\\lambda$ cannot reach 0: When $\\lambda = 0$, the support condition of the variational distribution (2) is not satisfied, which is required to guarantee that $L(\\boldsymbol{x})$ is a lower bound.", "claims_and_evidence": "- The paper claims the previous approximation is crude", "methods_and_evaluation_criteria": "- The mixtured approach makes sense to me", "theoretical_claims": "- I have checked Appendix A,B,C they all seems plausible to me.", "experimental_designs_or_analyses": "- The synthetic validation on GP samples and more complicated real problems make sense to me.\n- One slightly unconventional thing is the choice of synthetic function to optimize which seems not that common in MOBO, I would be interested in seeing its performance on at least one of the following commonly used benchmarks in contemporary literature:\n    - BraninCurrin\n    - DTLZ series", "relation_to_broader_scientific_literature": "The paper proposes a more accurate truncated Pareto frontier distribution approximation, which could be of interest in information-theoretic acquisition functions for Multi-Objective Bayesian Optimization community.", "essential_references_not_discussed": "All the essential references seem to have been discussed.", "other_strengths_and_weaknesses": "Strength\n- The approach of learning a Pareto frontier truncated conditional distribution is interesting and, to my best of knowledge, novel.\n- An extensive problem setting (decoupled setting, noisy observation, parallel query) has been discussed.\n\nWeakness\n- I think one issue is that the mixtured approximation, though could improve the original approximation, can still be too crude as there is only one parameter, in practice, it could happen is some region is overestimated, some region is underestimated, but this is not able to be faithfully characterized by this metric, hence its practical utility is not exactly clear. \n- While the approach makes sense theoretically, the practical empirical advantage is slightly puzzling me, as hypervolume decomposition is known to be complicated with high objective numbers, which means all of these information theoretical acquisition function, which relies on hypervolume decomposition, ideally is not expected to be leveraged there.", "other_comments_or_suggestions": "the paper in general is very well written, I do not have much further comments. \n\nOne only additional comment is it can be helpful to plot the evolvement of $\\lambda$ w.r.t BO iteration", "questions_for_authors": "- How the learned $\\lambda$ evolve over BO iteration, a plot is crucial to validate \n- What exactly is line 14-16 used for? it seems the BO has stopped at line 13 already.\n- Why $\\lambda$ cnnot reach $0$ in parameterization?", "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": "8uH9iZSYvz", "reviewer": "Reviewer_QnYc", "summary": "This paper considers a method for multi-objective Bayesian optimization based on entropy search. Many existing entropy search methods rely on a discrete approximation of the Pareto frontier (which may be continuous in many cases), which typically leads to over-truncation of the predictive distribution. The paper highlights this problem and proposes an improvement that leverages a mixture of two distributions that over/under-truncate, respectively. The paper proposes a variational lower bound, which can be optimized to jointly select the new candidate point to evaluate and the mixture parameter, and the paper gives two ways of approximating the lower bound with different bias/variance trade-offs. The authors evaluate the approach on synthetic problems", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your constructive comments.\n\n1) Population size of NSGA-II: As the reviewer mentioned, if the population size of NSGA-II increases, the approximation error between over truncation and the true truncation decreases. On the other hand, the continuous Pareto-frontier is the $L-1$-dimensional space, for which the required number of points to keep the sufficient density of approximation points increases exponentially when $L$ increases. In the linked figure ([link](https://anonymous.4open.science/r/AuthorResponse-0652/Fig28-hv.pdf)), we empirically investigate the truncation error by comparing the volume of under- and over- truncations. In this figure, we assume that the Pareto-frontier is the simplex ($\\sum_{i \\in [L]} f^i = 1$ and $f^i \\geq 0$), and approximation points are sampled uniformly in the simplex. We see that, when $L = 2$, the difference rapidly decreases, but for $L \\geq 3$, the large differences remain even when population size is $1,000$. Note that in this example, over truncation is closer to the true value than under truncation, but in general, it depends on the shape of the true Pareto frontier which is unknown.\n\n2) Truncation error and the sampling from GP: In our formulation, the over-truncation corresponds to $\\lambda = 0$. Therefore, the over truncation based lower bound is biased toward smaller value, because the bound is defined as the maximizer with respect to $\\lambda$ (Further, we prove that $\\lambda = 0$ never becomes the maximizer in the response #1 of reviewer 8xyH). This discussion holds even in the population $L(\\boldsymbol x, \\lambda)$ (i.e., before introducing the sample approximation), and therefore, increasing the number of sampling from the GPs does not solve the intrinsic bias.\n\n2) Realistic problem setting: The linked figure ([link](https://anonymous.4open.science/r/AuthorResponse-0652/Fig29-HPO.pdf)) shows the results on hyper-parameter optimization problems of a machine learning algorithm. These experiments were performed before the review available (so, baselines are same as the first submission). The problem setting is the class weight optimization for multi-class classification. We used LightGBM as a base model. We only focus on the optimization of the class weight parameters and thus the input dimension $d$ of BO is equal to the class size. The objective function is the test classification accuracy of each class (i.e., $L$ is also equal to the class size). Overall, the results on the four datasets indicate that PFEV has sufficient performance compared with other methods.\n\n3) Higher input dimension: See response #3 of reviewer SDSC.\n\n4) qLogNEHVI and Random: The linked figure ([link](https://anonymous.4open.science/r/AuthorResponse-0652/Fig30-31-qLogNEHVI.pdf)) shows results with qLogNEHVI and Random for GP-derived functions ($d = 3, L = 3,4,5$) and four benchmark functions. For qLogNEHVI, we used qLogNoisyExpectedHypervolumeImprovement of BoTorch. The reference point is defined by y_min - 0.1 (y_max - y_min), where y_min and y_max are vectors consisting of the minimum and the maximum of each dimension in training $\\boldsymbol{y}_i$, respectively. For qLogNEHVI, we used both the gradient descent (optimize_acqf function of BoTorch) and DIRECT for the GP-derived functions. We see that the gradient optimizer slightly improves the results, but DIRECT also has the similar performance. Overall, qLogNEHVI shows good performance, while PFEV also show comparable performance (particularly when comparing with the DIRECT version of qLogNEHVI). Although implementations are not exactly consistent (PFEV is GPy base and qLogNEHVI is BoTorch), we can see that performance in our results are not largely different from the well-known package. \n\n5) About DIRECT: See response #4 of review SDSC. Note that `5 random $\\boldsymbol x$' in the end of the second paragraph of Section 6 is about the initial points of BO (not the initial points of the acquisition function optimization). Sorry for confusing.\n\n6) Reference point of EHVI: In EHVI, the two reference points are required, shown as v_ref and w_ref in Shah and Ghahramani 2016. The worst point vector v_ref is defined by subtracting $10^{-4}$ from the vector consisting of the minimum value of each dimension of $\\boldsymbol y_i$ in the training data. On the other hand, the ideal point vector $w_ref$ is defined by adding $1$ to the vector consisting of the maximum value of each dimension of $\\boldsymbol{y}_i$. We will add explanation in the paper.\n\n7) Lower bound estimator: Throughout the paper, the lower bound estimator (8) is used, except for Section 6.3 in which the performance of two estimators (8) and (6) were compared. \n\n8) Implementation of HVKG: See response #1 of reviewer SDSC.\n\n9) Fig2 illustrates examples of under and over truncation in a three dimensional output space. We tried to show there exist large differences in the case of $L \\geq 3$.\n\nWe will also reflect other comments such as typos and references.", "claims_and_evidence": "Many of the claims are well-supported. However, there is a lack of evidence in a few areas:\n\n- For simplicity consider the case where only one sample from the GP is used (as opposed to using 10 in the paper). The issue of over-truncation issue will be most problematic when the approximate PF has very few points. This is easily avoidable but simply increasing the population size for NSGA-II. This would result in far less truncation/discretization error. Is the over-truncation problem still an issue if one uses a finer approximation of the PF (more points from NSGA-II). I could see there being a case made for this increasing computation cost (of NSGA-II and of the box decomposition), but no such argument is made and no evidence is provided to that end (and there are box decomposition algorithms that often are quite a bit faster (Lacour et al, 2017). \n\n- Is over-truncation still an issue if one uses more samples from the GP? If more samples are used then the acquisition function is integrated over different samples of the (discretized) PF. It is not clear from the paper whether over-truncation is only an issue if a small number of samples are used. Of course, even better performance would be achieve if more points on the PF were used in the discretization and more GP samples were used.\n\nResults on how much over-truncation matters when the number of points used to approximate the PF increases and number of GP samples increases seem critical to understanding the significance of this problem", "methods_and_evaluation_criteria": "The synthetic problems are a good start, but they are all low dimensional and they are all synthetic problems. It would be nice to see performance on more realistic examples and higher dimensional problems.", "theoretical_claims": "The derivations looked reasonable.", "experimental_designs_or_analyses": "The main issues with the empirical analysis is 1) lack of hard problems (higher dimensional and more realistic), 2) lack of baselines. At a minimum, random search and qLogEHVI (Ament et al, 2023) should be included as baselines.\n\nIn addition, why is DIRECT used to optimize the acquisition functions? Many of these are differentiable and gradient-based optimization is typically far more effective (Daulton et al, 2020) and is widely used in modern Bayesian optimization. 5 random starting points also likely leads to poor global optimization of the acquisition functions.\n\nHow are reference points chosen for evaluation and for EHVI? That should be documented as it likely has a large effect on the performance evaluation (i.e. how different parts of the Pareto frontier contribute to the overall hypervolume).\n\nWhich estimation of the lower bound is used in the experiments in the main text?", "relation_to_broader_scientific_literature": "This builds on work on information theoretic methods for multi-objective BO.", "essential_references_not_discussed": "Unexpected Improvements to Expected Improvement for Bayesian Optimization, Ament et al, 2023\nDifferentiable Expected Hypervolume Improvement for Parallel Multi-Objective Bayesian Optimization, Daulton et al, 2020", "other_strengths_and_weaknesses": "These are described above. The main issue is understanding the significance of the over truncation issue and understanding how the contribution of the work compares to the SOTA. Also, no code is provided and it appears the authors implemented all baselines by hand if they are in GPy? Using the implementations by the authors of many of the baselines in BoTorch would instill more confidence in the results.", "other_comments_or_suggestions": "In the abstract:\n- truncated -> truncated\n- truncation -> truncations\n\nFigure 2 is confusing, and I don't understand what is being depicted.", "questions_for_authors": "Questions are interspersed above.", "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": "PpaT78CHRl", "reviewer": "Reviewer_8xyH", "summary": "This work introduces a multi-objective Bayesian optimization estimation framework that utilizes an information-theoretic acquisition function. It presents an effective approach for conducting variational inference when the continuous Pareto frontier is not fully known. By employing a mixture of distributions with different supports, the method enhances the estimation of the target distribution, demonstrating strong performance across diverse experiments.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your constructive comments and suggestions.\n\n1) About $\\lambda$: We can prove $\\lambda = 0$ cannot be the maximizer, and thus, the optimization problem is well-defined. This can be derived from (8). First, from the definition, we see $\\theta_{MAP} \\in (0,1)$ (Note that $\\hat{p} \\in (0,1)$). When $\\lambda \\to 0$, the second term of (8) goes to $\\log 0$ (from the definition of $\\eta_{\\lambda}^{\\tilde{{\\cal F}}_S^*}$ shown after (4)). On the other hand, the first term is finite when $\\lambda \\to 0$. Therefore, if $\\lambda \\to 0$, we see (8) goes to $- \\infty$. Further, we already show that (8) is concave in Appendix C.\n\n2) Motivation of the lower bound approximation: The lower bound approximation can be written as (7). However, in (7), \n$p(\\boldsymbol{f_x} \\in {\\cal A}_O^{\\tilde{{\\cal F}}^*_S} \\mid \\tilde{{\\cal F}}^*)$ \nand \n\n$p(\\boldsymbol{f_x} \\in {\\cal A}_{U \\setminus O}^{\\tilde{{\\cal F}}^*_S} \\mid \\tilde{{\\cal F}}^*)$ \ncannot be calculated analytically. A naive approximation of these two $p$ are \n$I( \\tilde{\\boldsymbol{f_x}} \\in {\\cal A}_O^{\\tilde{{\\cal F}}^*_S} )$ \nand \n\n$I( \\tilde{\\boldsymbol{f_ x}} \\in {\\cal A}_{U \\setminus O}^{\\tilde{{\\cal F}}^*_S} )$, \nrespectively (line 245-246), which we used in the estimator called naive MC in Figure 6. Instead of this direct one sample estimation, we propose using approximation with the prior knowledge about \n$p(\\boldsymbol{f_x} \\in {\\cal A}_O^{\\tilde{{\\cal F}}^*_S} \\mid \\tilde{{\\cal F}}^*)$, \ni.e., this can be approximated by $\\hat{p}$. \n\nWe will also reflect other comments such as notations and claims about PI.", "claims_and_evidence": "Four claims are made by the authors:\n\n>    PFEV is the first approach to continuous space MOBO that is based on a general lower bound of MI\n\nI agree, but this lower bound of MI on MES concept has been proposed in Takeno 2022 in single-fidelity, so I won't consider it as a MAJOR contribution;\n\n>    We newly introduce an under truncation approximation for $p(\\boldsymbol{f}(x)\\mid\\mathcal{F}^\\ast). Further, we define the variational distribution as a mixture of distributions with the over and the under truncation, and show how to optimize the mixture weight\n\nThe claim is correct. However, I do have some concerns about the lower bound optimization part, which I will elaborate later;\n\n>    We also discuss properties and extensions of PFEV. For our MI lower bound, its relation with PI and a Monte-Carlo approximation are derived. We further discuss several extended settings such as parallel querying.\n\nI agree with the most of the claim, except the part involving PI. Although the authors show that the MI lower bound is an upper bound of PI AF, it does not explain many things. Since for acquisition function we only care finding the x that maximize them, the lower bound does not disclose any intrinsic relation between the optimal x derived from MI lower bound and optimal x derived from PI. Therefore, as I agree the results from Remark 3.2, I won't consider it as a major contribution. \n\n>    Through empirical evaluation on Gaussian process generated and benchmark functions, we demonstrate effectiveness of PFEV. We empirically observed that PFEV shows a particular difference from existing over truncation based methods when output dimension >= 3 , in which the difference of two truncation becomes more apparent.\n\nThe authors have clear evidence showing this claim.", "methods_and_evaluation_criteria": "Overall, I believe the authors clearly presented their proposed methods and corresponding contributions. The mathematical notations are well-defined, and the experiments used to validate their proposed method follow standard practices.", "theoretical_claims": "The main concern is about the design for maximizing the lower bound $\\max_{x, \\lambda\\in(0, 1]} L(x, \\lambda)$. The author did not show this is a well-defined optimization setting (optimum exists). What if the maximum happens at $\\lambda = 0$, in the sense that $p = q_O$?\n\nSome minor suggestions: \n\n- Line 121: \"If $q(f_x\\mid\\mathcal{F}^\\ast) = p(f_x\\mid\\mathcal{F}^\\ast)$\", should add \"for $\\mathcal{F}^\\ast$ almost everywhere\".\n\n- Equation (4): $L$ has been used for the number of objectives, please use another notation for the lower bound\n\n- Many approximation schemes are proposed between line 237 to line 265. It is a bit hard to follow their motivations. Could you explain more about them?", "experimental_designs_or_analyses": "The experimental designs are standard. I do not see any concerns in this part.", "relation_to_broader_scientific_literature": "This work extends previous research on solving MES with variational inference to a multi-objective setting. It introduces a novel approach to advancing information-theoretic multi-objective Bayesian optimization, making a meaningful contribution to the community.", "essential_references_not_discussed": "NaN", "other_strengths_and_weaknesses": "Strengths: \n\n- The work proposes a new way to solve multi-objective BO in the variational inference perspective;\n\n- The work is solid, with thorough discussion on previous works and well-defined mathematical notations;\n\n- The experiments are solid, showing clear advantages of their proposed method.\n\nWeaknesses: \n\n- The well-definedness of the optimization problem in Equation (4);\n\n- Lack of motivations for the approximation schemes in Section 3.3.", "other_comments_or_suggestions": "NaN", "questions_for_authors": "My concerns are listed in the weaknesses section. There is no other questions raised.", "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:

Meta-review: In this paper, the authors study multi-objective Bayesian optimization (MOBO) through the information gain of the Pareto-frontier. To calculate the information gain, a predictive distribution conditioned on the Pareto-frontier is needed. This distribution is a distribution truncated by the Pareto-frontier. However, it is usually impossible to obtain the entire Pareto-frontier in a continuous domain. The authors  consider an approximation of the truncated distribution by using a mixture distribution consisting of two possible approximate truncation obtainable from a subset of the Pareto-frontier. Since the optimal balance of the mixture is unknown beforehand, the authors propose optimizing the balancing coefficient through the variational lower bound maximization framework. The empirical evaluation demonstrates the effectiveness of the proposed method mostly when the number of objective functions is large. The reviewers have indicated the novelty of the proposed approach and the completeness of the experiments considered. They also highlight the thorough discussion on previous works and well-defined mathematical notations. They also say that the experiments are solid, showing clear advantages of their proposed method. The reviewers have also indicated small weaknesses of the proposed method related to the small dimensionality of the experiments and some lack of baselines that the authors should try to address. Overall, I think that this is not an outstanding paper. However, it will receive the attention of the community working on Bayesian optimization.
Decision: Accept (poster)