Multi-Objective Causal Bayesian Optimization
Authors: Shriya Bhatija
Paul-David Zuercher
Jakob Thumm
Thomas Bohné
Venue: ICML 2025 poster (2025)
Keywords: Causality
Multi-Objective Bayesian Optimisation
Primary area: general_machine_learning->causality
TL;DR: We incorporate causality into multi-objective Bayesian optimization, enabling decision-making in causal systems with multiple outputs.

Abstract: In decision-making problems, the outcome of an intervention often depends on the causal relationships between system components and is highly costly to evaluate. In such settings, causal Bayesian optimization (CBO) exploits the causal relationships between the system variables and sequentially performs interventions to approach the optimum with minimal data. Extending CBO to the multi-outcome setting, we propose *multi-objective Causal Bayesian optimization* (MO-CBO), a paradigm for identifying Pareto-optimal interventions within a known multi-target causal graph. Our methodology first reduces the search space by discarding sub-optimal interventions based on the structure of the given causal graph. We further show that any MO-CBO problem can be decomposed into several traditional multi-objective optimization tasks. Our proposed MO-CBO algorithm is designed to identify Pareto-optimal interventions by iteratively exploring these underlying tasks, guided by relative hypervolume improvement. Experiments on synthetic and real-world causal graphs demonstrate the superiority of our approach over non-causal multi-objective Bayesian optimization in settings where causal information is available.

Multi-Objective Causal Bayesian Optimization
Shriya Bhatija Paul-David Zuercher 23 Jakob Thumm
Thomas Bohné?
Abstract
(a) MO-CBO problem
MO-CBO solution
In decision-making problems, the outcome of an intervention often depends on the causal relationships between system components and is highly costly to evaluate. In such settings, causal Bayesian optimization (CBO) exploits the causal relationships between the system variables and sequentially performs interventions to ap- proach the optimum with minimal data. Extend- ing CBO to the multi-outcome setting, we pro- pose multi-objective causal Bayesian optimiza- tion (MO-CBO), a paradigm for identifying Pareto- optimal interventions within a known multi-target causal graph. Our methodology first reduces the search space by discarding sub-optimal in- terventions based on the structure of the given causal graph. We further show that any MO-CBO problem can be decomposed into several tradi- tional multi-objective optimization tasks. Our proposed MO-CBO algorithm is designed iden- tify Pareto-optimal interventions by iteratively ex- ploring these underlying tasks, guided by relative hypervolume improvement. Experiments on syn- thetic and real-world causal graphs demonstrate the superiority of our approach over non non-causal multi-objest oujeii-beecctve Bayesian optimization in settings where causal information is available.
BMR
Height
feasible region
Weight
BMI
CI
Aspini
Age
Pareto
front
Cancer
infeasible
Statin
PSA
Statin
Figure I. MO-CBO problem healthcare. (a) Causal graph where red, orange, and grey nodes depict outcome, manipulative, and non-manipulative variables, respectively. (b) The solution consists of interventions that yield optimal trade-offs between the targets.
by causal mechanisms, which can be exploited to approach optimal outcomes in targeted and cost-efficient manner. A well-established strategy for optimizing such expensive- to-evaluate black-box functions is Bayesian optimization (Shahriari et al., 2016), but cannot leverage the causal structure between its input variables. To this end, causal Bayesian optimization (CBO) (Aglietti al., 2020) was in- troduced to generalize Bayesian optimization to settings where causal information is available. While existing CBO variants focus on optimizing single objective, real-world systems often require the simultaneous optimization of mul- tiple outcome variables. Here, the aim is to establish optimal trade-offs between these variables instead of identifying the global optimum of a single objective. As an example, con- sider the graph in Figure (a), which depicts the causal relationships between prostate-specific antigen (PSA) and its risk factors (Ferro et al., 2015). For patients sensitive to Statin medications, the aim is to manipulate these risk factors to simultaneously minimize both the required Statin intake and PSA levels. Figure 1 (b) shows the Pareto front, i.e., the optimal trade-offs between Statin and PSA.
1. Introduction
Decision-making problems arise in various domains, such as healthcare, manufacturing, and public policy, and in- volve manipulating variables to obtain an outcome of in- terest. In many such domains, interventions are inherently costly, and practical applications are subject to budgetary constraints. Moreover, these systems are often governed
'Department of Computer Engineering, Technical University of Munich, Munich, Germany 'Department of Engineering, Univer- sity of Cambridge, Cambridge, United Kingdom "The Alan Turing Institute, London, United Kingdom. Correspondence to: Shriya Bhatija <shriya. bhatiaa.bbajaeee tum.de>.
We propose multi-objectiive causal Bayesian optimization (MO-CRO) to generalize CRO problems with multiple outcome variables. Figure 2 gives a high-level overview of our proposed methodology. Our key contributions are:
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
1.
We formally define MO-CBO as a new class of optimiza- tion problems.

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Multi-Objective Causal Bayesian Optimization
The MO-CBO problem [Sec.
Reducing the search space [Sec, 4.1]
Solving the local problems [Sec. 4.2]
Building the Pareto front [Sec. 4.31
X4
Vi
local problem
MO-CBO algorithm
Y+4
Pareto fronts of local problems
22
Xa
X1
Full search space has
X2
X3
[F)
2|x local problems
1. Select
local problem
local problem
16
Optimize (DGEMO)
local problem
Xa
Y2
X
Pareto front
Vi
Figure
Overview of MO-CBO methodology.
We present mathematical framework reduce the search space of MO-CBO problems based on the topology of the causal graph. It allows to discard sub-optimal interventions and focus exploration on possibly-optimal strategies.
iteration, while batch methods select multiple solutions simultancously for parallel evaluation. One of the most prominent single-point algorithms is ParEGO (Knowles, 2006), which randomly scalarizes the multi-objective prob- lem into single-objective one and chooses a sample that maximizes the expected improvement. As for batch meth- ods, MOEA/D (Zhang et al., 2010) builds on ParEGO to incorporate multiple scalarization weights and perform batch evaluation through MOEA/D (Zhang & Li, 2007). Moreover, TSEMO (Bradford et al., 2018) adopts Thomp- son sampling on the Gaussian process posterior as an acqui- sition function, optimizes multiple objectives with NSGA-II (Deb et al., 2002), and selects the next batch of samples by maximizing the hypervolume improvement. Recently, @NHVVI (Daulton al., 2021) was proposed as robust method that scales highly parallel evaluations of noisy objectives. DGEMO (Konakovic Lukovic al., 2020) is most relevant for our work, and employs a novel batch selec- tion strategy maintaining sample diversity in the input space. Specifically, it partitions the input space into so-called di- versity regions to guide the selection of diverse points in each batch. We use DGEMO in our MO-CBO algorithm to explore the possibly-optimes intervention strategies.
3. We propose an algorithm for the parallel exploration of these possibly-optimal intervention strategies, guided by a custom acquisition function.
We experimentally demonstrate on both synthetic and real-world MO-CBO problems that our method can sur- pass traditional multi-objective Bayesian optimization in scenarios with known causal structures, achieving more cost-effective, diverse, and accurate solutions.
To our knowledge, no other multi-objective optimization method exists in the literature that consider the causal structure. We prove that MO-CBO's reduced search space retains all solutions achievable by traditional multi-objective optimization, while in some cases containing superior, oth- erwise unattainable solutions. The empirical results confirm that our MO-CBO algorithm consistently matches, and in some scenarios exceeds, the performance of standard base- lines.
Causal Decision-Making Within this field, there is line of work focusing on multi-armed bandit problems and re- inforcement learning settings. Here, actions or arms cor- respond to interventions on an arbitrary causal graph with existing links between the agent's 's decisions and the received rewards. Lee & Bareinboim (2018) identify a set possibly- optimal arms that an agent should explore to maximize its expected reward in multi-armed bandit problem. More- over, Lee & Bareinboim (2019b) extend their previous work to scenarios with non-manipulative variables. Collectively, their findings represent the single-objective counterpart of our search space reduction.
1.1. Related Work
We combine multi-objective Bayesian optimization (MOBO) with techniques from causal inference to achieve MO-CBO. Our method lies within the field of causal decision-making seeking to leverage known causal structures to enable causally-ifformed decisions.
MOBO Multi-objective Bayesian optimization aims to efficiently optimize multiple, often conflicting, objective functions simultaneously. The existing algorithms can roughly categorized by their selection strategy: Single-poin point methods select and evaluate one candidate solution each
Causal deciision-makin also encompasses a growing body of research specifically focused on advancing CBO (Aglietti
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Multi-Objective Causal Bayesian Optimization
al., 2020). These advancements include extensions such as constrained CBO (Aglietti et al., 2023), time-dynamic CBO (Aglietti al., 2021), and various other variants (Bran- chini et al., 2023; Gultchin et al., 2023; Sussex et al., 2023; 2024; Ren & Qian, 2024; Zeitler& & Astudillo, 2024). How- ever, these methods are designed to optimize a single target variable, rendering them infeasible for applications with multiple objectives.
based on their performance properties in the input space. Formally, batch is chosen as follows:
B arg max HVI(
BC=
max ISRCK
6 (
min 6% (B) S 1, ISKSK
(2)
where B denotes the batch size and the functions (+)-) are defined as the number of elements from B that belong to RR. We refer to Konakovic Lukovic et al. (2020) for the complete selection algorithm.
2. Preliminaries
In this paper, random variables and their realizations are de- noted the upper and lower case Latin letters, respectively. Sets and vectors are written in bold. For a set X, its power set is denoted as P.X).
2.2. Causality
Graph Notation A graph g = (V, E) is defined by a finite vertex set V and an edge set E S V x V. contain- ing ordered pairs of distinct vertices. The subgraph of g restricted V C V is given by giv'l = (V', E(V")), where E the set of its parents, ancestors and descendants in G is de- noted as pa(V)g. an()),, and de(V),, respectively. Here, no vertex is a parent, an ancestor, or descendant of it- self. Conversely, with a capital letter, this notation is ex- tended to include the argument in the result, i.e., Pa(V)g = pa (V))) U (V}. Moreover, we define these relations for sets of variables V C V, i.e., pa(V")g Uvev (VV) and Pa(V') Uvev Pa(V)). Equivalent conventions apply to the ancestor and descendant relationships.
2.1. MOBO Notation
((i,j) E E i,j E V'). For V V,
MOBO simultaneously minimizes (or maximizes) a set of black-box objectives Sm X + R, where X is an arbitrary input space. It is designed to rely only on small number of function evaluations. Due to potential conflicts between objectives, MOBO aims to find trade-off solutions, known as Pareto optima (Miettinen, 1999):
Definition 2.1 (Pareto optimality). A point x E X is called Pareto-optimad if there is no other x' E X such that fx > fa(a') for all 1 < i < m and (x) > fi(x') for at least one 1 S S m. The set of Pareto-optimal points in X is called Pareto set, denoted P.. The Pareto front is the image of the Pareto set under the objective functions, given by P.).
Structural Causal Models Let (V, UV,U, U. F, P(U)) be a structural causal model (SCM) (Pearl, 2000) and G its associ- ated acyelic graph that encodes the underlying causal mech- anisms. Specifically, U a set of independent exogenous random variables distributed according to the probability distribution P(U). V is set of endogenous random vari- ables, and F = Jvev is set of deterministic functions such that V = fv (pa(V)g, UV), where UV C U is the set of exogenous variables affecting VE V. The set UV nuw consists of unobserved confounders between V, W E V. which are the exogenous variables influencing both V and W. Within V, there are three different types of variables be distinguished: Non mon-uuaiiee variables C that can- not be modified, treatment variables X which can be set to specific values, and output variables Y = (Yi...,, Y. which represent the outcome of interest. We consider only real-valued SCMS, where all endogenous variables have con- tinuous domains. For X, S X, CC(X.)g refers to the c-component of & (Tian & Pearl, 2002). which, in this con- text, is the maximal set of variables that includes X, and is connected via unobserved confounders. The joint distribu- tion of V, which is determined by P(U), is referred to as observational distribution and denoted P(V).
Py = (f(x)= =
At each iteration of a MOBO algorithm, prior data is used to fit a surrogate model of the objectives, for which Gaus- sian processes (Rasmussen, 2004) are predominantly used. Based on the surrogates, an approximation P, of the Pareto front computed. To select which point, batch of points, to evaluate next, acquisition function is used to assess the utility of those evaluations. The most commonly used acquisition function in MOBO is based on the hypervolume indicator H (Zitzler& & Thiele, 1999). The larger the hyper- volume, the better Py approximates the true Pareto front. The hypervolume improvement determines how much the hypervolume would increase if batch of samples B C X was added to the current approximation, and is given by
HVI(f(B), Py) = H(P, Uf(B)) - H(P)).
(1)
Since DGEMO is the backbone of our MO-CBO algorithm, we briefly describe its batch selection strategy. It consid- ers hypervolume improvement as well as sample diversity in the input space. To this end, the so called diversity re- gions RI...., Rx S X are constructed by using the cur- rent Pareto front approximation to group the optimal points
Interventions A set X, E P(X) is called an intervention set. The interventional domain of an intervention set
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Multi-Objective Causal Bayesian Optimization
given as x = Xxex, D() and describes the feasible values of x.. An intervention on X, involves replacing the structural equations fx with constant intervention value x, for all X E X,, This action is denoted with the do- operator do(X; = Xs), where the vector of intervention values is x, E D. The graph 9x2 represents this in- tervention and is obtained by removing the incoming edges into X,. The observational distribution of 9x, is denoted as P(V]do x xx)) and called interventional distribu- tion. For X, 2, no intervention is performed and the observational and interventional distributions coincide. The tuple (X,, X2) is referred to as an intervention set-value pair. Given two sets x, X, S X and x, E D)),), we write by the values of x, corresponding to X, nnx, X
3.1. Decomposition of MO-CBO Problems
We aim simplify the search space to navigate the discov- ery of Pareto-optimal intervention set-value pairs.
Definition 3.3 (Local MO-CBO problem). Let X, E P(X) be an intervention set. Then, the multi-objective optimiza- tion problem defined by the objective functions (x( :): D(X.) - R. X + (X, 1 S i S m, is called the local MO-CBO problem w.r.t. X.-
The Pareto set of the local MO-CBO problem w.r.t. X, E P(X) is denoted as (X,) and the associated Pareto front as Pl (X.). Each local MO-CBO problem corresponds to standard multi-objective optimization task, solvable with existing methods. The following proposition decom- poses MO-CBO problems into such local problems.
xs xx]]
3. The MO-CBO Problem
Proposition 3.4. Given (G, Y,X, C), let S C ((X) be non-empty collection of intervention sets. Then, it holds
In our setting, we assume that the causal relationships en- coded in 9 are known while the underlying parametrizations, i.e. F and P(U). can be unknown. This restricted informa- tion is denoted as (G, Y, X, C). The assumption is common within the CBO line of work and allows generalization across systems with the same causal structure.
[S| Pf(S) S U 8=1
(4)
A MO-CBO problem aims to identify intervention set-value pairs (X,, x,) that offer optimal trade-offs in minimizing (or, maximizing) all target variables in Y. The outcomes of an intervention dx, = x,) are captured as the expected values
Proof. See Appendix A. Core idea: We exploit that the space of all intervention set-value pairs is the union of the input spaces of each local problem.
Proposition 3.4 allows to match the Pareto-optimal interven- tion set-value pairs to the Pareto-optimal solutions from the local problems where the intervention set is fixed. Therefore, discovering P4 (P(X)) requires identifying Pareto- orrtial solutions of local MO-CBO problems with respect to all intervention sets X, E P(X).
[x(X, x.) ==
(3)
where P(Y:]do(X, (X, = x.)) denotes the interventional distri- bution of Y,, for all I,...,m. m. We write p(X.,,, x.) (X,, Mm x,))) for the vector notation. Next, we adopt the notion of Pareto optimality to inter- vention set-value pairs:
4. Solving MO-CBO Problems
In this section, we propose our methodology for solving MO-CBO problems, which has been outlined in Figure 2. In summary, we reduce the search space to subset S C (XX), solve the corresponding local MO-CBO problems w.r.t. each element in S, and extract only Pareto-optimal intervention set-value pairs construct the Pareto front Pg (P(()).
Definition 3.1 (Pareto-optimal intervention set- valuv pair). Given S C IP)), an intervention set-value pair (X,, x,) with X, E S. x, E D(X.) is called Pareto-optimal for S, there is no other intervention set-value pair (X,, xx) with X E S, x,_ E DD(X)) such that (X,,,, ) S (X,, for all < < m and pa (X,, XX,,' x x.) for least
one 1 < $ m.
4.1. Reducing the Search Space
Definition 3.2 (Pareto front for S). The space of all Pareto- optimal intervention set-value pairs for given S C (X) is called the Pareto set for S, denoted (S). The correspond- ing Pareto front for S, denoted PP5 (S)) is the m-dimensional image of (S) under the objectives μ4, 1 < <i< < m.
The complexity of solving the local MO-CBO problems w.r.t. all X, P(X) rises exponentially with the number of treatment variables, making this strategy impracticable for most tasks. This section proposes method to exploit the graph topology to identify a minimal subset S C P(X) with pe (I)())) = PS))). Hereby, we generalize the results from Lee & Bareinboim (2018) to the multi-objective set- ting. All proofs and derivations are given Appendix B. For now, we assume that there are no non-manipulative vari-
We define MO-CBO problems as identifying the Pareto set (X)))) which yields the optimal trade-offs among the objectives, represented by the Pareto front PP5 (P.()).
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Multi-Objective Causal Bayesian Optimization
ables, i.e., C Ø At the end of the section, we discuss the general case C * 23.
X3
Xi
Y
X4
X1
Yi
We first reduce the search space by disregarding intervention sets where some variables do not affect the targets:
X4
X2
Y2
X2
Y22
X3
Definition 4.1 (Minimal intervention set). A set X P(X) is called a minimal intervention set if there exists no sub- set X C X, such that for all X, E D(X,) it holds (X, X.) = (x,''' x [X 1 < i S m, for every SCM conforming to g.
Figure 3. Two causal graphs with X = (X1, X2, X3, Xa). Y (Yi, Y23. (a) No unobserved confounders. (b) The dashed bi- directed edge depicts an unobserved confounder between X, and Yi.
We denote the of minimal intervention sets with Mlg, YY The following proposition characterizes such sets in given causal graph G.
definitions from Lee & Bareinboim (2018) to the multi- objective setting. They aim to identify region, starting from Y. that is governed by unobserved confounders, along with its outside border that determines the realization of variables within the region.
Proposition 4.2. X, E I is a minimal intervention set if and only if holds X, S an an(a))))))) (Y)sx.'
Definition 4.5 (Minimal unobserved confounders' terri- tory). Let H = ([An(Y'g] A set of variables T in H. with Y S T, is called a UC-territory for 9 w.r.t. Y if De(T) T and CC(T) = T. The UC-territory T is said to be minimal, denoted T MUCT(G, Y). if no T' C T is UC-territory.
Proof. See Appendix B.I. Core idea: The "if" direction shows by contradiction that any non non-minimal intervention set cannot consist solely of the ancestors of Y. The "only if" direction straightforward to prove since variables without an ancestral relationship to Y are redundant intervene D
upon.
Definition 4.6 (Interventional border). Let us denote T = MUCT(G, Y). Then, B = pa(T) (T is called the interventional border for G w.r.t. Y, which we write as IBG
We adapt the notion of tion sets (Lee & Bareinboim, 2018) 1) for Pareto-optimalle Intuitively, a minimal intervention set is called possibly Pareto-optimal if it includes a Pareto-optimal intervention set-value pair whose outcome is unattainable with any other intervention set, for at least one SCM conforming to G.
minimal interven-
Example We illustrate these two concepts with the causal graphs from Figure 3. In Figure (a), there are unob- served confounders and thus, it holds CC(Y)g Y and De(Y); g Y. It follows MUCT(G, Y) = (Yr, Y2) and IB Y) X2). In Figure 3 (b), we construct the minimal UC-territory, starting from T = Y, as follows: Since Y, has an unobserved confounder with Xa. we update T = CC(Y)g = (Y,, Y2, Xa), and thereafter add all the descendants of X4, obtaining T {Yr, Y2, X1, Since there are more unobserved confounders between T and An( Y) T, the minimal UC-territory has been found and is given by MUCT(G, Y) = (Yi, Y2, X1, X4) along with IB Y) = [X2,
Definition 4.3 (Possibly Pareto-optimal minimal interven- tion set). A set X, E Mgg,y is called possibly Pareto- optimal if. for at least one SCM conforming to G, there exists X, E D(X,) such that (X,, xx) is Pareto-optimal for P(X), and for no X E Mgg. X,,x, E D(X;)) it holds p.(X. x S pa (X.,,, x.), for all Siiii m.
We denote the set of all possibly Pareto-optimal mini- mal intervention sets by Og,y. Next, we establish graph- theoretical criteria to identify such sets in a given causal graph. First, the proposition below considers a special case:
Interventional borders can fully determine possibly Pareto- optimal minimal intervention sets, which are described with the following two results.
Proposition 4.4. If no Y; is confounded with an(Y:) via unobserved confounders, then pa(Y) a G is the only possibly Pareto-optimal minimal intervention set.
Proposition 4.7. IB(Gx, Y) is a possibly Pareto-optimal minimal intervention fer any X, E P(X).
Proof. See Appendix B.2. Core idea: In the absence of
unobserved confounding between any Y; and its ancestors an the average effect of any intervention do(X, = x.) can be matched by intervening on pa(Y (a))..
Proof. See Appendix B.2. Core idea: We first prove in Proposition B.2 that IB(G, Y) is possibly Pareto-optimal minimal intervention set by constructing an SCM where do Y) = 0) is the single best intervention. This construction then easily extends to show that IB Y) can also represent the single optimal intervention.
D
To characterize possibly Pareto-optimal minimal interven-
tion sets in arbitrary graphs, we extend the following two
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Multi-Objective Causal Bayesian Optimization
Next, we can finally characterize possibly Pareto-optimal minimal intervention sets:
Algorithm 1 Our MO-CBO algorithm
Input: (g. Y, X, C). S E {Og.Y, Meg , P(X), data
D, batch size B, number of iterations N
Theorem 4.8. A set X, E P(X) is possibly Pareto-
Output: (S) (S)
optimal minimal intervention set if and only if it holds IB(9x, X,,
Initialize the dataset DD = D
for S = 1 to |S| do
Fit surrogates jú: (X, -) with Do. i 1.....W. m Approximate P!(X,) and (X,).) using ,i,..... Pm
Proof. See Appendix B.2. Core idea: The "if" statement is a special case of Proposition 4.7. We prove the "only if" direction by showing that intervening on I Y) is at least as optimal as intervening on X.,
end for
for n = 1 to N do
for s 1 to |S| do
Corollary 4.9. Let X, E F(X) and X = IB(9x, Y). For any x, E =Dxx)) there exist X E D(X') such that it holds μ (X,,, x S p(X,,, (.)) for all S sism m.
Select batch B, = (x))
60 via Equation (2)
end for
Select batch B, from (BI,.... Bist} via Equation (6)
Intervene on X; with Bg
Corollary 4.9 is a direct result from the proof of Theorem 4.8. In this setting, it is easy to construct an SCM, conforming to G. for which strict inequality holds in least one component. Finally, we show that it suffices to only consider possibly Pareto-optimal minimal intervention sets to solve MO-CBO problems.
Augment D, = -D U {(Xs, x5) p(X;, x,)) ) B Update surrogates (i,(Xa, -) with Dn, L..... m
b=1
Approximate P (X;)) and P) (Xx) using jud ; p.
end for
Compute P..)), (X.) from DN, s = 1,...,111 Compute (S) and (S))
Theorem 4.10. It holds P)(P(X)) = =P9(Os,x)
Proof. C: Assume P)(((X)) & (9(0).v). Then, there exists z E withh with z= u( ) for some intervention set-value pair (X,, such that z E (P(()) and z & () If x, E 0g.x, it follows that (X,, x.) is not Pareto-optimal for Dg.y, which is a contradiction since is Pareto-optimal for PXX). Conversely, if X, P(X))O0 (k))),g,, Y. set X = -IB Y) and from Corollary 4.9, infer that, for some SCM conforming to G, there exists x², E D(X)) with ,(X,,
In Figure (b), the intervention sets X2, and (X2, X33 satisfy the condition from Theorem 4.8, and thus, Oc.Y = ((X,,Xa))
X2, We now consider the more general case with C * Ø, where non-manipulative variables can be present. The definitions for the minimal intervention set and the pos- sibly Pareto-optimal minimal intervention set straight- forward extension. Lee & Bareinboim (2019a) propose projection G + g[Vic] which preserves the distri- bution of the underlying SCM. Given such a projection, we can identify the possibly Pareto-optimal minimal inter- vention sets in (G, Y, C) by applying Theorem 4.8 to (C[V Y.X).
4, xx) I ((X,,,x)) for all 1 < i S m, and ((X',, x²) < ((X,,, Xa), for at least one 1 < S m. This results in z & (P()),, which is a contradiction.
Then, there exists RT, with = (XX,,, Xs), such that z E (P,,))))) and z & (P()))). There exists some X, P(X)) (Og.v x. E D(X)) such that (X,, xx) is Pareto optimal and for which it holds x()) S. (((X..,,, for all 1 < i < m,
2: Assume x)) & ()()()).
4.2. Solving the Local Problems
and v(X', 8, x²) < (xX,,, x.). for at least one 1 S S m. Since X, is not possibly Pareto-optimal, we infer from
We propose our algorithm to solve MO-CBO problems', for which the procedure is summarized in Algorithm 1. assumes known causal graph (G, Y, X, C), prior data D, and set S E {0g,y, Mlg. Y IP IPX)) that specifies which local problems to consider. The idea is to alternately solve the local MO-CBO problems using the MOBO algorithm DGEMO.
Corollary 4.9 that for X" = -IB(G, Y) there exists x" E D(X") such that "XX", x") S x(X,, x,), for all S i m. Hence, it holds μ (X"", x E (P(P(X)), which is contradiction to E Y since X E Og,Y.
Using Theorem 4.10, we reduce the search space of MO- CBO problems to S = Og.y.
More specifically, the algorithm operates as follows: For each local MO-CBO problem w.r.t. X, E S, it first fits the surrogate model the objectives (x(,,, ). S S m, via independent Gaussian processes. Based on the means of
Example We illustrate the search space reduction with the causal graphs from Figure 3. Note that in both cases it holds F(X)- = 2 x = 16. In Figure 3 (a), there are no unobserved confounders, and it follows Og,y = pa(Y)g = (X1, X2).
'The full implementation of our algorithm is available ht https://gii thub. hoom//hriaaa yaBhat ja/MM-CCOOO
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Multi-Objective Causal Bayesian Optimization
Table The generational distances, averaged across 10 random seeds. Lower values indicate closer approximation to (I)()))..
Problem
ParEGO
MOEA/D-EGO
TSEMO
GNEHVI
DGEMO
MO-CBO (ours)
SYNTHETIC-1
0.30
0.38
0.16
0.27
0.14
0.14
SYNTHETIC-2
12.43
11.45
8.65
7.98
6.79
2.80
HEALTH
0.09
0.20
0.08
0.12
0.07
0.06
CREDIT APPROVAL
0.14
0.12
0.06
0.08
0.06
0.05
Table 2. The inverted generational distances, averaged across 10 random seeds. Lower values indicate more diverse coverage of solutions approximate to (P())).
Problem
ParEGO
MOEA/D-EGO
TSEMO
@NEHVI
DGEMO
MO-CBO (ours)
SYNTHETIC-1
3.63
3.24
2.82
2.45
2.57
1.40
SYNTHETIC-2
7.43
7.70
4.78
5.49
4.50
0.87
HEALTH
0.15
0.16
0.05
0.10
0.05
0.02
CREDIT APPROVAL
0.24
0.28
0.08
0.21
0.09
0.08
the Gaussian process posteriors. approximations of P! (X.) and P) (X.) are computed utilizing the Pareto discovery approach from DGEMO. After this initial step, the most promising intervention set is selected for batch evaluation at each iteration. The dataset is then augmented with the newly evaluated batch of samples. For the corresponding local problem, we again update the surrogate model and the Pareto set and front approximations. After completing all iterations, the algorithm identifies the final Pareto sets and fronts for each local MO-CBO problem using the collected objective function evaluations v. Thereafter, is easy construct (S)) and (S). see Section 4.3.
Overall, the proposed batch selection is designed to alter- nately advance the Pareto fronts (X1)...... (),, s)).
4.3. Building the Pareto Front
After Algorithm has computed the Pareto sets and Pareto fronts of the local problems, its final step is to simply extract Pareto-optimal points from U P! (X.). as justified by Proposition 3.4. This yields Pareto-optimal intervention set-value pairs (IP()) and their corresponding Pareto front Py (P((X)).
5. Experiments
Batch Selection For the local MO-CBO problem w.r.t. X. E S, let R, ..., Rx X C D(X.) denote the identified diversity regions from DGEMO, discussed in Sec- tion 2.1. Our algorithm seeks to balance the exploration of Pareto fronts from multiple local MO-CBO problems, but evaluating all B., 1, .,..., |8], during single it- eration, is inefficient strategy. Instead, select the batch with the most promising hypervolume improvement at each iteration. To this end, we introduce the term relative hypervolume improvement, defined as
We evaluate our MO-CBO algorithm with S = 0g. on the causal graphs shown in Figure 3 (a) (SYNTHETIC-1), Figure 3 (b) (SYNTHETIC-2), Figure (HEALTH), and an additional CREDIT APPROVAL example. We cover both syn- thetic and real-world scenarios. The full description of the underlying SCMs is given in Appendix C. We assume to have an initial dataset D {(X.,, x with K samples per intervention set. The batch size 5. For reproducibility, all experiments run across 10 random seeds, resulting in varying initializations of D.
RHVI(p(XX,, B.), (X), HVI(u(X,, B.). ((),) (XP))(x)) (5)
To the best of our knowledge, there exists no other multi-objective optimization method in the literature that can leverage the causal structure. As baselines, we there- fore apply some the most prominent MOBO algorithms such as ParEGO, MOEA/D-EGO, TSEMO, NEHVI, and DGEMO (see the literature review). They intervene on all treatment variables simultaneously and thus the objec- tive functions px(, :) D(X) - R, x p(X, x), == 1..... m. Notably, PC (P(()) contains at least as opti- mal outcomes as (X).
Baselines
As the name suggests, relative hypervolume improvement is normalized measure of improvement, enabling the sessment of batch evaluations across different intervention sets. Given B Bis we propose the following batch selection strategy for our MO-CBO algorithm:
B arg max RHVI(p(X.. B.), (Xa)))
(6)
B. B.C.B......B.sl

---

Multi-Objective Causal Bayesian Optimization
100
MO-CBO (ours)
MO-CBO (ours)
20
DGEMO
DGEMO
80
(IP(X)
G0000
()((()))
Y22
X 10
60
40
0
0
10
10
20
30
40
Y1
Y1
Figure SYNTHETIC-L Pareto front approximations from MO- CBO (ours) and DGEMO, Our method offers a higher coverage of (P.)))
Figure 5. SYNTHETIC-2 2. Pareto front approximations from MO- CBO (ours) and DGEMO. Our approach tightly fits (X)()
Evaluation We assess the quality of the resulting Pareto fronts by measuring their proximity to (P((X)) using the generational distance (GD) (Schutze et al., 2012). The GD is defined as the average distance from any point in the approximated front to its closest point on the ground- truth front. Moreover, we calculate the diversity of the identified solutions using the inverted generational distance (IGD) (Schutze et al., 2012). The IGD represents the av- erage distance from any point in the ground-truth front to its closest point on the approximated front. The mathemat- ical definitions are given in Appendix D. We present the performance metrics from our experiments in Table and Table 2.
SYNTHETIC-2 In this setting, an unobserved confounder exists between Yi and X4, placing SYNTHETIC-2 in the general case where hidden confounders may influence tar- get variables through their ancestors. Consequently, it holds Og.Y {[X2, X3), {Xi, X2, Xa)). The Pareto front ap- proximations illustrated in Figure 5, demonstrating that while the baseline method DGEMO fails to identify solu- tions on Pg (P))))), MO-CBO does indeed discover them. Similar observations are seen for the other baselines, see Appendix D. Further experiments reveal that only interven- tions on {X2, X33} can yield Pareto-optimal solutions in Pf (IP()) This can be explained follows: The baseline strategy disrupts the causal path X4 - X1 - Y,, letting the unobserved confounder influence Y1 without propagat- ing through the aforementioned path. In contrast, our ap- proach allows interventions on {X2, X3), preserving this causal structure. This distinction is crucial as the structural assignment of Yi includes the term -X1 X2 U/2, with U denoting the unobserved confounder (all structural equa- tions are specified in Appendix C). Not intervening X1, causes this term to always be negative, yielding lower func- tion values for Yi. However, if we do intervene on X1, it is positive with probability 0.5, causing higher values for Yi in the averaged outcomes.
We run each experiment until predefined cost budget is exhausted, assuming each intervention comes at certain cost. The cost structure is detailed in Appendix D. In this section, we will only present the visual results from the experiments with DGEMO, with corresponding plots for the other baselines provided in Appendix D.
5.1. Synthetic Problems
SYNTHETIC-1 As previously discussed in Section 4.1, holds Og.y = [[X,, X2 We observe that the gen- negligible for MO-CBO
erational distances in Table and all baseline algorithms. This is also supported by the Pareto front approximations shown in Figure 4 where solu- tions found by both MO-CBO and DGEMO closely match ()))). Similar results are observed for the other base- lines and are presented in Appendix D. Theoretically, these findings are expected, as xxX,xx x) = p(Og. Y. x (0g, x] guarantees that the baselines can reach the same solutions as MO-CBO. Furthermore, fers a better coverage of P9 (P(X),, a result confirmed by its lower inverted generational distance in Table 2. The improvement likely stems from avoiding unnecessary in- terventions on X3 and X4, allowing for more exploratory interventions on Og.y within the same budget.
5.2. Real-World Problems
HEALTH We revisit the causal graph from Figure 1. which is based on real-world causal relationships in the healthcare setting (Ferro et al., 2015). For patients sen- sitive Statin medication, one might aim to minimize both Statin usage and PSA levels simultaneously. There is only one possibly Pareto-optimal minimal intervention set, 0g {[BMI, Aspirin) J. Similarly to SYNTHETIC-L, both our MO-CBO algorithm and the MOBO baselines iden- tify Pareto-optimal solutions, while the baselines tend to produce sparser approximations of (P)()). See Figure for the results obtained using DGEMO.
observe that our method of-
8

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Multi-Objective Causal Bayesian Optimization
search space identify such solutions. Notably, have observed that traditional multi-objective optimization is sim- ply misspecified for causal systems and can therefore yield suboptimal outcomes. More specifically, our experimen- tal results reveal two distinct scenarios: In the absence of unobserved confounders between the targets and their ances- tors, both our MO-CBO algorithm and the MOBO baselines recover optimal solutions, albeit our method can offer a higher solution diversity. In the contrasting scenario, the MOBO baselines can lead to suboptimal solutions, while our method remains effective. These observations align with our theoretical findings.
7.0
MO-CBO (ours)
DGEMO
6.5
Pi (IP())
PSA
6.0
5.5
5.0
0.1
0.2
0.3
Statin
Figure HEALTH. Pareto front approximations from MO-CBO (ours) and DGEMO. Our approach yields better coverage of (P))).
The search space reduction in Section 4.1 requires prior causal knowledge, which is notable limitation of our ap- proach. In some domains, such knowledge is accessible through experimental studies (Blomqvist et al., 2020), and when unavailable, could potentially be inferred using meth- ods from causal discovery (Zanga et al., 2022). Moreover, in our current implementation, the surrogate model assumes independent outcomes, overlooking shared endogenous con- founders. Future work could enhance sample efficiency by integrating multi-task Gaussian processes to capture shared information across treatment variables. Other directions for future research include the adaptation of existing CBO variants to the multi-objective setting.
0.8
0.6
MO-CBO (ours)
0.4
DGEMO
(P))))
0.2
-2
0
loan duration
Figure 7. CREDIT APPROVAL, Pareto front approximations from MO-CBO (ours) and DGEMO. Our approach yiclds better cover- age of Pf (F(X)).
A Acknowledgements
This research was supported by the Bavarian State Ministry of Education, Science and the Arts, the Engineering and Physical Sciences Research Council [EP/S023917/1] and the Alan Turing Institute's ss studentship scheme.
CREDIT APPROVAL We consider causal system which models the credit approval probability function of de- mographic and financial variables. see Appendix D for the SCM specifications. The model is inspired by the German Credit UCI dataset (Murphy, 1994), with causal dependen- cies adapted from Karimi et al. (2020). Our objective is to maximize the probability credit approval and the re- ceived loan duration (measured as a deviation from the mean). There are no unobserved confounders, resulting in Og.y { (loan amount, income, savings)). Similarly to before, we observe that MO-CBO algorithm can yield more dense representation of 1P (P((P( (P)))) compared to some baselines.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which feel must be specifically highlighted here. We further emphasize that the health-related causal graph is a simplified, illustrative example with no ethical concerns or implications for clinical practice.
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6. Conclusion
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Multi-Objective Causal Bayesian Optimization
A. Decomposition of MO-CBO Problems
Recall the definition of the local MO-CBO problems.
Definition 3.3 (Local MO-CBO problem). Let X, E P(X) be an intervention set. Then, the multi-objective optimisation problem defined by the objective functions (4(X,, (X,, .) ((),) + IR, x, - (xX,,, X,), S m, is called local MO-CBO problem w.r.l. X,
For the local MO-CBO problem w.r.t. X, E P(X), we denote its Pareto set as (X.) and the associated Pareto front as P/ (
Proposition 3.4. Given (9, Y, X, C)2 les C P(X) be non-empty collection of intervention sets. Then, it holds
|S|
P)(S) S (X.)
a=1
Proof. Assume for contradiction that PS(S) & - Pj (X+). Then, there exists some E R such that E ()(S) and z & (X.) for all s = I,...: S|. For some intervention set X', E S and intervention value x, E D(X'), it holds z (WI(X, xx) ........ Pm x()). Let P (I........ (X)) be the Pareto sets of the associated local MO-CBO problems w.r.t. XI..., (ss) respectively. Since z $ P) (X.), follows x' $ P! (X.). i.e. x' is not Pareto-optimal in the local MO-CBO problem w.r.t. X Thus, there exists another intervention value x" E D(X,) such that μ4; (X, xX, x() 2 p4 (X,, x") for all and pu (X,, x > pu (X,; x") for least one i S m. In other words, the intervention set-value pair (X², x is not Pareto-optima for S since it is dominated by (X, x"). Therefore, (S) which is contradiction.
B. Reducing the Search Space
Lee & Barcinboim (2018) leverage the graph topology of an SCM to identify intervention sets that are redundant to consider in any optimisation scheme. Their formalism exploits the rules of do-caleulus to identify invariances and partial-orders among intervention sets, in order obtain those sets that could potentially yield optimal outcomes for given graph. To take advantage of their ideas for this paper, the relevant concepts and their theoretical properties must be extended to accommodate multi-target settings, which will be the focus of this section.
Let (V, F, P(U)) denote an SCM and G its associated acyclic graph that encodes the underlying causal mechanisms. Recall that we assume C = Ø, i.e. there are no non-manipulative variables. In this section, we require the notation [x(,),(XX=x..) [W]+ := E do = x.)] for sets X, S X, W V.
B.1. Equivalence of Intervention Sets
As first step, we establish invariances within P(X) in regards to the effects of intervention sets on the target variables. Recall the following definition from the main part of the paper.
Definition 4.1 (Minimal intervention set). A set X, E P(X) is called minimal intervention set if, for every SCM conforming to G. there exists no subset X C X, such that for all x, D(X.) holds all 1 < i < m.
-X,, x, [X!]), for
We denote the set of minimal intervention sets with Mlg, In other words, no subset of a minimal intervention set can achieve the same expected outcome on Y. Intervention sets, that are not minimal in the sense of Definition 4.1, are redundant to consider in any optimization task.
Proposition 4.2. X, E PXX) is a minimal intervention set if and only it holds X, S. Can(Y)gg (Y))z,
Proof. (If) Let x, E D(X.) be any intervention value. Assume that there is a subset X C X, such that for all SCMS conforming to G it holds [o,,, = xx)] EYyjjoo(X' X xx(X for all S S m. For the sake of contradiction, assume X, S an(Y) Consider an SCM with real-valued variables where each V with its own binary exogenous variable Uy with P(Uv = 1) = 0.5. Let the function of an endogenous variable be the sum of values of its parents. Then, there exists a directed path from X, \X; to some Y, without passing X
is associated
12

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Multi-Objective Causal Bayesian Optimization
Hence, setting W X, X to the values [[ X' - x, yields B|Y W w, X; x xx(x)))) [[Y, x, X )),, contradicting the assumption.
(Only if) Assume that X, g an (Y))x Then, for X = X, n (Y) it holds X C X, and by the third rule of do-calculus, for every x, D(X,) holds E[y, [XY,, x.)] = BIY, do [x]]] S m. This is
contradiction because X, was assumed to be minimal intervention set.
B.2. Partial-Orders among Intervention Sets
Recall the definition of possibly Pareto-optimal minimal intervention sets.
Definition 4.3 (Possibly Pareto-optimal minimal intervention set), A set X, E MMg.y is called possibly Pareto-optimal if, for at least one SCM conforming to G, there exists x, E (D(X,) such that (X,, is Pareto-optimal for P(X), and for no X E MMg, \ x,, E ((X") it holds y(X',, s: X2) Σ ( (X,, X,), for all 1 i m.
Characterizing such sets is the aim of this section. For simplicity, we first consider the special case in which G exhibits no unobserved confounders between Y; and any of its ancestors.
Proposition 4.4. If no Y, is confounded with an(Yi)g via unobserved confounders, then pa(Y) is the only possibly Pareto-optimal minimal intervention set.
Proof. Let X, = pa Y),, and let Pa(Y)g * X', be another minimal intervention set with x, E D(X.). Define Z = XX, 0 nX) and W = X 0 X)). Moreover, we choose an intervention value x E D(X.) such that it
dominates x D(X.) which is given by x [X]] = x[[ and x, [Z] B[Z]do(X', = xy)]. If x, is non-dominated,
define x = Xs. Then, for = 1,..., m holds
B|y do = =x]]] = [[]]do((X, nx, = x, Z = x114
(8)
S do n nx, = xX, Z = x. xx(
(9)
do[ n nx' = x, Z = (x W- = (()]
(10)
|o] n nx; =x,, W x((W)), ), Z [.]
(11)
EE|,/do(X, nx, =x,[X'] W = =x([W)]
(12)
= E[Y]do((X' = =x,].
(13)
where the inequality holds because x, (weakly) dominates x Note that the second and third equalities are derived through the third and second rules of do-calculus, respectively. The second rule of do-calculus assumes that Y; is not confounded with a via unobserved confounders. For X, pa( Y) it is possible to construct an SCM, conforming to G, such that strict inequality holds for some Y,, see the proof of Theorem B.2. This shows that pa YYg is the only possibly Pareto-optimal minimal intervention set.
We continue and study the more general case where unobserved confounders can be present between Y; and any of territory and
its ancestors. For this intent, we extend two existing concepts, called minimal unobserved-confone interventional border (Lee & Bareinhoim, 2018), the multi-objective setting. Using these notions, we derive results which can fully characterize possibly Pareto-optimal minimal intervention sets in the aforementioned scenario.
Definition 4.5 (Minimal unobserved confounders' territory). Let 71 = G[Ann[[An)aalaa gg]. A set of variables T in H. with Y C T, called a UC-territory for 9 w.r.t. Y if De(T) = T and CC(T) = T. The UC-territory T is said to be minimal, denoted T = (UCT(G, Y), if no T' C T is a UC-territory.
minimal UC-territory for & w.r.t. Y can be constructed by extending a set of variables, starting from Y, and iteratively updating the set with the c-component and descendants of the set. More intuitively, it the minimal subset of An( YYg that is governed by unobserved confounders, where at least one target Y, is adjacent to unobserved confounder.
Definition 4.6 (Interventional border). Let T MUCT(G, Y). Then, B pa () \T is called the interventional border for G w.r.t. Y, denoted as IB(G, Y).
We have already described these concepts in the main part. Before connecting the notion of minimal UC-territory and interventional border to possibly Pareto-optimal minimal intervention sets, we require the following proposition:
13

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Multi-Objective Causal Bayesian Optimization
Proposition B.1. Let T be a minimal UC-territory and B an interventional border for 9 w.r.t Y. Let X, C X be an intervention set and S (T X2) UB B. Then, for any X2 =D(xs) there exists E D(S) such that [E(; s)] I Eo = =x,)], for all i = 1, m. m.
Proof. (Case B C x Let x, E x be an intervention value. Then, by the third rule of do-calculus, it holds do x x,) [X n (T U B) x,
( U B))], S i m. Since X, n (TU S, by setting X T B), it follows ()Y, x.)] = do s]]l
(Case B - Xct E D(X,) be an intervention value. We define B' = S nS) - (X.,. - B (X. and W = (X, ns) xX, n (T)))). Moreover, let E D(S) such that it dominates s E D(S). which is given by s s(B'] (X, = x.)] and [X.] = x [U If non-dominated, we set
Then, for all i ,,..., m
holds
E[Vi/do(S= - s*)] = E|Vidoo (TUB) - s'[X. B' B' = s B'DI
(14)
S E|Y; n (TUB) s(X,) B' s s(B'i>
(15)
do n (TUB) =[X,] B' s(B'), W xxx,\]]
(16)
-E[Yi/do(X, n (TUB) s W x. [.[,),, B' s(B']
(17)
= do(,, (TUBB = s[X,), W = [x][]]
(18)
= [[[] xx)),
(19)
where the inequality holds because s is (weakly) dominated by s*. Furthermore, the second and third equalities are derived through the third and second rules of do-calculus, respectively.
The following proposition is a building block for characterizing possibly Pareto-optimal minimal intervention sets via interventional borders. The proof is similar to the one given by Lee & Bareinboim (2018)
Proposition B.2. IB, Y) is possibly Pareto-optimal minimal intervention set.
Proof. The intuition of this proof is to construct an SCM, conforming to G, for which the single best strategy involves intervening on IB, Y). Let T and B denote MUCT(G, Y) and (G, Y), respectively. Every exogenous variable in U shall be binary variable with its domain being {0, 1}. Let + denote the exclusive-or function and V the logical OR operator.
(Case T= Y) In this case, B corresponds to the parents of Y. Therefore, no target variable Y; is confounded with an(Y:) via unobserved confounders. Define an SCM such that
Each endogenous variable V E V is influenced by an exogenous variable Uy E V;
fv.
Vu + V pay; with P(UV: = 0) == 1, for all = 1.... m;
fx
O x pax) for X X and P(U = 0) 0.5 for every U E UVUUM U%).
By the third rule of do-calculus and by taking conditional expectations, it holds
do = 0)] E|Y, do Y)g 0)]
(20)
E[Y, do (Y:)g 0). U + PPU 0) EEY do(pa(Y)) 0), U% =0] P(U% 0) (21)
(22)
for every 1 sis S Meanwhile, all other interventions yield expectations greater than equal to 0.5 in at least one component. Therefore, B is a possibly Pareto-optimal minimal intervention set.
(Case T C Y) In this case, at least one target variable Y; has an unobserved confounder with its ancestors. As a first step, it will be shown that there exists an SCM, conforming to 7 = G[T U UB), where the intervention do do = 0) is the single best strategy. To achieve this, we first define individual SCMS for each unobserved confounder in H[T]]. and merge them into a single SCM where do(B 0) is indeed the best strategy. Let U
{U,}* be the set of unobserved confounders in H[T).
14

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Multi-Objective Causal Bayesian Optimization
Y1
X3
X
X
X
Y2
Y2
X6
X5
(a) g
(b) H1
(c) H2
Figure 8. Original causal graph and its color-coded subgraphs for each unobserved confounder.
Table 3. Values for M2 and M given X4 X5 0. The target variables are shown as bit sequences, Y and Y1, as well as binary
values, Y1 and Y2.
Mi
M2
M
U1
U2
x 2
x(1 1
((1)
((1)
X
(2)
(()
(2)
y(2)
yr(2)
Y1
Y2
Y1
Y
1
2
0
0
2
2
1010
1010
0
0
0
1
0
0
1
1
0110
0110
0
0
1
0
1
1
1
2
2
1001
1001
0
0
1
1
0
0
0101
0101
0
0
Given Uy E U", let B(i and R() denote its two children. We define an SCM Mj, where the graph structure is given by
H, H De ((BU) (BB,,OO))
n(n)) u (B (De ({ BL) R(()) })]
(23)
and all bidirected edges, except for U, are removed. order the structural equations for variables in Hy. the vertices will be labelled via colour coding: Let vertices in De (B()) \De (R()) be labelled as blue, De (R()), # \De (BG)) H as red, and De (B6)),, n n De (R())), as purple. All target variables are coloured as purple as well. Moreover, B6) and R(5) shall perceive U, and purple-coloured variables follows. For every blue and red vertex, the associated structural equation returns the common value of its parents of the same colour and returns 3 if coloured parents' values are not homogeneous. For every purple vertex, its corresponding equation returns 2 every blue, red and purple parent 0,1, and 2, respectively, and returns if 1,0,1, respectively.
parent coloured blue with value U, and red with value Uy respectively. The blue-, red- set to 3 if any of their parents in B is not O. Otherwise, their values determined as
Next, SCMS MI,..., Mr will be merged into one single SCM, that conforms to H. and which do(B 0) the single best intervention. Note that in My all variables can be represented with just two bits. To construct unified SCM, variables in T are represented with 2k bits, where M, takes the 2 11 and 2/thh bits. Every target variable Y; is represented sequence of bits and binarised follows. Y, is set to 0 if its 2j 1hhh and 27th bits 00, 01 10 for every and otherwise. Let P(U, 1) 0.5 for U, U'. Therefore, it holds Y; = 0 if do B 0) and Y; = 1 if do * 0). If any variable in T is intervened, then at least one SCM M, will be disrupted, resulting in an expectation larger than or equal to 0.5 for at least one target variable. In the multi-target setting, it may happen that some target variables do not occur any of the My''s 's. This happens if a target Y, has no parents in T, but only in B. For all such Y's,, we set fy; = u + V pay, with such, the newly constructed SCM enforces (doBBBBBBBB = 0)] Rs 0. Meanwhile, all other interventions yield expectations greater than or equal to 0.5
k.
P(U = 0) R
As last step, the previously defined SCM for H = G U B). will be extended to an SCM for G. However, we can ignore joint probability distributions for any exogenous variables only affecting endogenous variables outside of H. Setting structural equations for endogenous variables outside of H is redundant as well. For V E An(Y)g)T \T, we define the structural equations as fv = { u @ (9 pay). For E U,U', we set P(U = 0) = 0.5 if U's child(ren) is disjoint to T, and PU 0) otherwise. Note that do(B = 0) is still the single optimal intervention. Therefore, B possibly
15

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Multi-Objective Causal Bayesian Optimization
Pareto-optimal minimal intervention set.
In order to illustrate the construction an SCM where (G Y) 0) is the single best strategy, consider Figure showing exemplary graph and its colour-coded subgraphs, Hi and H2. for each unobserved confounder. Table presents the associated values for Mi and M2, well values for the target variables in the final SCM M. The next proposition generalizes the previous one.
Proposition 4.7. x Y) is a possibly Pareto-optimal minimal intervention set for any X, E E.))
Proof. Let X, be an intervention set. Let us denote T = MUCT(Sx,, Y) IB Y) and To = MUCT(G, Y).
Using the strategy from Theorem 4.7, we construct an SCM for G while ignoring unobserved confounders between T and To T. Let U' be the set of such unobserved confounders. Now, the SCM needs to be modified to ensure that do - 0) the single best intervention. Every U E U shall flip (i.e., + 1) the value of its endogenous child in T whenever U 1. Let PUU 0) 1, so that it holds doy 0)] 0. Intervening B expectations around 0.5 or above.
on any variable in T results in
Notably, Proposition 4.7 extends Proposition B.2 when X, * Ø. Note that, by iterating over all intervention sets X, E P(X), we can discover possibly Pareto-optimal minimal intervention sets in a given graph. The following theorem is an extension of the main result by Lee & Bareinboim (2018) to the scenario where multiple target variables are present. It shows that the aforementioned strategy suffices to find not some, but all, such sets.
Theorem 4.8. set X is a possibly Pareto-optimal minimal intervention set if and only if holds IB(),,) Y) = X.-
Proof. (If) This a special case of Proposition 4.7.
(Only if) Let X, be minimal intervention and x, DDXx)) intervention value. Denote T MUCT( Y), B IB x Y). To MUCT(G, Y) and Bo IB, C, Y). From Theorem B.I, know that POMIS intersects with An(Ba)o Bo and thus, it is possible to conclude X, C To U Bo Y. Note that it holds X, S An(B); since otherwise it would follow X, n T * 0,, which contradicts that X, is neither descendant of some variable nor confounded in Gx, Let n Moreover, we define an intervention value b* D(B) such that it dominates E D(B). which is given by b[B"] = [E[ = x.) and b b[X.] = x, [B]. If is non-dominated, we set b =
B' = (X, 0 B) and W X,\(X, B).
Then, for all i I..... m, holds
do b*)l doy n X, b*[x,] ] B' = b*[D')]
(24)
2 do Xx, b[X.), B' b[B'])]
(25)
do n X, b/X,). B' b/B'), W xx W])
(26)
Eooy n X, -b/X,/, W x. W)), B' bB]]
(27)
EE]y Xx, = =b[X,), W W W[)]
(28)
= doX, = =x)].
(29)
where the inequality holds because is (weakly) dominated by b'. Furthermore, the second and third equalities are derived through the third and second rules do-calculus, repectively.
Theorem 4.8 provides a necessary and sufficient condition for a set of variables to be a possibly Pareto-optimal minimal intervention set. The proof of the theorem gives the following corollary:
Corollary 4.9. Let X, P(X) and X IB( Y). For any x, E D(X,) there exist x, D(X') such that it holds x() & (X,,xx xx), for for all 1 < S m.
16

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Multi-Objective Causal Bayesian Optimization
C. MO-CBO Problems
In this section, we present the collection of synthetic and real-world causal graphs used to evaluate the performance of MO-CBO in comparison to standard MOBO baselines.
Cost of Interventions The MO-CBO algorithm requires evaluating the objective functions which inherently involves implementing interventions on the system. However, in practical scenarios, such interventions can be costly, making important prioritize sample efficiency and explicitly account for the cost of an intervention do(X, X2), denoted as cost(X,, X.). We use the convention cost(X., xx) xx,,x fixed cost budget, terminating once the budget is exhausted. To reflect practical constraints, every MO-CBO scenario includes a tailored cost structure that accurately represents the varying difficulty or expense of different interventions.
cost( (:)). Each experiment is conducted under
C.1. SYNTHETIC-1
We introduce the first synthetic MO-CBO problem in our experimental study, referred to as SYNTHETIC-I, 1, which is defined by the causal graph 9 and associated structural assignments presented in Figure 9. The interventional domains are specified as 1)), 2(2)) = [-1, 2] and D(x)), D(x,) = 1.. Moreover, all exogenous variables follow the standard normal distribution, and there are no unobserved confounders. All treatment variables X,, S i < 4, shall have fixed unit cost of
cost(X,, (X,, 24) for all 2; D(X;).
X3
X1
Yi
X1 exp((Xs Xx)/2) + Ux,
+ +x)) + +v, Y2 = (X1 +x2 10)² Uv
X2 = ((Xs - xx)/2)* + Ux2
X1 UXs
Ux, 10,,))
X4 = Ux4
X4
X2
Y2
Figure 9. SYNTHETIC-L 1. An SCM consisting of four treatment and two output variables, depicted with grey and red nodes, respectively. There unobserved confounders.
C.2. SYNTHETIC-2
SYNTHETIC-2 is the next MO-CBO problem of our experimental study, defined by the causal graph G and associated structural equations in Figure 10. The interventional domains are D(X,) [-2, 5]. D(X,) [- and D(X,) = [0. 5] for i 1, Moreover, the exogenous variables Ux,üUx follow Gaussian distribution, and there is unobserved confounder U influencing the target variable Y1 and its ancestor X4. All treatment variables X, 1 < < 4, have fixed unit cost of cost(X;, = 1 for all x; E D(X;).
X1 X1/2+ UX,
X₂ UR
X; -U
X4
X1
Yi
X4 U + UR,
Vi = In(1 + - U
Y2 sin(X3) - -x3 50
X2
Xs
U (-4,4), P(U = - P(U = 0.5
Ux,, Uv, ~ (00,0.5)
Figure SYNTHETIC-2. SCM consisting four treatment two output variables, depicted with grey and red nodes, respectively. It includes an unobserved confounder, denoted via the dashed bi di-directed edge, affecting one output and its ancestor.
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Multi-Objective Causal Bayesian Optimization
C.3. HEALTH
The MO-CBO problem HEALTH is defined by the causal graph and structural equations in Figure 11. This model originates from previous works of Ferro et al. (2015), and is based real-world causal relationships. It captures prostate-specifer antigen (PSA) levels in causal relation to its risk factors, such BMI, calorie intake (C1) and aspirin usage. The variable Aspirin indicates the daily aspirin regimen while Statin denotes a subject' statin medication. Additionally, PSA represents the total antigen level circulating in subject's blood, measured in ng/mL. For patients sensitive to Statin medications, the aim is to determine how to manipulate relevant risk factors to minimize both Statin and PSA. To this end, we treat both Statin and PSA as target variables. The treatment variables include BMI, Weight, CI, and Aspirin usage with interventional domains D(BMI) [20, 30). D (Weight)
[50, 100). D(ci) = [-100, 100] and ((Aspirin) = [o. 1]. We choose consider a specific age groups of interest, and define Usge as a Gaussian random variable with mean 65 and standard deviation 1, focusing on individuals close to the age of 65. The variables CI and Aspirin are set to have fixed unit cost, i.c. cost x) = 1
for X E fCI, Aspirin), I E D(X). Since BMI and weight are significantly harder to treat, we increase their cost to cost(X, (X, 2) = 3 for X E {BMI, weight), x E D
The single-objective version of HEALTH, aiming to minimize only PSA, has previously been used to demonstrate the applicability of CBO (Aglietti et al., 2020), as well as for several of its variants (e.g. Gultchin et al. (2023) and Aglietti et al. (2023)).
BMR
Height
CI Uei: Ue ~ 1000000, 100)
BMR 1500 + uMm: Unsight =,,(-,,))
Weight
height = 175 + 100hrigh Unright ~ (..
BMI
age Usge: Usge N(6,,
CI
weight = (BMR + 6.8age (11tghh//113+++111)7777
C1150/7716)
Aspirin
Age
ВMI weight/
(hte//eeeeiihoooee weight/(height/Ing
aspirin o( + 0. 0. 0.03BMI),
a(x) 146=
statin = o (113 + 0. 0.10age + 0.20BM1)
cancer a (2.2 0.05age + 0.01 0.04statin + 0.02aspirin)
Cancer
PSA = 6.8 0.0 0.15BMI 0.6statin + 0.55aspirin + cancer + Upa
Statin
PSA
Upa (00
Figure/ 11. HEALTH. An SCM with relations between variables such as age, BMI, aspirin and statin usage, and their effects on PSA levels (Gultchin et al., 2023). .) denotes uniform distribution and (a, b) a standard Gaussian distribution truncated between and 6. Red, orange, and grey nodes depict target, manipulative, and non- non-manipulative variables, respectively.
C.4. CREDIT APPROVAL.
The final MO-CBO problem in our experiments is called CREDIT APPROVAL, and is specified by the causal graph and structural equations shown in Figure 12. This problem models the probability of credit approval as a function of various demographic and financial variables, including age, gender, education, loan amount, loan duration, income, and savings. It is based on the German Credit UCI dataset (Murphy, 1994) and causal relationships adapted from in (Karimi et al., 2020). We treat both approval probability and loan duration as target variables. The treatment variables are education, loan amount, income and savings, with interventional domains given as: D(education) = 0.5,, 0.5), D(loan amount) = [-1, D(income) = 2- 1]. and (savings)) = [-5, 1]. Note that the variables age, education, loan amount, loan duration, income and savings are modelled as a deviations from their means. reduce complexity and focus our analysis, we fix the gender variable to 0 (c.g., male), diverging from the original specification by Karimi et al. (2020), where gender is defined as (0.55. Additionally, we assume that all other variables remain close to their mean values and reduce the level of observational noise compared to Karimi al. (2020). The variables loan amount and income are set to have fixed unit cost, i.e. cost( X, .r) = 1 for X E (loan amount, income), E D(X.. For education and savings, we increase the cost to cost(Xxz) x) = 2 for X E {education, savings}. x E D(X).
18

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Multi-Objective Causal Bayesian Optimization
age
gender
gender
age -35 Uages Une Ga0mma(10535
edu.
amount
edu. = -0.5 + o (-1+ 0.5gender + +o (0.lage) + Undu -1,
Una. ~ ~.(00,
amount 1 0.1. 5)(5 age) + gender Unmount, Unmonn NO0,4)
duration -1 0.0lage + 2gender + amount + dunaation
Udunation
(0,9))
duration
income
income = -4 + 0.1 35) + 2gender + gender edu. + Unnvosse Unncome ~ (0,44)
savings a--4 + 1.5 1K oiccmme Usavings Usaviggs ((,
savings
approval
approval (0.33 amount duration + income savings income savings))
(00.3(-mmount
Figure CREDIT APPROVAL. An SCM which models the probability of credit approval as a function various demographic and financial variables (Karimi al., 2020). Red, orange, and grey nodes depict target, manipulative, and non-manipulative variables, respectively.
D. Experiments
D.1. Hyperparameters
MO-CBO algorithm The DGEMO backbone of our MO-CBO algorithm has mostly the same hyperparameters as its original implementation from Konakovic Lukovic et al. (2020). The batch size is set to 5.
ParEGO We adopt a batch variant of ParEGO by using random scalarization weights in each iteration, with b being the batch size. Moreover, Chebyshev scalarization (Miettinen, 1999) and the CMA-ES algorithm (Hansen, 2023) are used to solve the scalarized single-objective problems with a = 0.5 initial standard deviation.
MOEA/D-EGO Following Konakovic Lukovic et al. (2020), our implementation of the MOEA/D-EGO baseline follows its original framework, with the key difference being the removal of FuzzyCM. Given the current computational efficiency of training Gaussian process models, they opt use them directly for prediction instead of relying faster but less accurate approximation techniques. As result, this version may offer improved performance due to the enhanced predictive accuracy. We employ simulated binary crossover and polynomial mutation for MOEA/D, with the remaining hyperparameters detailed in Konakovic Lukovic et al. (2020).
TSEMO For TSEMO, we largely adopt the same hyperparameter settings as in the original implementation. Specifically, we use 100 points for spectral sampling.
NEHVI We implement qNEHVI using the botorch library. We use optimization restarts, and 64 raw samples for acquisition maximization. Moreover, the acquisition function uses Sobol QMC sampler with 128 samples.
DGEMO For DGEMO, we retain the hyperparameter configuration from its original implementation.
D.2. Performance Metries
We require metrics to assess the quality of given Pareto front approximation. These metrics evaluate both the convergence of the approximated front the true front and the diversity the solutions across the performance space. For given optimisation problem, let A be the set of points from approximated Pareto front. If the ground-truth Pareto front is known, it is possible to evaluate how well A approximates it given the following two metrics. Let Z be the set of points on the true Pareto front.
Generational distance (GD) A common performance indicator for evaluating given Pareto front approximation is the so-called generational distance (Schutze et al., 2012). is the average distance from any point a, E A to its closest point in the Pareto front Z. Formally,
A
1/p
1
GD(A) =
d
(30)
A
1m1
19

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Multi-Objective Causal Bayesian Optimization
where de is the Euclidean distance from a; its nearest point in Z. We set p in experiments.
Inverted generational distance (IGD) The inverted generational distance measures the distance from any point z; E Z to its closest point in A (Schutze et al., 2012). Thereby, it serve indicator of the coverage given by the approximated front. Formally,
1/p
IGD(A)
1
d
(31)
7
where de is the Euclidean distance from 2, its nearest point in A. We p 2 in experiments.
The GD evaluates the convergence of the approximated Pareto front to the true front, whereas the IGD measures the diversity of the solutions across the output space. For both metrics, smaller values indicate a better approximation of the true Pareto front.
D.3. Runtime
All experiments were executed on machine equipped with an Apple M2 processor and 8GB of RAM. The average runtimes are reported in Table 4.
Table 4. Algorithm runtime comparison (scconds per iteration), averaged across 10 sceds.
Problem
ParEGO
MOEA/D-EGO
TSEMO
@NEHVI
DGEMO
MO-CBO (ours)
SYNTHETIC-1
14.39
0.62
0.20
14.23
4.35
3.43
SYNTHETIC-2
9.29
0.74
0.35
12.09
4.47
3,82
HEALTH
8.65
4.19
3.81
18.98
6.26
6.12
CREDIT APPROVAL
10.13
0.77
0.34
20.87
26.45
10.55
20

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Multi-Objective Causal Bayesian Optimization
D.4. Results
SYNTHETIC-1 We present the experimental results for the MO-CBO problem SYNTHETIC-I, Figure 13. We observe that MO-CBO algorithm consistently outperforms all MOBO baselines, yielding more diverse and well-distributed solutions across the target Pareto front (I)())) All methods were run until fixed cost budget of 150 was exhausted.
100
100
ParEGO
MOEA/D-ECE
80
60
40
40
0
5
10
15
0
5
10
15
100
TSEMO
NEHIV
125
100
75
50
40
0
5
10
15
0
10
15
100
100
DGEMO
80
60
40
0
5
10
15
0
5
10
15
(P)())
MO-CBO (ours)
Figure 13. SYNTHETIC-L 1. Comparison of Pareto front approximations produced by our MO-CBO algorithm and various MOBO baselines: ParEGO, MOEA/D-EGG TSEMO, (NEHVI, and DGEMO. The x rxaxss corresponds to objective Y1. the y-axis Y2.
21

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Multi-Objective Causal Bayesian Optimization
SYNTHETIC-2 We present the experimental results for the MO-CBO problem SYNTHETIC-2, Figure 14, We observe that the solutions identified by our MO-CRO algorithm tightly fit the target Pareto front py (P((X), and strictly dominate those found by the MOBO baselines. All methods were run until a fixed cost budget of 200 was exhausted.
ParEGO
MOEA/D- EGG
20
10
10
0-
0
0
20
40
10
30
50
TSEMO
@EHIV
20
10
10
0-
0
20
30
40
50
10
30
20
DGEMO
20
15
10
10
5-
0
0
10
20
30
40
50
10
20
30
(P)()))
MO-CBO (ours)
Figure 14. SYNTHETIC-2. Comparison of Pareto front approximations produced MO-CBO (ours) and various MOBO baselines: ParEGO, MOEA/D-EGO, TSEMO, (NEHVI, and DGEMO. The æ-axis corresponds to objective Yi, and the y-axis to Y2.
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Multi-Objective Causal Bayesian Optimization
HEALTH We present the experimental results for the MO-CBO problem HEALTH, Figure 15, We observe that the solutions identified by MO-CRO algorithm tightly fit the target Pareto front (P(())) and strictly dominate those found by the MOBO baselines, All methods were run until fixed cost budget of 120 was exhausted.
ParEGO
7.0
MOEA/D- EGO
6.5
6.5
6.0
6.0
5.5
5.5
5.0
5.0
0.1
0.2
0.3
0.1
0.2
0.3
7.0
TSEMO
7.0
@EHVV
6.5
6.5
6.0
6.0
5.5
5.5
5.0
5.0
0.1
0.2
0.3
0.1
0.3
7.0
DGEMO
7.0
6.5
6.5
6.0
6.0
5.5
5.0
5.0
0.1
0.2
0.3
0.1
0.2
0.3
(P(())
MO-CBO (ours)
Figure 15. HEALTH. Comparison of Pareto front approximations produced MO-CBO (ours) and various MOBO baselines: ParEGO. MOEA/D-EGO, TSEMO, (NEHVT, and DGEMO. The æ-axis corresponds to objective Yi, and the y-axis to Y2.
23

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Multi-Objective Causal Bayesian Optimization
We present experimental results for the MO-CBO problem CREDIT APPROVAL in Figure 16. Our MO-CBO algorithm consistently outperforms the MOBO baselines, producing solutions that both more diverse and better distributed across the target Pareto front. The experiments are executed until cost budget of 300 is reached.
CREDIT APPROVAL
ParEGO
0.8
MOEA/D-EGG EGG
0.8
0.7
0.6
0.6
0.4
0.5
-2
0
0.8
TSEMO
0.8
NEHVI
0.6
0.7
0.4
0.6
0.2
0.8
DGEMO
0.8
0.7
0.4
0.6
0.2
-2
(P)())
MO-CBO (ours)
Figure 16. CREDIT APPROVAL. Comparison of Pareto front approximations produced by MO-CBO (ours) and various MOBO baselines: ParEGO, MOEA/D-EGO, TSEMO, gNEVVI, and DGEMO. The z-axis corresponds to objective loan duration, and the y-axis to the approval probability.
24


Reviews:
[{"review_id": "BLuZg8WoXw", "reviewer": "Reviewer_fTDx", "summary": "This paper presents a unification of Multi-objective Bayesian Optimisation and Causal Bayesian Optimisation to causal settings in which dependence between variables is mediated by a causal graph, maintaining the same assumptions of a known causal graph but not the structural equations or exogenous distribution as CBO. This results in a procedure for performing Multi-objective Causal Bayesian Optimisation in such settings. The authors develop theory that allows them to reduce the search space down from the power set over manipulative variables to a smaller set of so-called \"possibly Pareto-optimal minimal intervention sets\", which motivates an algorithm that assesses only possibly Pareto-optimal minimal intervention sets to solve MOCBO problems. The performance of their method is validated in three experiments in which a non-causal Multi-objective Bayesian Optimisation procedure acts as a baseline.", "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": "Dear Reviewer fTDx,\n\nWe greatly appreciate your detailed feedback and your recognition of our contributions! We are happy to address your remaining questions below:\n\n# Proposition 4.2\nProposition 4.2 is based on Proposition 1 of Lee & Bareinboim (2018), which formalizes the concept of minimal intervention sets for graphs with a single target variable $Y$. We extend this result to causal graphs with multiple target variables $Y\\_1,…,Y\\_m$. \n\nYour question refers to the “if” direction of the proof. Let us first state the definition of minimal intervention sets along with the proposition.\n\n**Definition 4.1** (Minimal intervention set)**.** A set $\\mathbf{X}\\_s \\in \\mathbb{P}(\\mathbf{X})$ is called a *minimal intervention set* if there exists no subset $\\mathbf{X}\\_{s}' \\subset \\mathbf{X}\\_s$ such that for all $\\mathbf{x}\\_s \\in \\mathcal{D}(\\mathbf{X}\\_s)$ it holds $\\mu(\\mathbf{X\\_s},\\mathbf{x}\\_s) = \\mu(\\mathbf{X}\\_{s}',\\mathbf{x}\\_s[\\mathbf{X}\\_{s}'])$, $1 \\leq i \\leq m$, for every SCM conforming to $\\mathcal{G}$.\n \n**Proposition 4.2** Given a causal graph $\\mathcal{G}$, $\\mathbf{X}\\_s \\in \\mathbb{P}(\\mathbf{X})$ is a minimal intervention set if and only if it holds $\\mathbf{X}\\_s \\subseteq \\textnormal{an}(\\mathbf{Y})\\_{\\mathcal{G}\\_{\\overline{\\mathbf{X}}\\_s}}$.\n\n# Proof Strategy\n\nThe \"if\" direction of the proposition states: $\\mathbf{X}\\_s \\subseteq \\textnormal{an}(\\mathbf{Y})\\_{\\mathcal{G}\\_{\\overline{\\mathbf{X}}\\_s}} \\implies \\mathbf{X}\\_s$ is a minimal intervention set. To prove this, we use contraposition, meaning we will show the equivalent statement: $\\mathbf{X}\\_s$ is not a a minimal intervention set $\\implies$ $\\mathbf{X}\\_s \\not\\subseteq \\textnormal{an}(\\mathbf{Y})\\_{\\mathcal{G}\\_{\\overline{\\mathbf{X}}\\_s}}$. We show this contrapositive via contradiction: Suppose $\\mathbf{X}\\_s$ is not a minimal intervention set, but assume for contradition $\\mathbf{X}\\_s \\subseteq \\textnormal{an}(\\mathbf{Y})\\_{\\mathcal{G}\\_{\\overline{\\mathbf{X}}\\_s}}$. By definition 4.1, there exists a subset $\\mathbf{X}\\_{s}' \\subset \\mathbf{X}\\_s$ such that **for all SCMs conforming to** $\\mathcal{G}$, intervening on $\\mathbf{X}\\_{s}'$ yields the same outcome as intervening on $\\mathbf{X}\\_s$, i.e., $\\mu(\\mathbf{X}\\_s,\\mathbf{x}\\_s) = \\mu(\\mathbf{X}\\_{s}',\\mathbf{x}\\_s[\\mathbf{X}\\_{s}'])$, where $\\mathbf{x}\\_s [\\mathbf{X}\\_{s}']$ are the values of $\\mathbf{x}\\_s$ corresponding to $\\mathbf{X}\\_s \\cap \\mathbf{X}\\_{s}'$. We construct a specific SCM that conforms to $\\mathcal{G}$, meaning that we assign structural equations between the variables within $\\mathcal{G}$. Then, we leverage these structural equations along with the assumption $\\mathbf{X}\\_s \\subseteq \\textnormal{an}(\\mathbf{Y})\\_{\\mathcal{G}\\_{\\overline{\\mathbf{X}}\\_s}}$ to establish a contradiction, which is given by $\\mu(\\mathbf{X\\_s},\\mathbf{x}\\_s) > \\mu(\\mathbf{X}\\_{s}',\\mathbf{x}\\_s[\\mathbf{X}\\_{s}'])$, thereby breaking the equality. This invalidates our initial assumption, yielding $\\mathbf{X}\\_s \\not\\subseteq \\textnormal{an}(\\mathbf{Y})\\_{\\mathcal{G}\\_{\\overline{\\mathbf{X}}\\_s}}$. We have thus proven the contrapositive statement.\n\n**Generality.** Since $\\mathbf{X}_s \\not\\subseteq \\textnormal{an}(\\mathbf{Y})\\_{\\mathcal{G}\\_{\\overline{\\mathbf{X}}\\_s}}$ is a graph-topological property (i.e. it does not associate to any specific SCM), the above strategy completes the proof. Different SCMs cannot yield different graph-topological properties. The constructed SCM serves only as a tool to establish the desired result. We recognize that this aspect may not have been fully articulated in the original proof and have now revised it to improve clarity.  While our proof required some technical modifications to the one by Lee & Bareinboim (2018), the core idea remains unchanged.\n\nThank you for your time and extensive review of all our proofs. We believe your questions and remarks greatly improve the understanding of our derivations for future readers! We hope this clarification fully addresses your concerns, and we look forward to your response.\n\n## References\n\n Lee, S. and Bareinboim, E. Structural causal bandits: Where to intervene? In Advances in Neural Information Processing Systems, 2018.", "claims_and_evidence": "The theory and experiments support the claims made by the authors.", "methods_and_evaluation_criteria": "I think the experiments to evaluate their method are well-designed and demonstrate the value of the proposed method. They demonstrate this by considering settings in which there are and are not hidden confounders to highlight the overlap in and differences between MO-CBO and MOBO.", "theoretical_claims": "I went through all proofs. The only thing that struck me as potentially problematic (but more likely a misunderstanding on my part, hence I would appreciate clarification from the authors please) was the proof for Proposition 4.2. My concern is that the proof appears limited in generality under the assumption that all nodes are associated with their own fair binary exogenous variable, and that the structural equations are just sums over the values of the parent nodes. Why is it ok to consider this case only to prove what looks like a very general claim in Proposition 4.2?\n\nAlso, is Proposition 4.2 a new result? I am not sure that I have an especially well-calibrated sense of what is significant in this field, but this result seems like it deserves more attention than the authors are currently giving it?", "experimental_designs_or_analyses": "As discussed above, I think the experiments are selected well to highlight the value of this method over non-causal counterparts.", "relation_to_broader_scientific_literature": "The paper relates well to relevant prior literature in both causal decision-making and multi-objective Bayesian optimisation (MOBO). In particular, the method naturally extends upon the DGEMO algorithm for non-causal MOBO, and the benefits of this innovation are clearly demonstrated in the experiments.", "essential_references_not_discussed": "None that come to mind.", "other_strengths_and_weaknesses": "Further strengths are: notation is clear and defined well up front (I found it necessary to refer back to earlier parts of the work to look up notation, but could always find definitions clearly stated earlier); and I appreciated the two **Examples** the authors placed in Section 4 to help with intuition and clarity.", "other_comments_or_suggestions": "Page 4, just before Proposition 4.4, there is a typo: you write \"...which we **proof** in Appendix B.2...\" but you want \"...which we **prove** in Appendix B.2...\"", "questions_for_authors": "Please see **Theoretical claims** 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": "yL3mLD2hmd", "reviewer": "Reviewer_7sGg", "summary": "- Decision-making outcomes depend on causal relationships and evaluating them is costly.\n- Causal Bayesian optimization uses these relationships to find optimal interventions efficiently.\n- Multi-objective causal Bayesian optimization (MO-CBO) extends causal Bayesian optimization to identify Pareto-optimal interventions.\n- MO-CBO can decompose into several traditional multi-objective tasks and balance exploration across them.\n- MO-CBO is validated on synthetic and real data, outperforming non-causal methods when causal information is available.", "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": "Dear Reviewer 7sGg,\n\nThank you for your thoughtful feedback and for recognizing our contributions! We have incorporated all of your suggestions to enhance readability, include additional baselines, and provide a clearer formulation of the causal Pareto front for the community.\n\n# Baselines\nThank you for your suggested improvements to our experimental analysis! We now include the MOBO algorithms DGEMO, TSEMO, ParEGO, MOEA/D-EGO, qParEGO, and qNEHVI as baselines. Additionally, we added another real-world problem, where the SCM describes macro-economic relations and is based on real-world data (Höllig et al., 2023). The average GDs are (qParEGO and qNEHVI are still being implemented):\n\n|       |  DGEMO  |  TSEMO  |  ParEGO  |  MOEA/D-EGO |  MO-CBO (ours) |\n|----|:----:|:----:|:----:|:----:|:----:|\n| Synthetic-1 | **0.14** | 0.16 | 0.30 | 0.38 |**0.14**|\n|  Synthetic-2 | 7.55 | 7.78 | 12.50 | 11.95 |**1.85**|\n| Health | 0.07 | 0.07 | 0.09 | 0.20 |**0.06**|\n| Economics |  4.66  | 4.93 | 5.56 | 6.41 |**1.22** |\n\nAnalyzing both GD and IGD, we find that our method performs better than all baselines across all tasks.\n\n\n# Experiments\n**IGD and GD values at zero interventions.** Thank you for pointing this out - this is a x-ticks label error. Our algorithm is initialized with prior data that is used to compute a first estimate of the causal Pareto front before starting the optimization loop. We now label the start of the plots with len(prior data) = 5 to resolve this unclarity.\n\n**Similarity between MOBO and MO-CBO.** It is theoretically grounded from Proposition 4.4 that both methods will converge to the same results when no $Y\\_i$ is confounded with its ancestors via unobserved confounders, which is indeed the case for both Synthetic-1 and the Health example. In these scenarios, MOBO intervenes on all treatment variables $\\mathbf{X}$, while MO-CBO intervenes only on the parents of $\\mathbf{Y}$. Note that intervening on $\\mathbf{X}$ yields the same results as restricting the intervention to the parents of $\\mathbf{Y}$. As shown in Fig. 4, MO-CBO identifies a greater number of solutions by avoiding unnecessary interventions on elements outside the minimal intervention set defined by the parents of $\\mathbf{Y}$. This approach reduces costs while achieving broader coverage of the causal Pareto front. However, when some $Y\\_i$ is confounded with its ancestors via unobserved confounders, our method can establish superior results than traditional MOBO approaches as demonstrated in Fig. 6 for Synthetic-2. \n\nMoreover, we added the definitions of our metrics as well as the runtime results to the Appendix. All methods, including ours, have an average runtime per iteration between 0.3 and 14 seconds. The fastest method is TSEMO with 0.2 - 4 seconds per iteration (depending on the problem) while ours ranges between 3 and 6 seconds.\n\n# General Discussion\n**Number of independent Gaussian Processes.** The number of independent Gaussian processes (GP) $m$ is equal to the number of target variables, we describe this in the preliminaries of MO-CBO in Section 2. We train the GPs independently from each other, resulting in a linear increase of computational cost with $m$.\n\n**Proof Sketches.** Thank you for this suggestion - we think this greatly improves the overall understanding of our methodology! We have now included a description of the main idea of each proof within 1-2 sentences in the main body of the paper. For instance, for Proposition 3.4 we write: “The rigorous proof of Proposition 3.4 is given in Appendix A, and it exploits the fact that the space of all intervention set-value pairs is the union of the input spaces of each local problem. This allows to match the Pareto optimal intervention set-value pairs with the Pareto-optimal solutions from the local problems where the intervention set is fixed.”\n\n**Causal Pareto front.** In MO-CBO, the input space comprises all possible intervention set-value pairs. Based on proposition 3.4, the causal Pareto front is constructed by identifying the Pareto-optimal points across the solutions of local problems, each corresponding to a fixed intervention set. However, since *Pareto front* is a well-established concept in MOBO, we acknowledge that the phrase \"causal Pareto front\" can cause confusion. To clarify, the causal Pareto front simply represents the best attainable trade-offs *within in a causal system*, considering intervention set-value pairs as inputs. Based on your comment, we propose to rename the term to *Pareto front in causal systems*. \n\nWe have addressed all of your remaining comments as well as added the clarification on PSA (prostate specific antigen) and Statin (a type of medication) to the experiments section. Thanks again for your comments to enhance the clarity of our work! We look forward to your response.\n\n## Reference\nHöllig et al. Semantic meaningfulness: Evaluating counterfactual approaches for real-world plausibility and feasibility, in: xAI, 2023, pp. 636–659.", "claims_and_evidence": "- This research tackles an intriguing issue in causal Bayesian optimization.\n- The theoretical results strongly back the claims of this study, but I have some concerns regarding the experiments.", "methods_and_evaluation_criteria": "- Can you provide mathematical definitions of GD and IGD?\n- What are PSA and Stain in the Health problem?\n- How can we choose $m$ for the number of independent Gaussian processes?", "theoretical_claims": "- The authors can provide the proof sketches of the propositions and theorems.", "experimental_designs_or_analyses": "- All evaluations at intervention 0 should be identical across MO-CBO and MOBO, if the same random seeds are given for both methods.\n- I think the authors can add more baselines such as qNEHVI and ParEGO.\n- Why are the results of MO-CBO and MOBO similar in Synthetic 1?  Both just work well in this problem?  If there is a particular message of this problem, it might be able to be removed.\n- The results of the Health problem are also similar across two methods.  Why do both methods yield similar results?", "relation_to_broader_scientific_literature": "- This work aligns with causal Bayesian optimization and broader scientific literature on Bayesian optimization.", "essential_references_not_discussed": "- I don't think there are specific references not discussed.", "other_strengths_and_weaknesses": "Please see above.", "other_comments_or_suggestions": "- Are there additional real-world problems to consider? Including more could enhance the presentation of this work.\n- In Figure 2, there is Sec. ??.  It should be fixed.", "questions_for_authors": "- As $m$ becomes large, the computational cost of building a surrogate model increases.\n- Could you provide the run-time results for the proposed method?\n- How can the causal Pareto front be identified? Is there a difference between determining the standard Pareto front and the causal Pareto front?", "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": "shs7tO6Psj", "reviewer": "Reviewer_easG", "summary": "The paper considers the problem of multi-objective optimisation with knowledge of the causal graph of the underlying system, where the objectives are the interventional means of a set of target variables. The authors propose a bayesian optimisation solution and prove theoretically that knowledge of the causal graph provides advantages by reducing the search space for optimisation. The authors prove, using synthetic data, that their method is better than the baselines of no knowledge of causality.", "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": "Dear Reviewer easG, \n\nThank you for recognizing our contributions and for your insightful feedback pointing out the need to better discuss the applicability of our approach and adding further evaluations.\n\n# Applicability of MO-CBO\nRequiring prior knowledge about the causal graph is indeed a notable limitation of current causal optimization methods and of our work. We acknowledge that this limitation was not highlighted sufficiently in the initial version of our paper. To better communicate this, we now added a limitation section. \n\nFirstly, in many domains causal knowledge is readily available. For example, the effects of drugs are studied in experimental trials in the medical field. Knowledge about the effects, side-effects, and interactions is available and an example of causal relationships that can be used for CBO. Secondly, if causal information is not readily available, there exist methods to discover it, falling  under the field Causal Discovery (Zanga et al., 2023). Here, the corresponding methods aim to identify causal structures from data. \n\n**Health Problem.** We refer to the Health example as a real-world problem since the relations between the variables were derived from real-world analyses (Ferro et al., 2015). We agree that weight and BMI are not easy-to-treat variables. However, they are treatable under high costs. We increased the cost of intervention for the BMI and weight variables in our experiment and added a respective discussion. As a result, the distinction between MOBO and MO-CBO is even clearer, with MO-CBO demonstrating a significantly higher coverage of the causal Pareto front compared to MOBO. The IGD under MO-CBO is 0.02 whereas the IGD under MOBO is 0.05 in an output space ranging from values 0.05 to 0.4 for the Statin medication (thus the difference in IGD is not negligible) .\n\n**Economics Problem.** We include one more real-world problem, where the SCM describes macro-economic relations based on real-world data  (Höllig et al., 2023). Due to space limitations, we would like to point you to our response to reviewer 7sGg for preliminary results.\n\n# Graph Representation of Interventions \nRegarding your question if intervening on the system would render a different causal graph: An intervention on variables $\\mathbf{X}\\_s \\subseteq \\mathbf{X}$ involves replacing the structural equations $f\\_{X}$ with a constant $x$ for all $X \\in \\mathbf{X}\\_s$, denoted $\\text{do}(\\mathbf{X}\\_s = \\mathbf{x}\\_s)$. Thus, performing an intervention removes the influence of all variables on $\\mathbf{X}\\_s$. The graph $\\mathcal{G}\\_{\\overline{\\mathbf{X}}\\_s}$ represents this intervention and is obtained by removing incoming edges into $\\mathbf{X}\\_s$. Therefore, $\\mathcal{G}\\_{\\overline{\\mathbf{X}}\\_s}$ is used to denote the causal structure under intervention and to perform theoretical analyses regarding the effects of interventions. Importantly, the optimization problem itself remains unchanged. \n\n# Performance Metrics\nWe added the definitions of GD and IGD from Schulze et al. (2012). They are as follows:\nLet $\\mathbf{A}$ be the set of points from an approximated Pareto front and let $\\mathbf{Z}$ be the set of points on the true Pareto front. \n\n**GD.** The GD is the average distance from any point $\\mathbf{a}\\_i \\in \\mathbf{A}$ to its closest point in the Pareto front $\\mathbf{Z}$. Formally,\n             $$\\text{GD}(\\mathbf{A},\\mathbf{Z}) =  \\biggl( \\ \\frac{1}{|\\mathbf{A}|} \\sum\\_{i=1}^{|\\mathbf{A}|} d\\_i^p \\ \\biggr)^{1/p},$$\nwhere $d\\_i$ is the Euclidian distance from $\\mathbf{a}\\_i$ to its nearest point in $\\mathbf{Z}$; we set $p=2$ in our experiments.\n\n**IGD.** The IGD measures the average distance from any point $\\mathbf{z}\\_i \\in \\mathbf{Z}$ to its closest point in $\\mathbf{A}$. \nFormally, \n$$\\text{IGD}(\\mathbf{A},\\mathbf{Z}) = \\biggl( \\ \\frac{1}{|\\mathbf{Z}|} \\sum\\_{i=1}^{|\\mathbf{Z}|} \\hat{d\\_i}^p \\ \\biggr)^{1/p},$$ \nwhere $\\hat{d\\_i}$ is the Euclidian distance from $\\mathbf{z}\\_i$ to its nearest point in $\\mathbf{A}$.\n\nThe GD evaluates the convergence of the approximated Pareto front to the true front, whereas the IGD measures the diversity of the solutions across the output space.\n\nWe also implemented your remaining suggestions and included the definition of the hypervolume in the Appendix. We hope our explanations fully address your concerns, and we look forward to your response!\n\n## References\nZanga et al. 2022. A survey on causal discovery: Theory and practice.Int. J. of Approximate Reasoning, 151 (2022), 101–129.\n\nSchutze et al. (2012). Using the averaged hausdorff distance as a performance measure in evolutionary multiobjective optimization. IEEE T. on Evolutionary Computation, 16(4), 504–522.\n\nFerro et al. (2015). Use of statins and serum levels of prostate specific antigen. Acta Urológica Portuguesa, 32.\n\nHöllig et al. Semantic meaningfulness: Evaluating counterfactual approaches for real-world plausibility and feasibility, in: xAI, 2023, pp. 636–659.", "claims_and_evidence": "I think the claims are supported, with the exception of the applicability of which I talk in more detail in the weaknesses section.", "methods_and_evaluation_criteria": "I am unsure. I am not aware of the GD and IGD metrics to evaluate multi objective optimisation problems. I think it would be beneficial for the unknowledgeable readers like me to have a very short introduction on the appendix. In addition I would also include the definition of HVI and $\\mathcal{H}$ in the appendix.", "theoretical_claims": "The only proof I read in detail is the proof on the paper. If the rest of the proofs are of the same quality and clarity, I don’t think there should be any problem with those.", "experimental_designs_or_analyses": "The experimental analyses seem reasonable to me. I would not personally call the health problem a real-world problem since, in the end, it is using synthetic functions to evaluate the model. Additionally, I would not say “Weight” and “BMI” are easy variables to treat.", "relation_to_broader_scientific_literature": "This paper extends the research in two clear directions, one relating to the use of causality in Bayesian optimisation and the other relating to multiobjective Bayesian optimisation. I think the approach that the authors are proposing is sensible with respect to what has been already said in these two areas.", "essential_references_not_discussed": "Not that I am aware of.", "other_strengths_and_weaknesses": "Strengths:\n- I think the approach proposed by the authors is sensible and fills some of the gaps of the literature.\n- The theoretical analysis is serious and exploits some interesting characteristics of causality.\n\nWeaknesses: \n- I find the applicability of the method to be limited. Not in the sense that the method is not –theoretically– general enough, but in the sense that the conditions for the method to be applied are too constrained. When do such multi-objective optimisation problems arise in real world problems where in addition we have knowledge of the causal graph? \n- The authors could say that the health problem is one such possibility, but as I argued above, weight and BMI are not easily manipulable. But even beyond that, suppose that the health problem is a real world multiobjective optimisation with known causal graph. Then intervening on the system would render a different causal graph (where the arrows of the intervened variables are deleted) and the optimisation becomes a different problem? Maybe I’m missing something here but it would be interesting to understand this a bit better.", "other_comments_or_suggestions": "- There is a LaTeX compilation error on Figure 2 below  “Solve the local problems”.\n- Line 122, second column should be a subset symbol instead of a set membership symbol, I believe.\n- Line 124, second column, any reason for using lowercase $x$ instead of  $\\bigtimes$ (\\bigtimes) (or $\\Pi$ (\\Pi)) for the product space?\n- It seems this is intentional but I find it strange that in line 181 and 202, second column; 273 first column; $\\mu_i$ instead of just $\\mu$ but then you refer to “for all $i$”.", "questions_for_authors": "Please see weaknesses.", "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": "XOfSSNwC33", "reviewer": "Reviewer_8ruG", "summary": "The paper introduces a novel framework, MO-CBO, which integrates causal inference with multi-objective Bayesian optimization, addressing an underexplored research area. The theoretical characterization of Pareto-optimal intervention sets via causal graph topology is a notable contribution. However, several aspects of the paper require improvement.", "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": "Dear Reviewer 8ruG, \n\nThank you for your insightful suggestions to improve our work with a more extensive literature review, better defined limitations, and improved experimental evaluation! We hope that our response below will further reinforce your confidence in our work:\n\n# Extensions to MO-CBO\n\nOur contribution is the development of a multi-objective variant of CBO, MO-CBO, for which we theoretically and empirically show superior performance compared to traditional MOBO. We introduce MO-CBO as its own problem formulation and our paper aims to provide a methodology to address this new type of problem. This includes a decomposition of the MO-CBO task along with establishing graph characterizations to identify possibly Pareto-optimal sets to intervene upon. Our empirical validation supports the theoretical claims.\n\n**Combining MO-CBO with existing CBO variants.** There exist many CBO variants, each developed through its own dedicated research effort - examples include dynamic CBO, functional CBO, and constrained CBO. In this sense, our work stands as a distinct research contribution in its own right. We find it more appropriate to explore these integrations in future research.\n\n**Prior knowledge on the causal structures.** Requiring prior knowledge about the causal graph is indeed a notable limitation of causal Bayesian optimization methods, including our own work. To better communicate this, we now added a limitation section (see our response to reviewer easG for more details on the applicability of our method under causal knowledge requirements). We would like to emphasize that investigating how MO-CBO performs under partial knowledge of the causal structure should be the focus of a separate research effort aimed at developing graph-agnostic (MO-)CBO methods (e.g. see Mukherjee et al., (2024)).\n\n**Multi-task GPs.** Multi-task GPs are a possible technique to increase cost-effectiveness of MO-CBO. We had already discussed this at the end of the paper as a limitation of our work since we believe that an extensive study on multi-task GPs for MO-CBO is best dedicated to future work.\n\n# Literature Review\n\nWe do agree with you that our review of the current state of the field would benefit from a more thorough analysis of existing literature! Therefore, we included a review on multi-armed bandit problems as well as a more detailed description of existing CBO variants in order to give a better overview of the methods within causal decision-making. Moreover, we discuss the assumption of having fully vs. partial knowledge of the causal graph in the prior works.\n\n# Proofs \n\n**Intuition of proofs.** Thank you for pointing this out! We did restructure some of the proofs to enhance readability - see our response to reviewer fTDx for a more structured proof of Proposition 4.2. However, we still believe that maintaining mathematical rigor requires some level of technical detail. That said, to enhance readability, we have now included a description of the main idea of each proof within 1-2 sentences in the main body of the paper. For instance, for Proposition 3.4 we write: “The rigorous proof of Proposition 3.4 is given in Appendix A, and it exploits the fact that the space of all intervention set-value pairs is the union of the input spaces of each local problem. This allows to match the Pareto optimal intervention set-value pairs with the Pareto-optimal solutions from the local problems where the intervention set is fixed.”\n\n**Proof of Theorem 4.8.**  In Theorem 4.8 we show that any intervention on $\\mathbf{X}\\_s$ is weakly-dominated by some intervention on  $\\textnormal{IB}(\\mathcal{G}\\_{\\overline{\\mathbf{X}}\\_s}, \\mathbf{Y})$. Thus, we did not construct an SCM here. We assume you mean Proposition B.2 or Proposition 4.7? We are happy to answer any remaining questions regarding these proofs! \n\n# Experiments and Baselines\n\n**Additional real-world experiment.** Following your suggestion, we include one more real-world problem, where the SCM describes macro-economic relations based on real-world data (Höllig et al., 2023). Due to space limitations, we would like to point you to our response to reviewer 7sGg for preliminary results.\n\n**MOBO baselines.** We expand our ablation study to include the MOBO algorithms DGEMO, TSEMO, ParEGO, MOEA/D-EGO, qParEGO and qNEHVI. We find that our method performs better than all baselines across all tasks. For more details (including preliminary results) we refer to our response to reviewer 7sGg.\n\nThanks again for your thoughtful comments - they greatly contribute to improving the clarity of our work! We hope our explanations have addressed your concerns, and we look forward to hearing back.\n\n## References\nMukherjee et al. Graph Agnostic Causal Bayesian Optimisation. In NeurIPS 2024 Workshop on Bayesian Decision-making and Uncertainty.\n\nHöllig et al. Semantic meaningfulness: Evaluating counterfactual approaches for real-world plausibility and feasibility, in: xAI, 2023, pp. 636–659.", "claims_and_evidence": "yes", "methods_and_evaluation_criteria": "yes", "theoretical_claims": "yes", "experimental_designs_or_analyses": "yes", "relation_to_broader_scientific_literature": "Related to some extent", "essential_references_not_discussed": "yes", "other_strengths_and_weaknesses": "Strength: \n\nIntroduces a new class of optimization problems, MO-CBO, expanding causal Bayesian optimization to multi - objective scenarios, filling a research gap in multi-objective optimization considering causal structures.\n\nWeakness: \n\nAlthough it mentions the potential combination of existing CBO variants with MO-CBO in the future, the paper does not explore this aspect, lacking the full integration of existing research results and limiting the depth and breadth of the research.", "other_comments_or_suggestions": "None", "questions_for_authors": "a.\tThe paper lacks a thorough discussion of the current state of the field, making the motivation for this work somewhat unclear. A more explicit analysis of existing challenges and the necessity of this approach would strengthen the introduction.\nb.\tWhile the theoretical analysis (e.g., Propositions 3.4, 4.10) is rigorous, and the proofs in the appendix enhance the credibility of the results, the assumption of a fully known causal graph is restrictive and may not be practical in real-world scenarios. The paper should discuss how partial knowledge of the causal structure (e.g., unobserved confounders) could impact the algorithm’s performance. Additionally, the surrogate model assumes independent Gaussian processes for each objective, but a discussion on how to handle shared confounders or leverage multi-task learning would improve the methodological robustness.\nc.\tThe proof of section 4.1 derivations should be presented in a more symbolic and structured manner to enhance readability.\nd.\tWhile the paper is generally well-written, some sections lack clarity. For example, the construction of the structural causal model (SCM) in Appendix B.2 (Theorem 4.8 proof) is overly technical and difficult to follow. \ne.\tThe baseline comparisons appear to be limited to DGEMO (2020). To convincingly demonstrate the advantages of causal Bayesian optimization, the study should compare against more recent and efficient multi-objective optimization algorithms.\nf.\tFigure 7 presents experimental results comparing MO-CBO with MOBO on a real-world health application, where MO-CBO demonstrates superior performance. However, additional real-world applications should be explored to further substantiate the practical utility of MO-CBO.", "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: This manuscript considers the problem of multi-objective causal Bayesian optimization that operates by reducing the problem to solving a series of traditional multi-objective optimization problems, then solving these jointly while balancing exploration among this set. The authors perform a series of experiments on synthetic and real data demonstrating the performance of this approach.

The initial response to this work from the reviewers was somewhat mixed in terms of their scores. However, the reviewers agree that the considered setting is of interest to the ICML audience, and most reviewers agree that the paper is well written and the experiments well designed. The two "weak reject" reviews only point out minor weaknesses that I believe were adequately addressed by the author responses (and one of these two reviews actually doesn't really point out any weaknesses at all).

That said, no champion for the paper emerged during the discussion period, even from the reviewer with the highest rating. Thus I am only comfortable recommending weak acceptance here, supported by the apparent soundness of the approach and an assessment that it would be an interesting addition to the program.
Decision: Accept (poster)