Federated Causal Structure Learning with Non-identical Variable Sets
Authors: Yunxia Wang
Fuyuan CAO
Kui Yu
Jiye Liang
Venue: ICML 2025 poster (2025)
Keywords: Federated causal structure learning
privacy-preserving
non-identical variable sets
spurious dependencies
Primary area: general_machine_learning->causality
TL;DR: The paper aims to discover causal relationships over an integrated set of non-identical variables in a privacy-preserving manner, accounting for spurious dependencies and the varying importance of relationships between variables within a local graph.

Abstract: Federated causal structure learning aims to infer causal relationships from data stored on individual clients, with privacy concerns. Most existing methods assume identical variable sets across clients and present federated strategies for aggregating local updates. However, in practice, clients often observe overlapping but non-identical variable sets, and non-overlapping variables may introduce spurious dependencies. Moreover, existing strategies typically reflect only the overall quality of local graphs, ignoring the varying importance of relationships within each graph. In this paper, we study federated causal structure learning with non-identical variable sets, aiming to design an effective strategy for aggregating “correct” and “good” (non-)causal relationships across distributed datasets. Specifically, we first develop theories for detecting spurious dependencies, examining whether the learned relationships are “correct” or not. Furthermore, we define stable relationships as those that are both “correct” and “good” across multiple graphs, and finally design a two-level priority selection strategy for aggregating local updates, obtaining a global causal graph over the integrated variables. Experimental results on synthetic, benchmark and real-world data demonstrate the effectiveness of our method.

Federated Causal Structure Learning with Non-identical Variable Sets
Yunxia Wang' Fuyuan Cao
Kui Yu? Jiye Liang'
Abstract
presents challenging research task (Hernn & Robins, 2020; Wang et al., 2022; Zhu et al., 2024). With the rapid growth of data volume, data owners gradually refuse to share their personalized data to avoid privacy leakage (Cramer et al., 2015; Yang et al., 2019; Li et al., 2020), making the infer- ence of causal graphs from centralized data troublesome. Consequently, in recent years, federated causal structure learning (CSL) has been increasing attentions in uncovering the causal relationships between variables from decentral- ized data, with data-privacy protecting way.
Federated causal structure learning aims infer causal relationships from data stored individ- ual clients, with privacy concerns. Most exist- ing methods assume identical variable sets across clients and present federated strategies for aggre- gating local updates. However, in practice, clients often observe overlapping but non-identical vari- able sets, and non-overlapping variables may in- troduce spurious dependencies. Moreover, ex- isting strategies typically reflect only the overall quality of local graphs, ignoring the varying im- portance of relationships within each graph. In this paper, we study federated causal structure learning with non-identical variable sets, aiming to design an effective strategy for aggregating "correct" and "good" causal relationships across distributed datasets. Specifically, we first devel- op theories for detecting spurious dependencies, examining whether the learned causal and non- causal relationships are "correct" or not. Further- more, we define stable relationships as those that are both "correct" and "good" across multiple graphs, and finally design two-level priority s- election strategy for aggregating local updates, obtaining global causal graph over the integrat- ed variables. Experimental results on synthetic, benchmark and real-worke datasets demonstrate the effectiveness of our proposed method.
Existing federated CSL methods are usually classified into constraint-based (Li et al., 2024b; Huang et al., 2023; Guo et al., 2024), score-based (Mian et al., 2023; Ye et al., 2024), and continuous-optiminn Ng & Zhang, 2022; Gao al., 2023; Chengbo & Kai, 2024) methods. Almost all of them focus discovering causal relations among a set of identical variables, without consid- ering latent variables underlying the data. Additionally, the federated strategies presented by these methods typically reflect only the overall quality of locally discovered causal graphs in every iteration, ignoring the varying importance of relationships between variables within each graph. For instance, PERI (Mian et al., 2023) learned the final glob- al causal graph that minimizes the worst-case regret with respect to the locally discovered causal networks, using dis- tributed min-max regret optimization. FedCSL (Guo et al., 2024) first estimated the sample size of each client and then assigned sample-size-hee
based (Abyaneh al., 2022:
weights for the local results. Notears-ADMM (Ng & Zhang, 2022) relied on the ADMM (Boyd al., 201 1) optimization, FedDAG (Gao et al., 2023) employed the FedAvg (McMahan et al., 2017) technique, and FedCDH (Li et al., 2024b) constructed the summary statistics, all of which exchange statistics representing the whole local graphs.
1. Introduction
Causal discovery holds significant roles in various scientific fields, including biology, epidemiology, medicine, and eco- nomics (Replogle al., 2022; Cai et al., 2023; Squires et al., 2023; Feuerriegel et al.. 2024; Li et al., 2024a). Unveiling the causal relationships between random variables from data
In practice, often due to privacy, ethical, financial, and prac- tical concerns, the variables observed by different clients are not entirely identical but partially overlapping (Tillman & Spirtes, 2011), and the data collected by clients might be suitable for uncovering the relationships between different variables. For example, multiple hospitals, due to variations in professional specialties, medical resources, and technical proficiency, measure non-identical indicators and are suit- ed for diagnosing different types of discases. Therefore. is essential to study federated causal structure learning in
"School of Computer and Information Technology, Shanxi University, Taiyuan, China "School of Computer Science and In- formation Engincering. Hefei University of Technology, Hefei, China. Correspondence to: Fuyuan Cao <cfy @aay@sxxx..uuu edu.en.
Proceedings of the nnnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Federated Causal Structure Learning with Non-identical Variable Sets
tests "good" ones. Based on the above considerations, we define stable causal and non-causal relationships as those that are both "correct" and "good" across multiple graphs. And then we design two-level priority selection strategy (TPSS) in the context of non-identical variable sets, bridg- ing local learning on clients with global aggregation on the server. Extensive experiments on synthetic, benchmark, and real-world data validate the effectiveness of FedCDnv.
obbaming
Server
B
15
loarn data
B CCE E
ABCF
share
brosian back
dllent.o
eagreegate
(a) GroundTrulls (bb) lestted causel gragh over the set inlegrated variables
2. Formal Preliminaries
Figure 1. toy example to illustrate spurious dependencies caused by non-overlapping variable pairs. Here, F and F are not observed together by any client, so their relationship is unknown and depen- dencies between several variables adjacent to them are spurious, such as Ao-o E in elient er2 Co-oF in client Cm. and others.
When the causal sufficiency assumption (Spirtes et al., 2000) is violated, the underlying system is typically modeled us- ing Semi-Markov Causal Models (SMCMs) (Tian Pearl, 2002) or Maximal Ancestral Graphs (MAGs) (Richardson Spirtes, 2002), rather than Directed Acyclic Graphs (DAGs) (Spirtes et al., 2000). We use V = (O, to denote set of all variables describing the system, where O, L. and S represent disjoint sets of observed, latent, and selection vari- ables, respectively. In this work, we use MAGS to model the conditional independence relations among O, denoted by M = (0, E), where E - Oxx O is the set of edges including directed edges (-)), bi-directed edges (++), and undirected edges (-). For X;, X, E +, X; denotes that X; is direct cause of Xj. X,++X; denotes that the presence of latent confounders between X, and X,, and X; X,, indi- cates the presence of selection bias between them. MAGs with the exact set of conditional independencies are Markov equivalent, and the complete set represented as Partial Ancestral Graph (PAG). In a PAG, endpoints that can be either arrowheads (>) or tails (-) in different MAGs are de- noted with circle (o). In this work, we assume the presence of latent confounders but no selection bias.
the setting of non-identical variable sets, with an effective federated strategy.
However, designing such a federated strategy presents main challenges. The first arises from the presence of non-overlapping variable pairs-thoss that are never ob- served together by any client- may lead to spurious dependencies, as illustrated in Figure 1. Since no joint observations are available for these non-overlapping pairs, their relationships cannot be directly inferred from the in- put data. Moreover, the dependencies among overlapping pairs may be indirectly distort by non-overlapping ones: either variable in a non-overlapping pair may act an unob- served confounder or mediator for the other and its adjacent variables, thereby introducing spurious dependencies. The second challenge is how to evaluate the importance of causal relationships within each locally discovered causal graph and then aggregate the "good" ones across iterations. While some distributed causal discovery methods take into account non-identical variable sets (Cao et al., 2024; Huang et al., 2020; Triantafillou & Tsamardinos, 2015; Tillman et al., 2008; Tillman& & Spirtes, 2011), they ignore the issues men- tioned above and typically allow raw data to be shared.
Consider that there are m clients in total, denoted by C = {c , J. and one central server, denoted by S. Each client has its own local dataset D₃ and observed variables O₂ (k E [1,..., m)). Let n& and dę denote the number of samples and variables in client Ck. respectively. The nion of observed variables across m clients is denoted by the set O = U Oz. with d = |O] being the total num- ber of variables. Since the variable sets across clients are non-identical we categorize latent variables behind data into two types: absolute latent variables and relative latent variables. Absolute latent variables refer to the aforemen- tioned L. while relative latent variables exist in each client Ck. represented as L₁ OO,Og- In addition, we use non- overlapping variable pairs to describe pairs of variables that are not observed simultaneously by any client, and similarly use overlapping variable pairs to represent pairs that are observed simultaneously by at least one client.
To address the above key challenges, we propose a nov- el Federated Causal Discovery algorithm for Non-identical Variable sets, called FedCDny, aiming to design an effective federated strategy that aggregates both "correct" and "good" relationships during collaboratively learning, while preserv- ing data privacy. The strategy involve two factors. One is that detecting whether the learned relationships are "correc- I" or not. For this, we develop theories to first determine whether the relationship between each non-overlapping vari- able pair is definitively non-causal and then examine which dependencies among overlapping variables are affected. The other is that analyzing which relationships are "good" ones. Here, we consider the stability of the process for learning causal relationships, with the perspective that setting an in- terval of significance levels for conditional independence (CI) tests, and regarding relationships that remain stable CI
The core of our work is identify causal relationships on the integrated variable set O within the context of federat- ed learning, using individual datasets with latent variables.
2

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Federated Causal Structure Learning with Non-identical Variable Scts
The assumptions required for this paper are presented as Assumptions 2.1, 2.2 and 2.3.
II (X, is an overlapping variable pair and they are observed simultaneously by at least one client.
Assumption 2.1. Assume that Ps is the joint probability distribution over Oz. and M₁ is the MAG describing the causal relations among O₃ for the client Ck. then Ps and Mr are assumed to be faithful to each other.
For Category I, due to the absence of joint data for non- overlapping pairs, their relationships are initially considered to be non-definitlly non-causal ones, meaning it is unknown whether edges exist between variables of these pairs in the ground truth. To examine which of these relationships are definite non-causal, we introduce Theorem 3.1 as follows.
Assumption 2.2. All local datasets are uniformly sampled from the same causal DAG over V= = {0, S). and the probability distribution of samples can differ across different clients. In addition, any two local datasets D.. Dj (i,j E {1.... m.), 8 ( j) are not shared.
Theorem 3.1. Let (X,Y) be a non-overlapping variable pair (X, YEO) and A be the set of variables adjacent to X in G. If there exists A n AÇ = 0 & A + & A& + 0, and for each Y' E AAG (X, Y is not a non-overlapping variable pair (and vice versa), then the relationship between X and Y is considered definitively non-causal.
Assumption 2.3. Assume that the intersection of any two variable sets O₂ and Oj. observed by client e; and ej (i * j; i,j E ......))) is not empty, i.e., O, O, & 0.
Theorem 3.1 utilizes the properties of adjacency sets of vari- ables in non-overlapping variable pairs to examine whether the relationship between them definitively non-causal.
3. The Proposed FedCDnv Algorithm
In this section, we present FedCDnv, which consists of three key submodules. The first submodule focuses on developing theories to detect whether the relationship between each variable pair, both non-overlapping and overlapping ones, is definite (i.e.." "correct") causal or non-causal relationship. Based on the constructed theories, the second submodule aims to design federated strategy suited for the context of non-identical variable sets. The third submodule presents the implementation details of FedCDnv.
For Category II, we initially assume the causal and non- causal relationships among overlapping pairs are definite, or "correct". However, the presence of non-overlapping pairs may introduce spurious dependencies among overlapping pairs. For this, Lemmas 3.2 and 3.3 are introduced to detect which of these learned relationships are non-definite.
Lemma 3.2. Assume that X and Y (X,Y O) are ob- served simultaneously by at least one client and Zn, repre- sents union set of variables that appear in non non-overlapping variable pairs with non-deftnite, non-causal relationships. If X E Zn or Y E Zn, then the causal relationship between X and Y is considered to be non non-definite
3.1. Develop Theories
The setting of non-identical variable sets in federated causal discovery introduces two theoretical challenges. On the one hand, non-overlapping variable pairs may exist and their causal relationships cannot be theoretically identified due to the lack of their joint data. For this, our first focus is to determine whether there is definite non- non-causal rela- tionship between variables in each of these pairs. On the other hand, the presence of overlappgg variable pairs may cause spurious dependencies. Our second focus is to examine whether the discovered causal relationships among overlapping variable pairs are spurious.
In Lemma 3.2, if a non-overlapping variable pair cannot be determined to be definitively non-causal by Theorem 3.1, then the causal relationships between each variable in the pair and its adjacent variables, are considered non-definite.
Lemma 3.3. Assume that X and Y (X,Y E O) are ob- served simultaneously by at least one client and A is set of variables adjacent to X. in G. If there exists overlaplag variable pair (A, B) such that (A, B) C A U AG, then the causal relationship between X and Y is considered to be non-definite.
Assume that G2 = (Op, Ex) is a local causal graph learned from the dataset in client e (I E {1, , mm) with an ora- cle of conditional independence tests, where O₃ represents the variable set observed by c and E, S O2 x O₂ repre- sents the set of learned edges (including -,, ++, --))- Let G = E) be an integrated graph over O, obtained by aggregating adjacencies and arrowheads (one of the ori- entations) from all local graphs (i.e., 0 U Oz, E = UK= Ex). We first classify any two variables X, Y E O into one of the two following categories and then propose the following theories.
In Lemma 3.3, the adjacency between X and Y identified by the raw data may arise due to incomplete or incorrect sep- arating sets. In other words, the ideal separating coss-thosee that should render X and Y independent-mag contain pairs of variables that cannot be observed together by any client. Consequently, no valid separating set exists that can fully ensure the independence of X and Y.
3.2. Design the Federated Strategy
I (X, Y) is a non-overlapping variable pair, i.c., X, Y are not observed simultaneously by any client:
In this subsection, we design a federated strategy two- two- level priority selection strategy (TPSS), aiming to aggregate
3

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Federated Causal Structure Learning with Non-identical Variable Sets
level, which is within the range [a - 0,, a + 02]. This
provides empirical evidence supporting the intuition.
Step2: Present the details of TPSS. TPSS contains two levels of priority. The first level is to identify the inconsistent causal relationships caused by the non-identical variable sets. And the second level aims to evaluate the varying importance of relationships between variables.
Assume that (X,,X) (X,, X3 E O) is an overlapping vari- able puir. Let }Gur denote the set of local graphs where X; and X, are observed together, corresponding to the set of clients {cy). J. If X, and X, are adjacent in [Gy, J but not adjacent in the remaining graphs {Gwr \ (GGg) } (denoted as {G&, }), then TPSS begins to execute.
Figure 2. An example of p-value distributions between 40 nodes from different perspectives, with the left subplot showing frontal view and the right one displaying an adjusted angle. The gray plane represents the significance level (p-value) 0.05, with false positives and false negatives concentrated around a.
both "correct" and "good" relationships across different clients. It contains two steps as follows.
At the first level, we identify whether the inconsistent adja- cencies between X, and X, in {Gx; } and {Ge;, } are caused by non-identical variable sets across clients, and introduce Conditions 3.6 and 3.7 as follows.
Stepl: Define stable relationships. To represent both "cor- rect" and "good" relationships across multiple local causal graphs, we introduce the concept of stable relationships.
Condition 3.6. For variables X; X; with a bi-directed edge between them in client CN., if there exists a client owe, such that Ka # Wa, Xitooo E Ex,, and Y & Ow2 then Y is an unobserved confounder for client ck.-
Let G2 =0k, Eg) denote causal graph learned from the dataset of client e (k E (1,-.. m,)) and let (uft be the set of p-values obtained by conducting conditional independence (CT) tests on X, and X, (X,,,, E Oz),
Condition 3.7. For variables X., X3. which take forms such as X, X,0+X or X,toX; in client cg., their inconsistent adjacencies arise from non-identical observed variable sets, under the following condition: for any Gv, e [Gwe,] }. there exists a minimal separating set Z C Oue, such that X, 1L X, IZ holds in Dw., and for every Gv, E &(G,, J. the variables in Z. are never simultaneously observed-tlee is, there does not exist a subset of Ow, that contains all variables in Z, and X, X, $ Z.
conditioned on the separating sets in client Ck. Given an interval of significant level [a Q2-68 81, a 02] (0 < 0 < a, < 02 < 1-aa a, and a is the significant level), we define the stable causal and non-causal relationships in G2 as follows. Definition 3.4. stable causal relationship. In a causal graph Gk. if the relationship between X; and Xj is deter- mined to be definite causal, and the maximum of p-values oh for X; H.. D2. I (VZ C (x,, X,}) is lower than a- 81, we call the relationship between X, and X, a stable causal relationship in Ge.
If the first level of priority is not satisfied, then we suspect that the inconsistency of X, and X, in (Gx) ) is caused by inaccurate CI tests. Thus, at the second level of priority, we propose a way for evaluating varying importance of relationships between different variables. It involves two aspects: "correct" and "good" relationships for each client, which are represented as stable relationships; the impact of varying sample sizes across clients on these relationships.
Definition 3.5. stable non-causal relationship. In causal graph Gk., if the relation between X, and X, is de- termined to be definite non-causal, and there exists a set Z S O& (X,, X,} } such that X, L.D, X₃ Z with a corre- sponding p-value greater than a20 we call the relationship between X, and X, a stable non-causal relationship in Gg.
The definition of stable relationships involves two charac- teristics. First, stable relationship must be consistent with those in the ground truth (i.c., it is a definite relationship). If the relation between X; and X, deviates from the ground truth, will become unstable, thereby introducing errors. The second characteristic arises from an intuition that, in practice, when the p-value p is very small (c.g.. p S 0.001), the null hypothesis can be rejected, and when p very large (e.g.. p >> 0.5), there is insufficient evidence to reject the null hypothesis. Figure 2 illustrates the experimental distri- bution of false positives and false negatives relative to the confidence level a, which shows that most of false positives and false negatives are concentrated around the confidence
Specifically, for variables X, and X,. use 1 to represent the importance of stable relationships between them, w
(0 < wiy < 1) to indicate the importance of non-stable relationships relative to stable ones, and wo to account for the impact of the sample size of client Cyy, where cv E {cw)) The adjacency of X; and X, is determined by comparing vali, with O where vally is computed as follows.
valij
Σ (
w X Wag
(1)
ON
where w = my (nw denotes the sample size of client CK', and n' =E Eerrrf ny is the total sample size across

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Federated Causal Structure Learning with Non-identical Variable Scts
oe2 takes the values: 1, -1, w -wol." clients {ek }), and wiy" Specifically, 1, indicate the importance of stable causal and non-causal relationships, respectively, whereas wi and -uny reflect the importance of non-stable causal and non-causal relationships, respectively.
diants
Server
A B
P
B
grsh
Inlogpaaad gaoph
updittdd Craph
GrundTruhh
(b) integrated causel graphh during en intesattion
52' is calculated adaptively using p-values. Specifi- Here, "1 cally, when the p-value piy falls within [a 01, a.] or (a, 02], it is uniformly scaled the interval [0, 1] using (2).
Figure 3. An example for explaining the cases that rules described by Zhang (Zhang, 2008) are not applicable to the integrated graph Gg. Based on the rules, B and E' are oriented into B-+ in Go. which not consistent with E E-BB GroundTruth.
if
Poj ae' fa - 01. al
G0 Pgj
(2)
Ca' aa if p ev E (a, a + 02]
Obtaining Go serves two main purposes. First, it enables the determination of whether the relationship between each pair of variables is definite, upon which the status matrix stat is constructed (as explained below). Second, it allows for updating each local graph Pagø* as the orientation rules proposed by Zhang (Zhang, 2008) are not directly applicable to Go. as illustrated in Figure 3.
82
where Pij denotes the scaled p-value of X2 and X,. Then wij is obtained using (3).
PMy x H(w
C Waj
(3)
E Eayeloan Eloy) Péj" + Eaaelooy
The state matrix stat takes values from the set {-2, -1, 1, 2). each encoding a different type of relationship between two variables. Let G = (0, E)) denote the integrated graph and MM the ground truth. stat(i, = -2 represents a definite non- causal relationship between X, and Xj. indicating that they are not adjacent in either G or ... stat(i,j) = --1 denotes non-definite non-causal relationship between X, and Xj. meaning that they are not adjacent in G, but their relation in M is unknown- -hey may or may not be adjacent. stat(i,) j) = 2 represents a definite causal relationship between X; and Xj. meaning that they are adjacent in both G and M. Lastly, stal(i,, j) = denotes a non-definite causal relationship between X, and Xj. where they are adjacent in G. but is unknown whether they are adjacent in M..
where {ax } and {ae. } denote the sets of clients that iden- tify the relationship between X; and X, as non-stable and stable, respectively. If this relationship is identified as non- stable but the corresponding p- p-value does not fall within the Ei'
interval [a- - 01, a + 02], then p.j = PPjj
3.3. Present the FedCDnv Algorithm
FedCDnv outputs two graphs: FedG, a global causal graph over O; and FeddG, a subgraph of FedG that contains only the definite causal and non-causal relationships. The latter derived by further applying proposed theories to extract definite relations from FedG.
Next, Go is broadcast back to all clients, prompting each client to update its local causal graph, denoted as Pags* And in each client, p-values falling within the interval [a - f1, a + 02] are scaled to [0, 1] using (2), and then multiplied by -1, resulting in pat. is worth noting that scaling only the p-values within [a- a,-0,,a+0,, introduces a level of uncertainty that may help obscure precise statistical information, thereby enhancing privacy protection to some extent. The detailed algorithm is shown in Algorithm 2. Finally, Pag.* and i of each client are shared to S again. In S, FedCDny applies the proposed TPSS to aggregate ad- jacencies. For orientations, only the identified arrows are incorporated in the global graph. The detailed algorithm description is shown Algorithm 3.
3.3.1. OBTAINING FEDG
The algorithm description of FedCDny is presented in Algo- rithm 1, which involves two rounds of interaction between clients and the central server.
FedCDnv first applies an existing causal discovery method (e.g., FCI (Spirtes et al., 2000) or RFCI (Colombo et al., 2012)) to the local dataset of each client, obtaining causal graph over O₁, denoted by Pagō* and the corresponding set of p-values, denoted by f Each client then shares its local graph Pagó" and sample size ng with the central server S. At the server, the adjacencies and orientations from all local graphs (Pagō* 1 1k=1 are integrated into a global graph Go = (0, Eo). Specificallyy, for adjacencies, variable pair is considered adjacent if any client reports the adjacency between them. While for orientations (there are three types of orientations: circle 'o', tail '-', and arrow '>'), only the identified arrows are included in the global graph. For example, for X,o-oX,, X,o-XX, and X,- + Xx, the final relationship between X, and X, will be X,0
3.3.2. OBTAINING FEDDG
We apply the proposed theories in Section 3.1 to extract definite relationships from FedG, obtaining FeddG.
For each variable pair (X,, X,), FedCDnv initially divides it into one of two categories. For Category I, Theorem 3.1
5

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Federated Causal Structure Learning with Non-identical Variable Scts
Algorithm 1 FedCDnv
Algorithm 3 S-FedG
1: Input: (D1,... Dm) ) 01, 82, FCI parameters para
1: Input: (Pag. ). (p**), ). (nee). a, On, 02. stat
2: Output: FedG, FeddG
2: Output: FedG
3: for each client Ca E {er..... .mm) do
3: O + the set of integrated variables
4: (Pagō" pp*)====== C-InitPag(DDs a, para) /using FCF
4: for cach X,, E O do
5: ne <- obtain the sample size of client ck
5: if Conditions 3.6 and 3.7 are satisfied then
6: send Pagū" and ny to the central server S
6: X; and X, are adjacent in FedG
7: end for
else
8: Go + in S. transfer adjacencies and arrowheads of all graphs ([Pag"") ) into Go
8: for each client cv E {(e): where X, and Xy are jointly observed do
9: stat 4- in S. apply Theorem 3.1, Lemmas 3.2, and 3.3 to detect whether the learned relation between each variable pair definite or not, obtaining the status matrix stal 10: broadcast Go back to all clients
9:
compute W.y and way" 9i's based on Pag. {mx) stat, and pa end for vale is computed using (1)
10:
11:
11: for each client E {er.... , 4m do
12: if val., 0 then
12: PagS* C-UpdateeGGo, Dx, Pago
13:
X; and Xj are adjacent in FedG
13: pe -- scaled p-values in [a- 81, a + 02] using (2), multi-
14:
end if
plied -y
15: end if
14: send Pagi" and p* to the central server S
16: end for
15: end for
17:
transfer arrowheads from each Pag.* (Pag.* to FedG
16: FedG S-FedG (( ((Pag J. (p*)), {m=),, a, 01, 0z. stat)
17: FeddG + according to obtained stat, extracting definite rela-
tions from FedG
and then multiplied by -1, resulting in the transformed set of p-values, denoted by prt. This transformation reduces the risk of leaking precise statistical characteristics, therehy enhancing privacy to a certain extent.
Algorithm C-Upte
1: Input: Go, DE, Pagō"
2: Output: Pag.*
To further avoid data privacy leakage, several secure compu- tation techniques be considered in this paper. For exam- ple, secure multiparty computation (Cramer et al., 2015) is utilized to encrypt the exchanged graphs and sample sizes, enabling multiple clients to collectively compute function over their inputs while keeping the data private. Alternative- ly, homomorphic encryption (Acar et al., 2018) facilitates the processing of encrypted data through complex mathe- matical operations without compromising the encryption. Furthermore, to prevent the leakage of semantic information related to variables, methods such as assigning unique iden- tifiers to each variable can also be introduced. Also certain client can be selected as a proxy server to avoid the leakage of the graph structure (Gao et al., 2023). Further privacy protection efforts will be explored in the future works.
3: Os < obtain the set of observed variables from Pag.*
4: for each v-structure V = (X., X, in Go, where Y is the
collider
5: if C Oc. X,, Y are adjacent and so do X,, Y in Pago*
then
6: orient X,o+Y and Xjo+Y in Pag.
7:
end if
8:
Ar +- -adj(Y; Pag.*)
9:
for each B E A [X,, do
10:
if IZ C ABMB MB, X,) such that X, 1L BlZZ
X2 IL ( U Y then
orient Y B in Pagä 12: end
13: end for
14:
end for
applied directly. For Category II, Lemma 3.2 and Lemma 3.3 are called in sequence. All (non- )definite relationships represented by different values in slat. FedCDnv tracts the relationships with stat(i,j) -2 and stat(i,j) 2 from FedG as definite ones, obtaining FeddG.
3.4.2. COMMUNICATION COSTS
Assume that the client eæ observes de variables, and the server S integrates a total of d variables (d 2 dx. and "=" holding only when all m clients observe identical variable sets). For each client Cr. two rounds of communication with the server are involved: one for sending the initially learned graph and another for sharing the updated local graph. In the first round, the client sends a da de matrix (Pagū* ), a value 74g and receives a d x d matrix (Gu) from the server, which results in a communication cost of O + dl + 1). In the second round, the client sends the updated graphs Pagft and scaled statistical information pa* also de x de matrix, incurring a cost of O(24)). Therefore, the total communication cost for a single client et is O(32 +
3.4. Privacy and Costs Analysis
3.4. PRIVACY ANALYSIS
To protect data privacy in the federated setting, FedCDnv avoids sharing raw data between clients and the server. In- stead, structural information (i.e., local graphs Pagō", Pagst, and the global graph Go) and statistical characteristics (i.e., p and ng)) are exchanged. Notably, the p-values within the interval [a 01,a + 02] are first scaled using Eq. (2)
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Federated Causal Structure Learning with Non-identical Variable Sets
and the total cost across all m clients is oiΣ*=1 (3dg dl + 1)). Since d& S d, the overall communication cost is upper-bounded by O(4mdd + m)) which can be further approximated as O in large-scale settings.
0.75
0.75
Fl.edge
F1-aric
0.50
0.50
0.25
0.25
4. Experiments
0.00
0:01
Numbar nodes
Number nodes
In this section, we conduct extensive experiments to eval- uate the effectiveness of the proposed FedCDnv. First, we describe the experimental setting in Section 4.1 (with further details provided in Appendix 1. Then, in Section 4.2 and Appendices D.2-D.3, we compare FedCDnv with state-of- the-art methods from two perspectives: i) distributed CSL methods over non-identical variable sets, and ii) federated CSL, methods over identical variable sets. Next, in Section 4.3, we verify the effectiveness of the definitely causal and non-causal relationships identified by FedCDnv, and we perform sensitivity analysis of the stable relationships in Appendix D.4. Finally, in Appendix D.5, we evaluate the impact of the parameters listed in Table 4 on FedCDnv.
Figure 4. CDMiNi vs. FedCDnv
CSL (Guo et al., 2024).
Metrics. (1) We use precision, recall, and F1 score for returned edges and orientations, denoted pre-edge, rec- edge, F1-edge, pre-orie, rec-orie, and Fl-orie, to evaluate the returned causal graphs over non-identical variable sets. (2) We use pre-edge, rec-edge, and Fl-edge to evaluate causal skeletons learned by the federated CSL methods with identical variable sets. (3) To test the accuracy of definite causal and non-causal relationships returned by FedCDnv, we use the false discovery rate (FDR) for them, denoted as FDR-C, FDR-dC, FDR-nC, FDR-dnC.
4.1. Experimental Settings
Datasets. (1) Synthetic data. The underlying DAGs are generated using the Erdös-Rényi (Erdös & Rényi, 1959) graph model with the graph size n. Due to the presence of absolute latent variables, the corresponding MAG's graph size is set to d' n (1 - X) (0- < X S 1). That is, the total number of observed variables integrated across all clients is d (10)= = d). For each client, the number of observed variables is set to d& = d x 8, where 0 < 8 < 1 denotes the ratio of client-specific observed variables to the total integrated variables. (2) Benchmark data. We use 6 networks with ranging from 20 to 111 variables: Child, Alarm, Insurance, Barley, Child3, and Alarm3. (3) Real- world data. We use "Sachs" (Sachs al., 2005) evaluate the performance of the methods, randomly selecting 9 out of 11 variables as those observed by each client.
Parameters. For each invocation of FedCDnv, the default settings for each problem instance (set of datasets) are gener- ated using the values shown in Table of Appendix D. The default parameters are as follows: G2 tests of conditional in- dependence, the number of clients |C| 6, the significance level 0.05, 01 0.049, 82 0.45, À 0.15%, 8 0.85%, the sample size mg E [100, ns)] with ns = 2000.
4.2. Comparison of FedCDnv, FedCDnv-vote, CDUIOV, CD-MiNi, and FCI-base
4.2.1. FEDCDNV vs. CD-MINI
Since CD-MiNi is suitable for small-scale networks, we compare it with our method using synthetic data generated from networks containing 5 to 10 nodes, and the experimen- tal results are shown in Figure 4. The vertical error bars (i.e., whiskers) indicate the range of one standard deviation above and below the mean. This design is chosen for clarity in comparing multiple methods.
Baselines. We conduct two types of experiments. (1) We compare our method with distributed CSL methods over non-identical variable sets, including FedCDnv-vote, C- DUIOV (Cao et al., 2024), CD-MiNi (Huang et al., 2020), and FCI-base. FedCDnv- is variant of FedCDnv with the voting strategy. FCI-base is baseline constructed by integrating causal graphs discovered by FCI on each client, using a sample-size-based federated strategy. (2) We com- pare FedCDnv with the state-of-the-an federated CSL meth- ods in the setting of identical variable sets. Since FedCDav still assumes the existence of absolute latent variables, we focus only on evaluating the performance of the learned skeleton. This comparison is intended to serve as rough performance reference rather than strict evaluating. The compared methods include FedCDnv-vote, Notears-ADMM. (Ng & Zhang, 2022), FedPC (Huang et al., 2023), and Fed-
It can be observed that FedCDnv exhibits relatively superior performance compared with CD-MiNi. Specifically, for Fl-edge, while FedCDnv is slightly inferior to CD-MiNi in the network with 7 nodes and both are comparable in the network with 8 nodes, FedCDnv significantly outperforms CD-MiNi in remaining networks, i.e., networks with 5, 6, 9 and 10 nodes. As for the F1-orie metrie, FedCDnv performs better than CD-MiNi across all six cases displayed. This difference may be due to FedCDnv considering the inconsis- tent causal relationships caused by relative latent variables, which could remove certain spurious edges. Additionally, FedCDnv effectively addresses specific inconsistent adjacen-
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Federated Causal Structure Learning with Non-identical Variable Sets
Dm
CDUIOV
FCI-base
FedC
FedCDns-vote
CDULOV
100
1.00
1.00
1,00
0.75
0.75
H1-elte
Fl-ore
0.50
0.90
F1.
0.50
0.50
0.25
0.25
0.00
30
80
100
0.00
40
80
100
0.00
0.00
Number nodes
Number nodes
Bayesian networks
Bayesion nerworks
Figure Experimental results on synthetic data,
Figure 6. Experimental results on benchmark data.
cies by further considering the importance of relationships, thereby enhancing the reliability of causal discovery.
mance in the "Barley" and "Child3" networks, and performs slightly better than other methods in the remaining four networks. contrast, the performance of FedCDnv-vote and FCI-base comparable, but both are slightly inferior to FedCDnv. For Fl-orie, FedCDnv surpasses the compared al- gorithms by margin of 82% in four networks. Howev- "Insurance" and "Barley", its performance advantage is less pronounced. Furthermore, FedCDnv consistently outperforms FedCDnv-vote across all networks, demonstrat- ing the effectiveness of the proposed TPSS on benchmark datasets. Between FedCDnv-vote and FCI-base, their perfor- mances are relatively balanced; for example, FedCDnv-vote outperforms in networks like "Alarm", whereas FCI-base performs better in networks like "Insurance". In contrast, CDUIOV exhibits suboptimal performance in both edge and orientation identification. The reason for this occurrence is similar to what was discussed for synthetic data above.
4.2.2. FEDCDNV vs. FEDCDNV-VOTE vs. CDUTOV vs. FCI-BASE
Synthetic Data. Figure 5 presents the experimental results for F1-edge and Fi-orie of FedCDnv, FedCDnv-vote, C- DUIOV, and FCI-base on synthetic data. It is observed that FedCDnv exhibits the highest Fl-edge and Fl-orie across almost all networks, with its orientations being nearly 10% better than those of the compared methods.
Specifically, for Fl-edge, we observe that FedCDnv out- performs all other algorithms, followed by FedCDnv-vote and FCI-base, where FedCDnv-vote is slightly higher than FCI-base but not significantly, and CDUIOV performs the worst. For F1-orie, For F1-orie, FedCDnv again achieves the best performance across nearly all networks, except for the 10-node network. FedCDnv-vote's Fl-oric is slightly higher than that of FCI-base most networks, except in the 20-node network. CDUIOV shows a lower Fl-orie than all other methods. This inferior performance of CDUIOV on both Fl-edge and Fl-orie is likely due to its framework being specifically designed for intervention data and its sensitivity to the sample sizes of local datasets. In addi- tion, compared with FedeDuv-vote, the results indicate that the federated strategy proposed by FedCDnv is effective, supporting the consideration of both "correct" and "good" relationships. Finally, compared with other distributed meth- ods, the superior performance of FedCDnv synthetic data further demonstrates its effectiveness.
Real-world Data. Table presents the experimental results of FedCDnv, FedCDnv-vote, CDUIOV, and FCI-base on "Sachs". The results indicate that FedCDnv demonstrates significant advantage in identifying adjacencies between variables, as reflected by its superior performance in terms of pre-edge, rec-edge, and particularly Fl-edge. Notably, FedCDnv achieves nearly a 10% improvement in Fl-edge compared to the other methods. This performance gain can be attributed FedCDnv's 's ability to more effectively handle the heterogeneity and partial overlap in variable sets across clients. By designing two levels of TPSS, FedCDnv con- siders the impact of the presence of possible confounders and the varying importance of relationships, reducing spu- rious dependencies and enhancing the accuracy of learned edges. terms of learned orientations, while FedCDnv does not achieve the highest precision or recall individually, it attains the best Fl-orie. This may be due to the better performance of the edges identified by FedCDnv, which results in relatively high rec-orie. It is also worth noting that while CDUIOV achieves the highest Fl-orie value, this comes at the cost of significantly lower recall. This trade-off arises from its strong dependence on correctly identifying intervention targets.
Benchmark Data. Figure presents the results of FedCDnv, FedCDnv-vote, CDUIOV, and FCI-base on the benchmark networks. Here, six benchmarks are as the labels of the horizontal axis: 'C', 'A', 'T, 'B', 'C3', 'A3'. It observed that FedCDnv demonstrates relatively superior performance compared with the baseline algorithms. While FedCDnv's 's edge-learning performance is only slightly better than exist- ing methods, it shows a significant advantage in orientation identification across most networks.
Specifically, for Fl Feded FedCDny achieves optimal perfor-
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Federated Causal Structure Learning with Non-identical Variable Scts
Table 1. Experimental results in the "Sachs" datasct.
Algoritims
pre-edge
rec-edge
Fl-edge
pre-otte
rec-otie
Fl-orie
Dnv
0.6128 = 0,0.92 0.00% = 0,1322
0.6967 + 0,0316
0,3045 = 0.0431 0,1846 = 0.0713 0.2861 = 0,0840
ved
0.4643 + 0.0124
0.7647 = 1369
0.5778 + 0.0998
0.3851 4 0.0959 0.2260 = 0.0124 0.2544 + 0.0384
Sachs
CDUIOV
0.6000 = 0.0132
0.3529 =0.143
0.4444 0.143544
0.8843 0,0507 0.0858
0.0263 0.1211 + 0.0021
PCI-base
0.4643 + 0.0143
0.7722 = 0.0454
0.5823 + 0.1289
0.2157
+ 0.0750
0.2635 0.0689 0.2488 + 0.0689
variable pairs, extracting definite (or "correct") causal and non-causal relationships. Second, we consider the varying importance of causal relationships across different clients and define stable relationships as those that are not on- ly "correct" but also "good" within each client. To this end, we design a federated strategy-two-leeel priority se- lection strategy (TPSS) aggregate both "correct" and "good" causal relationships from individual updates, ob- taining the global causal graph over an integrated set of variables. Compared to existing haselines, FedCDnv's 's supe- rior performance demonstrates its effectiveness in federated settings with non-identical variable sets.
Table 2. Performance of definite causal and non-causal relations.
M
FDR-C
FDR-dC
FDR-aC
FDR-dnC
10
0.0000
0.0000=0.0000
0.2549=0. 1061
0.2270=20.1169
20
1.672411..1%6 0.0766:002255
1.0299=0.0291
0.0476:E0.0247
30
10.113 0,0681 1 0.1703
0.0169
0.0144 0.0058
0.00181
0.0016
40
0.11622:00099 0.0935=0.1176
0.0673+000093
30
0.0892 00.172
0.145%
0.0263 0.0055
0.0004
60
0,0974
0.1079=00118 180
0.0202==000045
0.019%=000007
80
0.0929:000600
0. 1.1431...100
0.012550000226
0.0009--0.0029
100
0.138001000
0.1351 1008
0.1655
0.009991..0022)
0.00192 0.0018
120
00.0619
0.0961
0800132=0.0019
0.0130=:000019
150
0.0069:10:0485
0.0537-10.0566
0.00333:000012
0.0132::0.0013
4.3. Performance of the Identified Definite Relations
In future work, we plan to explore three aspects. First, we aim to develop more effective methods for learning the causal graph from each clients dataset, which contains latent variables and limited sample sizes. Second, recognizing that the current theories for identifying definite relationships are not yet comprehensive, we intend to conduct more thor- ough theoretical analysis on each pair of variables. Third, we will also explore additional privacy protection efforts.
To evaluate the effectiveness of the proposed theories in iden- tifying definite relationships, we compare the FDR values between the returned causal and non-causal relationships and the returned definite causal and definite non-causal re- lationships. The experimental results are shown in Table 2, where [ denotes the number of variables in the underlying DAG. FDR-C and FDR-nC denote the FDR values of the returned causal and non-causal relationships, respectively, while FDR-dC and FDR-dnC represent the FDR values of the returned definite causal and definite non-causal relation- ships. A lower FDR value indicates better performance.
Impact Statement
This paper presents work whose goal is to discover causal relationships from distributed data stored across individual clients (e.g., hospitals), while preserving data privacy. It introduces a more realistic federated setting in which clients observe non-identical variable sets and contribute causal relationships with varying importance, thereby improving the practicality and reliability of causal discovery in real- world applications. While the method avoids direct data sharing, it involves the exchange of structural and statistical information, which may still pose risks of privacy leakage.
As shown in Table 2, compared to the identified causal and non-causal relationships, the performances of the extracted definite causal and non-causal relationships are superior in most networks. Specifically, in 6 out of networks, the FDR- dC value is lower than FDR-C, with an average reduction of 1%. In the remaining 3 networks, although the FDR-dC value is slightly higher, the increase is limited to just 0.5%. These results indicate that, on average, definite causal rela- tionships achieve lower FDR values. demonstrating better reliability. For definite non-causal relationships, FedCDnv shows improved performance in 90% of the networks, fur- ther confirming the effectiveness of the proposed theories. The consistently lower FDR values for definite relationship- s, whether causal or non-causal, empirically validate the effectiveness of extracting the "correct" relationships.
A Acknovledgments
This work supported by the National Natural Science Foundation of China (U24A20323, 62376145, 62376087), the Science and Technology Innovation Talent Team of Shanxi Province 202204051002016 and the Key Tech- nologies Program of Taihang Laboratory in Shanxi Province (THYF-JSZX-240107))
5. Conclusion and Future Work
We propose novel algorithm, FedCDnv, for discovering causal relationships over non-identical variable sets in fed- erated settings. In this work, we focus on two key aspects. First, we develop theories to examine spurious dependen- cies caused by the potential presence of non-overlapping
References
Abyaneh, A., Scherrer, N., Schwab, P., Bauer, S., Schölkopf, B., and Mehrjou, A. Fed-cd: federated causal discov-

---

Federated Causal Structure Learning with Non-identical Variable Sets
ery from interventional and observational data. arXiv preprint 1. 2022. doi: 10.48550/arXiv. 2211
I. S., and der Schaar, M. Causal machine learning for predicting treatment outcomes. Nature Medicine, 30(4): 958-968, 2024.
Acar, A., Aksu, H., Uluagac, A. S., and Conti, M. A survey on homomorphic eneryption schemes: theory and imple- mentation. ACM Computing Surveys (Csur), 51(4):1-35, 2018.
Gao, E., Chen, J., Shen, L., Liu, T., Gong, M., and Bondell, H. Feddag: Federated dag structure learning. Transac- tions on Machine Learning Research, 2023.
Guo, X., Yu, K., Liu, L., and Li. J. Fedesl: A scalable and accurate approach to federated causal structure learn- ing. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pp. 12235-12243, 2024.
Boyd, S., Parikh, N., Chu, E.,, Peleato, B., Eckstein, J., et al. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine learning, 3(1):1-122, 2011.
Hernn, M. A. and Robins, J. M. Causal inference: whut if. Cambridge University Press, United Kingdom, 2020.
Cai. L.. Gonzales, T., Wheeler, E., Kerrison, N. D.. Day, F. R., Langenberg, C., Perry, J. R., Brage, S., and Ware- ham, N. Causal associations between cardiorespiratory fitness and type? diabetes. Nature Communications, 14 (3904):1-13, 2023.
Huang, B., Zhang, K., Gong, M., and Glymour, C. Causal discovery from multiple data sets with non-identical vari- able sets. In Proceedings of the AAAI Conference Ar- tificial Intelligence, volume 34, pp. 10153-10161, 2020.
Cao, F., Wang, Y., Yu, K., and Liang, J. Causal discovery from unknown interventional datasets over overlapping variable sets. IEEE Transactions on Knowledge and Data Engineering, 36(12))7725-7744) 2024.
Huang. J.. Guo, X., Yu, K., Cao, F., and Liang, J. Towards privacy-aware causal structure learning in federated set- ting. IEEE Transactions on Big Data, 166):1225-5535, 2023.
Chengbo, Q. and Kai, Y. Interventional causal structure discovery over graphical models with convergence and optimality guarantees. arXiv preprini arXivv2408.044: 04819, 2024.
Li, B., Li, H., Xiong, R., Wu, A., Wu, F,, and Kuang, K. Learning shadow variable representation for treatment effect estimation under collider bias. Proceedings of the International Conference on Machine Learning, 2024a.
Chickering. D. M. Learning equivalence classes of bayesian- network structures. Journal of Machine Learning Re- search, 11445-498 2002.
Li, L., Ng. I., Luo, G., Huang, B., Chen, G., Liu, T., Gu, B., and Zhang, K. Federated causal discovery from het- erogeneous data. In Proceedings of the International Conference on Learning Representations, 2024b.
Colombo, D., Maathuis, M. H., Kalisch, M., and Richard- son, T. S. Learning high-dimensional directed acyelic graphs with latent and selection variables. The Annals of Statistics, pp. 294-321, 2012.
Li, T., Sahu, A. K., Talwalkar, A., and Smith, V. Federat- ed learning: challenges, methods, and future directions. IEEE Signal Processing Magazine, 37(3)500-60, 2020.
Cramer, R., Damgård, B., et al. Secure multiparty com- putation. Cambridge University Press, United Kingdom, 2015.
McMahan, H. B., Moore, E., Ramage, D., Hampson, S., and y Arcas, B. A. Communication-efiicce learning of deep networks from decentralized data. In International Conference on Artificial Intelligence and Statistics, pp. 1273-1282. PMLR, 2017.
Deng, C., Bello, K., Aragam, B., and Ravikumar, P. K. Optimizing notears objectives via topological swaps. In Proceedings of the International Conference on Machine Learning, pp. 7563-7595. PMLR, 2023.
Mian, O., Kaltenpoth, D., Kamp, M., and Vreeken, Noth- ing but regrets privacy preserving federated causal dis- covery. In International Conference Artificial Intelli- gence and Statistics, pp. 8263-8278. PMLR, 2023.
Dhir, A. and Lee, C. M. Integrating overlapping datasets using bivariate causal discovery. In Proceedings of the AAA/ Conference on Artificial Intelligence, volume 34, pp. 3781-3790, 2020.
Ng, and Zhang, K. Towards federated bayesian network structure learning with continuous optimization. In Inter- national Conference Artificial Intelligence and Statis- tics, pp. 8095-8111. PMLR, 2022.
Erdös, P. and Rényi, A. On random graphs i. Publicationes Mathematicae, 6(18):290-29) 1959.
Feuerriegel, S., Frauen, D., Melnychuk, V., Schweisthal, J., Hess, K., Curth, A., Bauer, S., Kilbertus, N., Kohane,
Replogle, J. M., Saunders, R. A., Pogson, A. N., Hussmann, J. A., Lenail, A., Guna, A., Mascibroda, L.., Wagner,
10

---

Federated Causal Structure Learning with Non-identical Variable Sets
E. J., Adelman, K., Lithwick-Yanal G., et al. Map- ping information-rich genotype-phenotype landscapes with genome-scale perturb-seq. Cell, 185(14))259-2575, 2022.
Yang, Q. Liu, Y., Chen, T., and Tong. Y. Federated ma- chine learning: concept and applications. Transactions Intelligent Systems and Technology, 10(2):1-1,, 2019.
Ye, Q., Amini, A., and Zhou, Q. Federated learning of generalized linear causal networks. IEEE Transactions Pattern Analysis and Machine Intelligence, 46( 10): 6623-6636 2024.
Richardson, T. and Spirtes, P. Ancestral graph markov models. The Annals of Statistics, 30(4):62--0300, 2002.
Rohekar, R. Y., Nisimov, S., Gurwicz, Y., and Novik, G. It- erative causal discovery in the possible presence of latent confounders and selection bias. In Advances in Neural Information Processing Systems, volume 34, pp. 2454- 2465, 2021.
Zhang, On the completeness of orientation rules for causal discovery in the presence of latent confounders and selection bias. Artificial Intelligence, 172(16-17); 1873-1896, 2008
Zheng, X., Aragam, B., Ravikumar, P. K., and Xing, P.
Sachs, K., Perez, O., Pe'er, D., Lauffenburger, D. A., and Nolan, G. Causal protein-signaling networks derived from multiparameter single-cell data. Science, 308(5721): 523- 529, 2005.
Dags with no tears: continuous optimization for structure learning. In Advances in Neural Information Processing Systems, volume 31, pp. 9492 9503, 2018.
Zhu, Y., He, Y., Ma, J., Hu, M., Li, S., and Li, Causal in- ference with latent variables: recent advances and future prospectives. In ACM SIGKDD Conference on Knowl- edge Discovery and Data Mining, pp. (677-6687, 2024.
Spirtes, P., Glymour, C., and Scheines, R. Causation, pre- diction, and search. The MIT press, New York, 2000.
Squires, C., Seigal, A., Bhate, S. S., and Uhler, C. Linear causal disentanglement via interventions. In Proceedings of the International Conference on Machine Learning, Pp. 32540-32560. PMLR, 2023.
Tian, J. and Pearl, J. A general identification condition for causal effects. In Proceedings of the AAAI Conference Artificial Intelligence, pp. 567-573, 2002.
Tillman, R. and Spirtes, P. Learning equivalence classes of acyclic models with latent and selection variables from multiple datasets with overlapping variables. In Interna- tional Conference Artificial Intelligence and Statistics, pp. 3-15, 2011.
Tillman, R. E. Structure learning with independent non- identically distributed data. Proceedings of the Interna- tional Conference on Machine Learning, pp. 1041-1048, 2009.
Tillman, R. E., Danks, D., and Glymour, C. Integrating lo- cally learned causal structures with overlapping variables. In Advances in Neural Information Processing Systems, volume 21, pp. 1665-1672, 2008
Triantafillou, and Tsamardinos, Constraint based causal discovery from multiple interventions over overlap- ping variable sets. Journal Machine Learning Research, 2015.
Wang, Y., Cao, F., Yu, K., and Liang, J. Efficient causal structure learning from multiple interventional datasets with unknown targets. In Proceedings of the AAA/ Confer- ence on Artificial Intelligence, volume 36, pp. 8594, 2022.

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Federated Causal Structure Learning with Non-identical Variable Scts
A. Related Work
In this section, we review and discuss related work from three perspectives. The first perspective involves the discussion of the existing federated causal discovery (FCD) methods. The second perspective reviews the causal discovery (CD) methods from distributed data with non-identical variable sets. And the third perspective presents a detailed discussion about the differences between the above two, that is FCD from decentralized data and CD in distributed setting.
A.1. Federated Causal Discovery from Decentralized Data
In recent years, to preserve data privacy, some FCD methods have been proposed to discover causal relationships among a set of observed variables from decentralized data. They are categorized into three classes: constraint- -aased, score- based and coninnoussoppimmizationn-ooooo
methods. Specifically, constraint- methods utilize conditional independence tests to discover causal relationships among variables in a context of federated learning, such as FedPC (Huang et al.. 2023), FedCSL (Guo et al., 2024), and FedCDH (Li et al., 2024b). FedPC introduces a layer-wise aggregation strategy to adapt PC into federated settings. FedCSL designs a federated local-to-global learning strategy, enabling it to scale to high-dimensional data. FedCDH is proposed to accommodate arbitrary causal models and heterogeneous data. For score-based methods, few have been published. DARLS (Ye et al., 2024) utilizes the distributed annealing strategy to search for the optimal graph, while PERI (Mian et al., 2023) aggregates the results of the local greedy equivalent search (GES) (Chickering, 2002) and chooses the worst-case regret for each iteration. Finally, pontinuous-oppmmizaionn--ot (Zheng et al., 2018; Deng et al., 2023) with federated strategies. NOTEARS-ADMI (Ng & Zhang, 2022) applies the ADMM (Boyd et al., 201 optimization to federated setting; FED-CD (Abyaneh et al., 2022) utilizes belief aggregation to address the problem of inferring causal structure from the mixed data of observational and interventional data; and FedDAG (Gao et al., 2023) learns an adjacency matrix to estimate the DAG and neural representation to approximate causal mechanisms.
methods mainly extend NOTEARS
All these methods focus on discovering causal relationships over set of identical variables and do not consider the presence of latent variables, which do not often fit real-world situations where variables measured in different clients are non-identical. In addition, most of them consider the overall quality of the learned causal graph, ignoring the varying importance of causal relationships between variables within this graph.
A.2. Causal Discovery from Distributed Data
Several methods have been proposed to discover causal relationships from distributed data with non-identical variable sets. Tillman et al. (Tillman et al., 2008) propose the ION algorithm, which discovers a minimal equivalence class of causal DAGs using local independence information on different subsets of variables. To mitigate contradictories resulting from statistical errors, Tillman et al. (Tillman & Spirtes, 2011) propose the IOD) algorithm to learn equivalence classes directly by employing statistical tests across datasets simultaneously (Tillman, 2009). Triantafillou et al. (Triantafillou & Tsamardinos, 2015) propose COmbINE, which accepts a collection of interventional datasets over overlapping variable sets under different experimental conditions, and then outputs summary of all causal models indicating the invariant and variant structural characteristics of all models that simultaneously fit all of the input data sets. To reduce the number of possible structures, Dhir et al. (Dhir & Lee, 2020) adapt and extend bivariate causal discovery algorithms to learn consistent causal structures from multiple datasets with non-identical variable sets. Huang et al. (Huang et al., 2020) propose CD-MiNi to identify causal relationships from multiple observations with non-identical variable sets, under the linearity and non- -anusauiiantty assumptions. Recently, Cao et al. (Cao et al., 2024) propose the CDUIOV algorithm, designed to discover causal relationships over overlapping variable sets from multiple interventional datasets with unknown intervention targets.
In general, the above algorithms allow the raw data to be shared and ignore the spurious dependencies caused by non- overlapping variables, resulting in the reliability of returned outputs not being guaranteed.
A.3. Differences between FCD from Decentralized Data and CD from Distributed Data
The differences between federated causal discovery (FCD) from decentralized data and causal discovery (CD) from distributed data mainly stem from data characteristics and application areas.
In a federated setting, data is scattered across multiple independent data sources or clients, and data is not shared between them. Therefore, federated causal discovery focuses on discovering causal relationships between variables from decentralized independent data sources, emphasizing protecting the data privacy and reducing communication costs. It is suitable for
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Federated Causal Structure Learning with Non-identical Variable Scts
scenarios that require high privacy protection, such as medical and health data (Chengbo & Kai, 2024). In contrast, although distributed data are stored across multiple nodes, these nodes are usually part of a centralized system, and data can flow and be shared between them. Therefore, causal discovery methods for distributed data focus more on the consistency and efficiency of the causal discovery framework. It is suitable for scenarios that require rapid processing and analysis of large amounts of data, such as e-commerce data.
B. Formal Preliminaries
B.1. Definitions and Key Concepts
When the causal sufficiency assumption (Spirtes al., 2000) is violated, the underlying system is typically modeled using SMCMs MAGs, rather than DAGs. Here, we first introduce the term mixed causal graphs to represent both SMCMs and MAGs. In a mixed causal graph G = (V,E), where V = {0, (,,, there are directed (-)), bi-directed (++), and undirected (-) edges. A directed edge X + Y denotes that X is a cause of Y. bi-directed edge X ++ Y indicates that there are unobserved confounders between X and Y, and an undirect edge X Y denotes the presence of selection biases between X and Y. In G. X is called an ancestor of Y and Y is called a descendant of X if either X = Y or there exists a directed path from X to Y. Let AY) denote the set of ancestors of Y in G. A directed cycle in G occurs when X- - Y and Y E (X). An almost directed cycle in G occurs when X ++ Y E & and either Y e Ang(X) or X E (Y). Directed cycles and almost directed cycles are allowed in mixed causal graphs.
We use the MAG M (O, E) to model the conditional independence relations among O and use X; M X3 Z and X; H.M M X,IZ to represent that X; and X3 are m-separated and m-connected (or conditionally independent and dependent) given a set Z, respectively. Here, m-separation is a natural extension of d-separation, which is presented as Definition B. Under the Causal Markov and Faithfulness assumptions (Spirtes et al., 2000), every m-separation present M corresponds to conditional independence and vice-versa.
Definition B.1. m-separation (Zhang, 2008; Triantafillou & Tsamardinos, 2015). In M, X, and X3 are m-separated by Z S O,X,, X₁), if every path between X; and Xj is blocked by Z. A path between X; and X; is blocked by Z if and only if the following conditions hold: 1) every non-collider J is not member of Z. 2) every collider T has descendant in Z.
In general, some orientations of MAGs are indistinguishable from observational data when using constraint- methods. MAGS with the same set of conditional independencies are considered Markov equivalent, and the complete set of Markov equivalent MAGs is represented by a Partial Ancestral Graph (PAG). In a PAG, endpoints that can be either arrowheads (>) or tails (-) > different MAGs are denoted with a circle (o), and the symbol serves as a wildcard to denote any of the three marks or o). There are four types of edges in a PAG: +, o-,, o-o and 4+,
B.2. Connections and Differences among DAGs, SMCMs, MAGs and PAGs
In this subsection, we describe the connections and differences among DAGs, SMCMs, MAGS, and PAGs, which instantiated as shown in Figure Before that, we first introduce inducing paths, as shown in Definition B.2, which play a pivotal role because adjacencies and non-adjacencies in MAGS can be interpreted as the existence absence of inducing paths in SMCMs.
Definition B.2. Inducing path (Zhang, 2008; Triantafillou & Tsamardinos, 2015). In mixed causal graph over V = OUCUS, let X, Y (X, Y E O) be any two vertices, and L, S be two disjoint sets of latent confounders and selection bias, respectively. A path p between X and Y is called an inducing path relative to (C, S) if every non-endpoint vertex on p is either in L or a collider, and every collider on p is an ancestor of either X or Y. or a member of S. When = S = 0. p is called a primitive inducing path between X and Y.
As shown in Figure 7, DAGs, SMCMs, MAGs, and PAGs are different types of causal graphs for describing system. Specifically, (a) represents a DAG over O S. where 0 (A C, L= {L}, and S = V. (b) and (c) are both mixed graphs. (b) represents SMCM over O, which is obtained by hiding the variable L in (a). and (c) represents MAG over O which is transformed by looking for inducing paths of every pair of variables in (b). (d) is a PAG over O, which represents the Markov equivalent class of MAGs with the exact set of conditional independencies.
In Figure 7(b), A and E are not adjacent but they are not m m-separated by any set, that is, A. EJ (B) [C] and A x B) C). Hence, the SMCM skeleton in (b) learned from the data is not unique. To overcome this obstacle, the
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Federated Causal Structure Learning with Non-identical Variable Sets
L
E
E
E
E
A
B
B
A
B
A
B
(a) DAG
(b) SMCM
(c) MAG
(c) PAG
Figure 7. A toy example graph for describing the connections and differences among DAGs, SMCMs, MAGs, and PAGs.
Table 3. Summary of notations
Notations
Meanings
my clients; central server
0.
integrated set of observed variables; the set variables observed by client oz
C: S
absolute latent variables; relative latent variables O'O2; selection variahles
Da: Paide
data stered in client cu: sample size of Da: number of variables in O₂
ME; Py
an underlying MAG describing the causal relationships ammon Oe: the joint probability distribution over OL
P2. X LDD YIZ
X and conditionaly independent given
D2; X set Z in Dp.
conditionally dependent given
Pagp Pag.."
an initial cansal graph leamndd from the data of client op: an updated causal graph over O₂
notation of MAGs is introduced. Here, Figure 7(c) is obtained from Figure 7(b) by looking for primitive inducing paths in (b). Specifically, the path A + B <> E is an inducing path in (b), and thus the necessary edge A - E' is added to (c). In addition, when an almost directed cycle B + E & B - E occurs in (b), the bi-directed edge B ++ E is removed from (c). Thus, converting a MAG into the corresponding SMCM is generally not possible, as we cannot know which edges correspond to direct causation or confounding, and which ones are due to primitive inducing paths. Finally, Figure 7(d) is learned using a causal discovery method (e.g., RFCI (Colombo et al., 2012) or ICD (Rohekar et al., 2021)) with an oracle of conditional independence tests. In (d), there are four types of edges: -, o-t, o-o, ++. Here, A +oO is identified as V-structure, and Bo->E can be identified using the rules described in (Zhang, 2008).
B.3. Summary of Notations
We summarize the important symbols and their meanings throughout the paper, as shown in Table
C. Proofs of the Proposed Theories
Assume that G& = (Op, E is local causal graph learned from the data of client ca (k E (1, .m) with an oracle of conditional independence tests. Assume that (O, E)) is an integrated graph formed by aggregating adjacencies and arrowheads (one the orientations) from all local graphs (i.c., UK- Oz, E = UK Ex), and M is the underlying MAG, representing the ground truth graph.
Theorem C.1. Let (X, Y) be a non-overlapping variable pair (X, Y E O) and & the set of variables adjacent to X in G. If there exists A A & A # á & A + 0. and for each Y' Aς, (X,Y") is not non-overlapping variable pair (and vice versa), then the relationship between X and Y is considered definitively non-causal.
Proof. Since X and Y are not observed simultaneously by any client, their relationship cannot be identified from any dataset. In other words, whether X and Y are adjacent in the ground truth MAG M remains unknown. In addition, assume that for each Y' E A& (X, Y") is not non-overlapping variable pair (and vice versa), and consider two possible relationships between X and Y in M. as follows.
(1) Assume that X and Y are not adjacent in M. The condition "AS n. -0 & 0 implics that either there are no paths between X and Y. or there exists paths that involve least two intermediary nodes, apart from X and Y. Therefore, the proposed conditions met in this case, and we conclude that the relationship between X and Y is non-causal.
(2) Assume that X and Y are adjacent in M. The condition "AQ n = 0" implies three possible cases.
(a) In the ground truth graph M, X adjacent only to Y, and vice versa. That is, X and are mutually adjacent in M. Since (X, Y) is a non-overiine non-overlapping variable pair, we learn A& and A& = from the data, leading to AC n A& 66.
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Federated Causal Structure Learning with Non-identical Variable Sets
(b) AA O,,X, Y such that X X (or Yo-X+ Since X and Y are never observed together by any client, A 0 (or (A]) and A& (A) (or f), which leads to A 0 AÇ = 00. (c) AA, E OV(Xy) Y such that Ao-XX+oY B. In this case, AX = {A, B} and A& = (B). However, if X and B form a non-overlapping variable pair, then A {4] and AC (B). and their intersection empty.
To avoid the cases (a) and (b), the condition that "A + 0 & AG + 0" is required. For case (c), the condition that "for each Y' E A4 (X. Y.) is not a non-overlapping variable pair (vice versa)" is presented.
Based on the above analysis, it is sufficient to prove that X and Y are not adjacent in the underlying graph .M using the mentioned conditions, thereby inferring the non-causal relationship between X and Y.
Lemma C.2. Assume that X and Y (X,Y Y E O) are observed simultaneously by at least one client and Zn represents a union set of variables that appear in non-overlapping variable pairs with non-definite, non-causal relationships. If X E Zn. or Y Zn then the causal relationship between X and Y is considered to be non-definite.
Proof. X and Y are observed simultaneously by least one client, implying that (X, Y) is not non-overlapping variable pair. Suppose there exists variable A E O {X; Y} such that X (or Y) and A form non-overlapping variable pair and their relationship is considered to be non-definite and non-causal, which implies that X E Zn or Y Zn. There are two cases regarding the relationship between X and Y.
Case 1: X and Y are adjacent in the ground truth M. The relationship between X and Y is inferred to be adjacent because no set Z exists such that X and Y are conditionally independent given Z. regardless of whether X and A are adjacent in M.
Case2: X and Y are not adjacent in the ground truth M. Since (X, (or AY) forms a non-overlgpiing variable pair with non-definite, non-causal relationship, X and Y may be learned as adjacent from local data, which conflicts with the non-adjacency between X and Y in M. This occurs because A could be a confounder for X and Y. leading to the learned adjacency between them. As a result, the causal relationship between X and Y is considered to be non-definite.
In summary, we conclude that if X E Zn or Y E Zn, then the causal relationship between X and Y is considered to be non-definite.
O
Lemma C.3. Assume that X and Y (X,Y E O) are observed simultaneously by at least one client and A& is a set of variables adjacent to X in G. If there exists a non-overlapping variable pair (A,, B) such that {A, B} C. A U AG, then the causal relationship between X and Y is considered to be non-definite.
Proof. Since X and Y are observed together by at least one client, their relationship can be learned from at least one local dataset. Assume that X and Y are not adjacent in the ground truth graph M, and then {A, B} S A U A& implies three possible cases regarding the structures of (A, B) and (X, Y) in M.
Casel: Both A and B are linked to both X and Y in M. In this case, the separating set that makes X and Y conditionally independent must be a subset of the adjacent set of either X or Y. If there exists a set Z. such that X and Y independent given Z. and Z contains {A, BJ. then based on the dataset Dy (k E I,.... -.m.,,)))) m)) we would learn that X and Y are not m-separated by Z. The reason is that Z S A U UQ, (A, B) C Z, but {A, B) & O& (k E 1,..., m)) Thus in this case, X and Y are adjacent in the learned causal graph Gk, which conflicts with the fact that X and Y are not adjacent in M.
Case2: Either A or B is linked to both X and Y in M. Suppose that A is adjacent to X, and B is adjacent to Y. Since the separating sets for X and Y are incomplete, we may learn that X and Y are adjacent in Gg. which would be inconsistent with their non-adjacency in M.
Case3: Neither A nor B is linked to X or Y. In this case, if (A, B} & A& UAG then the separating sets for and are complete. As a result, the learned causal relationship between X and Y consistent with the ground truth graph M.
In summary, when there exists non-overlapping variable pair (A, B) C A U AÇ the relationship between X and Y learned from the data may differ from those in the ground truth and is considered to be non-definite.
D. Experiments
This section provides a detailed overview of the experiments. Section D.I describes the problem attributes and presents the details of the baseline methods. Section D.2 displays additional experimental results on precision and recall, comparing
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Federated Causal Structure Learning with Non-identical Variable Sets
Table Test values and default values of problem attributes in generating experiments in each iteration of FedCInv.
Problem altribules
values
Default value
Number of variables generated the underlying DAG (PD
10, 20, 30, 40, 50, 60, 80, 100 13%, 10%, 13%, 20% 25%, 30%, 35%
30
Ratio of the number of latent variables to vartables in undertying DACs
15%
(A [V]
Number of clients (m)
(2. 6, 10, 15, 30F
Katio of the number of variables per dataset to the integrated variables
(95%, 90%, 83%, 80%, 73%, 70%, 65%, 60%E
85%
(6 TO
Interval of confidence level
0,, 9s] with varying 01, where
[0.001, 0.002, 0.003, 0.004, 0.005, 0.006, 0.007, 0,008,
[0.001, 0.5]
0.05 and o + 0g = 0.5
0.009)
Interval confidence level
01, a 02] with varying + 02. where
(0.5, 0.45, 0.4, 0.35, D.3, 0.25, 0.2, 0.15, D.I)
[0.001, 0.5]
=0.05 and a 02 0.001
Interval of Sample size per dataset (100, ns))
2000, 500, 1O00, 2000, 3000, 5000)
[100,2000]
Table 5. Description of benchmark BNs.
Network
Num Nodes
Num Edges
Max In/Out Degree
Min/Max JPC]
Type
Child
20
25
277
1/8
Medium
Insurance
52
1/9
Medium
Alarm
37
46
4/5
1/6
Medium
Barley
48
84
4/5
1/8
Medium
Child3
60
79
3/7
1/8
Large
Alarm3
111
149
4/5
1/6
Very-Larrge
FedCDnv, FedCDnv-vote, CDUIOV, CD-MiNi, and FCI-base. Section D.3 presents and analyzes the experimental results, comparing our method with existing FCD methods, including FedACD, FedPC, and Notears-ADMM. Section D.4 assesses the sensitivity of stable relationships the performance of FedCDnv. Finally, Section D.5 presents the impact of different parameters on the performance of FedCDnv.
D.1. More Details of Experimental Settings
First, for each invocation of the algorithm, the problem instance (i.e., the synthetic data) is generated using the parameters shown in Table 4. Then, Figure 5 shows the details of the benchmark networks used in the experiments. Finally, to comprehensively verify the effectivenees of the proposed method, we conduct two types of experiments. The descriptions of the all compared methods (including ours) are presented as follows.
For the first type of experiments, the compared methods include FedCDnv, FedCDnv-vote, CDUIOV (Cao et al., 2024), CD-MiNi (Huang et al., 2020), and FCI-base. FedCDnv is our proposed method. FedCDnv-vote is a variant of FedCDnv that incorporates a voting strategy. CDUIOV is a constraint-
method that discovers causal structures from interventional datasets with overlapping variable sets. CD-MiNi well-known SCM- method that identifies causal relationships over the integrated set of variables in linear, non-Gaussian settings. FCI-base is baseline method that integrates multiple causal structures learned by FCI using sample-size-based federated strategy in the context of federated learning.
For the second type of experiments, the compared methods include FedCDnv, FedCDnv-vote, Notcars- (Ng & Zhang, 2022), FedPC (Huang et al., 2023), and FedCSL (Guo et al., 2024). FedCDnv and FedCDnv-vote are our proposed methods, which differ in their federated strategies. Notears-ADMM, FedPC and FedCSL are based on the assumptions that all clients observe identical variable sets and that there are no latent caablees-assuuptin that differ from ours. Thus, the comparison is intended only provide a rough performance reference rather than a strict benchmarking. Consequently, we evaluate only the performance on the learned skeletons. Specifically, method in the federated learning setting. FedPC and FedCSL two constraint based federated CSL methods. FedPC presents layer-wise aggregation strategy for a seamless adaptation of the PC algorithm into the federated learning paradigm. FedCSL first infers the relative sample sizes held by each client, and then performs a weighted aggregation of the learned structures from each client using weights based on their sample sizes.
ADMMa is a continuous-optimmiztuue
D.2. Additional Details on the Comparison of FedCDnv, FedCDnv-vote, CDUIOV, CD-MiNi, and FCI-base
Figures 8, 9 present detailed experimental results for precision and recall of learned edges and orientations, comparing FedCDnv, FedCDnv-vote, CDUIOV, CD-MiNi, and FCI-base.
Figure& 8 presents the experimental results of FedCDnv and CD-MiNi on networks with 5-10 nodes. It is observed that
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Federated Causal Structure Learning with Non-identical Variable Scts
1,00
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Figure 8. The setting of non-identical variable sets: FedCDnv CD-MiNi.
compared with CD-MiNi, FedCDnv significantly outperforms CD-MiNi in terms of precision for learned edges (pre-edge) and orientations (pre-orie), with the pre-orie obtained by FedCDnv being nearly 30% better than CD-MiNi. However, the recall values of edges and orientations are almost comparable between FedCDnv and CD-MiNi. For the recall values of returned edges (rec-edge), FedCDnv outperforms CD-MiNi on some networks, while CD-MiNi performs better on others. As for the recall value of learned orientations (rec-orie), both methods are comparable, with FedCDnv slightly outperforming CD-MiNi. Therefore, considering the F1 scores for edges (Fl-edge) and orientations (Fl-orie), FedCDnv demonstrates superior performance. The reasons for the difference between the two methods are discussed in Section 4.2.
Figure 9 displays the detailed experimental results among FedCDny, FedCDnv-vote, CDUIOV, and FCI-base. It is observed that FedCDnv shows a significant performance advantage in learning orientations, with pre-orie and rec-orie values noticeably higher than those of the competing algorithms. In edge learning, FedCDnv outperforms the competing algorithms slightly. Specifically, the first two rows of Figure 9 show experimental results of five algorithms on networks with different scales. The results show that, for pre-edge, FedCDnv slightly outperforms other algorithms, while for rec-edge, the five algorithms comparable with little difference. As for the returned orientations, the pre-orie and rec-oric obtained by FedCDnv significantly outperform the competing algorithms in most networks. Next, we analyze the impact of varying numbers of clients on the algorithm's 's performance, shown in the third and fourth rows of Figure 9. Similar to the results on networks of different scale (i.e., the first two rows of Figure 9), FedCDnv clearly outperforms the competing algorithms in learning orientations. Furthermore, FedCDnv shows more notable superiority in precision of learned edges (pre-edge), outperforming the competing algorithms. Notably, CDUIOV shows more significant performance variation in pre-edge across different network scales and client numbers compared with the other algorithms. This may be due to the fact that CDUTOV requires intervention data, and the influence of different data sources is more pronounced. Finally, we analyze the experimental results of the five algorithms on the benchmark data, as shown in the last two rows of Figure 9, We observe that the advantages of FedCDny in terms of pre-edge and rec-edge are not very pronounced, but its performance in pre-orie and rec-orie is better overall, clearly outperforming other algorithms. Overall, the superior experimental results of FedCDnv demonstrate its effectiveness in causal structure learning with non-identical variable sets, when compared with other distributed CSL methods the same setting.
Moreover, based on Figure 9, we also analyze the effect of different network scales and numbers of clients on FedCDnv. For the former, shown in the first two rows of Figure 9, we observe general decreasing trend in the performance of FedCDnv as the number of nodes increases, except for pre-edge. For networks with 30 or more nodes, the value of pre-edge remains almost constant, while the value of rec-edge decreases as the number of nodes increases. Although the rec-edge obtained by FedCDnv reaches its lowest value for the network with 50 nodes, this may be due to errors in CI testing. As a result, the Fl-edge of FedCDnv decreases as the increasing of the network scale. Regarding the learned orientations, the value of pre-orie decreases significantly as the number of nodes increases, whereas rec-orie remains relatively stable, except
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Federated Causal Structure Learning with Non-identical Variable Sets
FedCDnv
FedCDnv-vote
CDUIOV
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0.75
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[M0-odge
rec so--edgge
F1-edge
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025-
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C3
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Bayesian networks
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Fl-orie
0.50
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Figure The setting of non-identical variable sets: FedCDnv vs. FedCDnv-vote vs. CDUIOV vs. FCI-base.
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Federated Causal Structure Learning with Non-identical Variable Scts
for networks with 50 nodes. Therefore, the overall Fl-orie also exhibits a decreasing trend. Regarding the impact of the number of clients on FedCDnv, we refer the results shown in the panels of the third and fourth rows of Figure 9. It can be observed that as the number of clients increases, the overall performance of FedCDnv in terms of both edges and orientations improves. Specifically, regarding edge learning performance, pre-edge shows minimal variation across different numbers of clients, while the value of rec-edge increases significantly. Consequently, the overall metric F1 Fl-edge also increases as the number of clients grows, For learned orientations, apart from the case where the number of clients is 2-in which case FedCDnv achieves relatively high values for pre-orie, rec-orie, and Fl-orie- the values of these three metries generally increase as the number of clients increases. This improvement in the metrics evaluating learned orientations is more gradual compared to the improvement in edge learning performance.
D.3. Compare FedCDnv, FevCDDv-,ote, Notears-ADMM, FedCSL, and FedPC
In this section, we compare the experimental performance of our method with that of existing FCD algorithms on the learned causal skeleton, assuming that multiple clients observe the entirely identical variables.
Synthetic Data. The first row of Figure 10 presents the experimental results of FedCDnv and other FCD methods on synthetic data with node counts ranging from 10 to 100. It is observed that, first, compared with FedCDnv-vote, the pre-edge value of FedCDny is significantly higher than that of FedCInv-vote in the presented networks, while the rec-edge values for the two methods are comparable. According to the Fl-edge metric, FedCDnv demonstrates better skeleton learning performance in networks with 10, 20, 50, 80, and 100 nodes. In the remaining networks, the performance of both methods is similar. These results indicate the effectiveness of the proposed federated strategy TPSS, In addition, compared with the other three FCD algorithms, the pre-edge value returned by FedCDnv is slightly inferior that of FedCSL but significantly better than FedPC and Notears-ADMM. Moreover, the rec-edge value returned by FedCDnv is significantly higher than the others, outperforming them by nearly 25%. Therefore, according to the Fl Fl-edge metric, FedCDnv outperforms significantly Notcars-ADMI in networks with fewer variables. As the number of variables increases, the performance gap between the two methods gradually narrows. Nevertheless, in networks with 80 and 100 nodes, FedCDnv still outperforms NoMM by approximately 5%. comparison with FedPC and FedCSI., FedCDnv obviously performs better. This can be attributed to the fact that, in this comparison, we assume the presence of absolute latent variables. Our method uses the FCI algorithm to learn the local causal graph for each client and aggregates the results in federated setting, producing PAG. In contrast, FedPC and FedCSL operate under the causal sufficiency assumption and use the CD algorithm (e.g., PC) to return the Markov equivalence class of DAGs. As result, the inconsistent assumptions lead to poorer skeleton learning performance in these two algorithms when applied to data with latent variables.
The second row of Figure 10 presents the experimental results of FedCDnv and other FCD algorithms on synthetic data with varying numbers of clients. The results show that FedCDnv consistently returns the highest pre-edge values across different client counts. Furthermore, as the number of clients increases, the pre-edge values of compared methods remain relatively stable, except for Notears-ADDMM, which experiences a significant drop when the number of clients reaches 30. Other algorithms' pre-edge values show only slight decline at 30 clients. As for the rec-edge, it can be seen that FedCDnv and FedCDnv-vote perform similarly, and both outperform FedPC and FedCSL. The reason for this, as mentioned above, is primarily differences in the underlying assumptions of the algorithms. It is also noteworthy that the rec-edge value obtained by Notears-ADMM increases with the number of clients and stabilizes after reaching 8 clients. In summary, from the Fl-edge metric, the performance of both FedCDnv and FedCDnv-vote remains stable as the number of clients increases, consistently outperforming the other three algorithms. In contrast, the performance of ADtears-ADMM gradually improves with the increasing number of clients. After reaching 8 or more clients, its performance stabilizes, but when the number of clients reaches 30, the Fl-edge value begins to decline. This suggests that Notears-ADMi may have an optimal range of client numbers.
Benchmark Data. The last row of Figure 10 shows the experimental results of FedCDny and other FCD methods on six benchmark networks. Specifically, FedCDnv performs similarly to FedCDnv-vote in four networks, while in the other two networks, "Insurance" and "Barley", FedCDnv significantly outperforms FedCDnv-vote. Compared with FedPC and FedCSL, FedCDny shows a clear advantage in five of the networks, but in the "Barley" network, FedCDnv performs slightly worse than FedCSL. The gap is mainly due to the pre-edge learned by FedCSL, which exceeds 70%, significantly higher than the approximately 50% achieved by other algorithms, making FedCSL the most optimal in the "Barley" network. Finally, it is also observed that Notears-ADMM. performs poorly across the benchmark networks. Although its performance in the "Barley" network is close to that of the other algorithms, in the remaining networks, Notcars-ADMM is significantly inferior to the other algorithms. This may be due to the unknown data generation mechanism in the benchmarks, which
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Federated Causal Structure Learning with Non-identical Variable Sets
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Bayesian networks
Figure 10. The setting of identical variahle Experimental results synthetic and henchmark data.
Table 6. Experimental results the "Sachs" dataset.
Algorithms
pre-odge
rec-edge
F1-edge
Sachs
FedCDnv
0.7222 4: 0.013
0.6500 1 0.1076
0
0.68
0.3846 = 0.0689
0.2500 : 0.0032
0,30M0 =: 0,02177
vodt
0,5152 = 0.0573
0.6500
0.6115
0.1231
0.3143 = 0.0522
0.2500 - 0.0032
0.2633 0.0332
Notears-Adme Admm
0.7731 + 0.0638
0.4091
4 0.0226
0.4304 0.1435
FedCSL
0.7692 + 0.0638
0,2500
+ 0.0096
0,3774
0.1131
FedPC
0,4250
0.1289
leads to slightly suboptimal performance of NoMcars-ADMM in certain networks.
Real-world Data. Table presents experimental results of FedCDnv and other FCD methods on the real-world data "Sachs", where the variable sets observed by all clients are assumed to identical. In addition, since FedCDnv assumes the existence of absolute latent variables, only compare the correctness of the learned skeleton against existing FCD methods. observed that, according to the overall Fl-edge metrie, FedCDnv shows the best performance. However, in terms of edge precision (pre-edge), Motears-ADMM performs the best, followed by FedCSL, and then FedCDnv, which lags behind Notears-ADMM by nearly 5%. Meanwhile, in terms of rec-edge, FedCDnv and FedCDnv-vote outperform Notears-ADMM by nearly 20% and FedCSL by nearly 40%. Overall, FedCDnv demonstrates the superior performance, highlighting the effectiveness of the proposed federated strategy on the "Sachs" dataset.
D.4. Sensitivity Analysis of Stable Relationships on the Performance of FedCDnv
For the parameter [a- 01, 02), we first evaluate the impact of varying 0 on the performance of FedCDnv, while keeping a + 8₂ 0.5. The experimental results are shown in the left subplot of Figure 11. can be observed that as the parameter a - 02 increases, the performance of the FedCDnv algorithm exhibits a specific pattern of variation. When a - 01 less than 0.003, the performance improves gradually. However, after a exceeds 0.003, its performance begins to
20

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Federated Causal Structure Learning with Non-identical Variable Sets
pre-edge
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1.00
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0.001 0.002 0.003 0.004 0.005 0.006
0.007
0.008 0.009
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0.45
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0.25
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0.15
0.1
a 0.5]
[0.001,
a 02]
Figure II. The performance FedCDnv with varying [a- 01, a + 621.
decline, reaching its lowest point at a 01 = 0.005 and 0.006. Subsequently, the performance rises again at a 01 = 0.007 and a - 01 0.008, but starts to decline once more when 0 reaches 0.009. In summary, the performance of FedCDnv fluctuates as a 0, changes, following a pattern of "rising, falling, rising again, and finally falling."
Similarly, we evaluate the impact of varying a + 02 while keeping a - 0 = 0.001, on the performance of FedCDnv. The experimental results are shown in the right subplot of Figure 11. It is observed that as the interval narrows, most metrics-rrec-dge. Fl-edge, pre-orie, rec-orie, and Fl Fl-orie-exhibit trend resembling a normal distribution: higher values in the middle and lower values on both ends, except for the pre-edge metric, which remains relatively stable. This indicates that appropriately chosen confidence interval can enhance the performance of FedCDnv, while intervals that are either too small or too large are less effective for establishing stable relationships.
In conclusion, the parameters 0 and 0g have a moderate impact (5%-10%) on the performance of FedCDny. It is notably that FedCDnv exhibits higher sensitivity to changes in a + 0z.
D.5. Effects of Different Parameters on the Performance of FedCDnv
In this subsection, we evaluate the impact of varying values of the parameters listed in Table 4 (i.e., problem attributes) on the performance of FedCDnv. During the evaluation each parameter, all other parameters are kept at their default values. The tested parameters include: the number of nodes in the underlying DAG ((V)). the number of clients (m), the proportion of each client's 's observed variables relative to the integrated variables (0), the proportion of absolute latent variables (A), the sample size per client ([100, nss] ranging from 100 to ns), and the interval of confidence level ([a - 01, 02] with 0.05). The experimental results and analysis for [V] and m are presented in Section D.2, while those for the confidence interval [a- - 01, + 0g] are given in Section D.4. Results for the remaining parameters are summarized in Figure 12.
For the parameter (left subfigure of Figure 12), the results show a general decreasing trend in the performance of FedCDnv for both edges and orientations as increases. Specifically, in terms of edge learning, the precision for edges (pre-edge) gradually declines, while the recall for edges (rec-edge) slightly increases with larger 8 values, resulting in a relatively stable F1-edge score. In contrast, for orientation learning, the precision (pre-orie), recall (rec-orie), and F1 score (F1-orie) all show consistent downward trend as ó increases. These observations indicate that lower overlapping rate among observed variables negatively affects the orientation learning capability of FedCDnv.
The impact of the parameter À on FedCDnv shown in the middle subfigure of Figure 12. The results indicate that as À increases from 0.05 to 0.2, the performance metrics for both edges and orientations exhibit a declining trend. However, when increases further from 0.2 to 0.35, the performance improves noticeably. Specifically, the precision for edges (pre-edge) remains relatively stable across different values, while the precision for orientations (pre-orie) gradually decreases as A rises to 0.2. Other metrics metrics--nccuuing rec-edge, F1-edge, rec-orie, and F1-orie-drey sharply as À approaches 0.2. Interestingly, we observe that from À = 0.25 to 0.35, the performance of FedCDnv in both edge and orientation learning improves significantly. Overall, the performance of FedCDnv tends to degrade as À increases.
Lastly, we investigate the impact of the sample size per client, n, E [100, ns] and ns E {200, 500, 1000, 2000, 3000, 5000), on the performance of FedCDnv, focusing on how the evaluated metrics vary with changes in ns. The experimental results are presented in the right subfigure of Figure 12. It is observed that as ns increases, the performance of FedCDnv improves
21

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Federated Causal Structure Learning with Non-identical Variable Sets
pre-edge
rec-edge
Fl-edge
pre-orie
rec-oric
Fl-oric
1.00
1.00
1.00
0.75
0.75
0.75
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95% 90% 85% 80% 75% 70% 68%
60%
5%
10%
15%
20%
25%
30%
35%
200
500
1000
2000
3000
5000
[100, ms]
Figure 12. Experimental results FedCDnv with different parameters.
significantly, with rec-edge, Fl-edge, rec-orie, and Fl-orie all exhibiting upward trends. In contrast, the pre-edge reaches its peak at nıs 2000 and then declines slightly as ns continues to increase. Similarly, pre-orie exhibits a trend similar to pre-edge, peaking at nss 1000 and maintaining relative stability as ns increases further.
22


Reviews:
[{"review_id": "HZGxVIBwxs", "reviewer": "Reviewer_7pKi", "summary": "This paper investigates federated causal structure learning, aiming to discover causal relationships between variables from data distributed across individual clients while considering privacy concerns. The paper addresses federated causal structure learning with non-identical variable sets and designs an effective strategy to aggregate \"correct\" and \"good\" relationships between variables during collaborative learning. Experimental results demonstrate that the proposed method is effective on synthetic, benchmark, and real-world data.", "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": "$\\textbf{Q1}$. Theorems 3.2 and 3.3 only indicate that the relationship between variables X and Y is uncertain, but do not analyze under what conditions their relationship can be determined.\n\n$\\textbf{R1}$. Thanks very much for your comment. As stated in line 114 of the manuscript, we initially assume that all relationships among overlapping variable pairs are definite (or \"correct\"). Theorems 3.2 and 3.3 are proposed to detect which of these relationships become non-definite due to the influence of non-overlapping variable pairs. Therefore, the relationships that satisfy the conditions of Theorems 3.2 and 3.3 are non-definite, while all other relationships remain definite.\n\n$\\textbf{Q2-1}$. How does the proposed method aggregate the results obtained from different servers? \n\n$\\textbf{R2-1}$. Thanks for your comment. There are two aggregation stages. \n\n(1) For the first stage, the server integrates the adjacencies and orientations of local causal graphs learned by all clients into a global graph. For adjacencies, a variable pair is considered adjacent if any client reports the adjacency between them. For orientations (there are three types of orientations: circle 'o', tail '-', and arrow '>'), only the identified arrows are included in the global graph.\n\n(2) For the second stage, the server applies the proposed two-level priority selection strategy to aggregate all updated local graphs. First, for adjacency aggregation: a) First Priority Level: Check whether adjacency conflicts arise due to non-overlapping variable pairs, using Lemma 3.4; b) Second Priority Level: If the conflict is due to statistical errors, the final adjacency is determined by comparing $val_{ij}$ (for $X_i$ and $X_j$) with 0. The computation of $val_{ij}$ depends on two factors: $w_{ij}^{c_k}$ and the sample size weight $w_{c_k}$. Second, for orientation aggregation, the process is similar to stage (1).\n\n$\\textbf{Q2-2}$. The paper discusses taking the union of variable sets and the union of edge sets. When the relationships between two variables differ, such as X $\\\\circ\\hspace{-0.4em}-\\hspace{-0.4em}\\circ\\$ Y, X $\\circ\\hspace{-0.43em}\\rightarrow$ Y, and X$\\rightarrow$Y, what does the union of edges look like?\n\n$\\textbf{R2-2}$. When the relationships between two variables differ across clients, only identified arrows are included. For example, for X $\\\\circ\\hspace{-0.4em}-\\hspace{-0.4em}\\circ\\$ Y, X $\\circ\\hspace{-0.43em}\\rightarrow$ Y, and X$\\rightarrow$Y, the final relationship between X and Y will be X $\\circ\\hspace{-0.43em}\\rightarrow$ Y.", "claims_and_evidence": "Yes. The claims are clear and convincing.", "methods_and_evaluation_criteria": "Yes. The proposed methods and evaluation criteria make sense for the problem of federated causal structure learning with non-identical variable sets.", "theoretical_claims": "Yes. I have carefully reviewed all the proofs.", "experimental_designs_or_analyses": "Yes. I have checked the soundness of all the experimental designs and the experimental analyses are valid.", "relation_to_broader_scientific_literature": "The key contributions of the paper related to the problem of causal discovery with non-identical variable sets.", "essential_references_not_discussed": "Yes. The related works are cited and discussed.", "other_strengths_and_weaknesses": "**Strengths:**\n\n1. This paper proposes a federated causal structure learning method to address the challenge of discovering causal relationships under non-identical variable sets.\n2. The effectiveness of the method is validated through experiments on synthetic data, benchmark data, and real-world data, demonstrating its applicability and robustness across various types of data.\n\n**Weaknesses:**\n\n1. Theorems 3.2 and 3.3 only indicate that the relationship between variables X and Y is uncertain, but do not analyze under what conditions their relationship can be determined.\n2. How does the proposed method aggregate the results obtained from different servers? The paper discusses taking the union of variable sets and the union of edge sets. When the relationships between two variables differ, such as Xo-oY, Xo->Y, and X->Y, what does the union of edges look like?", "other_comments_or_suggestions": "N/A", "questions_for_authors": "See the weaknesses 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": "AtnpuUqqFL", "reviewer": "Reviewer_od7H", "summary": "The paper introduces FedCDnv, a federated causal structure learning algorithm designed for scenarios where clients have non-identical but overlapping variable sets. The method introduces theoretical criteria to distinguish definite causal and non-causal relationships. A two-level priority selection strategy (PSS) is developed to aggregate both “correct” (definite) and “good” (stable) relationships from local causal graphs into a global causal graph.", "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": "$\\textbf{Responses for “Questions For Authors\" are as follows.}$\n\n$\\textbf{Q1}$. Could the authors provide a real-world motivating example where FCD is used/needed and there are non-overlapping variables observed by clients?\n\n$\\textbf{R1}$. Thanks for your comment. A real-world example where non-overlapping variables are observed can be found in the eICU Collaborative Research Database (eICU-CRD, link https://physionet.org/content/eicu-crd/2.0/) [1-3]. It is a multi-center intensive care unit database covering over 200,000 admissions to ICUs across the United States between 2014-2015. In the \"vitalPeriodic.csv\" file, different ICU stays (identified by \"patientunitstayid\") record partially overlapping physiological variables, as shown below:\n| vital-periodic-id | patient-unit-stay-id | sao2 | heart-rate | respiration | cvp | system-ic-systolic | system-ic-diastolic | system-ic-mean | pa-systolic | pa-diastolic | pa-mean | icp-st1 | icp-st2 | icp-st3 |\n|-------------------|----------------------|------|------------|-------------|-----|--------------------|---------------------|----------------|-------------|--------------|---------|---------|---------|---------|\n| 35511110          | 141945               | 100  | 76         | 17          |     |                    |                     |                |             |              |         | -1      | -0.81   | 0       |\n| 35417390          | 141945               | 89   | 87         | 24          |     |                    |                     |                |             |              |         | -1      | -0.69   | 0.1     |\n| 28818781          | 142000               | 96   | 108        | 30          | 8   | 134                | 48                  | 74             |             |              |         |         |         |         |\n| 28839535          | 142000               | 98   | 80         | 20          | 11  | 142                | 50                  | 82             |             |              |         |         |         |         |\n| 48431259          | 142035               | 98   | 98         | 29          | 17  | 116                | 60                  | 76             | 34          | 13           | 23      |         |         |         |\n| 48435712          | 142035               | 94   | 92         | 23          | 16  | 116                | 68                  | 82             | 34          | 14           | 25      |         |         |         |\n\nIn general, clinical practice requires access to up-to-date patient data from hospitals, which is often distributed and privacy-sensitive. Due to regulatory constraints such as HIPAA and institutional policies, hospitals cannot directly share raw and up-to-date patient data for collaborative analysis. This creates a critical need for federated causal structure learning with non-identical variables sets, which allows clients to collaboratively infer causal relationships over integrated variables while keeping data decentralized and secure.\n\n[1] Pollard, T., Johnson, A., Raffa, J., Celi, L. A., Badawi, O., & Mark, R. (2019). eICU Collaborative Research Database (version 2.0). PhysioNet. https://doi.org/10.13026/C2WM1R.\n\n[2] Nakayama LF, Restrepo D, Matos J, Ribeiro LZ, Malerbi FK, Celi LA, Regatieri CS. BRSET: A Brazilian Multilabel Ophthalmological Dataset of Retina Fundus Photos. PLOS Digit Health. 2024 Jul 11;3(7):e0000454. doi: 10.1371/journal.pdig.0000454. PMID: 38991014; PMCID: PMC11239107.\n\n[3] Goldberger, A., Amaral, L., Glass, L., Hausdorff, J., Ivanov, P. C., Mark, R., ... & Stanley, H. E. (2000). PhysioBank, PhysioToolkit, and PhysioNet: Components of a new research resource for complex physiologic signals. Circulation [Online]. 101 (23), pp. e215–e220.\n\n$\\textbf{Responses for “Other Comments Or Suggestions\" are as follows.}$\n\n$\\textbf{Q2-1}$. Figure 1 is never mentioned/referenced in the text.\n\n$\\textbf{R2-1}$. Thank you for pointing this out. We apologize for the oversight. Figure 1 is used in the fourth paragraph of the introduction to illustrate spurious dependencies, but it was not explicitly referenced in the text. In the revision, we will embed Figure 1 within the fourth paragraph of the introduction.\n\n$\\textbf{Q2-2}$. Line 294 Section 4.1 should read \"real-world data.\" \nThe error bars in the figures are hardly visible. I suggest either using error bands, or also displaying the whiskers (lines). \nTables 1 and 2 contain too many significant digits to be easily legible. I suggest reporting figures to 2 or max 3 decimal places.\n \n$\\textbf{R2-2}$. Thanks for your suggestions. We revised the errors and carefully checked the manuscript. In addition, for the error bars, we replaced them with whiskers (lines) to make them clearer. For Tables 1, 2 and 6, we retained three decimal places and revised them accordingly. Thanks again.", "claims_and_evidence": "Yes, the claims (detecting good and correct causal relationships) are backed up by theoretical statements with proofs.", "methods_and_evaluation_criteria": "Yes, standard causal discovery metrics (e.g., F1) are used for gauging the accuracy of the learned graph; and the datasets used (synthetic and real) make sense for the problem.", "theoretical_claims": "Yes, Theorem C.1. No issues were found.", "experimental_designs_or_analyses": "Yes, all the experiments in the main text and the parameter-sensitivity experiments in Appendix D.4.", "relation_to_broader_scientific_literature": "The work extends federated causal discovery to non-identical variable sets, addressing gaps in prior works (e.g., FedCSL, Notears-ADMM) that assume identical variables.", "essential_references_not_discussed": "None, to the best of my knowledge.", "other_strengths_and_weaknesses": "**Strengths**\n1. The problem of FCD with non-overlapping variables is novel and realistic one in practice.\n2. Theoretical justifications are provided (e.g.,correctness of identifying definite causal and non-causal relationships).\n3. Privacy and communication costs are given.\n\n**Weaknesses**\n1. The communication costs of a client $c_k$ scale with $O(d_k^2)$, where $d_k$ is the number of variables observed by the client.", "other_comments_or_suggestions": "1. Figure 1 is never mentioned/referenced in the text.\n2. Line 294 Section 4.1 should read \"real-**world** data.\"\n3. The error bars in the figures are hardly visible. I suggest either using error bands, or also displaying the whiskers (lines).\n4. Tables 1 and 2 contain too many significant digits to be easily legible. I suggest reporting figures to 2 or max 3 decimal places.", "questions_for_authors": "1. Could the authors provide a real-world motivating example where FCD is used/needed and there are non-overlapping variables observed by clients?", "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": "Ygj707T7Fx", "reviewer": "Reviewer_YBLn", "summary": "This paper proposes novel algorithm FedCDnv, a federated method for learning causal structure where different clients observe non-identical variable sets. It mainly addresses two challenges: 1) spurious dependencies introduced by non-overlapping variable pairs, which may lead to incorrect causal conclusions, and 2) Varying importance of (non-)causal relationships between different variables within a client, requiring a careful aggregation mechanism. It also develops theories to detect spurious dependencies, defining stable relationships as those that are both \"correct\" and \"good\" across graphs discovered by multiple clients, and bridging the local learning with federated aggregation. The experiments conducted on synthetic, benchmark, and real-world datasets support the claims.", "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": "$\\textbf{Responses for “Questions For Authors” are as follows.}$\n\n$\\textbf{Q1}$. Does the term “an oracle of conditional independence tests” refer to perfectly accurate CI tests?\n\n$\\textbf{R1}$. Yes, “an oracle of conditional independence tests” refers to fully accurate CI tests.\n\n$\\textbf{Q2}$. Could the authors clarify the process of Alg.3?\n\n$\\textbf{R2}$. Algorithm 3 is proposed for implementing the designed two-level priority selection strategy (PSS). The inputs include the updated local causal graph Pag$_u^{c_k}$, sample size $n_k$, and stability matrix R$_s^{c_k}$ of each client, while the output is the global causal graph FedG over the integrated variable set $O$.\n\nDuring aggregation, adjacencies are determined by comparing $val_{ij}$ (for $X_i$ and $X_j$) with 0. The computation of $val_{ij}$ depends on $w_{ij}^{c_k}$ and the sample size weight $w_{c_k}$. Specifically, if the relationship between $X_i$ and $X_j$ is stable, $w_{ij}^{c_k}$ is based solely on $w_{c_k}$. Otherwise, it is derived from the product of $w_{c_k}$ and the $w_{ij}$ value computed by each client (as Eq. (2)). Stability is encoded in the stability matrix, where each entry represents the stability score (2, -2, 1, -1) multiplied by the scaled p-value. Consequently, $w_{ij}$ is computed by dividing each entry by the corresponding stability score.\n\nFor orientation aggregation, only arrows are considered. If any local graph contains an arrow X→Y, it is incorporated into FedG as X $\\circ\\hspace{-0.43em}\\rightarrow$ Y.\n\n$\\textbf{Q3}$. How does the communication cost scale with an increasing number of clients? Have you explored optimizations for reducing overhead?\n\n$\\textbf{R3}$. We have theoretically analyzed the communication cost of the proposed FedCDnv algorithm, which is $O(4md^2)$ and increases linearly with the number of clients. We are aware that the stability matrix R$_s^{c_k}$ contributes to communication costs. In future work, we will explore optimizations to reduce this overhead.\n\n$\\textbf{Q4}$. Can FedCDnv be extended to handle intervention data, given that CDUIOV explicitly models interventions?\n\n$\\textbf{R4}$. Yes, FedCDnv can be extended to handle interventional data. CDUIOV learns causal structures over the integrated set of variables from interventional datasets across multiple domains. It assumes that within each domain, interventions are performed on an identical set of variables. CDUIOV aims to address inconsistencies in causal relationships caused by unknown intervention targets and non-overlapping variable pairs. Therefore, when each client holds multiple interventional datasets, FedCDnv can integrate these local structures into a global causal graph while preserving data privacy.\n\n$\\textbf{Responses for “Weaknesses” are as follows.}$\n\n$\\textbf{Q5-1}$. Alg.3 seems a bit misleading. \n\n$\\textbf{R5-1}$. Please see R2.\n\n$\\textbf{Q5-2}$. Limited real-world dataset evaluation, with most results relying on benchmarks.\n\n$\\textbf{R5-2}$. We have conducted experiments using the real-world Sachs dataset, which consists of measurements from 11 phosphorylated proteins and phospholipids in individual cells. The experimental results are presented in Table 1 and Table 6. \nIn addition, this study also has real-world applications. We found the eICU Collaborative Research Database (eICU-CRD)  (link https://physionet.org/content/eicu-crd/2.0/), a real-world dataset available on the PhysioNet website, as a motivating example for applying FedCDnv (See R2 of Reviewer 2 for details). \n\n$\\textbf{Q5-3}$. There is no discussion of worst-case performance, making it unclear how FedCDnv behaves under extreme heterogeneity.\n\n$\\textbf{R5-3}$. In the experimental section, we analyzed the worst-case performance of FedCDnv under increasing heterogeneity. When the observed variables differ across clients, the data distribution of each client also varies. We experimentally evaluated the impact of different values of $\\delta$ on FedCDnv's performance, where a lower $\\delta$ indicates fewer overlapping variables and greater variation in sample distributions. Fig. 12 shows that as $\\delta$  decreases from 95% to 60%, FedCDnv's performance drops by approximately 10%.", "claims_and_evidence": "Yes. The paper provides theoretical and empirical support for the claims where the proof support the detecting spurious dependencies and the experiment includes comparisons with distributed and federated CSL methods, testing on synthetic, benchmark, and real-world datasets.\nHowever, one potential weakness is that the paper does not deeply analyze or discuss the worst-case performance of FedCDnv, i.e., scenarios where FedCDnv underperforms or where its assumptions may not hold.", "methods_and_evaluation_criteria": "Yes. The proposed methods and the evaluation criteria make sense for the problem of FCD with non-identical variable sets, including multiple benchmark in synthetic and real-world datasets for the specific problem.", "theoretical_claims": "Yes, the proofs for the proposed theories appear logically sound where Theorem 3.1 formalizes conditions under which non-overlapping variable for non-causal relationships, Lemmas 3.2-3.3 formalize criteria for detecting spurious dependencies. Lemma 3.4 provides a heuristic rule for identifying unobserved confounders, though it might be better labeled as a \"proposition\" to reflect its heuristic nature.", "experimental_designs_or_analyses": "Yes, the experimental setup is sound as it include distributed and federated CSL methods and several benchmarks. An improvement would be testing on more real-world datasets to enhance generalizability.", "relation_to_broader_scientific_literature": "This work proposes a federated method for learning causal structure which extend the prior work by allowing different clients observe non-identical variable sets, considering the presence of latent variables, detecting spurious dependencies, and computing the varying importance of (non-)causal relationships between different variables within a client. While prior FCD methods, such as FedPC, FedCSL, NOtears-ADMM, assume identical variable sets. Distributed methods, such as CDUIOV and CD-MiNi, ignore the might incorrect causal conclusions caused by non-overlapping variable pairs. Both makes FedCDnv a significant contribution to handling real-world non-identical variables.", "essential_references_not_discussed": "The paper could cite: (1) Alternative methods for handling latent confounders in federated learning, such as \"Causal inference with latent variables: Recent advances and future prospectives.\" (2) Federated algorithms in other domains, such as \"scFed: federated learning for cell type classification with scRNA-seq\".", "other_strengths_and_weaknesses": "Strengths: \n- The problem studied in this paper is both novel and practically significant. The challenges arising from non-identical variable sets across clients in federated settings are nontrivial and require careful consideration. \n- The motivation for the study is clear and compelling, particularly regarding the spurious dependencies caused by non-overlapping variable pairs, which indeed require serious attention. (3) The underlying assumptions and theoretical proofs are well-established, and the experimental evaluation provided is thorough, covering two critical aspects comprehensively.\n\nWeaknesses: \n- Alg.3 seems a bit misleading. In line 9, $w_{ij}$ is calculated using Eq. (2), which requires scaled p-values. However, it is unclear where these scaled p-values are derived from, and no further detailed explanation is provided. \n- Limited real-world dataset evaluation, with most results relying on benchmarks.\n- There is no discussion of worst-case performance, making it unclear how FedCDnv behaves under extreme heterogeneity.", "other_comments_or_suggestions": "- Lines 273-274 in Page 5 \"… FedG as definite ones, obtaining FeddG\". The notation could be clarified for better readability.\n- The notation of $\\textbf{Z}$ and $Z$ in Alg. 2 is somewhat confusing, as $Z$ generally refers to a variable within the set $\\textbf{Z}$. Additionally, the presentation in line 4 of Alg. 2 could be improved for greater clarity. Line 198R: $\\frac{n_k}{n}$ should be corrected as $\\frac{n_{k’}}{n}$.\n- The capitalization format of section titles is inconsistent. For example, Section 3.4 should be titled \"Privacy and Costs Analysis\" for consistency. It is recommended that the authors thoroughly review the manuscript to ensure uniform formatting.", "questions_for_authors": "1. Does the term “an oracle of conditional independence tests” refer to perfectly accurate CI tests? \n2. Could the authors clarify the process of Alg.3?\n3. How does the communication cost scale with an increasing number of clients? Have you explored optimizations for reducing overhead?\n4. Can FedCDnv be extended to handle intervention data, given that CDUIOV explicitly models interventions?", "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": "PNM7D8skdT", "reviewer": "Reviewer_mhPQ", "summary": "The paper introduces FedCDnv, a novel algorithm for federated causal discovery where clients observe non-identical but overlapping variable sets. A key challenge in this scenario is the spurious dependencies introduced by non-overlapping variables. To address this, the paper proposes a two-level priority selection strategy that aggregates local graphs from each client to form a global causal graph. The paper demonstrates the effectiveness of FedCDnv through extensive experiments on synthetic, benchmark, and real-world datasets, showing improvements over existing methods.", "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": "$\\textbf{Responses for “Questions For Authors” are as follows.}$\n\n$\\textbf{R1}$. Our method handles distributed data with varying samples distributions, where the observed variable sets are non-identical. We also experimentally evaluate the impact of $\\delta$ on FedCDnv's performance. A lower $\\delta$ indicates fewer overlapping variables and greater variability in the sample distribution. Fig.12 shows that as $\\delta$ decreases from 95% to 60%, FedCDnv's performance drops by approximately 10%.\n\n$\\textbf{R2}$. The \"union\" operation is applied only in the first aggregation phase, where adjacencies and arrowheads of all local graphs are transferred to a global graph. This global graph is then used to update the local graph in each client, without altering their skeletons and can even learn new orientations by using Zhang’s [2008 AI] rules (as illustrated in Fig. 3). The experiments in Figures 4-6 show a significant improvement in F1-orie.\n\n$\\textbf{R3-1 for Theorem 3.1}$. Theorem 3.1 assumes that <X,Y> is a non-overlapping variable pair. Under case (2.c), if A o→ X ←o Y o→ B holds, then any dataset observing X will not observe Y, indicating that $A_X^{G}$ cannot be {A,Y}. \n\n$\\textbf{R3-2 for Lemma 3.2 (First)}$. Sorry for this misunderstanding. $Z_n$ is the set of variables that appear in non-overlapping variable pairs. We will clarify it in the revised version.\n\n$\\textbf{R3-2 for Lemma 3.2 (Second)}$. \n\nFor case (1): Since the ground truth is unknown, we must consider both case(1) and case(2). When the condition in Lemma 3.2 is satisfied, either case (1) or (2) occurs, but we do not know which one occurs, which prevents us from guaranteeing the correctness of the learned relationships. Therefore, Lemma 3.2 claims it is non-definite.\n\nFor case (2): Due to the unknown ground truth, as long as there exists any case where the learned relationship is non-definite, then the learned relationship is considered non-definite. Therefore, the situation that \"they are learned as non-causal\" no longer needs to be considered.\n\n$\\textbf{R3-3 for Lemma 3.3}$. The proof does not assume that X and Y are adjacent in the local graph. It only assumes that X and Y are not adjacent in the ground truth. Here, the learned non-causal relationship between X and Y does not satisfy the condition \"{A,B} $\\subseteq \\mathcal{A}_X^G \\cup \\mathcal{A}_Y^G$\" in Lemma 3.3, meaning the premise does not hold, and thus it cannot be discussed in Lemma 3.3.\n\n$\\textbf{R4}$. We use FDR to evaluate the reliability of FeddG by quantifying the proportion of false discoveries, where FeddG is the graph extracting only definite relationships from FedG. In contrast, recall is defined as True Positives / (True Positives + False Negatives). Since the denominator (True Positives + False Negatives) remains unchanged before and after extraction, the recall of FeddG (Rec-dC, Rec-dnC) is necessarily lower than (or equal with) that of FedG (Rec-C, Rec-nC). This is expected because FeddG is a subset of FedG, excluding non-definite relationships. The experimental results confirm this trend, as shown in the table below (Due to limited space, only a few are shown here).\n| nV  | Rec-C                | Rec-dC               | Rec-nC               | Rec-dnC              |\n|-----|----------------------|----------------------|----------------------|----------------------|\n| 20  | 0.60869 $\\pm$ 0.20393 | 0.41304 $\\pm$ 0.28344 | 0.99491 $\\pm$ 0.01685 | 0.97344 $\\pm$ 0.03836 |\n| 40  | 0.35476 $\\pm$ 0.03688 | 0.22619 $\\pm$ 0.02916 | 0.99567 $\\pm$ 0.00638 | 0.98617 $\\pm$ 0.00976 |\n| 60  | 0.54615 $\\pm$ 0.10288 | 0.40769 $\\pm$ 0.10383 | 0.99825 $\\pm$ 0.00261 | 0.99200 $\\pm$ 0.00784 |\n| 80  | 0.37650 $\\pm$ 0.05856 | 0.23132 $\\pm$ 0.07087 | 0.99868 $\\pm$ 0.00154 | 0.98734 $\\pm$ 0.00373 |\n| 100 | 0.58360 $\\pm$ 0.04306 | 0.34098 $\\pm$ 0.05933 | 0.99883 $\\pm$ 0.00158 | 0.98761 $\\pm$ 0.00652 |\n\n$\\textbf{R5}$. Thanks for your comment, but there is no [1] that you mentioned. \n\n$\\textbf{R6-1 for Weaknesses (lines 179-181)}$. Thanks for your comment. If the adjacency between $X_i$ and $X_j$ is not a bidirected edge but takes forms such as $X_i$ o—o $X_j$, $X_i$ o→ $X_j$ and $X_i$ ←o $X_j$, we provide a condition to address such cases. Specifically, if for every $G_{k_n} \\subseteq$ {$G_{k_n}$}, there exists $Z \\subseteq O_{k_n}$ such that $X⊥Y∣Z$ holds in $D_{k_n}$, and in every $G_{k_a}\\in$ {$G_{k_a}$}, the variables in $Z$ are never observed simultaneously ($X_i,X_j \\notin Z$), then the conflicting adjacency arises due to non-identical observed variable sets. We will clarify this in the revised version.\n\n$\\textbf{R6-2 for Weaknesses (line 200)}$. If $X_i$ and $X_j$ are adjacent in $G_{k'}$, $p_{ij}^{c_k'}$ represents the average p-value from independence tests between $X_i$ and $X_j$ across all possible conditioning sets in $c_k'$. If $X_i$ and $X_j$ are non-adjacent in $G_{k'}$, $p_{ij}^{c_k'}$ corresponds to the p-value associated with the separating set that renders them independent.", "claims_and_evidence": "Some claims in the paper are not fully supported by strong evidence. For example, the authors claim that FedCDnv can achieve federated causal discovery while \"preserving data privacy\" (line 93-94).\nThis claim seems over-claimed since the paper only mentions that \"FedCDnv exchanges structural information rather than raw data, protecting data privacy to a certain extent\", without any specific designs for the privacy concerns. \n\nAnother claim is that FedCDnv works even \"when the sample distributions differ across clients\" (Assumption 2.2, lines 101-105). However, the paper does not offer theoretical analysis or empirical results to support this point.", "methods_and_evaluation_criteria": "Methods: \nThe idea of aggregating both \"good\" and \"correct\" relationships using the concept of \"stable relationships\" is novel and well-motivated. However, some details need further explanation:\n* In Algorithm 1 (line 7) and lines 250-251, the paper claims to integrate all local PAGs into a global graph by taking the union of nodes and edges. However, there is no analysis of this union operation. Consider the following situations:\n  * If a client learns a wrong edge, it becomes part of the global graph. What effect does this have on the final result?\n  * When different clients report conflicting directions for an edge, how is the conflict resolved? Is it merged into a bi-directional edge, and how does this impact the outcome?\n* In Algorithm 2, lines 9–12, the paper applies an \"orientation rule\" on the client side. The rationale behind this rule is not sufficiently explained, and further analysis is needed to clarify its effectiveness and correctness.\n\nEvaluation Criteria:\nThe paper uses False Discovery Rate (FDR) to evaluate the effectiveness of FedCDnv in identifying the definite causal and non-causal relationships in Sec 4.3. It would be helpful to explain why FDR is chosen over other metrics and to also include results on recall for a more complete evaluation.", "theoretical_claims": "There are several issues with proofs of the theorem and lemmas in the paper:\n\nTheorem 3.1: There is an issue with case (2.c). In the provided example, the set A_X^{G} should be {A, Y} rather than {A, B}. This error questions the correctness of this case.\n\nLemma 3.2: This lemma has several problems. \n* First, the statement is unclear. It introduces Z_n as a set of \"variable pairs\" (like <X,Y>), yet later it states the \"single node\" X \\in Z_n. This inconsistency needs to be addressed.\n* Second, the proof is not well-explained. \n  * In case (1), the learned casual relationship A->B alligns with the gound truth and should be definite. However, the lemma claims it is non-definite. \n  * In case (2), the lemma says that X and Y \"might\" be adjacent, which suggests uncertainty, yet it does not clarify what happens if they are learned as non-causal. \n\nOverall, the proof of Lemma 3.2 is not convincing.\n  \n\nLemma 3.3: The proof seems to assume that X and Y are adjacent in the local graph, but this assumption is not mentioned in the lemma statement, which reduces clarity. \n\nLemma 3.4 appears to be correct.", "experimental_designs_or_analyses": "In Assumption 2.2, the paper claims that FedCDnv works when the sample distributions differ across clients. However, there is no empirical evidence to support this claim.\n\nIn Section 4.3, the authors use the false discovery rate (FDR) as an evaluation metric for definite relationships. It would be helpful to explain why FDR is chosen over other metrics and to also include results on recall for a more complete evaluation.\n\nThe reults about the performance across different numbers of clients are presented in appendix. These results could be moved to the main paper, as the number of clients is an important factor in federated leanring and deserves more emphasis.", "relation_to_broader_scientific_literature": "The paper studies federated causal discovery when clients observe non-identical but overlapping variable sets, which is a novel and well-motivated problem.\n\nThe idea of using \"stable relationship\" to aggregate both \"good\" and \"correct\" relationships from local causal graphs to form a global causal graph is novel and effective.\n\nThe experimental results indicate that FedCDnv outperforms state-of-the-art methods across varying numbers of nodes and clients, demonstrating the effectiveness of the proposed method.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "There are certain points that need to be clarified in the paper. \n* In Section 3.2.1 (lines 179-181), the paper states that: \"The first level is to determine whether the adjacency between X_i and X_j is caused by relative latent variables, which is detected by Lemma 3.4\". However, Lemma 3.4 only accounts for bidirected edges. What happens if the adjacency between X_i and X_j is not a bidirected edge?\n* In Section 3.2.1 (line 200), the definition of the p-value p_{ij}^{c^k'} is unclear.  Does it represent the average of all p-values from the independence tests between X_i and X_j across all possible conditioning sets in c_k'?\n* In Section 3.2.2 (lines 256-257), a brief description of \"the rules described by Zhang\" would be helpful. Specifically, it should clarify that these rules is about the orientation of edges in the causal graph.\n* In Section 4.1 (line 288), the term \"graph size\" is vague. It should be explicitly defined as the number of nodes in the graph.\n* The pseudocode for Algorithm 3 needs better formatting. The initial value of w_{ij}^{c^k'} is missing, and the first level of PSS is not included in the algorithm.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1. How does the method perform when the probability distribution of samples varies across clients?\n2. What is the rationale behind using the \"union\" operation to aggregate the local PAGs into a global graph?\n3. The issues mentioned in the Theoretical Claims part need to be addressed. (see the Theoretical Claims part for details) \n4. What recall results were obtained when evaluating the performance of the identified definite relationships?\n5. What's the performance of FedCDnv compared to the method proposed in [1] by Wang et al. (2023)? (see the Essential References Not Discussed part for details)\n6. Could you clarify the first two points mentioned in the Other Strengths and Weaknesses section?\n\nA clear and detailed responses to question 1-5 would help.", "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 paper addresses the problem of federated causal discovery with non-identical but overlapping variable sets, extending the scope of existing federated methods that typically assume identical variables across clients. All reviewers found the problem setting timely, and the methodological contributions reasonable. One reviewer raised concerns about the clarity and rigor of certain theoretical arguments but acknowledged the conceptual soundness of the approach. The authors provided clarifications in the rebuttal, which led to improved scores from other reviewers. Overall, the paper makes a meaningful contribution to federated causal discovery under more realistic settings, and I recommend acceptance.
Decision: Accept (poster)