Sample Quality Heterogeneity-aware Federated Causal Discovery through Adaptive Variable Space Selection

1 Introduction

Causal discovery (CD) aims to identify causal relationships among variables using observational data [He et al., 2021; Guo et al., 2022], with applications across diverse fields, including medicine [Anderson et al., 2023], trustworthy artificial intelligence [Huo et al., 2023a; Yu et al., 2011; Huo et al., 2023b] and computer science [Pearl, 2018; Zhang et al., 2023b]. For example, studying the causal relationships between different lifestyles (e.g., diet and exercise) and chronic diseases (e.g., cardiovascular diseases and diabetes) can help guide public health policies and recommendations. CD can be divided into two main categories: 1) combinatorial optimization-based methods, and 2) continuous optimization-based methods [Yu et al., 2020; Vowels et al., 2022]. The former heuristically evaluates the goodness-of-fit between structural combinations of variables and the dataset used to learn optimal causal relationships. The latter employs gradient descent to optimize a weight adjacency matrix to fit causal relationships among variables.

∗Corresponding Author

Existing CD methods are always performed at a centralized site where all data are stored [Sheng et al., 2024; Xiang et al., 2023]. In practice, data are often distributed across multiple parties (e.g., patient data across hospitals [McMahan et al., 2017; Liu et al., 2024a; Zhang et al., 2023a]), and cannot be moved to a central location due to data privacy concerns [Yang et al., 2019; Liu et al., 2022; Zhong et al., 2023; Miao et al., 2024; Liu et al., 2024b]. As a result, each data owner holds a relatively small amount of samples, which severely limits the performance of existing CD methods. To address this issue, federated causal discovery (FCD) [Ng and Zhang, 2022] has emerged to uncover the underlying causal relationships among variables from decentralized data in a privacy-preserving manner. Existing FCD methods train models on the entire variable space of each client's local data. The entire model parameters from each client are involved in federated learning (FL) model aggregation at the FL server. However, for each FL client, the sample quality varies across different variable spaces. The sample quality heterogeneity issue results in data from different clients being suitable for learning different causal relationships among variables.

For example (Figure 1), when identifying the causal relationship between X1 and X2, only samples from clients 1 and 3 in the variable space of X1 and X2 exhibit "good" quality (i.e., can correctly determine the causal relationship between X1 and X2). In contrast, samples from other clients in the variable space of X1 and X2 are of "bad" quality (i.e., cannot correctly learn the causal relationship between X1 and X2). Similarly, when determining the causal relationship between variables X3 and X4, only samples from clients 2 and 5 in the variable space of X3 and X4 are of "good" quality, while samples from other clients in the variable space of X3 and X4 are of "bad" quality.

Existing FCD methods cannot deal with the sample quality heterogeneity issue since they train models on the entire variable space of each local data. To bridge this gap, we propose a first-of-its-kind Federated Adaptive Causal Discovery (FedACD) method. At the core of FedACD is a novel strategy to adaptively select the causal relationships learned under the "good" variable space (i.e., the space with high-quality samples) at each FL client, and only send them to the server. The causal relationships learned under the "bad" variable space (i.e., the space with low-quality samples) are masked. This way, each FL client only communicates the optimal learning results with the FL server during each round of training, thereby achieving accurate FCD. Under reasonable assumptions, theoretical analysis proves the superiority of FedACD over existing methods. Extensive experiments on benchmark Bayesian network data, synthetic Non-IID data, and realworld data demonstrate significant advantages of FedACD against five state-of-the-art methods, improving the F1 score by 19.65% and reducing the Structural Hamming Distance by 23.57% compared to the best baseline on average.

2 RelatedWork

The Fusion Problem inCausal Discovery. Although FCD and the fusion problem in causal discovery [Mooij et al., 2020; Perry et al., 2022; Huang et al., 2020] share similarities in integrating causal relationships frommultiple datasets, they have distinct characteristics and challenges. FCD focuses on learning causal relationships from decentralized data while preserving privacy and reducing communication costs, which are not primary concerns in the fusion problem. Additionally, in the fusion problem, the causal relationships between variables in the ground truth causal graph are typically required to satisfy specific assumptions, and not any arbitrary directed acyclic graph structure is applicable.

Federated Causal Discovery. Notable FCD methods include NOTEARS-ADMM [Ng and Zhang, 2022], FEDCD [Abyaneh et al., 2022], DARLS [Ye et al., 2022], PERI [Mian et al., 2023], FedDAG [Gao et al., 2023], FedPC [Huang et al., 2023a] and FedCSL [Guo et al., 2024]. Specifically, NOTEARS-ADMM directly applies the distributed optimization algorithm ADMM [Boyd et al., 2011] to optimize the NOTEARS method [Zheng et al., 2018]. FEDCD is designed for a federated learning scenario that contains both observational and interventional data. As for DARLS and PERI, the former utilizes the distributed annealing strategy [Arshad and Silaghi, 2004] to search for the optimal causal graph, while the latter aggregates the results of the local greedy equivalent search [Chickering, 2002] and chooses the worst-case regret for each iteration. FedDAG adopts a two-level structure for each local model, where the first level learns causal relationships by communicating with the server, and the second level approximates variable relationships at each client for handling data heterogeneity. FedPC proposed a layer-wise aggregation strategy to adapt PC [Spirtes et al., 2000] into FL settings. To address the scalability and accuracy limitations of existing methods, FedCSL designs a federated local-to-global learning strategy and a highly privacypreserving weighted aggregation scheme, respectively.

However, these existing studies are not designed to deal with the FL client sample quality heterogeneity issue, which limits their applicability in practice. To the best of our knowledge, FedACD is the first FCD method to bridge this gap.

3 Preliminaries

In this work, we consider a horizontal FL setting, where different clients have large overlaps in the variable space but little overlap in the sample space. Let C = {ck}k∈{1,2,...,m} = {c1, c2, ..., cm} be a set of m clients, X = {Xi}i∈{1,2,...,d} = {X1,X2, ...,Xd} be a set of d variables at each client, and Dck ∈ Rnck ×d represent the local dataset owned by client ck. Here, nck is the number of samples in Dck . Causal relationships over X are often represented by a causal directed acyclic graph (DAG) [Guo et al., 2023]. In a causal DAG, if there is a direct edge Xi1 → Xi2 (i1, i2 ∈ {1, 2, ..., d}), Xi1 is a direct cause of Xi2 , and Xi2 is a direct effect of Xi1 .

FCD aims to identify a causal DAG G from all local datasets {Dck }k∈{1,2,...,m} in a privacy-preserving manner. The causal relationship between variables at different clients satisfies the following Assumption 1.

Assumption 1 (Invariant Causal DAG [Gao et al., 2023]). All local datasets are uniformly sampled from the same causal DAG G, and the probability distribution of samples for the same variable space can differ across different clients.

In this paper, if there is Xi1 → Xi2 or Xi2 → Xi1 , we say that Xi1 and Xi2 are causal neighbors to each other. We use CN ck i to represent the causal neighbor of Xi learned at client ck. Under the faithfulness and causal sufficiency assumptions [Pearl, 1988], if Xi1 ∈ CN ck i2 or Xi2 ∈ CN ck i1 hold at client ck, Xi1 ̸⊥⊥ Xi2 |Z always holds, where Z ⊆ X \ {Xi1 ,Xi2 }. We use ̸⊥⊥ (or ⊥⊥) to represent the dependence (or independence) relation. We employ the G2 test [Spirtes et al., 2000] for discrete data and Fisher's Z test [Pena, 2008] for continuous data to conduct conditional independence (CI) tests for identifying causal relationships between variables. Assume that ρ is the p-value returned by CI tests, and α is a given significance level. Under the null hypothesis of "H0 : Xi1 ⊥⊥ Xi2 |Z", for a CI test of Xi1 and Xi2 given Z, Xi1 ⊥⊥ Xi2 |Z holds if and only if ρ > α.

When conducting CI tests for two variables under all subsets consisting of their respective causal neighbors, if there exists a CI test that makes ρ ∈ (α, 1] hold, then the CI test with a larger p-value is more reliable [Ramsey, 2016]. Thus, we make the following assumption.

Assumption 2. Given a local dataset Dck , if for ∀Z1 (⊆ CN ck i1 or ⊆ CN ck i2 ), there exists CI tests that makes Xi1 ⊥ ⊥ Xi2 |Z hold, with the maximum p-value across all CI tests denoted as ˆ p(Xi1 ,Xi2 ), and similarly if for ∀Z2 (⊆ CN ck i3 or ⊆ CN ck i4 ), there exists CI tests that makes Xi3 ⊥⊥ Xi4 |Z hold, with the maximum p-value denoted as ˆ p(Xi3 ,Xi4 ), then the test result between Xi1 and Xi2 is more reliable than that between Xi3 and Xi4 if ˆ p(Xi1 , Xi2 ) > ˆ p(Xi3 , Xi4 ).

4 The Proposed FedACDMethod

As shown in Figure 2, the proposed FedACD method consists of two phases: 1) federated causal skeleton learning (FCSL, details in Section 4.1) and 2) federated skeleton orientation (FSO, details in Section 4.2). Due to space limit, the detailed pseudo-code of FedACD is provided in Appendix B, and its time complexity is analyzed in Appendix C.

4.1 Federated Causal Skeleton Learning (FCSL)

As shown in Figure 2, Phase 1 is an iterative process. Specifically, FedACD first constructs a complete undirected graph over X = {Xi}i∈{1,2,...,d} and sends it to each client. Then, at each client, it utilizes the conditional independence (CI) tests with the size of the conditioning set as l (start with 0) to remove the false causal edges. Next, it adaptively masks causal edges that may be deleted by mistake and sends all masked causal skeletons to the server for aggregation. Finally, it determines whether the aggregated causal skeleton meets the convergence condition: if so, Phase 1 ends; otherwise, it increments the size of l for the next iteration.

Phase 1-1: Learning Causal Skeleton.

Since the sample space at different clients almost does not overlap in the horizontal FL scenario, the causal skeletons learned at each client may differ greatly, resulting in poor aggregation results. Therefore, we aggregate and update the causal skeletons learned at each client by limiting the size of the condition set for CI tests.

At client ck (k ∈ {1, 2, ...,m}), for two variables Xi1 and Xi2 (Xi1 ,Xi2 ∈ X) that are causal neighbors, FedACD utilizes CI tests to determine whether they are conditionally independent given the conditioning set Z (Z ⊆ CN ck i1 or Z ⊆ CN ck i2 ) with |Z| = l. If Xi1 ⊥⊥ Xi2 |Z holds, the causal edge between Xi1 and Xi2 is removed. By performing the above operations at each client, we obtain m causal skeletons and their corresponding adjacency matrices {Ack l }k∈{1,2,...,m}. Ack l (i1, i2) = 1 indicates that on the local dataset Dck , Xi1 and Xi2 are conditionally dependent given any Z (Z ⊆ CN ck i1 or Z ⊆ CN ck i2 ) with |Z| = l. On the contrary, Ack l (i1, i2) = 0 means that there exists a Z (Z ⊆ CN ck i1 or Z ⊆ CN ck i2 ) with |Z| = l that makes Xi1 and Xi2 conditionally independent. Since the causal skeleton is an undirected graph, Ack l (i1, i2) = Ack l (i2, i1) holds for ∀i1, i2, k, l.

Phase 1-2: Adaptively Masking the Causal Skeleton.

As illustrated in Figure 1, the sample quality heterogeneity issue implies that the local datasets {Dck }k∈{1,2,...,m} from different clients might be suitable for learning different causal relationships among variables. Therefore, we design an adaptive strategy to mask the causal edges that may be erroneously removed from {Ack l }k∈{1,2,...,m}, which has been learned in Phase 1-1 with "-1", rendering them ineffective in subsequent aggregation phase. The detailed process is as follows.

Let r ∈ [0, 1] denote the masking rate for each adjacency matrix Ack l , tck represent the number of causal edges removed on the adjacency matrix Ack l in Phase 1-1, and E ck j (j ∈ {1, 2, ..., tck }) denote the j-th causal edge removed on Ack l . Further, E ck j (a) and E ck j (b) represent the two variables linked by this edge. In Phase 1-1, there might be multiple conditioning sets that can make two variables E ck j (a) and E ck j (b) conditionally independent. However, the CI test with a larger p-value is considered more reliable [Ramsey, 2016]. Therefore, we only record the test results with the maximum p-value among all CI tests that indicate the conditional independence of two variables, and denote the maximum pvalue as ˆ p(E ck j (a), E ck j (b)). According to Assumption 2, the smaller ˆ p(E ck j (a), E ck j (b)), the more likely outcome is that E ck j is a mistakenly deleted causal edge. Conversely, a larger value of ˆ p(E ck j (a), E ck j (b)) implies that there is no causal connection between E ck j (a) and E ck j (b) in the ground truth. Thus, we need to mask the causal edges that have been removed from Ack l learned in Phase 1-1 with low ˆ p(·, ·). Let the indices of the edges that need to be masked in Ack l be stored in the vector Φck , we have:

where ˚ Bottom⌈r∗tck⌉ is used to obtain the indices corresponding to the bottom ⌈r∗tck ⌉ elements in a vector sorted in descending order. We use Ack l to denote the adjacencymatrix after masking Ack l :

Phase 1-3: Aggregating All Masked Causal Skeletons.

After obtaining all the masked matrices {Ack l }k∈{1,2,...,m} in Phase 1-2, the FL server aggregates and updates these matrices. Since a value of "-1" in a cell of the masked matrices indicates that the corresponding learning result is unreliable, such masked causal relationships are invalid during aggregation. In other words, each element in Ack l only participates in voting if its value is not "-1". Let the aggregated adjacency matrix be Al, we have:

where x represents the number of adjacency matrices in {Ack l }k∈{1,2,...,m} where the causal relationship betweenXi1 and Xi2 is masked, i.e.,

and y denotes the number of adjacency matrices in {Ack l }k∈{1,2,...,m} where there exists a causal edge between Xi1 and Xi2 , i.e.,

Phase 1-4: Iterating Phases 1-1 to 1-3 until Convergence.

For the newly aggregated causal skeleton Al, the server first determines whether it has converged (i.e., whether any erroneous causal edges have not been removed). Specifically, the server checks whether there are variables in Al whose number of causal neighbors is greater than l. If such variables exist, it means that new CI tests, where the size of the conditioning set is greater than l, can be performed to remove possible erroneous causal edges further; otherwise, it means that Al has converged and no more CI test is needed. The convergence condition is formalized as:

If Eq. (6) holds, l is incremented by 1, and the server sends Al as the new initial causal skeleton to each client to repeat Phases 1-1 to 1-3; otherwise, the optimal causal skeleton is returned.

Analytical Evaluation of FCSL

Here, under reasonable assumptions, we theoretically show the superiority of FedACD over existing methods in Phase 1. Let ´ A represent the adjacency matrix corresponding to the causal skeleton in the ground truth. According to the adaptive masking strategy in Eqs. (1) and (2), at client ck, the probability of not being masked in {E ck j }j∈{1,2,...,tck} is (1 - r), i.e., P(Ack l (E ck j (a), E ck j (b)) ̸= -1). Thus, when P( ´ A(E ck j (a), E ck j (b)) = 0) ≥ (1-r) 2 holds, in Phase 1-2, the proportion of correctly removed E ck j in all unmasked E ck j is 50% or more in Ack l . Therefore, we have Theorem 1.

For existing voting-based FCD methods (e.g., FedPC), to ensure that each client has correctly removed causal edges in more than half of all removed causal edges in each round of iteration, P( ´ A(E ck j (a), E ck j (b)) = 0) ≥ 12 must be satisfied, whereas our method only needs to satisfy P( ´ A(E ck j (a), E ck j (b)) = 0) ≥ (1-r) 2 . According to the aggregation rule in Eq. (3), we have the following Theorem 2.

Theorem 2. Given ∀Xi1 ,Xi2 ∈ X, in {Ack l }k∈{1,2,...,m} containing m matrices, if P(Ack l (i1, i2) = 0| ´ A(i1, i2) = 0) > 1-P(Ack l (i1,i2)=-1| ´ A(i1,i2)=0) 2 holds, the false causal edge between Xi1 and Xi2 can be correctly removed in Phase 1-3; if P(Ack l (i1, i2) = 1| ´ A(i1, i2) = 1) ≥ 1-P (Ack l (i1,i2)=-1| ´ A(i1,i2)=1) 2 holds, the true causal edge between Xi1 and Xi2 does not be discarded in Phase 1-3.

Existing voting-based FCD methods (e.g., FedPC) require P (Ack l (i1, i2) = 1| ´ A(i1, i2) = 1) ≥ 1 2 and P(Ack l (i1, i2) = 0| ´ A(i1, i2) = 0) > 12 to ensure that the true causal edge between Xi1 and Xi2 does not be discarded while the false causal edge between Xi1 and Xi2 can be correctly removed in each round of aggregation. In contrast, based on Theorem 2, FedACD only requires P (Ack l (i1, i2) = 0| ´ A(i1, i2) = 0) > 1-P(Ack l (i1,i2)=-1| ´ A(i1,i2)=0) 2 and P(Ack l (i1, i2) = 1| ´ A(i1, i2) = 1) ≥ 1-P(Ack l (i1,i2)=-1| ´ A(i1,i2)=1) 2 to achieve optimal performance, which greatly improves the accuracy of the aggregated causal skeleton in practice. The proofs of Theorems 1 and 2 are given in Appendix A.

4.2 Federated Skeleton Orientation (FSO)

As shown in Figure 2, after obtaining the optimal causal skeleton, Phase 2 orients the undirected edges at each client for learning m causal DAGs, and then aggregates all causal DAGs at the server side to produce the final causal DAG.

Phase 2-1: Orienting Undirected Edges at Each Client.

Let A∗ represent the adjacency matrix corresponding to the optimal causal skeleton obtained in Phase 1. The server first sends A∗ to each client, then the score-and-search strategy is adopted to greedily orient the undirected edges in A∗ for obtaining a causal DAG with the highest score at each client. Let G ck denote the causal DAG learned at client ck, and "G ck (i1, i2) = 1" denotes that there is an edge from Xi1 to Xi2 in G ck . For discrete datasets, we utilize a Bayesian score, BDeu [Scutari, 2016], and a search procedure, hillclimbing [G´amez et al., 2011], to implement the above scoreand-search process. Here, the BDeu score for the causal DAG

G ck learned on client dataset Dck is defined as:

where P(Dck |β, G ck ) denotes the probability of the local dataset Dck given the equivalent sample size parameter β and the causal DAG G ck . It measures how well the causal DAG predicts the observed discrete data. P(β, G ck ) represents the prior probability of β. It serves as a regularization term and influences the strength of prior beliefs about the density of the causal DAG.

For continuous datasets, we use an information-theoretic score, BIC [Watanabe, 2013], to calculate the fitting score between G ck and Dck . The BIC score for G ck learned on Dck is defined as:

where ˆL(G ck ,Dck ) denotes the ability of the causal DAG to explain the observed continuous dataset Dck . A higher likelihood indicates a better fit. µ represents the average density of G ck . The penalty term "µ · ln(nck )" discourages overly dense causal DAGs, favoring sparser causal DAGs to reduce the risk of overfitting.

Phase 2-2: Aggregating All Causal DAGs at the Server.

FedACD sends all causal DAGs {G c1 , G c2 , · · · , G cm } learned in Phase 2-1 back to the server to compute the aggregated causal DAG G ∗ as:

where ⊕ represents the element-wise addition of matrices. Finally, we compare the elements at corresponding positions on the diagonal of matrix G ∗ for obtaining the final causal DAG. Specifically, if G ∗ (i1, i2) > G ∗ (i2, i1), then there exists a directed edge from Xi1 to Xi2 . If G ∗ (i1, i2) ≤ G ∗ (i2, i1) and G ∗ (i2, i1) ̸= 0, there exists a directed edge fromXi2 to Xi1 ; Otherwise, there is no edge between Xi1 and Xi2 . To summarize, we have:

4.3 Privacy and Cost Analysis

Privacy Preservation Capability of FedACD

In Phases 1-1 and 1-2, all CI tests requiring raw data are performed by the respective FL clients, and the adaptive masking strategy is locally executed by each client based on the pvalues obtained from the CI tests. In Phase 2-1, the score-andsearch process, which requires raw data, is also performed locally by each client. As a result, FedACD only exchanges structural information represented by the adjacency matrices throughout the entire FL process, without exposing clients' raw data or any statistical information from the CI tests.

To further mitigate the risk of inferring local data from the learned structural information, FedACD can be combined with the following techniques. (1) Additive homomorphic encryption (Paillier's scheme [Paillier, 1999]) can be applied in the aggregation process of Phase 1-3 and Phase 2-2 to prevent leakage of the sample quality heterogeneity reflected in the masking information on each client. (2) To prevent the leakage of semantic information about each variable during communication between clients and the server, we incorporate an easily implementable privacy protection strategy [Huang et al., 2023a] into FedACD. Specifically, the remote server instructs each client to assign unique identifiers (e.g., "1", "2", "3", etc.) to the variable semantics following their alphabetical order. If multiple variables share the same initial letter, they are further sorted by subsequent letters. Each client then sends only these assigned identifiers to the remote server for aggregation, protecting the variable semantics.

Communication Cost of FedACD

Communication is a critical bottleneck in FL. Therefore, developing communication-efficient methods for FL model training is essential. Here, we argue that FedACD introduces relatively low communication overhead. In Phase 1, during each iteration, all clients need to send a masked adjacency matrix Ack l of size d ∗ d to the server. The server, in turn, sends back the aggregated adjacency matrix Al to each client, requiring O(m|Ack l |+m|Al|) = O(md2 +md2 ) = O(md2 ) information cost. Assume that Phase 1 requires L (usually L ≤ 3) iterations in total. Therefore, Phase 1 incurs a total communication cost of O(md2L). In Phase 2, firstly, the adjacency matrix A∗ corresponding to the final causal skeleton needs to be sent from the server to each client. Then, the adjacency matrices G ck learned by each client are sent back to the server to obtain the final causal DAG through an aggregation strategy, resulting in a total information cost of O(m|A∗ | + m|G ck |) = O(md2 + md2 ) = O(md2 ).

5 Experimental Evaluation

5.1 Experiment Setting

Datasets. We utilize the following three types of datasets.

• Benchmark BN datasets. We use three benchmark BN datasets: Child with 20 variables, Insurance with 27 variables and Alarm with 37 variables, and each dataset contains 5,000 samples [Tsamardinos et al., 2006].
• Synthetic Non-IID datasets. We employ the publicly available code from [Gao et al., 2023] to generate two batches of Non-IID datasets, distributed among a total of 5,000 samples across {3, 5, 10, 15, 20} clients, with each client's data originating from a different distribution. The ground truth for each batch of data across different clients is consistent, comprising 20 variables with 40 edges and 30 variables with 60 edges, respectively.
• Real-world datasets. We also compare the proposed method with the baselines on two non-overlapping networks of sizes {8, 20} from the lung cancer geneexpression dataset, REGED [Statnikov et al., 2015]. For each network, we generate {3, 5, 10, 15, 20} distinct environments (i.e., FL clients) to create a horizontal FL scenario, with a total of 1,000 samples.

Evaluation metrics. We adopt the Structural Hamming Distance (SHD) (the lower the better) and F1 score (the higher the better) [Huang et al., 2023b] to evaluate the learned causal DAGs in FL settings. We have also adopted other metrics (details can be found in Appendix D).

Comparison methods. FedACD is compared with five state-of-the-art FCD methods: 1) NOTEARS-ADMM [Ng and Zhang, 2022], 2) NOTEARS-MLP-ADMM [Ng and Zhang, 2022], 3) GS-FedDAG [Gao et al., 2023], 4) ASFedDAG [Gao et al., 2023] and 5) FedPC [Huang et al., 2023a]. We adopt these baselines partly because of their publicly available implementations.

Implementation details of the FedACD algorithm and the baselines are provided in Appendix F.

5.2 Results on Benchmark Bayesian Network Data

In this section, we report the comparison results between FedACD and the baselines on benchmark Bayesian network datasets in terms of SHD and F1 score.

From Figure 3, it can be observed that FedACD achieves the lowest SHD value and the highest F1 score in most scenarios, which validates its superiority. In particular, on the Child and Alarm datasets, when the number of clients reaches 20, FedACD shows significantly superior performance compared to other baselines. This is attributed to the fact that, with a larger number of clients, the allocated sample size per client becomes smaller. In such scenarios, the sample quality varies significantly across different variable spaces at each client, yet existing FCD methods cannot discern which variable spaces exhibit better sample quality. In contrast, FedACD's adaptive masking strategy renders the learning results from samples in low-quality variable spaces at each client ineffective, while preserving the results from samples in high-quality variable spaces.

Compared with the best baseline, FedPC, the F1 score of FedACD is 42% higher on the Child dataset when the number of clients is 20. The F1 scores achieved by FedACD are significantly higher than those achieved by NOTEARSADMM, NOTEARS-MLP-ADMM, GS-FedDAG and ASFedDAG, since it is hard for them to select a suitable threshold to prune false causal edges. As a result, the causal DAGs learned by these four baselines often contain a larger number of false causal edges. In addition, as the number of clients increases, all algorithms experience significant performance degradation, whereas FedACD maintains good stability.

5.3 Results on Synthetic Non-IIDData

In real-world scenarios, clients' local datasets are often NonIID. Therefore, in this section, we also evaluate the performance of all algorithms on synthesized Non-IID data. The results are shown in Figure 4. It can be observed that regardless of the number of clients, FedACD consistently achieves the lowest SHD values and the highest F1 scores. This indicates that the adaptive masking strategy designed in FedACD is well suited to the horizontal FL setting with Non-IID local data. This is because in such scenarios, the significant differences in sample quality across variable spaces at each client are obvious, and FedACD is designed to address this issue.

5.4 Results on Real-World Data

The experimental results on the two real-world datasets, REGED with 8 nodes and 20 nodes [Statnikov et al., 2015], are presented in Figure 5. It can be observed that under REGED with 8 nodes, FedACD achieves the lowest SHD value when the number of clients is 3, 10, 15 and 20. In addition, FedACD achieves the highest F1 score under REGED with 20 nodes, except when the number of clients is 20. Compared to other baselines, our method achieves more stable performance under REGED with 8 nodes with different numbers of clients.

5.5 Ablation Study

In this section, we conduct ablation experiments to validate the effectiveness of the proposed adaptive masking strategy. Specifically, we first develop a variant of FedACD, denoted as "FedACD w/o masking", which maintains a constant masking rate of 0 throughout the entire learning process. We then compare FedACD with "FedACD w/o masking" using three benchmark BN datasets, Child, Insurance and Alarm, across {3,5,10,15,20} clients. The experimental results are presented in Figure 6. It can be observed that FedACD consistently achieves higher F1 scores and lower SHD values than "FedACD w/o masking". This demonstrates the effectiveness of our proposed adaptive masking strategy under a horizontal FL setting, especially when dealing with potentially significant differences in sample quality across different variable spaces at each client.

5.6 Sensitivity Analysis

In this section, we conduct sensitivity analysis on the masking rate r of FedACD using the benchmark datasets. Specifically, we first generate three batches of data using Child, Insurance and Alarm BNs, respectively. For each dataset, there are 5,000 samples in total, allocated evenly across 10 clients. Then, we test FedACD on these datasets by varying the masking rate from 0.1 to 0.9, and record the F1 scores obtained. The experimental results are presented in Figure 7.

It can be observed that the F1 score achieved by FedACD increases with the increase of the masking rate r when r ≤ 0.6. However, when r ≥ 0.7, the F1 score achieved by FedACD drops significantly. The reason behind this phenomenon lies in Theorem 1, which indicates that when P( ´ A(E ck j (a), E ck j (b)) = 0) is fixed, a larger masking rate r implies a higher probability of successfully removing a false causal edge in Phase 1-2. However, when the masking rate becomes excessively large (e.g., r ≥ 0.7), ∃i1, i2 such that "∑mk=1 Ack l (i1, i2) ̸= -m" holds, leading to the ineffectiveness of the aggregation strategy in Eq. (3). Consequently, the performance of FedACD experiences a noticeable decline. In our experiments, the masking rate r for FedACD is set to 0.6 on all types of datasets.

6 Conclusions and FutureWork

In the horizontal FL scenarios, the sample quality under different variable spaces at each FL client can be different. As a result, the local data of each client might only be suitable for learning the causal relationships among certain variables, while the effectiveness of learning causal relationships among other variables may be limited. Existing FCD methods cannot effectively handle this sample quality heterogeneity issue. Our proposed FedACD bridges this crucial gap by adaptively selecting and sending only the optimal causal relationships learned under the "good" variable space (i.e., the space with high-quality samples) at each FL client to the server, while masking the causal relationships learned under the "bad" variable space (i.e., the space with low-quality samples) at each FL client, during FL model training. In this way, each client only communicates the optimal learning results to the server, achieving accurate FCD. Theoretical analysis and extensive experimental results demonstrate the efficacy of FedACD.

In the future, we plan to explore FCD in more challenging scenarios, such as data containing hidden variables and data combining observational and interventional sources.

Acknowledgments

This work was supported in part by the National Science and Technology Major Project (under grant 2020AAA0106100); the National Natural Science Foundation of China (under grant 62376087); the China Scholarship Council (under grant 202306690023); the National Natural Science Foundation of China (under grant 92367202); the National Research Foundation Singapore and DSO National Laboratories under the AI Singapore Programme (No. AISG2-RP-2020-019); and the RIE 2020 Advanced Manufacturing and Engineering (AME) Programmatic Fund (No. A20G8b0102), Singapore.