FedIGL: Federated Invariant Graph Learning for Non-IID Graphs
Authors: Lingren Wang
Wenxuan Tu
Jiaxin Wang
Xiong Wang
Jieren Cheng
Jingxin Liu
Venue: NeurIPS 2025 poster ()
Keywords: Federated Graph Learning
Invariant Learning
Primary area: general_machine_learning

Abstract: Federated Graph Learning (FGL) effectively facilitates cross-domain graph model training by enabling decentralized learning across multiple domains, while ensuring data privacy through local data storage and communication of model updates instead of raw data. Existing approaches usually assume shared generic knowledge (e.g., prototypes, spectral features) via aggregating local structures statistically to alleviate structural heterogeneity. However, imposing overly strict assumptions about the presumed correlation between structural features and the global objective often fails in generalizing to local tasks, leading to suboptimal performance. To tackle this issue, we propose a **Fed**erated **I**nvariant **G**raph **L**earning (**FedIGL**) framework based on invariant learning, which effectively disrupts spurious correlations and further mines the invariant factors across different distributions. Specifically, a server-side global model is trained to capture client-agnostic subgraph patterns shared across clients, whereas client-side models specialize in client-specific subgraph patterns. Subsequently, without compromising privacy, we propose a novel Bi-Gradient Regularization strategy that introduces gradient constraints to guide the model in identifying client-agnostic and client-specific subgraph patterns for better graph representations. Extensive experiments on graph-level clustering and classification tasks demonstrate the superiority of FedIGL against its competitors.

FedIGL: Federated Invariant Graph Learning for Non-IID Graphs
Lingren Wang'
Wenxuan Tu Jiaxin Wang? Jieren Cheng Jingxin Liu
Xiong Wang?
School of Information and Communication Engineering, Hainan University
Schooo of Computer Science and Technology, Hainan University School of Cyberspace Security, Hainan University (twx, 992730) Ohainanu. edu. cn
Abstract
Federated Graph Learning (FGL) effectively facilitates cross-domain graph model training by enabling decentralized learning across multiple domains, while ensur- ing data privacy through local data storage and communication of model updates instead of raw data. Existing approaches usually assume shared generic knowledge (e.g., prototypes, spectral features) via aggregating local structures statistically to al- leviate structural heterogeneity. However, imposing overly strict assumptions about the presumed correlation between structural features and the global objective often fails in generalizing to local tasks, leading to suboptimal performance. To tackle this issue, we propose a Federated Invariant Graph Learning (FedIGL) framework based on invariant learning, which effectively disrupts spurious correlations and further mines the invariant factors across different distributions. Specifically, a server-side global model is trained to capture client-agnostic subgraph patterns shared across clients, whereas client-side models specialize in client-specific sub- graph patterns. Subsequently, without compromising privacy, we propose a novel Bi-Gradient Regularization strategy that introduces gradient constraints to guide the model in identifying client-agnostic and client-specific subgraph patterns for better graph representations. Extensive experiments on graph-level clustering and classification tasks demonstrate the superiority of FedIGL against its competitors.
Introduction
Graph Neural Networks (GNNs) research 50 33
38 36 g is rapidly growing due to the
ability of GNNs learn representations from graph-structured data. In practice, centralizing large amounts of real-world graph data for training is prohibitive due to privacy concerns and regulatory restrictions [26] 47 335 Federated Graph Learning (FGL), growing distributed learning paradigm, offers potential solution to this challenge while preserving data privacy [24, 23), Nonetheless, the non-IID problem remains a major challenge in FGL, graph data from different distributions usually vary significantly [40].
Existing approaches typically rely on the assumption that generic knowledge learned from training on non-IID clients can be effectively reconstructed across clients to enable collaborative training [3 The shared knowledge includes consensus prototypes [22 48 generic spectral knowledge (32
and structure encoder parameters 311. which are introduced as shared knowledge representations to facilitate the execution of graph-level learning tasks. Despite the enormous success, existing methods overly rely on statistical correlation, misleadingly assuming that the robust representations learned from prior knowledge are widely applicable. The correlation between generic knowledge
"Conesponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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and the target is not necessarily task-related, and such spurious correlations embedded in the learned representations often fail generalize real-world scenarios. Furthermore, these approaches typically upload the learned knowledge or prototypes to the server, which may lead to potential data leakage. Therefore, it is critical to promote inter-client negotiation in the framework of federated graph learning without compromising data privacy.
An intuitive solution is to exploit factors that remain con- sistently stable and effective across clients to mitigate the impact of spurious correlations that merely reflect statisti- cal commonality. In other words, the global model should be capable of identifying invariant factors across clients, Empirical observations indicate that graphs with differ- ent distributions often share common subgraph patterns (54). even when their distributions differ significantly, as shown in Fig. [] This observation inspires the following idea: if can identify subgraph patterns that are shared across different distributions, these common patterns can serve as a foundation for inter-client collaboration and im- prove the generalization of the global model. In contrast, distribution-specife patterns should be retained locally each client to prevent them from negatively impacting the global representation. This insight prompts to consider two fundamental questions: (1) How to discover invariant subgraph patterns across different distributions in FGL? (2) How can one extract invariant subgraph patterns a crivac-prreeevvngg manner, considering that FL prohibits data sharing across clients? To the best of our knowledge, both questions remain largely unexplored.
MUTAG
COLLAB
AIDS
ENZYMES
Figure 1: Illustration of shared sub-
graphs in different distributions in TU- Datasets (30), where the red nodes high- light the common structure.
To tackle these challenges, we propose a Federated Invariant Graph Learning (FedIGL). which aims to identify invariant subgraph patterss shared across clients while preserving local privacy. To address the first question, inspired by invariant graph learning (20, (7). which focuses on improving generalization out-of-distribution graphs, design Federated Subgraph Generator (FSG) to extract client-agnostic and client-specific subgraphs. The generated client-agnostic and client- specific subgraphs used to promote negotiation among clients and maintain the heterogeneity inherent to each client, respectively. To address the second question, we introduce novel Bi- Gradient Regularization strategy that imposes consistency and diversity constraints on gradients. is effectively guides the generator to learn disentangled subgraph patterns while ensuring data privacy not compromised. After obtaining the specitc subgraphs, we design local model for each client that is excluded from collaborative training and is trained solely on these subgraphs. FedIGL is encouraged to discover invariant subgraph patterns across data distributions, thereby mitigating client drift caused by graph heterogeneity.
To the best of our knowledge, this is the first work leveraging invariant learning in federated graph learning to enhance generalization under non-lID client settings.
We propose the Bi-Gradien Regularization strategy, which can coordinate the clients to learn disentangled subgraph patterns without compromising data privacy.
We conduct extensive experiments verify both our theoretical results and the superiority of FedIGL, which consistently outperforms existing approaches on graph-level classification and clustering tasks.
Related Work
Federated Graph Learning. FGL enables distributed training GNNs across multiple parties, facilitating collaborative learning on graph-structured data without compromising data privacy [2] Due to significant differences client distribution and graph structure across domains, low inter-graph similarity hinders unified processing 44 Existing methods mitigate structural heterogeneity by leveraging shared, pre-trained representations from cross-domain client models [42.. Examples include prototype-based structures 40 53). spectral
43
46
19
13 5

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feature alignment [32], and shared structural encoder parameters [3] FedSSP [32 shares gener- alized spectral knowledge with a personalized module to adapt client-specific graph structures, while FedGCN [22] leverages multi-source clustering to generate global consensus representations, enhancing its ability to handle complex graph structures. Despite their success, most methods rely heavily on assumed shared knowledge, which limits adaptability to diverse distributions. Since this knowledge is learned via pre-trained shared parameters unrelated to task causality, distribution shifts
can degrade representation quality and harm model performance [55 39 17
Invariant Graph Learning (IGL). Invariant Learning is class of learning methods focused on distribution generalization or robust modeling 2 [ Its main idea is to learn representations or predictive functions that remain stable and effective across different environments data distributions. As previously discussed, although graphs from different distributions are heterogeneous, they share certain common subgraph patterns. Building upon these findings, IGL. has emerged as prominent research direction in recent years (2) 47 [29]. The main idea of IGL. is designing a subgraph generator to partition a graph into two components: the invariant subgraph, which captures structures that are consistent across different distributions, and the environment-speen
subgraph, which represents structures that are present only in particular distributions. In this paper, we extend invariant graph learning to federated learning, where each client with distinct distribution is treated as separate environment.
3 Preliminaries
Federated Learning (FL). Given local dataset (r, Ps for each client k, where Ps denotes a client-specific data distribution, the goal of standard FI. approaches [27 [21] is to learn global model that minimizes the empirical risk across all client distributions, defined as:
arg min E {(felz).))
(I)
where (()) is task-specific loss function and (o(-) is the global model parameterized by 0. In practice, FL methods typically decompose this global objective into a weighted sum of local empirical losses and perform independent optimization on each client. However, independently optimizing local objectives often leads suboptimal convergence 41), particularly when client distributions exhibit sigmificant heterogeneity. This is because, under non-lII conditions, the local updates from different
clients may follow divergent optimization directions 44
25
28
16
Methodology
In this section, introduce proposed method in detail, whose framework is shown in Fig. First, we define our optimization objective. Then, we present the FSG identifying client-invariant subgraph patterns. Finally, we propose the Bi-Gradient Regularization strategy for objective optimization and provide its theoretical analysis. Algorithmic details can be found in Appendix A
4.1 Problem Formulation
In the federated optimization objective in Eq. 1 when client data are not identically distributed, each client's optimization direction tends push the global model along different update trajectories [40]. This leads to gradient conflicts, hindering convergence to a globally optimal solution. To mitigate this, our goal to guide the global model to focus on features that shared across all client distributions, while excluding distribution-specife features from global optimization [27]6 o.. This strategy helps prevent conflicting updates and promotes more stable convergence. We therefore aim to incorporate client- agnostic subgraphs, namely invariant subgraphs, into global model training to better accommodate distributional shifts (20). In contrast, client-specific subgraphs, which capture environment-specine patterns, are processed by dedicated local models that remain discrepancies to each client. In this paper, we decouple the model into two distinct components: global model (-), which participates in federated aggregation, and local model fo(:), which remains client-specific.

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Federated Subgraph Generator
Gv
fr(Grilenn)
fv(Ge
No
V
cient-speclfc
Vi
cliant
Variant encode branch
Downstream Tank
global
Vo
G
albent
V
Diversity Regularization
tm
fr(Og; Pmm)
clint clent-aghoss agnosto
Vil
Invariant emoodo
Locol
Vil Ieo
Vol
Gv
obent
M(
Go
cliont
h spec
client spectic
clant-spacifi
Consistency Regularization
client
plient
(a) Federated Invariant Graph Learning Framework
(b) Bi-Gradient Regularization strategy
Figure 2: Architecture illustration of FedIGL. The left box (a) represents the process by which clients obtain client-agnostic and client-specific subgraphs through the federated subgraph generator (FSG). The yellow and blue boxes denote parameters that are globally shared and not shared, respectively. The right box (b) shows that diversity regularization penalizes overly similar gradients in the variant encoder. Consistency regularization encourages stability and agreement in the optimization trajectory of the invariant encoder across rounds.
Based on this formulation, the optimization objective of FedIGL is defined as follows:
min
Σ (z,y)) (fg(:)))) fa(x; 6))) v)]
(2)
6 (0
where 0% denotes the local model parameters of client k, ((::)) is the loss function, and h(::)) is fusion function, such as concatenation or addition, used to integrate the outputs of the global and local models for downstream tasks. We decompose the overall optimization objective into two stages: global model optimization and local model optimization, which will be elaborated upon in the following sections.
4.2 Discovering Invariant and Variant Subgraphs
For the global model, we first employ FSG to decompose each graph into client-agnostic and client- specific subgraphs. These subgraphs are then encoded separately, and their resulting representations are combined and fed into the downstream task for training.
Similar to prior work 115 [44] implement the FSG using graph neural network (GNN). Given graph G with n nodes and its adjacency matrix A = {0, 1 where As,; represents that there exists an edge between node i and j. and Ai.j 0 otherwise. The FSG first generates mask matrix M E Ruxa for A:
M,J = (CP(CCCTTT(( Z GNN(G),
where MLP(-) is multilayer perceptron and CONCAT(::)) is the concatenation operation, Z; denotes the representation of the i-th node in graph G. We use MLP instead of the inner product to generate the matrix M, since in federated learning. the inner product may leak structural information by enabling edge reconstruction [22]. Then, we can obtain the adjacency A, and Ay corresponding to the client-agnostic subgraph 1₁ and the client-specific subgraph Gy:
A TM C A), Av = A A Ar.
where (-), select the elements in M whose size is the top percent t, and è a hyperparameter. After obtaining G₁ and Gy, we adopt two branches, invariant and variant encode branch, to encode the obtained agnostic and client-specffi subgraphs representation, respectively:
h iav (G7;;mm)))) h var (G(fv(Gv;
(5)

---

where R(-) is used to obtain the graph-level representation, and f(:) is the graph encoder. Then, we use the sum of h and tar as the representation of graph G to train the global model. The global model loss in client k is then defined as:
c Loohaa (SSG Dinu, Owar) = E(h ire
ear y),
(6)
where FFSG is the parameters of the FSG.
For the client model, we define private encoder f-(-) for each client to encode the client-specific subgraph. Then, we take Gy from the subgraph generator as the input of the local model, and we can get the client-specific subgraph representation in client l:
h spec
R (fv))
should be emphasized that h spee and haar though derived from the same client-specific subgraph, encoded using distinct encoders. After obtaining the client-specific subgraph representation, use the sum of hnn and heee as the overall graph representation to train the local model in client k. The local model loss is defined as:
Clocal E(h inv + h spec, y).
(8)
Note that we fix the global model when training the local model. That is, h obtained from the global model does not participate in gradient calculation and only optimizes the local model in minimizing Liocel.
4.3 Bi-Gradient Regularization strategy
After introducing the overall FedIGL framework, we now provide detailed insights into the optimiza- tion strategies for the global and local models. In this section, we focus on how to optimize the global model achieve disentangled and effective feature learning across distributed clients. We begin with the global optimization objective for a specific client l:
min C globall (OFSG, Oinos Over).
In this objective, directly minimizing Cytobal for each client independently would degenerate to conventional federated learning objectives. This approach fails to enforce collaboration among clients for learning dlient-invarian and client-specific representations. Accordingly, we delve deeper into optimizing the global model with respect this objective.
Assume the invariant subgraph generator is ideal, such that it can extract the same semantic subgraph pattern from different distributed graphs. Then, the feature representations of such subgraphs should be consistent across all clients. Structural and feature differences can be reflected by GNN gradients, proved in (48). Consequently. if the encoder fı is applied uniformly across clients, the optimization gradients with respect to iin should also align across any clients k. k':
V Oino Lk global Von. Ck global,
(10)
where Vo... Lk alobal represents the gradient of invariant branch encoder f, when optimizing the global loss Eq. 6) for client k. Similarly, each client-specific subgraph captures features unique to each distribution. Therefore, the feature difference should drive different optimization directions for the variant branch encoder fv with the same parameters. More precisely, the gradient between any clients should satisfy:
Vo... Lk
Viver
E,
(11)
where is predefined margin enforcing gradient diversity across clients for the variant branch. This inequality indicates that the optimization direction of heterogeneous features from different clients on the same encoder should have significant differences.
Building upon these observations, we design a novel Bi-Gradient Regularization strategy. The central idea is to regulate the update directions of the invariant and variant encoders, such that 1) the invariant branch encoder gradients across clients are encouraged to be consistent; 2) the variant branch encoder gradients enforced to be diverse (repulsion beyond margin E).
5

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We achieve this by using the global aggregated gradients from the previous round as reference directions. In each round t, all clients' current gradients are compared with the previous global aggregation gradients, and regularization is applied accordingly. The total loss for the global model is then defined as:
Liotal = Ligbbbll + 1l| Vor.. Lytobal
Gin ll2 + 3 max(0, V 86a. folobol
Boar 112): (12)
where À and / hyperparameters controlling the strength of alignment and diversity regularization, V Vou! and V D:=! the aggregated gradients with the FedAvg [ method. The second term, consis- tency regularization, encourages stability and agreement in the invariant encoder's optimization trajectory across rounds. The third term, diversity regularization, penalizes overly similar gradients the variant encoder, ensuring meaningful divergence between client-specific representations.
4.4 Theoretical Analysis
We provide a theoretical analysis of the FedIGL. framework with invariant and variant branch encoders with bi-gradient regularization. Our objective to demonstrate that (i) the invariant encoder achieves convergence with gradient alignment regularization; (ii) the variant encoder yields distinguishable representations across clients due repulsive regularization.
Let C = f1....,K' }} be the set clients, and D2 denote the local dataset of client k. Let = (Olav, var. (G) denote the parameters of f1. fv and FSG, respectively. Define the task loss of client k: as (0). We denote the gradients as:
S = Von. (O')),
h1 V (0'),
g
9
h-1
Rk
1-1
The total optimization objective round is:
K
min
Cx(0) + Ryns + Rvars
(13)
where Rin and Rvar are the consistency and diversity regularization, respectively. In the federated setting, clients do not have access to the current global mean gradients g', ht at round when performing local updates. Therefore, we compute the regularization terms using the previous round's statisties g²-1, - which are broadcast Nevertheless, the theoretical analysis below evaluates the variance and convergence behavior with respect the true global means g' and ht as is standard in federated optimization literature.
clients at the start of each communication round.
Proposition 4.1 (Convergence of Invariant Encoder). Assume each L (f) is convex and L-smooth, and the learning rate 1, S I/L. Under the FEDAVG scheme with gradient alignment Ry the global invariant parameters 0% satisfy:
{0(
min E || Venn (globbr(("))) (0') S
(14)
T
Proposition 4.2 (Representation Separation for Variant Encoder). Let h1, denote the gradient of the variant encoder for client k. When R. is minimized, then for all K + k', the representations remain sufficiently separated:
li4l 1
(15)
This guarantees per-client distinguishabiliyy the variant branch.
Theorem 4.1 (Global Objective Convergence with Gradient Regularization). Let the total loss be (0) + Rim + Ryar If each Lx(0) is convex and L-smooth, and 17 = ((1 (T), then:
T min E [ (0' 13) < <O
(16)
T
Indicating convergence. Furthermore, the learned representation satisfies: (i) invariant alignment across clients in 0mn and (ii) per-client diversity in Owar

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Based on the above formulation, we provide theoretical justification for the effectiveness of our Bi-gradient regularization strategy. Specifically, Proposition shows that the gradient alignment regularization term Rim reduces the variance of client gradients in the invariant encoder, thus promoting convergence in federated optimization. Proposition 4.2 demonstrates that the repulsive regularization Ryar enforces diversity among the variant gradients, enabling the model to capture client-specific characteristics. Together, these results support the disentanglement of invariant and variant subgraph patterns in federated setting. Theorem 4.1 further establishes the convergence guarantee of our overall optimization procedure under bi-gradient regularization, quantifying the trade-off between alignment and diversity in gradient space. The proof of the above proposition and theorem see Appendix
5 Experiments
In this section, conduct extensive experiments graph-level classification and clustering tasks in various cross-dataset and cross-domain scenarios to validate the superiority of FedIGL. The following research questions need to be validated. (RQ1) Can FedIGL achieve better performance compared to SOTA baselines? (RQ2) Does FedIGL. converge under the constraints bi- ri-gradient optimization? (RQ3) How does each of the strategies we propose contribute to the final performance? (RQ4) How about the hyperparameter sensitivity FedIGL?
5.1 Experiment Setup
Benchmark Datasets. We employed a total of 19 diverse datasets across multiple domains to conduct comprehensive evaluations both classification and clustering tasks. These domains include Small Molecules (e.g.. MUTAG, BZR, COX2, DHFR, PTC_MR, AIDS, BZR_MD and NCII), Bioinformatics (e.g., DD, PROTEINS, OHSU, and Peking 1). Synthetic (SYNTHETIC), Social Networks (e.g., COLLAB. IMDBMULTI, and IMDB- HIDB-BINARY) and Computer Vision (c.g.. Letter-high, Letter-low, and Letter-med). Regarding classification tasks, We follow the settings in 332. which include six distinct experimental designs: (1) cross-dataset setting utilizing seven small molecule datasets (SM), and (2)-(6) settings that incorporate both cross-dataset and cross- domain aspects, based on datasets from two different domains (BIO-SM, SM-CV) and three different domains (BIO-SM-SN, BIO-SN-CY SM-SN-CV For clustering tasks, we adopt the protocols in [22]. including five types of non-IID settings: (1) 2 clusters within the same domain (SM), (2) 3 clusters within the same domain (SN), (3) 15 clusters within the same domain (CV), (4) clusters across two domains (SM-BIO), and (5) 2 clusters across three domains (SM-BIO-SY). The dataset and experimental implementation details are provided in Appendix C.I
Baseline Methods. In both classification and clustering tasks, we compare FedIGL with two classical federated learning methods, FedAvg (27] and FedProx (2T). Additionally, we include four state-of-
the-art federated graph learning methods: FedSage [52], GCFL 48] FedStar B1. and FedSSP 32
For clustering tasks specifically, we also compare with FedGCN
22
Implementation Details. To ensure fair comparisons, all methods, including FedIGL and baselines, implemented in PyTorch and executed on the same NVIDIA GeForce RTX 3090 GPU. For graph-level structure embeddings, we use a three-layer Graph Isomorphism Network (GIN) [51 with hidden dimension of 64 and batch size of 128 (34). Model optimization is performed using the Adam optimizer with a learning rate of le-3, Dropout is set to 0.5 and weight decay to 5e-4 to improve generalization.
5.2 Experimental Results
Performance Comparison (RQ1). Tab. and Tab, 2 present comparison of the performance of FedIGL. against SOTA methods on graph-level classification and clustering tasks. In the classification task, FedIGL achieves the best results in out of 6 classification settings and ranks second in the remaining one, demonstrating the most robust overall performance. Notably, under the single-domaii SM setting, FedIGL improves over the FedSSP about a 4.3% relative gain. Moreover, although FedSSP slightly outperforms FedIGL on SM-CV, FedIGI shows more pronounced advantages in the more challenging multi-domain scenarios, indicating better generalization under stronger distribution shifts. In the clustering task, FedIGL ranks among the top methods across the five clustering settings,
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demonstrating more stable clustering quality under non-lID setting. Notably, on SN and CV. FedGCN achieves a slight success on specific tasks but remains suggesting that the two methods emphasize cluster alignment consistency and clustering decision correctness, respectively. Existing methods often perform well on specific tasks but remain vulnerable to spurious correlations, which limits their generalization.
Table I: Comparison with ofte--of-the-ar methods on one cross-dataset and five cross-domain settings for classification tasks. The best is marked with boldface and the second best is with underline.
Single-domain SM
Double-domain BIO-SM
Multi-Domain BIO-SN-CV
Methods
SM-CV
BIO-SM-SN
SM-SN-CV
FedAvg (ASTAT17)
74.12
67.82
1.63
81.21
67.31 2.56
70.93
2.91
75.33 + 1.06
FedProx (arXiv
69.35
67.27
417
70.02
63.89 4.33
69.32
67.15
2.25
FedSage (NeurIPS2I)
75.61 1,16
72.60
3.18
76.23
70.84
+:88
69.69 1.11
73.36
0.86
GCFL (NeurIPS21)
77.71 1.53
72.05
2.20
72.64 0.71
70.43
1.39
67.91
2.15
71.79
1: 0.21
FedStar (AAA123)
78.63
2.11
72.71
1.22
78.84= 1.07
72.60
244
69.51
+ 0.84
75.94
0.40
FedSSP (NeurlPS24)
79.62 + 2.23
73.66
2.3
84,29 + 0.68
72.37
2.18
2
79.12
1.23
Fed (ours)
83.07 # 1.76
77.02 # 1.32
83.14 + 0.28
75.25
1.13
78.50
0.44
79.23
1.28
Table 2: Comparison with state-of-the-ar methods on three cross-dataset and two cross-domain settings for clustering tasks. Please note that the FGL methods marked with the symbol have been adapted from classification clustering tasks, was done in previous studies. The best is marked with boldface and the second best is with underline.
FedAvg
FedProx
FedSage
GCFL
FedStar' AAAI23
FedSSP" NeurIPS24
FedGCN
FedIGL
Domain
Metrie
ASTAT17
arXiv18
NeurIPS21
NeurIPS21
AAA125
Ours
ACC
35.3
69.4
55.6
61.1 1
58.9 2.4
05
75.9 4
77.0
NMI
1.
8.4
12.2
8.7
12.0
12
24.9
25.7
SM
ARI
8.4
9.5
7.6
9,4224
0.1
34.1
31.1
36.3
F1
12.7
48,2
1..2
1.
49.
67.1 68.2
69.4
ACC
61.5
78.4
53.3
52.1
51.7
73.6
66.6
73.7 0.
NMI
15.6
11.6
14.8
12.5
13.7
11.8 34
30.4 6.6 23.1
SN
ARI
10.3
2273
11.6
13.2
1.
31.9
34.1 22
FI
58.1
$3.7
4..
52.3
50.7
64.9
50.7
24 6277
ACC
337
38.6
10.1
13.7
12.4
24
34.6 39.2 L.1
NMI
25.9
12.1
2.7
30.5
17.7 24
12.0 22.4
34.2 1.4 27.3 1.8
CV
ARI
10.5
12.7 2.1
13.6 1.8
24
15.3
333 33.1
19.3 à 188 38.1 #
FI
19.7
54.5
2.5
10.4
13.2 1.4
11.6 1.9
31.6 + 36,2 2 2.4
ACC
24.3
6..1
27.4
6..1
1.
72.3=21
69,2 0.6
73.6
NMI
11.8
2.3
5.2
4.
5.3
11.3 1.2
23.0 24
SM-BIO
ARI
7.2
1.7
4.2
3.2
3.8
2.14
17.5
30.7 25 0.9
FI
48.7
2..
49.9
4.5
51.7
59.4
ACC 56.1
68.3
1.
59.1
$7.9
0.
68.6
76.7
NMI
12.5
8.0
1.9
14.4
15.7
1.
23.
22.7
SM-BIO-SY
ARI
0.
8.9
17.6
13.7
34.8 1..1
3.
34,6
F1
1.71
2..7
1.7
$2.3
52.
69.7
67.5
Convergence Analysis (RQ2). We visualize the graph classification testing loss with re- spect to the communication rounds to show the convergence FedIGL clients under the non- IID setting, as shown in Fig. Despite the inherent heterogeneity of data across clients, method demonstrates a smooth and mono- tonic decrease in the global objective over communication rounds. This behavior aligns with the theoretical guarantees of federated optimization. In particular, introduction of bi- di-gradient regularization further stabilizes the learning dynamics by mitigating the diver- gence caused by client drift, leading to faster and more consistent convergence. More non-Ill settings in Appendix
Clert
Cherl
Communication Rounds
Communiication Rounds
(a) SM
(b) SM-BIO
Figure 3: The loss curves of each clisnt on the SM and SM-BIO of non-lID setting.
C.4
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Ablation Study (RQ3). We per- form ablation studies to assess how the proposed Bi-Gradient Regulariza- tion strategy contributes to the over- all performance. Tab. [ reports re- sults of FedIGL and its variants under non-IID settings: (i) removing both consistency regularization (CR) and diversity regularization (DR), (ii) en- abling only one of them (CR or DR), and (iii) enabling both. An intuitive observation is that FedIGL. performs best when CR and DR are enabled si- multaneously. Each component on its own still delivers consistent gains across most datasets. On the SM-BIO-SY clustering bench- mark, introducing CR alone leads to negligible gains over the baseline. In contrast, enabling DR alone yields noticeable improvement of 2.54%. This indicates that under multi-domain shift, encouraging representation diversity better mitigates domain bias and leads to more sta- ble clustering. Overall, the empirical trends align with our theoretical analysis, supporting the design of jointly integrating CR and DR. Additional results are provided in Appendix
Table 3: The ablation study covers both classification and clustering tasks. A checkmark (V) indicates inclusion of the strategy, while cross (X) indicates its exclusion. Our non-IlI settings include single-domain, double-domain, and multi-domain scenarios, corresponding to the SM, SM-BIO, EM-BIO-ST for the classification task and SM-BIO-SY for the clustering task, respectively.
SM-B10
SM-BIO-SN (SV)
Clnsaitcation
Clustering
Clastiication
Clustering
Claesiication
Clustecing
78.21
72.45
74.67
69.73
70,56
68.33
79.
73.04
75.83
70.34
71.24
68.36
75.30
70.57
83,07
77.04
77.02
73.71
75.25
76.71
HlyperPParamete Study (RQ4). We investi- gate the sensitivity of several hyper-parameters in our method, including the invariance regu- larization strength X, variance regularization strength 3, divergence parameter e, and invari- ant subgraph ratio т. Under the SM dataset sel- ting, we evaluate the performance of FedIGL across various hyperparameter combinations on classification and clustering tasks, as shown in Fig. 4 The results indicate that the optimal val- ues of X, 3, and E differ by task. For classifica- tion, À = 0.05, B 0.2, and = 0.1 provide the best performance balance; for clustering, A = 0.2, 8 = 0.25, and ε 0.15 perform best. This likely reflects distinct requirements for graph representations, suggesting that client subgraph invariance and variability are influ- enced by downstream tasks. The parameter T governs the proportion of invariant subgraphs; large I may include excessive variant struc- tures, while a small 7 may limit structural capture. In our experiments, 7 == 0.25 balances shared and client-specific structures effectively in both tasks. Additionally, we conduct further hyper-parameter sensitivity analyses under non-IID settings in Appendix C.3
SM
0.700
0.765
0.100
0.750
0.755
Gessitoation
Caotering
0.70
Classiiicatio
0.780
6.2
0.1
6.2
0.3
0.4
SM
0,760
0.82
0.705
Clasuitiation
0.750
0.530
0.750
Clustering
0.725
007995
Clastring
Figure 4: Hyperparameter Sensitivity analysis on SM setting. The x-axis represents the four hyper- parameters: X, B, s, and 7, with the left and right y-axis representing the classification and clustering accuracy, respectively.
6 Conclusion
In this work, we present a novel Federated Invariant Graph Learning framework from fresh perspec- tive, aimed at capturing invariant subgraph structures to mitigate client distribution shifts. We propose a Bi-Gradient Regularization strategy applying consistency regularization to the invariant subgraph encoder and diversity regularization the variant one, which enhances graph representation quality, stability and model performance. Overall, as pioneering study, FedIGL provides valuable insights for addressing the graph structural differences associated with client distributional heterogeneity and is supported by extensive experimental and theoretical analysis. While our approach intuitively protects client privacy by avoiding the sharing of prototype structures, future work will further explore stringent measures for model privacy preservation.

---

Acknowledgments
This work was supported by the National Natural Science Foundation of China (NSFC) (Grant No. 62562026, 62506102) (02), the Key Research and Development Program of Hainan Province (Grant No. 2DYF2024GX/S014, ZDYF2023GXISI633 Project (No. Qhyb2023- 104), and the Natural Science Foundation of Hainan University (Grant No. XI2400009401).
the Hainan Province Graduate Innovation Research
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A
Algorithm
Algorithm Optimizatiin process FedIGL
Input: Maximum epoch T'; Number of clients K; The distributed Non-IID datasets {Da) nyper-parameters t, X, 3,
Output: Model trained with FedIGL..
Initialize s@ Pinv: ) for i 1 to T do
for k to K do
Obtain Gr, Gv for each graph G Da with by Eq. (4
Obtain subgraph representation h h haar for G Gv.
Calculate C glohas for client with X, E by Eq. [2). Update (Orsa, Oin.: )ear) ) by Stochastic Gradient Descent. Fix the parameters of {0FsG,
varr).
Obtain client-specific subgraph representation hapee with Gy.
Calculate Liooal for client k by Eq. (8).
Update local model f for client k by Stochastic Gradient Descent. end for Aggregate {0rsc, Oinos Ovar }. end for
B Proofs in Section 4.4
Lemma B.1 (Gradient Variance Reduction for Invariant Branch), Suppose each Lx(0) is L-smooth and the initial variance of local invariant gradients satisfies Bligh g'll" 112 S 0². Then, applying the regularization Rim leads to exponential decay in gradient variance:
Eligi g" (1 (ur) o²,
assuming 0 < 17 1/L and A1 O.
Proof. Using L-smoothness, we apply the standard gradient descent update:
g = v -
Applying the regularization Rus effectively forces each o to align with the global average g' thereby reducing the variance. The gradient variance evolves as:
Ellak g'l < (1 o
B.1 Proofs of Proposition 4.1
Proof. With convexity and smoothness, we apply standard convergence results for Stochastic Gradient Descent (SGD) with variance reduction. By Lemma B.1 the variance of of decreases time, which helps stabilize the FedAvg updates. Using the descent lemma and unbiased gradients, we have:
Cgloba (0'+1) < (O") 1 Ve., Cgloba + nj² Co?.
Averaging over T rounds gives the O(1/T) rate.
B.2 Proofs of Proposition 4.2
Proof. The penalty max(0, c 1 - 12 pushes each hj to be at least away from the mean. If ||A' 1i the penalty is active, increasing the loss. At the optimum, this penalty is zero, implying that | - M 2 assuming that the mean of all clients' gradients remains close to the previous gradients. And thus pairwise || - hi. 2 e. Consequently, the absolute difference between the gradients of any two clients exceeds the penalty term.
21

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B.3 Proofs of Theorem 4.1
Proof. The total loss includes smooth convex functions and squared penalties. Using standard convergence bounds for smooth objectives with gradient regularization and diminishing step size " = ((1vT), we get the convergence rate of (I/vVT) for gradient norms. The variance reduction and margin-enforeing terms ensure stable updates for both branches.
Response: On Computational Complexity. The per-round per-client computation in FedIGL mainly involves two parts: 1. The Federated Subgraph Generator (FSG), which consists -lyyyrrr GNN with complexity O (VV(d) to encode node features, and edge-wise MLP scorer with complexity O El(e) for selecting invariant edges; 2. The dual-branch GNN encoder, each branch having L2 layers, resulting in a total complexity of ( (V]))) Here [V| and [E] denote the number of nodes and edges in the client's 's local graph, respectively; d the feature dimensionality of each node; and L1, L2 represent the number of GNN layers in the Federated Subgraph Generator and dual-branch encoder, respectively. Hence, the overall per-client cost per round is: ((L) + (2) d + [V|d) + Eidd), which remains linear thus comparable standard GNN-based federated learning methods. The extra overhead from edge scoring is lightweight and only applied once per round.
graph size and
C Additional Experiments
C.1 Experiment Dataset
Evaluation Metries In classification tasks, we employ Accuracy (ACC) to assess the performance of the method. Regarding clustering tasks, we utilize widely-adopted clustering result evaluation metrics, namely Accuracy (ACC), Adjusted Rand Index (ARI), Normalized Mutual Information (NMI), and F1 Score (F1). These metrics provide malti-faceted evaluations of the clustering results. Specifically, larger values of these metrics correspond to better performance. They imply more efficient data partitioning and more accurate capture of the underlying data structure, thereby demonstrating the superiority of the clustering method in organizing data and uncovering its inherent characteristics.
Table 4: A superscript "1" in the upper-right comeer indicates that the dataset is only used for classification tasks, "2" indicates that dataset is only used for clustering tasks, and the absence of superscript indicates that the dataset is used for both classification and clustering tasks.
Datasets
Domain
Classes
Graphs
A.Nodes
A.Edges
MUTAG
2
188
17.93
19.79
BZR
2
405
35.75
38.36
COX2
2
467
41.22
43.45
DHFR
2
756
42.43
44.54
SM
PTC MR
2
344
14.29
14.69
AIDS
2
2000
15.69
16.20
NCII
2
4110
29.87
32.30
BZR BZR_MDD
2
306
21.30
225.06
DD
2
1178
284.32
715,66
PROTEINS
2
1113
39.06
72.82
BIO
OHSU
2
79
82.01
199.66
Peking 1
2
85
39.31
77.35
SYNTHETIC'
SY
2
300
100.00
196.00
COLLAB
3
5000
74.49
2457.78
IMDB-MULTI
SN
3
1500
13.00
65.94
2
1000
19.77
96.53
Letter-high
15
2250
4.67
4,50
Letter-low
CV
15
2250
4.68
3.13
Letter-med
15
2250
4.67
3.21
22

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C.2 Ablation Study
We present additional ablation experiments across multiple non-III) settings, as shown in Tab The results demonstrate that the combination two optimization strategies significantly outperforms the use of individual strategies, thereby validating the effectiveness of our proposed optimization framework. Notably, our method not only excels in specific settings but also exhibits consistent performance across wide range of scenarios, highlighting its robustness and adaptability to varying conditions. The proposed approach effectively handles diverse datasets and domain configurations, yielding high-quality graph representations that deliver superior performance in the classification tasks.
Table 3: Ablation study of key components, namely Consistency Regularization (CR) and Diversity Regularization (DR), of FedIGL on double-domain and multi-domain settings (SM-CV, BIO-SN-CY and SM-SN-CV) in the classification.
CR
DR
SM-CV
BIO-SN-CV
SM-SN-CV
x
78.35
73.89
69.82
X
79.76
74.95
72.31
X
V
80.43
75.11
71.69
V
83.14
78.50
79.23
C.3 Hyper-Parameter Study
We investigate the sensitivity of several hyperparameters in our method, including the invariance regularization strength X, the variance regularization strength 8, the divergence parameter e, and the invariant subgraph ratio т. The hyperparameter tuning results across the non-IlI settings presented in 5ig[5] with the following key observations:()) The value of T is primarily influenced by cross-domain distribution shifts, rather than downstream tasks, with the optimal range identified between [0.2, 0.3]. (2) The value of À is notably task-dependent. For clustering tasks, the optimal range lies within [0.2, 0.3], while for classification tasks, is within [0.1, 0.2]. (3) The optimal values for 3 and are found within the ranges of [0.2. 0.3] and [0.1, 0.2], respectively. These results provide crucial theoretical insights into hyperparameter optimization and serve as a strong foundation for adapting our model to heterogeneous data distributions in real-world applications.
SM-BIO
SM-BIO
SM-BIO
0.725
0.72
5.750
0.25
0.72
0.09
Classitcation
Chastering
0.54
Stattaiing
0.475
0.700
0.700
Clsstiicatoon
Chattering
0.66
0.700
E3725
Chaserrng
0.3
6.6
6.4
83
Figure 5: Hyperparameter Sensitivity analysis. The x-axis represents the four hyperparameters: X, 3, c, and т, with the left and right y-axis representing the classification and clustering accuracy, respectively. The dashed lines in the figure represent the highest test accuracy of the baseline method under the settings of the SM-BIO.
C.4 Convergence Analysis
Fig. 0 provides additional information on the relationship between graph classification loss and communication rounds in three non-IID settings. The experimental results indicate that, as the commu- nication rounds progress, each client's 's loss curve exhibits smooth, consistent decline, substantiating the effectiveness of our method in promoting model convergence. Further convergence experiments under the non-IlI setting reinforce the superiority of the proposed bi-gradient regularization strategy in cross-dataset and cross-domain scenarios, demonstrating enhanced training stability and stronger generalization capability.
23

---

3.0
BIO-SN-CV
3,0
SM-BIO-SN
3.0
SM-SN-CV
2.5
Died
Dient
Ciest
Cienkh
Chere
2.5
Clon
Cäenn
Chon1
Clam
Clen
2.5
Chem
Cánn
Clent
Clon 10
Clon 11
Test
1.5
Test
1.5
1.0
1.0
1.0
0.5
0.5
1000
Communication Rounds
Communication Rounds
Communication Rounds
Figure 6: Loss trends of individual clients under three dataset settings: BIO-SN-CV, SM-BIO-SN, and SM-SN-CV.
24


Reviews:
[{"review_id": "oVEqzhexxC", "reviewer": "Reviewer_DuEb", "summary": "The paper introduces FedIGL, a federated graph-learning framework that aims to improve generalization under non-IID client distributions by (i) splitting each client graph into “invariant” and “variant” subgraphs with a federated subgraph generator, and (ii) applying a Bi-Gradient Regularization (gradient alignment for invariant features, gradient repulsion for variant features).", "questions": "1.  How sensitive is FedIGL to early-round noise? Would a warm-up or EMA of historical gradients improve stability?\n\n1. Please specify exactly how graphs were split into clients (by dataset, by class, or random). Performance can vary drastically with partition severity.\n\n1. Given recent gradient-inversion attacks on GNNs, can the authors quantify information leakage under their gradient-sharing scheme? Differential-privacy noise or secure aggregation would strengthen the story.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths：\n1. End-to-end implementation with a reasonable experimental breadth (19 datasets).\n1. High-level idea is easy to grasp; figures help.\n1. First explicit pairing of gradient alignment/repulsion within a subgraph-mask framework for FGL.\n\nWeaknesses：\n1. Baseline selection omits strong personalized-FL and IRM baselines (FedDyn, Ditto, FedRep, IRM-GNN).\n1. Key hyper-parameters (client splits, local epochs, comm. cost) undocumented, hurting reproducibility.\n1. Incremental combination of known ideas (IRM-style alignment + gradient diversity) yields limited novelty; impact likely modest without stronger theory or privacy guarantees.", "was_revised": false, "final_justification": "", "rebuttal": "**W1. More Baseline.** Thank your suggestions. To ensure fair comparison, we benchmarked against SOTA federated graph learning methods. Following your suggestion, we additionally conducted experiments with the referenced baselines on the SM and SM-BIO datasets. Table 1 demonstrates that FedIGL achieves significantly better performance than all baselines across both classification and clustering tasks. Note that GIL [1] is IRM-GNN without a federated framework; we apply FedAvg for parameter aggregation to ensure a consistent evaluation setup.\n\n### Table 1\n| Dataset | FedDyn | Ditto  | FedRep | GIL* | OURS   |\n|-|-|-|-|-|-|\n| SM | 76.87 ± 1.54   | 72.14 ± 01.87  | 73.43 ± 0.85  | 73.75 ± 2.09  | 83.07 ± 1.76 |\n| SM-BIO | 70.87 ± 2.14  | 67.48 ± 2.05   | 65.67 ± 1.77   | 71.12 ± 1.40   | 77.02 ± 1.32 |\n\n[ 1 ] Learning Invariant Graph Representations for Out-of-Distribution Generalization. Neurips, 2022.  \n\n**W2. & Q2. Experimental Details.** Thanks for your suggestions. We would like to clarify that our data partition protocols strictly follow standard baselines to ensure fair comparison. Specifically, for the classification tasks, we adopt the client-level non-IID partition from GCFL [2], and for clustering tasks, we follow FedGCN[3] Details of dataset splits and configurations can be found in Appendix C.1. Specifically, **Dataset Partition:** As downstream tasks differ in the number of classes, we assign a complete dataset to each client, naturally introducing a high level of non-IIDness. **Local Epochs:** Following prior work, each client is trained for 500 local epochs. We also provide a convergence analysis, shown in Fig.3. **comm. cost:** On the SM dataset with 7 clients, FedIGL incurs 0.2315 MB of communication overhead per client per round, compared to 0.2643 MB for GCFL, indicating a moderate reduction in communication cost. In summary, more details of the experimental setup will be provided in the final version.\n\n[ 2 ] Federated graph classification over non-iid graphs. NeuraIPS, 2021.  \n[ 3 ] Federated graph-level clustering network. AAAI, 2025.   \n\n**W3. Technical Contributions.** Thank your comments. We would like to further clarify the novelty and technical contributions of our work. Our main contributions can be summarized as follows:  \n**(i) Novel Perspective.** Prior works that impose overly strict assumptions about the presumed correlation between structural features and the global objective often fail to generalize to local tasks.  Our work focuses on identifying and leveraging client-invariant, discriminative factors to reduce the effect of spurious correlations. To the best of our knowledge, this is the first work to introduce invariant subgraph structures into FGL to address the distribution shifts challenge.       \n**(ii) Technical Contributions and Theoretical Analysis.** We propose a bi-gradient regularization framework that fundamentally advances beyond the classical IRM paradigm and directly addresses the optimization challenges of federated learning under non-IID distributions. Unlike existing methods that merely alleviate sub-optimal results caused by gradient heterogeneity, our approach introduces a structural disentanglement strategy that explicitly decomposes each graph into invariant and variant subgraphs. These are jointly optimized via two complementary regularizers: gradient consistency ensures global alignment, while gradient diversity drives personalized adaptation. This design not only mitigates conflicts arising from structural heterogeneity but also achieves collaborative optimization with enhanced personalization, marking a clear step forward in federated graph learning.   \nWe provide a theoretical analysis in Section 3.4 to formally support the proposed mechanism. Specifically, we prove that the introduced regularizers contribute to the convergence of the global optimization objective and effectively enhance the discriminability of local structural representations. We have provided detailed theoretical proofs in Appendix A.   \n**(iii) Extensive Experiments and Privacy Preservation.** In contrast to most existing methods that are typically validated on a single task, we conduct extensive experiments on both classification and clustering tasks under non-IID settings. Results against advanced baselines validate the generalizability of FedIGL.Additionally, unlike prior approaches that commonly share global prototypes or full structural representations, potentially exposing sensitive local information, FedIGL shares only a subset of subgraph structures with high structural similarity across clients. Such information is inherently less reconstructable, offering a promising degree of privacy protection. Furthermore, we provide additional experimental validation in our response to Question 3.  \nIn summary, FedIGL prioritizes simplicity and efficiency over overly complex design. \n\n**Q1. Training Setup.** To ensure fair comparison, our training protocol strictly follows prior works, focusing instead on the core framework design. We conducted additional experiments measuring early-round accuracy variance (T = 5, 10) in SM datasets. Table 2 shows that FedIGL exhibits low accuracy variance at both round 5 and round 10, with only minor fluctuations, indicating strong training stability. Furthermore, the incorporation of EMA yields only marginal improvements in both variance and accuracy, suggesting that FedIGL is inherently stable, even without additional smoothing techniques.\n\n### Table 2\n| Method       | Variance (T=5) | Variance (T=10) | ACC (%) |\n|--------------|----------------|------------------|---------|\n| FedIGL       | 5.83         | 5.29           | 83.07 ± 1.76   |\n| FedIGL+EMA   | 4.91         | 4.56          | 84.15 ± 1.20   |\n\n**Q3. Privacy Analysis.**\nWe conduct additional experiments to evaluate the privacy-preserving capability of FedIGL, incorporating both gradient inversion attacks via DLG (Deep Leakage from Gradients) [4,5] and formal privacy guarantees through DP [6].  \n**(i)** We employ DLG to reconstruct node features from shared gradients and use the MSE between the original and reconstructed features as a proxy for privacy leakage. As reported in Table 3, FedIGL achieves an MSE of 0.60 on SM and 0.52 on BZR, which are slightly higher than those of the strongest baselines. Given that all features are normalized to the [0,1] range, these values suggest that attackers face substantial difficulty in accurately recovering the original inputs. Furthermore, unlike in vision domains, graph data comprises both node attributes and topological structure, and the discrete nature of graph connectivity further increases the difficulty of faithful reconstruction.  \n\n### Table 3 \n| Dataset | FedIGL (+DLG) | GCFL (+DLG) |\n|:--------|:--------------:|:-----------:|\n| SM      |     0.63       |    0.59     |\n| BZR     |     0.52       |    0.50     |\n\n**(ii)** We further evaluate the performance of FedIGL on the SM dataset by injecting differential privacy noise under varying privacy budgets $\\varepsilon$ $\\in$ {$\\infty$, 3, 1, 0.5, 0.1}. As shown in Table 4, FedIGL consistently achieves the highest accuracy under all privacy settings, reaching 81.47% even at the strictest budget of $\\varepsilon$ = 0.1, and significantly outperforming competing methods. These results indicate that FedIGL effectively mitigates the performance degradation caused by differential privacy and demonstrates strong privacy-preserving capability.  \n### Table 4\n|     Methond    | GCFL | FedStar | FedIGL |\n|:-:|:-:|:-:|:-:|\n|  ACC （ε = $\\infty$)   |   77.61 ± 1.79    |     75.39 ± 2.03  |    83.07 ± 1.76        |\n| ACC (w/ DP, $\\varepsilon$=3) |   75.78 ± 1.30  | 72.16 ± 1.67    |    82.09 ± 2.03  |\n| ACC (w/ DP, $\\varepsilon$=1) |   75.61 ± 2.19  |71.49 ± 1.12     |    81.32 ± 1.19      |\n| ACC (w/ DP, $\\varepsilon$=0.5) |   74.17 ± 1.41   | 72.02 ± 0.89     |   81.81 ± 1.03      |\n| ACC (w/ DP, $\\varepsilon$=0.1) |   74.08 ± 3.02  | 71.75  ± 1.99     |    81.47 ± 1.35      |\n\nThese results demonstrate the privacy-preserving potential of FedIGL, which can be attributed to the proposed bi-gradient regularization mechanism that limits shared information to invariant substructure encoders, thereby avoiding the exposure of complete graph structures or gradients. \n\n[ 4 ] Deep Leakage from Gradient. NeurIPS, 2019.  \n[ 5 ] Hiding in Plain Sight: Disguising Data Stealing Attacks in Federated Learning.  ICLR,  2024.  \n[ 6 ] Privacy-Preserving Personalized Federated Prompt Learning for Multimodal Large Language Models. ICLR, 2025."}, {"review_id": "UrAy09wbtv", "reviewer": "Reviewer_n8up", "summary": "This work develops a novel framework called FedIGL, aimed at mitigating data heterogeneity in federated graph learning. Rather than relying on statistical aggregation of structural information, the proposed approach employs a Bi-Gradient Regularization strategy that enables clients to effectively disentangle invariant and client-specific representations. Experimental results demonstrate that FedIGL exhibits promising performance across various graph datasets.", "questions": "1. Eq. (12) lacks sufficient detail regarding the optimization of losses. A more explanation of their optimization would improve clarity and reproducibility.\n2. How does the proposed framework protect privacy in both the global and local models?\n3. What do the metrics in Table 1 represent?\n4. How the heterogeneous data distribution are defined for both clustering and classification？\n5. There a missing reference at line 83.", "limitations": "Yes", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths \n1. The paper offers an interesting perspective and appears to be well-executed. The methodology is both rigorous and novel, with each module clearly and intuitively explained. The model design effectively emphasizes invariant subgraph patterns in addressing structural heterogeneity in federated graph learning.\n2. Extensive experiments and theoretical analysis strengthen the empirical findings, suggesting that FedIGL exhibits strong potential and competitive performance in relevant tasks.\n\nWeaknesses \n1. While the FSG module appears well-designed, further clarification is needed on how its separation mechanism aligns with the learning objectives of downstream tasks in federated settings. In particular, it remains unclear how the global and local loss functions are coordinated within the proposed framework to ensure consistent and effective optimization.\n2. The explanation of the experimental setup is insufficient, particularly with respect to how labels are partitioned in classification tasks and how heterogeneous data distributions are defined and constructed in both clustering and classification settings.\n3. While the use of subgraph gradients may offer implicit privacy benefits, the lack of a formal privacy analysis limits the validity of comparisons with established privacy-preserving techniques. The authors are encouraged to provide a rigorous privacy evaluation to strengthen their claims.", "was_revised": true, "final_justification": "I thank the authors for their detailed rebuttal. I believe my concerns have been well addressed, and their clarifications have improved my understanding and appreciation of this work, so I will keep my positive rating.", "rebuttal": "**W1. Global and Local Loss.** Thank you for your comments. The objective of standard Federated Learning methods is to learn a global model that minimizes the empirical risk across all client distributions. FedIGL achieves consistent and effective optimization by systematically coordinating global and local loss functions through structural disentanglement and bi-gradient regularization. Specifically, each input graph is partitioned into a client-agnostic invariant subgraph and a client-specific variant subgraph via the proposed FSG module. The global loss is imposed on the invariant representation, which is optimized through cross-client aggregation, with the consistency regularizer $R_{inv}$ reducing gradient variance across clients to ensure convergence to a consistent global objective (refer to Proposition 3.1, Section 3.4). Meanwhile, the local loss is computed solely on the client-specific subgraph, with its encoder remaining strictly local and personalized. The diversity regularizer $R_{var}$ encourages representational separation per-client (refer to Proposition 3.2), thereby preserving local adaptation. This design allows the invariant encoder to achieve global alignment while the variant encoder supports personalization, ensuring stable and collaborative optimization in structurally heterogeneous federated graph learning.\n\n**W2. Confusing in classification tasks.** Thanks. Following prior work[1], each client is assigned a separate dataset. In intra-domain scenarios, clients are associated with different datasets within the same domain, each exhibiting distinct class priors or label distributions, thereby introducing label-level heterogeneity. In cross-domain scenarios, clients are assigned datasets from entirely different domains (e.g., small molecules, bioinformatics, and computer vision), where substantial differences exist in label semantics and structural characteristics. In each of the settings, a client owns one of the corresponding datasets and randomly splits it into three parts: 80% for training, 10% for validation, and 10% for testing. \n\n**W3. & Q2. Privacy Analysis.** Thanks. The architecture of FedIGL is intrinsically aligned with privacy-preserving objectives. All training operations, including updates to both the invariant and variant encoders, are executed locally on the client side. The variant encoder is strictly confined to the client and remains entirely excluded from any communication. Moreover, the gradients shared during training pertain only to the invariant subgraph encoder, which captures structurally similar patterns across clients. The limited informativeness of shared gradients under non-IID settings substantially reduces the risk of reconstructing client-specific graph data. In contrast to methods that transmit prototypes, full model parameters, or complete client updates, FedIGL restricts the shared content to structurally coarse and partially informative components, thereby substantially narrowing the attack surface and enhancing resilience to gradient-based privacy attacks. We further evaluate the utility of FedIGL under differential privacy by injecting noise into the model on the SM dataset, across a range of privacy budgets $\\varepsilon \\in {\\infty, 3, 1, 0.5, 0.1}$. As shown in Table 1, FedIGL consistently achieves the highest accuracy under all privacy levels, maintaining 81.47% even at the strictest setting of $\\varepsilon = 0.1$, and significantly outperforming both GCFL and FedStar [3]. These results confirm that FedIGL preserves high model utility even under stringent privacy constraints.\n### Table 1\n|     Methond    | GCFL | FedStar | FedIGL |\n|:-:|:-:|:-:|:-:|\n|  ACC （ε = $\\infty$)   |   77.61 ± 1.79    |     75.39 ± 2.03  |    83.07 ± 1.76        |\n| ACC (w/ DP, $\\varepsilon$=3) |   75.78 ± 1.30  | 72.16 ± 1.67    |    82.09 ± 2.03  |\n| ACC (w/ DP, $\\varepsilon$=1) |   75.61 ± 2.19  |71.49 ± 1.12     |    81.32 ± 1.19      |\n| ACC (w/ DP, $\\varepsilon$=0.5) |   74.17 ± 1.41   | 72.02 ± 0.89     |   81.81 ± 1.03      |\n| ACC (w/ DP, $\\varepsilon$=0.1) |   74.08 ± 3.02  | 71.75  ± 1.99     |    81.47 ± 1.35      |\n\n**Q1. Confusing in Eq.(12).** Thanks. Eq. (12) defines a bi-gradient regularization objective for global model aggregation. It is important to clarify that the local training loss on each client is not omitted. Specifically, in each communication round, the server compares the gradients uploaded by clients with the previously aggregated global gradients. The gradients of the invariant branch are regularized to be consistent across clients, promoting the generalization of the global model, while the variant branch gradients are encouraged to diverge across clients, preserving personalization for local models. Meanwhile, each client optimizes its task-specific loss on local data to update the variant encoder, while the global model integrates consistency and diversity regularizers to guide distributed representation learning, fostering synergy between local personalization and global generalization.\n\n**Q3. & Q5. Improving Writing.** Thank you for your suggestion. The metric reported in Table 1 is accuracy, and we will explicitly clarify this in the table caption in the final version of the manuscript. Furthermore, we will ensure that all missing citations are properly added and will carefully revise the manuscript to correct any remaining inconsistencies in metric reporting and formatting.\n\n**Q4. Definition of Heterogeneous Data Distributions.** Thanks. We follow the experimental settings established in prior work [1, 2], utilizing datasets from diverse domains such as small molecules, bioinformatics, and computer vision. These datasets are used to construct heterogeneous client distributions that reflect both label distribution shifts, classes, and domain-level variations. For both classification and clustering tasks, each client is assigned a distinct partition of the data. In single-domain scenarios, clients are sampled from different datasets within the same domain, introducing heterogeneity in label spaces or clustering structures. In cross-domain settings, clients originate from entirely different domains, thereby exhibiting significant disparities in label semantics, feature distributions, and underlying graph topologies. The non-IID settings and dataset details for both classification and clustering tasks are provided in Appendix C.\n \n[ 1 ] Federated graph classification over non-iid graphs, NeuraIPS, 2021.  \n[ 2 ] Federated graph-level clustering network, AAAI, 2025.  \n[ 3 ] Federated learning on non-iid graphs via structural knowledge sharing, AAAI, 2023."}, {"review_id": "5lWpiH2tNr", "reviewer": "Reviewer_KuhH", "summary": "This paper presents Federated Invariant Graph Learning (FedIGL), a novel framework that effectively designed to address the challenge of non-IID data distributions in federated graph learning. Specifically, FedIGL leverages invariant learning to suppress spurious correlations and extract consistent subgraph patterns across non-iid client distributions. Moreover, a Bi-Gradient Regularization strategy is introduced to disentangle shared and client-specific representations by coordinating global and local models. Experiments on graph-level benchmarks show that FedIGL consistently improves performance on both clustering and classification tasks.", "questions": "Please check the weaknesses.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\n1. This paper addresses the limitations of data heterogeneity in federated graph learning by disentangle client-invariant and client-specific sub-graphs, offering a new approach for non-IID graph data.\n2. The framework description and formula derivations are generally clear, and the flowchart aids in understanding the overall process.\n3. Comprehensive experiments show statistically significant results, ablation studies and parameter analysis validate the necessity of each module.\n\nWeaknesses:\n1. While the Bi-Gradient Regularization strategy effectively separates invariant and client-specific subgraph patterns, it remains unclear how this separation aligns with the overall optimization objectives of federated learning?\n2. In Equation (11), ε is introduced as a margin to enforce gradient diversity across clients. Could the authors clarify its valid range or provide guidance on how this parameter should be selected in practice?\n3. In Fig. 2, the annotations for $f_V(G_V;\\theta_{\\text{var}})$ and $f_I(G_V; \\theta_{\\text{inv}})$ are missing. Providing these would improve the clarity and interpretability of the illustration.\n4. A clearer description of the experimental setup would enhance transparency. In particular, it is important to clarify how the non-IID configurations differ between the classification and clustering tasks. What motivates these design choices, and how are the datasets partitioned across clients in each case?\n5. The scalability analysis with respect to the number of clients is insufficient. How does the proposed framework perform as the number of clients increases?\n6. A more comprehensive discussion of privacy is needed. Could the sharing of gradients during training lead to potential information leakage? Moreover, how does the proposed method compare to existing privacy-preserving baselines in terms of its ability to protect sensitive client information?", "was_revised": true, "final_justification": "As all of my concerns have been addressed;I keep my rating of accept.", "rebuttal": "**Q1. Clarification of the Optimization Objective.** We appreciate your comment. Most existing works achieve this by decomposing the global objective into a weighted sum of local empirical losses and performing independent optimization on each client. However, under non-IID conditions, local updates from different clients may follow divergent optimization directions, impairing convergence and model generalization. Our proposed Bi-Gradient Regularization strategy facilitates decoupled and efficient feature learning across distributed clients to optimize the global model objective. Specifically, we first employ the FSG module to disentangle client-agnostic and client-specific subgraphs. Then, the gradient alignment mechanism minimizes the variance of gradients across clients on the invariant encoder, promoting consistency in the shared subgraph representation and guiding the global model toward a unified optimization direction. In contrast, the diversity of gradients on the variant encoder preserves client-specific structural information, preventing the dilution of personalized knowledge during aggregation. Moreover, FedIGL effectively fosters collaboration among clients to uncover invariant subgraph patterns across heterogeneous distributions, thereby mitigating client drift induced by structural heterogeneity.\n\n**Q2. Sensitivity Analysis of ε.** We appreciate your comment. In our work, ε is introduced as a lower bound on the gradient differences in the variant branch across clients, encouraging meaningful diversity in learning directions and thereby preserving client-specific personalized features. We tested ε values in the range from 0.1 to 0.3 and found that they yielded the best performance, as shown in Fig. 4 of the manuscript.\n\n**Q3.  Better Understanding.** Thank you for your suggestion. In our framework diagram, $f_V(G_V; \\theta_{\\text{var}})$ and $f_I(G_V; \\theta_{\\text{inv}})$ represent the variant and invariant subgraph encoders, respectively. We will enhance the clarity of this notation and include corresponding explanations in the figure caption of the final version.\n\n**Q4. Implementation Details.** Thanks for your comments. (1) The settings are motivated by real-world federated scenarios, where clients naturally hold heterogeneous data due to differences in data sources, tasks, or domains. This setup also aligns with widely adopted benchmarks [1–2]. (2) In both classification and clustering tasks, each client is assigned a distinct dataset to simulate highly non-IID settings.  In classification tasks, the non-IID nature primarily stems from differences in the number of classes, label distributions, and domain semantics across client datasets. In clustering tasks, non-IIDness arises from variations in the number of clusters, clustering structures, and domain-specific graph properties.\n\n**Q5. Scalability Analysis.** Thank you for your valuable comments. We appreciate the reviewer’s suggestion regarding scalability. To evaluate the scalability of our proposed FedIGL framework, we conducted additional experiments on the SM dataset by varying the number of participating clients while keeping the total dataset size fixed. Specifically, we tested configurations with 5, 10, and 20 clients under consistent non-IID conditions. As summarized in Table 1, FedIGL achieves stable accuracy across various client numbers. Although performance slightly decreases as the number of clients increases, the overall results remain stable and reliable.\n### Table 1\n| Number of Clients |   5   |   10   |   20   |\n|-------------------|-------|--------|--------|\n| ACC ( % )           | 83.07 ± 1.76 | 83.11 ± 1.06  | 82.78 ± 0.97  |\n\n**Q6. Privacy Analysis.** Thanks. Compared to existing FGL methods that share global prototypes or the full set of encoder parameters, FedIGL transmits only the gradients of partial invariant components. More precisely, the client-specific subgraph structures remain entirely local and are never involved in communication. Moreover, the shared invariant subgraph encoder captures structural patterns that are highly similar across clients. Under non-IID conditions, such partial and redundant information makes it difficult to reconstruct a client’s complete local graph. This design helps reduce potential privacy leakage risks while preserving effective collaboration. We further evaluate the utility of FedIGL under differential privacy by injecting noise into model updates on the SM dataset, across a range of privacy budgets $\\varepsilon$ $\\in$ {$\\infty$, 3, 1, 0.5, 0.1}. As shown in Table 2, FedIGL consistently achieves the highest accuracy under all privacy levels, maintaining 81.47% even at the strictest setting of $\\varepsilon = 0.1$, and significantly outperforming both GCFL [2] and FedStar [3]. Empirical evidence indicates that FedIGL maintains strong model performance despite operating under strict privacy constraints.\n### Table 2\n|     Methond    | GCFL | FedStar | FedIGL |\n|:-:|:-:|:-:|:-:|\n|  ACC （ε = $\\infty$)   |   77.61 ± 1.79    |     75.39 ± 2.03  |    83.07 ± 1.76        |\n| ACC (w/ DP, $\\varepsilon$=3) |   75.78 ± 1.30  | 72.16 ± 1.67    |    82.09 ± 2.03  |\n| ACC (w/ DP, $\\varepsilon$=1) |   75.61 ± 2.19  |71.49 ± 1.12     |    81.32 ± 1.19      |\n| ACC (w/ DP, $\\varepsilon$=0.5) |   74.17 ± 1.41   | 72.02 ± 0.89     |   81.81 ± 1.03      |\n| ACC (w/ DP, $\\varepsilon$=0.1) |   74.08 ± 3.02  | 71.75  ± 1.99     |    81.47 ± 1.35      |\n\n[ 1 ] Federated graph classification over non-iid graphs. NeuralPS, 2021.  \n[ 2 ] Federated graph-level clustering network. AAAI, 2025.  \n[ 3 ] Federated learning on non-iid graphs via structural knowledge sharing. AAAI, 2023.  \n[ 4 ] Privacy-Preserving Personalized Federated Prompt Learning for Multimodal Large Language Models. ICLR, 2025."}, {"review_id": "eQxGS0JepN", "reviewer": "Reviewer_hKG3", "summary": "This work focuses on federated invariant graph learning for non-IID graphs. Unlike traditional federated graph learning approaches, the proposed FedIGL incorporates invariant learning to mitigate structural heterogeneity across clients. The Bi-Gradient Regularization strategy is designed to optimize graph representations, thereby improving performance on clustering and classification tasks.", "questions": "a. Some formula symbols are not clearly defined. For example, the meanings of f_V and f_I in Eq. (5) are insufficiently explained and require further clarification. \nb. How dose Eq. (16) contributes to the analysis of global and local convergence behavior within the proposed framework? A more detailed explanation would strengthen the theoretical analysis.\nc. How is the framework implemented in scenarios where clients possess different numbers of label categories in downstream classification tasks?\nd. Eq. (1) lacks appropriate citations.", "limitations": "Please see the weaknesses and questions.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Pros:\na. The paper investigates an important yet challenging research topic in the domain of federated graph learning. The framework design is clear and well-presented.\nb. The authors propose a novel federated graph embedding method that alleviates the data heterogeneity issue from the perspective of invariant learning. \nc. Both theoretical analysis and empirical results demonstrate the effectiveness and superiority of the proposed method compared to advanced baseline methods.\n\nCons:\na. In Section 3.2, FSG is employed to identify invariant and variant subgraphs. However, it remains unclear how the subgraph generator distinguishes invariant subgraphs, and what mechanism are in place to ensures the effectiveness of this process？\nb. The manuscript lacks sufficient clarity regarding implementation details, particularly in downstream scenarios where the number of label categories may vary across clients.\nc. The time complexity of the proposed method is not thoroughly analyzed, and there is limited discussion on computational efficiency. This makes it somewhat difficult to assess the scalability of the approach, especially in large-scale settings.", "was_revised": true, "final_justification": "After reading the author's response, most of my concerns have been well addressed. I maintain my score.", "rebuttal": "**W1. The FSG Analysis.** Thanks. Following prior work [3], we integrate an invariant subgraph generator into the federated learning framework. This component identifies invariant and variant subgraphs by analyzing gradient information during training. Specifically, we compute the gradients of the local loss concerning structural and attribute dimensions and use them to assign importance scores to edges. Edges with low gradient variance across clients, which reflect stable contributions to the learning objective, are selected to form the invariant subgraph $G_I$ that captures globally shared structural patterns. In contrast, edges with high gradient sensitivity are included in the variant subgraph $G_V$ to preserve client-specific information. To ensure the reliability of this partitioning, we adopt a gradient-based selection strategy controlled by a tunable threshold parameter $\\varepsilon$ (refer to Eq. 12), which defines the minimum divergence required to identify variant nodes. Moreover, a rigorous theoretical analysis is presented in Section 3.4.\n\n**W2. & Q3. Implementation Details.** Thanks for your comments. We clarify that our data partitioning strictly follows established baselines to ensure fair and consistent comparisons. As downstream tasks involve varying class structures, each client is assigned a full dataset, thereby introducing strong non-IID conditions. For classification, we adopt the client-level non-IID protocol from GCFL [1]. Details regarding dataset descriptions and splits are provided in Appendix C.\n\n**W3. Time Complexity Analysis.** Thanks for your comments. The per-round and per-client computation in FedIGL mainly involves two parts: **(1)** The Federated Subgraph Generator (FSG), which consists of a $L_1$-layer GNN with complexity $O\\big(L_1 (|E| d + |V| d^2)\\big)$ to encode node features, and an edge-wise MLP scorer with complexity $O(|E| d^2)$for selecting invariant edges; **(2)**  The dual-branch GNN encoder, each branch having $L_2$ layers, resulting in a total complexity of $O\\big(L_2 (|E| d + |V| d^2)\\big)$. Here $|V|$ and $|E|$ denote the number of nodes and edges in the client's local graph, respectively; \\(d\\) is the feature dimensionality of each node; and $L_1, L_2$ represent the number of GNN layers in the Federated Subgraph Generator and dual-branch encoder, respectively. Hence, the overall per-client cost per round is: $O\\big( (L_1 + L_2)(|E| d + |V| d^2) + |E| d^2 \\big)$, which remains linear in the graph size and thus comparable to standard GNN-based federated learning methods. The extra overhead from edge scoring is lightweight and only applied once per round. In terms of communication efficiency, FedIGL only transmits partial gradients of the invariant encoder, significantly reducing bandwidth consumption compared to approaches that synchronize full models or personalized components. We will further investigate the scalability of FedIGL on larger-scale graph datasets in future work, aiming to enhance its effectiveness and practicality in real-world federated environments.\n\n**Q1. Formula Interpretation.** Thanks for your comments. In this paper, $F_I$ denotes the Invariant Feature Encoder, which is designed to extract globally shared representations from client-invariant subgraphs, thereby enhancing the generalization ability of the global model. In contrast, $F_V$ refers to the Variant Feature Encoder, which captures client-specific information to preserve local characteristics and improve the performance of personalized models. Clearer and more interpretable annotations will be provided in the final version to support better understanding.\n\n**Q2. Convergence Behavior.** We appreciate the reviewer's interest in the theoretical grounding of our method. Equation (16) formally quantifies the trade-off between convergence of the invariant encoder (refer to Proposition 3.1) and representation separation for the variant encoder (refer to Proposition 3.2), thereby establishing a convergence guarantee under the Bi-Gradient Regularization strategy. Specifically, the gradient consistency regularization $R_{inv}$ aligns the optimization directions of clients in the invariant encoder, enabling more stable training in federated settings. Meanwhile, $R_{var}$ enhances client-specific diversity by encouraging representation separation, thereby preserving local adaptability. These two regularization terms strike a balance between global invariance and local personalization. Furthermore, the learned representation is jointly optimized to ensure invariant alignment across clients via $\\theta_{\\text{inv}}$, while preserving per-client diversity through $\\theta_{\\text{var}}$. As the detailed proofs are already provided in Appendix A. Furthermore, as shown in Figure 3, our method demonstrates a smooth and monotonic reduction in the global objective across rounds, empirically validating the convergence behavior. \n\n**Q4. Writing Improvements.** Following your suggestions. All missing citations and writing issues will be addressed and corrected in the final version.\n\n[ 1 ] Federated graph classification over non-iid graphs. NeuralPS. 2021.  \n[ 2 ] Federated graph-level clustering network. AAAI. 2025.  \n[ 3 ] Graph Invariant Learning with Subgraph Co-mixup for Out-of-Distribution Generalization. AAAI. 2024."}]

Author rebuttal:

Meta-review: The authors present Federated Invariant Graph Learning (FedIGL), a framework that combines a global model for collaborative training with client-specific models to capture both shared and local subgraph patterns. They also introduce a Bi-Gradient Regularization strategy to disentangle invariant and spurious correlations, resulting in superior performance on graph clustering and classification tasks.

This paper received generally positive reviews, with final scores ranging from borderline accept to accept. The authors made a strong effort during the rebuttal to address reviewers’ concerns, and most reviewers acknowledged the responses as satisfactory (though reviewer DuEb did not provide a final justification, his/her initial weak accept score was already positive).

Overall, the AC considers the paper suitable for acceptance, given the favorable consensus and effective resolution of reviewer feedback.
Decision: Accept (poster)