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Federated Granger Causality Learning For Interdependent Clients With State Space Representation
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Proposes a federated framework in which clients with Kalman-filter state-space models and server jointly learn cross-client Granger causality (off-diagonal blocks of the state matrix) by sharing only low-dimensional states, with convergence and differential privacy analyses.
Problem
Learning cross-client Granger causality in geographically distributed industrial processes with high-dimensional measurements that cannot be centralized, due to bandwidth, compute, and privacy constraints.
Approach
Clients augment their Kalman-filter models with a linear ML term (theta_m times measurements) that encodes cross-client causality, while the server estimates off-diagonal state-matrix blocks from shared low-dimensional states; the iterative client-server loop is reformulated as a linear recurrence with sublinear/linear convergence conditions.
Key results
- On synthetic data, the server loss L_s stays small (order 1e-3 to 1e-2) across perturbations of causality; under reversed causality the server loss is highest, indicating the server detects topology change.
- On real ICS data, the federated estimate of A differs from the centralized estimate by Frobenius norm 0.8140 (HAI) and 3.0816 (SWaT) at P=2.
- Federated communication saves 144 bytes per round on HAI and 176 bytes per round on SWaT.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| increase perturbation to the trained cross-client causality | Our method · Synthetic causality perturbation | server loss L_s: 0.003 -> 0.007 -> 0.010 -> .013 | Helped |
| alter the trained causality topology | Our method · Synthetic network topology | server loss L_s: 0.003 -> 0.033 -> 0.006 -> 0.008 | Mixed |
| infer topology changes using augmented client loss alone | Our method · Synthetic network topology | augmented client loss (L_2)_a | No clear effect |
| increase measurement dimension D | Our method · Synthetic scalability | server loss L_s: 0.0027, 0.0061, 0.0090, 0.0084 | Mixed |
| increase number of clients M | Our method · Synthetic scalability | server loss L_s: 0.0003, 0.0001, 0.0026, 0.0004 | No clear effect |
| use federated rather than centralized learning | Granger causality learning · HAI | data saved per communication round: 144 bytes | Helped |
| use federated rather than centralized learning | Granger causality learning · SWaT | data saved per communication round: 176 bytes | Helped |
| remove client augmentation (ablation) | Federated framework · Synthetic topology reversing | server loss L_s: 10^{-5} -> 0.033 | Hurt |
| remove the server model (ablation) | Federated framework · Synthetic topology reversing | server loss L_s | Hurt |
| replace iterative optimization with pre-trained client models (ablation) | Federated framework · Synthetic topology reversing | server loss L_s: 0.065 vs 0.033 | Hurt |
Surprising findings
- The client augmented-model loss does not show a clear trend with changes in network topology, so inference based on client loss alone can yield false alarms of causality change.
Limitations
- Linear ML augmentation and linear state-space assumptions limit expressiveness.
- Convergence results rely on strong assumptions (convexity, Lipschitz smoothness) that the explicit loss function does not satisfy in general.
- Client-loss-based detection of causality changes is unreliable in experiments.
Experimental setup
Datasets: HAI SWaT synthetic linear state space data
Benchmarks: HAI SWaT synthetic linear state space data
Metrics: L2 loss of augmented client server loss L_s Frobenius norm error of A-hat data volume saved per communication round
Baselines: no client augmentation no server model pre-trained client model centralized oracle Kalman filter
Builds on: Kalman filter Granger causality vertical federated learning
Closest papers by meaning
- Federated Causal Structure Learning with Non-identical Variable Sets (ICML 2025 · similarity 0.86)
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- Root Cause Analysis in Microservice Using Neural Granger Causal Discovery (AAAI 2024 · similarity 0.86)
- Jacobian Regularizer-based Neural Granger Causality (ICML 2024 · similarity 0.85)
- FedCSL: A Scalable and Accurate Approach to Federated Causal Structure Learning (AAAI 2024 · similarity 0.85)
- GCAD: Anomaly Detection in Multivariate Time Series from the Perspective of Granger Causality (AAAI 2025 · similarity 0.85)
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