Federated Causal Representation Learning in State-Space Systems for Decentralized Counterfactual Reasoning

Nazal Mohamed; Ayush Mohanty; Nagi Gebraeel

cs.LGFeb 23, 2026

Federated Causal Representation Learning in State-Space Systems for Decentralized Counterfactual Reasoning

Authors: Nazal Mohamed, Ayush Mohanty, Nagi Gebraeel

Organizations: Georgia Institute of Technology, Atlanta, GA

Abstract

Networks of interdependent industrial assets (clients) are tightly coupled through physical processes and control inputs, raising a key question: how would the output of one client change if another client were operated differently? This is difficult to answer because client-specific data are high-dimensional and private, making centralization of raw data infeasible. Each client also maintains proprietary local models that cannot be modified. We propose a federated framework for causal representation learning in state-space systems that captures interdependencies among clients under these constraints. Each client maps high-dimensional observations into low-dimensional latent states that disentangle intrinsic dynamics from control-driven influences. A central server estimates the global state-transition and control structure. This enables decentralized counterfactual reasoning where clients predict how outputs would change under alternative control inputs at others while only exchanging compact latent states. We prove convergence to a centralized oracle and provide privacy guarantees. Our experiments demonstrate scalability, and accurate cross-client counterfactual inference on synthetic and real-world industrial control system datasets.

Figures & tables

Figure 1: A simplified view of the computation inside the client $m$

Panel of Figure 2 — Figure 2: Global loss at the server and losses at the client from different models (augmented client loss: $L_{m,a}^{k}$ , proprietary client loss: $\frac{1}{T}\sum_{t=1}^{T}{||{r^{t}_{m,c}}_{2}||}^{2}$ are plotted against the number of iterations ( $k$ ).

Panel of Figure 3 — Figure 3: Disentanglement penalty at the server and $\delta_{d}$ at the clients vs. number of iterations.

Panel of Figure 4 — Figure 4: Prediction error on the validation data for different models and the centralized oracle. ACM: Augmented Client Model, PCM: Proprietary Client Model

Figure 5: Comparison between Penalty Method (PM) and Augmented Lagrangian (AL) for the two client system.

L_{s}

\mathcal{D}

, Local loss (client residual)

L_{m,a}

and

\delta d_{m}

vs Number of iterations (

k

) are plotted. Note that we defined

\delta d_{m}:=\lVert\phi_{m}-\frac{1}{T}\sum_{t=1}^{T}\left(\sum_{n\neq m}\hat{B}_{mn}u_{n}^{t-1}\right)\rVert_{2}

$D_{m}=16$		$D_{m}=32$		$D_{m}=64$		$D_{m}=128$
$L_{s}$	$\mathcal{D}$	$L_{s}$	$\mathcal{D}$	$L_{s}$	$\mathcal{D}$	$L_{s}$	$\mathcal{D}$
0.7649	0.0034	1.0987	0.0041	1.4243	0.0046	1.1805	0.0047

Table 1:

L_{s}

and

\mathcal{D}

by scaling measurement dim.

D_{m}

$M=2$		$M=4$		$M=8$		$M=16$
$L_{s}$	$\mathcal{D}$	$L_{s}$	$\mathcal{D}$	$L_{s}$	$\mathcal{D}$	$L_{s}$	$\mathcal{D}$
0.0744	0.0013	0.0412	0.0010	0.1714	0.0036	0.3825	0.0069

Table 2: Server loss

L_{s}

by scaling number of clients

M

Panel of Figure 6 — Figure 6: Tranfer Function characteristics

Figure 7: HAI dataset

Causal Representation Learning

Causal representation learning seeks latent features that reveal underlying causal variables and mechanisms for robust prediction, intervention, and transfer.

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Root Cause Analysis Heterogeneous Federated Learning Industrial Anomaly Detection

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Root cause analysis (RCA) in networked industrial systems, such as supply chains and power networks, is notoriously difficult due to unknown and dynamically evolving interdependencies among geographically distributed clients. These clients represent heterogeneous physical processes and industrial assets equipped with sensors that generate large volumes of nonlinear, high-dimensional, and heterogeneous IoT data. Classical RCA methods require partial or full knowledge of the system's dependency graph, which is rarely available in these complex networks. While federated learning (FL) offers a natural framework for decentralized settings, most existing FL methods assume homogeneous feature spaces and retrainable client models. These assumptions are not compatible with our problem setting. Different clients have different data features and often run fixed, proprietary models that cannot be modified. This paper presents a federated cross-client interdependency learning methodology for feature-partitioned, nonlinear time-series data, without requiring access to raw sensor streams or modifying proprietary client models. Each proprietary local client model is augmented with a Machine Learning (ML) model that encodes cross-client interdependencies. These ML models are coordinated via a global server that enforces representation consistency while preserving privacy through calibrated differential privacy noise. RCA is performed using model residuals and anomaly flags. We establish theoretical convergence guarantees and validate our approach on extensive simulations and a real-world industrial cybersecurity dataset.

Abstract

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