FedAvg-Based CTMC Hazard Model for Federated Bridge Deterioration Assessment

Takato Yasuno

cs.LGFeb 22, 2026

FedAvg-Based CTMC Hazard Model for Federated Bridge Deterioration Assessment

Authors: Takato Yasuno

Organizations: 1Yachiyo Engineering Co., Ltd., Tokyo, Japan

Abstract

Bridge periodic inspection records contain sensitive information about public infrastructure, making cross-organizational data sharing impractical under existing data governance constraints. We propose a federated framework for estimating a Continuous-Time Markov Chain (CTMC) hazard model of bridge deterioration, enabling municipalities to collaboratively train a shared benchmark model without transferring raw inspection records. Each User holds local inspection data and trains a log-linear hazard model over three deterioration-direction transitions -- Good $\to$ Minor, Good $\to$ Severe, and Minor $\to$ Severe -- with covariates for bridge age, coastline distance, and deck area. Local optimization is performed via mini-batch stochastic gradient descent on the CTMC log-likelihood, and only a 12-dimensional pseudo-gradient vector is uploaded to a central server per communication round. The server aggregates User updates using sample-weighted Federated Averaging (FedAvg) with momentum and gradient clipping. All experiments in this paper are conducted on fully synthetic data generated from a known ground-truth parameter set with region-specific heterogeneity, enabling controlled evaluation of federated convergence behaviour. Simulation results across heterogeneous Users show consistent convergence of the average negative log-likelihood, with the aggregated gradient norm decreasing as User scale increases. Furthermore, the federated update mechanism provides a natural participation incentive: Users who register their local inspection datasets on a shared technical-standard platform receive in return the periodically updated global benchmark parameters -- information that cannot be obtained from local data alone -- thereby enabling evidence-based life-cycle planning without surrendering data sovereignty.

Figures & tables

Figure 1: Computation flow of the proposed FedAvg-CTMC framework for bridge deterioration benchmark estimation.

Figure 2: Four-panel scale comparison for 500, 2,000, and 4,000 Users. Top-left : NLL convergence curves. Top-right : Aggregated gradient norm in log scale. Bottom-left : Beta estimation MAE per transition (grouped bar). Bottom-right : Final-round NLL and $\|\bar{\mathbf{g}}\|$ with wall-clock time.

Metric	500	2,000	4,000
Transition pairs (total)	70,487	278,627	552,844
Users per round	50	200	400
Avg samples/round	$\sim$ 6,000	$\sim$ 28,000	$\sim$ 55,000
Round 1 Avg NLL	3.42	3.57	3.55
Round 50 Avg NLL	0.775	0.763	0.766
Round 50 $\\|\bar{\mathbf{g}}\\|$	0.354	0.175	0.140

Table 1: Scale comparison results (50 rounds, 10% participation rate).

Figure 3: Learning curves for the 4,000-User run (50 rounds, $\rho=10\%$ ). Top-left : Average NLL per communication round. Top-right : Aggregated gradient norm $\|\bar{\mathbf{g}}_{r}\|$ in log scale. Bottom-left : Trajectories of 8 $\beta$ coefficients for transitions $0\to 1$ and $0\to 2$ ; dashed lines show ground-truth values. Bottom-right : Trajectories of 4 $\beta$ coefficients for transition $1\to 2$ ; dashed lines show ground-truth values.

Transition	Coef	True	Learned	Error
$0\to 1$	$\beta_{0}$	$-2.000$	$-1.273$	$+0.727$
	$\beta_{1}$	$+0.500$	$-0.207$	$-0.707$
	$\beta_{2}$	$-0.300$	$-0.526$	$-0.226$
	$\beta_{3}$	$+0.100$	$-0.187$	$-0.287$
$0\to 2$	$\beta_{0}$	$-4.000$	$-2.666$	$+1.334$
	$\beta_{1}$	$+0.300$	$-1.302$	$-1.602$

Table 2: Learned vs. ground-truth

\boldsymbol{\beta}

(4,000-User run, round 50).

Scenario ( $z_{1},z_{2},z_{3}$ )	State 0	State 1
Young, far, small $(0.2,0.8,0.1)$	${\to}0$ : 0.570, ${\to}1$ : 0.398, ${\to}2$ : 0.032	${\to}1$ : 0.630, ${\to}2$ : 0.370
Mid-age, near, medium $(0.5,0.3,0.5)$	${\to}0$ : 0.535, ${\to}1$ : 0.438, ${\to}2$ : 0.027	${\to}1$ : 0.646, ${\to}2$ : 0.354
Old, near, large $(0.9,0.1,0.9)$	${\to}0$ : 0.562, ${\to}1$ : 0.425, ${\to}2$ : 0.013	${\to}1$ : 0.724, ${\to}2$ : 0.276

Table 3: Transition+ probabilities from the global model (4,000-User run,

\Delta t=3

yr).

Figure 4: 3 $\times$ 3 covariate-pair heatmaps of benchmark transition probabilities ( $\Delta t=3$ yr, 4,000-User model). Rows : transitions $0\to 1$ / $0\to 2$ / $1\to 2$ . Columns : covariate pairs $(z_{1},z_{2})$ / $(z_{1},z_{3})$ / $(z_{2},z_{3})$ ; the remaining covariate is fixed at 0.5. Colour scale is shared within each row; white contour lines mark iso-probability levels.

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Convergence-Aware Pareto Selection of Covariate Scaling Transformations for Markov Deterioration Hazard Models: Evidence from Bridge Inspection Data

Maximum likelihood estimation of Markov deterioration hazard models for infrastructure asset management is sensitive to the numerical conditioning of explanatory covariates, yet production pipelines often adopt a single scaling convention without systematic justification. We study four covariate scaling transformations---baseline max scaling, min-max scaling, z-score scaling, and Box-Cox transformation with a training-derived positivity shift---applied to an Exponential Hazard Markov (EHM) model estimated via L-BFGS-B on bridge inspection transition data.

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