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Geographically Regularized AUC-Maximizing Personalized Federated Learning
Mayu Hiraishi, Kensuke Tanioka, Toshio Shimokawa
TL;DR
Privacy constraints and institutional heterogeneity complicate clinical model development across healthcare institutions. GrAUC-PFL combines personalized federated learning, direct AUC optimization, and geographic graph regularization; simulations and a COVID-19 application report improved discriminative performance, with results varying across settings and counties.
Problem
Privacy and governance requirements can restrict patient-level data sharing, and institutional heterogeneity may limit the adequacy of a single global model.
Source: §1 · p. 1, §2.1 · p. 2Method
GrAUC-PFL keeps patient-level data local, directly optimizes a smooth AUC surrogate, and uses geographic graph regularization to encourage neighboring institutions’ coefficient vectors to be similar.
Source: §1 · p. 2, §3.1 · p. 4, §3.2 · pp. 5–6Results
Simulations and a real-world COVID-19 application demonstrated improved discriminative performance, with GrAUC-PFL achieving the highest mean AUC across counties but being outperformed by FedAvg in some counties.
Source: §7 · p. 18, §6 · p. 18Takeaways & Limitations
The findings support the potential benefit of geographic information sharing for personalized models, particularly where geographically related institutions have similar characteristics.
Source: §6 · p. 18, §7 · p. 18The approach assumes geographically proximate institutions tend to share similar model parameters, which may not always hold in practice.
Source: §7 · p. 18
Abstract
from arXiv · showhide
Accurate diagnostic and risk-prediction models are important for supporting clinical decision-making during infectious disease outbreaks. However, privacy and governance requirements may restrict patient-level data sharing across healthcare institutions, and data distributions often vary. Moreover, AUC is widely used to evaluate discriminative performance, motivating its direct optimization in model development. We propose geographically regularized AUC-maximizing personalized federated learning (GrAUC-PFL), which directly optimizes a smooth pairwise AUC surrogate to learn personalized models while keeping patient-level data local and accounting for institutional heterogeneity. Graph-based regularization encourages geographically neighboring institutions to have similar coefficient vectors while retaining a personalized models. Simulations and a real-data application suggest improved discriminative performance, particularly when geographically neighboring institutions have similar data-generating characteristics.
1 Introduction
Privacy and institutional heterogeneity complicate pooled clinical modeling, while geographic proximity may provide useful information for personalized learning. GrAUC-PFL combines direct AUC optimization, geographic graph regularization, and an ADMM-based algorithm.
1 Introduction
- Privacy and regulatory requirements can restrict patient-level data sharing, while institutional heterogeneity may make a single global model inadequate.Differences in patient populations, clinical practice, and evaluation criteria contribute to that heterogeneity.Source: §1 · p. 1
- GrAUC-PFL directly optimizes a smooth pairwise AUC surrogate and uses geographic graph regularization to encourage neighboring institutions’ model parameters to be similar while retaining personalized models.Geographic proximity provides a structural prior distinct from relationships inferred from local model updates or parameter similarities.Source: §1 · p. 2
- The framework uses an ADMM-based algorithm to handle the combination of a non-decomposable AUC objective and graph-based regularization.The algorithm extends the optimization framework of Liu et al. [2025].Source: §1 · p. 2
2 Related works
Personalized federated learning retains institution-specific models while using information from other institutions, addressing limits of a single global model. AUC measures positive-versus-negative ranking, and a logistic surrogate makes its optimization differentiable.
2.1 Personalized federated learning (PFL)
- A single global model may perform poorly across heterogeneous institutions, whereas personalized federated learning retains local models while leveraging information from others.Prior approaches share information through global models, client clusters, or relationships inferred from model updates; this paper instead uses geographic proximity as a prior.Source: §2.1 · pp. 2–3
Definition of AUC
- AUC is the probability that a randomly selected positive sample receives a higher score than a randomly selected negative sample.Under the linear scoring model, AUC is expressed through pairwise score comparisons and corresponds to the standardized Mann–Whitney U-statistic.Source: Definition of AUC · pp. 3–4
Surrogate function of AUC maximization
- Because the AUC indicator is discontinuous and unsuitable for gradient-based optimization, the study uses logistic loss as a smooth surrogate that preserves pairwise ranking structure.The surrogate enables gradient-based optimization.Source: Surrogate function of AUC maximization · p. 4
3 Geographically regularized AUC-maximizing personalized federated learning (GrAUC-PFL)
GrAUC-PFL trains institution-specific models from local data while using geographic relationships to share information and directly optimize AUC. Its graph-based penalty and ADMM updates support this objective while preserving personalized coefficients.
3.1 Framework of GrAUC-PFL
- Institutions retain local datasets and personalized models, while the server coordinates communication and graph constraints using model-related information.Patient-level data remain local during iterative exchanges.Source: §3.1 · pp. 4–5
3.2 Optimization problem of GrAUC-PFL
- The objective combines a pairwise logistic surrogate for AUC with a distance-weighted L1 penalty on coefficient differences along a minimum spanning tree.The MST connects all institutions with Z −1 edges; the penalty can set some neighboring coefficient differences to zero, forming coefficient clusters.Source: §3.1 · p. 4, §3.2 · pp. 5–6
3.3 Reformulation of GrAUC-PFL
- The MST incidence structure and Kronecker product express neighboring institutions’ coefficient differences as linear operators, then an auxiliary δ separates coupled terms for ADMM.The auxiliary variable represents edgewise coefficient differences, enabling subproblem decomposition.Source: §3.3 · pp. 6–7
3.4 Update β
- The β update approximates the loss with a quadratic majorizer, enabling local gradient calculations and reducing the update cost from O((Zp)^3) to O((Zp)^2).The global component is computed at the server; institution-specific loss gradients are computed locally.Source: §3.4 · p. 8
Update on server
The update procedure splits computation between server and institutions, then selects tuning parameters using stratified cross-validation. Server-side auxiliary-variable updates support the graph-regularized optimization.
Update at local institutions
- With H chosen as in Eq. (14), the β update avoids direct matrix inversion by reducing the inverse to a scalar multiple of the identity matrix.The update is divided between a server-side component and local calculations at institutions.Source: Update at local institutions · p. 9
3.5 Update δ
- The server updates δ with soft thresholding after β, using the edgewise coefficient differences and weights in the regularization term.δ represents the differences between connected institutions’ coefficients.Source: §3.5 · p. 9
3.6 Update γ
- The server updates γ using the deviation from the constraint, and Algorithm 1 iterates the β, δ, and γ updates.Source: §3.6 · p. 10
3.7 Selection of tuning parameters
- Stratified K-fold cross-validation selects λ, ρ, and wdist by maximizing mean validation AUC across institutions.Positive and negative proportions are approximately maintained in each fold.Source: §3.7 · p. 10
4 Numerical simulation
The simulations vary whether geographic groups align with mean structure, whether institutional training sizes differ, and whether heterogeneity is unrelated to geography. GrAUC-PFL performs best in geographically aligned settings, while its advantage narrows or reverses in other scenarios.
4.1 Simulation design
- The four simulations varied geographic alignment of mean structure, institutional training sizes, and whether institutional mean structure was unrelated to geography.Simulation 4 examined continuous institutional heterogeneity without a group structure.Source: §4.1 · p. 10, Simulation 2 · p. 12, Simulation 3 · p. 12
- Performance was evaluated using institution-level test AUC averaged across institutions, against local individual models and a shared-coefficient FedAvg model.The individual method independently maximized the AUC surrogate at each institution.Source: §4.1 · p. 11
4.2 Simulation results
- GrAUC-PFL achieved the highest mean AUC across all Simulation 1 patterns at both training sizes.Its performance improved as mean-structure differences increased, while FedAvg showed no notable change; at 300 training samples, individual models approached GrAUC-PFL, especially at higher prevalence.Source: Figure 3 · p. 13, §4.2 · p. 14, Figure 4 · p. 14
- In Simulation 2, GrAUC-PFL performed better in almost all patterns, while FedAvg consistently had the lowest performance.Estimation accuracy increased with higher signal and prevalence.Source: §4.2 · p. 15
- In Simulation 3, GrAUC-PFL was slightly better than individual AUC maximization at higher signal levels, while individual models performed well in other scenarios.The differences between methods in those other scenarios were not particularly large.Source: §4.2 · p. 15
- In Simulation 4, FedAvg was nearly tied with GrAUC-PFL at prevalence 0.3 and 0.25, and slightly outperformed it in the remaining patterns.These results were for 50 training samples per institution.Source: §4.2 · p. 15
5 Real data application
The real-data application evaluates GrAUC-PFL on county-level COVID-19 hospitalization data and finds higher mean AUC than the compared methods. County-level results vary, while estimated coefficient patterns cluster geographically.
5.1 Data description and Experimental Setting
- The application used 11,628 complete cases from a 100,000-record subset of public U.S. COVID-19 surveillance data, with hospitalization as the outcome.The analysis used six covariates and 18 counties; counties were treated as units, and geographic proximity was represented using a minimum spanning tree.Source: §5.1 · p. 15
- County training and test sets were created using outcome-stratified sampling, and tuning parameters were selected by stratified three-fold cross-validation.Approximately 20% of each outcome class was allocated to training, with adjustments for counties with few inpatients; performance was evaluated using mean county AUC.Source: §5.1 · p. 15
5.2 Results
- GrAUC-PFL achieved higher mean AUC than the compared methods, though county-level comparisons varied.It performed better in half of the counties, while FedAvg was better in Dallas, Orange, Nassau, and New York.Source: §5.2 · p. 16, Table 4 · p. 17, Table 5 · p. 17
- The coefficient heatmap's dendrogram grouped counties into East Coast, West Coast, and Florida clusters, with adjacent regions within the same state also grouped.The East Coast group included Nassau, Suffolk, New York, Kings, and Queens; the West Coast group included Riverside, Maricopa, San Diego, Los Angeles, and Orange; Florida included Broward and Palm Beach.Source: §5.2 · p. 17, Figure 6 · p. 17
6 Discussion
In simulations, GrAUC-PFL performed best when neighboring institutions shared similar data distributions, including with varied training sample sizes. In real COVID-19 data, it had the highest mean county AUC, though some county-level comparisons favored FedAvg.
6 Discussion
- GrAUC-PFL had superior performance when geographically neighboring institutions shared similar distributions, including when training sample sizes varied across institutions.When proximity did not reflect distribution similarity, it performed comparably to the individual method without substantial performance degradation in the simulations.Source: §6 · p. 18
- In the COVID-19 application, GrAUC-PFL achieved the highest mean AUC across counties and outperformed the compared methods in Los Angeles, Santa Clara, and Riverside, where prevalence was relatively low.It generally had comparable AUC where individual models performed best, while FedAvg outperformed it in some counties, including Nassau and Orange.Source: §6 · p. 18
7 Conclusion
The paper concludes that GrAUC-PFL directly maximizes AUC while incorporating geographic similarity in personalized federated models. Its key limitation is the assumption that nearby institutions tend to share similar model parameters.
7 Conclusion
- GrAUC-PFL combines direct AUC maximization with geographic similarity in a personalized federated learning framework.The framework leverages information from geographically related institutions without disclosing patient-level data externally.Source: §7 · p. 18
- Simulations and a real-data application demonstrated improved discriminative performance.Source: §7 · p. 18
- The method assumes geographically proximate institutions tend to share similar model parameters, an assumption that may not always hold.The authors identify alternative similarity measures and alternative AUC surrogate losses as directions for future work.Source: §7 · p. 18
A Proof of Lemma 1
The proof bounds the Hessian of the institution-level objective and uses that bound to establish Lipschitz continuity of its gradient.
A Proof of Lemma 1
- The proof derives the institution-level gradient and Hessian from the logistic surrogate and then establishes the Hessian's positive-semidefinite properties.Source: A Proof of Lemma 1 · pp. 21–22
- Averaging pairwise terms yields a Hessian bound expressed through the largest eigenvalue of Q_z.The proof uses positive-semidefinite ordering to bound the eigenvalues of the Hessian.Source: A Proof of Lemma 1 · pp. 21–22
- The institution-level gradient is μ_z-Lipschitz continuous, with L equal to the maximum μ_z across institutions.Source: A Proof of Lemma 1 · p. 22
B Proof of Theorem 2
The proof constructs a majorizing function for updating β and establishes conditions under which the update decreases the augmented Lagrangian. It also identifies a positive-definiteness condition for H, which makes the surrogate function convex.
B Proof of Theorem 2
- The proof derives the descent inequality by combining the smooth-function inequality with vector identities and bounds on quadratic terms.It applies the Rayleigh quotient theorem when simplifying the bound.Source: B Proof of Theorem 2 · pp. 23–24
- Eq. (17) ensures that the augmented Lagrangian decreases at each β update when the stated condition holds.The proof uses this decrease to establish the update behavior in Theorem 2.Source: B Proof of Theorem 2 · pp. 23–24
- H must be positive definite for the constructed function in Eq. (17) to be a valid upper-bounding function.The proof connects this requirement to convexity of the surrogate function.Source: B Proof of Theorem 2 · p. 24
- The eigenvalues of H are r − ρτςk(Ω⊤Ω), so r > ρτςmax(Ω⊤Ω) makes H positive definite.The proof identifies this as the condition for convexity of the surrogate function.Source: B Proof of Theorem 2 · p. 24
C Results of numerical simulations
The numerical simulations vary training sample sizes across institutions and compare settings with 50 and 300 training samples per institution. The supplied captions describe these settings but do not report simulation outcomes.
C Results of numerical simulations
- Simulation 2 uses different institution-level training sample sizes from Nz,train = {30, 50, 100, 200, 300}.Figure 7 presents the results for this varying-sample-size setting.Source: Figure 7 · p. 26
- Simulation 3 reports separate results for Nz,train = 50 and Nz,train = 300.The results are shown in Figures 8 and 9, respectively.Source: Figure 8 · p. 27, Figure 9 · p. 28
- Simulation 4 contrasts Nz,train = 50 in the left-column plots with Nz,train = 300 in the right-column plots.Figure 10 presents the results for these two sample-size settings.Source: Figure 10 · p. 29