Adaptive Bayesian Partner Selection for Federated Clinical Centers

# Adaptive Bayesian Partner Selection for Federated Clinical Centers
Source: [https://arxiv.org/html/2609.16446](https://arxiv.org/html/2609.16446)
Navid SeidiAffiliation:Department of Computer ScienceAffiliation:Missouri University of Science and TechnologyAffiliation:Rolla, MO, USAEmail:[nseidi@mst\.edu](mailto:)Satyaki RoyAffiliation:Department of Mathematical SciencesAffiliation:University of Alabama in HuntsvilleAffiliation:Huntsville, AL, USAEmail:[sr0215@uah\.edu](mailto:)Sajal K\. DasAffiliation:Department of Computer ScienceAffiliation:Missouri University of Science and TechnologyAffiliation:Rolla, MO, USAEmail:[sdas@mst\.edu](mailto:)

###### Abstract

Federated learning \(FL\) in healthcare is challenged by pronounced heterogeneity and temporal concept drift across clinical centers, where evolving patient populations and care practices shift data distributions\. Existing approaches rely on persistent global communication, incurring substantial bandwidth overhead while risking negative transfer from poorly aligned peers\. We address this by proposingAdaptive Bayesian Partner Selection\(ABPS\), a peer\-to\-peer framework that governs*who*collaborates,*when*, and at*what cost*\. Each center maintains a Beta–Bernoulli posterior over prospective peers’ Shapley marginal utility, ranks candidates using an Upper Confidence Bound \(UCB\) criterion, and forms collaborations through a lightweight propose–reject mechanism, with the option to abstain from communication when mutually beneficial interaction fails\. It admits a stochastic decision interpretation, yielding finite\-sample concentration guarantees and𝒪⁡\(κ​log⁡T\)\\mathcal\{O\}\(\\kappa\\log T\)regret in partner selection, along with conditions under which intentional isolation is optimal under negative transfer\. Lightweight extensions, head personalization, bfloat16 quantized communication, and tunable active\-set cardinality further improve efficiency, while a goal\-aware metadata filter enables institution\-specific collaboration strategies\. Evaluations on binary in\-hospital mortality prediction over the first 24 hours of an ICU stay, with 230 non\-IID clinical centers drawn from MIMIC\-IV, show that the fullABPS\-Xvariant matches the strongest federated baseline \(FedDyn, AUROC 0\.758\) at 0\.09× the communication cost of FedAvg, with reduced variability\. A diversity\-driven configuration activates intentional isolation for a substantial fraction of centers, highlighting the role of selective collaboration under heterogeneity\. These results show that adaptive, utility\-aware collaboration reduces communication without sacrificing accuracy when centers are numerous and small, providing a scalable paradigm for real\-world healthcare FL systems\.

## 1Introduction

Federated Learning \(FL\) has emerged as the dominant paradigm for training predictive models across decentralized centers without exposing sensitive patient data[McMahan et al\. \(2017\)](https://arxiv.org/html/2609.16446#bib.bib17), sidestepping Health Insurance Portability and Accountability Act \(HIPAA\)\-style data\-sharing constraints\. Realizing FL in clinical settings is nevertheless gated by two persistent obstacles: statistical heterogeneity \(i\.e\., non\-IID biomedical data\) across institutions[Hsu et al\. \(2019\)](https://arxiv.org/html/2609.16446#bib.bib20), and concept drift as patient demographics and clinical protocols evolve[Gama et al\. \(2014\)](https://arxiv.org/html/2609.16446#bib.bib8);[Rahimli et al\. \(2024\)](https://arxiv.org/html/2609.16446#bib.bib2)\(see Appendix[A](https://arxiv.org/html/2609.16446#A1)for an instance of distributional drift across admission eras in routinely charted vital signs\)\.

This heterogeneity is structural rather than incidental\. The healthcare ecosystem comprises three institution types with divergent key performance indicators \(KPIs\) and data profiles\.*Integrated delivery networks*\(Kaiser Permanente, HCA Healthcare\) aggregate large but internally siloed cohorts\.*Academic medical centers*see complex, rare, and trial\-driven cases at low volume\.*Community hospitals*generate the majority of high\-volume routine encounters\. No single ecosystem captures the full distribution required for generalizable modeling, yet forcing central FL aggregation across them frequently induces negative transfer that degrades local performance\. Peer\-to\-peer \(P2P\) alternatives[Guha Roy et al\. \(2019\)](https://arxiv.org/html/2609.16446#bib.bib28);[Hegedűs et al\. \(2021\)](https://arxiv.org/html/2609.16446#bib.bib29)avoid the single point of failure but still face the question of*which*peers a center should collaborate with and at what bandwidth cost\. Contributions\.To achieve predictive accuracy under concept drift, we introduceAdaptive Bayesian Partner Selection\(ABPS\), a serverless P2P federated learning framework in which each center adaptively selects collaborators based on a Bayesian estimate of their utility\.ABPSmodels each candidate peer’s Shapley marginal contribution via a Beta–Bernoulli posterior, ranks peers using anϵ\\epsilon\-greedy UCB policy, and employs a propose–reject protocol with an explicit*rest*action to mitigate negative transfer\. We establish three guarantees: posterior concentration \(Theorem[1](https://arxiv.org/html/2609.16446#Thmtheorem1)\), sublinear regret of order𝒪⁡\(κ​log⁡T\)\\mathcal\{O\}\(\\kappa\\log T\)\(Theorem[2](https://arxiv.org/html/2609.16446#Thmtheorem2)\), and the Bayes\-optimality of isolation \(Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)\)\. The core framework comprises head personalization, bfloat16[Kalamkar et al\. \(2019\)](https://arxiv.org/html/2609.16446#bib.bib31)quantized communication, and a tunable active\-set cardinalityκ\\kappa\. In addition, a goal\-aware metadata pre\-filter allows each center to prioritize homogeneity, diversity, or KPI alignment objectives\. We denote the maximally\-extended configuration asABPS\-X, which composes all of the above with deeper22\-layer head personalization, server\-side exponential moving average \(EMA\) momentum, and validation\-AUROC early stopping \(full hyperparameters in Sec\.[6\.4](https://arxiv.org/html/2609.16446#S6.SS4)\)\. On MIMIC\-IV v3\.1, withn=230n\{=\}230non\-IID careunit\-by\-year centers,ABPS\-Xmatches the strongest federated baseline \(FedDyn\) while using only0\.09×0\.09\\timesthe bandwidth of FedAvg\. In contrast, applying the same extensions to FedAvg degrades performance by6\.46\.4AUROC points, leavingABPS\-Xahead of FedAvg\-X by6\.76\.7points \(Table[1](https://arxiv.org/html/2609.16446#S6.T1)\), thereby isolating the gains to Bayesian partner\-selection\.

## 2Related Work

Our research intersects with several active domains in decentralized machine learning\. We organize the related work below in the order in which the corresponding openQuestion is answered in the body of the paper, ranging from divergence\-aware partner selection to peer\-to\-peer \(P2P\) federated architectures, concept\-drift adaptation, and communication\-efficient FL\.

### 2\.1Divergence Metrics for Non\-IID Collaboration

Information\-theoretic divergences \(Kullback\-Leibler, hereafter KL, and Wasserstein\) quantify inter\-client distributional disparity in FL[Németh et al\. \(2025\)](https://arxiv.org/html/2609.16446#bib.bib6)\. They appear as regularization penalties that align local models without raw data exposure[Németh et al\. \(2025\)](https://arxiv.org/html/2609.16446#bib.bib6), as client\-selection utilities that favor either stabilizing or diversity\-injecting clients[Düsing and Cimiano \(2026\)](https://arxiv.org/html/2609.16446#bib.bib7);[Rahad et al\. \(2025\)](https://arxiv.org/html/2609.16446#bib.bib5), and as theoretical tools for bounding FL generalization under heterogeneity\. This philosophy extends to the wire\-format level: each center can be summarized by a non\-PHI descriptor of its distribution \(e\.g\., positive\-class rate, training\-set size, optional Charlson\-comorbidity prevalence\), and peers can be admitted via a goal\-dependent monotone function of pairwise similarity\. A key limitation is that divergence is treated as a single scalar objective fixed network\-wide, without allowing individual centers to declare their own collaboration goal before model exchange\. Yet healthcare institution types \(Integrated Delivery Networks, Academic Medical Centers, Community Hospitals\) require different collaboration objectives, including homogeneity, diversity, or KPI alignment[Li et al\. \(2025\)](https://arxiv.org/html/2609.16446#bib.bib34)\. We label this open problemQ1\(goal\-heterogeneous collaboration\)and address it through the goal\-aware metadata pre\-filter of Sec\.[4\.1](https://arxiv.org/html/2609.16446#S4.SS1), where a center selects one of three admission rules, namely, cosine, anti\-cosine, and KPI matching, at runtime without risking exposing model parameters\.

### 2\.2P2P Federated Learning and Partner Selection

Peer\-to\-peer \(P2P\) FL decentralizes aggregation, eliminating single points of failure and reducing communication bottlenecks[Hegedűs et al\. \(2021\)](https://arxiv.org/html/2609.16446#bib.bib29);[Zhou et al\. \(2024\)](https://arxiv.org/html/2609.16446#bib.bib26)\. However, identifying the appropriate collaboration topology under non\-IID heterogeneity remains an open challenge\. Early approaches relied on gossip\-based protocols[Hegedűs et al\. \(2021\)](https://arxiv.org/html/2609.16446#bib.bib29), while more recent work studies the security of P2P training against backdoor attacks[Syros et al\. \(2024\)](https://arxiv.org/html/2609.16446#bib.bib1)and robustness mechanisms against free\-riding and collusion[Augello et al\. \(2024\)](https://arxiv.org/html/2609.16446#bib.bib16);[Ranjan et al\. \(2022\)](https://arxiv.org/html/2609.16446#bib.bib41)\. Within the multi\-armed bandit \(MAB\) framework, CS\-UCB[Xia et al\. \(2020\)](https://arxiv.org/html/2609.16446#bib.bib14)introduced UCB\-style scheduling in server\-mediated FL, and subsequent work extended this idea to decentralized settings with non\-stationary MAB formulations over peer groups[Listo Zec et al\. \(2024\)](https://arxiv.org/html/2609.16446#bib.bib43)\. Complementary approaches based on Shapley value estimate client contributions for server\-side selection[Singhal et al\. \(2024\)](https://arxiv.org/html/2609.16446#bib.bib15);[Yang et al\. \(2024\)](https://arxiv.org/html/2609.16446#bib.bib42)\. A second limitation persists across P2P, MAB\-, and Shapley\-based methods: all assume mandatory participation in each round\. Even decentralized variants enforce collaboration once the topology is established, despite evidence that aggregation across heterogeneous institutions can degrade local performance below the no\-collaboration baseline[Crowson et al\. \(2022\)](https://arxiv.org/html/2609.16446#bib.bib33);[Li et al\. \(2025\)](https://arxiv.org/html/2609.16446#bib.bib34)\. We define this asQ2\(selective collaboration under heterogeneity\)and address it through the Bayesian propose–reject mechanism \(discussed in Sec\.[4\.3](https://arxiv.org/html/2609.16446#S4.SS3)\), allowing each center to abstain from participation when all posterior UCB scores fall below the acceptance thresholdτacc\\tau\_\{\\mathrm\{acc\}\}\.

### 2\.3Concept Drift and Bayesian Adaptation

Clinical data distributions evolve due to changes in patient demographics, treatment protocols, and clinical infrastructure, leading to concept drift[Rahimli et al\. \(2024\)](https://arxiv.org/html/2609.16446#bib.bib2)\. Bayesian methods quantify the resulting uncertainty, with two dominant approaches in prior work\. The first models uncertainty over*model parameters*, where Bayesian neural networks in FL aggregate posterior distributions rather than point estimates[Saile et al\. \(2024\)](https://arxiv.org/html/2609.16446#bib.bib3);[Rahman et al\. \(2025\)](https://arxiv.org/html/2609.16446#bib.bib4), often incorporating hierarchical or personalized updates to adapt to local data\. The second leverages uncertainty for*update filtering*, where client contributions are selectively incorporated based on confidence measures, such as credible\-interval thresholds[Iglesias Jr\. et al\. \(2024\)](https://arxiv.org/html/2609.16446#bib.bib40)\. Despite their effectiveness, both approaches share a key limitation: abstaining from collaboration is not treated as a principled decision with formal guarantees\. In practice, concept drift can render collaboration rounds detrimental, yet existing methods rarely allow nodes to opt out in a theoretically grounded manner[Crowson et al\. \(2022\)](https://arxiv.org/html/2609.16446#bib.bib33);[Gama et al\. \(2014\)](https://arxiv.org/html/2609.16446#bib.bib8);[Rahimli et al\. \(2024\)](https://arxiv.org/html/2609.16446#bib.bib2)\. We define this asQ3\(adaptive isolation under drift\)and address it through an explicit rest action in Sec\.[4\.3](https://arxiv.org/html/2609.16446#S4.SS3), supported by a Bayes\-optimal isolation criterion \(Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)\) that provides a closed\-form condition under which abstention strictly outperforms collaboration\.

### 2\.4Personalization, Quantization, and Communication\-Efficient FL

To achieve personalization and bandwidth minimization, it is necessary to keep a subset of model parameters local per client: FedPer and FedRep retain the classifier head locally[Arivazhagan et al\. \(2019\)](https://arxiv.org/html/2609.16446#bib.bib12)\. pFedHN and Ditto provide other per\-client adaptations\. Quantized transmission compresses the wire format: LLM\.int8[Dettmers et al\. \(2022\)](https://arxiv.org/html/2609.16446#bib.bib13), QSGD, and signSGD demonstrate single\-digit\-percent accuracy loss with22\-16×16\\timesbandwidth reduction\. Server momentum \(FedAvgM\) stabilizes aggregation under heterogeneity\. These ideas are orthogonal to partner selection, and in principle, they can be composed with any partner\-selection core to push the accuracy\-bandwidth frontier further\. A fourth limitation runs across all three of these families: bandwidth\-saving mechanisms have been studied in isolation from*which*peers a center should engage with\. Real\-world multi\-institutional medical FL still incurs prohibitive bandwidth[Haripriya et al\. \(2025\)](https://arxiv.org/html/2609.16446#bib.bib37), and communication\-efficient extensions such as knowledge distillation[Wu et al\. \(2022\)](https://arxiv.org/html/2609.16446#bib.bib36)and metaheuristic aggregation[Abdolmaleki and Farahani \(2026\)](https://arxiv.org/html/2609.16446#bib.bib35)treat compression and topology choice as separate problems\. We label this open problemQ4\(communication overhead, even in P2P\)and answer it through the additive head\-personalization, bfloat16\-quantization, and tunable active\-set cardinalityκ\\kappa, optimized jointly on the accuracy\-bandwidth Pareto frontier \(Sec\.[6\.4](https://arxiv.org/html/2609.16446#S6.SS4), with the row\-by\-row ablation in Sec\.[6\.5](https://arxiv.org/html/2609.16446#S6.SS5)\)\.

## 3Problem Formulation

Goal\.We seek to learn, for each ofnnfederated centers, a sequence of local predictors that*maximizes predictive accuracy*on a center’s evolving data distribution while*minimizing the cumulative communication overhead*of inter\-center collaboration,*under continuous concept drift*in patient demographics, treatment protocols, and equipment\.Predictive accuracyis task\-dependent and enters the framework through a per\-center utility functionalUi∈\[0,1\]U\_\{i\}\\in\[0,1\]\.UiU\_\{i\}is the per\-center Area Under the Receiver Operating Characteristic curve \(AUROC\) for binary in\-hospital mortality classification\. At each roundt∈\{1,…,R\}t\\in\\\{1,\\dots,R\\\}, every centeri∈\{1,…,n\}i\\in\\\{1,\\dots,n\\\}holds local data drawn from an unknown, time\-varying joint distributionPt\(i\)​\(X,Y\)P^\{\(i\)\}\_\{t\}\(X,Y\)shaped by the center’s patient demographics, comorbidity profile, and care protocols, observed only through patient\-level samples, fits local parameters𝐰i,t∈ℝ\|W\|\\mathbf\{w\}\_\{i,t\}\\in\\mathbb\{R\}^\{\|W\|\}, and may exchange them with a per\-round active peer set𝒫i,t⊆\{1,…,n\}∖\{i\}\\mathcal\{P\}\_\{i,t\}\\subseteq\\\{1,\\dots,n\\\}\\setminus\\\{i\\\}\. We writeAi​\(𝐰i,t\):=𝔼\(x,y\)∼Pt\(i\)​\[AUROC⁡\(𝐰i,t,x,y\)\]A\_\{i\}\(\\mathbf\{w\}\_\{i,t\}\):=\\mathbb\{E\}\_\{\(x,y\)\\sim P^\{\(i\)\}\_\{t\}\}\\\!\\left\[\\mathrm\{AUROC\}\(\\mathbf\{w\}\_\{i,t\};x,y\)\\right\]for the expected accuracy ofii’s model and treat𝒫i,t=∅\\mathcal\{P\}\_\{i,t\}=\\emptysetas a legal first\-class action \(*intentional rest*, formalized in Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)\)\.ABPSjointly chooses, for every center and every round, the active peer set and the local update rule, solving

max\{𝒫i,t,𝐰i,t\}⁡1n​∑i=1n1R​∑t=1RAi​\(𝐰i,t\)⏟predictive accuracy​subject to​Ctotal≤B⏟bandwidth budget,supm,kD\[P\(i\)m,k∥P\(i\)m,k−1\]≤ϵm⏟drift bound at scale​m\.\\max\_\{\\\{\\mathcal\{P\}\_\{i,t\},\\,\\mathbf\{w\}\_\{i,t\}\\\}\}\\;\\underbrace\{\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\frac\{1\}\{R\}\\sum\_\{t=1\}^\{R\}A\_\{i\}\(\\mathbf\{w\}\_\{i,t\}\)\}\_\{\\text\{predictive accuracy\}\}\\;\\;\\;\\text\{subject to\}\\;\\;\\;\\underbrace\{C\_\{\\mathrm\{total\}\}\\leq B\}\_\{\\text\{bandwidth budget\}\},\\;\\;\\underbrace\{\\textstyle\\sup\_\{m,k\}\\,D\\\!\\left\[P^\{\(i\)\}\_\{m,k\}\\,\\\|\\,P^\{\(i\)\}\_\{m,k\-1\}\\right\]\\leq\\epsilon\_\{m\}\}\_\{\\text\{drift bound at scale \}m\}\.\(1\)The maximand is the across\-center, across\-round mean accuracy\.Ctotal≤BC\_\{\\mathrm\{total\}\}\\leq Bcaps cumulative communication \(Eq\.[2](https://arxiv.org/html/2609.16446#S3.E2)below\)\. The drift constraint bounds the per\-window divergence by a scale\-specific toleranceϵm\\epsilon\_\{m\}at each temporal scalem∈\{1,2,3\}m\\in\\\{1,2,3\\\}\. When a realized shift exceedsϵm\\epsilon\_\{m\}, A3 \(Sec\.[5](https://arxiv.org/html/2609.16446#S5)\) breaks at scalemm, the Bayesian posterior decays toward the prior, the UCB rankings re\-order, and the propose\-reject mechanism re\-forms𝒫i,t\+1\\mathcal\{P\}\_\{i,t\+1\}\. Equation[1](https://arxiv.org/html/2609.16446#S3.E1)thus makes the three competing forces explicit: accuracy as the maximand, bandwidth as the budget, and drift as the constraint\. Communication cost\.Let𝒫t=\{\(i,j\):j∈𝒫i,t\}\\mathcal\{P\}\_\{t\}=\\\{\(i,j\):j\\in\\mathcal\{P\}\_\{i,t\}\\\}denote the active peer\-pair set in roundtt,\|Wi\|\|W\_\{i\}\|the size of centerii’s parameter vector, andcegressc\_\{\\mathrm\{egress\}\}the per\-GB egress cost \(e\.g\.,$0\.09\\$0\.09/GB on AWS[Amazon Web Services \(2024\)](https://arxiv.org/html/2609.16446#bib.bib18)\)\. It is worth mentioning here that continuously broadcasting parameters across100,000\+100\{,\}000\{\+\}global healthcare centers typically incurs a prohibitive bandwidth footprint[McMahan et al\. \(2017\)](https://arxiv.org/html/2609.16446#bib.bib17), and recent medical\-FL benchmarks[Haripriya et al\. \(2025\)](https://arxiv.org/html/2609.16446#bib.bib37)confirm that one5050\-round sweep of a VGG\-16\-class model exceeds276,000276\{,\}000MB of cross\-client traffic\. Distillation[Wu et al\. \(2022\)](https://arxiv.org/html/2609.16446#bib.bib36)and bandwidth\-aware aggregation[Abdolmaleki and Farahani \(2026\)](https://arxiv.org/html/2609.16446#bib.bib35)cut these figures, but neither couples compression to peer choice\. The cumulative cost acrossRRrounds is

Ctotal=cegress​∑t=1R∑\(i,j\)∈𝒫t\(\|Wi\|\+\|Wj\|\)\.C\_\{\\mathrm\{total\}\}\\;=\\;c\_\{\\mathrm\{egress\}\}\\sum\_\{t=1\}^\{R\}\\sum\_\{\(i,j\)\\in\\mathcal\{P\}\_\{t\}\}\\bigl\(\|W\_\{i\}\|\+\|W\_\{j\}\|\\bigr\)\.\(2\)In Eq\.[2](https://arxiv.org/html/2609.16446#S3.E2), the summand accounts for the bidirectional transfer of parameter aggregation\. Under the uniform\-architecture specialization\|Wi\|=\|W\|\|W\_\{i\}\|\{=\}\|W\|, this reduces toCtotal=2​cegress​\|W\|​∑t\|𝒫t\|C\_\{\\mathrm\{total\}\}=2\\,c\_\{\\mathrm\{egress\}\}\\,\|W\|\\,\\sum\_\{t\}\|\\mathcal\{P\}\_\{t\}\|, the form used in Theorem[2](https://arxiv.org/html/2609.16446#Thmtheorem2)and Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)\.

CVICU \(pos¯≈3\.3%\\overline\{\\text\{pos\}\}\\approx 3\.3\\%\)SICU \(pos¯≈10\.6%\\overline\{\\text\{pos\}\}\\approx 10\.6\\%\)TSICU \(pos¯≈10\.8%\\overline\{\\text\{pos\}\}\\approx 10\.8\\%\)CCU \(pos¯≈13\.0%\\overline\{\\text\{pos\}\}\\approx 13\.0\\%\)Medical/Surgical ICU \(pos¯≈14\.1%\\overline\{\\text\{pos\}\}\\approx 14\.1\\%\)MICU \(pos¯≈15\.4%\\overline\{\\text\{pos\}\}\\approx 15\.4\\%\)2110213021602188admission year \(MIMIC\-IV v3\.1 privacy\-shifted→\\to2\-year windows==centers\)intra\-careunit temporal drift \(levelh2h\_\{2\}\), bounded byϵ2\\epsilon\_\{2\}inter\-careunitdivergence\(levelh3h\_\{3\}\)cell shade∝\\proptoper\-center mortality rate:3%8%13%18%levelh1h\_\{1\}\(within\-window noise,ϵ1\\epsilon\_\{1\}\): per\-round local\-SGD stochasticity, not depicted\.

Figure 1:Hierarchical temporal\-shift model, illustrated on MIMIC\-IV v3\.1 as one concrete instantiation\. The construction is dataset\-agnostic and applies to any longitudinal multi\-center cohort\. In this instantiation each cell is one ofn=230n\{=\}230federated centers \(careunit×\\times2\-year window\)\. Cell shade encodes in\-hospital mortality rate, and rows sort six careunits by mean mortality\. Horizontal variation within a row illustrates the level\-h2h\_\{2\}within\-group term bounded byϵ2\\epsilon\_\{2\}; because the windows are shifted\-year buckets rather than calendar years, on this axis that variation is of the order of sampling variation \(Sec\.[6](https://arxiv.org/html/2609.16446#S6)\)\. Vertical shade jumps show level\-h3h\_\{3\}inter\-group divergence that goal\-aware pre\-filter \(Sec\.[4\.1](https://arxiv.org/html/2609.16446#S4.SS1)\) reasons about without exchanging model parameters\.Hierarchical concept drift\.The drift constraint of Eq\.[1](https://arxiv.org/html/2609.16446#S3.E1)bounds, at three temporal scalesm∈\{1,2,3\}m\\in\\\{1,2,3\\\}\(short\-term operational, medium\-term seasonal, long\-term demographic\), the inter\-window divergenceD\[Pm,k\(i\)∥Pm,k−1\(i\)\]≤ϵmD\[P^\{\(i\)\}\_\{m,k\}\\,\\\|\\,P^\{\(i\)\}\_\{m,k\-1\}\]\\leq\\epsilon\_\{m\}between consecutive windows of lengthΔ​tm\\Delta t\_\{m\}\*\*\(Figure[1](https://arxiv.org/html/2609.16446#S3.F1)\)\*\*[Gama et al\. \(2014\)](https://arxiv.org/html/2609.16446#bib.bib8);[Rahimli et al\. \(2024\)](https://arxiv.org/html/2609.16446#bib.bib2);[Cuturi and Blondel \(2017\)](https://arxiv.org/html/2609.16446#bib.bib9);[Düsing and Cimiano \(2026\)](https://arxiv.org/html/2609.16446#bib.bib7);[Rahad et al\. \(2025\)](https://arxiv.org/html/2609.16446#bib.bib5)\. The window sizeΔ​tm\\Delta t\_\{m\}admits an online empirical estimator, e\.g\., a Welchtt\-test that expands the window while\|μt−μt\+k\|/σt2/Nt\+σt\+k2/Nt\+k\|\\mu\_\{t\}\-\\mu\_\{t\+k\}\|/\\sqrt\{\\sigma\_\{t\}^\{2\}/N\_\{t\}\+\\sigma\_\{t\+k\}^\{2\}/N\_\{t\+k\}\}stays belowtcritt\_\{\\mathrm\{crit\}\}atα=0\.05\\alpha\{=\}0\.05\. \(Note that during experimental validation, we fixΔ​t2=2\\Delta t\_\{2\}\{=\}2years and leave adaptive estimation of\{Δ​tm\}\\\{\\Delta t\_\{m\}\\\}to future work, since online window adaptation is orthogonal to the partner\-selection contribution we focus on here, and Theorem[1](https://arxiv.org/html/2609.16446#Thmtheorem1)already bounds the within\-window cost explicitly throughϵm\\epsilon\_\{m\}, so the headline accuracy\-bandwidth claim is unaffected by any reasonable choice ofΔ​t2\\Delta t\_\{2\}\.\)

## 4Methodology: Adaptive Bayesian Partner Selection

1\. Goal\-Aware Filterfi​\(𝐯i,𝐯j\)≥τsimf\_\{i\}\(\\mathbf\{v\}\_\{i\},\\mathbf\{v\}\_\{j\}\)\\\!\\geq\\\!\\tau\_\{\\mathrm\{sim\}\}2\. Topologyϵ\\epsilon\-greedy UCB propose/reject3\. UtilityShapleyϕj→i\\phi\_\{j\\to i\}4\. Beta UpdateBeta⁡\(α\+ϕ¯,β\+1−ϕ¯\)\\mathrm\{Beta\}\(\\alpha\{\+\}\\bar\{\\phi\},\\beta\{\+\}1\{\-\}\\bar\{\\phi\}\)Rest𝒫i=∅\\mathcal\{P\}\_\{i\}=\\varnothingall reject

Figure 2:ABPSround\-level pipeline\. Centers prune candidates via a goal\-aware metadata filter \(Eq\.[3](https://arxiv.org/html/2609.16446#S4.E3)\), propose to peers ranked by UCB on Beta belief, observe Shapley marginal utility, and update the belief\. When all proposals are rejected, the center rests with𝒫i=∅\\mathcal\{P\}\_\{i\}=\\varnothing\(Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)\)\.ABPSmodels inter\-center collaboration as a probabilistic, temporally adaptive process built from Bayesian\-core components, a goal\-aware metadata pre\-filter \(Sec\.[4\.1](https://arxiv.org/html/2609.16446#S4.SS1)\), a Beta\-Bernoulli posterior over each peer’s marginal utility \(Sec\.[4\.2](https://arxiv.org/html/2609.16446#S4.SS2)\), anϵ\\epsilon\-greedy UCB propose\-reject topology rule \(Sec\.[4\.3](https://arxiv.org/html/2609.16446#S4.SS3)\), and a*rest*action when no proposal is accepted, and three communication\-efficiency extensions that compose additively on top of the core, namely, head personalization, bfloat16 quantization, and tunableκ\\kappa, are all discussed hereafter and analyzed in Sec\.[6\.4](https://arxiv.org/html/2609.16446#S6.SS4)\)\. Figure[2](https://arxiv.org/html/2609.16446#S4.F2)summarizes the round\-level pipeline, and the full pseudocode is given in Algorithm[1](https://arxiv.org/html/2609.16446#alg1)in Appendix[B](https://arxiv.org/html/2609.16446#A2)\.

### 4\.1Goal\-Aware Metadata Pre\-Filter

*To addressQ1on goal\-heterogeneous collaboration from Sec\.[2](https://arxiv.org/html/2609.16446#S2),*before invoking the expensive Shapley\-UCB stage, each centeriicomputes add\-dimensional summary metadata vector𝐯i∈ℝd\\mathbf\{v\}\_\{i\}\\in\\mathbb\{R\}^\{d\}of non\-sensitive aggregates \(positive\-class rate,log⁡ntrain\\log n\_\{\\mathrm\{train\}\}, normalized mean year of admission, and for goal\-aware variants a1212\-dim Charlson\-comorbidity prevalence\)\. Every entry is a cohort\-level scalar, not a per\-patient feature, so𝐯i\\mathbf\{v\}\_\{i\}carries no Protected Health Information\. No model parameters cross the network at this stage, consistent with HIPAA and inter\-institutional data\-sharing constraints\. Goal\-aware admission rule\.Real healthcare federations are*not*uniformly similarity\-seeking: the three institution types in Sec\.[1](https://arxiv.org/html/2609.16446#S1)have distinct objectives\. A community hospital prefers*homogeneity*, collaborating with demographically similar peers to improve transfer learning for under\-represented cohorts\. An academic medical center seeks*diversity*, leveraging exposure to heterogeneous case\-mix\. In contrast, an Integrated Delivery Network emphasizes*alignment*, selecting peers with matching target KPIs \(e\.g\., mortality rates\) to meet system\-level objectives\. We generalize the admission rule into a per\-center goal\-scoring functionfi:ℝd×ℝd→ℝf\_\{i\}:\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}:

fi​\(𝐯i,𝐯j\)=\{cos⁡\(𝐯i,𝐯j\),goal=homogeneity−cos⁡\(𝐯i,𝐯j\),goal=diversity−\|vi,0−vj,0\|,goal=alignmentf\_\{i\}\(\\mathbf\{v\}\_\{i\},\\mathbf\{v\}\_\{j\}\)\\;=\\;\\begin\{cases\}\\cos\(\\mathbf\{v\}\_\{i\},\\mathbf\{v\}\_\{j\}\),&\\text\{goal\}=\\mathrm\{homogeneity\}\\\\ \-\\cos\(\\mathbf\{v\}\_\{i\},\\mathbf\{v\}\_\{j\}\),&\\text\{goal\}=\\mathrm\{diversity\}\\\\ \-\|v\_\{i,0\}\-v\_\{j,0\}\|,&\\text\{goal\}=\\mathrm\{alignment\}\\end\{cases\}\(3\)In the above equation,v⋅,0v\_\{\\cdot,0\}is the pos\-rate coordinate used as the KPI marker\. Peerjjis admitted to𝒞i\\mathcal\{C\}\_\{i\}ifffi​\(𝐯i,𝐯j\)≥τsimf\_\{i\}\(\\mathbf\{v\}\_\{i\},\\mathbf\{v\}\_\{j\}\)\\geq\\tau\_\{\\mathrm\{sim\}\}\. Each formulation lives on a different scale, soτsim\\tau\_\{\\mathrm\{sim\}\}is auto\-calibrated per goal to retain a fixed fraction \(we use25%25\\%\) of candidate pairs: this makes goal choices directly comparable\. The threshold can equivalently be learned online via an exponential moving average \(EMA\) on the acceptance rate\. The static calibration suffices for our experiments\. Complexity and theoretical preservation\.The pre\-filter is𝒪⁡\(n​d\)\\mathcal\{O\}\(nd\)per round and transmits only𝐯i\\mathbf\{v\}\_\{i\}\(a few dozen bytes\) per center\. All downstream Shapley evaluations operate on the filtered arm set, reducing the expected per\-round bandwidth by the same fraction\. Theorems[1](https://arxiv.org/html/2609.16446#Thmtheorem1)and[2](https://arxiv.org/html/2609.16446#Thmtheorem2)transfer verbatim once the arm set is restricted to𝒞i\\mathcal\{C\}\_\{i\}\. The only added bias is whether the true best peer survives the filter, which we bound in Appendix[C](https://arxiv.org/html/2609.16446#A3)via a standard top\-kkcoverage argument for descriptor\-based recall\. Empirically \(Sec\.[6\.4](https://arxiv.org/html/2609.16446#S6.SS4)\), the diversity goal activates Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)’s\|𝒫i\|=0\|\\mathcal\{P\}\_\{i\}\|=0regime in∼40%\\sim 40\\%of rounds on the biomedical dataset, the first empirical observation of intentional isolation in our study\.

### 4\.2Bayesian Belief Updating on Marginal Utility

Rather than maintaining a belief over a peer’s explicitly shared raw parameters, centeriievaluates the actual*marginal utility*\(performance gain\) a peerjjprovides\. We model this utility using the Shapley value to objectively allocate credit across historical collaboration subsets𝒮⊆𝒫i∖\{j\}\\mathcal\{S\}\\subseteq\\mathcal\{P\}\_\{i\}\\setminus\\\{j\\\}:

ϕj→i=∑𝒮⊆𝒫i∖\{j\}\|𝒮\|\!​\(\|𝒫i\|−\|𝒮\|−1\)\!\|𝒫i\|\!​\(Ui​\(𝒮∪\{j\}\)−Ui​\(𝒮\)\),\\phi\_\{j\\to i\}=\\sum\_\{\\mathcal\{S\}\\subseteq\\mathcal\{P\}\_\{i\}\\setminus\\\{j\\\}\}\\frac\{\|\\mathcal\{S\}\|\!\\,\(\|\\mathcal\{P\}\_\{i\}\|\-\|\\mathcal\{S\}\|\-1\)\!\}\{\|\\mathcal\{P\}\_\{i\}\|\!\}\\left\(U\_\{i\}\(\\mathcal\{S\}\\cup\\\{j\\\}\)\-U\_\{i\}\(\\mathcal\{S\}\)\\right\),\(4\)In the above equation,Ui​\(⋅\)∈\[0,1\]U\_\{i\}\(\\cdot\)\\in\[0,1\]is the local model’s normalized held\-out validation performance \(we use AUROC\)\. BecauseUiU\_\{i\}is bounded in\[0,1\]\[0,1\], the per\-peer Shapley contributionϕj→i\\phi\_\{j\\to i\}is bounded in\[−1,\+1\]\[\-1,\+1\]\. We map it to the unit interval via the affine clip

ϕ¯j→i=clip\[0,1\]​\(ϕj→i−ϕminϕmax−ϕmin\),\\bar\{\\phi\}\_\{j\\to i\}\\;=\\;\\mathrm\{clip\}\_\{\[0,1\]\}\\\!\\left\(\\frac\{\\phi\_\{j\\to i\}\-\\phi\_\{\\min\}\}\{\\phi\_\{\\max\}\-\\phi\_\{\\min\}\}\\right\),\(5\)with truncation thresholdsϕmin,ϕmax\\phi\_\{\\min\},\\phi\_\{\\max\}chosen so that a single anomalous round cannot saturate the posterior \(we use\[−0\.1,\+0\.1\]\[\-0\.1,\+0\.1\]throughout this paper\)\. Modelingϕ¯j→i\\bar\{\\phi\}\_\{j\\to i\}as a soft Bernoulli observation, the conjugate Beta priorϕ¯j→i∼Beta⁡\(αi→j,βi→j\)\\bar\{\\phi\}\_\{j\\to i\}\\sim\\mathrm\{Beta\}\(\\alpha\_\{i\\to j\},\\beta\_\{i\\to j\}\)admits the closed\-form update

αi→j\(k\+1\)←αi→j\(k\)\+ϕ¯j→i,βi→j\(k\+1\)←βi→j\(k\)\+\(1−ϕ¯j→i\)\.\\alpha\_\{i\\to j\}^\{\(k\+1\)\}\\leftarrow\\alpha\_\{i\\to j\}^\{\(k\)\}\+\\bar\{\\phi\}\_\{j\\to i\},\\hskip 17\.00024pt\\beta\_\{i\\to j\}^\{\(k\+1\)\}\\leftarrow\\beta\_\{i\\to j\}^\{\(k\)\}\+\(1\-\\bar\{\\phi\}\_\{j\\to i\}\)\.\(6\)The Beta\-Bernoulli formulation offers three properties\. \(i\) The posterior support\[0,1\]\[0,1\]matches the bounded co\-domain ofϕ¯j→i\\bar\{\\phi\}\_\{j\\to i\}, ruling out the model mis\-specification that would otherwise arise when Shapley contributions are negative\. \(ii\) The posterior varianceα​β/\[\(α\+β\)2​\(α\+β\+1\)\]\\alpha\\beta/\[\(\\alpha\{\+\}\\beta\)^\{2\}\(\\alpha\{\+\}\\beta\{\+\}1\)\]admits the Hoeffding\-style concentration exploited in Theorem[1](https://arxiv.org/html/2609.16446#Thmtheorem1)\. \(iii\) The upper\-confidence bound\(in Sec\.[4\.3](https://arxiv.org/html/2609.16446#S4.SS3)\) for partner ranking takes the standard UCB1 form and inherits sublinear regret \(Theorem[2](https://arxiv.org/html/2609.16446#Thmtheorem2)\)\.

### 4\.3Collaborative Topology Formulation

*RecallQ2\(on selective collaboration under heterogeneity\) andQ3\(on adaptive isolation under drift\) from Sec\.[2](https://arxiv.org/html/2609.16446#S2)\.*The propose\-reject protocol below operationalizes both, with the explicit*rest*action carrying through to the optimality result of Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)\. The global time horizonTTis partitioned into discrete interaction intervals\{t1,…,tK\}\\\{t\_\{1\},\\dots,t\_\{K\}\\\}\. Collaboration decisions among the centers are governed by a propose\-and\-reject protocol approximating a stable\-marriage solution, augmented with anϵ\\epsilon\-greedy Upper Confidence Bound \(UCB\) strategy[Auer et al\. \(2002\)](https://arxiv.org/html/2609.16446#bib.bib10)\. During each periodtkt\_\{k\}, centeriicomputes a utility\-based UCB score for every candidate peerjj:

UCBj→i​\(tk\)=μ^i→j​\(tk\)\+γ​2​log⁡tkni→j​\(tk\)\+1,\\mathrm\{UCB\}\_\{j\\to i\}\(t\_\{k\}\)\\;=\\;\\hat\{\\mu\}\_\{i\\to j\}\(t\_\{k\}\)\\;\+\\;\\gamma\\sqrt\{\\frac\{2\\log t\_\{k\}\}\{n\_\{i\\to j\}\(t\_\{k\}\)\+1\}\},\(7\)Here,μ^i→j​\(tk\)=αi→j\(tk\)/\(αi→j\(tk\)\+βi→j\(tk\)\)\\hat\{\\mu\}\_\{i\\to j\}\(t\_\{k\}\)=\\alpha\_\{i\\to j\}^\{\(t\_\{k\}\)\}/\\big\(\\alpha\_\{i\\to j\}^\{\(t\_\{k\}\)\}\+\\beta\_\{i\\to j\}^\{\(t\_\{k\}\)\}\\big\)is the posterior mean of the Beta belief \([6](https://arxiv.org/html/2609.16446#S4.E6)\),ni→j​\(tk\)n\_\{i\\to j\}\(t\_\{k\}\)is the number of past observations fromjj, andγ\>0\\gamma\>0is the exploration weight\. With probability1−ϵ1\-\\epsilon, centeriisequentially proposes collaboration to peers sorted by descendingUCBj→i\\mathrm\{UCB\}\_\{j\\to i\}\. With probabilityϵ\\epsilon, it purposefully queries a uniformly random unproven neighbor\. A receiving peerjjevaluates the proposal and accepts only if its own UCB oniiexceeds the thresholdτacc\\tau\_\{\\mathrm\{acc\}\}\. Ifjjrejects,iiextends the proposal to the next highest\-ranked peer until eitherκ\\kappaacceptances accrue or the candidate list is exhausted\. Critically, if all proposals are rejected, centerii*rests*: its active peer set𝒫i\\mathcal\{P\}\_\{i\}collapses to∅\\varnothingand no parameters are exchanged this round\. This isolation mechanism is a deliberate design feature, not a system failure\. In highly heterogeneous networks, forcing connections with dissimilar nodes degrades local performance via negative transfer\. Choosing isolation whenever the expected gain falls below the threshold helps protect local utility and conserves bandwidth \(Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)\)\. Resting is per\-round and self\-correcting\. The next round re\-runs the propose\-reject loop with an updated posterior and a UCB exploration radius that grows intt, and theϵ\\epsilon\-greedy swap admits a uniformly\-random peer with probabilityϵ\\epsilonregardless of UCB ranking, so collaboration resumes the moment any peer’s UCB rises aboveτacc\\tau\_\{\\mathrm\{acc\}\}or drift makes a previously\-low\-utility peer informative again\.

## 5Theoretical Guarantees and Complexity Analysis

We analyzeABPSalong posterior concentration \(Theorem[1](https://arxiv.org/html/2609.16446#Thmtheorem1)\), partner\-selection regret \(Theorem[2](https://arxiv.org/html/2609.16446#Thmtheorem2)\), and Bayes\-optimality of isolation \(Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)\)\. Proofs are in Appendix[C](https://arxiv.org/html/2609.16446#A3)\. The analysis rests on three assumptions\.A1bounded utility,Ui∈\[0,1\]U\_\{i\}\\in\[0,1\], soϕ¯j→i∈\[0,1\]\\bar\{\\phi\}\_\{j\\to i\}\\in\[0,1\]via the affine clip of Eq\.[5](https://arxiv.org/html/2609.16446#S4.E5)\(see Sec\.[4\.2](https://arxiv.org/html/2609.16446#S4.SS2)\)\.A2conditional independence of theϕ¯\\bar\{\\phi\}observations givenμi→j⋆\\mu^\{\\star\}\_\{i\\to j\}, justified by independent local stochastic gradient descent draws and disjoint mini\-batches per round\.A3quasi\-stationary drift\|𝔼⁡\[ϕ¯j→i\(k\)\]−μi→j⋆\|≤ϵm\|\\mathbb\{E\}\[\\bar\{\\phi\}^\{\(k\)\}\_\{j\\to i\}\]\-\\mu^\{\\star\}\_\{i\\to j\}\|\\leq\\epsilon\_\{m\}within any windowΔ​tm\\Delta t\_\{m\}, binding the theory to the hierarchical formulation of Sec\.[3](https://arxiv.org/html/2609.16446#S3)\. For the filtered\-arm\-set refinement of Theorem[2](https://arxiv.org/html/2609.16446#Thmtheorem2), we additionally assumeA4that the metadata descriptor𝐯i\\mathbf\{v\}\_\{i\}isLL\-Lipschitz informative about utility,\|μi→j⋆−μi→j′⋆\|≤L​‖𝐯j−𝐯j′‖\|\\mu^\{\\star\}\_\{i\\to j\}\-\\mu^\{\\star\}\_\{i\\to j^\{\\prime\}\}\|\\leq L\\,\\\|\\mathbf\{v\}\_\{j\}\-\\mathbf\{v\}\_\{j^\{\\prime\}\}\\\|\(see Appendix[C\.2](https://arxiv.org/html/2609.16446#A3.SS2)for the precise statement and use\)\.

### 5\.1Convergence of the Beta Posterior

###### Theorem 1\(Beta\-belief concentration\)\.

Letϕ¯j→i\(1\),…,ϕ¯j→i\(K\)∈\[0,1\]\\bar\{\\phi\}\_\{j\\to i\}^\{\(1\)\},\\dots,\\bar\{\\phi\}\_\{j\\to i\}^\{\(K\)\}\\in\[0,1\]be the clipped Shapley observations collected by centeriifor peerjjacrossKKrounds inside a single drift window, with empirical meanμ¯K=1K​∑k=1Kϕ¯j→i\(k\)\\bar\{\\mu\}\_\{K\}=\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\bar\{\\phi\}\_\{j\\to i\}^\{\(k\)\}and posterior meanμ^K=α0\+K​μ¯Kα0\+β0\+K\\hat\{\\mu\}\_\{K\}=\\frac\{\\alpha\_\{0\}\+K\\bar\{\\mu\}\_\{K\}\}\{\\alpha\_\{0\}\+\\beta\_\{0\}\+K\}\. Under A1\-A3, for anyδ∈\(0,1\)\\delta\\in\(0,1\)with probability at least1−δ1\-\\delta

\|μ^K−μi→j⋆\|≤log⁡\(2/δ\)2​K⏟Hoeffding\+α0\+β0α0\+β0\+K⏟prior decay\+ϵm⏟drift\.\\big\|\\hat\{\\mu\}\_\{K\}\-\\mu^\{\\star\}\_\{i\\to j\}\\big\|\\;\\leq\\;\\underbrace\{\\sqrt\{\\frac\{\\log\(2/\\delta\)\}\{2K\}\}\}\_\{\\text\{Hoeffding\}\}\\;\+\\;\\underbrace\{\\frac\{\\alpha\_\{0\}\+\\beta\_\{0\}\}\{\\alpha\_\{0\}\+\\beta\_\{0\}\+K\}\}\_\{\\text\{prior decay\}\}\\;\+\\;\\underbrace\{\\epsilon\_\{m\}\\vphantom\{\\sqrt\{\\frac\{\\log\(2/\\delta\)\}\{2K\}\}\}\}\_\{\\text\{drift\}\}\.\(8\)Henceμ^K→𝑝μi→j⋆\\hat\{\\mu\}\_\{K\}\\xrightarrow\{p\}\\mu^\{\\star\}\_\{i\\to j\}asK→∞K\\to\\inftywithϵm→0\\epsilon\_\{m\}\\to 0\.

The proof \(Appendix[C\.1](https://arxiv.org/html/2609.16446#A3.SS1)\) decomposes the error into Hoeffding noise \(𝒪⁡\(1/K\)\\mathcal\{O\}\(1/\\sqrt\{K\}\)\), prior decay \(𝒪⁡\(1/K\)\\mathcal\{O\}\(1/K\)\), and within\-window drift bounded byϵm\\epsilon\_\{m\}, the last of which motivates the Welch\-ttwindow selection of Sec\.[3](https://arxiv.org/html/2609.16446#S3)\.

### 5\.2Regret of the UCB Partner Selection Bandit

We bound the regret of the per\-center partner\-selection problem viewed as aκ\\kappa\-armed bandit[Lattimore and Szepesvári \(2020\)](https://arxiv.org/html/2609.16446#bib.bib39)over the candidate pool𝒞i\\mathcal\{C\}\_\{i\}\. Letμ\(1\)⋆≥μ\(2\)⋆≥…\\mu^\{\\star\}\_\{\(1\)\}\\geq\\mu^\{\\star\}\_\{\(2\)\}\\geq\\dotsdenote the ordered true utilities ofii’s candidate peers and define the suboptimality gapsΔj=μ\(1\)⋆−μ\(j\)⋆\\Delta\_\{j\}=\\mu^\{\\star\}\_\{\(1\)\}\-\\mu^\{\\star\}\_\{\(j\)\}\.

###### Theorem 2\(Sublinear partner\-selection regret\)\.

Under assumptions A1\-A2 and a stationary window \(ϵm=0\\epsilon\_\{m\}=0\), the cumulative regret ofABPSwith exploration parameterγ=2\\gamma=\\sqrt\{2\}andϵ\\epsilon\-greedy exploration probabilityϵ∈\[0,1\)\\epsilon\\in\[0,1\)overTTrounds satisfies

ℛ\(T\)≤κ∑j:Δj\>0\(8​log⁡TΔj\+Δj\(1\+π23\)\)\+ϵT⋅𝔼j∼Unif⁡\(𝒞i\)\[Δj\]\.\\mathcal\{R\}\(T\)\\;\\leq\\;\\kappa\\sum\_\{j:\\Delta\_\{j\}\>0\}\\\!\\left\(\\frac\{8\\log T\}\{\\Delta\_\{j\}\}\+\\Delta\_\{j\}\\\!\\left\(1\+\\frac\{\\pi^\{2\}\}\{3\}\\right\)\\right\)\\;\+\\;\\epsilon\\,T\\cdot\\mathbb\{E\}\_\{j\\sim\\mathrm\{Unif\}\(\\mathcal\{C\}\_\{i\}\)\}\[\\Delta\_\{j\}\]\.\(9\)Settingϵ=𝒪⁡\(1/T\)\\epsilon=\\mathcal\{O\}\(1/\\sqrt\{T\}\)recovers the standard𝒪⁡\(log⁡T\)\\mathcal\{O\}\(\\log T\)regret rate of UCB1 up to the factorκ\\kappa\.

The proof \(Appendix[C\.2](https://arxiv.org/html/2609.16446#A3.SS2)\) bounds the greedy phase via canonical UCB1 analysis[Auer et al\. \(2002\)](https://arxiv.org/html/2609.16446#bib.bib10)scaled byκ\\kappaand the exploratory phase via theϵ​T⋅𝔼⁡\[Δj\]\\epsilon T\\cdot\\mathbb\{E\}\[\\Delta\_\{j\}\]term, with a filtered\-arm\-set refinement under A4 that improves the constants when the goal\-aware pre\-filter is active\. In stationary windows,ABPStherefore matches an oracle that always proposes to theκ\\kappabest peers\.

### 5\.3Optimality of Intentional Isolation

*RecallQ3\(adaptive isolation under drift\) from Sec\.[2](https://arxiv.org/html/2609.16446#S2)\.*The lemma below is the formal optimality guarantee that the rest of the action of Sec\.[4\.3](https://arxiv.org/html/2609.16446#S4.SS3)promised\.

###### Lemma 1\(Intentional isolation dominates forced collaboration\)\.

LetVirestV\_\{i\}^\{\\mathrm\{rest\}\}denote the expected one\-step utility of centeriiwhen it rests \(𝒫i=∅\\mathcal\{P\}\_\{i\}=\\varnothing\) andVicoll​\(𝒫\)V\_\{i\}^\{\\mathrm\{coll\}\}\(\\mathcal\{P\}\)the expected utility when it forcibly collaborates with peer set𝒫≠∅\\mathcal\{P\}\\neq\\varnothing\. Suppose the per\-peer expected marginal utility satisfiesμi→j⋆<cneg\\mu^\{\\star\}\_\{i\\to j\}<c\_\{\\mathrm\{neg\}\}for everyj∈𝒫j\\in\\mathcal\{P\}, wherecnegc\_\{\\mathrm\{neg\}\}is the negative\-transfer threshold defined byVicoll​\(\{j\}\)=VirestV\_\{i\}^\{\\mathrm\{coll\}\}\(\\\{j\\\}\)=V\_\{i\}^\{\\mathrm\{rest\}\}whenμi→j⋆=cneg\\mu^\{\\star\}\_\{i\\to j\}=c\_\{\\mathrm\{neg\}\}\. Then

Virest\>Vicoll​\(𝒫\),V\_\{i\}^\{\\mathrm\{rest\}\}\\;\>\\;V\_\{i\}^\{\\mathrm\{coll\}\}\(\\mathcal\{P\}\),\(10\)and the bandwidth saved by resting is exactlyCirest=2​\|𝒫\|⋅cegress⋅\|W\|C\_\{i\}^\{\\mathrm\{rest\}\}=2\|\\mathcal\{P\}\|\\cdot c\_\{\\mathrm\{egress\}\}\\cdot\|W\|per round \(cf\. eq\. \(1\)\)\.

The proof \(Appendix[C\.3](https://arxiv.org/html/2609.16446#A3.SS3)\) combines Shapley efficiency with the affine clip of Eq\.[5](https://arxiv.org/html/2609.16446#S4.E5), which fixescneg=−ϕmin/\(ϕmax−ϕmin\)=0\.5c\_\{\\mathrm\{neg\}\}=\-\\phi\_\{\\min\}/\(\\phi\_\{\\max\}\-\\phi\_\{\\min\}\)=0\.5for\(ϕmin,ϕmax\)=\(−0\.1,\+0\.1\)\(\\phi\_\{\\min\},\\phi\_\{\\max\}\)=\(\-0\.1,\+0\.1\), exactlyτacc\\tau\_\{\\mathrm\{acc\}\}\.ABPSapproximates this oracle by resting whenever every UCB falls belowτacc\\tau\_\{\\mathrm\{acc\}\}, and Theorem[1](https://arxiv.org/html/2609.16446#Thmtheorem1)guarantees convergence to the oracle asK→∞K\\to\\infty\.

### 5\.4Computational Complexity

Letρ∈\(0,1\]\\rho\\in\(0,1\]denote the expected top\-kkcoverage of the goal\-aware pre\-filter \(Sec\.[4\.1](https://arxiv.org/html/2609.16446#S4.SS1)\), i\.e\. the fraction of peers that survive theτsim\\tau\_\{\\mathrm\{sim\}\}threshold\. Whenτsim\\tau\_\{\\mathrm\{sim\}\}is calibrated to retain a prescribed keep\-fraction,ρ\\rhoequals that target \(we useρ=0\.25\\rho=0\.25\)\. Define the post\-filter candidate size\|𝒞i\|≈ρ⁡\(n−1\)\|\\mathcal\{C\}\_\{i\}\|\\approx\\rho\(n\-1\)\. The per\-round computational cost at each center is strictly bounded and decomposes into four components\. \(1\)Goal\-aware filter:Evaluatingfi​\(𝐯i,𝐯j\)f\_\{i\}\(\\mathbf\{v\}\_\{i\},\\mathbf\{v\}\_\{j\}\)for each of then−1n\-1peers usingdd\-dimensional metadata incurs a cost of𝒪⁡\(n​d\)\\mathcal\{O\}\(nd\), whered≤15d\\leq 15in our implementation \(3 base features and 12 comorbidity indicators\); \(2\)UCB ranking and propose–reject:Ranking candidates in the filtered set\|𝒞i\|≈ρ​n\|\\mathcal\{C\}\_\{i\}\|\\approx\\rho nand traversing the ordered list requires𝒪⁡\(ρ​n​log⁡\(ρ​n\)\)\\mathcal\{O\}\(\\rho n\\log\(\\rho n\)\)time requires aρ\\rho\-factor reduction compared to the unfiltered case; \(3\)Shapley credit assignment:The exact computation of this involves evaluating2\|𝒫i\|2^\{\|\\mathcal\{P\}\_\{i\}\|\}coalitions\. Since\|𝒫i\|≤κ\|\\mathcal\{P\}\_\{i\}\|\\leq\\kappaby design, the cost is𝒪⁡\(2κ\)\\mathcal\{O\}\(2^\{\\kappa\}\), independent ofnn\. Forκ\>5\\kappa\>5, we instead employ Truncated Monte Carlo \(TMC\) Shapley[Ghorbani and Zou \(2019\)](https://arxiv.org/html/2609.16446#bib.bib19), which achieves anε\\varepsilon\-accurate estimate in𝒪⁡\(κ​log⁡κ/ε2\)\\mathcal\{O\}\(\\kappa\\log\\kappa/\\varepsilon^\{2\}\)samples; and \(4\)Beta–Bernoulli update:Posterior updates require only closed\-form scalar operations, yielding𝒪⁡\(1\)\\mathcal\{O\}\(1\)complexity\. The total per\-round computation at centeriiis therefore𝒪⁡\(n​d\+ρ​n​log⁡\(ρ​n\)\+2κ\)\\mathcal\{O\}\(nd\+\\rho n\\log\(\\rho n\)\+2^\{\\kappa\}\)\. For the defaultρ=0\.25\\rho=0\.25andκ=1\\kappa=1this collapses to𝒪⁡\(n​d\)\\mathcal\{O\}\(nd\),*independent of the number of model parameters\|W\|\|W\|*\. Communication\.Each round transmits \(i\) the metadata vector𝐯i\\mathbf\{v\}\_\{i\}once at filter time \(a few dozen bytes per center,*no model weights*\) and \(ii\) the active\-peer aggregation for theκ\\kappapeers that survive both the filter and the UCB threshold, costing2​κ​\|W\|2\\kappa\|W\|per active center per round\. With personalization \(Sec\.[6\.4](https://arxiv.org/html/2609.16446#S6.SS4)\), only the shared trunk is transmitted\. With quantization, each\|W\|\|W\|is reduced to its bfloat16 footprint\. When\|𝒫i\|=0\|\\mathcal\{P\}\_\{i\}\|=0\(intentional isolation, empirically realized under the diversity goal\), the communication collapses to zero for that center\-round\.

## 6Experimental Evaluation

We evaluateABPSalong four axes: \(i\) predictive accuracy under realistic non\-IID partitioning of MIMIC\-IV, a publicly available database sourced from the Beth Israel Deaconess Medical Center \(BIDMC\) electronic health record[Johnson et al\. \(2023\)](https://arxiv.org/html/2609.16446#bib.bib21), \(ii\) cumulative transmitted bytes as a direct proxy for communication cost, \(iii\) the empirical realization of the intentional\-isolation property of Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1), and \(iv\) an ablation decomposing the contribution of each framework extension\. Code, sbatch scripts, and per\-seed per\-method result JSONs will accompany the camera\-ready submission\.

### 6\.1Dataset and Federation Setup

The task at hand is the binary in\-hospital mortality prediction over the first 24 hours of an ICU stay[Johnson et al\. \(2023\)](https://arxiv.org/html/2609.16446#bib.bib21)\. After applying the standard age filter \(18≤age≤9518\\leq\\text\{age\}\\leq 95\), the cohort contains roughly76,00076\{,\}000ICU stays\. The patient features: demographics \(age, gender, race\), admission context \(admission type, location, insurance\), Charlson\-style comorbidity binaries, discussed in Sec\.[6\.3](https://arxiv.org/html/2609.16446#S6.SS3), derived from ICD\-10/ICD\-9 codes, and aggregated first\-24h vitals \(heart rate, systolic/diastolic/mean BP, respiration rate, SpO2, temperature\) summarized as\{mean,min,max\}\\\{\\mathrm\{mean\},\\min,\\max\\\}\. Continuous features are standardized*per center*on the local training split to respect federated isolation\. Our experiments rely on acareunit\-by\-yearpartitioning of MIMIC\-IV, where a center is defined as the Cartesian product of a care unit and a consecutive two\-year admission window\. The care units include CVICU, CCU, MICU, Medical/Surgical ICU, SICU, and TSICU, yieldingn=230n=230centers spanning the shifted temporal range 2110 to 2191 in MIMIC\-IV v3\.1\. MIMIC\-IV de\-identifies dates by a single random offset per subject, applied uniformly to all of that subject’s events[Johnson et al\. \(2023\)](https://arxiv.org/html/2609.16446#bib.bib21), so within\-patient order is exact, but the shifted\-year windows are not aligned with calendar time: across the 71,008 stays in the partition, Cramér’sVVbetween the assigned window and the publishedanchor\_year\_groupfield is 0\.045, and mean window purity is 0\.335 against a chance value of 0\.336\. We therefore treat the windows as a partitioning device rather than a calendar axis, and the empirical evidence of drift over calendar time in Appendix[A](https://arxiv.org/html/2609.16446#A1)uses the published era field\. What the construction provides is a deterministic partition into many small centers whose outcome distributions differ: the mean pairwise Jensen\-Shannon divergence between the Bernoulli mortality distributions of centers from different care units is 0\.0066 nats, against 0\.0009 nats within a care unit, and mortality rates vary across units \(approximately3%3\\%in CVICU versus15%15\\%in MICU\), aligning with the IDN, AMC, and Community Hospital framing in Sec\.[1](https://arxiv.org/html/2609.16446#S1)\. The pipeline also supports Dirichlet\(α\\alpha\) label\-skew partitions[Hsu et al\. \(2019\)](https://arxiv.org/html/2609.16446#bib.bib20)and uniform IID splits, but the careunit\-by\-year setting is used for all primary results\. A FedProx\-Synthetic\(α,β\)\(\\alpha,\\beta\)generator[Li et al\. \(2020\)](https://arxiv.org/html/2609.16446#bib.bib22)matching the MIMIC feature schema drives implementation tests but is not used for headline numbers\.

### 6\.2Baselines

We compare against ten methods grouped by purpose\.*Reference anchors:*Centralizedpools all data into one Multi\-Layer Perceptron \(MLP\) \(non\-federated upper bound, in the Table[1](https://arxiv.org/html/2609.16446#S6.T1)caption\),Local\-onlytrains each center independently \(no\-collaboration lower bound\), andFedAvg[McMahan et al\. \(2017\)](https://arxiv.org/html/2609.16446#bib.bib17)averages weights every round\.*Centralized non\-IID FL:*FedProx[Li et al\. \(2020\)](https://arxiv.org/html/2609.16446#bib.bib22)adds a proximal regularizer,FedDyn[Acar et al\. \(2021\)](https://arxiv.org/html/2609.16446#bib.bib23)aligns local objectives with the global stationary point, andMOON[Li et al\. \(2021\)](https://arxiv.org/html/2609.16446#bib.bib24)maximizes agreement between local and global representations\.*Decentralized P2P FL:*DeceFL[Yuan et al\. \(2023\)](https://arxiv.org/html/2609.16446#bib.bib25)provably converges to the centralized optimum,DeFTA[Zhou et al\. \(2024\)](https://arxiv.org/html/2609.16446#bib.bib26)is a plug\-and\-play decentralized FedAvg with trust\-based reweighting \(the closest peer\-selection competitor\), andWPFed[Ye et al\. \(2024\)](https://arxiv.org/html/2609.16446#bib.bib27)uses Locality\-Sensitive Hashing \(LSH\) similarity filtering plus weighted neighbor selection, mirroring the metadata pre\-filter of Sec\.[4\.1](https://arxiv.org/html/2609.16446#S4.SS1)but lacking the Bayesian posterior update\.*Bayesian FL:*BNN\+FL[Saile et al\. \(2024\)](https://arxiv.org/html/2609.16446#bib.bib3)replaces the MLP with a Bayesian neural network and aggregates posterior moments \(ABPS’s novelty is Bayesianizing the*utility*signal rather than the weights\)\. BrainTorrent[Guha Roy et al\. \(2019\)](https://arxiv.org/html/2609.16446#bib.bib28), Gossip Learning[Hegedűs et al\. \(2021\)](https://arxiv.org/html/2609.16446#bib.bib29), KL\-FedDis[Rahad et al\. \(2025\)](https://arxiv.org/html/2609.16446#bib.bib5), and Peer\-Driven Reputation FL[Seidi et al\. \(2025\)](https://arxiv.org/html/2609.16446#bib.bib30)are conceptual antecedents discussed in Sec\.[2](https://arxiv.org/html/2609.16446#S2)but not run as baselines\.

### 6\.3Implementation and Metrics

The local model is a two\-hidden\-layer MLP \(128→64128\\to 64, ReLU, dropout 0\.2\) optimized with Adam \(η=10−3\\eta=10^\{\-3\}, weight decay10−510^\{\-5\}\)\. Each round runsKlocal=2K\_\{\\mathrm\{local\}\}=2local epochs with batch size 128 overK=50K=50rounds \(or up toK=100K=100for theABPS\-Xvariant with validation\-AUROC early stopping at patience 10\)\. BaseABPSusesκ=3\\kappa=3,ϵ=0\.1\\epsilon=0\.1,γ=2\\gamma=\\sqrt\{2\},τacc=0\.5\\tau\_\{\\mathrm\{acc\}\}=0\.5, the 3\-dim base metadata vector\[positive\-class rate,log⁡ntrain,year¯/2030\]\[\\text\{positive\-class rate\},\\log n\_\{\\mathrm\{train\}\},\\overline\{\\text\{year\}\}/2030\], andτsim=0\\tau\_\{\\mathrm\{sim\}\}=0\(no filtering\)\. The goal\-aware variants enrich metadata with a 12\-dim per\-center prevalence vector over Charlson Comorbidity Index \(CCI\)[Charlson et al\. \(1987\)](https://arxiv.org/html/2609.16446#bib.bib32)chronic\-condition categories \(binary presence per patient, averaged over local training cohort, no per\-patient leakage\) and auto\-calibrateτsim\\tau\_\{\\mathrm\{sim\}\}to retain25%25\\%of peer pairs per goal\. Shapley contributions are computed exactly when\|𝒫i\|≤5\|\\mathcal\{P\}\_\{i\}\|\\leq 5and via truncated Monte\-Carlo withB=8B=8permutations otherwise\. We report mean AUROC across centers \(across\-seed std over55seeds in\{11,22,33,44,55\}\\\{11,22,33,44,55\\\}\) and cumulative transmitted bytes counted once per undirected edge\. Experiments were run on a Simple Linux Utility for Resource Management \(SLURM\)[Yoo et al\. \(2003\)](https://arxiv.org/html/2609.16446#bib.bib38)\-managed High\-Performance Computing \(HPC\) cluster, each SLURM task on a single NVIDIA Tesla V100\-SXM2 GPU \(32 GB\) with 8 CPU cores and 32 GB RAM, under PyTorch 2\.5\.1 \(CUDA 12\.1\) and Python 3\.11\. Then=230n\{=\}230\-center sweep \(60 array tasks: 12 configurations×\\times5 seeds\) finishes in 1 to 3 wall\-clock hours, and the headlineABPS\-Xsweep \(10 tasks at 100 rounds with early stopping\) finishes in 3 to 9 minutes per task\. Reproduction of table cells and figures requires a singlesbatchof the two sweep scripts released with the supplementary material and JSON outputs\.

### 6\.4Headline Result: Accuracy vs\. Bandwidth

Table 1:In\-hospital mortality prediction on MIMIC\-IV \(n=230n\{=\}230centers, 50 rounds, 5 seeds\)\. AUROC is the mean across centers \(mean±\\pmstandard deviation\)\. Bandwidth is a cumulative parameter exchange relative to FedAvg=1\.00×=1\.00\\times\. As a non\-federated upper\-bound reference \(not a fair federated comparator\)\.*RecallQ4\(communication overhead, even in P2P\) from Sec\.[2](https://arxiv.org/html/2609.16446#S2)\.*Table[1](https://arxiv.org/html/2609.16446#S6.T1)and the Pareto frontier of Figure[3](https://arxiv.org/html/2609.16446#S6.F3)are the empirical answer\. Table[1](https://arxiv.org/html/2609.16446#S6.T1)reports final\-round mean AUROC over 5 seeds, and Figure[3](https://arxiv.org/html/2609.16446#S6.F3)plots the same data as an accuracy\-bandwidth Pareto frontier\. All federated methods gain∼15\\sim 15AUROC points over the local\-only lower bound \(0\.587±0\.0130\.587\\pm 0\.013\) and close most of the gap to the non\-federated Centralized upper bound \(0\.827±0\.0070\.827\\pm 0\.007\)\. BaseABPS\(withκ=3\\kappa\{=\}3and no personalization or quantization\) already achieves AUROC0\.753±0\.0150\.753\\pm 0\.015, statistically indistinguishable from FedAvg \(0\.755±0\.0190\.755\\pm 0\.019\), FedProx \(0\.753±0\.0190\.753\\pm 0\.019\), and FedDyn \(0\.758±0\.0170\.758\\pm 0\.017\)\. It transmits1\.50×1\.50\\timesFedAvg’s bandwidth because aκ=3\\kappa\{=\}3mesh has more unique edges than FedAvg’s star, matching the theoretical accounting in Sec\.[5\.4](https://arxiv.org/html/2609.16446#S5.SS4)\. Three extensions independently move the Pareto frontier:*\(P\)*head personalization[Arivazhagan et al\. \(2019\)](https://arxiv.org/html/2609.16446#bib.bib12)boosts AUROC to0\.757±0\.0150\.757\\pm 0\.015at the same bandwidth \(the per\-center head specializes to careunit mortality base\-rates, e\.g\., CVICU∼3%\\sim 3\\%vs MICU∼15%\\sim 15\\%\)\.*\(Q\)*bfloat16[Dettmers et al\. \(2022\)](https://arxiv.org/html/2609.16446#bib.bib13)halves bandwidth with no accuracy loss \(0\.753±0\.0150\.753\\pm 0\.015at0\.75×0\.75\\times\)\.*\(κ=1\\kappa\{=\}1\)*collapsing the active set to a single partner halves the mesh edges again and, combined with \(P\) and \(Q\), yields theABPS\+P\+Q\+κ=1\\kappa\{=\}1row:0\.748±0\.0120\.748\\pm 0\.012at0\.25×0\.25\\times, already Pareto\-dominating federated baselines\. Adding the goal\-aware pre\-filter, 2\-layer personalization, server momentum, and validation\-AUROC early stopping yields fullABPS\-X:0\.758±0\.0100\.758\\pm 0\.010\(matching FedDyn\) at0\.09×0\.09\\timesFedAvg\.

Figure 3:Results on MIMIC\-IV careunit\-by\-year \(n=230n=230centers, 5 seeds\)\.\(a\)Accuracy ranking: mean AUROC with across\-seed std, color\-coded by method family \(blue: star\-topology FL, green: decentralized, orange:ABPS\+P/Q, red:ABPS\-X, purple/brown: goal\-aware, olive: local\-only\)\. The dashed line denotes the non\-federated centralized upper bound \(off\-axis\)\.\(b\)Accuracy vs\. bandwidth Pareto \(log\-scale bandwidth\):ABPS\-X\(dark\-red star\) matches FedDyn’s AUROC \(0\.7580\.758\) at0\.09×0\.09\\timesFedAvg bandwidth\. Red arrows show the ablation pathABPS→\\\!\\to\\\!\+P→\\\!\\to\\\!\+Q→\\\!\\to\\\!\+P\+Q\+κ=1\\kappa\{=\}1→\\\!\\to\\\!ABPS\-X, with each step improving the trade\-off\.Fair comparator and ABPS\-X\.Applying P\+Q to FedAvg \(FedAvg\+P\+Q\) yields0\.758±0\.0170\.758\\pm 0\.017at0\.50×0\.50\\times, a modest improvement, but cannot reachκ=1\\kappa\{=\}1because the star topology has no mechanism for selecting a single best peer\. The fullABPS\-Xvariant adds 2\-layer personalization, server\-side EMA momentumβ=0\.5\\beta\{=\}0\.5, 15\-dim goal\-aware metadata under the homogeneity goal, and validation\-AUROC early stopping \(patience 10 of 100\), reaching0\.758±0\.010\\mathbf\{0\.758\\pm 0\.010\}at0\.09×\\mathbf\{0\.09\\times\}FedAvg with tighter variance than FedDyn\. The same\-extension FedAvg\-X fair comparator drops to0\.691±0\.0080\.691\\pm 0\.008,6\.76\.7AUROC points behindABPS\-X: 2\-layer personalization over\-adapts each private head when the star topology averages across all 230 peers, whereasABPS’sκ=1\\kappa\{=\}1Bayesian selection supplies the one\-peer constraint under which personalization helps\. Empirical isolation does not activate here \(𝔼⁡\[\|𝒫i\|\]≈1\.00\\mathbb\{E\}\[\|\\mathcal\{P\}\_\{i\}\|\]\\approx 1\.00whenκ=1\\kappa\{=\}1\), but the goal\-aware filter of the next subsection engages Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)’s rest regime in up to41%41\\%of rounds\.

### 6\.5Empirical Validation of Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)\(Intentional Isolation\)

The diversity\-goal experiment below is the empirical activation of the rest regime promised by Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)to address*Q3\(adaptive isolation under drift\)\.*\. The isolation lemma predicts that when every peer’s true expected utility falls below the negative\-transfer thresholdcnegc\_\{\\mathrm\{neg\}\}, the optimal action is𝒫i=∅\\mathcal\{P\}\_\{i\}=\\varnothingand the resulting bandwidth is zero\. We validate this prediction on MIMIC\-IV through the goal\-aware filter of Sec\.[4\.1](https://arxiv.org/html/2609.16446#S4.SS1): by varying the admission rule, we directly control which peers survive to the UCB\-Shapley stage, which in turn controls whether the UCB thresholds admit proposals\. We runABPS\+P\+Q\+κ=1\\kappa\{=\}1on the fulln=230n\{=\}230federation under each of the three goals in Eq\.[3](https://arxiv.org/html/2609.16446#S4.E3)withτsim\\tau\_\{\\mathrm\{sim\}\}auto\-calibrated to retain25%25\\%of peers per center, and record the per\-round fraction of centers with\|𝒫i\|=0\|\\mathcal\{P\}\_\{i\}\|\{=\}0\.Homogeneity and alignmentadmit peers with similar \(or KPI\-matched\) marginals, so isolation rarely activates \(𝔼⁡\[\|𝒫i\|\]=0\.99\\mathbb\{E\}\[\|\\mathcal\{P\}\_\{i\}\|\]\{=\}0\.99\), giving AUROC0\.7420\.742and0\.7390\.739at0\.25×0\.25\\timesbandwidth\.Diversityadmits dissimilar peers, many crossing the negative\-transfer threshold, so the UCB falls belowτacc\\tau\_\{\\mathrm\{acc\}\}and rest activate for41%41\\%of centers \(𝔼⁡\[\|𝒫i\|\]=0\.59\\mathbb\{E\}\[\|\\mathcal\{P\}\_\{i\}\|\]\{=\}0\.59\)\. Bandwidth drops to0\.16×0\.16\\timesFedAvg, the lowest observed, while AUROC decreases to0\.6940\.694\. The bandwidth reduction tracks isolation within5%5\\%, the collaborate\-to\-rest transition is sharp at the Hoeffding ratelog⁡\(2/δ\)/\(2​K\)\\sqrt\{\\log\(2/\\delta\)/\(2K\)\}, and this is the first setting where Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)’s\|𝒫i\|=0\|\\mathcal\{P\}\_\{i\}\|\{=\}0regime activates at scale on real clinical data\. Ablations:Reading Table[1](https://arxiv.org/html/2609.16446#S6.T1):ABPS\+P improves AUROC by\+0\.4\+0\.4at the same bandwidth, while \+Q maintains performance at0\.75×0\.75\\times\. The combinedABPS\+P\+Q\+κ=1\\kappa\{=\}1setting preserves AUROC at just0\.25×0\.25\\timesbandwidth\. In contrast, FedAvg\+P\+Q reaches0\.758±0\.0170\.758\\pm 0\.017at0\.50×0\.50\\timesbandwidth but cannot operate atκ=1\\kappa\{=\}1, indicating that the additional2×2\\timesreduction is due to Bayesian Shapley\-UCB selection\. The three goal\-aware rows \(Eq\.[3](https://arxiv.org/html/2609.16446#S4.E3)\) ablate the pre\-filter \(Sec\.[6\.5](https://arxiv.org/html/2609.16446#S6.SS5)\)\. Hyperparametersϵ\\epsilon,γ\\gamma, and truncated\-MCBBare fixed from preliminary synthetic runs\.

## 7Conclusion and Future Work

We presentedABPS, an adaptive Bayesian P2P federated\-learning framework that replaces raw parameter aggregation with Shapley\-based marginal\-utility evaluation, formalizes*intentional isolation*as a Bayes\-optimal action under negative transfer, and composes with three communication\-efficiency extensions \(head personalization, bfloat16 quantization, tunableκ\\kappa\) plus a goal\-aware metadata pre\-filter\. On MIMIC\-IV withn=230n\{=\}230careunit\-by\-year centers, the fullABPS\-Xvariant matches the strongest federated baseline \(FedDyn\) at0\.09×0\.09\\timesFedAvg bandwidth, while applying the same extensions to FedAvg drops6\.46\.4AUROC points \(isolating the gain to the Bayesian selection itself\), and the diversity goal activates Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)’s rest regime for41%41\\%of centers per round, a first on real clinical data\. Limitations and future work\.The evaluation is on a single dataset and binary clinical task, the regret bound degrades with unmeasuredϵm\\epsilon\_\{m\}, and we provide no formal\(ϵ,δ\)\(\\epsilon,\\delta\)\-DP, survival, or Byzantine guarantees\. Natural follow\-ups include \(i\) a Gaussian\-mechanism DP wrapper on𝐯i\\mathbf\{v\}\_\{i\}for certifiable privacy, \(ii\) multi\-modal pipelines fusing EHR with medical imaging, \(iii\) online Welch\-ttestimation of\{Δ​tm\}\\\{\\Delta t\_\{m\}\\\}together with a drifting\-bandit regret analysis, and \(iv\) porting to time\-to\-event outcomes \(matching the Cox\-style framing of[Seidi et al\. \(2025\)](https://arxiv.org/html/2609.16446#bib.bib30)\)\. Scope of the accuracy claim\.The matched\-accuracy result holds for federations of many small centers\. On a 40\-center partition of the same cohort built from the publishedanchor\_year\_grouperas, with a median of roughly 2,000 stays per center, FedAvg and FedDyn reach 0\.813 and 0\.805 AUROC after 100 rounds whileABPS\-Xpeaks at 0\.793, althoughABPS\-Xreaches each intermediate AUROC target \(0\.750, 0\.770, 0\.785\) with four to six times fewer bytes\. Where centers are large, each local model is already well estimated and aggregating across all of them is close to optimal; where centers are numerous and small, indiscriminate aggregation carries more harmful transfer and selective exchange is competitive\.

Broader impact\.Cutting per\-center bandwidth tenfold at matched accuracy on federations of many small centers lowers the entry barrier for resource\-constrained sites that disproportionately serve under\-represented populations\. Residual privacy and goal\-misuse risks \(metadata re\-identification under auxiliary information, and goal\-aware filtering used to entrench rather than correct bias\) are addressed by the recommended DP wrapper and governance over goal declarations, documented in full in the NeurIPS Reproducibility Checklist\.

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## Appendix AEmpirical Evidence of Concept Drift on MIMIC\-IV

To support the concept\-drift premise empirically, we compared the distribution of two routinely charted ICU vital signs, heart rate and respiratory rate, between two non\-overlapping calendar eras of MIMIC\-IV v3\.1\. The era of each stay is the publishedanchor\_year\_groupfield of the patient, which is the true three\-year admission era and is unaffected by the per\-subject date shift; we use the earliest era,20082008to20102010, and the latest,20202020to20222022\. For each ICU stay we take the mean of all charted values of the vital during the stay, so that each stay contributes one observation, and we remove stay means that lie more than three standard deviations from the per\-era mean\. Heart rate has29,78629\{,\}786stays in the earlier era and10,74010\{,\}740in the later one, with means of84\.984\.9and83\.983\.9beats per minute; respiratory rate has29,88529\{,\}885and10,74210\{,\}742stays, with means of19\.319\.3and19\.419\.4breaths per minute and a markedly narrower spread in the later era \(standard deviation4\.24\.2against5\.75\.7\)\. Welch two\-samplett\-tests givet=5\.45t=5\.45,p=5×10−8p=5\\times 10^\{\-8\}for heart rate andt=−2\.07t=\-2\.07,p=0\.038p=0\.038for respiratory rate\. The shifts are small in the mean but significant at this sample size, and the change in spread for respiratory rate indicates a change in charting or patient mix rather than a mean shift alone, which is the kind of distributional drift the per\-window treatment ofPt\(i\)P^\{\(i\)\}\_\{t\}in Sec\.[3](https://arxiv.org/html/2609.16446#S3)is designed to absorb\.

Figure 4:Histograms of per\-stay mean heart rate \(panel a\) and respiratory rate \(panel b\) for ICU stays in two non\-overlapping eras of MIMIC\-IV v3\.1,20082008to20102010and20202020to20222022, defined by the publishedanchor\_year\_groupfield\. Stay means beyond three standard deviations from the per\-era mean are removed\. Vertical dashed lines mark the per\-era means\. Welchtt\-tests givet=5\.45t=5\.45,p=5×10−8p=5\\times 10^\{\-8\}for heart rate andt=−2\.07t=\-2\.07,p=0\.038p=0\.038for respiratory rate, evidencing distributional shift across eras in routinely collected measurements and motivating the per\-window adaptive treatment ofPt\(i\)P^\{\(i\)\}\_\{t\}in Sec\.[3](https://arxiv.org/html/2609.16446#S3)\.
## Appendix BAlgorithm Pseudocode

The complete round\-level pseudocode forABPSis given in Algorithm[1](https://arxiv.org/html/2609.16446#alg1), deferred from the main text to save space\. The algorithm operationalizes the four Bayesian\-core components of Sec\.[4](https://arxiv.org/html/2609.16446#S4)\(goal\-aware metadata pre\-filter,ϵ\\epsilon\-greedy UCB ranking, propose\-reject topology with explicit rest, and the Beta\-Bernoulli posterior update\), and is referenced from the regret proof in Appendix[C\.2](https://arxiv.org/html/2609.16446#A3.SS2)\(lines 13 and 16\)\.

Algorithm 1Adaptive Bayesian Partner Selection \(ABPS\)1:centers

\{1,…,n\}\\\{1,\\dots,n\\\}, rounds

\{t1,…,tK\}\\\{t\_\{1\},\\dots,t\_\{K\}\\\}, rank threshold

κ\\kappa, exploration weight

γ\\gamma, exploration probability

ϵ\\epsilon, acceptance threshold

τacc\\tau\_\{\\mathrm\{acc\}\}, similarity threshold

τsim\\tau\_\{\\mathrm\{sim\}\}, Beta prior

\(α0,β0\)\(\\alpha\_\{0\},\\beta\_\{0\}\)
2:Updated local parameters

θi\\theta\_\{i\}and posterior beliefs

\(αi→j,βi→j\)\(\\alpha\_\{i\\to j\},\\beta\_\{i\\to j\}\)
3:Initialization:

4:foreach center

iido

5:Initialize local model

θi\\theta\_\{i\}and metadata vector

𝐯i\\mathbf\{v\}\_\{i\}
6:foreach center

j≠ij\\neq ido

7:

\(αi→j,βi→j\)←\(α0,β0\)\(\\alpha\_\{i\\to j\},\\beta\_\{i\\to j\}\)\\leftarrow\(\\alpha\_\{0\},\\beta\_\{0\}\),

ni→j←0n\_\{i\\to j\}\\leftarrow 0
8:endfor

9:endfor

10:foreach round

k=1,…,Kk=1,\\dots,Kdo

11:foreach center

iido

12:

𝒞i←\{j:fi​\(𝐯i,𝐯j\)≥τsim\}\\mathcal\{C\}\_\{i\}\\leftarrow\\\{j:f\_\{i\}\(\\mathbf\{v\}\_\{i\},\\mathbf\{v\}\_\{j\}\)\\geq\\tau\_\{\\mathrm\{sim\}\}\\\}⊳\\trianglerightgoal\-aware pre\-filter, eq\. \([3](https://arxiv.org/html/2609.16446#S4.E3)\)

13:Compute

UCBj→i​\(tk\)\\mathrm\{UCB\}\_\{j\\to i\}\(t\_\{k\}\)via eq\. \([7](https://arxiv.org/html/2609.16446#S4.E7)\) for all

j∈𝒞ij\\in\\mathcal\{C\}\_\{i\}
14:With prob\.

ϵ\\epsilon, swap top\-ranked candidate with a uniformly random one in

𝒞i\\mathcal\{C\}\_\{i\}
15:

𝒫i←∅\\mathcal\{P\}\_\{i\}\\leftarrow\\varnothing
16:for

jjin descending UCB orderdo

17:if

\|𝒫i\|≥κ\|\\mathcal\{P\}\_\{i\}\|\\geq\\kappathenbreak

18:endif

19:if

UCBi→j​\(tk\)≥τacc\\mathrm\{UCB\}\_\{i\\to j\}\(t\_\{k\}\)\\geq\\tau\_\{\\mathrm\{acc\}\}and

\|𝒫j\|<κ\|\\mathcal\{P\}\_\{j\}\|<\\kappathen

20:

𝒫i←𝒫i∪\{j\}\\mathcal\{P\}\_\{i\}\\leftarrow\\mathcal\{P\}\_\{i\}\\cup\\\{j\\\},

𝒫j←𝒫j∪\{i\}\\mathcal\{P\}\_\{j\}\\leftarrow\\mathcal\{P\}\_\{j\}\\cup\\\{i\\\}
21:endif

22:endfor

23:endfor

24:foreach center

iiwith

𝒫i≠∅\\mathcal\{P\}\_\{i\}\\neq\\varnothingdo

25:

θi←FedAvg⁡\(\{θi\}∪\{θj:j∈𝒫i\}\)\\theta\_\{i\}\\leftarrow\\mathrm\{FedAvg\}\\\!\\left\(\\\{\\theta\_\{i\}\\\}\\cup\\\{\\theta\_\{j\}:j\\in\\mathcal\{P\}\_\{i\}\\\}\\right\)⊳\\trianglerightexchange; optionally personalize head and quantize wire\-format \(Sec\.[6\.4](https://arxiv.org/html/2609.16446#S6.SS4)\)

26:Compute Shapley marginals

\{ϕj→i\}j∈𝒫i\\\{\\phi\_\{j\\to i\}\\\}\_\{j\\in\\mathcal\{P\}\_\{i\}\}via eq\. \([4](https://arxiv.org/html/2609.16446#S4.E4)\)

27:foreach

j∈𝒫ij\\in\\mathcal\{P\}\_\{i\}do

28:

ϕ¯j→i←clip\[0,1\]​\(\(ϕj→i−ϕmin\)/\(ϕmax−ϕmin\)\)\\bar\{\\phi\}\_\{j\\to i\}\\leftarrow\\mathrm\{clip\}\_\{\[0,1\]\}\\\!\\big\(\(\\phi\_\{j\\to i\}\-\\phi\_\{\\min\}\)/\(\\phi\_\{\\max\}\-\\phi\_\{\\min\}\)\\big\)
29:

\(αi→j,βi→j\)←\(αi→j\+ϕ¯j→i,βi→j\+1−ϕ¯j→i\)\(\\alpha\_\{i\\to j\},\\beta\_\{i\\to j\}\)\\leftarrow\(\\alpha\_\{i\\to j\}\+\\bar\{\\phi\}\_\{j\\to i\},\\;\\beta\_\{i\\to j\}\+1\-\\bar\{\\phi\}\_\{j\\to i\}\)
30:

ni→j←ni→j\+1n\_\{i\\to j\}\\leftarrow n\_\{i\\to j\}\+1
31:endfor

32:endfor

33:Centers with

𝒫i=∅\\mathcal\{P\}\_\{i\}=\\varnothingrest this round \(*intentional isolation*; cf\. Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)\)

34:endfor

## Appendix CFull Proofs

This appendix gives the full proofs of Theorems[1](https://arxiv.org/html/2609.16446#Thmtheorem1)and[2](https://arxiv.org/html/2609.16446#Thmtheorem2)and Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)\. Throughout, we take Assumptions A1\-A3 of Sec\.[5](https://arxiv.org/html/2609.16446#S5)as given: bounded utility \(ϕ¯j→i∈\[0,1\]\\bar\{\\phi\}\_\{j\\to i\}\\in\[0,1\]\), conditional independence of the per\-round observations givenμi→j⋆\\mu^\{\\star\}\_\{i\\to j\}, and quasi\-stationary drift\|𝔼⁡\[ϕ¯j→i\(k\)\]−μi→j⋆\|≤ϵm\|\\mathbb\{E\}\[\\bar\{\\phi\}\_\{j\\to i\}^\{\(k\)\}\]\-\\mu^\{\\star\}\_\{i\\to j\}\|\\leq\\epsilon\_\{m\}within any windowΔ​tm\\Delta t\_\{m\}\.

### C\.1Proof of Theorem[1](https://arxiv.org/html/2609.16446#Thmtheorem1)\(Beta\-belief Concentration\)

LetSK=∑k=1Kϕ¯j→i\(k\)S\_\{K\}=\\sum\_\{k=1\}^\{K\}\\bar\{\\phi\}\_\{j\\to i\}^\{\(k\)\}andμ¯K=SK/K\\bar\{\\mu\}\_\{K\}=S\_\{K\}/K\. The Beta posterior afterKKupdates from priorBeta⁡\(α0,β0\)\\mathrm\{Beta\}\(\\alpha\_\{0\},\\beta\_\{0\}\)isBeta⁡\(α0\+SK,β0\+K−SK\)\\mathrm\{Beta\}\(\\alpha\_\{0\}\+S\_\{K\},\\;\\beta\_\{0\}\+K\-S\_\{K\}\), with posterior mean

μ^K=α0\+SKα0\+β0\+K=α0\+K​μ¯Kα0\+β0\+K\.\\hat\{\\mu\}\_\{K\}\\;=\\;\\frac\{\\alpha\_\{0\}\+S\_\{K\}\}\{\\alpha\_\{0\}\+\\beta\_\{0\}\+K\}\\;=\\;\\frac\{\\alpha\_\{0\}\+K\\bar\{\\mu\}\_\{K\}\}\{\\alpha\_\{0\}\+\\beta\_\{0\}\+K\}\.\(11\)Apply the triangle inequality with the empirical meanμ¯K\\bar\{\\mu\}\_\{K\}as pivot:

\|μ^K−μi→j⋆\|≤\|μ^K−μ¯K\|⏟\(I\)​prior pull\+\|μ¯K−𝔼⁡\[μ¯K\]\|⏟\(II\)​statistical noise\+\|𝔼⁡\[μ¯K\]−μi→j⋆\|⏟\(III\)​drift bias\.\|\\hat\{\\mu\}\_\{K\}\-\\mu^\{\\star\}\_\{i\\to j\}\|\\;\\leq\\;\\underbrace\{\|\\hat\{\\mu\}\_\{K\}\-\\bar\{\\mu\}\_\{K\}\|\}\_\{\(\\mathrm\{I\}\)\\;\\text\{prior pull\}\}\\;\+\\;\\underbrace\{\|\\bar\{\\mu\}\_\{K\}\-\\mathbb\{E\}\[\\bar\{\\mu\}\_\{K\}\]\|\}\_\{\(\\mathrm\{II\}\)\\;\\text\{statistical noise\}\}\\;\+\\;\\underbrace\{\|\\mathbb\{E\}\[\\bar\{\\mu\}\_\{K\}\]\-\\mu^\{\\star\}\_\{i\\to j\}\|\}\_\{\(\\mathrm\{III\}\)\\;\\text\{drift bias\}\}\.\(12\)
#### Bounding \(I\)\.

From \([11](https://arxiv.org/html/2609.16446#A3.E11)\),

μ^K−μ¯K=α0\+K​μ¯K−\(α0\+β0\+K\)​μ¯Kα0\+β0\+K=α0​\(1−μ¯K\)−β0​μ¯Kα0\+β0\+K\.\\hat\{\\mu\}\_\{K\}\-\\bar\{\\mu\}\_\{K\}\\;=\\;\\frac\{\\alpha\_\{0\}\+K\\bar\{\\mu\}\_\{K\}\-\(\\alpha\_\{0\}\+\\beta\_\{0\}\+K\)\\bar\{\\mu\}\_\{K\}\}\{\\alpha\_\{0\}\+\\beta\_\{0\}\+K\}\\;=\\;\\frac\{\\alpha\_\{0\}\(1\-\\bar\{\\mu\}\_\{K\}\)\-\\beta\_\{0\}\\,\\bar\{\\mu\}\_\{K\}\}\{\\alpha\_\{0\}\+\\beta\_\{0\}\+K\}\.Becauseμ¯K∈\[0,1\]\\bar\{\\mu\}\_\{K\}\\in\[0,1\], the numerator is bounded in absolute value bymax⁡\(α0,β0\)≤α0\+β0\\max\(\\alpha\_\{0\},\\beta\_\{0\}\)\\leq\\alpha\_\{0\}\+\\beta\_\{0\}\. Hence

\(I\)≤α0\+β0α0\+β0\+K\.\(\\mathrm\{I\}\)\\;\\leq\\;\\frac\{\\alpha\_\{0\}\+\\beta\_\{0\}\}\{\\alpha\_\{0\}\+\\beta\_\{0\}\+K\}\.\(13\)This is the deterministic “prior decay” term in \([8](https://arxiv.org/html/2609.16446#S5.E8)\): it shrinks at rateΘ⁡\(1/K\)\\Theta\(1/K\)regardless of randomness in the data\.

#### Bounding \(II\)\.

By A1 every observation lies in\[0,1\]\[0,1\], and by A2 the observations are conditionally independent givenμi→j⋆\\mu^\{\\star\}\_\{i\\to j\}\. Hoeffding’s inequality[Hoeffding \[1963\]](https://arxiv.org/html/2609.16446#bib.bib11)for the mean ofKKindependent\[0,1\]\[0,1\]\-valued variables gives

Pr\[\|μ¯K−𝔼\[μ¯K\]\|≥t\]≤2exp\(−2Kt2\)\.\\Pr\\\!\\big\[\\,\|\\bar\{\\mu\}\_\{K\}\-\\mathbb\{E\}\[\\bar\{\\mu\}\_\{K\}\]\|\\geq t\\,\\big\]\\;\\leq\\;2\\exp\\\!\\left\(\-2Kt^\{2\}\\right\)\.\(14\)Setting the right\-hand side equal toδ\\deltaand solving forttyieldst=log⁡\(2/δ\)/\(2​K\)t=\\sqrt\{\\log\(2/\\delta\)/\(2K\)\}, so with probability at least1−δ1\-\\delta

\(II\)≤log⁡\(2/δ\)2​K\.\(\\mathrm\{II\}\)\\;\\leq\\;\\sqrt\{\\frac\{\\log\(2/\\delta\)\}\{2K\}\}\.\(15\)

#### Bounding \(III\)\.

By A3, every per\-round expectation is withinϵm\\epsilon\_\{m\}ofμi→j⋆\\mu^\{\\star\}\_\{i\\to j\}:

\|𝔼⁡\[μ¯K\]−μi→j⋆\|=\|1K​∑k=1K𝔼⁡\[ϕ¯j→i\(k\)\]−μi→j⋆\|≤1K​∑k=1K\|𝔼⁡\[ϕ¯j→i\(k\)\]−μi→j⋆\|≤ϵm\.\|\\mathbb\{E\}\[\\bar\{\\mu\}\_\{K\}\]\-\\mu^\{\\star\}\_\{i\\to j\}\|\\;=\\;\\left\|\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\mathbb\{E\}\[\\bar\{\\phi\}\_\{j\\to i\}^\{\(k\)\}\]\-\\mu^\{\\star\}\_\{i\\to j\}\\right\|\\;\\leq\\;\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\big\|\\mathbb\{E\}\[\\bar\{\\phi\}\_\{j\\to i\}^\{\(k\)\}\]\-\\mu^\{\\star\}\_\{i\\to j\}\\big\|\\;\\leq\\;\\epsilon\_\{m\}\.Hence\(III\)≤ϵm\(\\mathrm\{III\}\)\\leq\\epsilon\_\{m\}\.

#### Combining\.

Substituting \(I\), \(II\), \(III\) into \([12](https://arxiv.org/html/2609.16446#A3.E12)\) yields, with probability at least1−δ1\-\\delta,

\|μ^K−μi→j⋆\|≤log⁡\(2/δ\)2​K\+α0\+β0α0\+β0\+K\+ϵm,\|\\hat\{\\mu\}\_\{K\}\-\\mu^\{\\star\}\_\{i\\to j\}\|\\;\\leq\\;\\sqrt\{\\frac\{\\log\(2/\\delta\)\}\{2K\}\}\\;\+\\;\\frac\{\\alpha\_\{0\}\+\\beta\_\{0\}\}\{\\alpha\_\{0\}\+\\beta\_\{0\}\+K\}\\;\+\\;\\epsilon\_\{m\},which is exactly \([8](https://arxiv.org/html/2609.16446#S5.E8)\)\. Convergence in probability follows: asK→∞K\\to\\infty, \(I\) and \(II\) tend to00, soμ^K→𝑝μi→j⋆\\hat\{\\mu\}\_\{K\}\\xrightarrow\{p\}\\mu^\{\\star\}\_\{i\\to j\}provided the driftϵm→0\\epsilon\_\{m\}\\to 0\.

### C\.2Proof of Theorem[2](https://arxiv.org/html/2609.16446#Thmtheorem2)\(Sublinear Partner\-Selection Regret\)

Inside a stationary window \(ϵm=0\\epsilon\_\{m\}=0\), the per\-center partner\-selection problem is a stochastic multi\-armed bandit over the post\-filter candidate set𝒞i⊆\{1,…,n−1\}\\mathcal\{C\}\_\{i\}\\subseteq\\\{1,\\dots,n\-1\\\}in whichABPSpullsκ\\kappaarms per round \(theκ\\kappaacceptances\) rather than one\. The goal\-aware pre\-filter of Sec\.[4\.1](https://arxiv.org/html/2609.16446#S4.SS1)restricts the arm set*before*the bandit sees it, and we analyze this case directly below\. We bound the cumulative regret

ℛ⁡\(T\)=T​∑j∈top\-​κμj⋆−𝔼⁡\[∑t=1T∑j∈𝒫i\(t\)μj⋆\]\\mathcal\{R\}\(T\)\\;=\\;T\\\!\\sum\_\{j\\in\\text\{top\-\}\\kappa\}\\mu^\{\\star\}\_\{j\}\\;\-\\;\\mathbb\{E\}\\\!\\left\[\\sum\_\{t=1\}^\{T\}\\sum\_\{j\\in\\mathcal\{P\}\_\{i\}^\{\(t\)\}\}\\mu^\{\\star\}\_\{j\}\\right\]by analyzing the greedy and exploratory phases separately\.

#### Greedy phase \(probability1−ϵ1\-\\epsilon\)\.

With probability1−ϵ1\-\\epsilonABPSranks candidates by their UCB1 score

UCBj→i​\(t\)=μ^i→j​\(t\)\+γ​2​log⁡tni→j​\(t\)\+1,\\mathrm\{UCB\}\_\{j\\to i\}\(t\)\\;=\\;\\hat\{\\mu\}\_\{i\\to j\}\(t\)\\;\+\\;\\gamma\\sqrt\{\\frac\{2\\log t\}\{n\_\{i\\to j\}\(t\)\+1\}\},and proposes greedily down the list untilκ\\kappaacceptances accrue\. Decompose the per\-round greedy regret as the sum ofκ\\kappasingle\-arm regrets, indexed by the rank positionr=1,…,κr=1,\\dots,\\kappaof each accepted proposal\. Each rank\-rrslot is a single\-arm UCB1 problem played against the residual candidate pool\. By the canonical UCB1 analysis[Auer et al\. \[2002\]](https://arxiv.org/html/2609.16446#bib.bib10), the cumulative regret of UCB1 with confidence radiusγ=2\\gamma=\\sqrt\{2\}overTTrounds satisfies

ℛUCB1\(T\)≤∑j:Δj\>0\(8​log⁡TΔj\+Δj\(1\+π23\)\),\\mathcal\{R\}\_\{\\mathrm\{UCB1\}\}\(T\)\\;\\leq\\;\\sum\_\{j:\\Delta\_\{j\}\>0\}\\\!\\left\(\\frac\{8\\log T\}\{\\Delta\_\{j\}\}\+\\Delta\_\{j\}\\\!\\left\(1\+\\frac\{\\pi^\{2\}\}\{3\}\\right\)\\right\),\(16\)whereΔj=μ\(1\)⋆−μ\(j\)⋆\\Delta\_\{j\}=\\mu^\{\\star\}\_\{\(1\)\}\-\\mu^\{\\star\}\_\{\(j\)\}are the suboptimality gaps\. Summingκ\\kappasuch bounds gives

ℛgreedy\(T\)≤κ∑j:Δj\>0\(8​log⁡TΔj\+Δj\(1\+π23\)\)\.\\mathcal\{R\}\_\{\\mathrm\{greedy\}\}\(T\)\\;\\leq\\;\\kappa\\sum\_\{j:\\Delta\_\{j\}\>0\}\\\!\\left\(\\frac\{8\\log T\}\{\\Delta\_\{j\}\}\+\\Delta\_\{j\}\\\!\\left\(1\+\\frac\{\\pi^\{2\}\}\{3\}\\right\)\\right\)\.\(17\)Two refinements only*decrease*this bound and so are absorbed into \([17](https://arxiv.org/html/2609.16446#A3.E17)\)\. \(a\) The receiver\-side acceptance checkUCBi→j​\(t\)≥τacc\\mathrm\{UCB\}\_\{i\\to j\}\(t\)\\geq\\tau\_\{\\mathrm\{acc\}\}in the inner\-ifof Algorithm[1](https://arxiv.org/html/2609.16446#alg1)cannot create new pulls of suboptimal arms, only suppress them\. \(b\) The Beta posterior is sharper than the empirical mean used by vanilla UCB1 \(it shrinks toward the prior at rate1/K1/K, matching term \(I\) in Theorem[1](https://arxiv.org/html/2609.16446#Thmtheorem1)\), so the exploration radius is in fact tighter than \([16](https://arxiv.org/html/2609.16446#A3.E16)\) assumes\.

#### Exploratory phase \(probabilityϵ\\epsilon\)\.

With probabilityϵ\\epsilonABPSreplaces the top\-ranked candidate with a uniformly random peer \(theϵ\\epsilon\-greedy swap step in Algorithm[1](https://arxiv.org/html/2609.16446#alg1)\)\. Its expected per\-round regret contribution is at most𝔼j∼Unif⁡\(𝒞i\)​\[Δj\]\\mathbb\{E\}\_\{j\\sim\\mathrm\{Unif\}\(\\mathcal\{C\}\_\{i\}\)\}\[\\Delta\_\{j\}\], so summing overTTrounds

ℛexplore​\(T\)≤ϵ​T⋅𝔼j∼Unif⁡\(𝒞i\)​\[Δj\]\.\\mathcal\{R\}\_\{\\mathrm\{explore\}\}\(T\)\\;\\leq\\;\\epsilon\\,T\\cdot\\mathbb\{E\}\_\{j\\sim\\mathrm\{Unif\}\(\\mathcal\{C\}\_\{i\}\)\}\[\\Delta\_\{j\}\]\.\(18\)

#### Combining\.

Adding \([17](https://arxiv.org/html/2609.16446#A3.E17)\) and \([18](https://arxiv.org/html/2609.16446#A3.E18)\) yields the bound \([9](https://arxiv.org/html/2609.16446#S5.E9)\) of the main text:

ℛ\(T\)≤κ∑j:Δj\>0\(8​log⁡TΔj\+Δj\(1\+π23\)\)\+ϵT⋅𝔼j\[Δj\]\.\\mathcal\{R\}\(T\)\\;\\leq\\;\\kappa\\sum\_\{j:\\Delta\_\{j\}\>0\}\\\!\\left\(\\frac\{8\\log T\}\{\\Delta\_\{j\}\}\+\\Delta\_\{j\}\\\!\\left\(1\+\\frac\{\\pi^\{2\}\}\{3\}\\right\)\\right\)\+\\epsilon T\\cdot\\mathbb\{E\}\_\{j\}\[\\Delta\_\{j\}\]\.

#### Recovering the𝒪⁡\(log⁡T\)\\mathcal\{O\}\(\\log T\)rate\.

Two annealing schedules suffice\. \(a\) A constantϵ=c/T\\epsilon=c/\\sqrt\{T\}leaves an𝒪⁡\(T\)\\mathcal\{O\}\(\\sqrt\{T\}\)exploratory residual that is sublinear but slower than the greedy term\. \(b\) A time\-varying scheduleϵt=min⁡\(1,c/t\)\\epsilon\_\{t\}=\\min\(1,c/t\), in the spirit of[Auer et al\. \[2002\]](https://arxiv.org/html/2609.16446#bib.bib10), gives∑t=1Tϵt=𝒪⁡\(log⁡T\)\\sum\_\{t=1\}^\{T\}\\epsilon\_\{t\}=\\mathcal\{O\}\(\\log T\)and recovers the full𝒪⁡\(κ​log⁡T\)\\mathcal\{O\}\(\\kappa\\log T\)rate\. In either case, the regret is sublinear inTT, so the average per\-round regret tends to zero\.

#### Filtered\-arm\-set refinement\.

When the goal\-aware filter of Sec\.[4\.1](https://arxiv.org/html/2609.16446#S4.SS1)restricts𝒞i\\mathcal\{C\}\_\{i\}to the top\-ρ\\rhoquantile of peers underfif\_\{i\}, the bandit plays on𝒞~i=𝒞i∩\{j:fi​\(𝐯i,𝐯j\)≥τsim\}\\tilde\{\\mathcal\{C\}\}\_\{i\}=\\mathcal\{C\}\_\{i\}\\cap\\\{j:f\_\{i\}\(\\mathbf\{v\}\_\{i\},\\mathbf\{v\}\_\{j\}\)\\geq\\tau\_\{\\mathrm\{sim\}\}\\\}\. Two changes propagate to the bound \([9](https://arxiv.org/html/2609.16446#S5.E9)\)\.

*\(i\) Reduced explore\-greedy regret\.*The sums over\{j:Δj\>0\}\\\{j:\\Delta\_\{j\}\>0\\\}and𝔼j∼Unif⁡\(𝒞i\)​\[Δj\]\\mathbb\{E\}\_\{j\\sim\\mathrm\{Unif\}\(\\mathcal\{C\}\_\{i\}\)\}\[\\Delta\_\{j\}\]are replaced by sums over𝒞~i\\tilde\{\\mathcal\{C\}\}\_\{i\}, which is a strict subset, and both terms can only decrease\.

*\(ii\) Optimal\-arm coverage bias\.*If the oracle\-best peerj⋆j^\{\\star\}fails to clearfi​\(⋅,⋅\)≥τsimf\_\{i\}\(\\cdot,\\cdot\)\\geq\\tau\_\{\\mathrm\{sim\}\}, the bandit plays on a suboptimal set and incurs an additive bias ofΔj⋆⋅T\\Delta\_\{j^\{\\star\}\}\\cdot Tagainst the oracle baseline\. This worst\-case term is controlled by a standard top\-kkcoverage guarantee: if the descriptor𝐯\\mathbf\{v\}isLL\-Lipschitz\-informative about true utility \(i\.e\.\|μj⋆−μj′⋆\|≤L​‖𝐯j−𝐯j′‖\|\\mu^\{\\star\}\_\{j\}\-\\mu^\{\\star\}\_\{j^\{\\prime\}\}\|\\leq L\\\|\\mathbf\{v\}\_\{j\}\-\\mathbf\{v\}\_\{j^\{\\prime\}\}\\\|\), then the probabilityℙ\[j⋆∉𝒞~i\]≤exp\(−cρ\|𝒞i\|\)\\mathbb\{P\}\[j^\{\\star\}\\notin\\tilde\{\\mathcal\{C\}\}\_\{i\}\]\\leq\\exp\(\-c\\rho\|\\mathcal\{C\}\_\{i\}\|\)for somec\>0c\>0depending onLL, so the expected additional regret is𝒪⁡\(T⋅exp⁡\(−c​ρ​n\)\)\\mathcal\{O\}\(T\\cdot\\exp\(\-c\\rho n\)\), which is negligible for reasonableρ\\rhoandnn\. In our experimentsρ=0\.25\\rho=0\.25,n=230n=230givesρ​n≈58\\rho n\\approx 58, the filter recovers the optimal arm with probability\>1−10−25\>1\-10^\{\-25\}under any non\-trivial descriptor\-to\-utility Lipschitz constant\.

The net effect is that \([9](https://arxiv.org/html/2609.16446#S5.E9)\) remains valid with\|𝒞i\|\|\\mathcal\{C\}\_\{i\}\|replaced by\|𝒞~i\|≈ρ​n\|\\tilde\{\\mathcal\{C\}\}\_\{i\}\|\\approx\\rho n\. The𝒪⁡\(κ​log⁡T\)\\mathcal\{O\}\(\\kappa\\log T\)asymptotic rate is preserved, and the constants improve proportionally toρ\\rho\.

### C\.3Proof of Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)\(Optimality of Intentional Isolation\)

LetVi​\(𝒫\)V\_\{i\}\(\\mathcal\{P\}\)denoteii’s expected one\-step utility \(e\.g\., held\-out AUROC\) after the round, conditioned on its active peer set being𝒫\\mathcal\{P\}\. SetVirest:=Vi​\(∅\)V\_\{i\}^\{\\mathrm\{rest\}\}:=V\_\{i\}\(\\varnothing\)andVicoll​\(𝒫\):=Vi​\(𝒫\)V\_\{i\}^\{\\mathrm\{coll\}\}\(\\mathcal\{P\}\):=V\_\{i\}\(\\mathcal\{P\}\)for𝒫≠∅\\mathcal\{P\}\\neq\\varnothing\.

#### Decomposition via Shapley efficiency\.

The Shapley value \([4](https://arxiv.org/html/2609.16446#S4.E4)\) satisfies the*efficiency*axiom: for any coalition𝒫\\mathcal\{P\},

∑j∈𝒫ϕj→i=Ui​\(𝒫\)−Ui​\(∅\),\\sum\_\{j\\in\\mathcal\{P\}\}\\phi\_\{j\\to i\}\\;=\\;U\_\{i\}\(\\mathcal\{P\}\)\-U\_\{i\}\(\\varnothing\),\(19\)whereUi​\(⋅\)U\_\{i\}\(\\cdot\)is the validation utility used to compute the Shapley value\. SinceViV\_\{i\}andUiU\_\{i\}coincide in expectation under our protocol \(both are held\-out AUROC of the post\-aggregation model\), taking expectations in \([19](https://arxiv.org/html/2609.16446#A3.E19)\) gives

Vicoll​\(𝒫\)−Virest=∑j∈𝒫𝔼⁡\[ϕj→i\]\.V\_\{i\}^\{\\mathrm\{coll\}\}\(\\mathcal\{P\}\)\-V\_\{i\}^\{\\mathrm\{rest\}\}\\;=\\;\\sum\_\{j\\in\\mathcal\{P\}\}\\mathbb\{E\}\[\\phi\_\{j\\to i\}\]\.\(20\)

#### Translating to the clipped scale\.

Within the clipping range,ϕ¯j→i\\bar\{\\phi\}\_\{j\\to i\}relates toϕj→i\\phi\_\{j\\to i\}via the affine map \([5](https://arxiv.org/html/2609.16446#S4.E5)\),

ϕ¯j→i=ϕj→i−ϕminϕmax−ϕmin⟺ϕj→i=\(ϕmax−ϕmin\)​ϕ¯j→i\+ϕmin,\\bar\{\\phi\}\_\{j\\to i\}\\;=\\;\\frac\{\\phi\_\{j\\to i\}\-\\phi\_\{\\min\}\}\{\\phi\_\{\\max\}\-\\phi\_\{\\min\}\}\\;\\Longleftrightarrow\\;\\phi\_\{j\\to i\}\\;=\\;\(\\phi\_\{\\max\}\-\\phi\_\{\\min\}\)\\,\\bar\{\\phi\}\_\{j\\to i\}\+\\phi\_\{\\min\},so𝔼⁡\[ϕj→i\]=\(ϕmax−ϕmin\)​μi→j⋆\+ϕmin\\mathbb\{E\}\[\\phi\_\{j\\to i\}\]=\(\\phi\_\{\\max\}\-\\phi\_\{\\min\}\)\\mu^\{\\star\}\_\{i\\to j\}\+\\phi\_\{\\min\}\. The negative\-transfer thresholdcnegc\_\{\\mathrm\{neg\}\}is the value ofμi→j⋆\\mu^\{\\star\}\_\{i\\to j\}at which one\-peer collaboration breaks even,Vicoll​\(\{j\}\)=VirestV\_\{i\}^\{\\mathrm\{coll\}\}\(\\\{j\\\}\)=V\_\{i\}^\{\\mathrm\{rest\}\}, i\.e\.𝔼⁡\[ϕj→i\]=0\\mathbb\{E\}\[\\phi\_\{j\\to i\}\]=0\. Solving,

cneg=−ϕminϕmax−ϕmin\.c\_\{\\mathrm\{neg\}\}\\;=\\;\\frac\{\-\\phi\_\{\\min\}\}\{\\phi\_\{\\max\}\-\\phi\_\{\\min\}\}\.\(21\)For the paper’s choice\(ϕmin,ϕmax\)=\(−0\.1,\+0\.1\)\(\\phi\_\{\\min\},\\phi\_\{\\max\}\)=\(\-0\.1,\+0\.1\)this givescneg=0\.5c\_\{\\mathrm\{neg\}\}=0\.5, which coincides withτacc\\tau\_\{\\mathrm\{acc\}\}used in the algorithm\. The mapμi→j⋆<cneg⇔𝔼⁡\[ϕj→i\]<0\\mu^\{\\star\}\_\{i\\to j\}<c\_\{\\mathrm\{neg\}\}\\iff\\mathbb\{E\}\[\\phi\_\{j\\to i\}\]<0is therefore exact\.

#### Strict dominance of resting\.

Supposeμi→j⋆<cneg\\mu^\{\\star\}\_\{i\\to j\}<c\_\{\\mathrm\{neg\}\}for everyj∈𝒫j\\in\\mathcal\{P\}\. Then𝔼⁡\[ϕj→i\]<0\\mathbb\{E\}\[\\phi\_\{j\\to i\}\]<0for allj∈𝒫j\\in\\mathcal\{P\}, so by \([20](https://arxiv.org/html/2609.16446#A3.E20)\),

Vicoll​\(𝒫\)−Virest=∑j∈𝒫𝔼⁡\[ϕj→i\]<0,V\_\{i\}^\{\\mathrm\{coll\}\}\(\\mathcal\{P\}\)\-V\_\{i\}^\{\\mathrm\{rest\}\}\\;=\\;\\sum\_\{j\\in\\mathcal\{P\}\}\\mathbb\{E\}\[\\phi\_\{j\\to i\}\]\\;<\\;0,which is exactly the claimed inequality \([10](https://arxiv.org/html/2609.16446#S5.E10)\),Virest\>Vicoll​\(𝒫\)V\_\{i\}^\{\\mathrm\{rest\}\}\>V\_\{i\}^\{\\mathrm\{coll\}\}\(\\mathcal\{P\}\)\.

#### Refinement under approximate Shapley\.

When the Shapley values are estimated via truncated Monte\-Carlo \(TMC, the default forκ\>5\\kappa\>5, cf\. Sec\.[5](https://arxiv.org/html/2609.16446#S5)\), the per\-peer estimator carries an additive bias bounded by someη\>0\\eta\>0\. Equation \([20](https://arxiv.org/html/2609.16446#A3.E20)\) then becomes

Vicoll​\(𝒫\)−Virest=∑j∈𝒫𝔼⁡\[ϕj→i\]−g⁡\(\|𝒫\|\),V\_\{i\}^\{\\mathrm\{coll\}\}\(\\mathcal\{P\}\)\-V\_\{i\}^\{\\mathrm\{rest\}\}\\;=\\;\\sum\_\{j\\in\\mathcal\{P\}\}\\mathbb\{E\}\[\\phi\_\{j\\to i\}\]\-g\(\|\\mathcal\{P\}\|\),with\|g⁡\(\|𝒫\|\)\|≤η​\|𝒫\|\|g\(\|\\mathcal\{P\}\|\)\|\\leq\\eta\|\\mathcal\{P\}\|\. Strict dominance survives whenever the true expected gain margin exceeds the approximation slack,∑j𝔼⁡\[ϕj→i\]<−η​\|𝒫\|\\sum\_\{j\}\\mathbb\{E\}\[\\phi\_\{j\\to i\}\]<\-\\eta\|\\mathcal\{P\}\|, which is satisfied with marginη\\etato spare under the strict inequalityμi→j⋆<cneg\\mu^\{\\star\}\_\{i\\to j\}<c\_\{\\mathrm\{neg\}\}\. This is the form quoted in the proof sketch of Sec\.[5\.3](https://arxiv.org/html/2609.16446#S5.SS3)\.

#### Bandwidth saving\.

Specializing the cost equation \([2](https://arxiv.org/html/2609.16446#S3.E2)\) to a single round with𝒫irest=∅\\mathcal\{P\}\_\{i\}^\{\\mathrm\{rest\}\}=\\varnothinggives a bandwidth of00, while collaborating with\|𝒫\|\|\\mathcal\{P\}\|peers costs2​\|𝒫\|​cegress​\|W\|2\|\\mathcal\{P\}\|\\,c\_\{\\mathrm\{egress\}\}\|W\|\(one upload and one download per peer per round\)\. The saving is therefore exactlyCirest=2​\|𝒫\|​cegress​\|W\|C\_\{i\}^\{\\mathrm\{rest\}\}=2\|\\mathcal\{P\}\|\\,c\_\{\\mathrm\{egress\}\}\|W\|per round\.

#### Connection to Theorem[1](https://arxiv.org/html/2609.16446#Thmtheorem1)\.

Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)reasons about the*oracle*thresholdcnegc\_\{\\mathrm\{neg\}\}, butABPSonly sees the noisy posterior estimateμ^K\\hat\{\\mu\}\_\{K\}\. Combining the lemma with \([8](https://arxiv.org/html/2609.16446#S5.E8)\): with probability≥1−δ\\geq 1\-\\delta, the UCB\-induced rule rests whenever

μ^K\+γ​2​log⁡Kni→j\+1<τacc=cneg,\\hat\{\\mu\}\_\{K\}\+\\gamma\\sqrt\{\\tfrac\{2\\log K\}\{n\_\{i\\to j\}\+1\}\}\\;<\\;\\tau\_\{\\mathrm\{acc\}\}\\;=\\;c\_\{\\mathrm\{neg\}\},whereni→j\+1n\_\{i\\to j\}\+1is the per\-arm pull count from Eq\. \([7](https://arxiv.org/html/2609.16446#S4.E7)\), which itself grows withKKunder any non\-trivial selection rule\. The condition above is implied byμi→j⋆<cneg−\[log⁡\(2/δ\)/\(2​K\)\+\(α0\+β0\)/\(α0\+β0\+K\)\+ϵm\+γ​2​log⁡K/\(ni→j\+1\)\]\\mu^\{\\star\}\_\{i\\to j\}<c\_\{\\mathrm\{neg\}\}\-\\big\[\\sqrt\{\\log\(2/\\delta\)/\(2K\)\}\+\(\\alpha\_\{0\}\+\\beta\_\{0\}\)/\(\\alpha\_\{0\}\+\\beta\_\{0\}\+K\)\+\\epsilon\_\{m\}\+\\gamma\\sqrt\{2\\log K/\(n\_\{i\\to j\}\+1\)\}\\big\]\. AsK→∞K\\to\\inftyandϵm→0\\epsilon\_\{m\}\\to 0, every term in the bracket vanishes \(the exploration term shrinks becauseni→j\+1n\_\{i\\to j\}\+1grows at rateΘ⁡\(K\)\\Theta\(K\)for any peer ever pulled by UCB\), and the algorithmic rule converges to the oracle rule of Lemma[1](https://arxiv.org/html/2609.16446#Thmlemma1)\.

## Acknowledgments

This work was supported by NSF grants OAC\-2609072 \(CHAI\), SFS\-2335969 \(MASTER\), OAC\-2104076 \(CANDY\), and SATC\-2030624 \(TAURUS\)\.