Opinion Dynamics-based Coalition Formation for Federated Learning in Heterogeneous IoT Systems

# Opinion Dynamics-based Coalition Formation for Federated Learning in Heterogeneous IoT Systems
Source: [https://arxiv.org/html/2609.19695](https://arxiv.org/html/2609.19695)
Mohammed El HanjriAnas AbouaomarEmail:[anas\_abouaomar@um5\.ac\.ma](mailto:[email protected])Affiliation:ENSIAS, Mohammed V University in Rabat, MoroccoHamidou TembineEmail:[hamidou\.tembine@uqtr\.ca](mailto:[email protected])Affiliation:Department of Electrical and Computer Engineering, School of Engineering, University of Quebec at Trois\-Rivieres, Quebec, CanadaAbdellatif KobbaneEmail:[abdellatif\.kobbane@ensias\.um5\.ac\.ma](mailto:[email protected])Affiliation:ENSIAS, Mohammed V University in Rabat, Morocco

###### Abstract

Federated learning \(FL\) enables privacy\-preserving, on\-device training across heterogeneous Internet\-of\-Things \(IoT\) deployments such as smart\-city water\-metering networks, where each smart meter observes a household\-specific consumption time series\. Under such statistical heterogeneity, the standard Federated Averaging \(FedAvg\) aggregation averages dissimilar local models into a single global model that may fail to capture client\-specific patterns\. We address this by forming client coalitions directly in the local\-weight space and aggregating at the coalition level\. Extending a prior weight\-driven coalition\-formation scheme, we model coalition formation as a Hegselmann–Krause \(HK\) bounded\-confidence opinion\-dynamics process acting on the local weights, and develop variants of the HK interaction based on Euclidean\-distance and cosine\-similarity confidence criteria\. The framework is applied to short\-term water\-consumption forecasting with local Long Short\-Term Memory \(LSTM\) models and evaluated against FedAvg, Per\-FedAvg, FedProx, and FedAvg with Euclidean\-distance or cosine\-similarity coalition formation\. Experiments on a real smart\-metering dataset of water consumption show that the proposed HK\-based coalition formation produces stable, endogenous coalition structures within at most ten inner iterations, incurs no additional client\-side computation or communication compared to FedAvg, and reduces the average MAE by up to 54% relative to FedAvg, 39% relative to FedProx, and 24% relative to Per\-FedAvg, while achieving the highest global accuracy \(83–85%\)\.

###### keywords

Federated Learning, Internet of Things, Data Heterogeneity, Coalition Formation, Hegselmann–Krause Opinion Dynamics, Water Consumption Forecasting, Smart Cities

## 1Introduction

Federated learning \(FL\) has become a standard paradigm for training models collaboratively across large populations of distributed, privacy\-sensitive Internet\-of\-Things \(IoT\) devices[McMahan et al\. \(2017\)](https://arxiv.org/html/2609.19695#bib.bib29);[Kairouz et al\. \(2021\)](https://arxiv.org/html/2609.19695#bib.bib7)\. By keeping raw data on each device and exchanging only model parameters, FL preserves privacy and reduces communication relative to centralized training, which makes it attractive for smart\-city and edge deployments[Pandya et al\. \(2023\)](https://arxiv.org/html/2609.19695#bib.bib9);[Jiang et al\. \(2020\)](https://arxiv.org/html/2609.19695#bib.bib14);[Singh et al\. \(2022\)](https://arxiv.org/html/2609.19695#bib.bib10)\. Its effectiveness in practice, however, is limited by statistical heterogeneity: IoT devices observe data from distinct, non\-identical distributions, so their local objectives conflict and the standard Federated Averaging \(FedAvg\) aggregation blends dissimilar local models into a single global model that is a poor compromise for individual devices[Li et al\. \(2020\)](https://arxiv.org/html/2609.19695#bib.bib31)\. Improving the efficiency of FL under such heterogeneity is the problem this paper addresses\.

We study this problem through a representative heterogeneous IoT use case: short\-term load forecasting \(STLF\) of water consumption in smart cities\. Networked smart meters record household\-level consumption that is highly heterogeneous, shaped by behavioral, demographic, and environmental factors that differ widely across households[Kavya et al\. \(2023\)](https://arxiv.org/html/2609.19695#bib.bib3);[Drogkoula et al\. \(2023\)](https://arxiv.org/html/2609.19695#bib.bib4), and the data are privacy\-sensitive, since fine\-grained traces can reveal occupancy and activity patterns\. Each device trains a local Long Short\-Term Memory \(LSTM\) forecaster, well suited to such series through its gated memory of long\-range temporal dependencies[Nasser et al\. \(2020\)](https://arxiv.org/html/2609.19695#bib.bib5);[Bezzar et al\. \(2022\)](https://arxiv.org/html/2609.19695#bib.bib6)\. This setting exhibits exactly the non\-IID structure that challenges FL and serves as the testbed for our experiments; the proposed method itself is independent of the forecasting model and operates only on the exchanged weights\.

A natural response to heterogeneity is to aggregate not across all devices at once, but within groups of compatible clients whose models can be combined without mutual interference; the advantage of such personalized or group\-wise models over a single shared model is well documented for load forecasting[Reguieg et al\. \(2023\)](https://arxiv.org/html/2609.19695#bib.bib8);[Xu et al\. \(2024\)](https://arxiv.org/html/2609.19695#bib.bib23)\. The key question is how to form these groups without access to the private data: a privacy\-compatible signal is the local model itself, since clients with similar data tend to learn similar weights\. This idea has been exploited through weight\-driven coalition formation with a fixed number of distance\-based coalitions[El Hanjri et al\. \(2024\)](https://arxiv.org/html/2609.19695#bib.bib33)\.

In this paper we generalize that scheme by casting coalition formation as a bounded\-confidence opinion\-dynamics process on the local weights\. Specifically, we apply the Hegselmann–Krause \(HK\) model[Rainer and Krause \(2002\)](https://arxiv.org/html/2609.19695#bib.bib32)to the post\-training local models: each client is an agent whose “opinion” is its weight vector, and agents reinforce one another only when their models are mutually compatible\. Running this interaction to its fixed points yields an endogenous partition, since the number and membership of coalitions emerge from the data rather than being fixed in advance, and it naturally isolates atypical clients as outliers instead of forcing them into a coalition\. We study different notions of compatibility: a Euclidean confidence ball, a cosine\-similarity threshold, and an asymmetric cosine\-confidence bound, motivated by the observation that the direction of high\-dimensional weights is often more informative than their absolute distance\. In this paper the contributions are as follows\.

- •We formulate heterogeneity\-aware FL aggregation as a coalition\-formation problem on the local weights, and show how a HK bounded\-confidence interaction yields coalitions whose number and membership are endogenous and adaptive\.
- •We develop three instantiations of the interaction \(Euclidean, cosine\-similarity, and asymmetric cosine\-confidence\) and a coalition aggregation algorithm that combines coalition barycenters into the shared model, with a complexity analysis showing no additional client\-side computation or communication relative to FedAvg\.
- •We instantiate the framework for short\-term water\-consumption forecasting with local LSTM models and evaluate it on a smart\-city dataset against FedAvg with Euclidean coalitions, FedAvg with cosine coalitions, FedProx[Li et al\. \(2020\)](https://arxiv.org/html/2609.19695#bib.bib31), and Per\-FedAvg[Reguieg et al\. \(2023\)](https://arxiv.org/html/2609.19695#bib.bib8), reporting coalition structure, convergence, and forecasting accuracy\.

The remainder of the paper is organized as follows\. Section[2](https://arxiv.org/html/2609.19695#S2)reviews related work\. Section[3](https://arxiv.org/html/2609.19695#S3)presents the system model and the problem formulation\. Section[4](https://arxiv.org/html/2609.19695#S4)develops the proposed HK\-based coalition\-formation framework\. Sections[5](https://arxiv.org/html/2609.19695#S5)and[6](https://arxiv.org/html/2609.19695#S6)describe the experimental setup and discuss the results, and the conclusion and percpectives in Section[7](https://arxiv.org/html/2609.19695#S7)\.

## 2Related Work

FL was introduced as a communication\-efficient, privacy\-preserving alternative to centralized training[McMahan et al\. \(2017\)](https://arxiv.org/html/2609.19695#bib.bib29);[Hard et al\. \(2018\)](https://arxiv.org/html/2609.19695#bib.bib12), and surveys document its breadth and open problems[Kairouz et al\. \(2021\)](https://arxiv.org/html/2609.19695#bib.bib7), including smart\-city applications[Pandya et al\. \(2023\)](https://arxiv.org/html/2609.19695#bib.bib9);[Jiang et al\. \(2020\)](https://arxiv.org/html/2609.19695#bib.bib14)and privacy\-preserving IoT deployments[Singh et al\. \(2022\)](https://arxiv.org/html/2609.19695#bib.bib10)\. A central obstacle is statistical heterogeneity: when client data are non\-IID, FedAvg converges slowly and the resulting global model may deviate substantially from individual client optima\. FedProx[Li et al\. \(2020\)](https://arxiv.org/html/2609.19695#bib.bib31)adds a proximal term to the local objective to limit client drift, improving stability under heterogeneity\. Federated approaches have also been applied to energy and demand management at the edge[Rezazadeh and Bartzoudis \(2022\)](https://arxiv.org/html/2609.19695#bib.bib1);[El Hanjri et al\. \(2026\)](https://arxiv.org/html/2609.19695#bib.bib15), with incentive mechanisms studied to encourage participation[Khan et al\. \(2020\)](https://arxiv.org/html/2609.19695#bib.bib13)\. Our work targets the same heterogeneity obstacle but, rather than regularizing a single global model, restructures the aggregation itself around coalitions of compatible clients\.

A complementary line of work produces multiple or client\-specific models\. Personalized FL adapts the shared model to each client, for instance the meta\-learning\-based Per\-FedAvg, evaluated against FedAvg under Dirichlet\-skewed data in[Reguieg et al\. \(2023\)](https://arxiv.org/html/2609.19695#bib.bib8), whereas decentralized, heterogeneity\-aware schemes have been proposed for smart\-home energy and comfort prediction[Xu et al\. \(2024\)](https://arxiv.org/html/2609.19695#bib.bib23)\. Clustering ideas, classical in edge and sensor networks \(e\.g\.kk\-means grouping of devices[El Mezouary et al\. \(2016\)](https://arxiv.org/html/2609.19695#bib.bib11)\), have been transposed to FL to aggregate within groups of similar clients\. Closest to this paper, the scheme of[El Hanjri et al\. \(2024\)](https://arxiv.org/html/2609.19695#bib.bib33)forms coalitions directly in the local\-weight space using a distance between models and a fixed number of coalitions\. The present paper differs in that the coalitions are not prescribed: they emerge as fixed points of a bounded\-confidence interaction, and we additionally introduce cosine\-based compatibility, which is invariant to weight magnitude\.

Beyond fixed\-size coalitions, a growing body of recent work clusters FL clients directly from their model parameters\. WSCC[Tian et al\. \(2022\)](https://arxiv.org/html/2609.19695#bib.bib16)applies affinity propagation to the cosine distance between client weight vectors, so that the number of clusters is determined automatically rather than prescribed; FedClust[Islam et al\. \(2024\)](https://arxiv.org/html/2609.19695#bib.bib17)forms clusters in a one\-shot manner from strategically selected partial weights of the locally trained models, avoiding both the long stabilization phase of iterative clustered FL and a predefined number of groups\. Game\-theoretic formulations instead treat grouping as an explicit coalition game: DualGFL[Chen et al\. \(2025\)](https://arxiv.org/html/2609.19695#bib.bib18)couples a lower\-level hedonic game, in which clients form coalitions according to preference profiles, with an upper\-level auction in which coalitions bid for participation in training\. These works confirm that weight similarity is a reliable, privacy\-compatible proxy for data similarity, but they rely on generic clustering heuristics or hand\-designed utility functions\. In contrast, we obtain the partition as the set of fixed points of a bounded\-confidence interaction, a class of dynamics whose convergence and clustering properties are supported by a mature theory[Bernardo et al\. \(2024\)](https://arxiv.org/html/2609.19695#bib.bib19), which yields coalitions that are simultaneously endogenous, outlier\-aware, and free of any auxiliary clustering machinery\.

Bounded\-confidence models from opinion dynamics describe how agents update their positions by averaging only over peers within a confidence range\. The HK model[Rainer and Krause \(2002\)](https://arxiv.org/html/2609.19695#bib.bib32)is the canonical such model: agents repeatedly move to the mean of the opinions lying within a confidence bound, and the dynamics settle into clusters of consensus separated by gaps\. This clustering\-by\-confidence behavior is a natural fit for coalition formation in FL, where each client holds a model \(an opinion\) and only compatible models should be aggregated together\. We are, to our knowledge, applying the HK interaction to the local weights of an FL system to drive coalition formation, and we extend it from the classical Euclidean ball to cosine\-similarity and asymmetric cosine\-confidence neighborhoods suited to high\-dimensional model parameters\.

The use case in which we evaluate the framework, short\-term load and demand forecasting, has its own substantial literature\. Machine\-learning models, and deep recurrent networks in particular, are now standard for this task\. Marino et al\.[Marino et al\. \(2016\)](https://arxiv.org/html/2609.19695#bib.bib24)used sequence\-to\-sequence LSTM architectures for building energy load, with strong results at fine temporal resolution but limited gains at coarser \(hourly\) granularity\. Evolutionary and feature\-selection strategies have been used to tune LSTM forecasters[Almalaq and Zhang \(2018\)](https://arxiv.org/html/2609.19695#bib.bib25);[Bouktif et al\. \(2018\)](https://arxiv.org/html/2609.19695#bib.bib26), though their accuracy often degrades when transferred to new datasets\. To cope with consumption variability, some works group users with similar profiles or pool heterogeneous series to broaden diversity[Stephen et al\. \(2015\)](https://arxiv.org/html/2609.19695#bib.bib27);[Shi et al\. \(2017\)](https://arxiv.org/html/2609.19695#bib.bib28)\. In the water domain, LSTM\- and micro\-service\-based systems have been proposed for urban demand prediction[Nasser et al\. \(2020\)](https://arxiv.org/html/2609.19695#bib.bib5);[Kavya et al\. \(2023\)](https://arxiv.org/html/2609.19695#bib.bib3);[Drogkoula et al\. \(2023\)](https://arxiv.org/html/2609.19695#bib.bib4), and time\-series analysis has been applied to household\-level consumption[Bezzar et al\. \(2022\)](https://arxiv.org/html/2609.19695#bib.bib6)\. These forecasting approaches are predominantly centralized; the privacy and scalability constraints they face are precisely what motivate the federated, heterogeneity\-aware formulation studied here\.

Federated formulations of short\-term load forecasting itself have emerged very recently\. In[Rahman et al\. \(2025a\)](https://arxiv.org/html/2609.19695#bib.bib20), the authors personalize FL\-based load prediction under non\-IID metering data by adapting per\-client learning rates through meta\-learning, and a follow\-up work extends federated forecasting to multihop smart\-metering networks with limited connectivity[Rahman et al\. \(2025b\)](https://arxiv.org/html/2609.19695#bib.bib22)\. Bose and Kim[Bose and Kim \(2023\)](https://arxiv.org/html/2609.19695#bib.bib21)instead keep selected personalization layers of the forecasting model local to each client, so that shared layers capture common temporal structure while private layers absorb heterogeneity across buildings\. These approaches validate FL as a practical substrate for consumption forecasting, but they personalize each client individually rather than restructuring the aggregation itself, and they focus on electricity; group\-level aggregation for water\-consumption forecasting, as studied here, remains largely unexplored\.

In summary, FL restores privacy but struggles with statistical heterogeneity, and existing remedies either regularize a single global model \(FedProx\), personalize per client \(Per\-FedAvg\), or cluster clients with a fixed group structure \(weight\-driven coalitions[El Hanjri et al\. \(2024\)](https://arxiv.org/html/2609.19695#bib.bib33)\); deep\-learning forecasters, meanwhile, are accurate but conventionally centralized\. This paper unifies the coalition idea with bounded\-confidence opinion dynamics, yielding an endogenous, outlier\-aware partition of clients and a family of compatibility geometries, and evaluates it on smart\-city water\-consumption forecasting as a representative heterogeneous IoT use case\.

## 3System Model and Problem Formulation

We consider an FL system over a heterogeneous IoT network, instantiated on a smart\-city water\-metering scenario\. A set of edge clients collaboratively trains forecasting models under the coordination of an edge server, without sharing raw data\. We keep the local learner generic and treat the LSTM forecaster and the water\-consumption data as the concrete instantiation used in the experiments \(Section[5](https://arxiv.org/html/2609.19695#S5)\); the focus here is the FL protocol and the statistical heterogeneity that makes a single shared model inadequate\. The overall architecture is shown in Figure[1](https://arxiv.org/html/2609.19695#S3.F1), and the notation is summarized in Table[1](https://arxiv.org/html/2609.19695#S3.T1)\.

![Refer to caption](https://arxiv.org/html/2609.19695v1/Picture1.png)Figure 1:Network architecture: a single edge \(MEC\) server orchestrates federated training overNNIoT clients \(smart meters\); each client trains a local model on its private data and exchanges only model parameters\.Table 1:Summary of notation\.### 3\.1System architecture

Let𝒩=\{1,…,N\}\\mathcal\{N\}=\\\{1,\\dots,N\\\}be the set of clients, each an IoT device \(in the case study, a household smart meter\) holding a private dataset and training locally\. A single MEC server[Filali et al\. \(2020\)](https://arxiv.org/html/2609.19695#bib.bib2)orchestrates and aggregates; it holds no data\. Training runs overTTrounds: at roundttthe server selects an active subset𝒮t⊆𝒩\\mathcal\{S\}^\{t\}\\subseteq\\mathcal\{N\}with\|𝒮t\|=S≤N\|\\mathcal\{S\}^\{t\}\|=S\\leq N\(reflecting connectivity and energy constraints\), and only those clients compute and communicate\.

Each clientiitrains a modelf⁡\(⋅,𝝎\)f\(\\cdot;\\bm\{\\omega\}\)with parameters𝝎∈ℝd\\bm\{\\omega\}\\in\\mathbb\{R\}^\{d\}; all clients share the same architecture, so the parameter vectors are directly comparable\. In roundtt, clientiistarts from the shared model𝜽t−1\\bm\{\\theta\}^\{t\-1\}and runsEElocal SGD epochs \(learning rateη\\eta\) on its empirical riskFi​\(𝝎\)=1mi​∑n=1miℓ⁡\(f⁡\(𝐱i\(n\),𝝎\),yi\(n\)\)F\_\{i\}\(\\bm\{\\omega\}\)=\\frac\{1\}\{m\_\{i\}\}\\sum\_\{n=1\}^\{m\_\{i\}\}\\ell\(f\(\\mathbf\{x\}\_\{i\}^\{\(n\)\};\\bm\{\\omega\}\),y\_\{i\}^\{\(n\)\}\)with squared\-error lossℓ\\ell,

𝝎it,e\+1=𝝎it,e−η∇Fi\(𝝎it,e\),e=0,…,E−1,𝝎it,0=𝜽t−1,\\bm\{\\omega\}\_\{i\}^\{t,e\+1\}=\\bm\{\\omega\}\_\{i\}^\{t,e\}\-\\eta\\,\\nabla F\_\{i\}\\\!\\bigl\(\\bm\{\\omega\}\_\{i\}^\{t,e\}\\bigr\),\\quad e=0,\\dots,E\-1,\\quad\\bm\{\\omega\}\_\{i\}^\{t,0\}=\\bm\{\\theta\}^\{t\-1\},\(1\)producing the local model𝝎it≜𝝎it,E\\bm\{\\omega\}\_\{i\}^\{t\}\\triangleq\\bm\{\\omega\}\_\{i\}^\{t,E\}\. It uploads only𝝎it\\bm\{\\omega\}\_\{i\}^\{t\}, and the server aggregates\. Under FedAvg[McMahan et al\. \(2017\)](https://arxiv.org/html/2609.19695#bib.bib29),

𝜽t=∑i∈𝒮tmiMt​𝝎it,Mt=∑i∈𝒮tmi\.\\bm\{\\theta\}^\{t\}=\\sum\_\{i\\in\\mathcal\{S\}^\{t\}\}\\frac\{m\_\{i\}\}\{M^\{t\}\}\\,\\bm\{\\omega\}\_\{i\}^\{t\},\\qquad M^\{t\}=\\sum\_\{i\\in\\mathcal\{S\}^\{t\}\}m\_\{i\}\.\(2\)Equation \([2](https://arxiv.org/html/2609.19695#S3.E2)\) is the baseline our coalition\-aware aggregation replaces\. Only model parameters ever leave a device, and the server keeps updates only transiently for aggregation, which preserves data locality under an honest\-but\-curious server\.

### 3\.2Local data and learning task

Clientiiholds a private dataset𝒟i=\{\(𝐱i\(n\),yi\(n\)\)\}n=1mi\\mathcal\{D\}\_\{i\}=\\\{\(\\mathbf\{x\}\_\{i\}^\{\(n\)\},y\_\{i\}^\{\(n\)\}\)\\\}\_\{n=1\}^\{m\_\{i\}\},mi=\|𝒟i\|m\_\{i\}=\|\\mathcal\{D\}\_\{i\}\|, drawn from a client\-specific distribution𝒫i\\mathcal\{P\}\_\{i\}, withM=∑imiM=\\sum\_\{i\}m\_\{i\}\. In the case study the data are univariate water\-consumption series mapped to fixed\-length look\-back windows𝐱i\(n\)\\mathbf\{x\}\_\{i\}^\{\(n\)\}with one\-step\-ahead targetsyi\(n\)y\_\{i\}^\{\(n\)\}\(windowing details in Section[5](https://arxiv.org/html/2609.19695#S5)\)\. The forecasterffis an LSTM, suited to consumption data through its gated memory of temporal dependencies[Nasser et al\. \(2020\)](https://arxiv.org/html/2609.19695#bib.bib5);[Bezzar et al\. \(2022\)](https://arxiv.org/html/2609.19695#bib.bib6); the proposed framework is agnostic to this choice and operates solely on the parameter vectors𝝎\\bm\{\\omega\}\.

### 3\.3Statistical heterogeneity

Statistical heterogeneity is the central difficulty addressed here\. Clients are non\-IID \(𝒫i≠𝒫j\\mathcal\{P\}\_\{i\}\\neq\\mathcal\{P\}\_\{j\}fori≠ji\\neq j\), combining distributional heterogeneity \(distinct data\-generating processes, so the local minimizers𝝎i⋆=arg⁡min𝝎​ℱi​\(𝝎\)\\bm\{\\omega\}\_\{i\}^\{\\star\}=\\arg\\min\_\{\\bm\{\\omega\}\}\\mathcal\{F\}\_\{i\}\(\\bm\{\\omega\}\)of the population risksℱi\\mathcal\{F\}\_\{i\}differ across clients\) and quantity skew through unequalmim\_\{i\}\. We quantify distributional heterogeneity by a standard bounded\-dissimilarity condition: there existζ≥0\\zeta\\geq 0,β≥1\\beta\\geq 1with

1N​∑i∈𝒩‖∇Fi​\(𝝎\)−∇F​\(𝝎\)‖22≤ζ2\+β2​‖∇F​\(𝝎\)‖22,∀𝝎,\\frac\{1\}\{N\}\\sum\_\{i\\in\\mathcal\{N\}\}\\bigl\\\|\\nabla F\_\{i\}\(\\bm\{\\omega\}\)\-\\nabla F\(\\bm\{\\omega\}\)\\bigr\\\|\_\{2\}^\{2\}\\;\\leq\\;\\zeta^\{2\}\+\\beta^\{2\}\\,\\bigl\\\|\\nabla F\(\\bm\{\\omega\}\)\\bigr\\\|\_\{2\}^\{2\},\\qquad\\forall\\bm\{\\omega\},\(3\)whereζ=0\\zeta=0is the IID case and largeζ\\zetasignals conflicting local optima\. The global objective is the sample\-weighted aggregate

min𝝎∈ℝd⁡F⁡\(𝝎\)=∑i∈𝒩pi​ℱi​\(𝝎\),pi=miM,\\min\_\{\\bm\{\\omega\}\\in\\mathbb\{R\}^\{d\}\}\\;F\(\\bm\{\\omega\}\)=\\sum\_\{i\\in\\mathcal\{N\}\}p\_\{i\}\\,\\mathcal\{F\}\_\{i\}\(\\bm\{\\omega\}\),\\qquad p\_\{i\}=\\frac\{m\_\{i\}\}\{M\},\(4\)which FedAvg targets through \([2](https://arxiv.org/html/2609.19695#S3.E2)\)\. The premise of this work is that under largeζ\\zetaa single minimizer of \([4](https://arxiv.org/html/2609.19695#S3.E4)\) is a poor compromise, which motivates solving it at the level of coalitions of compatible clients \(Sections[3](https://arxiv.org/html/2609.19695#S3)–[4](https://arxiv.org/html/2609.19695#S4)\)\. Beyond \([3](https://arxiv.org/html/2609.19695#S3.E3)\), the later discussion assumes only standard conditions \(ρ\\rho\-smoothFiF\_\{i\}, unbiased bounded\-variance stochastic gradients, a common parameterization\); we make no convergence claim beyond what these support\.

FedAvg returns a single model𝜽\\bm\{\\theta\}that minimizes the aggregate objective \([4](https://arxiv.org/html/2609.19695#S3.E4)\)\. When clients are heterogeneous, the local minimizers𝝎i⋆\\bm\{\\omega\}\_\{i\}^\{\\star\}are spread out and the aggregate minimizer is a compromise that may be far from each of them: for any𝜽\\bm\{\\theta\},

∑i∈𝒩pi​‖𝝎i⋆−𝜽‖22≥∑i∈𝒩pi​‖𝝎i⋆−𝝎¯⋆‖22=Varp​\(𝝎⋆\),𝝎¯⋆=∑ipi​𝝎i⋆,\\sum\_\{i\\in\\mathcal\{N\}\}p\_\{i\}\\,\\bigl\\\|\\bm\{\\omega\}\_\{i\}^\{\\star\}\-\\bm\{\\theta\}\\bigr\\\|\_\{2\}^\{2\}\\;\\geq\\;\\sum\_\{i\\in\\mathcal\{N\}\}p\_\{i\}\\,\\bigl\\\|\\bm\{\\omega\}\_\{i\}^\{\\star\}\-\\bar\{\\bm\{\\omega\}\}^\{\\star\}\\bigr\\\|\_\{2\}^\{2\}\\;=\\;\\mathrm\{Var\}\_\{p\}\\\!\\bigl\(\\bm\{\\omega\}^\{\\star\}\\bigr\),\\qquad\\bar\{\\bm\{\\omega\}\}^\{\\star\}=\\sum\_\{i\}p\_\{i\}\\,\\bm\{\\omega\}\_\{i\}^\{\\star\},\(5\)i\.e\. no single model can be closer to all local optima than their weighted dispersion\. This dispersion grows with the heterogeneity constantζ\\zetain \([3](https://arxiv.org/html/2609.19695#S3.E3)\): largerζ\\zetaimplies more conflicting local gradients and hence a largerVarp​\(𝝎⋆\)\\mathrm\{Var\}\_\{p\}\(\\bm\{\\omega\}^\{\\star\}\)\. Two further effects compound this\. First, with partial participation the per\-round average in \([2](https://arxiv.org/html/2609.19695#S3.E2)\) is taken over a sampled subset𝒮t\\mathcal\{S\}^\{t\}, so its expectation matches \([4](https://arxiv.org/html/2609.19695#S3.E4)\) but its variance is inflated by heterogeneity, slowing and destabilizing convergence\. Second, a few strongly atypical clients \(outliers violating \([3](https://arxiv.org/html/2609.19695#S3.E3)\)\) can drag𝜽\\bm\{\\theta\}away from the bulk of clients\. A single global model is therefore a structurally poor target whenζ\\zetais large; some form of client grouping is needed so that aggregation is performed only among compatible clients\.

### 3\.4Local\-weight similarity metrics

Grouping clients requires a measure of how similar their learning problems are\. The data distributions𝒫i\\mathcal\{P\}\_\{i\}are not observable at the server under the privacy model of Section[3\.1](https://arxiv.org/html/2609.19695#S3.SS1), but the post\-training local models𝝎it\\bm\{\\omega\}\_\{i\}^\{t\}are exactly the quantities already exchanged\. Because each𝝎it\\bm\{\\omega\}\_\{i\}^\{t\}results from local optimization toward𝝎i⋆\\bm\{\\omega\}\_\{i\}^\{\\star\}, clients with similar distributions tend to produce nearby weights, while dissimilar clients diverge in parameter space\. The local weights are thus a natural, privacy\-compatible proxy for distributional similarity, requiring no additional disclosure\. We compare two geometries onℝd\\mathbb\{R\}^\{d\}: the Euclidean distance

dE​\(𝝎it,𝝎jt\)=‖𝝎it−𝝎jt‖2,d\_\{E\}\(\\bm\{\\omega\}\_\{i\}^\{t\},\\bm\{\\omega\}\_\{j\}^\{t\}\)=\\bigl\\\|\\bm\{\\omega\}\_\{i\}^\{t\}\-\\bm\{\\omega\}\_\{j\}^\{t\}\\bigr\\\|\_\{2\},\(6\)which captures absolute deviation, and the cosine similarity

cosim⁡\(𝝎it,𝝎jt\)=⟨𝝎it,𝝎jt⟩‖𝝎it‖2​‖𝝎jt‖2,\\mathrm\{cosim\}\(\\bm\{\\omega\}\_\{i\}^\{t\},\\bm\{\\omega\}\_\{j\}^\{t\}\)=\\frac\{\\langle\\bm\{\\omega\}\_\{i\}^\{t\},\\bm\{\\omega\}\_\{j\}^\{t\}\\rangle\}\{\\\|\\bm\{\\omega\}\_\{i\}^\{t\}\\\|\_\{2\}\\,\\\|\\bm\{\\omega\}\_\{j\}^\{t\}\\\|\_\{2\}\},\(7\)which captures directional \(angular\) agreement and is invariant to the magnitude of the weight vectors\. The two need not agree: high\-dimensional neural weights can differ substantially in norm \(e\.g\. because of quantity skew or unequal local progress\) while pointing in similar directions\. In such cases cosine similarity may identify clients with aligned learning dynamics that Euclidean distance would separate, which motivates studying both geometries within the same framework \(Section[4](https://arxiv.org/html/2609.19695#S4)\)\.

### 3\.5Coalition\-formation problem statement

We replace the single\-centroid aggregation of FedAvg by aggregation over a partition of the active clients into coalitions of similar local models\. At roundtt, letΠt=\{𝒞1t,…,𝒞Kt\}\\Pi^\{t\}=\\\{\\mathcal\{C\}\_\{1\}^\{t\},\\dots,\\mathcal\{C\}\_\{K\}^\{t\}\\\}be a partition of𝒮t\\mathcal\{S\}^\{t\}intoKKdisjoint coalitions, and let𝐛kt=1\|𝒞kt\|​∑i∈𝒞kt𝝎it\\mathbf\{b\}\_\{k\}^\{t\}=\\frac\{1\}\{\|\\mathcal\{C\}\_\{k\}^\{t\}\|\}\\sum\_\{i\\in\\mathcal\{C\}\_\{k\}^\{t\}\}\\bm\{\\omega\}\_\{i\}^\{t\}denote the barycenter of coalition𝒞kt\\mathcal\{C\}\_\{k\}^\{t\}\. The coalition\-formation objective is to chooseΠt\\Pi^\{t\}so that within\-coalition dissimilarity is small, e\.g\.

minΠt∑k=1K∑i∈𝒞ktd\(𝝎it,𝐛kt\),\\min\_\{\\Pi^\{t\}\}\\;\\sum\_\{k=1\}^\{K\}\\sum\_\{i\\in\\mathcal\{C\}\_\{k\}^\{t\}\}d\\\!\\bigl\(\\bm\{\\omega\}\_\{i\}^\{t\},\\mathbf\{b\}\_\{k\}^\{t\}\\bigr\),\(8\)whered⁡\(⋅,⋅\)d\(\\cdot,\\cdot\)is the chosen dissimilarity, derived from \([6](https://arxiv.org/html/2609.19695#S3.E6)\) or \([7](https://arxiv.org/html/2609.19695#S3.E7)\)\. Aggregation is then carried out at the coalition level rather than over all clients at once, so that dissimilar clients no longer average into one another\. Directly optimizing \([8](https://arxiv.org/html/2609.19695#S3.E8)\) is a combinatorial clustering problem, and a fixedKKwith hard assignment does not adapt the number or membership of coalitions to the evolving weights or to outliers\. This motivates the bounded\-confidence formulation of Section[4](https://arxiv.org/html/2609.19695#S4), in which coalitions emerge from a HK interaction on the local weights: each client aggregates only with peers inside a confidence neighborhood, so the partition forms endogenously and atypical clients remain isolated rather than forced into a coalition\.

## 4Proposed HK\-based Coalition Formation Framework

We now address the coalition\-formation problem \([8](https://arxiv.org/html/2609.19695#S3.E8)\) by letting coalitions emerge from a bounded\-confidence interaction on the local weights, rather than fixing their number or membership in advance\. We first state the general mechanism and its rationale \(Section[4\.1](https://arxiv.org/html/2609.19695#S4.SS1)\), then instantiate it with a Euclidean confidence ball \(Section[4\.2](https://arxiv.org/html/2609.19695#S4.SS2)\), a cosine\-similarity threshold \(Section[4\.3](https://arxiv.org/html/2609.19695#S4.SS3)\), and an asymmetric cosine\-confidence bound \(Section[4\.4](https://arxiv.org/html/2609.19695#S4.SS4)\), and finally give the coalition aggregation algorithm \(Section[4\.5](https://arxiv.org/html/2609.19695#S4.SS5)\)\. This extends the weight\-driven coalition scheme of[El Hanjri et al\. \(2024\)](https://arxiv.org/html/2609.19695#bib.bib33)by replacing its fixed\-KKassignment with an opinion\-dynamics process whose fixed points define the coalitions\.

### 4\.1Coalitions as a bounded\-confidence opinion dynamics

Following the problem statement of Section[3\.5](https://arxiv.org/html/2609.19695#S3.SS5), we treat each client’s local model as an opinion in parameter space and let opinions interact only when they are mutually compatible\. This is the HK bounded\-confidence model of opinion dynamics[Rainer and Krause \(2002\)](https://arxiv.org/html/2609.19695#bib.bib32)applied to the local weights: an agent revises its position toward the average of the agents inside its confidence neighborhood, so consensus forms within groups of compatible agents while incompatible agents stay apart\. Used on local models, this yields an endogenous partition, where compatible clients converge to a common position \(a coalition\) and atypical clients remain isolated, which is exactly the adaptive behavior that a fixed\-KKhard assignment cannot provide\. Figure[2](https://arxiv.org/html/2609.19695#S4.F2)summarizes the resulting pipeline\.

Figure 2:Proposed HK\-based coalition\-formation framework within one federated roundtt\. Each client trains a local LSTM on its private data and uploads only the resulting weights𝝎it\\bm\{\\omega\}\_\{i\}^\{t\}\. The edge \(MEC\) server then \(1\) runs the bounded\-confidence HK interaction on the local weights, \(2\) extracts the endogenous coalitions\{𝒞kt\}\\\{\\mathcal\{C\}\_\{k\}^\{t\}\\\}, leaving atypical clients isolated as outliers, and \(3\) aggregates the coalition barycenters𝐛kt\\mathbf\{b\}\_\{k\}^\{t\}, with equal weight per coalition, into the next shared model𝜽t\\bm\{\\theta\}^\{t\}, which is broadcast back to the clients\.Concretely, at federated roundttthe local models\{𝝎it\}i∈𝒮t\\\{\\bm\{\\omega\}\_\{i\}^\{t\}\\\}\_\{i\\in\\mathcal\{S\}^\{t\}\}obtained after local training \(Section[3\.1](https://arxiv.org/html/2609.19695#S3.SS1)\) are used to initialize an inner HK iteration indexed bys=0,1,…s=0,1,\\dots:

𝒛i\(0\)=𝝎it,i∈𝒮t\.\\bm\{z\}\_\{i\}^\{\(0\)\}=\\bm\{\\omega\}\_\{i\}^\{t\},\\qquad i\\in\\mathcal\{S\}^\{t\}\.\(9\)Given a confidence neighborhood𝒩i​\(𝒛\(s\)\)⊆𝒮t\\mathcal\{N\}\_\{i\}\(\\bm\{z\}^\{\(s\)\}\)\\subseteq\\mathcal\{S\}^\{t\}\(defined per variant below, and always withi∈𝒩ii\\in\\mathcal\{N\}\_\{i\}\), each opinion is updated to the mean of its neighbors,

𝒛i\(s\+1\)=1\|𝒩i​\(𝒛\(s\)\)\|​∑j∈𝒩i​\(𝒛\(s\)\)𝒛j\(s\),i∈𝒮t\.\\bm\{z\}\_\{i\}^\{\(s\+1\)\}=\\frac\{1\}\{\\bigl\|\\mathcal\{N\}\_\{i\}\(\\bm\{z\}^\{\(s\)\}\)\\bigr\|\}\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\(\\bm\{z\}^\{\(s\)\}\)\}\\bm\{z\}\_\{j\}^\{\(s\)\},\\qquad i\\in\\mathcal\{S\}^\{t\}\.\(10\)The iteration is run until the opinions stabilize, i\.e\. untilmaxi⁡‖𝒛i\(s\+1\)−𝒛i\(s\)‖2≤δ\\max\_\{i\}\\\|\\bm\{z\}\_\{i\}^\{\(s\+1\)\}\-\\bm\{z\}\_\{i\}^\{\(s\)\}\\\|\_\{2\}\\leq\\deltafor a small toleranceδ\>0\\delta\>0\(or a maximum number of inner stepssmaxs\_\{\\max\}is reached\); let𝒛i⋆\\bm\{z\}\_\{i\}^\{\\star\}denote the resulting profile\. The coalitions are the groups of clients that have reached the same limiting opinion,

i∼j⟺‖𝒛i⋆−𝒛j⋆‖2≤δ,i\\sim j\\;\\Longleftrightarrow\\;\\bigl\\\|\\bm\{z\}\_\{i\}^\{\\star\}\-\\bm\{z\}\_\{j\}^\{\\star\}\\bigr\\\|\_\{2\}\\leq\\delta,\(11\)and the partitionΠt=\{𝒞1t,…,𝒞Ktt\}\\Pi^\{t\}=\\\{\\mathcal\{C\}\_\{1\}^\{t\},\\dots,\\mathcal\{C\}\_\{K^\{t\}\}^\{t\}\\\}is the set of equivalence classes of∼\\sim\. The number of coalitionsKtK^\{t\}is not prescribed: it is determined by the confidence parameter and by the configuration of the local weights, and may vary across rounds\. A single application of \([10](https://arxiv.org/html/2609.19695#S4.E10)\) \(smax=1s\_\{\\max\}=1\) recovers the one\-step weight\-averaging interaction; iterating to a fixed point is what produces well\-separated coalitions\.

The three variants below differ only in how the confidence neighborhood𝒩i\\mathcal\{N\}\_\{i\}is defined, i\.e\. in the geometry used to decide which opinions are mutually compatible, building directly on the two geometries motivated in Section[3\.4](https://arxiv.org/html/2609.19695#S3.SS4)\.

### 4\.2Euclidean HK interaction

The first variant measures compatibility by the Euclidean distance \([6](https://arxiv.org/html/2609.19695#S3.E6)\) and admits as neighbors all opinions within a confidence ball of radiusε\>0\\varepsilon\>0:

𝒩i​\(𝒛\(s\)\)=\{j∈𝒮t:‖𝒛j\(s\)−𝒛i\(s\)‖2≤ε\}\.\\mathcal\{N\}\_\{i\}\(\\bm\{z\}^\{\(s\)\}\)=\\Bigl\\\{\\,j\\in\\mathcal\{S\}^\{t\}\\;:\\;\\bigl\\\|\\bm\{z\}\_\{j\}^\{\(s\)\}\-\\bm\{z\}\_\{i\}^\{\(s\)\}\\bigr\\\|\_\{2\}\\leq\\varepsilon\\Bigr\\\}\.\(12\)This is the classical HK ball: clients whose models are close in absolute terms reinforce one another\. The radius

ε\\varepsiloncontrols granularity \(small

ε\\varepsilonyields many tight coalitions, large

ε\\varepsilonmerges them\), and its appropriate scale depends on the magnitude of the weights, which is the main limitation addressed by the cosine variant next\.

In our implementation, the Euclidean variant operates on normalized weight vectors: each opinion is first rescaled to unit norm,

𝒛^i\(s\)=𝒛i\(s\)/‖𝒛i\(s\)‖2\\hat\{\\bm\{z\}\}\_\{i\}^\{\(s\)\}=\\bm\{z\}\_\{i\}^\{\(s\)\}/\\bigl\\\|\\bm\{z\}\_\{i\}^\{\(s\)\}\\bigr\\\|\_\{2\}, before evaluating \([12](https://arxiv.org/html/2609.19695#S4.E12)\)\. This removes the dependence of the distance scale on the weight magnitudes: distances between unit vectors are bounded,

‖𝒛^i−𝒛^j‖2∈\[0,2\]\\bigl\\\|\\hat\{\\bm\{z\}\}\_\{i\}\-\\hat\{\\bm\{z\}\}\_\{j\}\\bigr\\\|\_\{2\}\\in\[0,2\], and satisfy

‖𝒛^i−𝒛^j‖22=2​\(1−cosim⁡\(𝒛i,𝒛j\)\)\\bigl\\\|\\hat\{\\bm\{z\}\}\_\{i\}\-\\hat\{\\bm\{z\}\}\_\{j\}\\bigr\\\|\_\{2\}^\{2\}=2\\bigl\(1\-\\mathrm\{cosim\}\(\\bm\{z\}\_\{i\},\\bm\{z\}\_\{j\}\)\\bigr\), so a confidence radius has a consistent geometric meaning across rounds and model dimensions\. It also makes the common threshold grid

\{0\.5,0\.75,0\.9\}\\\{0\.5,0\.75,0\.9\\\}used in Section[6](https://arxiv.org/html/2609.19695#S6)directly comparable across the two geometries \(with a slight abuse of notation, we denote the Euclidean radius by

τ\\tauas well\): both criteria then operate on a bounded, norm\-independent scale and differ only in whether compatibility is expressed as a distance ball or as an angular alignment\. Note that the two parameters act in opposite directions: smaller

τ\\tauyields finer coalitions in the Euclidean case, whereas larger

τ\\taudoes so in the cosine case\.

### 4\.3Cosine\-similarity HK interaction

The second variant measures compatibility by the cosine similarity \([7](https://arxiv.org/html/2609.19695#S3.E7)\) and admits as neighbors the opinions whose direction is sufficiently aligned, through a thresholdτ∈\[−1,1\]\\tau\\in\[\-1,1\]:

𝒩i​\(𝒛\(s\)\)=\{j∈𝒮t:cosim⁡\(𝒛i\(s\),𝒛j\(s\)\)≥τ\}\.\\mathcal\{N\}\_\{i\}\(\\bm\{z\}^\{\(s\)\}\)=\\Bigl\\\{\\,j\\in\\mathcal\{S\}^\{t\}\\;:\\;\\mathrm\{cosim\}\\bigl\(\\bm\{z\}\_\{i\}^\{\(s\)\},\\bm\{z\}\_\{j\}^\{\(s\)\}\\bigr\)\\geq\\tau\\Bigr\\\}\.\(13\)which is equivalent to;1−cosim⁡\(zi\(s\),zj\(s\)\)≤1−τ1\-\\operatorname\{cosim\}\\bigl\(z\_\{i\}^\{\(s\)\},z\_\{j\}^\{\(s\)\}\\bigr\)\\leq 1\-\\tau, perfectly aligned opinions \(cosim=1\\operatorname\{cosim\}=1\) are always neighbors, andi∈𝒩ii\\in\\mathcal\{N\}\_\{i\}holds automatically sincecosim⁡\(zi\(s\),zi\(s\)\)=1\\operatorname\{cosim\}\\bigl\(z\_\{i\}^\{\(s\)\},z\_\{i\}^\{\(s\)\}\\bigr\)=1\. Because cosine similarity is invariant to the norm of the weight vectors, this neighborhood groups clients with aligned learning directions even when their models differ in magnitude \(e\.g\. from quantity skew or unequal local progress\), a situation in which the Euclidean ball \([12](https://arxiv.org/html/2609.19695#S4.E12)\) would wrongly separate them\. This makes the cosine variant better suited to high\-dimensional neural weights, where direction is often more informative than absolute distance\.

### 4\.4Asymmetric cosine\-confidence model

In practice the pairwise cosine similarities among local models tend to concentrate near11, since the weights are correlated across clients\. A single thresholdτ\\tauthen offers limited control in that concentrated regime\. We therefore generalize \([13](https://arxiv.org/html/2609.19695#S4.E13)\) to a two\-sided confidence bound with independent lower and upper boundsεℓ,εr\\varepsilon\_\{\\ell\},\\varepsilon\_\{r\}with−1≤εℓ≤εr≤1\-1\\leq\\varepsilon\_\{\\ell\}\\leq\\varepsilon\_\{r\}\\leq 1:

𝒩i​\(𝒛\(s\)\)=\{j∈𝒮t:εℓ≤cosim⁡\(𝒛i\(s\),𝒛j\(s\)\)≤εr\}\.\\mathcal\{N\}\_\{i\}\(\\bm\{z\}^\{\(s\)\}\)=\\Bigl\\\{\\,j\\in\\mathcal\{S\}^\{t\}\\;:\\;\\varepsilon\_\{\\ell\}\\leq\\mathrm\{cosim\}\\bigl\(\\bm\{z\}\_\{i\}^\{\(s\)\},\\bm\{z\}\_\{j\}^\{\(s\)\}\\bigr\)\\leq\\varepsilon\_\{r\}\\Bigr\\\}\.\(14\)The lower boundεℓ\\varepsilon\_\{\\ell\}excludes weakly aligned \(dissimilar\) clients, as before, while the upper boundεr\\varepsilon\_\{r\}allows excluding near\-collinear models when desired; the one\-sided threshold variant \([13](https://arxiv.org/html/2609.19695#S4.E13)\) is recovered by settingεr=1\\varepsilon\_\{r\}=1\(so thatτ=εℓ\\tau=\\varepsilon\_\{\\ell\}\)\. The asymmetric bound gives finer coalitions in the high\-similarity regime by carving the narrow interval where most pairwise similarities lie\. In the experiments we use the one\-sided threshold form, i\.e\.εr=1\\varepsilon\_\{r\}=1, withτ∈\{0\.5,0\.75,0\.9\}\\tau\\in\\\{0\.5,0\.75,0\.9\\\}\(Section[6](https://arxiv.org/html/2609.19695#S6)\)\.

### 4\.5Coalition aggregation algorithm

Once the partitionΠt\\Pi^\{t\}is obtained from the converged HK profile \([11](https://arxiv.org/html/2609.19695#S4.E11)\), aggregation is performed at the coalition level\. For each coalition𝒞kt\\mathcal\{C\}\_\{k\}^\{t\}we compute its barycenter from the original local models \(not the smoothed opinions\), preserving the information learned locally,

𝐛kt=1\|𝒞kt\|​∑i∈𝒞kt𝝎it,\\mathbf\{b\}\_\{k\}^\{t\}=\\frac\{1\}\{\|\\mathcal\{C\}\_\{k\}^\{t\}\|\}\\sum\_\{i\\in\\mathcal\{C\}\_\{k\}^\{t\}\}\\bm\{\\omega\}\_\{i\}^\{t\},\(15\)and form the next shared model as the average of the coalition barycenters,

𝜽t=1Kt​∑k=1Kt𝐛kt\.\\bm\{\\theta\}^\{t\}=\\frac\{1\}\{K^\{t\}\}\\sum\_\{k=1\}^\{K^\{t\}\}\\mathbf\{b\}\_\{k\}^\{t\}\.\(16\)Aggregating barycenters rather than individual models, together with weighting coalitions equally in \([16](https://arxiv.org/html/2609.19695#S4.E16)\), limits the influence of any single heterogeneous or outlier client on𝜽t\\bm\{\\theta\}^\{t\}: an isolated client forms a singleton coalition and contributes at most1/Kt1/K^\{t\}, instead of the potentially larger sample\-weighted share it would receive under FedAvg \([2](https://arxiv.org/html/2609.19695#S3.E2)\)\. When all clients fall into a single coalition, \([16](https://arxiv.org/html/2609.19695#S4.E16)\) reduces to a plain \(uniform\) average, recovering standard aggregation as a special case\. The overall procedure is summarized in Algorithm[1](https://arxiv.org/html/2609.19695#alg1)\.

Algorithm 1FedHK: Hegselmann–Krause Coalition Formation for Heterogeneous FLInitialize global model

𝜽0\\bm\{\\theta\}^\{0\}; set confidence parameter \(

ε\\varepsilon,

τ\\tau, or

\(εℓ,εr\)\(\\varepsilon\_\{\\ell\},\\varepsilon\_\{r\}\)\), tolerance

δ\\delta, inner cap

smaxs\_\{\\max\}\. each round

t=1,2,…,Tt=1,2,\\dots,TServer selects active set

𝒮t⊆𝒩\\mathcal\{S\}^\{t\}\\subseteq\\mathcal\{N\}and broadcasts

𝜽t−1\\bm\{\\theta\}^\{t\-1\}\. each client

i∈𝒮ti\\in\\mathcal\{S\}^\{t\}in parallel

𝝎it←ClientUpdate​\(i,𝜽t−1\)\\bm\{\\omega\}\_\{i\}^\{t\}\\leftarrow\\textsc\{ClientUpdate\}\(i,\\bm\{\\theta\}^\{t\-1\}\)EElocal SGD epochs, Eq\. \([\\ENDFOR\\STATE](https://arxiv.org/html/2609.19695#S3.E1)\)HK interaction:\\REPEAT\\FOR

𝒛i\(0\)←𝝎it\\bm\{z\}\_\{i\}^\{\(0\)\}\\leftarrow\\bm\{\\omega\}\_\{i\}^\{t\},

∀i∈𝒮t\\forall i\\in\\mathcal\{S\}^\{t\};

s←0s\\leftarrow 0\. each client

i∈𝒮ti\\in\\mathcal\{S\}^\{t\}Form

𝒩i​\(𝒛\(s\)\)\\mathcal\{N\}\_\{i\}\(\\bm\{z\}^\{\(s\)\}\)via Eq\. \([12](https://arxiv.org/html/2609.19695#S4.E12)\), \([13](https://arxiv.org/html/2609.19695#S4.E13)\), or \([\\STATE](https://arxiv.org/html/2609.19695#S4.E14)\)\.

𝒛i\(s\+1\)←1\|𝒩i​\(𝒛\(s\)\)\|​∑j∈𝒩i​\(𝒛\(s\)\)𝒛j\(s\)\\bm\{z\}\_\{i\}^\{\(s\+1\)\}\\leftarrow\\frac\{1\}\{\|\\mathcal\{N\}\_\{i\}\(\\bm\{z\}^\{\(s\)\}\)\|\}\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\(\\bm\{z\}^\{\(s\)\}\)\}\\bm\{z\}\_\{j\}^\{\(s\)\}Eq\. \([\\ENDFOR\\STATE\\UNTIL](https://arxiv.org/html/2609.19695#S4.E10)\)

s←s\+1s\\leftarrow s\+1\.

maxi⁡‖𝒛i\(s\)−𝒛i\(s−1\)‖2≤δ\\max\_\{i\}\\\|\\bm\{z\}\_\{i\}^\{\(s\)\}\-\\bm\{z\}\_\{i\}^\{\(s\-1\)\}\\\|\_\{2\}\\leq\\deltaor

s≥smaxs\\geq s\_\{\\max\}Extract coalitions

Πt=\{𝒞1t,…,𝒞Ktt\}\\Pi^\{t\}=\\\{\\mathcal\{C\}\_\{1\}^\{t\},\\dots,\\mathcal\{C\}\_\{K^\{t\}\}^\{t\}\\\}from

𝒛⋆\\bm\{z\}^\{\\star\}via Eq\. \([\\FOR](https://arxiv.org/html/2609.19695#S4.E11)\)\. each coalition

𝒞kt\\mathcal\{C\}\_\{k\}^\{t\}𝐛kt←1\|𝒞kt\|​∑i∈𝒞kt𝝎it\\mathbf\{b\}\_\{k\}^\{t\}\\leftarrow\\frac\{1\}\{\|\\mathcal\{C\}\_\{k\}^\{t\}\|\}\\sum\_\{i\\in\\mathcal\{C\}\_\{k\}^\{t\}\}\\bm\{\\omega\}\_\{i\}^\{t\}Eq\. \([\\ENDFOR\\STATE](https://arxiv.org/html/2609.19695#S4.E15)\)

𝜽t←1Kt​∑k=1Kt𝐛kt\\bm\{\\theta\}^\{t\}\\leftarrow\\frac\{1\}\{K^\{t\}\}\\sum\_\{k=1\}^\{K^\{t\}\}\\mathbf\{b\}\_\{k\}^\{t\}Eq\. \([\\ENDFOR\\STATE](https://arxiv.org/html/2609.19695#S4.E16)\)return

𝜽T\\bm\{\\theta\}^\{T\}
\\STATE\\FOR\\STATE\\FOR\\STATE\\COMMENT\\STATE\\COMMENT\\STATE\\STATE\\COMMENT\\COMMENT

In terms of complexity, beyond standard FedAvg, each round adds the HK interaction\. One inner step computes pairwise compatibilities over the active set, costingO⁡\(S2​d\)O\(S^\{2\}d\)forS=\|𝒮t\|S=\|\\mathcal\{S\}^\{t\}\|participating clients and parameter dimensiondd; with at mostsmaxs\_\{\\max\}inner steps the per\-round overhead isO⁡\(smax​S2​d\)O\(s\_\{\\max\}S^\{2\}d\)\. Since the HK iteration and coalition extraction run on the server over already\-received models, no extra client computation or communication is incurred relative to FedAvg, so the communication cost per round is unchanged\.

## 5Experimental Setup

### 5\.1Data preprocessing

We evaluate the framework on the Smart Water Meter Consumption Time Series dataset from the city of Alicante, Spain[\(2023\)](https://arxiv.org/html/2609.19695#bib.bib30), which provides hourly consumption measurements forNtotal=1,099N\_\{\\text\{total\}\}=1\{,\}099residential clients\. Because residential consumption profiles vary widely across households, the dataset is a suitable testbed for heterogeneity\-aware FL\. We apply a three\-stage preprocessing pipeline: active\-client filtering, federated cohort sampling, and temporal aggregation\.

For the first stage, letC=\{1,…,Ntotal\}C=\\\{1,\\dots,N\_\{\\text\{total\}\}\\\}be the full client set andx¯i\\bar\{x\}\_\{i\}the mean hourly consumption of clientiiover the observation period, with global meanμC=1\|C\|​∑i∈Cx¯i\\mu\_\{C\}=\\frac\{1\}\{\|C\|\}\\sum\_\{i\\in C\}\\bar\{x\}\_\{i\}\. To retain clients with consistently measurable consumption, we keep those whose mean exceeds the global average by at least60%60\\%,

Cactive=\{i∈C:x¯i≥1\.6​μC\},C\_\{\\text\{active\}\}=\\bigl\\\{\\,i\\in C\\;:\\;\\bar\{x\}\_\{i\}\\geq 1\.6\\,\\mu\_\{C\}\\,\\bigr\\\},\(17\)which yieldsCactive=188C\_\{\\text\{active\}\}=188clients and excludes predominantly zero or minimal\-consumption clients\.

Of the active clients,184184with at least two years of recorded data form the federated poolCFLC\_\{\\text\{FL\}\}\. At each communication round the server samples, uniformly without replacement, an active subset ofS=50S=50clients from this pool \(Section[3\.1](https://arxiv.org/html/2609.19695#S3.SS1)\), which keeps the experiments computationally tractable and mitigates selection bias\.

Finally, hourly residential consumption is highly sparse and zero\-inflated\. For each clienti∈CFLi\\in C\_\{\\text\{FL\}\}with hourly series𝐳\(i\)=\(z1\(i\),…,zT\(i\)\)\\mathbf\{z\}^\{\(i\)\}=\(z\_\{1\}^\{\(i\)\},\\dots,z\_\{T\}^\{\(i\)\}\), we aggregate consumption into non\-overlappingk=24k=24\-hour windows,

yj\(i,k\)=∑t=\(j−1\)​k\+1min⁡\(j⋅k,T\)zt\(i\),j=1,…,⌊T/k⌋,y\_\{j\}^\{\(i,k\)\}=\\sum\_\{t=\(j\-1\)k\+1\}^\{\\min\(j\\cdot k,\\,T\)\}z\_\{t\}^\{\(i\)\},\\qquad j=1,\\dots,\\lfloor T/k\\rfloor,\(18\)which markedly reduces sparsity \(Table[2](https://arxiv.org/html/2609.19695#S5.T2)\)\. The aggregated daily series are then normalized and converted into the supervised one\-step\-ahead form of Section[3\.2](https://arxiv.org/html/2609.19695#S3.SS2)using sliding look\-back windows of lengthL=12L=12time steps, and split chronologically into80%80\\%training and20%20\\%test partitions\.

Table 2:Dataset characteristics before and after temporal aggregation\.
### 5\.2Baselines

We compare the two coalition variants of Section[4](https://arxiv.org/html/2609.19695#S4)against two heterogeneity\-oriented FL baselines:

- •FedAvg with Euclidean coalitions: the proposed framework with the Euclidean confidence ball \([12](https://arxiv.org/html/2609.19695#S4.E12)\)\.
- •FedAvg with cosine coalitions: the proposed framework with the cosine\-similarity threshold \([13](https://arxiv.org/html/2609.19695#S4.E13)\)\.
- •FedProx[Li et al\. \(2020\)](https://arxiv.org/html/2609.19695#bib.bib31): FedAvg with a proximal term that limits client drift under heterogeneity\.
- •Per\-FedAvg[Reguieg et al\. \(2023\)](https://arxiv.org/html/2609.19695#bib.bib8): a meta\-learning\-based personalized FL method that adapts the shared model to each client\.

All methods share the same LSTM architecture, optimizer and round budget so that differences are attributable to the aggregation/personalization mechanism alone\.

### 5\.3Implementation and hyperparameters

All experiments were run on an ASUS TUF A15 \(AMD Ryzen 7 6800H at 4\.7 GHz, 16 GB RAM, NVIDIA RTX 3070 Ti\)\. The federated pool consists of the184184clients with at least two years of recorded data; at each of theT=20T=20communication rounds the server samplesS=50S=50of them uniformly at random \(Section[3\.1](https://arxiv.org/html/2609.19695#S3.SS1)\)\. Each selected client runsE=5E=5local epochs of Adam \(η=10−3\\eta=10^\{\-3\}, batch size3232, MSE loss\) on its own series, split chronologically into80%80\\%training and20%20\\%test data, with a look\-back window ofL=12L=12time steps\. The HK interaction operates on the post\-training local weights \(Section[4\.1](https://arxiv.org/html/2609.19695#S4.SS1)\), unit\-normalized in the Euclidean case \(Section[4\.2](https://arxiv.org/html/2609.19695#S4.SS2)\), and is evaluated with both the Euclidean and the cosine compatibility geometries, each with confidence thresholdsτ∈\{0\.5,0\.75,0\.9\}\\tau\\in\\\{0\.5,0\.75,0\.9\\\}; the inner iteration stabilizes within at most ten steps in all settings \(Section[6\.1](https://arxiv.org/html/2609.19695#S6.SS1)\)\. The full hyperparameter configuration is summarized in Table[3](https://arxiv.org/html/2609.19695#S5.T3)\.

Table 3:Hyperparameter configuration\.ComponentHyperparameterValueDataAggregation windowkk2424hActive\-client threshold1\.6​μC1\.6\\,\\mu\_\{C\}Look\-back window lengthLL1212time stepsTrain/test split ratio80%80\\%/20%20\\%LSTM modelInput size11LSTM layers \(units\)22\(3232,1616\)Dropout rate0\.20\.2Read\-out layersDense→→116\\\!\\to\\\!8\\\!\\to\\\!1, ReLUOutput activationReLU \(non\-negative predictions\)Local trainingOptimizerAdamLocal learning rateη\\eta1×10−31\\times 10^\{\-3\}Batch size3232Local epochsEE55LossMSEFederated protocolTotal clientsNtotalN\_\{\\text\{total\}\}1,0991\{,\}099Client pool \(≥2\\geq 2years of data\)184184Clients per roundSS5050\(randomly sampled\)Communication roundsTT5050HK coalitionOpinion vectorlocal weights𝝎it\\bm\{\\omega\}\_\{i\}^\{t\}Compatibility geometriesEuclidean, cosineConfidence thresholdsτ\\tau\{0\.5,0\.75,0\.9\}\\\{0\.5,0\.75,0\.9\\\}
### 5\.4Evaluation metrics

Forecasting accuracy is reported with the Mean Squared Error \(MSE\) and the Mean Absolute Error \(MAE\),

MSE=1P​∑i=1P\(zi−z^i\)2,MAE=1P​∑i=1P\|zi−z^i\|,\\mathrm\{MSE\}=\\frac\{1\}\{P\}\\sum\_\{i=1\}^\{P\}\\bigl\(z\_\{i\}\-\\hat\{z\}\_\{i\}\\bigr\)^\{2\},\\qquad\\mathrm\{MAE\}=\\frac\{1\}\{P\}\\sum\_\{i=1\}^\{P\}\\bigl\|z\_\{i\}\-\\hat\{z\}\_\{i\}\\bigr\|,\(19\)whereziz\_\{i\}andz^i\\hat\{z\}\_\{i\}are the actual and predicted values andPPis the number of predictions\. The MSE on the held\-out test set, tracked as a function of the communication round, is used for the convergence comparison across methods, while the MAE, averaged over the client cohort, is used for the final head\-to\-head accuracy comparison\.

## 6Experimental Results and Discussion

### 6\.1Coalition structure under HK dynamics

Figure[3](https://arxiv.org/html/2609.19695#S6.F3)shows the HK opinion trajectories of the local weights for the Euclidean \(left\) and cosine \(right\) interactions, with thresholdsτ∈\{0\.5,0\.75,0\.9\}\\tau\\in\\\{0\.5,0\.75,0\.9\\\}\. In all six settings the opinions stabilize within at most ten inner iterations, which supports the small inner capsmaxs\_\{\\max\}used in the complexity analysis \(Section[4\.5](https://arxiv.org/html/2609.19695#S4.SS5)\)\. The threshold sets how fine the partition is, in the same way for both geometries\. Withτ=0\.5\\tau=0\.5\(Figures[3a](https://arxiv.org/html/2609.19695#S6.F3.sf1)and[3b](https://arxiv.org/html/2609.19695#S6.F3.sf2)\) almost all clients merge into one large coalition, while a few atypical clients never interact and stay isolated\. Withτ=0\.75\\tau=0\.75\(Figures[3c](https://arxiv.org/html/2609.19695#S6.F3.sf3)and[3d](https://arxiv.org/html/2609.19695#S6.F3.sf4)\) the population splits into two well\-separated coalitions, again with a few isolated clients\. Withτ=0\.9\\tau=0\.9\(Figures[3e](https://arxiv.org/html/2609.19695#S6.F3.sf5)and[3f](https://arxiv.org/html/2609.19695#S6.F3.sf6)\) several smaller coalitions appear, along with more singletons\. The isolated clients are households with unusual consumption \(e\.g\. very large occupancy or non\-residential use\)\. The bounded\-confidence rule treats them as outliers instead of forcing them into a coalition, as a fixed\-KKscheme would\. This is the intended behavior \(Section[4\.1](https://arxiv.org/html/2609.19695#S4.SS1)\): the number of coalitions is endogenous and is controlled by a single parameter\.

![Refer to caption](https://arxiv.org/html/2609.19695v1/HK051_euclidean.png)\(a\)Euclidean,τ=0\.5\\tau=0\.5
![Refer to caption](https://arxiv.org/html/2609.19695v1/HK051_cosine.png)\(b\)Cosine,τ=0\.5\\tau=0\.5
![Refer to caption](https://arxiv.org/html/2609.19695v1/HK0751_euclidean.png)\(c\)Euclidean,τ=0\.75\\tau=0\.75
![Refer to caption](https://arxiv.org/html/2609.19695v1/HK0751_cosine.png)\(d\)Cosine,τ=0\.75\\tau=0\.75
![Refer to caption](https://arxiv.org/html/2609.19695v1/HK091_euclidean.png)\(e\)Euclidean,τ=0\.9\\tau=0\.9
![Refer to caption](https://arxiv.org/html/2609.19695v1/HK091_cosine.png)\(f\)Cosine,τ=0\.9\\tau=0\.9

Figure 3:HK opinion trajectories of the local weights under the Euclidean \(left\) and cosine \(right\) interactions for confidence thresholdsτ∈\{0\.5,0\.75,0\.9\}\\tau\\in\\\{0\.5,0\.75,0\.9\\\}\. The permissive threshold \(a\)–\(b\) yields a single large coalition with a few isolated outliers; the intermediate threshold \(c\)–\(d\) yields two well\-separated macro\-coalitions; the tightest threshold \(e\)–\(f\) yields a finer partition with more coalitions and singletons\. All settings converge within at most ten inner iterations\.
### 6\.2Convergence comparison

Figure[4](https://arxiv.org/html/2609.19695#S6.F4)compares the average held\-out MSE of the methods across the communication rounds: FedAvg, Per\-FedAvg, FedProx, the fixed\-KKcoalition scheme of[El Hanjri et al\. \(2024\)](https://arxiv.org/html/2609.19695#bib.bib33)\(FL \+ Coalition, Euclidean and cosine\), and the two proposed HK variants \(HK\-FL\)\. Three points stand out\. First, the HK\-FL variants are the best throughout training: they start lower, decay faster, and reach the lowest final MSE \(about0\.0100\.010–0\.0110\.011, versus0\.0140\.014–0\.0150\.015for all baselines; Table[4](https://arxiv.org/html/2609.19695#S6.T4)\)\. Dissimilar clients stop averaging into one another early, so useful shared models appear sooner\. Second, the fixed\-KKbaselines track FedAvg closely: the gain of HK\-FL comes from the endogenous partition, not from coalition averaging itself\. Third, FedProx improves slightly on FedAvg, while Per\-FedAvg is noisier and does not; both end well above HK\-FL, so neither regularization nor personalization alone handles the heterogeneity\. Figure[5](https://arxiv.org/html/2609.19695#S6.F5)shows the accuracy per round and confirms this ordering: HK\-FL plateaus at8383–85%85\\%, the other baselines at7575–79%79\\%, and FedAvg lowest, with the cosine variant slightly above the Euclidean one\.

![Refer to caption](https://arxiv.org/html/2609.19695v1/lossF.png)Figure 4:Average held\-out MSE per communication round for FedAvg, Per\-FedAvg, FedProx, the fixed\-KKcoalition scheme of[El Hanjri et al\. \(2024\)](https://arxiv.org/html/2609.19695#bib.bib33)\(FL \+ Coalition, Euclidean and cosine\), and the proposed HK\-based variants \(HK\-FL, Euclidean and cosine\)\.![Refer to caption](https://arxiv.org/html/2609.19695v1/accuracyF.png)Figure 5:Global\-model accuracy per communication round for the same methods as in Figure[4](https://arxiv.org/html/2609.19695#S6.F4)\. The proposed HK\-FL variants reach the highest accuracy, with the cosine variant slightly above the Euclidean one\.
### 6\.3Comparison across methods

Figure[6](https://arxiv.org/html/2609.19695#S6.F6)and Table[4](https://arxiv.org/html/2609.19695#S6.T4)report the final Mean Absolute Error \(MAE\), averaged over the evaluated clients \(mean±\\pmstd\)\. The HK\-FL variants obtain the two lowest errors: the cosine variant cuts the average MAE by about54%54\\%versus FedAvg \(3\.533\.53versus7\.627\.62\),39%39\\%versus FedProx, and24%24\\%versus Per\-FedAvg\. Replacing the fixed\-KKpartition with the endogenous HK partition lowers the MAE from6\.656\.65to4\.164\.16\(Euclidean\) and from4\.784\.78to3\.533\.53\(cosine\), which isolates the contribution of the bounded\-confidence mechanism\. Two more patterns are worth noting\. First, cosine beats Euclidean within each scheme, supporting the argument of Section[3\.4](https://arxiv.org/html/2609.19695#S3.SS4)that the direction of high\-dimensional weights is more informative than their distance\. Second, the HK variants have small cross\-client spreads \(±1\.39\\pm 1\.39and±1\.52\\pm 1\.52, versus±3\.82\\pm 3\.82for FedProx and±3\.75\\pm 3\.75for Per\-FedAvg\): the gain is shared across the cohort, which we attribute to outliers being isolated instead of contaminating the shared model\.

![Refer to caption](https://arxiv.org/html/2609.19695v1/barplotF.png)Figure 6:Average MAE over the evaluated clients \(error bars:±\\pmone standard deviation across clients\) for the compared methods\.Table 4:Final forecasting error: MAE \(mean±\\pmstd across clients\) and final held\-out MSE \(approximate values read from Figure[4](https://arxiv.org/html/2609.19695#S6.F4)\); lower is better\.
### 6\.4Client\-level forecasting accuracy

Figure[7](https://arxiv.org/html/2609.19695#S6.F7)shows 60 days of actual and predicted daily consumption for three representative clients\. All methods capture the routine consumption level\. The single\-model baselines \(FedAvg, FedProx\) smooth toward a common mean, while the HK\-FL forecasts follow the day\-to\-day changes of each household more closely, such as the high\-consumption episode around days 25–35 of user 97\. This matches the design: averaging only within compatible coalitions preserves client\-specific structure that a global average smooths out\. Extreme one\-day spikes \(about800800liters for user 97 and1,5001\{,\}500liters for user 715\) are missed by every method; such events are hard to predict from past consumption alone, and this limitation is not specific to coalition formation\.

![Refer to caption](https://arxiv.org/html/2609.19695v1/user97_daily_60d.png)\(a\)Client user 97
![Refer to caption](https://arxiv.org/html/2609.19695v1/user108_daily_60d.png)\(b\)Client user 108
![Refer to caption](https://arxiv.org/html/2609.19695v1/user715_daily_60d.png)\(c\)Client user 715

Figure 7:Actual versus predicted daily water consumption over 60 days for three representative clients, comparing FedAvg, FedProx, Per\-FedAvg, and the proposed HK\-FL variants\.In summary, HK coalition formation on the local weights yields stable, interpretable partitions controlled by a single threshold \(Section[6\.1](https://arxiv.org/html/2609.19695#S6.SS1)\), and outperforms FedAvg, FedProx, Per\-FedAvg, and the fixed\-KKscheme in convergence, final loss, accuracy, and MAE \(Sections[6\.2](https://arxiv.org/html/2609.19695#S6.SS2)–[6\.3](https://arxiv.org/html/2609.19695#S6.SS3)\), while isolating outliers instead of letting them distort the shared model\. The fixed\-KKcomparison shows that the endogenous partition drives the gain, and the consistent edge of the cosine geometry supports direction\-based compatibility for high\-dimensional weights\.

## 7Conclusion and Perspectives

We addressed statistical heterogeneity in FL for heterogeneous IoT systems by forming client coalitions directly in the local\-weight space and aggregating at the coalition level\. Casting coalition formation as an HK bounded\-confidence interaction on the local weights yields a partition whose number and membership are endogenous, and which isolates atypical clients as outliers rather than forcing them into a coalition\. We instantiated the interaction with three compatibility geometries \(a Euclidean confidence ball, a cosine\-similarity threshold, and an asymmetric cosine\-confidence bound\) and gave a coalition aggregation algorithm that adds no client computation or communication over FedAvg\.

Instantiated for short\-term water\-consumption forecasting with local LSTM models on the Alicante smart\-meter dataset, the framework produced stable, interpretable coalition structures and outperformed FedAvg, FedProx, Per\-FedAvg, and a fixed\-KKweight\-driven coalition baseline: the HK variants converged faster to the lowest held\-out MSE and highest accuracy, and the cosine variant reduced the cohort\-averaged MAE by about54%54\\%relative to FedAvg while explicitly surfacing atypical clients as outliers\. We report these as empirical observations on a single dataset and do not claim general superiority of any single method\.

Several directions remain\. On the methodological side, promising extensions include a sensitivity analysis of the confidence parameters and an adaptive schedule for them, a theoretical study of the convergence of coalition\-level aggregation under the bounded\-heterogeneity condition, the integration of secure aggregation or differential privacy on the uploaded updates, and the use of coalition\-specific models for personalization rather than a single global model\.

#### Acknowledgements

This work is supported by the Alkhawarizmi AI Project \(grant number: Alkhawarizmi/2020/34\)\.

## Declarations

Ethics declaration:Not applicable\.

Conflict of interest:The authors declare that they have no conflict of interest\.

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