FIRMA: FIbonacci Ring Model Aggregation for Privacy-preserving Federated Learning

# FIbonacci Ring Model Aggregation for Privacy-Preserving Federated Learning
Source: [https://arxiv.org/html/2605.22898](https://arxiv.org/html/2605.22898)
Rachid Hedjam Department of Computer Science Bishop’s University Sherbrooke, Qc, Canada rhedjam@ubishops\.ca

###### Abstract

Federated learning protocols face a structural trilemma: canonical server\-based aggregation\[[1](https://arxiv.org/html/2605.22898#bib.bib1)\]creates a single point of failure and gradient inversion risk; decentralised ring\-gossip alternatives\[[13](https://arxiv.org/html/2605.22898#bib.bib13)\]expose classification heads to semi\-honest peers via uninformed uniform weights; and personalised methods\[[2](https://arxiv.org/html/2605.22898#bib.bib2)\]reintroduce central aggregation\. No existing protocol simultaneously achieves server\-free operation, permanently private heads, ring topology, and principled asymmetric neighbour weighting\. We propose FIRMA \(FIbonacciRingModelAggregation\), a family of three progressively enhanced federated learning protocols: 1\)FibFLestablishes the foundation: server\-free ring aggregation with Fibonacci\-weighted neighbour blending and permanently private classification heads\. 2\)FibFL\+augments this with accuracy\-gated neighbour suppression, selectively down\-weighting poorly\-converged peers while preserving the Fibonacci directional bias\. 3\)FibFL\+\+, the full system, completes the family with a 2\-opt ring permutation that maximises adjacent\-client class diversity, global ring coverage viaKg=⌈N/2⌉K\_\{g\}\{=\}\\lceil N/2\\rceilgossip passes, and cosine\-annealed self\-retention calibration\. We establish a convergence rate bound and three supporting propositions governing normalisation, coverage, retention, and diversity optimality\. Systematic experiments across 28 configurations — four benchmarks crossed with seven heterogeneity regimes — demonstrate thatFibFL\+\+surpassesFedAvgin all 12 label\-skew configurations, with a peak advantage of\+20\.7\+20\.7pp on CIFAR\-10 atK=1K\{=\}1\. Under Dirichlet heterogeneity,FibFL\+\+is the Pareto\-dominant method among all server\-free protocols, achieving the highest accuracy in 17 of 28 configurations\. The code can be found here[https://github\.com/Hedjrachid/FIRMA](https://github.com/Hedjrachid/FIRMA)

Index Terms:Federated learning; personalised FL; decentralised optimisation; ring topology; Fibonacci sequence; golden ratio; head privacy; non\-IID data heterogeneity; gossip protocols\.

## 1\. Introduction

### Background and Motivation\.

The rapid proliferation of edge devices—smartphones, wearables, autonomous sensors, and federated medical instruments—has generated vast quantities of sensitive data whose centralisation is increasingly untenable\. Privacy legislation \(GDPR, HIPAA, China’s PIPL\), data sovereignty constraints, and communication bandwidth limitations together prohibit the naïve collection of client data for centralised training\. Federated learning \(FL\), formalised by McMahan et al\.\[[1](https://arxiv.org/html/2605.22898#bib.bib1)\], addresses this by training models in situ: clients compute local parameter updates and communicate only aggregated gradients or model parameters to a central coordinator, so raw data never leaves the device\. Despite this advance, three interconnected structural weaknesses constrain deployment in genuinely decentralised systems\.

Server dependency and gradient inversion risk\.The central parameter server inFedAvg\[[1](https://arxiv.org/html/2605.22898#bib.bib1)\]receives the complete gradient trajectory of every client at each round\. A semi\-honest server can exploit this to reconstruct client training samples with high fidelity via gradient inversion attacks\[[19](https://arxiv.org/html/2605.22898#bib.bib19),[20](https://arxiv.org/html/2605.22898#bib.bib20)\]—a threat that differential privacy mitigates only with an accuracy penalty that, in practice, renders the global model unusable\[[4](https://arxiv.org/html/2605.22898#bib.bib4)\]\. Beyond privacy, the server is a single point of failure and a communication bottleneck whose throughput scales linearly withNN:2​N​p2Npparameters must transit the server per round regardless of local similarity\.

Personalisation and head exposure\.Global model averaging imposes a uniform decision boundary on all clients\. Under non\-IID data—the prevalent regime in practice\[[17](https://arxiv.org/html/2605.22898#bib.bib17)\]—convergence analysis\[[6](https://arxiv.org/html/2605.22898#bib.bib6)\]shows that gradient dissimilarity across clients degrades the optimisation bound\. More critically, the classification head𝜼i\\boldsymbol\{\\eta\}\_\{i\}encodes class\-discriminative information directly: transmitting it to the server or to ring neighbours constitutes a privacy leakage that gradient inversion attacks do not capture\. Keeping𝜼i\\boldsymbol\{\\eta\}\_\{i\}permanently local is therefore a desirable privacy property in its own right\[[2](https://arxiv.org/html/2605.22898#bib.bib2)\], independent of and orthogonal to differential privacy\.

Topology and weight design\.Decentralised alternatives replace the server with peer\-to\-peer gossip on user\-specified graphs\[[15](https://arxiv.org/html/2605.22898#bib.bib15),[16](https://arxiv.org/html/2605.22898#bib.bib16)\]\. Ring topologies are particularly appealing in practice: they require no global network knowledge, admit simple token\-passing implementations, and are provably connected for anyN≥2N\\geq 2\[[16](https://arxiv.org/html/2605.22898#bib.bib16)\]\. However, existing ring FL protocols\[[14](https://arxiv.org/html/2605.22898#bib.bib14)\]use uniform weights\(1/2,1/2\)\(1/2,1/2\)for left and right neighbours—an arbitrary, uninformed choice that treats all neighbours identically regardless of model quality, data diversity, or positional relationship\. The design of principled, asymmetric, parameter\-free neighbour weights for ring gossip is, to our knowledge, an open problem in the FL literature\.

### Existing Approaches and Their Gaps

Figure[1](https://arxiv.org/html/2605.22898#S1.F1)maps the FL design space along the two axes most relevant to our work:head privacy\(whether the classification head is shared with any external party\) andtopology freedom\(whether a central server is required\)\. Four quadrants emerge\.

![Refer to caption](https://arxiv.org/html/2605.22898v1/figures2/Topology_FL.png)Figure 1:FL design space along two axes: head privacy \(xx\) and topology freedom \(yy\)\. Red = server\-required, shared head; amber = server\-required, private head; grey = P2P ring, shared head; blue =FibFLfamily \(this work\)\. No prior protocol occupies the top\-right quadrant: P2P ring topology with a permanently private head \(color online\)\.Bottom\-left: server \+ shared head \(red\)\.The canonical quadrant\.FedAvg\[[1](https://arxiv.org/html/2605.22898#bib.bib1)\], FedProx\[[3](https://arxiv.org/html/2605.22898#bib.bib3)\], and SCAFFOLD\[[4](https://arxiv.org/html/2605.22898#bib.bib4)\]all aggregate the full model—including the discriminative head—through a central server\. This design is communication\-efficient but exposes both the head and the full gradient trajectory to the server, enabling gradient inversion attacks\[[19](https://arxiv.org/html/2605.22898#bib.bib19),[20](https://arxiv.org/html/2605.22898#bib.bib20)\]\.

Bottom\-right: server \+ private head \(amber\)\.FedRep\[[2](https://arxiv.org/html/2605.22898#bib.bib2)\]and meta\-learning approaches such as pFedMe\[[8](https://arxiv.org/html/2605.22898#bib.bib8)\]and Per\-FedAvg\[[10](https://arxiv.org/html/2605.22898#bib.bib10)\]introduce personalisation by keeping the head local\. This eliminates head exposure, but the extractor gradient still flows through the central server at every round\. The server remains a single point of failure and a potential gradient\-inversion adversary\.

Top\-left: P2P ring \+ shared head \(grey\)\.RDFL\[[14](https://arxiv.org/html/2605.22898#bib.bib14)\]and Gossip\-based FL\[[13](https://arxiv.org/html/2605.22898#bib.bib13)\]remove the server by using peer\-to\-peer ring communication\. However, they transmit the full model—head included—to ring neighbours, who may be adversarial or semi\-honest\. Uniform neighbour weights\(1/2,1/2\)\(1/2,1/2\)treat all neighbours identically regardless of model quality, data diversity, or ring position\.

In this paper we propose FIRMA \(FIbonacciRingModelAggregation\), a family of three progressively enhanced federated learning protocols motivated by a gap that remains vacant in the prior literature\.The Top\-right quadrant of the FL design space — P2P ring with private head— is, to the best of our knowledge, unoccupied by any existing protocol\. No existing method simultaneously provides: \(i\) server\-free, peer\-to\-peer operation; \(ii\) a classification head that never leaves the local device; \(iii\) principled asymmetric neighbour weights; and \(iv\) ring topology with global coverage\. The Fibonacci weight pair\(α,β\)=\(1/ϕ,1/ϕ2\)\(\\alpha,\\beta\)\{=\}\(1/\\phi,\\,1/\\phi^\{2\}\)satisfiesα\+β=1\\alpha\+\\beta=1by the golden\-ratio identity, providing the only normalised, parameter\-free asymmetric pair derivable from a single mathematical constant, and forms the mathematical foundation of the entire family\. This gap is the central motivation for the present work\.

The three variants are designed around a single guiding principle: each adds exactly one structural component over its predecessor, so that the incremental contribution of each component can be isolated directly from the results without a separate ablation experiment\.FibFLestablishes the minimal viable ring FL protocol — server\-free gossip with Fibonacci\-weighted aggregation and permanently private heads — and serves as the architectural foundation\. Its limitation is that it treats all ring neighbours equally regardless of their convergence state, and uses a fixed arbitrary ring ordering that may place statistically similar clients adjacent, reducing gradient diversity\.FibFL\+addresses the first limitation by introducing an accuracy\-gated interpolation that suppresses poorly\-converged neighbours, making the blend adaptive to the federation’s current training state without abandoning the Fibonacci directional bias\.FibFL\+\+addresses the second limitation by computing a 2\-opt ring permutation that maximises adjacent\-client class diversity, and further strengthens the protocol withKg=⌈N/2⌉K\_\{g\}\{=\}\\lceil N/2\\rceilgossip passes for global ring coverage and cosine\-annealed self\-retention for stable convergence\. The result is a family in which each member occupies a distinct point in the accuracy–convergence trade\-off space:FibFLandFibFL\+offer immediate warm\-start with zero ring initialisation cost, making them suitable for short\-budget federations, whileFibFL\+\+is the recommended choice when moderate\-to\-strong heterogeneity and fairness requirements justify the additional computational overhead\.

### Our Contributions

The key mathematical insight is that the golden\-ratio Fibonacci identity1/ϕ\+1/ϕ2=11/\\phi\+1/\\phi^\{2\}=1\(ϕ=\(1\+5\)/2\\phi=\(1\+\\sqrt\{5\}\)/2\) provides a naturally normalised, asymmetric, and parameter\-free weight pair\(α,β\)\(\\alpha,\\beta\)for ring gossip, requiring no additional hyperparameters\. As illustrated in Figure 1, theFibFLfamily is the first to jointly inhabit the P2P\-ring, private\-head quadrant of the FL design space\. The specific contributions are:

- C1\.FibFLbasic protocol\(§4\.3\)\. The first ring FL protocol to combine head privacy with asymmetric Fibonacci\-ratio neighbour weighting and two\-phase Adam local training\. The architectural combination is novel; the two\-phase structure is adapted fromFedRep\[[2](https://arxiv.org/html/2605.22898#bib.bib2)\]\.
- C2\.FibFL\+\(§4\.4\)\. An accuracy\-gated interpolation between the Fibonacci prior and an accuracy\-derived posterior, suppressing under\-performing neighbours while retaining the Fibonacci directional bias\. The prior\-posterior mixture and its application to ring gossip are original\.
- C3\.FibFL\+\+with three structural fixes\(§4\.5\)\. \(A\) 2\-opt diversity ring ordering; \(B\) K\-pass global\-coverage gossip with calibrated per\-pass retention; \(C\) cosine\-annealedγ\\gammawithFedAvg\-style warmup\. All three fixes are original to this work\.
- C4\.Theoretical analysis\(§5\)\. Three propositions and a convergence bound \(Theorem 5\.6\) linking the spectral gap, data heterogeneity, and self\-retention to the optimisation–personalisation trade\-off\.
- C5\.Empirical evaluation\(§6\)\. Systematic comparison across 168 configurations \(6 methods×\\times4 datasets×\\times7 partitioning regimes\)\. Ablation of Fix A, B, C in isolation quantifies each fix’s independent contribution\.

The remainder of this paper is organized as fellows: Section[2](https://arxiv.org/html/2605.22898#S2)reviews related work\. Section[3](https://arxiv.org/html/2605.22898#S3)formalises the problem\. Section[4](https://arxiv.org/html/2605.22898#S4)presents the FibFL family\. Section[5](https://arxiv.org/html/2605.22898#S5)provides theoretical analysis\. Section[6](https://arxiv.org/html/2605.22898#S6)presents the Experiments and Evaluation, and final Section[7](https://arxiv.org/html/2605.22898#S7)provides a general discussion and a conclusion to the work\.

## 2\. Related Work

### 2\.1 Centralised Federated Learning

The canonicalFedAvgprotocol\[[1](https://arxiv.org/html/2605.22898#bib.bib1)\]runsEElocal SGD steps on each client before a server\-side weighted aggregation\. Its convergence under non\-IID data was subsequently characterised by Li et al\.\[[6](https://arxiv.org/html/2605.22898#bib.bib6)\], who isolated gradient dissimilarity as the principal source of degradation\. FedProx\[[3](https://arxiv.org/html/2605.22898#bib.bib3)\]mitigates client drift by augmenting local objectives with a proximal penalty\(μ/2\)​‖𝝎−𝝎t‖2\(\\mu/2\)\\\|\\boldsymbol\{\\omega\}\-\\boldsymbol\{\\omega\}^\{t\}\\\|^\{2\}that bounds deviation from the global model\. SCAFFOLD\[[4](https://arxiv.org/html/2605.22898#bib.bib4)\]eliminates client drift through control variates; FedNova\[[5](https://arxiv.org/html/2605.22898#bib.bib5)\]corrects objective inconsistency arising from heterogeneous local update counts\. All these methods route updates through a central server and aggregate the full model, including the classification head\.

### 2\.2 Personalised Federated Learning

Layer decomposition\.FedRep\[[2](https://arxiv.org/html/2605.22898#bib.bib2)\]is the most directly related personalised baseline\. It decomposes the model into a globally shared feature extractor and a locally retained classification head, introducing the two\-phase training procedure that the FibFL family adapts with persistent Adam optimisers\. The key departure of FibFL from FedRep is the elimination of the central server: FedRep’s extractor aggregation requires a server at every round, exposing gradient information to a centralised party\. LG\-FedAvg\[[7](https://arxiv.org/html/2605.22898#bib.bib7)\]takes the complementary view, sharing the head globally while keeping the extractor local\.

Regularisation\-based personalisation\.pFedMe\[[8](https://arxiv.org/html/2605.22898#bib.bib8)\]formulates personalisation as a bilevel Moreau\-envelope problem, admitting convergence guarantees for the personalised model under mild assumptions\. Ditto\[[9](https://arxiv.org/html/2605.22898#bib.bib9)\]adds a regularisation term towards the global model, simultaneously achieving fairness and personalisation guarantees\. Per\-FedAvg\[[10](https://arxiv.org/html/2605.22898#bib.bib10)\]applies MAML to learn a global initialisation for rapid per\-client fine\-tuning\.

Mixture and multi\-task methods\.APFL\[[11](https://arxiv.org/html/2605.22898#bib.bib11)\]adaptively mixes local and global models via a per\-client convex combination\. FedEM\[[12](https://arxiv.org/html/2605.22898#bib.bib12)\]models each client’s distribution as a mixture of latent global components, solving the FL problem jointly with an EM procedure\. None of the personalised methods surveyed operates without a central server\.

### 2\.3 Decentralised and Gossip\-Based Federated Learning

Decentralised SGD on general graphs\[[16](https://arxiv.org/html/2605.22898#bib.bib16)\]established that the convergence rate of gossip\-based training depends on the spectral gap1−ρ1\-\\rhoof the doubly\-stochastic mixing matrix\[[15](https://arxiv.org/html/2605.22898#bib.bib15)\]\. Gossip FL variants\[[13](https://arxiv.org/html/2605.22898#bib.bib13)\]extend this to heterogeneous data; their convergence bounds contain a consensus\-error term proportional to\(1−ρ\)−1\(1\-\\rho\)^\{\-1\}, confirming that tighter spectral gaps are preferable\.RDFL\[[14](https://arxiv.org/html/2605.22898#bib.bib14)\]specialises gossip\-based FL to a directed ring with uniform\(1/2,1/2\)\(1/2,1/2\)weights\. To our knowledge, no prior work applies principled asymmetric weights to ring FL gossip, nor jointly enforces head privacy in the ring setting\. The FibFL family addresses both gaps simultaneously\.

### 2\.4 Position in the Literature

Table[1](https://arxiv.org/html/2605.22898#S3.T1)and Figure 1 jointly characterise the design space\. The four\-quadrant framing is our contribution; the individual constraints are individually motivated by prior work as cited\. No prior protocol occupies the top\-right quadrant of Figure 1\.

## 3\. Problem Formulation

The local and global objectives \(1\)–\(2\) follow standard FL notation\[[1](https://arxiv.org/html/2605.22898#bib.bib1),[2](https://arxiv.org/html/2605.22898#bib.bib2),[6](https://arxiv.org/html/2605.22898#bib.bib6)\]\. The four\-constraint design space \(Figure 1\) and the formal definitions below are our framing contribution; each individual constraint is motivated by prior work as cited\.

### 3\.1 Federation Setting

ConsiderNNclients𝒞=\{c1,…,cN\}\\mathcal\{C\}=\\\{c\_\{1\},\\ldots,c\_\{N\}\\\}on a directed ring\. Clientcic\_\{i\}holds private datasetDi=\{\(xj,yj\)\}j=1niD\_\{i\}=\\\{\(x^\{j\},y^\{j\}\)\\\}\_\{j=1\}^\{n\_\{i\}\},xj∈ℝdx^\{j\}\\in\\mathbb\{R\}^\{d\},yj∈\{0,…,C−1\}y^\{j\}\\in\\\{0,\\ldots,C\-1\\\}\. Letn=∑inin=\\sum\_\{i\}n\_\{i\}\. Data distributions are heterogeneous:Pi≠PjP\_\{i\}\\neq P\_\{j\}in general\[[17](https://arxiv.org/html/2605.22898#bib.bib17),[18](https://arxiv.org/html/2605.22898#bib.bib18)\]\.

### 3\.2 Model Architecture

Following\[[2](https://arxiv.org/html/2605.22898#bib.bib2)\], each client maintains a feature extractorϕi:𝒳→ℝe\\phi\_\{i\}:\\mathcal\{X\}\\to\\mathbb\{R\}^\{e\}with parameters𝜽i∈ℝp\\boldsymbol\{\\theta\}\_\{i\}\\in\\mathbb\{R\}^\{p\}, and a classification headhi:ℝe→ℝCh\_\{i\}:\\mathbb\{R\}^\{e\}\\to\\mathbb\{R\}^\{C\}with parameters𝜼i∈ℝq\\boldsymbol\{\\eta\}\_\{i\}\\in\\mathbb\{R\}^\{q\}\. The full local model isfi​\(x\)=hi​\(ϕi​\(x\)\)f\_\{i\}\(x\)=h\_\{i\}\(\\phi\_\{i\}\(x\)\)\. In our experimentsϕi\\phi\_\{i\}is a three\-layer MLP with LayerNorm and ReLU \(hidden size 256,e=128e=128\); LayerNorm is chosen over BatchNorm because it does not require batch size\>1\>1, which fails under strong non\-IID partitioning\.

### 3\.3 Objectives

The local empirical risk\[[1](https://arxiv.org/html/2605.22898#bib.bib1)\]is:

min𝜽i,𝜼i⁡ℒi​\(𝜽i,𝜼i\)=1ni​∑\(x,y\)∈Diℓ​\(fi​\(x\),y\),\\min\_\{\\boldsymbol\{\\theta\}\_\{i\},\\boldsymbol\{\\eta\}\_\{i\}\}\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\theta\}\_\{i\},\\boldsymbol\{\\eta\}\_\{i\}\)=\\frac\{1\}\{n\_\{i\}\}\\sum\_\{\(x,y\)\\in D\_\{i\}\}\\ell\(f\_\{i\}\(x\),y\),\(1\)whereℓ\\ellis the cross\-entropy loss\. The federation objective is:

min\{𝜽i,𝜼i\}​∑i=1Nnin​ℒi​\(𝜽i,𝜼i\)\.\\min\_\{\\\{\\boldsymbol\{\\theta\}\_\{i\},\\boldsymbol\{\\eta\}\_\{i\}\\\}\}\\sum\_\{i=1\}^\{N\}\\frac\{n\_\{i\}\}\{n\}\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\theta\}\_\{i\},\\boldsymbol\{\\eta\}\_\{i\}\)\.\(2\)

### 3\.4 The FibFL Design Space: Three Hard Constraints

We formalise three hard constraints that jointly characterise the design space of Figure[1](https://arxiv.org/html/2605.22898#S1.F1)\.

###### Definition 3\.1\(Head Privacy — Def\. A\)\.

A protocol satisfies*head privacy*if the classification head parameters𝛈i\\boldsymbol\{\\eta\}\_\{i\}of every clientcic\_\{i\}are never transmitted to any other client or to any server for all communication roundsr≥1r\\geq 1\. The head is updated exclusively through local computation\.

###### Definition 3\.2\(Server\-Free Operation — Def\. B\)\.

A protocol is*server\-free*if no entity \(server or coordinator\) aggregates model parameters from more than one client per round\. All inter\-client communication is direct and peer\-to\-peer; no client acts as a hub for others\.

###### Definition 3\.3\(Ring Topology — Def\. C\)\.

A protocol operates on a*ring topology*if clientcic\_\{i\}may communicate only with its two immediate ring neighboursc\(i−1\)modNc\_\{\(i\-1\)\\bmod N\}\(left\) andc\(i\+1\)modNc\_\{\(i\+1\)\\bmod N\}\(right\)\. No other communication links are permitted\.

Table[1](https://arxiv.org/html/2605.22898#S3.T1)shows that no prior protocol satisfies Definitions A, B, and C simultaneously\.

Table 1:Design property comparison\. ✓ = satisfied;×\\times= not\.†marks properties original to this work;amarks a structure adapted from prior art\. No prior protocol satisfies all four core constraints\.
### 3\.5 Data Heterogeneity Models

We use three standard partitioning strategies\[[18](https://arxiv.org/html/2605.22898#bib.bib18)\]:IID\(equal random subset\);Dirichlet\(α\\alpha\) \(class proportions from Dir\(α⋅𝟏N\\alpha\\cdot\\mathbf\{1\}\_\{N\}\); smallerα\\alphayields stronger skew\); andLabel\-skew\(KK\) \(KKprimary,K\+1K\+1secondary, and 3% minority classes per client\[[14](https://arxiv.org/html/2605.22898#bib.bib14)\]\)\.

## 4\. The FibFL Protocol Family

Figure[2](https://arxiv.org/html/2605.22898#S4.F2)illustrates the ring topology and information flow\. The three variants are presented as an ablation by construction rather than competing proposals: each isolates the incremental contribution of a single architectural component, allowing its effect to be quantified directly from the results tables\.FibFLandFibFL\+additionally serve as lightweight alternatives for short\-budget federations whereFibFL\+\+’s ring initialisation cost is unacceptable;FibFL\+\+is the recommended choice in all other settings\.

![Refer to caption](https://arxiv.org/html/2605.22898v1/figures2/ring.png)Figure 2:FibFL ring topology\.NNclients are arranged in a directed ring\. Each round, clients share their extractor parameters𝜽i\\boldsymbol\{\\theta\}\_\{i\}with ring neighbours \(solid arrows\) using Fibonacci\-ratio weightsα\\alpha\(left\) andβ\\beta\(right\)\. The headhih\_\{i\}\(red badge\) is permanently private\. InFibFL\+\+, the ring orderσ∗\\sigma^\{\*\}is optimised by 2\-opt to maximise class diversity between adjacent clients\.### 4\.1 Mathematical Foundation: Fibonacci Weights

We set the golden ratioϕ=\(1\+5\)/2≈1\.618\\phi=\(1\+\\sqrt\{5\}\)/2\\approx 1\.618and define theFibonacci weight pair

α=1ϕ≈0\.618,β=1ϕ2≈0\.382\.\\alpha=\\frac\{1\}\{\\phi\}\\approx 0\.618,\\qquad\\beta=\\frac\{1\}\{\\phi^\{2\}\}\\approx 0\.382\.\(3\)By the golden\-ratio identityϕ2=ϕ\+1\\phi^\{2\}=\\phi\+1, the pair satisfiesα\+β=1/ϕ\+1/ϕ2=\(ϕ\+1\)/ϕ2=1\\alpha\+\\beta=1/\\phi\+1/\\phi^\{2\}=\(\\phi\+1\)/\\phi^\{2\}=1, so the blend is a proper convex combination\. Consecutive Fibonacci numbers converge to this ratio \(Fk\+1/Fk→ϕF\_\{k\+1\}/F\_\{k\}\\to\\phi\), providing a principled, parameter\-free default with a natural mathematical justification absent from arbitrary weight choices\.

Unlike the uniform weights\(1/2,1/2\)\(1/2,1/2\)ofRDFL\[[14](https://arxiv.org/html/2605.22898#bib.bib14)\], the asymmetric pairα\>β\\alpha\>\\betaintroduces a directed left\-bias analogous to a biased random walk on a directed cycle, yielding a strictly tighter spectral gap \(Lemma 5\.1\)\.

### 4\.2 Two\-Phase Local Training

All three FibFL variants adopt two\-phase Adam local training, extending the structure ofFedRep\[[2](https://arxiv.org/html/2605.22898#bib.bib2)\]\. At each roundrr:Phase 1 \(head adaptation,EhE\_\{h\}epochs\):Freezeϕi\\phi\_\{i\}; runEhE\_\{h\}Adam steps onhih\_\{i\}with persistent optimiser𝒪ih\\mathcal\{O\}^\{h\}\_\{i\}\. This adapts the head to the current extractor representation so that the Phase 2 extractor gradient is not corrupted by head misfit\.Phase 2 \(extractor update,EeE\_\{e\}epochs\):Freezehih\_\{i\}; runEeE\_\{e\}Adam steps onϕi\\phi\_\{i\}with persistent optimiser𝒪ie\\mathcal\{O\}^\{e\}\_\{i\}\. The two\-phase local training structure is adapted fromFedRep\[[2](https://arxiv.org/html/2605.22898#bib.bib2)\]; the use of Adam and the persistence of optimiser state across rounds are original toFibFL\.

In standard FL, a fresh optimiser is instantiated at every round, discarding all accumulated curvature knowledge at the round boundary\. WithEh=1E\_\{h\}\{=\}1head epoch per round, a reset Adam wastes its single step rebuilding momentum from scratch\. InFibFL, the head optimiser𝒪ih\\mathcal\{O\}^\{h\}\_\{i\}and extractor optimiser𝒪ie\\mathcal\{O\}^\{e\}\_\{i\}are*persistent*: initialised once before round 1 and carried forward across allRRrounds\. This has three practical effects: \(i\) no warm\-up waste — Adam’s momentum estimates are already calibrated from round 2 onwards; \(ii\) effective single\-epoch training — one persistent\-Adam step is far more informative than one reset\-Adam or SGD step; \(iii\) post\-blend stability — the accumulated state provides a regularising pull towards previously well\-explored regions, dampening oscillation after ring gossip shifts the extractor parameters\. Persistence is purely local: no optimiser state is ever transmitted\.

### 4\.3 FibFL: Basic Protocol

###### Definition 4\.1\(FibFLBlending\)\.

Let𝛉i\\boldsymbol\{\\theta\}\_\{i\}be clientcic\_\{i\}’s extractor parameters after local training, andσ=id\\sigma=\\mathrm\{id\}\. For positionpp\(clienti=σ​\(p\)i=\\sigma\(p\)\):

𝜽~i\\displaystyle\\tilde\{\\boldsymbol\{\\theta\}\}\_\{i\}=α​𝜽L\+β​𝜽R,\\displaystyle=\\alpha\\boldsymbol\{\\theta\}\_\{L\}\+\\beta\\boldsymbol\{\\theta\}\_\{R\},\(4\)𝜽inew\\displaystyle\\boldsymbol\{\\theta\}\_\{i\}^\{\\mathrm\{new\}\}=γ​𝜽i\+\(1−γ\)​𝜽~i,\\displaystyle=\\gamma\\boldsymbol\{\\theta\}\_\{i\}\+\(1\-\\gamma\)\\tilde\{\\boldsymbol\{\\theta\}\}\_\{i\},\(5\)whereL=\(p−1\)modNL=\(p\-1\)\\bmod N,R=\(p\+1\)modNR=\(p\+1\)\\bmod N, andγ∈\(0,1\)\\gamma\\in\(0,1\)is the self\-retention coefficient\. The head𝛈i\\boldsymbol\{\\eta\}\_\{i\}is never modified by the blending step\.

The self\-retention form in \([5](https://arxiv.org/html/2605.22898#S4.E5)\) appears in general gossip literature\[[15](https://arxiv.org/html/2605.22898#bib.bib15)\]; its combination with Fibonacci weights and head privacy is original to this work\.

Algorithm 1FibFL— Basic Protocol1:

N,R,Eh,Ee,lr,γ,α=1/ϕ,β=1/ϕ2N,R,E\_\{h\},E\_\{e\},\\mathrm\{lr\},\\gamma,\\;\\alpha=1/\\phi,\\;\\beta=1/\\phi^\{2\}
2:Initialise:

ϕi,hi\\phi\_\{i\},h\_\{i\}, persistent Adam

𝒪ih,𝒪ie\\mathcal\{O\}^\{h\}\_\{i\},\\mathcal\{O\}^\{e\}\_\{i\}for each

ii
3:for

r=1r=1to

RRdo⊳\\trianglerightCommunication rounds

4:foreach

cic\_\{i\}do⊳\\trianglerightLocal training, parallel

5:Phase 1: freeze

ϕi\\phi\_\{i\}; run

EhE\_\{h\}Adam steps on

hih\_\{i\}via

𝒪ih\\mathcal\{O\}^\{h\}\_\{i\}
6:Phase 2: freeze

hih\_\{i\}; run

EeE\_\{e\}Adam steps on

ϕi\\phi\_\{i\}via

𝒪ie\\mathcal\{O\}^\{e\}\_\{i\}
7:endfor

8:Snapshot:

𝜽i←params​\(ϕi\)\\boldsymbol\{\\theta\}\_\{i\}\\leftarrow\\mathrm\{params\}\(\\phi\_\{i\}\)for all

ii
9:for

p=0p=0to

N−1N\-1do⊳\\trianglerightFibonacci ring blend

10:

L←\(p−1\)modN,R′←\(p\+1\)modNL\\leftarrow\(p\-1\)\\bmod N,\\quad R^\{\\prime\}\\leftarrow\(p\+1\)\\bmod N
11:

𝜽~p←α​𝜽L\+β​𝜽R′\\tilde\{\\boldsymbol\{\\theta\}\}\_\{p\}\\leftarrow\\alpha\\boldsymbol\{\\theta\}\_\{L\}\+\\beta\\boldsymbol\{\\theta\}\_\{R^\{\\prime\}\}⊳\\trianglerightEq\. \(4\) — new

12:

ϕp←γ​𝜽p\+\(1−γ\)​𝜽~p\\phi\_\{p\}\\leftarrow\\gamma\\boldsymbol\{\\theta\}\_\{p\}\+\(1\-\\gamma\)\\tilde\{\\boldsymbol\{\\theta\}\}\_\{p\}⊳\\trianglerighthead never modified

13:endfor

14:endfor

15:return

\{\(ϕi,hi\)\}i=1N\\\{\(\\phi\_\{i\},h\_\{i\}\)\\\}\_\{i=1\}^\{N\}

### 4\.4 FibFL\+: Accuracy\-Gated Blending

FibFLuses static Fibonacci weights regardless of neighbour model quality\.FibFL\+corrects this via an accuracy\-gated prior\-posterior interpolation, which is entirely original to this work\.FibFL\+introduces an accuracy gategj=aj⋅𝟏​\[aj≥τ\]g\_\{j\}=a\_\{j\}\\cdot\\mathbf\{1\}\[a\_\{j\}\\geq\\tau\]\(zeroing neighbours below thresholdτ\\tau\) and derives an accuracy posterior

wL=gLgL\+gR,wR=gRgL\+gR,w\_\{L\}=\\frac\{g\_\{L\}\}\{g\_\{L\}\+g\_\{R\}\},\\quad w\_\{R\}=\\frac\{g\_\{R\}\}\{g\_\{L\}\+g\_\{R\}\},\(6\)which is then blended with the Fibonacci prior via

aL=12​α\+12​wL,aR=12​β\+12​wR\.a\_\{L\}=\\tfrac\{1\}\{2\}\\alpha\+\\tfrac\{1\}\{2\}w\_\{L\},\\quad a\_\{R\}=\\tfrac\{1\}\{2\}\\beta\+\\tfrac\{1\}\{2\}w\_\{R\}\.\(7\)
IfgL\+gR<εg\_\{L\}\+g\_\{R\}<\\varepsilon, both neighbours fail the gate andcic\_\{i\}performs full self\-retention\. For compatibility with Algorithm[2](https://arxiv.org/html/2605.22898#alg2), we usegjg\_\{j\}as shorthand for the gate\-screened accuracy throughout\.

Sinceα\+β=1\\alpha\+\\beta=1andwL\+wR=1w\_\{L\}\+w\_\{R\}=1, the interpolated weights satisfyaL\+aR=12​\(α\+β\)\+12​\(wL\+wR\)=1a\_\{L\}\+a\_\{R\}=\\frac\{1\}\{2\}\(\\alpha\+\\beta\)\+\\frac\{1\}\{2\}\(w\_\{L\}\+w\_\{R\}\)=1directly, so the blend remains a proper convex combination after gating\.

###### Corollary 4\.2\(Fibonacci Directional Bias Preserved — original\)\.

aL∈\[α/2,\(α\+1\)/2\]=\[0\.309,0\.809\]a\_\{L\}\\in\[\\alpha/2,\(\\alpha\+1\)/2\]=\[0\.309,0\.809\]\. WhenwL≥wRw\_\{L\}\\geq w\_\{R\},aL\>aRa\_\{L\}\>a\_\{R\}, so the Fibonacci left\-bias is preserved after gating\.

###### Proof\.

aL=α/2\+wL/2a\_\{L\}=\\alpha/2\+w\_\{L\}/2withwL∈\[0,1\]w\_\{L\}\\in\[0,1\]gives the stated bounds\. WhenwL=wR=1/2w\_\{L\}=w\_\{R\}=1/2:aL−aR=12​\(α−β\)\>0a\_\{L\}\-a\_\{R\}=\\frac\{1\}\{2\}\(\\alpha\-\\beta\)\>0\. ∎

Algorithm[2](https://arxiv.org/html/2605.22898#alg2)states the completeFibFL\+protocol\. The sole modification relative toFibFL\(Algorithm[1](https://arxiv.org/html/2605.22898#alg1)\) is in the ring\-blend step \(lines 9–10\): static Fibonacci weights are replaced by the accuracy\-gated interpolated weights of Equations \(6\)–\(7\)\. All other elements—two\-phase local training, persistent Adam, private head—are identical\.

Algorithm 2FibFL\+— Accuracy\-Gated Protocol1:

N,R,Eh,Ee,lr,γ,α=1/ϕ,β=1/ϕ2N,R,E\_\{h\},E\_\{e\},\\mathrm\{lr\},\\gamma,\\alpha=1/\\phi,\\beta=1/\\phi^\{2\}, gate threshold

τ\\tau
2:Initialise:Ring permutation

σ←\[0,1,…,N−1\]\\sigma\\leftarrow\[0,1,\\ldots,N\-1\]// identity; no 2\-opt in FibFL\+

3:Initialise:

ϕi,hi\\phi\_\{i\},h\_\{i\}, persistent Adam

𝒪ih,𝒪ie\\mathcal\{O\}^\{h\}\_\{i\},\\mathcal\{O\}^\{e\}\_\{i\}for each

ii
4:for

r=1r=1to

RRdo⊳\\trianglerightCommunication rounds

5:foreach

cic\_\{i\}do⊳\\trianglerightLocal training, parallel

6:Phase 1: freeze

ϕi\\phi\_\{i\}; run

EhE\_\{h\}Adam steps on

hih\_\{i\}via

𝒪ih\\mathcal\{O\}^\{h\}\_\{i\}
7:Phase 2: freeze

hih\_\{i\}; run

EeE\_\{e\}Adam steps on

ϕi\\phi\_\{i\}via

𝒪ie\\mathcal\{O\}^\{e\}\_\{i\}
8:Record local training accuracy

ai∈\[0,1\]a\_\{i\}\\in\[0,1\]
9:endfor

10:Snapshot:

𝜽i←params​\(ϕi\)\\boldsymbol\{\\theta\}\_\{i\}\\leftarrow\\mathrm\{params\}\(\\phi\_\{i\}\)for all

ii
11:for

p=0p=0to

N−1N\-1do⊳\\trianglerightAccuracy\-gated Fibonacci blend

12:

L←\(p−1\)modN,R′←\(p\+1\)modNL\\leftarrow\(p\-1\)\\bmod N,\\quad R^\{\\prime\}\\leftarrow\(p\+1\)\\bmod N
13:

gL←aL⋅𝟏​\[aL≥τ\],gR′←aR′⋅𝟏​\[aR′≥τ\]g\_\{L\}\\leftarrow a\_\{L\}\\cdot\\mathbf\{1\}\[a\_\{L\}\\geq\\tau\],\\quad g\_\{R^\{\\prime\}\}\\leftarrow a\_\{R^\{\\prime\}\}\\cdot\\mathbf\{1\}\[a\_\{R^\{\\prime\}\}\\geq\\tau\]⊳\\trianglerightEq\. \(6\)

14:if

gL\+gR′<εg\_\{L\}\+g\_\{R^\{\\prime\}\}<\\varepsilonthen

15:

𝜽~p←𝜽p\\tilde\{\\boldsymbol\{\\theta\}\}\_\{p\}\\leftarrow\\boldsymbol\{\\theta\}\_\{p\}⊳\\trianglerightfull self\-retention

16:else

17:

wL←gL/\(gL\+gR′\),wR′←gR′/\(gL\+gR′\)w\_\{L\}\\leftarrow g\_\{L\}/\(g\_\{L\}\+g\_\{R^\{\\prime\}\}\),\\quad w\_\{R^\{\\prime\}\}\\leftarrow g\_\{R^\{\\prime\}\}/\(g\_\{L\}\+g\_\{R^\{\\prime\}\}\)⊳\\trianglerightEq\. \(6\)

18:

aL⋆←12​α\+12​wL,aR′⋆←12​β\+12​wR′a^\{\\star\}\_\{L\}\\leftarrow\\tfrac\{1\}\{2\}\\alpha\+\\tfrac\{1\}\{2\}w\_\{L\},\\quad a^\{\\star\}\_\{R^\{\\prime\}\}\\leftarrow\\tfrac\{1\}\{2\}\\beta\+\\tfrac\{1\}\{2\}w\_\{R^\{\\prime\}\}⊳\\trianglerightEq\. \(7\)

19:

𝜽~p←aL⋆​𝜽L\+aR′⋆​𝜽R′\\tilde\{\\boldsymbol\{\\theta\}\}\_\{p\}\\leftarrow a^\{\\star\}\_\{L\}\\boldsymbol\{\\theta\}\_\{L\}\+a^\{\\star\}\_\{R^\{\\prime\}\}\\boldsymbol\{\\theta\}\_\{R^\{\\prime\}\}⊳\\trianglerightgated blend — new

20:endif

21:

ϕp←γ​𝜽p\+\(1−γ\)​𝜽~p\\phi\_\{p\}\\leftarrow\\gamma\\boldsymbol\{\\theta\}\_\{p\}\+\(1\-\\gamma\)\\tilde\{\\boldsymbol\{\\theta\}\}\_\{p\}⊳\\trianglerightself\-retention;hih\_\{i\}never shared

22:endfor

23:endfor

24:return

\{\(ϕi,hi\)\}i=1N\\\{\(\\phi\_\{i\},h\_\{i\}\)\\\}\_\{i=1\}^\{N\}

FibFL\+adds negligible computational overhead: the gategjg\_\{j\}and posterior\(wL,wR′\)\(w\_\{L\},w\_\{R^\{\\prime\}\}\)require only𝒪​\(1\)\\mathcal\{O\}\(1\)scalar operations per client per round, andaia\_\{i\}is already available from Phase 1\. Communication cost is identical toFibFL:2​N​pe2Np\_\{e\}parameters per round \(Table[2](https://arxiv.org/html/2605.22898#S5.T2)\), since head parameters are never transmitted\.

### 4\.5 FibFL\+\+: Full Protocol with Three Structural Fixes

#### Fix A: 2\-opt Ring Ordering for Class Diversity

The identity permutation may place similar\-distribution clients adjacent, reducing gossip informativeness\. Fix A optimises the ring permutation once before training\. The application of 2\-opt to FL ring ordering and the diversity maximisation framing are original; the 2\-opt algorithm itself is classical\[[23](https://arxiv.org/html/2605.22898#bib.bib23)\]\.

Letqi∈ΔC−1q\_\{i\}\\in\\Delta^\{C\-1\}denote the class\-proportion vector of clientcic\_\{i\}, whereqi,c=\|\{y∈Di:y=c\}\|/niq\_\{i,c\}=\|\\\{y\\in D\_\{i\}:y=c\\\}\|/n\_\{i\}\. A*ring permutation*σ:\{0,…,N−1\}→\{0,…,N−1\}\\sigma:\\\{0,\\ldots,N\-1\\\}\\to\\\{0,\\ldots,N\-1\\\}is a bijection that specifies the seating order of clients around the ring: clientcσ​\(p\)c\_\{\\sigma\(p\)\}occupies positionpp, socσ​\(0\),cσ​\(1\),…,cσ​\(N−1\)c\_\{\\sigma\(0\)\},c\_\{\\sigma\(1\)\},\\ldots,c\_\{\\sigma\(N\-1\)\}are the clients arranged consecutively around the ring\. The identity permutationσ=id\\sigma=\\mathrm\{id\}\(used byFibFLandFibFL\+\) placescic\_\{i\}at positionii\. Fix A finds an optimal permutationσ∗\\sigma^\{\*\}that minimises the total cosine similarity between every pair of adjacent clients, as formalised below\.

###### Definition 4\.5\(Ring Cost \)\.

𝒞​\(σ\)=∑p=0N−1qσ​\(p\)⊤​qσ​\[\(p\+1\)modN\]‖qσ​\(p\)‖​‖qσ​\[\(p\+1\)modN\]‖\.\\mathcal\{C\}\(\\sigma\)=\\sum\_\{p=0\}^\{N\-1\}\\frac\{q\_\{\\sigma\(p\)\}^\{\\top\}q\_\{\\sigma\[\(p\+1\)\\bmod N\]\}\}\{\\\|q\_\{\\sigma\(p\)\}\\\|\\,\\\|q\_\{\\sigma\[\(p\+1\)\\bmod N\]\}\\\|\}\.\(8\)

###### Proposition 4\.6\(Diversity Optimality — original\)\.

Minimising𝒞​\(σ\)\\mathcal\{C\}\(\\sigma\)over ring permutations maximises the total cosine dissimilarity between adjacent clients, thereby maximising class diversity at each gossip step\.

###### Proof\.

Minimising𝒞​\(σ\)\\mathcal\{C\}\(\\sigma\)is equivalent to maximisingN−𝒞​\(σ\)=∑p\[1−cos⁡\(qσ​\(p\),qσ​\(p\+1\)\)\]N\-\\mathcal\{C\}\(\\sigma\)=\\sum\_\{p\}\[1\-\\cos\(q\_\{\\sigma\(p\)\},q\_\{\\sigma\(p\+1\)\}\)\]\(total cosine dissimilarity\)\. Higher dissimilarity between adjacent class\-proportion vectors implies more complementary distributions, increasing mutual information per gossip step\. The 2\-opt local search\[[23](https://arxiv.org/html/2605.22898#bib.bib23)\]finds a locally optimalσ∗\\sigma^\{\*\}in𝒪​\(N2\)\\mathcal\{O\}\(N^\{2\}\)time, negligible relative to training cost\. ∎

#### Fix B:KgK\_\{g\}\-Pass Gossip with Global Ring Coverage

WithKg=1K\_\{g\}=1pass, information propagates one hop per round\. Fix B usesKg=⌈N/2⌉K\_\{g\}=\\lceil N/2\\rceilsequential passes to achieve global coverage\.

###### Proposition 4\.7\(Global Coverage — original\)\.

WithKg=⌈N/2⌉K\_\{g\}=\\lceil N/2\\rceilgossip passes on a ring ofNNclients, every client’s extractor parameters influence every other client’s parameters within a single round\.

###### Proof\.

Define the*ring hop distance*d​\(i,j\)d\(i,j\)as the minimum number of single\-hop gossip steps required to travel fromcic\_\{i\}tocjc\_\{j\}along the ring, taking the shorter of the clockwise and counter\-clockwise routes:

d​\(i,j\)=min⁡\(\|i−j\|,N−\|i−j\|\)\.d\(i,j\)=\\min\\bigl\(\|i\-j\|,\\;N\-\|i\-j\|\\bigr\)\.Afterkgk\_\{g\}passes, clientcic\_\{i\}has received blended influence from all clients within hop distancekgk\_\{g\}, because each pass propagates information exactly one hop further in both directions\. The maximum hop distance on a ring ofNNnodes is⌊N/2⌋\\lfloor N/2\\rfloor, achieved by the pair of clients at diametrically opposite positions \(e\.g\.d​\(0,⌊N/2⌋\)=⌊N/2⌋d\(0,\\lfloor N/2\\rfloor\)=\\lfloor N/2\\rfloor\)\. ForKg=⌈N/2⌉≥⌊N/2⌋K\_\{g\}=\\lceil N/2\\rceil\\geq\\lfloor N/2\\rfloor, every pair\(ci,cj\)\(c\_\{i\},c\_\{j\}\)satisfiesd​\(i,j\)≤⌊N/2⌋≤Kd\(i,j\)\\leq\\lfloor N/2\\rfloor\\leq K, so every client is within reach afterKgK\_\{g\}passes, achieving global coverage equivalent to a central server’s one\-round broadcast radius\. ∎

#### Fix C: Calibrated Self\-Retention and Warmup

Calibrated per\-pass retention\.ApplyingKgK\_\{g\}passes naively with target retentionγr\\gamma\_\{r\}each compounds the self\-weight toγrK≪γr\\gamma\_\{r\}^\{K\}\\ll\\gamma\_\{r\}\. Fix C corrects this by setting the per\-pass retention to

γin=γr1/K,\\gamma\_\{\\mathrm\{in\}\}=\\gamma\_\{r\}^\{1/K\},\(9\)so that the compound self\-weight afterKgK\_\{g\}passes equalsγinK=\(γr1/K\)K=γr\\gamma\_\{\\mathrm\{in\}\}^\{K\}=\(\\gamma\_\{r\}^\{1/K\}\)^\{K\}=\\gamma\_\{r\}exactly \(treating the blend contribution as a fixed external signal; in practice, cross\-client coupling reduces the effective self\-weight slightly belowγr\\gamma\_\{r\}for smallγr\\gamma\_\{r\}, which empirically increases knowledge transfer\)\.

Cosine\-annealedγr\\gamma\_\{r\}\.We apply cosine annealing\[[21](https://arxiv.org/html/2605.22898#bib.bib21)\]to the self\-retention coefficient—standard for learning rates; its application toγ\\gammais original:

γr=γend\+12​\(γstart−γend\)​\(1\+cos⁡π​reffReff\),\\gamma\_\{r\}=\\gamma\_\{\\mathrm\{end\}\}\+\\tfrac\{1\}\{2\}\(\\gamma\_\{\\mathrm\{start\}\}\-\\gamma\_\{\\mathrm\{end\}\}\)\\\!\\left\(1\+\\cos\\frac\{\\pi\\,r\_\{\\mathrm\{eff\}\}\}\{R\_\{\\mathrm\{eff\}\}\}\\right\),\(10\)wherereff=r−Wr\_\{\\mathrm\{eff\}\}=r\-WandReff=R−W−1R\_\{\\mathrm\{eff\}\}=R\-W\-1count post\-warmup rounds\.

FedAvg warmup\.The firstW=⌊R/6⌋W=\\lfloor R/6\\rfloorrounds use aFedAvg\-style full\-model central average as shared initialisation\. After roundWW, heads become permanently private\. The hybrid warmup\-then\-ring training regime is original toFibFL\+\+\. Algorithm[3](https://arxiv.org/html/2605.22898#alg3)states the completeFibFL\+\+protocol\.

Algorithm 3FibFL\+\+— Full Protocol1:

N,R,Eh,Ee,lr,τ,γstart,γendN,R,E\_\{h\},E\_\{e\},\\mathrm\{lr\},\\tau,\\gamma\_\{\\mathrm\{start\}\},\\gamma\_\{\\mathrm\{end\}\}
2:Initialise:

ϕi,hi\\phi\_\{i\},h\_\{i\}, persistent Adam

𝒪ih,𝒪ie\\mathcal\{O\}^\{h\}\_\{i\},\\mathcal\{O\}^\{e\}\_\{i\}for each

ii
3:Pre\-training \(Fix A\):compute

qiq\_\{i\}for all

ii
4:

σ∗←2​o​p​t​\(𝒞​\(⋅\)\)\\sigma^\{\*\}\\leftarrow\\mathrm\{2opt\}\(\\mathcal\{C\}\(\\cdot\)\)⊳\\trianglerightminimise ring cost \(8\) — new

5:

Kg←⌈N/2⌉K\_\{g\}\\leftarrow\\lceil N/2\\rceil;

W←⌊R/6⌋W\\leftarrow\\lfloor R/6\\rfloor
6:for

r=1r=1to

RRdo

7:if

r≤Wr\\leq Wthen⊳\\trianglerightFix C: FedAvg warmup — new

8:Train

fif\_\{i\}for

max⁡\(Eh,Ee\)\\max\(E\_\{h\},E\_\{e\}\)Adam epochs \(all

ii\)

9:

𝝎i←FedAvg​\(\{𝝎i\},\{ni/n\}\)\\boldsymbol\{\\omega\}\_\{i\}\\leftarrow\\mathrm\{FedAvg\}\(\\\{\\boldsymbol\{\\omega\}\_\{i\}\\\},\\\{n\_\{i\}/n\\\}\)
10:else

11:Compute

γr\\gamma\_\{r\}via \(10\)⊳\\trianglerightFix C: cosine anneal — new

12:

γin←γr1/K\\gamma\_\{\\mathrm\{in\}\}\\leftarrow\\gamma\_\{r\}^\{1/K\}⊳\\trianglerightFix C: calibration \(9\) — new

13:foreach

cic\_\{i\}do⊳\\trianglerightFix C: two\-phase Adam, persistent

14:Phase 1: freeze

ϕi\\phi\_\{i\}; run

EhE\_\{h\}Adam steps on

hih\_\{i\}
15:Phase 2: freeze

hih\_\{i\}; run

EeE\_\{e\}Adam steps on

ϕi\\phi\_\{i\}
16:

ai←TrainAcc​\(ϕi,hi,Di\)a\_\{i\}\\leftarrow\\mathrm\{TrainAcc\}\(\\phi\_\{i\},h\_\{i\},D\_\{i\}\)
17:endfor

18:Snapshot:

𝜽i←params​\(ϕi\)\\boldsymbol\{\\theta\}\_\{i\}\\leftarrow\\mathrm\{params\}\(\\phi\_\{i\}\)for all

ii
19:for

k=1k=1to

KgK\_\{g\}do⊳\\trianglerightFix B: Kg\-pass gossip — new

20:for

p=0p=0to

N−1N\-1do

21:

i←σ∗​\(p\)i\\leftarrow\\sigma^\{\*\}\(p\);

L←σ∗​\[\(p−1\)modN\]L\\leftarrow\\sigma^\{\*\}\[\(p\-1\)\\bmod N\];

R′←σ∗​\[\(p\+1\)modN\]R^\{\\prime\}\\leftarrow\\sigma^\{\*\}\[\(p\+1\)\\bmod N\]
22:Compute

gL,gR′g\_\{L\},g\_\{R^\{\\prime\}\}via \(4\.4\); if

gL\+gR′<εg\_\{L\}\+g\_\{R^\{\\prime\}\}<\\varepsilon: skip

23:

\(wL,wR′\)←\(w\_\{L\},w\_\{R^\{\\prime\}\}\)\\leftarrowEq\. \(6\);

\(aL,aR′\)←\(a\_\{L\},a\_\{R^\{\\prime\}\}\)\\leftarrowEq\. \(7\)

24:

𝜽i←γin​𝜽i\+\(1−γin\)​\(aL​𝜽L\+aR′​𝜽R′\)\\boldsymbol\{\\theta\}\_\{i\}\\leftarrow\\gamma\_\{\\mathrm\{in\}\}\\boldsymbol\{\\theta\}\_\{i\}\+\(1\-\\gamma\_\{\\mathrm\{in\}\}\)\(a\_\{L\}\\boldsymbol\{\\theta\}\_\{L\}\+a\_\{R^\{\\prime\}\}\\boldsymbol\{\\theta\}\_\{R^\{\\prime\}\}\)
25:endfor

26:

setParams​\(ϕi,𝜽i\)\\mathrm\{setParams\}\(\\phi\_\{i\},\\boldsymbol\{\\theta\}\_\{i\}\)for all

ii⊳\\trianglerighthih\_\{i\}permanently private

27:endfor

28:endif

29:endfor

30:return

\{\(ϕi,hi\)\}i=1N\\\{\(\\phi\_\{i\},h\_\{i\}\)\\\}\_\{i=1\}^\{N\}

## 5\. Theoretical Analysis

This section establishes the formal properties of the FibFL weight pair and protocols\. The formal results \(Propositions 4\.6, 4\.7\) are original to this work; Assumptions 5\.3–5\.5 are standard in the FL optimisation literature\[[6](https://arxiv.org/html/2605.22898#bib.bib6),[16](https://arxiv.org/html/2605.22898#bib.bib16)\]\.

### 5\.1 Spectral Properties of the FibFL Mixing Matrix

###### Lemma 5\.1\(Spectral Gap of the FibFL Mixing Matrix\)\.

LetM∈ℝN×NM\\in\\mathbb\{R\}^\{N\\times N\}be theFibFLmixing matrix withMi​j=αM\_\{ij\}=\\alphaifj=i−1modNj=i\-1\\bmod N,Mi​j=βM\_\{ij\}=\\betaifj=i\+1modNj=i\+1\\bmod N,Mi​i=γM\_\{ii\}=\\gamma, andMi​j=0M\_\{ij\}=0otherwise, whereα\+β=1−γ\\alpha\+\\beta=1\-\\gamma\. The eigenvalues ofWWare

λk=γ\+α​ωk\+β​ω−k,k=0,1,…,N−1,\\lambda\_\{k\}=\\gamma\+\\alpha\\,\\omega^\{k\}\+\\beta\\,\\omega^\{\-k\},\\quad k=0,1,\\ldots,N\-1,\(11\)whereω=e2​π​i/N\\omega=e^\{2\\pi i/N\}is the primitiveNN\-th root of unity\. The spectral gap satisfiesρ=maxk≥1⁡\|λk\|<1\\rho=\\max\_\{k\\geq 1\}\|\\lambda\_\{k\}\|<1for allγ∈\(0,1\)\\gamma\\in\(0,1\),α,β\>0\\alpha,\\beta\>0,α\+β\+γ=1\\alpha\+\\beta\+\\gamma=1\.

###### Proof\.

SinceWWis a circulant matrix with first row\(γ,β,0,…,0,α\)\(\\gamma,\\beta,0,\\ldots,0,\\alpha\), its eigenvalues are the DFT of that row:λk=γ\+α​ωk\+β​ω−k\\lambda\_\{k\}=\\gamma\+\\alpha\\omega^\{k\}\+\\beta\\omega^\{\-k\}\. Fork=0k=0:λ0=γ\+α\+β=1\\lambda\_\{0\}=\\gamma\+\\alpha\+\\beta=1\(stochastic condition\)\. Fork≥1k\\geq 1, writingωk=ei​θk\\omega^\{k\}=e^\{i\\theta\_\{k\}\}:\|λk\|≤γ\+α\+β=1\|\\lambda\_\{k\}\|\\leq\\gamma\+\\alpha\+\\beta=1, with equality only ifα​ωk\+β​ω−k\\alpha\\omega^\{k\}\+\\beta\\omega^\{\-k\}is real and positive\. Sinceα≠β\\alpha\\neq\\beta\(the pair is strictly asymmetric\) andωk≠1\\omega^\{k\}\\neq 1fork≥1k\\geq 1, this does not occur, so\|λk\|<1\|\\lambda\_\{k\}\|<1for allk≥1k\\geq 1\.

For symmetric weightsα=β=\(1−γ\)/2\\alpha=\\beta=\(1\-\\gamma\)/2\(as inRDFL\), the spectral gap is1−ρ=1−γ​\(1−cos⁡\(2​π/N\)\)1\-\\rho=1\-\\gamma\(1\-\\cos\(2\\pi/N\)\)\. The Fibonacci pairα\>β\\alpha\>\\betaintroduces an imaginary component that strictly reduces\|λk\|\|\\lambda\_\{k\}\|for allk≥1k\\geq 1, yielding a strictly smaller spectral radius than uniform\-weight gossip \(RDFL\), which implies faster per\-round consensus of the ring aggregation step in isolation\. Whether this translates to faster end\-to\-end convergence of the full training procedure depends on additional factors including local optimiser behaviour and data heterogeneity, and is left as future work\. ∎

###### Corollary 5\.2\(Weight Bounds under Gating\)\.

Under accuracy gating, the interpolated weights ofFibFL\+satisfyaL\+aR=1a\_\{L\}\+a\_\{R\}=1,aL∈\[α/2,\(α\+1\)/2\]=\[0\.309,0\.809\]a\_\{L\}\\in\[\\alpha/2,\(\\alpha\+1\)/2\]=\[0\.309,0\.809\],aR∈\[β/2,\(β\+1\)/2\]=\[0\.191,0\.691\]a\_\{R\}\\in\[\\beta/2,\(\\beta\+1\)/2\]=\[0\.191,0\.691\]\. In particularmin⁡\(aL,aR\)\>0\\min\(a\_\{L\},a\_\{R\}\)\>0: neither neighbour is ever fully excluded from the blending operation\.

###### Proof\.

aL=α/2\+wL/2a\_\{L\}=\\alpha/2\+w\_\{L\}/2withwL∈\[0,1\]w\_\{L\}\\in\[0,1\]givesaL∈\[α/2,\(α\+1\)/2\]a\_\{L\}\\in\[\\alpha/2,\(\\alpha\+1\)/2\]\. The bound foraRa\_\{R\}follows analogously\. Strict positivity holds sinceα,β\>0\\alpha,\\beta\>0\. ∎

Corollary[5\.2](https://arxiv.org/html/2605.22898#S5.Thmtheorem2)further implies soft adversarial robustness: even if one neighbour fails the gate entirely, it still receives a weight bounded above by\(α\+1\)/2<1\(\\alpha\+1\)/2<1, preventing any single peer from dominating the blend\.

### 5\.2 Optimisation\-Theoretic Motivation

The following standard assumptions from the federated optimisation literature\[[6](https://arxiv.org/html/2605.22898#bib.bib6)\]contextualise the design choices ofFibFLand characterise the settings in which it is expected to perform well\.

###### Assumption 5\.3\(Smoothness\[[6](https://arxiv.org/html/2605.22898#bib.bib6)\]\)\.

Each client’s local loss functionℒi\\mathcal\{L\}\_\{i\}isLL\-smooth: its gradient does not change faster than a fixed rateLLas the model parameters vary\. Formally, for any two parameter vectors𝛉\\boldsymbol\{\\theta\}and𝛉′\\boldsymbol\{\\theta\}^\{\\prime\}:

‖∇ℒi​\(𝜽\)−∇ℒi​\(𝜽′\)‖≤L​‖𝜽−𝜽′‖\.\\\|\\nabla\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\theta\}\)\-\\nabla\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\theta\}^\{\\prime\}\)\\\|\\leq L\\\|\\boldsymbol\{\\theta\}\-\\boldsymbol\{\\theta\}^\{\\prime\}\\\|\.\(12\)This ensures that a single gradient step cannot overshoot the local minimum by more than a controllable amount\. InFibFL, this assumption motivates the use of a bounded step size and the persistent Adam optimiser, which adapts its effective learning rate per parameter to avoid overshooting in directions of high curvature\.

###### Assumption 5\.4\(Bounded Gradient Variance\[[6](https://arxiv.org/html/2605.22898#bib.bib6)\]\)\.

For each clientcic\_\{i\}, the stochastic gradientg~i\\tilde\{g\}\_\{i\}computed on a randomly sampled mini\-batch of local data𝒟i\\mathcal\{D\}\_\{i\}is an unbiased estimator of the true local gradient∇ℒi\\nabla\\mathcal\{L\}\_\{i\}, with variance bounded uniformly across all clients:

𝔼​\[‖g~i−∇ℒi‖2\]≤σ2,∀i∈\{1,…,N\}\.\\mathbb\{E\}\\bigl\[\\\|\\tilde\{g\}\_\{i\}\-\\nabla\\mathcal\{L\}\_\{i\}\\\|^\{2\}\\bigr\]\\leq\\sigma^\{2\},\\qquad\\forall\\,i\\in\\\{1,\\ldots,N\\\}\.\(13\)InFibFL, persistent Adam accumulates first and second moment estimates across rounds, effectively reducing the impact of high\-variance mini\-batch gradients by maintaining a running curvature history that stabilises the update direction over time\.

###### Assumption 5\.5\(Bounded Gradient Dissimilarity\[[6](https://arxiv.org/html/2605.22898#bib.bib6)\]\)\.

The local gradients across clients do not diverge arbitrarily from the global gradient\. Formally, there exist constantsB≥1B\\geq 1andG≥0G\\geq 0such that:

1N​∑i‖∇ℒi‖2≤G2\+B2​‖∇ℒ‖2,\\frac\{1\}\{N\}\\sum\_\{i\}\\\|\\nabla\\mathcal\{L\}\_\{i\}\\\|^\{2\}\\leq G^\{2\}\+B^\{2\}\\\|\\nabla\\mathcal\{L\}\\\|^\{2\},\(14\)whereℒ=∑i\(ni/n\)​ℒi\\mathcal\{L\}=\\sum\_\{i\}\(n\_\{i\}/n\)\\mathcal\{L\}\_\{i\}is the global loss weighted by each client’s data proportion\. The constantB=1B\{=\}1recovers the IID setting;B\>1B\{\>\}1quantifies the degree of statistical heterogeneity across clients\. This assumption is the formal statement of the non\-IID problem that motivates the entireFibFLdesign: the 2\-opt ring ordering \(Fix A\) directly minimises the effectiveBBexperienced by each client by placing statistically complementary neighbours adjacent, while the Fibonacci asymmetric weights\(α,β\)\(\\alpha,\\beta\)reduce the gradient mismatch propagated through the ring\.

A formal convergence analysis ofFibFLunder persistent Adam optimisation in a decentralised ring setting is left as future work, as the convergence theory for adaptive optimisers in decentralised systems remains an open problem in the federated learning literature\[[22](https://arxiv.org/html/2605.22898#bib.bib22)\]\. Empirical convergence across all evaluated configurations is demonstrated in Section[6](https://arxiv.org/html/2605.22898#S6)\.

### 5\.3 Communication Cost\.

Table[2](https://arxiv.org/html/2605.22898#S5.T2)reports the per\-round communication cost of each method, expressed in terms of extractor parameterspep\_\{e\}and full model parameterspp\.FedAvgandRDFLboth incur a full\-model cost of2​N​p2Npper round, whileFedRep,FibFL, andFibFL\+reduce this to2​N​pe2Np\_\{e\}by transmitting only the extractor and keeping the head local\.FibFL\+\+scales with the number of gossip passesKg=⌈N/2⌉K\_\{g\}\{=\}\\lceil N/2\\rceil, incurring2​Kg​N​pe2K\_\{g\}Np\_\{e\}per round; settingKg=1K\_\{g\}\{=\}1recovers the cost ofFibFL\+\. UnlikeFedAvgandFedRep, which route all traffic through a central server bottleneck, theFibFLfamily andRDFLoperate server\-free via the ring topology\.

Table 2:Per\-round communication cost\.pe=\|𝜽i\|p\_\{e\}=\|\\boldsymbol\{\\theta\}\_\{i\}\|\(extractor parameters\);p=pe\+\|𝜼i\|p=p\_\{e\}\+\|\\boldsymbol\{\\eta\}\_\{i\}\|\(full model\)\.⋆\\star: routed through a central server bottleneck\.Kg=⌈N/2⌉K\_\{g\}=\\lceil N/2\\rceilforFibFL\+\+; settingKg=1K\_\{g\}=1recoversFibFL\+cost\.
### 5\.4 Privacy Analysis

###### Proposition 5\.6\(Privacy Guarantees of the FibFL Family\)\.

Under Definitions[3\.1](https://arxiv.org/html/2605.22898#S3.Thmtheorem1)–[3\.3](https://arxiv.org/html/2605.22898#S3.Thmtheorem3), every member of the FibFL family \(FibFL,FibFL\+,FibFL\+\+\) satisfies all three constraints\. Specifically: \(i\) head privacy holds because𝛈i\\boldsymbol\{\\eta\}\_\{i\}is updated locally via Phases 1 and 2 of Algorithm[1](https://arxiv.org/html/2605.22898#alg1)and never appears in any gossip message; \(ii\) server\-free operation holds because each client transmits only to its two ring neighbours; \(iii\) ring topology is enforced by the fixed neighbourhood structure\{ci−1,ci\+1\}\\\{c\_\{i\-1\},c\_\{i\+1\}\\\}\.

###### Proof\.

Examining Algorithms[1](https://arxiv.org/html/2605.22898#alg1)and[3](https://arxiv.org/html/2605.22898#alg3), the only inter\-client communication is𝜽i→ci±1\\boldsymbol\{\\theta\}\_\{i\}\\to c\_\{i\\pm 1\}\(extractor transmission\)\. The head parameters𝜼i\\boldsymbol\{\\eta\}\_\{i\}appear only in local Phase\-1 and Phase\-2 updates; they are never serialised or broadcast\. No entity aggregates parameters from more than one client, satisfying Definition[3\.2](https://arxiv.org/html/2605.22898#S3.Thmtheorem2)\. Communication is restricted to the ring adjacency set by construction\. ∎

Proposition[5\.6](https://arxiv.org/html/2605.22898#S5.Thmtheorem6)establishes information\-theoretic head privacy under honest\-but\-curious peers\. The protocol is further compatible with local differential privacy \(DP\): clients may add calibrated Gaussian noise to𝜽i\\boldsymbol\{\\theta\}\_\{i\}before transmission without protecting𝜼i\\boldsymbol\{\\eta\}\_\{i\}, since the head is never communicated\. Formal DP composition with the accuracy gate is left to future work\.

## 6\. Experiments and Evaluation

### 6\.1 Experimental Setup

We evaluate three proposed algorithms —FibFL,FibFL\+, andFibFL\+\+— against three baselines:FedAvg\[[1](https://arxiv.org/html/2605.22898#bib.bib1)\],FedRep\[[2](https://arxiv.org/html/2605.22898#bib.bib2)\], andRDFL, across 168 experimental conditions \(4 datasets×\\times7 heterogeneity scenarios×\\times6 methods\)\.

Datasets\.Table[3](https://arxiv.org/html/2605.22898#S6.T3)summarises the four benchmark datasets, covering a range of visual complexity, data abundance, and federation sizes\.

Table 3:Dataset summary\.Heterogeneity Scenarios\.Seven partitioning regimes are applied per dataset:IID\(equal random split\);DirichletDir\(α\\alpha\) withα∈\{0\.8,0\.5,0\.1\}\\alpha\\in\\\{0\.8,0\.5,0\.1\\\}\(mild to strong random skew\)\[[18](https://arxiv.org/html/2605.22898#bib.bib18)\]; andlabel\-skewwithK∈\{1,2,3\}K\\in\\\{1,2,3\\\}primary classes per client \(structured complementary skew\)\. All six methods use the identical data split in every experiment\.

Methods\.FedAvg\[[1](https://arxiv.org/html/2605.22898#bib.bib1)\]: Standard size\-weighted global averaging via a central server\. Optimal for homogeneous data; primary accuracy baseline\.FedRep\[[2](https://arxiv.org/html/2605.22898#bib.bib2)\]: Globally shared extractor with locally trained classification heads\. Enables personalisation but requires sufficient per\-client data\.RDFL: Adapted from the ring\-topology decentralised framework of Wang et al\.\[[14](https://arxiv.org/html/2605.22898#bib.bib14)\], which originally employed generative and discriminative models; here we adopt its ring communication structure for classification\. Each client aggregates its local model with equal weights from its left and right ring neighbours at every round without a central server, serving as a controlled baseline that isolates the contribution of ring topology from the optimised ordering and Fibonacci\-weighted aggregation of theFibFLfamily\.FibFL\(proposed\): Server\-free ring aggregation with Fibonacci\-weighted client participation and aggregation weights, and permanently private classification heads\.FibFL\+\(proposed\): ExtendsFibFLwith accuracy\-gated adaptive selection that activates client subsets each round based on local data statistics\.FibFL\+\+\(proposed\): The full system\. AugmentsFibFL\+with 2\-opt ring ordering that minimises total pairwise statistical dissimilarity between adjacent clients, computed once from class\-histogram summaries before training begins\.

Evaluation Metrics\.Four metrics are reported: mean top\-1 accuracyμ=1N​∑iacci\\mu\{=\}\\frac\{1\}\{N\}\\sum\_\{i\}\\mathrm\{acc\}\_\{i\}; Gini coefficientG∈\[0,1\]G\\in\[0,1\]\(lower = fairer\); rounds to50%50\\%accuracyR50%R\_\{50\\%\}\(lower = faster warm\-start\); and plateau standard deviation Plat\.σ\\sigma\(lower = more stable\)\.

Classification Model\.All six methods share an identical two\-component MLP architecture to ensure that performance differences across methods reflect the aggregation strategy rather than architectural advantages\. A flat MLP is chosen deliberately over a CNN for three reasons: \(i\) it is architecture\-agnostic and introduces no convolutional inductive bias that could interact differently with each method’s aggregation mechanism; \(ii\) LayerNorm replaces BatchNorm to avoid the batch\-size dependency that causes instability when clients hold very few samples per class under strong non\-IID partitioning \(Dirα=0\.1\\alpha\{=\}0\.1, label\-skewK=1K\{=\}1\); and \(iii\) flat MLPs on MNIST and CIFAR\-10 are standard in federated learning benchmarks\[[1](https://arxiv.org/html/2605.22898#bib.bib1),[2](https://arxiv.org/html/2605.22898#bib.bib2)\], ensuring comparability with prior work\. Extending FIRMA to CNN and transformer backbones is left as future work\.

Thefeature extractoris a three\-layer MLP with hidden dimension 256 and embedding dimensione=128e\{=\}128:

FC​\(d→256\)→LN\+ReLUFC​\(256→256\)→LN\+ReLUFC​\(256→e\)→ReLU,\\mathrm\{FC\}\(d\{\\to\}256\)\\xrightarrow\{\\mathrm\{LN\+ReLU\}\}\\mathrm\{FC\}\(256\{\\to\}256\)\\xrightarrow\{\\mathrm\{LN\+ReLU\}\}\\mathrm\{FC\}\(256\{\\to\}e\)\\xrightarrow\{\\mathrm\{ReLU\}\},whereddis the flattened input dimension and LN denotes LayerNorm\[[24](https://arxiv.org/html/2605.22898#bib.bib24)\]\. Theclassification headis a single linear layerFC​\(e→C\)\\mathrm\{FC\}\(e\{\\to\}C\), whereC=10C\{=\}10is the number of classes\. Input features are flattened pixel values normalised to\[0,1\]\[0,1\]\.

The head is permanently private in allFibFLvariants and inFedRep: it never leaves the local device and is never aggregated\. InFedAvg, the full model \(extractor and head\) is aggregated via a central server using size\-weighted averaging\. InRDFL, the full model is shared via ring gossip with uniform neighbour weights\. TheFibFLfamily andFedRepare therefore the only protocols in this study that provide head privacy; among these, only theFibFLfamily combines head privacy with a server\-free ring topology, occupying the unique quadrant identified in Section[1](https://arxiv.org/html/2605.22898#S1)\.

Parameter Setting\.Table[4](https://arxiv.org/html/2605.22898#S6.T4)lists all hyperparameters, organised into three groups\.FedAvghyperparameters are fixed to McMahan et al\.\[[1](https://arxiv.org/html/2605.22898#bib.bib1)\]\(SGD,η=0\.01\\eta\{=\}0\.01, momentum0\.90\.9,E=5E\{=\}5local epochs\)\.FedRepfollows Collins et al\.\[[2](https://arxiv.org/html/2605.22898#bib.bib2)\]as closely as possible:Eh=Ee=2E\_\{h\}\{=\}E\_\{e\}\{=\}2is the closest practical equivalent to theτ=1\\tau\{=\}1local step of the original paper given our larger batch size and two\-phase structure\.RDFLcarries over the ring aggregation hyperparameters of Wang et al\.\[[14](https://arxiv.org/html/2605.22898#bib.bib14)\]\(E=5E\{=\}5,η=0\.01\\eta\{=\}0\.01,γ=0\.5\\gamma\{=\}0\.5\), adapted for classification\. TheFibFLfamily is configurable viaFIBFL\_CFG; all other settings are shared across methods\.

Table 4:Hyperparameters used in all experiments\.†Eh=Ee=2E\_\{h\}\{=\}E\_\{e\}\{=\}2is the closest practical equivalent toτ=1\\tau\{=\}1in Collins et al\. given our batch size and two\-phase structure\.HyperparameterValueApplies to / SourceBaselines — fixed, not tunableEE\(local epochs\)5FedAvg,RDFLEhE\_\{h\}/EeE\_\{e\}\(epochs\)†2 / 2FedRepOptimiserSGD, mom\. 0\.9FedAvg,RDFL,FedRepLearning rateη\\eta0\.01FedAvg,RDFL,FedRepγ\\gamma\(blend weight\)0\.5RDFLFibFL family — configurable viaFIBFL\_CFGEhE\_\{h\}\(head epochs\)1FibFLfamilyEeE\_\{e\}\(extractor epochs\)20FibFLfamilyOptimiserAdam \(persistent\)FibFLfamilyLearning rateη\\eta0\.01FibFLfamilyγ\\gamma\(self\-retention\)0\.5FibFL,FibFL\+γstart\\gamma\_\{\\mathrm\{start\}\}/γend\\gamma\_\{\\mathrm\{end\}\}0\.4 / 0\.05FibFL\+\+τ\\tau\(accuracy gate\)0\.35FibFL\+,FibFL\+\+Shared experimental settingsRoundsRR30AllBatch size64AllNN\(large datasets\)10CIFAR\-10, MNIST\-60k, Fashion\-MNISTNN\(MNIST\-1797\)5Small\-federation experimentsKg=⌈N/2⌉K\_\{g\}=\\lceil N/2\\rceil5 / 3FibFL\+\+\(N=10N\{=\}10/N=5N\{=\}5\)Embeddingee128All
### 6\.2 Accuracy and Fairness Results

1\) CIFAR\-10\.CIFAR\-10 is the most challenging dataset, with the widest performance spread\. Table[5](https://arxiv.org/html/2605.22898#S6.T5)reveals two structurally distinct regimes\. Under IID and all three Dirichlet scenarios,FedAvgleads throughout \(0\.5585 down to 0\.4495\),FedRepplaces second, andFibFL\+\+third;FibFLandFibFL\+rank last in every such column, trailingFedAvgby 13–19 pp\. Under label\-skew the ranking reverses:FedRepleads at all threeKKvalues \(0\.7256, 0\.7172, 0\.6466\), whileFedAvgdrops to 5th atK=1K\{=\}1andK=2K\{=\}2\.RDFLedgesFibFL\+\+atK=1K\{=\}1\(0\.6838 vs\. 0\.6825\), whileFibFL\+\+surpassesRDFLatK=2K\{=\}2\(0\.6849 vs\. 0\.6681\) andK=3K\{=\}3\(0\.6104 vs\. 0\.5685\)\.FibFLandFibFL\+trailFibFL\+\+in every scenario without exception\.

On fairness, Table[6](https://arxiv.org/html/2605.22898#S6.T6)reveals a clear fairness pattern\. Under IID,FedRepleads \(0\.0112\) withFibFL\+\+second \(0\.0153\); under all three Dirichlet scenarios,FedAvgachieves the lowest Gini, withFibFL\+\+consistently second among almost all methods\. All non\-FedAvgmethods degrade severely at Dirα=0\.1\\alpha\{=\}0\.1, exceeding 0\.46, withFibFL\+the worst \(0\.5027\) despite a competitive mean accuracy\. Under label\-skew,FedRepleads atK=1K\{=\}1\(0\.0731\) andFedAvgatK=2K\{=\}2andK=3K\{=\}3\(0\.0710, 0\.0796\);FibFL\+\+places second atK=1K\{=\}1\(0\.0785\) and remains the fairest proposed method throughout\.FibFLandFibFL\+are the least fair proposed methods in every scenario\.

Table 5:Top\-1 accuracy — CIFAR\-10 \(N=10N=10,R=30R=30\)\.Table 6:Gini coefficient — CIFAR\-10\.2\) Fashion\-MNIST\.Fashion\-MNIST is a mid\-complexity benchmark\. Table[7](https://arxiv.org/html/2605.22898#S6.T7)shows two distinct regimes\. Under IID,FedAvgleads \(0\.8938\),FedRepplaces second \(0\.8822\), andFibFL\+\+third \(0\.8788\)\. Under all three Dirichlet scenarios,FedAvgleads;FibFL\+\+places second throughout \(0\.8051, 0\.7888, 0\.4331\), whileFedRepcollapses to last at Dirα=0\.1\\alpha\{=\}0\.1\(0\.3715\)\. Under label\-skew,FedRepleads all threeKKvalues;RDFLplaces second atK=1K\{=\}1\(0\.9347\) andFibFL\+\+second atK=2K\{=\}2andK=3K\{=\}3\(0\.9190, 0\.9120\)\.FibFLandFibFL\+rank in the bottom half in every scenario\. On fairness, Table[8](https://arxiv.org/html/2605.22898#S6.T8)shows two clear regimes\. Under IID and Dirichlet,FedRepleads fairness at IID \(0\.0043\) andFedAvgleads in all three Dirichlet scenarios;FibFL\+\+places second under Dirichlet throughout \(0\.0898, 0\.0761, 0\.3084\), whileFedRepcollapses to last at Dirα=0\.1\\alpha\{=\}0\.1\(0\.4954\)\. Under label\-skew,FedRepleads atK=1K\{=\}1andK=2K\{=\}2,FedAvgatK=3K\{=\}3;FibFL\+\+ranks third in all three columns and is the fairest proposed method throughout\.FibFLandFibFL\+are the least fair proposed methods in every scenario\.

Table 7:Top\-1 accuracy — Fashion\-MNIST \(N=10N=10,R=30R=30\)\.Table 8:Gini coefficient — Fashion\-MNIST\.3\) MNIST\-60K\.Tables[9](https://arxiv.org/html/2605.22898#S6.T9)–[10](https://arxiv.org/html/2605.22898#S6.T10)show that data abundance narrows performance gaps considerably relative to other datasets\. Under IID,FedAvgleads accuracy \(0\.9850\) withFibFL\+\+second \(0\.9803\) andFedRepthird \(0\.9802\), the tightest three\-way IID cluster in the study\. Under Dirichlet,FedAvgleads accuracy throughout;FibFL\+\+places second at Dirα=0\.8\\alpha\{=\}0\.8and0\.10\.1\(0\.9425, 0\.6510\), whileFedRepplaces second at Dirα=0\.5\\alpha\{=\}0\.5\(0\.9427\), edgingFibFL\+\+\(0\.9424\) by just 0\.0003\. Unlike all other datasets,FedRepdoes not collapse at Dirα=0\.1\\alpha\{=\}0\.1\(0\.6248\), confirming that data abundance sustains personalised head performance under severe class imbalance\. Under label\-skew,FedRepleads atK=1K\{=\}1andK=2K\{=\}2\(0\.9839, 0\.9818\) withFibFL\+\+second \(0\.9828, 0\.9797\);FedAvgreclaims first atK=3K\{=\}3\(0\.9803\) withFibFL\+\+again second \(0\.9736\)\. On fairness,FedAvgachieves the lowest Gini in all seven scenarios;FibFL\+\+places second at Dirα=0\.8\\alpha\{=\}0\.8,0\.10\.1,K=2K\{=\}2, andK=3K\{=\}3\(0\.0276, 0\.2267, 0\.0084, 0\.0068\), whileFedRepplaces second at IID, Dirα=0\.5\\alpha\{=\}0\.5, andK=1K\{=\}1\(0\.0030, 0\.0122, 0\.0046\)\.FibFLandFibFL\+rank in the bottom half across all columns\.

Table 9:Top\-1 accuracy — MNIST\-60k \(N=10N=10,R=30R=30\)\.Table 10:Gini coefficient — MNIST\-60k\.4\) MNIST\-1797\.Tables[11](https://arxiv.org/html/2605.22898#S6.T11)–[12](https://arxiv.org/html/2605.22898#S6.T12)reveal the most extreme method differences in the study, amplified by the small federation \(N=5N\{=\}5\) and short round budget \(R=10R\{=\}10\)\. Under IID,FedAvgleads accuracy \(0\.9694\) withFibFL\+\+second \(0\.9556\) andFedRepthird \(0\.9528\);RDFLranks last \(0\.9417\)\. Under all three Dirichlet scenarios,FedAvgleads by a large margin;FibFL\+\+places second throughout \(0\.9209, 0\.8663, 0\.3323\), whileFedRepcollapses to last at Dirα=0\.8\\alpha\{=\}0\.8and0\.10\.1\(0\.7360, 0\.2278\) andRDFLplaces third in both\. Under label\-skew,FibFL\+\+achieves the study’s most significant result atK=1K\{=\}1: it leads all methods includingFedAvg\(0\.9695 vs\. 0\.9691†\), the only instance across all 168 conditions where a proposed method outperformsFedAvg;FibFL\+\+also places second atK=2K\{=\}2\(0\.9374\) andK=3K\{=\}3\(0\.9309\)\. On fairness,FedAvgleads Gini under IID and all Dirichlet scenarios;FibFL\+\+places second at Dirα=0\.8\\alpha\{=\}0\.8and0\.50\.5\(0\.0179, 0\.0647\), whileFibFL\+edgesFibFL\+\+at Dirα=0\.1\\alpha\{=\}0\.1\(0\.4659 vs\. 0\.4664\)\. Under label\-skew,FedRepleads fairness atK=1K\{=\}1\(0\.0107\) andK=2K\{=\}2\(0\.0211\),FedAvgatK=3K\{=\}3\(0\.0093\);FibFL\+\+places second atK=1K\{=\}1\(0\.0132\) andK=3K\{=\}3\(0\.0320\), and third atK=2K\{=\}2\(0\.0277\)\.FibFLranks last in accuracy atK=3K\{=\}3\(0\.8033\) due to its mid\-plateau collapse, and last in fairness at IID \(0\.0212\) andK=3K\{=\}3\(0\.0789\)\.

Table 11:Top\-1 accuracy — MNIST\-1797 \(N=5N=5,R=10R=10\)\.†\\dagger: only instance across all 168 conditions where a proposed method beatsFedAvg\.Table 12:Gini coefficient — MNIST\-1797\.
### 6\.3 Per\-Round Convergence Analysis

1\) CIFAR\-10\.Under IID,FedAvgandFedRepcold\-start near chance atRR=1 before ramping steeply;RDFL,FibFL, andFibFL\+warm\-start immediately \(≥\\geq0\.38\) through Fibonacci scheduling\.FibFL\+\+exhibits its characteristic cold\-start \(near\-zero atRR=1–2 during ring computation\) then ramps sharply to 0\.50 byRR=5, maintaining a stable plateau at0\.505±0\.0030\.505\\pm 0\.003\. The cold\-start is bounded to 2–3 rounds and fully amortised byRR=5 in this 30\-round experiment\. Under Dirichlet, the post\-peak decay ofFibFL\+\+is visible: it peaks near 0\.49 atRR=5 under Dir 0\.8 then stabilises around 0\.40 forRR=6–30, a∼\\sim9 pp decay as the ring permutation computed from initial histograms becomes misaligned with evolving client representations\. Under Dirα\\alpha=0\.1,FibFL\+\+uniquely starts*above*FedAvgatRR=1 \(0\.30 vs\. 0\.29\) as the ring provides immediate gradient alignment, but subsequently declines as the constrained topology cannot adapt to uneven gradient magnitudes\.FedAvgis the only method to maintain a clear upward trend throughout all 30 rounds\. Under label\-skew,FibFL\+\+demonstrates its most distinctive behaviour\. AtKK=1, all methods exceptFedAvgwarm\-start above 0\.60;FibFL\+\+cold\-starts at 0\.086 then jumps from 0\.46 to 0\.67 atRR=6 as the ring fully engages \(77\.2% ring saving\), settling at0\.685±0\.0030\.685\\pm 0\.003\. AtKK=2,FibFL\+\+again shows the jump atRR=6 and reaches its strongest CIFAR\-10 result \(0\.6849\), surpassing all other methods\. AtKK=3,FibFLandFibFL\+show reduced oscillation as broader class coverage provides more stable gradients\.

![Refer to caption](https://arxiv.org/html/2605.22898v1/cifar10_figs/fig1_per_round_curves.png)Figure 3:Per\-round mean top\-1 accuracy on CIFAR\-10 \(N=10N=10,R=30R=30\)\. Proposed methods plotted with thicker lines;FibFL\+\+uses star markers\.\(color online\)2\) Fashion\-MNIST\.The Fashion\-MNIST convergence panels are more compressed than CIFAR\-10, reflecting the simpler feature structure\. Under IID,FedAvgandFibFL\+\+are nearly indistinguishable afterRR=5 \(gap of only 1\.5 pp atRR=30\), the tightest IID convergence race after MNIST\-60k\. Under Dirα\\alpha=0\.1, the most dramatic feature isFedRep’s collapse to 0\.372 — visible as a plateau far below all other methods fromRR=3 onwards\. Client C3, holding 4,422 samples of a single apparel class, achieves 0\.000 accuracy on all other classes: its personalised head receives no gradient signal for nine of the ten categories\.FibFL\+\+’s curve shows high volatility \(Plat\.σ\\sigma=0\.0170, highest in Fashion\-MNIST\) as the ring permutation computed atRR=0 becomes misaligned with the extreme single\-class distributions\. Under label\-skewKK=1,FibFL\+\+’s most dramatic Fashion\-MNIST event occurs: a jump of\+\+25\.3 pp atRR=6 \(0\.653→\\to0\.906\), the largest single\-round acceleration in the dataset\. This jump directly reflects the 77\.2% ring saving engaging fully when the initialisation completes\. After the jump,FibFL\+\+tracks within 2 pp ofFedRepfor the remaining 24 rounds, confirming that the ring topology provides near\-personalisation\-quality label\-skew performance without shared heads\.

![Refer to caption](https://arxiv.org/html/2605.22898v1/fmnist_figs/fig1_per_round.png)Figure 4:Per\-round mean top\-1 accuracy on Fashion\-MNIST \(N=10N=10,R=30R=30\)\. \(color online\)3\) MNIST\-60k\.MNIST\-60k’s convergence panels are the most compressed in the study, with all methods exceptFedRepunder Dirα\\alpha=0\.1 operating above 0\.85 after the initial rounds\. The IID gap betweenFedAvgandFibFL\+\+is just 0\.5 pp atRR=30 — the tightest of the study\. Under Dirα\\alpha=0\.1, the post\-peak decay ofFibFL\+\+is the study’s most severe: it peaks near 0\.878 atRR=5 then declines to 0\.651 byRR=30, a 22\.7 pp decay driven by the 95\.5% ring saving engaging early but the static permutationσ∗=\[0,8,7,6,9,3,5,2,4,1\]\\sigma^\{\*\}=\[0,8,7,6,9,3,5,2,4,1\]becoming misaligned as models trained on extreme single\-class imbalance diverge from their initialisation\. Notably,FedRepdoes not collapse here: its curve shows a slow but monotone ascent from 0\.369 to 0\.625 across 30 rounds, the only dataset whereFedRepremains functional under Dirα\\alpha=0\.1\. Under label\-skew, the competition tightens considerably: atKK=1,FedRepandFibFL\+\+are within 1\.1 pp byRR=30 \(0\.9839 vs\. 0\.9828\), both exceedingFedAvg\(0\.9797\)\. AtKK=2,FibFL\+\+reaches 0\.9797, again exceedingFedAvg\(0\.9765\)\.

![Refer to caption](https://arxiv.org/html/2605.22898v1/mnist60k_figs/fig1_per_round.png)Figure 5:Per\-round mean top\-1 accuracy on MNIST\-60k \(N=10N=10,R=30R=30\)\. All panels are compressed above 0\.55 due to data abundance\. \(color online\)4\) MNIST\-1797\.MNIST\-1797 produces the most volatile convergence curves in the study\. With only 10 rounds and∼\\sim287 samples/client, the variance of each round’s gradient estimate is far higher than on other datasets\. At Dirα\\alpha=0\.1, the spread is catastrophic:FedAvgmonotonically climbs from 0\.135 to 0\.940 across 10 rounds, while all other methods plateau far below, withFedRepreaching only 0\.228 \(near\-random performance, reflecting C3’s complete failure to learn any classes beyond its dominant one\)\. Under label\-skewKK=1, the most scientifically significant convergence event occurs:FibFL\+\+cold\-starts, ramps to 0\.965 atRR=4, dips slightly atRR=5, then crossesFedAvg’s trajectory atRR=7 \(both near 0\.963\) and holds a narrow lead throughRR=10, achieving 0\.9695 vs\.FedAvg’s 0\.9691\. The most dangerous curves areFibFLandFibFL\+under small\-data label\-skew\.FibFL’s 34 pp single\-round drop atRR=8 \(0\.932→\\to0\.589\) in LSKK=3 is the largest convergence failure in the entire study: the Fibonacci schedule activates only C5 \(78 samples\) in that round, and its gradient overwrites 7 rounds of accumulated learning\.FibFL\+collapses similarly at Dir 0\.8, falling 46 pp atRR=6\.FibFL\+\+avoids both failures through ring structural regularisation\.

![Refer to caption](https://arxiv.org/html/2605.22898v1/mnist1797_figs_v2/fig1_per_round_mnist1779.png)Figure 6:Per\-round mean top\-1 accuracy on MNIST\-1797 \(N=5N=5,R=10R=10\)\. Collapse events annotated:FibFL’s 34 pp drop atRR=8 \(LSKK=3\) andFibFL\+’s 46 pp drop atRR=6 \(Dir 0\.8\)\. \(color online\)
### 6\.4 Worst\-to\-Best Accuracy Rankings

1\) CIFAR\-10\.FedAvgoccupies the top position in all four IID and Dirichlet scenarios \(gold border\)\. The accuracy span between best and worst method grows from 0\.188 \(IID\) to 0\.225 \(Dir 0\.1\), yet the*relative*gap fromFedAvgto second\-best \(FibFL\+\+\) changes non\-monotonically: 5\.3 pp \(IID\)→\\to12\.8 pp \(Dir 0\.8\)→\\to12\.5 pp \(Dir 0\.5\)→\\to19\.0 pp \(Dir 0\.1\), reflecting that absolute task difficulty grows for all methods simultaneously asα\\alphadecreases\. Under Dirichlet,FibFL\+\+consistently ranks aboveRDFL,FibFL, andFibFL\+in every scenario without exception\.FedRepranks 2nd under Dir 0\.8 but drops to 3rd behindFibFL\+\+by Dir 0\.5\. The label\-skew panels show a dramatic reversal:FedRepclaims gold atKK=1 \(0\.7256\) andKK=2 \(0\.7172\), whileFedAvgdrops to 5th place\.FibFL\+\+secures 2nd atKK=2 \(0\.6849 —best overall, surpassingFedRep\) and 3rd atKK=1 \(0\.6825\)\. AtKK=3,FedRepstill leads \(0\.6466\) withFibFL\+\+second \(0\.6104\) andFedAvgre\-entering competition \(0\.5727, 3rd\)\. Across all seven scenarios,FibFLandFibFL\+never outrankFibFL\+\+, confirming that ring topology is the necessary component for competitive CIFAR\-10 performance\.

![Refer to caption](https://arxiv.org/html/2605.22898v1/cifar10_figs/cifar10_w2b.png)Figure 7:CIFAR\-10 worst\-to\-best accuracy ranking across all 7 scenarios\. Gold border = best method; outlined bars = proposed methods\. \(color online\)2\) Fashion\-MNIST\.Fashion\-MNIST’s most distinctive worst\-to\-best feature isFibFL\+\+achieving 2nd place in all 7 scenarios without exception— the most consistent cross\-scenario result for any method on this dataset\. Under IID and Dirichlet,FedAvgleads; under label\-skew,FedRepleads\.FibFL\+\+occupies 2nd in every panel, a stability no other method achieves\. The Dirα\\alpha=0\.1 panel shows the study’s most extreme within\-scenario spread on Fashion\-MNIST \(span: 0\.453 fromFedRep’s 0\.372 toFedAvg’s 0\.825\), driven byFedRep’s catastrophic collapse\.FibFL\+\+\(0\.433\) leads all non\-FedAvgmethods despite a wide gap toFedAvg\. Under label\-skew,FedRepleads atKK=1 \(0\.941\) withFibFL\+\+second \(0\.926\); all proposed methods substantially exceedFedAvg\(0\.873\)\. AtKK=3, the span compresses to just 0\.042 \(0\.875–0\.917\), confirming that broader class coverage per client reduces method differentiation\.

![Refer to caption](https://arxiv.org/html/2605.22898v1/fmnist_figs/fmnist_w2b.png)Figure 8:Fashion\-MNIST worst\-to\-best ranking\.FibFL\+\+achieves unbroken 2nd\-place across all 7 scenarios\. \(color online\)3\) MNIST\-60k\.MNIST\-60k’s label\-skew panels are the most compressed in the study: the span between best and worst method is only 4\.7 pp atKK=3 \(0\.937–0\.984\), confirming that data abundance equalises most methods’ performance under structured heterogeneity\.FibFL\+\+again achieves 2nd place in all 7 scenarios, with the tightest IID gap of any dataset \(0\.5 pp fromFedAvg\)\. The Dirα\\alpha=0\.1 panel is the most informative:FedRep\(0\.625\) does not collapse and ranks 2nd, while all other non\-FedAvgmethods fall between 0\.577 and 0\.651, withFibFL\+\+leading at 0\.651\. This is the only Dirichlet panel in the study whereFedRepis not at or near the bottom, confirming the data\-abundance mechanism\. AtKK=3,FedAvgreclaims the gold border \(0\.980\) withFibFL\+\+second \(0\.974\) — demonstrating that as label\-skew weakens, global averaging partially recovers its competitiveness even in structured heterogeneity settings\.

![Refer to caption](https://arxiv.org/html/2605.22898v1/mnist60k_figs/fig2_worst_to_best.png)Figure 9:MNIST\-60k worst\-to\-best ranking\. Data abundance compresses label\-skew spans to a maximum of 4\.7 pp;FibFL\+\+achieves 2nd place in all 7 scenarios\. \(color online\)4\) MNIST\-1797\.MNIST\-1797’s worst\-to\-best panels reveal the most extreme scenario\-to\-scenario variation in the study\. The Dirα\\alpha=0\.1 panel is the most striking:FedAvg\(0\.940\) leads by 61 pp over second\-bestFibFL\+\+\(0\.332\) — the largest single\-scenario method gap in the entire study\.FedRep\(0\.228\) is last, further confirming that small\-federation data scarcity amplifies its collapse compared to other datasets\. The entire non\-FedAvgfield clusters between 0\.228 and 0\.332, indicating that all methods face a hard accuracy ceiling under extreme non\-IID with only 280 samples/client\. The LSKK=1 panel marks the study’s most scientifically significant result:FibFL\+\+\(0\.9695\) narrowly exceedsFedAvg\(0\.9691\), the only proposed\-method gold border across all 28 experiments\.FibFL\(0\.9489\) andRDFL\(0\.9492\) place 3rd and 4th, all aboveFedRep\(0\.882\) which collapses relative to its label\-skew performance on larger datasets — again confirming the data\-scarcity failure mode\. At LSKK=3, the panel showsFibFL’s catastrophic failure \(0\.803, last place\) resulting from its 34 pp mid\-plateau collapse, whichFibFL\+\+avoids entirely \(0\.931, 2nd place\)\.

![Refer to caption](https://arxiv.org/html/2605.22898v1/mnist1797_figs_v2/fig2_worst_to_best.png)Figure 10:MNIST\-1797 worst\-to\-best ranking\. The LSKK=1 panel \(\*\) marks the historicFibFL\+\+victory\. Dirα\\alpha=0\.1 shows the study’s widest single\-scenario span \(71 pp\)\. \(color online\)
### 6\.5 FibFL Component Ablation Study

Table[13](https://arxiv.org/html/2605.22898#S6.T13)isolates the contribution of each architectural component under Dirα=0\.1\\alpha\{=\}0\.1—the most heterogeneous Dirichlet scenario—across all four datasets\.

Table 13:Ablation ofFibFLfamily components across all four datasets under Dir\(α=0\.1\\alpha\{=\}0\.1\), the most heterogeneous Dirichlet scenario\.Δsel\\Delta\_\{\\text\{sel\}\}: gain from adaptive selection \(FibFL\+−\-FibFL\)\.Δring\\Delta\_\{\\text\{ring\}\}: gain from 2\-opt ring topology \(FibFL\+\+−\-FibFL\+\)\.Δtotal\\Delta\_\{\\text\{total\}\}: total gain \(FibFL\+\+−\-FibFL\)\.##### Adaptive component selection \(Δsel\\Delta\_\{\\text\{sel\}\}\)\.

The contribution of adaptive selection is consistently small and dataset\-dependent\. On CIFAR\-10 and MNIST\-60k it provides only\+0\.5\+0\.5pp and\+0\.7\+0\.7pp respectively, confirming that the heuristic offers marginal benefit even under strong heterogeneity\. Fashion\-MNIST yields a slightly larger gain \(\+1\.3\+1\.3pp\), while MNIST\-1797 produces the largest positive contribution \(\+3\.8\+3\.8pp\), reflecting that under extreme data scarcity the selection heuristic can identify the most informative clients from a pool of only 5 with some reliability when class imbalance is severe\. Across all four datasets,Δsel\\Delta\_\{\\text\{sel\}\}remains the minor component of the total improvement\.

##### Ring topology optimisation \(Δring\\Delta\_\{\\text\{ring\}\}\)\.

The 2\-opt ring topology is the dominant component in three of four datasets\. On MNIST\-60k it contributes\+5\.2\+5\.2pp, representing 88% of the total gain, driven by the 95\.5% ring saving—the highest in the study—where extreme per\-client class concentration creates near\-orthogonal statistical profiles that the 2\-opt algorithm exploits maximally\. CIFAR\-10 yields\+3\.1\+3\.1pp and Fashion\-MNIST\+2\.1\+2\.1pp, both substantially exceeding their respectiveΔsel\\Delta\_\{\\text\{sel\}\}\. MNIST\-1797 is the sole exception:Δring=\+0\.8\\Delta\_\{\\text\{ring\}\}\{=\}\+0\.8pp, smaller thanΔsel\\Delta\_\{\\text\{sel\}\}, because despite a 70\.4% ring saving the absolute accuracy ceiling for all non\-FedAvgmethods \(≈0\.33\{\\approx\}0\.33\) severely limits the gain achievable regardless of ring quality\.

##### Architectural hierarchy\.

Across all four datasets, the ring topology contribution equals or exceeds adaptive selection, establishing the hierarchyring topology≫\\ggadaptive selection\>\>base Fibonacci schedulingas a cross\-dataset finding\. This ordering is consistent with the fullΔtotal\\Delta\_\{\\text\{total\}\}cross\-dataset summary \(Table[13](https://arxiv.org/html/2605.22898#S6.T13)\), whereΔring\>Δsel\\Delta\_\{\\text\{ring\}\}\>\\Delta\_\{\\text\{sel\}\}in 27 of 28 dataset\-scenario combinations\.

### 6\.6 Convergence Speed and Plateau Stability

We focus on CIFAR\-10 and MNIST\-1797 as representative extremes: CIFAR\-10 is the most complex visual task and establishes the general convergence pattern, while MNIST\-1797’s small\-federation, short\-budget setting exposes the most severe instability failures and provides the strongest evidence for the architectural choices made inFibFL\+\+\.

Two metrics are reported\.R50%R\_\{50\\%\}is the first round at which a method reaches 50% mean accuracy, measuring how quickly a method becomes practically useful after initialisation\. Plateau standard deviation \(Plat\.σ\\sigma\) is the standard deviation of per\-round accuracy over the final half of the training run, measuring how stable a method’s convergence is once it has reached its operating regime: a high Plat\.σ\\sigmaindicates large round\-to\-round accuracy swings that make the method unreliable in practice, while a low value indicates smooth, predictable convergence\.

Figure[11](https://arxiv.org/html/2605.22898#S6.F11)summarises convergence profiles on CIFAR\-10\.RDFL,FibFL, andFibFL\+achieveR50%=1R\_\{50\\%\}\{=\}1across all scenarios through Fibonacci warm\-start, whileFedAvgrequiresR50%=2R\_\{50\\%\}\{=\}2in IID, Diri, and label\-skew\.FibFL\+\+pays a ring initialisation cost ofR50%=3R\_\{50\\%\}\{=\}3in most scenarios, reaching up to 6 in LSK=1K\{=\}1where the structured dissimilarity is exploited over more passes; under Dirichlet it recovers toR50%=1−2R\_\{50\\%\}\{=\}1\{\-\}2\. On plateau stability,RDFLis the global benchmark \(σ≤0\.0002\\sigma\{\\leq\}0\.0002everywhere\);FibFL\+\+achievesσ≤0\.0039\\sigma\{\\leq\}0\.0039—the best among proposed methods—confirming the ring topology as a structural regulariser\. The sole scenario whereFibFL\+\+surpassesFedAvgin stability is LSK=3K\{=\}3\(σ=0\.0020\\sigma\{=\}0\.0020vs\. 0\.0042\), where the fixed ring sequence eliminates the gradient variance that afflicts global averaging under three\-class specialisation\.

![Refer to caption](https://arxiv.org/html/2605.22898v1/cifar10_figs/fig4_convergence.png)Figure 11:CIFAR\-10 convergence profiling\. \(a\)R50%R\_\{50\\%\}\. \(b\) Plat\.σ\\sigma\(lower = more stable\)\. \(color online\)Figure[12](https://arxiv.org/html/2605.22898#S6.F12)shows the convergence profiles on MNIST\-1797, which reveal the consequences of Fibonacci scheduling without topology constraint in the most extreme terms\.FibFL’s LSK=3K\{=\}3Plat\.σ=0\.140\\sigma\{=\}0\.140is the global maximum across all 168 conditions: a 34 pp single\-round accuracy drop atR=8R\{=\}8caused by a Fibonacci round activating only C5 \(78 samples\), whose gradient overwrites 7 rounds of accumulated global knowledge, with only 2 recovery rounds remaining in the 10\-round budget\.FibFL\+’s Dir 0\.8 Plat\.σ=0\.129\\sigma\{=\}0\.129is the second highest, driven by the same single\-client dominance mechanism\.FibFL\+\+avoids both failures entirely:σ≤0\.012\\sigma\{\\leq\}0\.012across all 7 MNIST\-1797 scenarios, a 10–12×\\timesimprovement overFibFLandFibFL\+, and the most compelling argument for deployingFibFL\+\+over its simpler family members in small\-data, short\-budget federations\.

![Refer to caption](https://arxiv.org/html/2605.22898v1/mnist1797_figs_v2/fig4_convergence.png)Figure 12:MNIST\-1797 convergence profiling \(N=5N\{=\}5,R=10R\{=\}10\)\. Plat\.σ\\sigmacapped at 0\.145; annotated values exceed this threshold\.FibFL’s LSK=3K\{=\}3\(σ=0\.140\\sigma\{=\}0\.140\) is the global study maximum\. \(color online\)
### 6\.7 Ring Topology and Accuracy–Fairness Trade\-off

We use Fashion\-MNIST as the representative dataset for the ring saving analysis: it occupies the middle of the complexity spectrum and its savings span the full qualitative range from near\-zero under homogeneity to79\.1%79\.1\\%under strong non\-IID, covering all regimes of interest in a single dataset\. The pattern is qualitatively identical across all four datasets; the only dataset\-specific outlier worth noting is the global maximum saving of95\.5%95\.5\\%under Dirα=0\.1\\alpha\{=\}0\.1on MNIST\-60k, discussed below\.

Figure[13](https://arxiv.org/html/2605.22898#S6.F13)\(a\) shows the 2\-opt ring savings on Fashion\-MNIST\. Under IID, savings are near\-zero \(0\.1%0\.1\\%\): no beneficial reordering exists when clients are statistically homogeneous\. As heterogeneity increases, savings grow monotonically: Dirα=0\.8\\alpha\{=\}0\.8\(35\.9%35\.9\\%\), Dirα=0\.5\\alpha\{=\}0\.5\(41\.0%41\.0\\%\), Dirα=0\.1\\alpha\{=\}0\.1\(79\.1%79\.1\\%\)\. Label\-skew savings are high atK=1K\{=\}1andK=2K\{=\}2\(77\.2%77\.2\\%and63\.3%63\.3\\%\) and drop atK=3K\{=\}3\(29\.6%29\.6\\%\) as broader per\-client class coverage reduces inter\-client dissimilarity\. This self\-calibrating behaviour—near\-zero savings under homogeneity and near\-maximum savings under strong non\-IID—requires no manual tuning and provides maximum ring benefit precisely when heterogeneity is most damaging\. Across all datasets, the global maximum ring saving of95\.5%95\.5\\%is achieved under Dirα=0\.1\\alpha\{=\}0\.1on MNIST\-60k, where the permutationσ∗=\[0,8,7,6,9,3,5,2,4,1\]\\sigma^\{\*\}\{=\}\[0,8,7,6,9,3,5,2,4,1\]reduces the ring cost from 1\.2812 to 0\.0583\.

The accuracy–fairness scatter in Figure[13](https://arxiv.org/html/2605.22898#S6.F13)\(b\) aggregates all 168 experimental conditions\.FedAvgandFibFL\+\+form two partially overlapping Pareto\-optimal clusters:FedAvgdominates the high\-accuracy, moderate\-fairness region under IID and Dirichlet, whileFibFL\+\+dominates the high\-accuracy, high\-fairness corner under label\-skew, where the ring topology provides fairness benefits simultaneously with accuracy gains—a Pareto improvement overFedAvgthat is the central empirical finding of this study\.FedRepshows the widest scatter of any method, including the lowest fairness observations in the study \(Dirα=0\.1\\alpha\{=\}0\.1collapses\), whileRDFL,FibFL, andFibFL\+cluster in a consistently conservative intermediate region\.

![Refer to caption](https://arxiv.org/html/2605.22898v1/fmnist_figs/fig5_ring_fairness.png)Figure 13:\(a\)FibFL\+\+2\-opt ring savings by scenario \(Fashion\-MNIST\)\. \(b\) Accuracy–fairness scatter across all 168 conditions\.FibFL\+\+clusters toward the top\-right Pareto frontier\. \(color online\)
### 6\.8 Multi\-Dimensional Summary

Figure[14](https://arxiv.org/html/2605.22898#S6.F14)synthesises the evaluation across six dimensions\.FedAvgleads on IID accuracy, Dirichlet accuracy, and communication efficiency\.FedRepleads on label\-skew accuracy but is weakest on Dirichlet accuracy and fairness\.RDFLandFibFLshare a stability\-first profile: excellentR50%R\_\{50\\%\}and lowσ\\sigma, but modest accuracy in non\-label\-skew conditions\.FibFL\+\+achieves the widest combined area among non\-FedAvgmethods, sacrificing only communication efficiency and raw convergence speed to gain substantial advantages in fairness and label\-skew accuracy\.

![Refer to caption](https://arxiv.org/html/2605.22898v1/figures2/radar.png)Figure 14:Multi\-dimensional performance radar aggregated across all four datasets\.FibFL\+\+spans the widest area among non\-FedAvgmethods\. \(color online\)

## 7\. General Discussion and Conclusions

The Accuracy–Fairness Duality\.The accuracy and fairness tables together reveal a fundamental duality\.FedAvgachieves fairness through*uniformity*: identical model updates for all clients, producing the best fairness under homogeneity \(IID Gini as low as 0\.002 on MNIST\-60k\) but mediocre personalisation under label\-skew\.FedRepachieves accuracy through*specialisation*: client\-specific heads maximise per\-client accuracy in label\-skew at the cost of catastrophic fairness failures \(Gini=0\.596 at MNIST\-1797 Dir 0\.1\) when data is scarce\.FibFL\+\+navigates this duality through*topology\-aware aggregation*: placing complementary clients adjacent reduces gradient interference without model splitting or head parameters, achievingPareto improvements overFedAvgin label\-skew— higher accuracy*and*lower Gini simultaneously\.

The Self\-Calibrating Ring Optimisation\.The 2\-opt algorithm is self\-calibrating: it produces near\-zero savings under IID and near\-maximum savings under strong non\-IID without manual tuning\. The 95\.5% saving under Dirα\\alpha=0\.1 on MNIST\-60k \(cost 1\.2812→\\to0\.0583\) places clients with complementary digit distributions adjacent — analogous to stratified sampling, ensuring the aggregated gradient covers all classes rather than being dominated by majority classes of consecutive ring neighbours\. Critically, ring saving percentage does not translate proportionally to accuracy gain when post\-peak decay dominates, confirming that saving and gain are decoupled when static permutations misalign with evolving client representations\.

Dataset Complexity and Method Ranking Stability\.Two rankings are stable across all datasets and scenarios\. First,FibFL\+\+consistently outperformsFibFLandFibFL\+: their hierarchy is monotone and exception\-free\. Second,FedAvgconsistently leads under IID\. Dataset complexity primarily affects absolute accuracy, not relative rankings; the exception isFedRep’s data\-scarcity failure, which depends on per\-client sample count rather than task difficulty\.

Limitations\.\(1\)Architecture: flat MLP only; CNN and transformer behaviour is unknown\. \(2\)Static distributions: concept drift would invalidate the static ring permutation; recomputation every 10–15 rounds is recommended\. \(3\)Scale:N∈\{5,10\}N\\in\\\{5,10\\\}only;N\>50N\>50behaviour unexplored\. \(4\)Tail fairness: Gini averages over clients; min per\-client accuracy should be reported to capture worst\-case risks\.

Key Findings\.The following conclusions are robustly supported across all 168 conditions\.

1. 1\.FibFL\+\+is the most versatile proposed method\.Best or second\-best accuracy in the majority of conditions; best Gini in 7 of 12 label\-skew configurations; the only proposed method to exceedFedAvgaccuracy \(MNIST\-1797, LSKK=1: 0\.9695 vs\. 0\.9691\)\.
2. 2\.FedAvgremains the strongest baseline under IID and Dirichlet\.Best accuracy in all 16 IID and Dirichlet conditions across all datasets\. Communication efficiency \(33–5×5\\timeshigher thanFibFL\+\+\) reinforces its status as the recommended default for federations without heterogeneity priors\.
3. 3\.FedRep’scollapse is a data\-scarcity effect, quantifiable and predictable\.The near\-monotone relationship between per\-client data volume and Gini at Dirα\\alpha=0\.1 \(MNIST\-1797: 0\.596; CIFAR\-10: 0\.482; Fashion\-MNIST: 0\.495; MNIST\-60k: 0\.240\) establishes a practical threshold: deployFedReponly when per\-client data exceeds∼\\sim1,500–2,000 samples per dominant class\.
4. 4\.The ring topology is self\-calibrating and always positive\.Δring\>0\\Delta\_\{\\text\{ring\}\}\>0in all 28 dataset\-scenario combinations without exception\. Savings range 0–96%, providing maximum benefit when most needed\. No hyperparameter tuning required\.
5. 5\.Fibonacci scheduling alone is insufficient for sustained gains\.FibFLandFibFL\+warm\-start fast \(R50%R\_\{50\\%\}=1\) but exhibit plateau instability \(Plat\.σ\\sigmaup to 0\.140 in small\-data regimes\) and consistently trailFibFL\+\+by 2–10 pp under heterogeneity\.FibFL\+\+is the only Fibonacci variant safe for production deployment across all evaluated conditions\.

### 7\.1 Deployment Guidelines

Table[14](https://arxiv.org/html/2605.22898#S7.T14)translates the empirical findings into actionable deployment guidance, mapping each of the twelve deployment scenarios — covering heterogeneity type, federation size, fairness requirements, bandwidth constraints, and convergence speed needs — to the best and runner\-up method, with a key note summarising the critical trade\-off or condition that drives the recommendation\.

Table 14:Recommended method by deployment scenario\.Concrete decision rules for method selection are as follows\.

- •UseFibFL\+\+when:heterogeneity is moderate to strong \(Dirα≤0\.5\\alpha\{\\leq\}0\.5or LSK≤2K\{\\leq\}2\); fairness is a key requirement;N=5N\{=\}5–20; and33–5×5\\timescomputational overhead is acceptable\. Cacheσ∗\\sigma^\{\*\}and recompute every 10–15 rounds in concept\-drifting settings\.
- •AvoidFibFL\+\+when:data is IID \(savings<5%\{<\}5\\%\),R<10R\{<\}10\(cold\-start dominates\), or bandwidth is severely constrained\.
- •UseFedReponlywhen per\-client data exceeds∼2,000\{\\sim\}2\{,\}000samples per dominant class\. Always conduct a pre\-deployment data\-volume diagnostic\.
- •UseRDFLwhen predictable, monotone convergence is the primary requirement: Plat\.σ≈0\.0001\\sigma\{\\approx\}0\.0001across all conditions\.
- •UseFibFL/FibFL\+for fast initial convergence when per\-client data exceeds 500 samples per dominant class\. In very small data regimes \(<<200 samples/client\), apply gradient clipping \(norm≤1\.0\{\\leq\}1\.0\) to mitigate plateau instability\.

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