A Unified Framework for Fair and Personalized Decentralized Learning under Communication Constraints

# A Unified Framework for Fair and Personalized Decentralized Learning under Communication Constraints
Source: [https://arxiv.org/html/2608.26493](https://arxiv.org/html/2608.26493)
## A Unified Framework for Fair and Personalized Decentralized Learning under Communication ConstraintsThanks:K\. S\. Tharakan and C\. Fischione are with the School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Stockholm, Sweden\. Email: \{tharakan, carlofi\}@kth\.se\.

###### Abstract

Decentralized learning systems aim to collaboratively train models across multiple clients without relying on a central coordinator\. While decentralization improves scalability, privacy, and robustness, it also exacerbates three fundamental challenges: statistical heterogeneity across clients, fairness in client\-level performance, and stringent communication constraints\. This raises a natural question:*how fair can decentralized learning be under limited communication?*We address this question by presenting a unified framework for decentralized learning under communication constraints, bringing together graph\-based personalization, agnostic fairness, and compressed event\-triggered communication\. Specifically, we propose a new algorithm DMFL\-SQ, a decentralized multi\-task learning algorithm that couples personalized model training over a communication graph with an agnostic mixture fairness objective, while reducing communication through sparsification, quantization, and event\-triggered synchronization\. We establish convergence guarantees for general non\-convex objectives and show that DMFL\-SQ achieves an𝒪\(T−1/2\)\\mathcal\{O\}\(T^\{\-1/2\}\)rate in expected squared Moreau\-envelope stationarity despite sparse, quantized, and event\-triggered communication\. We further derive PAC\-Bayes generalization guarantees for the fairness\-aware mixture objective\. Experiments on CIFAR\-10 and the real heterogeneous MUSMET EEG dataset demonstrate that DMFL\-SQ substantially reduces communication while maintaining predictive performance and improving fairness across clients\. Together, our theoretical and empirical results show that personalization, fairness, and communication efficiency can be jointly achieved in decentralized learning while preserving the dominant convergence rate\.

###### Index Terms:

decentralized learning, fairness, multi\-task federated learning, sparse communication\.

## IIntroduction

Federated learning \(FL\) enables multiple clients to collaboratively train machine learning models without directly sharing their local data\[[1](https://arxiv.org/html/2608.26493#bib.bib1),[2](https://arxiv.org/html/2608.26493#bib.bib2)\]\. In the classical FL setting, a central server coordinates the training process by collecting local model updates from participating clients and aggregating them into a single global model\. This paradigm has enabled large\-scale learning across distributed data sources, including mobile devices, edge sensors, and networked intelligent systems\[[3](https://arxiv.org/html/2608.26493#bib.bib3),[4](https://arxiv.org/html/2608.26493#bib.bib4)\]\. However, practical deployments of FL face several fundamental challenges\. Client data are often statistically heterogeneous, making a single global model suboptimal for many users\. Optimizing only the average performance may also lead to unfair outcomes, where clients with scarce, noisy, or non\-representative data experience disproportionately high loss\[[5](https://arxiv.org/html/2608.26493#bib.bib5)\]\. In addition, frequent communication of high\-dimensional model updates creates a severe bottleneck in bandwidth\-limited, energy\-constrained, and unreliable networks\.

A natural way to address statistical heterogeneity is through personalized or multi\-task learning\. Instead of enforcing a single shared model across all clients, each client learns a personalized model while still benefiting from collaboration with related clients\. In graph\-based personalized learning, clients are represented as nodes of a communication or similarity graph, and neighboring models are encouraged to remain close through a graph\-based regularization mechanism\. This allows the learning process to exploit statistical similarity among clients while preserving local adaptability\. Such formulations have been studied in multi\-task and personalized federated learning\[[6](https://arxiv.org/html/2608.26493#bib.bib6),[7](https://arxiv.org/html/2608.26493#bib.bib7),[8](https://arxiv.org/html/2608.26493#bib.bib8),[3](https://arxiv.org/html/2608.26493#bib.bib3)\]\. Nevertheless, most existing personalized FL methods either rely on centralized coordination or do not explicitly address fairness across clients\.

Fairness\-aware federated learning has emerged as an important direction for mitigating disparities in client\-level performance\. Rather than optimizing only the average loss, fairness\-aware approaches seek to improve the performance of underrepresented or high\-loss clients\. Methods based on distributionally robust optimization, agnostic federated learning, and conditional value\-at\-risk \(CVaR\) aim to control the worst\-case or upper\-tail client risk, thereby improving robustness to heterogeneous client distributions\[[9](https://arxiv.org/html/2608.26493#bib.bib9),[10](https://arxiv.org/html/2608.26493#bib.bib10),[11](https://arxiv.org/html/2608.26493#bib.bib11)\]\. However, these methods typically assume centralized aggregation and frequent global synchronization\. Their integration into fully decentralized networks with limited communication remains largely unexplored\.

Communication efficiency is another central challenge in decentralized learning\[[12](https://arxiv.org/html/2608.26493#bib.bib12),[13](https://arxiv.org/html/2608.26493#bib.bib13)\]\. Unlike server\-based FL, fully decentralized systems require clients to exchange information directly with their neighbors over a communication graph\. This removes the need for a central coordinator and improves robustness to server failures, but it also makes communication more complex\. Each client must repeatedly transmit model updates or gradients over bandwidth\-limited links, and these updates are often high\-dimensional\. Compression, sparsification, quantization, and event\-triggered communication can substantially reduce this cost\[[14](https://arxiv.org/html/2608.26493#bib.bib14),[15](https://arxiv.org/html/2608.26493#bib.bib15)\]\. However, aggressive communication reduction may introduce additional bias, network disagreement, and instability, especially in non\-convex learning problems\. Establishing rigorous convergence guarantees under decentralized graph coupling, fairness regularization, and compressed event\-triggered communication is therefore technically challenging\.

Despite substantial progress on personalization, fairness, and communication efficiency, these components are usually studied in isolation\. To the best of our knowledge, existing work does not provide a unified theoretical and algorithmic framework that simultaneously captures graph\-based personalization, agnostic fairness, and compressed decentralized communication under general non\-convex objectives, and DMFL\-SQ introduces a new integrated decentralized algorithmic framework\. This paper addresses the following question:*how fair can decentralized learning be under limited communication?*We answer this question by developing a unified framework for decentralized learning under communication constraints, bringing together three practical components: graph\-based personalization to handle statistical heterogeneity, agnostic fairness to control client\-level performance disparities, and sparsified, quantized, and event\-triggered communication to reduce bandwidth consumption\.

DMFL\-SQ introduces a new integrated decentralized algorithmic framework\. Existing compressed decentralized optimization methods, such as SPARQ\-SGD\[[14](https://arxiv.org/html/2608.26493#bib.bib14)\], CHOCO\-SGD\[[13](https://arxiv.org/html/2608.26493#bib.bib13)\], and related variants, primarily target consensus optimization or a shared global model\. In contrast,DMFL\-SQoptimizes a graph\-regularized personalized objective in which client models need not agree\. Similarly, agnostic federated learning methods address worst\-client or mixture\-risk robustness, but are typically studied in centralized or server\-assisted settings and do not account for sparse\-quantized event\-triggered neighbor communication over a decentralized graph\[[16](https://arxiv.org/html/2608.26493#bib.bib16),[17](https://arxiv.org/html/2608.26493#bib.bib17),[18](https://arxiv.org/html/2608.26493#bib.bib18)\]\.

In contrast,DMFL\-SQtackles decentralized multi\-task learning with graph\-based personalization and agnostic client\-level fairness\. This coupling creates new analytical challenges because the fairness envelope is generally nonsmooth, the optimization variable is the full personalized model collection, and communication compression introduces stale neighbor\-copy errors into the graph\-coupling direction\. The proposed algorithm addresses these challenges by using scalar fairness\-consensus to identify the active agnostic mixture component and by communicating only sparse\-quantized model\-copy innovations when the event trigger is activated\. The accompanying analysis establishes stationarity of the fairness\-aware objective while preserving the standard𝒪\(T−1/2\)\\mathcal\{O\}\(T^\{\-1/2\}\)non\-convex stationarity rate, and explicitly accounting for stochastic\-gradient noise, stale neighbor\-copy error, compression, and event\-triggering\. We also derive PAC\-Bayes generalization guarantees for the agnostic mixture objective, thereby characterizing both optimization and statistical aspects of fairness\-aware decentralized learning\.

### I\-ARelated Work

A straightforward implementation of FL with stochastic gradient descent requires frequent exchange of high\-dimensional gradient or model\-update vectors, which may contain millions of parameters\. This creates substantial communication overhead, particularly in wireless edge networks with limited radio resources\. One common approach to mitigate this bottleneck is to compress the gradient information before transmission\[[13](https://arxiv.org/html/2608.26493#bib.bib13)\]\. A simple example is sign\-based compression, commonly known as SignSGD, where only the sign of each gradient coordinate is communicated\[[19](https://arxiv.org/html/2608.26493#bib.bib19)\]\. These methods improve scalability and avoid a single point of failure, but most existing decentralized optimization algorithms focus on learning a single global model or minimizing an average objective\. In contrast, this work studies a fully decentralized setting in which clients learn personalized models while jointly accounting for fairness and communication constraints\.

In decentralized settings, Assran et al\.\[[20](https://arxiv.org/html/2608.26493#bib.bib20)\]and Tatarenko and Touri\[[21](https://arxiv.org/html/2608.26493#bib.bib21)\]analyze stochastic gradient\-push methods for non\-convex objectives, where the main focus is on approximating distributed averaging over directed networks rather than reducing communication through gradient compression\. In contrast, Tang et al\.\[[12](https://arxiv.org/html/2608.26493#bib.bib12)\]study unbiased stochastic compression mechanisms for exchanging gradient or model information across decentralized nodes\. Personalized FL addresses statistical heterogeneity by allowing clients to learn task\-specific models adapted to their local data\[[6](https://arxiv.org/html/2608.26493#bib.bib6),[7](https://arxiv.org/html/2608.26493#bib.bib7),[22](https://arxiv.org/html/2608.26493#bib.bib22)\]\. These methods improve local adaptation under heterogeneous data, but they generally do not incorporate agnostic client\-level fairness objectives or compressed event\-triggered communication in fully decentralized networks\.

Communication reduction has been widely studied through sparsification\[[23](https://arxiv.org/html/2608.26493#bib.bib23)\], quantization\[[24](https://arxiv.org/html/2608.26493#bib.bib24)\], and error\-feedback correction\[[25](https://arxiv.org/html/2608.26493#bib.bib25)\]\. Event\-triggered communication further reduces bandwidth consumption by allowing clients to communicate only when local updates are sufficiently informative\. SPARQ\-SGD\[[14](https://arxiv.org/html/2608.26493#bib.bib14)\], for example, combines sparsification, quantization, and event\-triggered updates for decentralized optimization\. Recent works have also established non\-convex convergence guarantees for decentralized SGD under compression, network constraints, and related communication limitations\[[26](https://arxiv.org/html/2608.26493#bib.bib26),[27](https://arxiv.org/html/2608.26493#bib.bib27),[28](https://arxiv.org/html/2608.26493#bib.bib28)\]\. However, these methods typically optimize an average objective or a single global model, and do not analyze the joint effect of graph\-based personalization, agnostic fairness, and compressed decentralized communication\.

The above directions address important aspects of distributed learning, but largely in isolation: personalization handles statistical heterogeneity, fairness\-aware FL controls client\-level disparities, and compressed decentralized optimization reduces communication, summarized in Table[I](https://arxiv.org/html/2608.26493#S1.T1)\. In contrast, DMFL\-SQ provides a unified framework that jointly incorporates graph\-based personalization, agnostic fairness, and sparsified, quantized, event\-triggered decentralized communication, while providing both nonconvex convergence and PAC\-Bayes generalization bounds for the resulting fairness\-aware mixture objective\.

TABLE I:Literature ReviewFeature\[[29](https://arxiv.org/html/2608.26493#bib.bib29)\]\[[13](https://arxiv.org/html/2608.26493#bib.bib13)\]\[[6](https://arxiv.org/html/2608.26493#bib.bib6)\]\[[18](https://arxiv.org/html/2608.26493#bib.bib18)\]\[[14](https://arxiv.org/html/2608.26493#bib.bib14)\]\[[30](https://arxiv.org/html/2608.26493#bib.bib30)\]\[[17](https://arxiv.org/html/2608.26493#bib.bib17)\]ProposedFully decentralized✓✓✓✓Personalized✓✓Fairness\-aware objective✓✓Compressedcommunication✓✓✓✓✓Event\-triggeredcommunication✓✓Non\-convexconvergence guarantee✓✓✓Generalizationguarantee✓✓
### I\-BContributions and Organization

The main contributions of this paper are summarized as follows\.

- •Unified decentralized fair multi\-task learning framework:We formulate decentralized learning under communication constraints as a unified problem involving graph\-based personalization, agnostic fairness, and compressed event\-triggered communication\. This provides a single framework for jointly studying personalization, client\-level fairness, and bandwidth efficiency in fully decentralized networks\.
- •Communication\-efficient decentralized algorithm:We proposeDMFL\-SQ, a decentralized multi\-task learning algorithm that combines local stochastic updates, graph\-based model coupling, sparsified and quantized communication, and event\-triggered synchronization\. The algorithm reduces communication overhead while preserving fairness\-aware collaboration among heterogeneous clients\.
- •Non\-convex convergence guarantees:We establish finite\-time convergence guarantees for general non\-convex objectives\. The analysis explicitly separates the optimization error, communication\-induced neighbor\-copy residual, compression residual error, and event\-triggering error, and shows thatDMFL\-SQachieves an𝒪\(T−1/2\)\\mathcal\{O\}\(T^\{\-1/2\}\)rate in expected squared Moreau\-envelope stationarity, matching the standard stochastic nonconvex optimization rate while accounting for decentralized graph coupling, fairness selection, and compressed, event\-triggered communication\.
- •PAC\-Bayes generalization analysis:We derive PAC\-Bayes generalization bounds for the fairness\-aware mixture objective\. The analysis accounts for the complexity of the mixture class and quantifies the statistical cost of controlling both worst\-client and general client\-mixture risks\.
- •Empirical validation on controlled and real heterogeneous data:We evaluateDMFL\-SQon CIFAR\-10 under controlled non\-iid partitions and on the real MUSMET EEG dataset, which provides naturally heterogeneous musician\-specific data\. The results demonstrate the trade\-off among accuracy, fairness, personalization, and communication efficiency, showing that fairness\-aware decentralized learning can improve performance equity across clients while substantially reducing communication\.

Overall, this paper shows that fairness\-aware decentralized learning does not necessarily require sacrificing asymptotic optimization efficiency\. When graph\-based personalization, agnostic fairness, and compressed decentralized communication are jointly designed, fair and personalized models can be learned under stringent communication constraints while preserving the standard non\-convex convergence behavior of stochastic gradient methods\.

An outline of the remainder of the paper is as follows\. The system model and problem formulation are described in Section[II](https://arxiv.org/html/2608.26493#S2)\. Section[III](https://arxiv.org/html/2608.26493#S3)discusses the DMFL\-SQ algorithm\. Section[IV](https://arxiv.org/html/2608.26493#S4)presents the PAC\-Bayes generalization analysis, while Section[V](https://arxiv.org/html/2608.26493#S5)establishes the convergence guarantees\. The simulation results and conclusions are described in Section[VI](https://arxiv.org/html/2608.26493#S6)and Section[VII](https://arxiv.org/html/2608.26493#S7), respectively\.

## IISystem Model and Problem Formulation

We consider a decentralized FL system comprisingnnclients connected through a connected undirected communication graph𝒢=\(𝒱,ℰ\)\\mathcal\{G\}=\(\\mathcal\{V\},\\mathcal\{E\}\)\. Each clienti∈𝒱i\\in\\mathcal\{V\}maintains a personalized local modelwi∈ℝdw\_\{i\}\\in\\mathbb\{R\}^\{d\}and exchanges information only with its neighborsj∈𝒩ij\\in\\mathcal\{N\}\_\{i\}, as illustrated in Fig\.[1](https://arxiv.org/html/2608.26493#S2.F1)\. LetA=\[ai​j\]A=\[a\_\{ij\}\]denote the symmetric nonnegative graph\-weight matrix, whereai​j\>0a\_\{ij\}\>0if\(i,j\)∈ℰ\(i,j\)\\in\\mathcal\{E\}andai​j=0a\_\{ij\}=0otherwise\. These weights define the graph\-coupling regularizer that encourages neighboring personalized models to remain similar while preserving client\-specific solutions\.

Node11Node22Node33w1,F1​\(w1\)w\_\{1\},\\;F\_\{1\}\(w\_\{1\}\)𝒟1∼P1\\mathcal\{D\}\_\{1\}\\sim P\_\{1\}𝒟2∼P2\\mathcal\{D\}\_\{2\}\\sim P\_\{2\}w2,F2​\(w2\)w\_\{2\},\\;F\_\{2\}\(w\_\{2\}\)𝒟3∼P3\\mathcal\{D\}\_\{3\}\\sim P\_\{3\}w3,F3​\(w3\)w\_\{3\},\\;F\_\{3\}\(w\_\{3\}\)Decentralized graph coupling viaai​ja\_\{ij\}ℛ𝒢​\(𝐰\)=α2​∑\(i,j\)∈ℰai​j​‖wi−wj‖2\\mathcal\{R\}\_\{\\mathcal\{G\}\}\(\\mathbf\{w\}\)=\\frac\{\\alpha\}\{2\}\\sum\_\{\(i,j\)\\in\\mathcal\{E\}\}a\_\{ij\}\\\|w\_\{i\}\-w\_\{j\}\\\|^\{2\}Agnostic fairness envelopeΨ⁡\(𝐰\)=supλ∈Λ∑i=1nλi​Fi​\(wi\)\\displaystyle\\Psi\(\\mathbf\{w\}\)=\\sup\_\{\\lambda\\in\\Lambda\}\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}F\_\{i\}\(w\_\{i\}\)Scalar max\-consensus on\(f^it,i\)\(\\widehat\{f\}\_\{i\}^\{t\},i\)f^1t\\widehat\{f\}\_\{1\}^\{t\}f^2t\\widehat\{f\}\_\{2\}^\{t\}f^3t\\widehat\{f\}\_\{3\}^\{t\}Sparse\-quantized event\-triggered model\-copy updatesit\+1=wit\+1−w^it,cit=𝒞⁡\(sit\+1\)\\displaystyle s\_\{i\}^\{t\+1\}=w\_\{i\}^\{t\+1\}\-\\widehat\{w\}\_\{i\}^\{t\},\\hskip 8\.19447ptc\_\{i\}^\{t\}=\\mathcal\{C\}\(s\_\{i\}^\{t\+1\}\)Transmit if‖sit\+1‖≥ϑt\\\|s\_\{i\}^\{t\+1\}\\\|\\geq\\vartheta\_\{t\}; neighbors update their reconstructed copies usingcitc\_\{i\}^\{t\}Local personalized update at every round:wit\+1=wit−γt​hitw\_\{i\}^\{t\+1\}=w\_\{i\}^\{t\}\-\\gamma\_\{t\}h\_\{i\}^\{t\}Fig\. 1:Illustration of DMFL\-SQ\. Each client maintains a personalized modelwiw\_\{i\}, performs local learning at every round, and communicates only sparse\-quantized event\-triggered model\-copy innovations\. Neighboring models are softly coupled throughℛ𝒢​\(𝐰\)\\mathcal\{R\}\_\{\\mathcal\{G\}\}\(\\mathbf\{w\}\), while the active fairness component is selected through scalar max\-consensus over local loss estimates\.Each clientiihas access to a local dataset𝒟i=\{\(xi,k,yi,k\)\}k=1mi\\mathcal\{D\}\_\{i\}=\\\{\(x\_\{i,k\},y\_\{i,k\}\)\\\}\_\{k=1\}^\{m\_\{i\}\}generated from an underlying distributionℙi\\mathbb\{P\}\_\{i\}, where the client distributions\{ℙi\}i=1n\\\{\\mathbb\{P\}\_\{i\}\\\}\_\{i=1\}^\{n\}may differ across clients, capturing statistical heterogeneity \(non\-iid data\)\. The expected local loss is defined as

Fi​\(wi\)=𝔼\(x,y\)∼ℙi​\[ℓi​\(wi,x,y\)\],F\_\{i\}\(w\_\{i\}\)=\\mathbb\{E\}\_\{\(x,y\)\\sim\\mathbb\{P\}\_\{i\}\}\[\\ell\_\{i\}\(w\_\{i\};x,y\)\],\(1\)whereℓi​\(⋅\)\\ell\_\{i\}\(\\cdot\)is a potentially non\-convex loss function that is assumed to beLL\-smooth with respect to the model parameter\.

Unlike standard FL, which enforces a single shared global model, we adopt a*multi\-task*formulation in which each client learns a personalized modelwiw\_\{i\}while collaborating with neighboring nodes\. Let𝐰=\[w1,…,wn\]\\mathbf\{w\}=\[w\_\{1\},\\dots,w\_\{n\}\]denote the collection of client\-specific models\. Collaboration among neighboring clients is modeled through the graph\-Laplacian regularizer

ℛ𝒢​\(𝐰\)=α2​∑\(i,j\)∈ℰai​j​‖wi−wj‖2,\\mathcal\{R\}\_\{\\mathcal\{G\}\}\(\\mathbf\{w\}\)=\\frac\{\\alpha\}\{2\}\\sum\_\{\(i,j\)\\in\\mathcal\{E\}\}a\_\{ij\}\\\|w\_\{i\}\-w\_\{j\}\\\|^\{2\},\(2\)whereα\>0\\alpha\>0controls the strength of graph\-based inter\-client coupling andai​j≥0a\_\{ij\}\\geq 0are graph\-regularization weights\.

To incorporate*fairness*, we adopt an*agnostic mixture\-based formulation*\[[18](https://arxiv.org/html/2608.26493#bib.bib18)\]\. We consider a mixture distributionPλ=∑i=1nλi​ℙiP\_\{\\lambda\}=\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}\\mathbb\{P\}\_\{i\}, whereλ∈Λ⊆Δn\\lambda\\in\\Lambda\\subseteq\\Delta\_\{n\}andΔn\\Delta\_\{n\}denotes the probability simplex\. We define the agnostic fairness envelope as

Ψ⁡\(𝐰\):=supλ∈Λ∑i=1nλi​Fi​\(wi\)\.\\displaystyle\\Psi\(\\mathbf\{w\}\):=\\sup\_\{\\lambda\\in\\Lambda\}\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}F\_\{i\}\(w\_\{i\}\)\.\(3\)The proposed decentralized fair multi\-task objective is then

min\{wi\}i=1n⁡ℒ⁡\(𝐰\):=∑i=1nFi​\(wi\)\+ℛ𝒢​\(𝐰\)\+ρ​Ψ​\(𝐰\),\\min\_\{\\\{w\_\{i\}\\\}\_\{i=1\}^\{n\}\}\\;\\mathcal\{L\}\(\\mathbf\{w\}\)\\;:=\\;\\sum\_\{i=1\}^\{n\}F\_\{i\}\(w\_\{i\}\)\\;\+\\;\\mathcal\{R\}\_\{\\mathcal\{G\}\}\(\\mathbf\{w\}\)\\;\+\\;\\rho\\,\\Psi\(\\mathbf\{w\}\),\(4\)whereρ≥0\\rho\\geq 0controls the trade\-off between aggregate performance and worst\-case \(fairness\) risk across clients\. The envelopeΨ⁡\(𝐰\)\\Psi\(\\mathbf\{w\}\)corresponds to the worst\-case mixture loss over clients, as studied in agnostic federated learning\[[18](https://arxiv.org/html/2608.26493#bib.bib18)\]\. The formulation in \([4](https://arxiv.org/html/2608.26493#S2.E4)\) can be viewed as a penalized version of the pure min\-max problem, allowing a continuous trade\-off between average accuracy and fairness\. Settingρ=0\\rho=0recovers standard decentralized multi\-task learning, while larger values ofρ\\rhoincreasingly emphasize worst\-case client performance\. Unlike classical FL objectives that learn a single global model, the formulation in \([4](https://arxiv.org/html/2608.26493#S2.E4)\) yields personalized modelswiw\_\{i\}that adapt to heterogeneous data while maintaining fairness across the network\.

We study this problem under general*non\-convex*local objectivesFi​\(⋅\)F\_\{i\}\(\\cdot\)and develop a decentralized stochastic algorithm that achieves communication efficiency through sparsified, quantized, and event\-triggered updates\. The proposed algorithm,DMFL\-SQ, admits convergence guarantees to first\-order stationary points under general non\-convex objectives\.

### II\-AAssumptions

We make the following standard assumptions commonly used in decentralized stochastic optimization and federated learning analyses\.

###### Assumption 1\(Smoothness\)\.

Each local objectiveFi:ℝd→ℝF\_\{i\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}isLL\-smooth, i\.e\., for allu,v∈ℝdu,v\\in\\mathbb\{R\}^\{d\},

‖∇Fi​\(u\)−∇Fi​\(v\)‖≤L​‖u−v‖\.\\\|\\nabla F\_\{i\}\(u\)\-\\nabla F\_\{i\}\(v\)\\\|\\leq L\\\|u\-v\\\|\.\(5\)

###### Assumption 2\(Independent Loss Selection and Unbiased Stochastic Gradients\)\.

Letℱt\\mathcal\{F\}\_\{t\}denote the algorithmic history available before the mini\-batches at roundttare sampled\. At every round, the collection of gradient mini\-batches\{Bi,gradt\}i=1n\\\{B\_\{i,\\mathrm\{grad\}\}^\{t\}\\\}\_\{i=1\}^\{n\}is conditionally independent of the collection of selection mini\-batches\{Bi,selt\}i=1n\\\{B\_\{i,\\mathrm\{sel\}\}^\{t\}\\\}\_\{i=1\}^\{n\}givenℱt\\mathcal\{F\}\_\{t\}\.

The stochastic gradient computed fromBi,gradtB\_\{i,\\mathrm\{grad\}\}^\{t\}satisfies

𝔼\[git\|ℱt,λ^t\]=∇Fi\(wit\),\\mathbb\{E\}\\left\[g\_\{i\}^\{t\}\\,\\middle\|\\,\\mathcal\{F\}\_\{t\},\\widehat\{\\lambda\}^\{\\,t\}\\right\]=\\nabla F\_\{i\}\(w\_\{i\}^\{t\}\),\(6\)and

𝔼\[‖git−∇Fi\(wit\)‖2\|ℱt,λ^t\]≤σi2\.\\mathbb\{E\}\\left\[\\left\\\|g\_\{i\}^\{t\}\-\\nabla F\_\{i\}\(w\_\{i\}^\{t\}\)\\right\\\|^\{2\}\\,\\middle\|\\,\\mathcal\{F\}\_\{t\},\\widehat\{\\lambda\}^\{\\,t\}\\right\]\\leq\\sigma\_\{i\}^\{2\}\.\(7\)Letσ¯2:=1n​∑i=1nσi2\\bar\{\\sigma\}^\{2\}:=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\sigma\_\{i\}^\{2\}denote the average stochastic\-gradient variance\.

###### Assumption 3\(Communication Graph and Coupling Weights\)\.

The communication graph𝒢=\(𝒱,ℰ\)\\mathcal\{G\}=\(\\mathcal\{V\},\\mathcal\{E\}\)is undirected and connected\. The graph\-coupling weights satisfyai​j=aj​i≥0a\_\{ij\}=a\_\{ji\}\\geq 0, withai​j\>0a\_\{ij\}\>0only if\(i,j\)∈ℰ\(i,j\)\\in\\mathcal\{E\}, andai​j=0a\_\{ij\}=0otherwise\. Moreover, the weighted degree is uniformly bounded, i\.e\.,

d𝒢:=max⁡∑j∈𝒩ii∈𝒱⁡ai​j<∞\.d\_\{\\mathcal\{G\}\}:=\\max\_\{i\\in\\mathcal\{V\}\}\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}a\_\{ij\}<\\infty\.LetL𝒢L\_\{\\mathcal\{G\}\}denote the weighted graph Laplacian associated with\{ai​j\}\\\{a\_\{ij\}\\\}\. Then the graph regularizer

ℛ𝒢​\(𝐰\)=α2​∑\(i,j\)∈ℰai​j​‖wi−wj‖2\\mathcal\{R\}\_\{\\mathcal\{G\}\}\(\\mathbf\{w\}\)=\\frac\{\\alpha\}\{2\}\\sum\_\{\(i,j\)\\in\\mathcal\{E\}\}a\_\{ij\}\\\|w\_\{i\}\-w\_\{j\}\\\|^\{2\}is smooth with constantLR=α​λmax​\(L𝒢\)L\_\{R\}=\\alpha\\lambda\_\{\\max\}\(L\_\{\\mathcal\{G\}\}\)\.

###### Assumption 4\(Contractive Compressor\)\.

The possibly randomized compression operator𝒞:ℝd→ℝd\\mathcal\{C\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}satisfies the contraction property: there existsω∈\(0,1\]\\omega\\in\(0,1\]such that

𝔼​‖𝒞⁡\(v\)−v‖2≤\(1−ω\)​‖v‖2,∀v∈ℝd,\\mathbb\{E\}\\\|\\mathcal\{C\}\(v\)\-v\\\|^\{2\}\\leq\(1\-\\omega\)\\\|v\\\|^\{2\},\\quad\\forall v\\in\\mathbb\{R\}^\{d\},\(8\)where the expectation is taken over the randomness of the compressor\.

Assumptions[1](https://arxiv.org/html/2608.26493#Thmassmon1)\-[4](https://arxiv.org/html/2608.26493#Thmassmon4)control the regularity and stochastic moments of the local learning directions, the graph\-based personalized coupling, and compressed communication\. Assumption[3](https://arxiv.org/html/2608.26493#Thmassmon3)ensures that the graph\-coupling direction is well defined and that errors arising from stale reconstructed neighbor models can be controlled through the communication residual\. Unlike consensus\-based decentralized SGD, DMFL\-SQ does not impose model consensus through a mixing step; personalization is instead maintained through the graph regularizerℛ𝒢\\mathcal\{R\}\_\{\\mathcal\{G\}\}\.

## IIIDMFL\-SQ Algorithm

We now present the proposed decentralized multi\-task fair FL with sparsified and quantized communication, referred to asDMFL\-SQ, also summarized in Algorithm[1](https://arxiv.org/html/2608.26493#alg1)\. The method enables each client to perform local stochastic updates, exchange compressed information with its neighbors only when necessary, and promote fairness across heterogeneous clients via the agnostic mixture\-based objective in \([4](https://arxiv.org/html/2608.26493#S2.E4)\)\.

At iterationtt, each clientiidraws two conditionally independent mini\-batches,Bi,seltB\_\{i,\\mathrm\{sel\}\}^\{t\}andBi,gradtB\_\{i,\\mathrm\{grad\}\}^\{t\}\. The first mini\-batch is used to estimate the local loss for fairness\-component selection:

f^it=1\|Bi,selt\|​∑z∈Bi,seltℓi​\(wit,z\)\.\\widehat\{f\}\_\{i\}^\{t\}=\\frac\{1\}\{\|B\_\{i,\\mathrm\{sel\}\}^\{t\}\|\}\\sum\_\{z\\in B\_\{i,\\mathrm\{sel\}\}^\{t\}\}\\ell\_\{i\}\(w\_\{i\}^\{t\};z\)\.\(9\)The second mini\-batch is used to compute the stochastic gradient

git=1\|Bi,gradt\|​∑z∈Bi,gradt∇ℓi​\(wit,z\)\.g\_\{i\}^\{t\}=\\frac\{1\}\{\|B\_\{i,\\mathrm\{grad\}\}^\{t\}\|\}\\sum\_\{z\\in B\_\{i,\\mathrm\{grad\}\}^\{t\}\}\\nabla\\ell\_\{i\}\(w\_\{i\}^\{t\};z\)\.\(10\)Letℱt\\mathcal\{F\}\_\{t\}denote the algorithmic history available before the selection and gradient mini\-batches at roundttare sampled\. SinceBi,gradtB\_\{i,\\mathrm\{grad\}\}^\{t\}is conditionally independent of\{Bk,selt\}k=1n\\\{B\_\{k,\\mathrm\{sel\}\}^\{t\}\\\}\_\{k=1\}^\{n\}givenℱt\\mathcal\{F\}\_\{t\}, we have

𝔼\[git\|ℱt,𝐟^t,λ^t\]=∇Fi\(wit\)\.\\mathbb\{E\}\\\!\\left\[g\_\{i\}^\{t\}\\,\\middle\|\\,\\mathcal\{F\}\_\{t\},\\widehat\{\\mathbf\{f\}\}^\{\\,t\},\\widehat\{\\lambda\}^\{t\}\\right\]=\\nabla F\_\{i\}\(w\_\{i\}^\{t\}\)\.\(11\)For any mixture vectorλ∈Λ\\lambda\\in\\Lambda, define the population and empirical mixture losses, respectively, as

Φλ​\(𝐰\):=∑i=1nλi​Fi​\(wi\),Φ^λt:=∑i=1nλi​f^it\.\\Phi\_\{\\lambda\}\(\\mathbf\{w\}\):=\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}F\_\{i\}\(w\_\{i\}\),\\qquad\\widehat\{\\Phi\}\_\{\\lambda\}^\{\\,t\}:=\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}\\widehat\{f\}\_\{i\}^\{t\}\.\(12\)For a general mixture classΛ\\Lambda, the empirical fairness mixture is selected according toλ^t∈arg⁡maxλ∈Λ⁡Φ^λt\\widehat\{\\lambda\}^\{\\,t\}\\in\\arg\\max\_\{\\lambda\\in\\Lambda\}\\widehat\{\\Phi\}\_\{\\lambda\}^\{\\,t\}\. For the full\-simplex caseΛ=Δn\\Lambda=\\Delta\_\{n\}, this reduces toit⋆∈argmaxi∈\[n\]f^it,λ^it=𝟙\{i=it⋆\}i\_\{t\}^\{\\star\}\\in\\arg\\max\_\{i\\in\[n\]\}\\widehat\{f\}\_\{i\}^\{t\},\\widehat\{\\lambda\}\_\{i\}^\{t\}=\\mathbbm\{1\}\\\{i=i\_\{t\}^\{\\star\}\\\}\. The stochastic fairness direction implemented by clientiiis defined asξ^it=λ^it​git\\widehat\{\\xi\}\_\{i\}^\{t\}=\\widehat\{\\lambda\}\_\{i\}^\{t\}g\_\{i\}^\{t\}\. We use the notationξ^it\\widehat\{\\xi\}\_\{i\}^\{t\}to emphasize that this direction is constructed from mini\-batch loss estimates and is not necessarily a population subgradient of the fairness envelopeΨ\\Psi\.

To quantify the discrepancy between empirical fairness selection and the population fairness envelope, define the population fairness\-selection gap

εΨ,t:=Ψ⁡\(𝐰t\)−Φλ^t​\(𝐰t\)≥0\.\\varepsilon\_\{\\Psi,t\}:=\\Psi\(\\mathbf\{w\}^\{t\}\)\-\\Phi\_\{\\widehat\{\\lambda\}^\{\\,t\}\}\(\\mathbf\{w\}^\{t\}\)\\geq 0\.\(13\)ForΛ=Δn\\Lambda=\\Delta\_\{n\}, this becomesεΨ,t=maxi∈\[n\]⁡Fi​\(wit\)−Fit⋆​\(wit⋆t\)\\varepsilon\_\{\\Psi,t\}=\\max\_\{i\\in\[n\]\}F\_\{i\}\(w\_\{i\}^\{t\}\)\-F\_\{i\_\{t\}^\{\\star\}\}\(w\_\{i\_\{t\}^\{\\star\}\}^\{t\}\)\. Thus,εΨ,t=0\\varepsilon\_\{\\Psi,t\}=0when the empirically selected client is also a population\-active client\. In general, exact scalar max\-consensus guarantees agreement on the empirical maximizer but does not imply that the selected client maximizes the population loss\.

Letδt:=supλ∈Λ\|Φ^λt−Φλ​\(𝐰t\)\|\\delta\_\{t\}:=\\sup\_\{\\lambda\\in\\Lambda\}\\left\|\\widehat\{\\Phi\}\_\{\\lambda\}^\{t\}\-\\Phi\_\{\\lambda\}\(\\mathbf\{w\}^\{t\}\)\\right\|\. BecauseΛ⊆Δn\\Lambda\\subseteq\\Delta\_\{n\},δt≤maxi∈\[n\]⁡\|f^it−Fi​\(wit\)\|\\delta\_\{t\}\\leq\\max\_\{i\\in\[n\]\}\\left\|\\widehat\{f\}\_\{i\}^\{t\}\-F\_\{i\}\(w\_\{i\}^\{t\}\)\\right\|\. Sinceλ^t\\widehat\{\\lambda\}^\{\\,t\}maximizesΦ^λt\\widehat\{\\Phi\}\_\{\\lambda\}^\{t\}overΛ\\Lambda, the population fairness\-selection gap satisfies

εΨ,t≤2​δt\.\\varepsilon\_\{\\Psi,t\}\\leq 2\\delta\_\{t\}\.\(14\)Thus, the empirically selected fairness mixture becomes increasingly accurate in terms of the population fairness objective as the uniform loss\-estimation error decreases\.

Algorithm 1DMFL\-SQ: Decentralized Multi\-Task Fair Learning with Sparse\-Quantized Event\-Triggered Communication1:Input:stepsizes

\{γt\}\\\{\\gamma\_\{t\}\\\}, trigger thresholds

\{ϑt\}\\\{\\vartheta\_\{t\}\\\}, graph weights

\{ai​j\}\\\{a\_\{ij\}\\\}, compression operator

𝒞\\mathcal\{C\}, fairness weight

ρ\\rho, graph regularization weight

α\\alpha, fairness\-consensus rounds

KΨK\_\{\\Psi\}\.

2:Initialize

wi0w\_\{i\}^\{0\}for all clients

i∈\{1,…,n\}i\\in\\\{1,\\ldots,n\\\}\.

3:Initialize the locally stored transmitted copies

w^i0=wi0\\widehat\{w\}\_\{i\}^\{0\}=w\_\{i\}^\{0\}∀\\forallii, the neighbor copies

w^i→j0=wi0\\widehat\{w\}\_\{i\\to j\}^\{0\}=w\_\{i\}^\{0\}∀\\forall\(i,j\)∈ℰ\(i,j\)\\in\\mathcal\{E\}\.

4:for

t=0,…,T−1t=0,\\ldots,T\-1do

5:foreach client

i=1,…,ni=1,\\ldots,nin paralleldo

6:Sample conditionally independent mini\-batches

Bi,seltB\_\{i,\\mathrm\{sel\}\}^\{t\}and

Bi,gradtB\_\{i,\\mathrm\{grad\}\}^\{t\}and compute

f^it=1\|Bi,selt\|​∑z∈Bi,seltℓi​\(wit,z\),\\widehat\{f\}\_\{i\}^\{t\}=\\frac\{1\}\{\|B\_\{i,\\mathrm\{sel\}\}^\{t\}\|\}\\sum\_\{z\\in B\_\{i,\\mathrm\{sel\}\}^\{t\}\}\\ell\_\{i\}\(w\_\{i\}^\{t\};z\),git=1\|Bi,gradt\|​∑z∈Bi,gradt∇ℓi​\(wit,z\)\.g\_\{i\}^\{t\}=\\frac\{1\}\{\|B\_\{i,\\mathrm\{grad\}\}^\{t\}\|\}\\sum\_\{z\\in B\_\{i,\\mathrm\{grad\}\}^\{t\}\}\\nabla\\ell\_\{i\}\(w\_\{i\}^\{t\};z\)\.
7:endfor

8:Run

KΨK\_\{\\Psi\}scalar max\-consensus rounds on

\(f^it,i\)\(\\widehat\{f\}\_\{i\}^\{t\},i\)to obtain

it⋆∈arg⁡maxi∈\[n\]⁡f^it\.i\_\{t\}^\{\\star\}\\in\\arg\\max\_\{i\\in\[n\]\}\\widehat\{f\}\_\{i\}^\{t\}\.
9:Set

λ^it=𝟙\{i=it⋆\}\\widehat\{\\lambda\}\_\{i\}^\{t\}=\\mathbbm\{1\}\\\{i=i\_\{t\}^\{\\star\}\\\},

ξ^it=λ^it​git\\widehat\{\\xi\}\_\{i\}^\{t\}=\\widehat\{\\lambda\}\_\{i\}^\{t\}g\_\{i\}^\{t\}\.

10:foreach client

i=1,…,ni=1,\\ldots,nin paralleldo

11:Compute the graph\-coupling direction

rit=α​∑j∈𝒩iai​j​\(wit−w^j→it\)\.r\_\{i\}^\{t\}=\\alpha\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}a\_\{ij\}\\left\(w\_\{i\}^\{t\}\-\\widehat\{w\}\_\{j\\to i\}^\{t\}\\right\)\.
12:Form the local stochastic direction

hit=git\+rit\+ρ​ξ^it=\(1\+ρ​λ^it\)​git\+rit\.h\_\{i\}^\{t\}=g\_\{i\}^\{t\}\+r\_\{i\}^\{t\}\+\\rho\\widehat\{\\xi\}\_\{i\}^\{t\}=\\left\(1\+\\rho\\widehat\{\\lambda\}\_\{i\}^\{t\}\\right\)g\_\{i\}^\{t\}\+r\_\{i\}^\{t\}\.
13:Update the personalized model

wit\+1=wit−γt​hit\.w\_\{i\}^\{t\+1\}=w\_\{i\}^\{t\}\-\\gamma\_\{t\}h\_\{i\}^\{t\}\.
14:Compute the model\-copy innovation

sit\+1=wit\+1−w^it,s\_\{i\}^\{t\+1\}=w\_\{i\}^\{t\+1\}\-\\widehat\{w\}\_\{i\}^\{t\},where

w^it\\widehat\{w\}\_\{i\}^\{t\}denotes the most recent transmitted copy of

wiw\_\{i\}reconstructed by its neighbors\.

15:if

‖sit\+1‖≥ϑt\\\|s\_\{i\}^\{t\+1\}\\\|\\geq\\vartheta\_\{t\}then

16:Compress and transmit

cit=𝒞⁡\(sit\+1\)c\_\{i\}^\{t\}=\\mathcal\{C\}\(s\_\{i\}^\{t\+1\}\)to all neighbors

j∈𝒩ij\\in\\mathcal\{N\}\_\{i\}\.

17:Update the locally stored transmitted copy

w^it\+1=w^it\+cit\.\\widehat\{w\}\_\{i\}^\{t\+1\}=\\widehat\{w\}\_\{i\}^\{t\}\+c\_\{i\}^\{t\}\.
18:else

19:Set

cit=0c\_\{i\}^\{t\}=0and

w^it\+1=w^it\.\\widehat\{w\}\_\{i\}^\{t\+1\}=\\widehat\{w\}\_\{i\}^\{t\}\.
20:endif

21:endfor

22:foreach client

i=1,…,ni=1,\\ldots,nin paralleldo

23:For every received

cjtc\_\{j\}^\{t\}from

j∈𝒩ij\\in\\mathcal\{N\}\_\{i\}, update

w^j→it\+1=w^j→it\+cjt\.\\widehat\{w\}\_\{j\\to i\}^\{t\+1\}=\\widehat\{w\}\_\{j\\to i\}^\{t\}\+c\_\{j\}^\{t\}\.
24:If no message is received from neighbor

jj, set

w^j→it\+1=w^j→it\.\\widehat\{w\}\_\{j\\to i\}^\{t\+1\}=\\widehat\{w\}\_\{j\\to i\}^\{t\}\.
25:endfor

26:endfor

27:Output:personalized models

\{wiT\}i=1n\\\{w\_\{i\}^\{T\}\\\}\_\{i=1\}^\{n\}\.

Fairness\-consensus implementation:ForΛ=Δn\\Lambda=\\Delta\_\{n\}, DMFL\-SQ runsKΨK\_\{\\Psi\}rounds of scalar max\-consensus over the pairs\(f^it,i\)\(\\widehat\{f\}\_\{i\}^\{t\},i\), using a fixed lexicographic tie\-breaking rule\. IfKΨ≥diam⁡\(𝒢\)K\_\{\\Psi\}\\geq\\operatorname\{diam\}\(\\mathcal\{G\}\), all clients recover the same empirical maximizerit⋆∈arg⁡maxi∈\[n\]⁡f^iti\_\{t\}^\{\\star\}\\in\\arg\\max\_\{i\\in\[n\]\}\\widehat\{f\}\_\{i\}^\{t\}\. Exact max\-consensus therefore eliminates disagreement among clients regarding the empirical fairness component\. It does not, by itself, guarantee thatit⋆i\_\{t\}^\{\\star\}is an active maximizer of the population fairness envelope\. The resulting statistical discrepancy is quantified byεΨ,t\\varepsilon\_\{\\Psi,t\}in \([13](https://arxiv.org/html/2608.26493#S3.E13)\) and is explicitly retained in the convergence analysis\.

Each client maintains reconstructed copies of its neighbors’ most recently communicated models\. Letw^j→it\\widehat\{w\}\_\{j\\to i\}^\{t\}denote the latest copy of clientjj’s model available at clientii\. These copies are used to evaluate the graph\-coupling term in the personalized objective\. Specifically, clientiicomputes

rit=α​∑j∈𝒩iai​j​\(wit−w^j→it\),r\_\{i\}^\{t\}=\\alpha\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}a\_\{ij\}\\left\(w\_\{i\}^\{t\}\-\\widehat\{w\}\_\{j\\to i\}^\{t\}\\right\),\(15\)which is a communication\-efficient approximation of∇wiℛ𝒢​\(𝐰t\)\\nabla\_\{w\_\{i\}\}\\mathcal\{R\}\_\{\\mathcal\{G\}\}\(\\mathbf\{w\}^\{t\}\)\. Although the graph regularizerℛ𝒢​\(𝐰\)\\mathcal\{R\}\_\{\\mathcal\{G\}\}\(\\mathbf\{w\}\)is defined using the current neighboring models, a decentralized event\-triggered implementation cannot accesswjtw\_\{j\}^\{t\}at every round\. Therefore, clientiievaluates the graph\-coupling direction using the latest reconstructed neighbor copyw^j→it\\widehat\{w\}\_\{j\\to i\}^\{t\}\. The resulting discrepancywjt−w^j→itw\_\{j\}^\{t\}\-\\widehat\{w\}\_\{j\\to i\}^\{t\}is explicitly captured by the communication residualℰt\\mathcal\{E\}^\{t\}and is accounted for in the convergence analysis\. The local stochastic direction implemented by clientiiis

hit=git\+rit\+ρ​ξ^it=\(1\+ρ​λ^it\)​git\+rit\.h\_\{i\}^\{t\}=g\_\{i\}^\{t\}\+r\_\{i\}^\{t\}\+\\rho\\widehat\{\\xi\}\_\{i\}^\{t\}=\\left\(1\+\\rho\\widehat\{\\lambda\}\_\{i\}^\{t\}\\right\)g\_\{i\}^\{t\}\+r\_\{i\}^\{t\}\.\(16\)Clientiithen updates its personalized model according to

wit\+1=wit−γt​hit\.w\_\{i\}^\{t\+1\}=w\_\{i\}^\{t\}\-\\gamma\_\{t\}h\_\{i\}^\{t\}\.\(17\)After the local model update, clientiichecks whether its current model has changed sufficiently relative to the last version reconstructed by its neighbors\. Letw^it\\widehat\{w\}\_\{i\}^\{t\}denote this last transmitted copy and define the model\-copy innovation assit\+1=wit\+1−w^its\_\{i\}^\{t\+1\}=w\_\{i\}^\{t\+1\}\-\\widehat\{w\}\_\{i\}^\{t\}\.

Clientiicommunicates only if‖sit\+1‖≥ϑt\\\|s\_\{i\}^\{t\+1\}\\\|\\geq\\vartheta\_\{t\}, whereϑt\\vartheta\_\{t\}is a time\-varying trigger threshold\. If the trigger condition is satisfied, clientiisends the sparse\-quantized updatecit=𝒞⁡\(sit\+1\)c\_\{i\}^\{t\}=\\mathcal\{C\}\(s\_\{i\}^\{t\+1\}\)to all neighbors and updates its locally stored transmitted copy asw^it\+1=w^it\+cit\\widehat\{w\}\_\{i\}^\{t\+1\}=\\widehat\{w\}\_\{i\}^\{t\}\+c\_\{i\}^\{t\}\. If the trigger condition is not satisfied, no message is sent, andcit=0c\_\{i\}^\{t\}=0,w^it\+1=w^it\\widehat\{w\}\_\{i\}^\{t\+1\}=\\widehat\{w\}\_\{i\}^\{t\}\.

Upon receivingcjtc\_\{j\}^\{t\}from a neighborjj, clientiiupdates its local reconstructed copy asw^j→it\+1=w^j→it\+cjt\\widehat\{w\}\_\{j\\to i\}^\{t\+1\}=\\widehat\{w\}\_\{j\\to i\}^\{t\}\+c\_\{j\}^\{t\}\. If no message is received from neighborjj, the copy is kept unchanged:w^j→it\+1=w^j→it\\widehat\{w\}\_\{j\\to i\}^\{t\+1\}=\\widehat\{w\}\_\{j\\to i\}^\{t\}\. The event trigger in Algorithm 1 controls only communication and does not suppress local learning\. Every client updates its personalized model at each iteration\. When the trigger is inactive, the client skips broadcasting its model\-copy innovation, while retaining the updated local model\. This mechanism reduces communication while preserving personalization\. Unlike consensus\-based decentralized SGD, DMFL\-SQ does not average the client models after every local update\. Instead, neighboring models are softly coupled through the graph regularizerℛ𝒢​\(𝐰\)\\mathcal\{R\}\_\{\\mathcal\{G\}\}\(\\mathbf\{w\}\), while sparse, quantized, and event\-triggered communication maintains sufficiently accurate neighbor model copies\. Therefore, client\-specific variation is controlled by the personalized objective itself rather than being suppressed by an explicit gossip\-averaging step\.

Communication accounting\.LetPPdenote the number of model parameters,kitk\_\{i\}^\{t\}the number of nonzero coordinates transmitted by clientiiat roundtt,bqb\_\{q\}the quantization precision, anddegi=\|𝒩i\|\\deg\_\{i\}=\|\\mathcal\{N\}\_\{i\}\|its graph degree\. Define the trigger indicatorχit=𝟏\{∥sit\+1∥≥ϑt\}\\chi\_\{i\}^\{t\}=\\mathbf\{1\}\\\{\\\|s\_\{i\}^\{t\+1\}\\\|\\geq\\vartheta\_\{t\}\\\}\. The model\-copy communication cost at roundttis

Bmodel,t\\displaystyle B\_\{\\mathrm\{model\},t\}=∑i=1ndegi\[btrig\+χit\(kitbq\+min\{kit⌈log2P⌉,P\}\\displaystyle=\\sum\_\{i=1\}^\{n\}\\deg\_\{i\}\\bigg\[b\_\{\\mathrm\{trig\}\}\+\\chi\_\{i\}^\{t\}\\bigg\(k\_\{i\}^\{t\}b\_\{q\}\+\\min\\\!\\left\\\{k\_\{i\}^\{t\}\\lceil\\log\_\{2\}P\\rceil,P\\right\\\}\+bscale\+bhdr\)\],\\displaystyle\+b\_\{\\mathrm\{scale\}\}\+b\_\{\\mathrm\{hdr\}\}\\bigg\)\\bigg\],\(18\)where the terms account for the quantized values, coordinate indices or a binary mask, the quantizer scale, message header, and trigger metadata, respectively\. If silence indicates an inactive trigger, we setbtrig=0b\_\{\\mathrm\{trig\}\}=0\. The factordegi\\deg\_\{i\}accounts for delivery to all neighbors; for an undirected graph,∑idegi=2​\|ℰ\|\\sum\_\{i\}\\deg\_\{i\}=2\|\\mathcal\{E\}\|\.

The scalar fairness\-consensus step incurs

BΨ,t=2​KΨ​\|ℰ\|​\(bloss\+⌈log2⁡n⌉\),B\_\{\\Psi,t\}=2K\_\{\\Psi\}\|\\mathcal\{E\}\|\\left\(b\_\{\\mathrm\{loss\}\}\+\\lceil\\log\_\{2\}n\\rceil\\right\),\(19\)since the pair\(f^it,i\)\(\\widehat\{f\}\_\{i\}^\{t\},i\)is exchanged in both directions over every edge during each of theKΨK\_\{\\Psi\}max\-consensus rounds\. Hence, the total communication overTTrounds and the average communication per client are

Btot=∑t=0T−1\(Bmodel,t\+BΨ,t\),B¯client=Btotn\.B\_\{\\mathrm\{tot\}\}=\\sum\_\{t=0\}^\{T\-1\}\\left\(B\_\{\\mathrm\{model\},t\}\+B\_\{\\Psi,t\}\\right\),\\qquad\\overline\{B\}\_\{\\mathrm\{client\}\}=\\frac\{B\_\{\\mathrm\{tot\}\}\}\{n\}\.\(20\)
Overall, DMFL\-SQ updates each personalized model using a stochastic approximation of the full objective, while communication is used only to maintain reconstructed neighbor copies for evaluating the graph regularizer\. Unlike consensus\-based decentralized SGD, the client models are not averaged after every local update\. Instead, personalization is preserved because inter\-client coupling is controlled softly throughℛ𝒢​\(𝐰\)\\mathcal\{R\}\_\{\\mathcal\{G\}\}\(\\mathbf\{w\}\), and the sparse\-quantized event\-triggered mechanism reduces communication by updatingw^j→it\\widehat\{w\}\_\{j\\to i\}^\{t\}only when necessary\.

## IVPAC\-Bayes Generalization for DMFL\-SQ

We establish an algorithm\-agnostic PAC\-Bayes generalization guarantee for the fairness\-aware agnostic mixture objective under possibly non\-convex client losses\. We subsequently complement this result with a corollary that quantifies the one\-step empirical mixture\-risk perturbation induced by stale and compressed neighbor\-model information\. Considernnclients with mutually independent datasets, where clientiihas access to𝒟i=\{zi,k\}k=1mi\\mathcal\{D\}\_\{i\}=\\\{z\_\{i,k\}\\\}\_\{k=1\}^\{m\_\{i\}\}, whose samples are drawn iid fromℙi\\mathbb\{P\}\_\{i\}, withℙi≠ℙj\\mathbb\{P\}\_\{i\}\\neq\\mathbb\{P\}\_\{j\}in general\. Letℓi​\(wi,z\)∈\[0,1\]\\ell\_\{i\}\(w\_\{i\};z\)\\in\[0,1\]denote the loss at clientii\. Define the population and empirical risks as

Fi​\(wi\):=𝔼z∼ℙi​\[ℓi​\(wi,z\)\],F^i​\(wi\):=1mi​∑k=1miℓi​\(wi,zi,k\)\.\\displaystyle F\_\{i\}\(w\_\{i\}\):=\\mathbb\{E\}\_\{z\\sim\\mathbb\{P\}\_\{i\}\}\[\\ell\_\{i\}\(w\_\{i\};z\)\],\\widehat\{F\}\_\{i\}\(w\_\{i\}\):=\\frac\{1\}\{m\_\{i\}\}\\sum\_\{k=1\}^\{m\_\{i\}\}\\ell\_\{i\}\(w\_\{i\};z\_\{i,k\}\)\.
Let𝐰:=\(w1,…,wn\)\\mathbf\{w\}:=\(w\_\{1\},\\dots,w\_\{n\}\)denote the collection of client models\. To capture fairness\-aware objectives, we consider agnostic mixture risks defined over a set of mixture weightsΛ⊆Δn\\Lambda\\subseteq\\Delta\_\{n\}, whereΔn\\Delta\_\{n\}is the probability simplex:

Ψ⁡\(𝐰\):=supλ∈Λ∑i=1nλi​Fi​\(wi\),Ψ^​\(𝐰\):=supλ∈Λ∑i=1nλi​F^i​\(wi\)\.\\displaystyle\\Psi\(\\mathbf\{w\}\):=\\sup\_\{\\lambda\\in\\Lambda\}\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}F\_\{i\}\(w\_\{i\}\),\\vskip 5\.69046pt\\widehat\{\\Psi\}\(\\mathbf\{w\}\):=\\sup\_\{\\lambda\\in\\Lambda\}\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}\\widehat\{F\}\_\{i\}\(w\_\{i\}\)\.We emphasize that no convexity is assumed onFiF\_\{i\}\.

### IV\-APAC\-Bayes Generalization Bound

For a fixed mixture vectorλ∈Λ\\lambda\\in\\Lambda, definemλ:=\(∑i=1nλi2mi\)−1\.m\_\{\\lambda\}:=\\left\(\\sum\_\{i=1\}^\{n\}\\frac\{\\lambda\_\{i\}^\{2\}\}\{m\_\{i\}\}\\right\)^\{\-1\}\.For a general compact mixture class, definemeff:=infλ∈Λmλ\.m\_\{\\mathrm\{eff\}\}:=\\inf\_\{\\lambda\\in\\Lambda\}m\_\{\\lambda\}\.

###### Theorem 1\(PAC\-Bayes Generalization for Agnostic Mixtures\)\.

Assume thatℓi​\(wi,z\)∈\[0,1\]\\ell\_\{i\}\(w\_\{i\};z\)\\in\[0,1\]for all clientsi∈\[n\]i\\in\[n\]\. LetFi​\(wi\):=𝔼z∼ℙi​\[ℓi​\(wi,z\)\]F\_\{i\}\(w\_\{i\}\):=\\mathbb\{E\}\_\{z\\sim\\mathbb\{P\}\_\{i\}\}\[\\ell\_\{i\}\(w\_\{i\};z\)\],F^i​\(wi\):=1mi​∑k=1miℓi​\(wi,zi,k\)\\widehat\{F\}\_\{i\}\(w\_\{i\}\):=\\frac\{1\}\{m\_\{i\}\}\\sum\_\{k=1\}^\{m\_\{i\}\}\\ell\_\{i\}\(w\_\{i\};z\_\{i,k\}\), and define the population and empirical agnostic mixture risksΨ⁡\(𝐰\):=supλ∈Λ∑i=1nλi​Fi​\(wi\)\\Psi\(\\mathbf\{w\}\):=\\sup\_\{\\lambda\\in\\Lambda\}\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}F\_\{i\}\(w\_\{i\}\),Ψ^​\(𝐰\):=supλ∈Λ∑i=1nλi​F^i​\(wi\)\\widehat\{\\Psi\}\(\\mathbf\{w\}\):=\\sup\_\{\\lambda\\in\\Lambda\}\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}\\widehat\{F\}\_\{i\}\(w\_\{i\}\)\. LetΠ\\Pibe any data\-independent prior over the stacked model collection𝐰\\mathbf\{w\}, and letQQbe any posterior over the same space\. LetτPB\>0\\tau\_\{\\mathrm\{PB\}\}\>0be any deterministic PAC\-Bayes inverse\-temperature parameter chosen independently of the observed client datasets\. Then, for anyδ∈\(0,1\)\\delta\\in\(0,1\), the following statements hold with probability at least1−δ1\-\\deltaover the draw of all client datasets\{𝒟i\}i=1n\\\{\\mathcal\{D\}\_\{i\}\\\}\_\{i=1\}^\{n\}\.

Case 1: Full simplex\.IfΛ=Δn\\Lambda=\\Delta\_\{n\}, then, for any fixedτPB\>0\\tau\_\{\\mathrm\{PB\}\}\>0, with probability at least1−δ1\-\\delta, the following holds simultaneously for all posteriorsQQ:

𝔼𝐰∼Q​\[Ψ​\(𝐰\)\]\\displaystyle\\mathbb\{E\}\_\{\\mathbf\{w\}\\sim Q\}\\\!\\left\[\\Psi\(\\mathbf\{w\}\)\\right\]≤𝔼𝐰∼Q​\[Ψ^​\(𝐰\)\]\+KL\(Q∥Π\)\+lnn\+ln\(1/δ\)τPB\\displaystyle\\leq\\mathbb\{E\}\_\{\\mathbf\{w\}\\sim Q\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}\)\\right\]\+\\frac\{\\mathrm\{KL\}\(Q\\\|\\Pi\)\+\\ln n\+\\ln\(1/\\delta\)\}\{\\tau\_\{\\mathrm\{PB\}\}\}\+τPB8​mmin,\\displaystyle\+\\frac\{\\tau\_\{\\mathrm\{PB\}\}\}\{8m\_\{\\min\}\},\(21\)wheremmin:=mini∈\[n\]⁡mim\_\{\\min\}:=\\min\_\{i\\in\[n\]\}m\_\{i\}\.

Case 2: General compact mixture class\.IfΛ⊆Δn\\Lambda\\subseteq\\Delta\_\{n\}is compact, then, for anyϵ\>0\\epsilon\>0and any fixedτPB\>0\\tau\_\{\\mathrm\{PB\}\}\>0, with probability at least1−δ1\-\\delta, the following holds simultaneously for all posteriorsQQ:

𝔼𝐰∼Q​\[Ψ​\(𝐰\)\]≤\\displaystyle\\mathbb\{E\}\_\{\\mathbf\{w\}\\sim Q\}\\\!\\left\[\\Psi\(\\mathbf\{w\}\)\\right\]\\leq\{\}𝔼𝐰∼Q​\[Ψ^​\(𝐰\)\]\+ϵ\+τPB8​meff\\displaystyle\\mathbb\{E\}\_\{\\mathbf\{w\}\\sim Q\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}\)\\right\]\+\\epsilon\+\\frac\{\\tau\_\{\\mathrm\{PB\}\}\}\{8m\_\{\\mathrm\{eff\}\}\}\+KL\(Q∥Π\)\+ln\|𝒩ϵ\(Λ,∥⋅∥1\)\|\+ln\(1/δ\)τPB,\\displaystyle\+\\frac\{\\mathrm\{KL\}\(Q\\\|\\Pi\)\+\\ln\\left\|\\mathcal\{N\}\_\{\\epsilon\}\(\\Lambda,\\\|\\cdot\\\|\_\{1\}\)\\right\|\+\\ln\(1/\\delta\)\}\{\\tau\_\{\\mathrm\{PB\}\}\},\(22\)wheremeff:=infλ∈Λ\(∑i=1nλi2/mi\)−1m\_\{\\mathrm\{eff\}\}:=\\inf\_\{\\lambda\\in\\Lambda\}\\left\(\\sum\_\{i=1\}^\{n\}\{\\lambda\_\{i\}^\{2\}\}/\{m\_\{i\}\}\\right\)^\{\-1\}\.

###### Proof:

See Appendix[A](https://arxiv.org/html/2608.26493#A1)\. ∎The first case corresponds to the agnostic mixture class used in our main fairness objective\. In this setting, the supremum overΔn\\Delta\_\{n\}is attained at a vertex, so the additional complexity cost is onlyln⁡n\\ln n\. The second case shows how the result extends to a general compact mixture class, where the price of uniformity overΛ\\Lambdaappears through the covering numberln\|𝒩ε\(Λ,∥⋅∥1\)\|\\ln\|\\mathcal\{N\}\_\{\\varepsilon\}\(\\Lambda,\\\|\\cdot\\\|\_\{1\}\)\|\.

We next complement this algorithm\-agnostic generalization result with a corollary that isolates the one\-step empirical mixture\-risk perturbation caused by stale, sparse, quantized, and event\-triggered neighbor\-model exchange\. Its proof uses the communication\-residual bound established later in Lemma[6](https://arxiv.org/html/2608.26493#Thmlemma6)\.

###### Corollary 1\(One\-Step Effect of Stale and Compressed Neighbor Copies\)\.

Let𝐝t=𝐫t−∇ℛ𝒢​\(𝐰t\)\\mathbf\{d\}^\{t\}=\\mathbf\{r\}^\{t\}\-\\nabla\\mathcal\{R\}\_\{\\mathcal\{G\}\}\(\\mathbf\{w\}^\{t\}\)denote the graph\-coupling approximation error, and define the exact\-neighbor shadow update𝐰ext\+1:=𝐰t−γt​\(𝐡t−𝐝t\)\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{t\+1\}:=\\mathbf\{w\}^\{t\}\-\\gamma\_\{t\}\(\\mathbf\{h\}^\{t\}\-\\mathbf\{d\}^\{t\}\)\. Thus,𝐰ext\+1\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{t\+1\}uses the same stochastic gradients and fairness selection as DMFL\-SQ, but evaluates the graph\-coupling direction using the current neighbor models\. Since𝐰t\+1−𝐰ext\+1=−γt​𝐝t\\mathbf\{w\}^\{t\+1\}\-\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{t\+1\}=\-\\gamma\_\{t\}\\mathbf\{d\}^\{t\}, suppose that each empirical lossF^i\\widehat\{F\}\_\{i\}isGlipG\_\{\\mathrm\{lip\}\}\-Lipschitz with respect towiw\_\{i\}\. Then,

\|𝔼⁡\[Ψ^​\(𝐰t\+1\)\]−𝔼⁡\[Ψ^​\(𝐰ext\+1\)\]\|\\displaystyle\\left\|\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}^\{t\+1\}\)\\right\]\-\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{t\+1\}\)\\right\]\\right\|≤Glip​γt​n​K𝒢​𝔼​\[ℰt\],\\displaystyle\\leq G\_\{\\mathrm\{lip\}\}\\gamma\_\{t\}\\sqrt\{nK\_\{\\mathcal\{G\}\}\\,\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]\},\(23\)whereEtE^\{t\}is the average weighted communication residual defined in \([42](https://arxiv.org/html/2608.26493#S5.E42)\), and Lemma[5](https://arxiv.org/html/2608.26493#Thmlemma5)gives‖𝐝t‖2≤n​KG​Et\\\|\\mathbf\{d\}^\{t\}\\\|^\{2\}\\leq nK\_\{G\}E^\{t\}\.

Letτ\\taube sampled uniformly from\{0,…,T−1\}\\\{0,\\ldots,T\-1\\\}, independently of the algorithmic randomness\. Under the finite\-horizon stepsizeγt=γ/T\\gamma\_\{t\}=\\gamma/\\sqrt\{T\},

\|𝔼⁡\[Ψ^​\(𝐰τ\+1\)\]−𝔼⁡\[Ψ^​\(𝐰exτ\+1\)\]\|≤Glip​γ​n​K𝒢T​\(1T​∑t=0T−1𝔼⁡\[ℰt\]\)1/2\.\\displaystyle\\left\|\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}^\{\\tau\+1\}\)\\right\]\-\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{\\tau\+1\}\)\\right\]\\right\|\\leq\\frac\{G\_\{\\mathrm\{lip\}\}\\gamma\\sqrt\{nK\_\{\\mathcal\{G\}\}\}\}\{\\sqrt\{T\}\}\\left\(\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]\\right\)^\{1/2\}\.\(24\)
By Lemma[6](https://arxiv.org/html/2608.26493#Thmlemma6),

1T​∑t=0T−1𝔼⁡\[ℰt\]≤2χ0​ω​T​\(𝔼⁡\[ℰ0\]\+χ1​Bh​γ2\+χ2​ϑ02​\(1\+log⁡T\)\)\.\\displaystyle\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]\\leq\\frac\{2\}\{\\chi\_\{0\}\\omega T\}\\left\(\\mathbb\{E\}\[\\mathcal\{E\}^\{0\}\]\+\\chi\_\{1\}B\_\{h\}\\gamma^\{2\}\+\\chi\_\{2\}\\vartheta\_\{0\}^\{2\}\(1\+\\log T\)\\right\)\.\(25\)Consequently,

\|𝔼⁡\[Ψ^​\(𝐰τ\+1\)\]−𝔼⁡\[Ψ^​\(𝐰exτ\+1\)\]\|\\displaystyle\\left\|\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}^\{\\tau\+1\}\)\\right\]\-\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{\\tau\+1\}\)\\right\]\\right\|≤Glip​γ​2​n​K𝒢χ0​ω​T​\(𝔼⁡\[ℰ0\]\+χ1​Bh​γ2\+χ2​ϑ02​\(1\+log⁡T\)\)1/2\.\\displaystyle\\qquad\\leq\\frac\{G\_\{\\mathrm\{lip\}\}\\gamma\\sqrt\{2nK\_\{\\mathcal\{G\}\}\}\}\{\\sqrt\{\\chi\_\{0\}\\omega\}\\,T\}\\left\(\\mathbb\{E\}\[\\mathcal\{E\}^\{0\}\]\+\\chi\_\{1\}B\_\{h\}\\gamma^\{2\}\+\\chi\_\{2\}\\vartheta\_\{0\}^\{2\}\(1\+\\log T\)\\right\)^\{1/2\}\.\(26\)Hence, the one\-step empirical mixture\-risk perturbation induced by stale, sparse, quantized, and event\-triggered neighbor\-model exchange decays as𝒪⁡\(log⁡TT\)\\mathcal\{O\}\\\!\\left\(\\frac\{\\sqrt\{\\log T\}\}\{T\}\\right\)\.

## VConvergence Analysis

Before proving the convergence of Algorithm[1](https://arxiv.org/html/2608.26493#alg1), we first summarize the regularity properties of the agnostic fairness envelopeΨ⁡\(𝐰\)\\Psi\(\\mathbf\{w\}\)and the resulting full personalized objective\. Although the agnostic fairness envelope

Ψ⁡\(𝐰\)=supλ∈ΛΦλ​\(𝐰\),Φλ​\(𝐰\)=∑i=1nλi​Fi​\(wi\),\\Psi\(\\mathbf\{w\}\)=\\sup\_\{\\lambda\\in\\Lambda\}\\Phi\_\{\\lambda\}\(\\mathbf\{w\}\),\\qquad\\Phi\_\{\\lambda\}\(\\mathbf\{w\}\)=\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}F\_\{i\}\(w\_\{i\}\),\(27\)is generally nonsmooth, it inherits weak convexity from the local objectives\. In particular, since eachFiF\_\{i\}isLL\-smooth, every mixture objectiveΦλ\\Phi\_\{\\lambda\}isLL\-weakly convex with respect to the stacked model variable𝐰\\mathbf\{w\}\. Because the pointwise supremum of functions sharing the same weak\-convexity constant is also weakly convex,Ψ\\PsiisLL\-weakly convex\.

Consequently, the objective function isκ\\kappa\-weakly convex withκ=\(1\+ρ\)​L,\\kappa=\(1\+\\rho\)L,becauseℛ𝒢\\mathcal\{R\}\_\{\\mathcal\{G\}\}is convex\. We therefore analyze Algorithm[1](https://arxiv.org/html/2608.26493#alg1)through the Moreau envelope ofℒ\\mathcal\{L\}\. This framework permits changes in the active fairness component between successive iterates and explicitly accommodates the empirical fairness\-selection gapεΨ,t\\varepsilon\_\{\\Psi,t\}defined in \([13](https://arxiv.org/html/2608.26493#S3.E13)\)\.

ForKΨ≥diam⁡\(𝒢\)K\_\{\\Psi\}\\geq\\operatorname\{diam\}\(\\mathcal\{G\}\), scalar max\-consensus ensures that all clients agree on the same maximizer of the empirical mini\-batch losses\. However, this empirical maximizer need not coincide with a population maximizer ofΨ\\Psi\. The resulting discrepancy is quantified by the fairness\-selection gapεΨ,t\\varepsilon\_\{\\Psi,t\}defined above and is explicitly controlled in the subsequent Moreau\-envelope convergence analysis\.

We next state the additional assumptions required for the Moreau\-envelope and communication\-residual analysis\.

###### Assumption 5\(Bounded Second Moment of the Exact\-Neighbor Population Direction\)\.

Define

Vit:=\(1\+ρλ^it\)∇Fi\(wit\)\+∇wiℛ𝒢\(𝐰t\),𝐕t:=\(V1t,…,Vnt\)\.V\_\{i\}^\{t\}:=\\left\(1\+\\rho\\widehat\{\\lambda\}\_\{i\}^\{t\}\\right\)\\nabla F\_\{i\}\(w\_\{i\}^\{t\}\)\+\\nabla\_\{w\_\{i\}\}\\mathcal\{R\}\_\{\\mathcal\{G\}\}\(\\mathbf\{w\}^\{t\}\),\\mathbf\{V\}^\{t\}:=\(V\_\{1\}^\{t\},\\ldots,V\_\{n\}^\{t\}\)\.\(28\)There exists a constantH\>0H\>0, independent ofttandTT, such that𝔼⁡\[‖𝐕t‖2\|ℱt\]≤H2\\mathbb\{E\}\\left\[\\\|\\mathbf\{V\}^\{t\}\\\|^\{2\}\\,\\middle\|\\,\\mathcal\{F\}\_\{t\}\\right\]\\leq H^\{2\}\.

###### Assumption 6\(Fairness\-Selection Accuracy\)\.

LetεΨ,t=Ψ⁡\(𝐰t\)−Φλ^t​\(𝐰t\)\\varepsilon\_\{\\Psi,t\}=\\Psi\(\\mathbf\{w\}^\{t\}\)\-\\Phi\_\{\\widehat\{\\lambda\}^\{\\,t\}\}\(\\mathbf\{w\}^\{t\}\)denote the population fairness\-selection gap defined in \([13](https://arxiv.org/html/2608.26493#S3.E13)\)\. There exists a deterministic nonnegative sequence\{ϵΨ,t\}t≥0\\\{\\epsilon\_\{\\Psi,t\}\\\}\_\{t\\geq 0\}such that

𝔼⁡\[εΨ,t\|ℱt\]≤ϵΨ,t\.\\mathbb\{E\}\\left\[\\varepsilon\_\{\\Psi,t\}\\,\\middle\|\\,\\mathcal\{F\}\_\{t\}\\right\]\\leq\\epsilon\_\{\\Psi,t\}\.\(29\)Moreover, there exists a constantBΨ\>0B\_\{\\Psi\}\>0, independent ofTT, such that, for every training horizonT≥1T\\geq 1,

1T​∑t=0T−1ϵΨ,t≤BΨT\.\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\epsilon\_\{\\Psi,t\}\\leq\\frac\{B\_\{\\Psi\}\}\{\\sqrt\{T\}\}\.\(30\)

By \([14](https://arxiv.org/html/2608.26493#S3.E14)\), Assumption[6](https://arxiv.org/html/2608.26493#Thmassmon6)admits an explicit sufficient condition\. Suppose that, conditional onℱt\\mathcal\{F\}\_\{t\}, the selection\-loss samples are independent and their centered losses areσsel\\sigma\_\{\\rm sel\}\-sub\-Gaussian, and letbt:=mini⁡\|ℬi,selt\|b\_\{t\}:=\\min\_\{i\}\|\\mathcal\{B\}\_\{i,\\mathrm\{sel\}\}^\{t\}\|\. Standard maximal concentration gives𝔼⁡\[εΨ,t∣ℱt\]≤2​σsel​2​log⁡\(2​n\)/bt\\mathbb\{E\}\[\\varepsilon\_\{\\Psi,t\}\\mid\\mathcal\{F\}\_\{t\}\]\\leq 2\\sigma\_\{\\rm sel\}\\sqrt\{2\\log\(2n\)/b\_\{t\}\}\. Thus,bt≥c⁡\(t\+1\)​log⁡\(2​n\)b\_\{t\}\\geq c\(t\+1\)\\log\(2n\)implies

T−1​∑t=0T−1𝔼⁡\[εΨ,t∣ℱt\]≤4​σselT​2c,T^\{\-1\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\varepsilon\_\{\\Psi,t\}\\mid\\mathcal\{F\}\_\{t\}\]\\leq 4\\frac\{\\sigma\_\{\\rm sel\}\}\{\\sqrt\{T\}\}\\sqrt\{\\frac\{2\}\{c\}\},\(31\)verifying Assumption[6](https://arxiv.org/html/2608.26493#Thmassmon6)\. For losses whose range has widthRR, one may takeσsel≤R/2\\sigma\_\{\\rm sel\}\\leq R/2\.

###### Assumption 7\(Stepsize and Trigger Schedule\)\.

For a training horizonTT, DMFL\-SQ uses

γt=γT,ϑt=ϑ0t\+1,t=0,…,T−1,\\gamma\_\{t\}=\\frac\{\\gamma\}\{\\sqrt\{T\}\},\\qquad\\vartheta\_\{t\}=\\frac\{\\vartheta\_\{0\}\}\{\\sqrt\{t\+1\}\},\\qquad t=0,\\ldots,T\-1,whereγ\>0\\gamma\>0andϑ0\>0\\vartheta\_\{0\}\>0are constants\. Hence,

∑t=0T−1γt​ϑt2=γ​ϑ02T​∑t=0T−11t\+1=𝒪⁡\(log⁡TT\)\.\\sum\_\{t=0\}^\{T\-1\}\\gamma\_\{t\}\\vartheta\_\{t\}^\{2\}=\\frac\{\\gamma\\vartheta\_\{0\}^\{2\}\}\{\\sqrt\{T\}\}\\sum\_\{t=0\}^\{T\-1\}\\frac\{1\}\{t\+1\}=\\mathcal\{O\}\\left\(\\frac\{\\log T\}\{\\sqrt\{T\}\}\\right\)\.After normalization by∑t=0T−1γt=γ​T\\sum\_\{t=0\}^\{T\-1\}\\gamma\_\{t\}=\\gamma\\sqrt\{T\}, this contributes

𝒪⁡\(log⁡TT\)\\mathcal\{O\}\\left\(\\frac\{\\log T\}\{T\}\\right\)to the final stationarity bound\.

### V\-AWeak Convexity and Moreau\-Envelope Regularity

Recall thatΨ⁡\(𝐰\)=supλ∈ΛΦλ​\(𝐰\),Φλ​\(𝐰\)=∑i=1nλi​Fi​\(wi\)\\Psi\(\\mathbf\{w\}\)=\\sup\_\{\\lambda\\in\\Lambda\}\\Phi\_\{\\lambda\}\(\\mathbf\{w\}\),\\Phi\_\{\\lambda\}\(\\mathbf\{w\}\)=\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}F\_\{i\}\(w\_\{i\}\), whereΛ⊆Δn\\Lambda\\subseteq\\Delta\_\{n\}is nonempty and compact\.

###### Lemma 1\(Weak Convexity of the Fairness Envelope and Objective Function\)\.

Suppose that eachFiF\_\{i\}isLL\-smooth\. Then, for everyλ∈Λ\\lambda\\in\\Lambda, the mixture objectiveΦλ\\Phi\_\{\\lambda\}isLL\-weakly convex\. Consequently, the fairness envelopeΨ\\PsiisLL\-weakly convex, and the objective functionℒ⁡\(𝐰\)=∑i=1nFi​\(wi\)\+ℛ𝒢​\(𝐰\)\+ρ​Ψ​\(𝐰\)\\mathcal\{L\}\(\\mathbf\{w\}\)=\\sum\_\{i=1\}^\{n\}F\_\{i\}\(w\_\{i\}\)\+\\mathcal\{R\}\_\{\\mathcal\{G\}\}\(\\mathbf\{w\}\)\+\\rho\\Psi\(\\mathbf\{w\}\)isκ\\kappa\-weakly convex withκ=\(1\+ρ\)​L\\kappa=\(1\+\\rho\)L\.

###### Proof\.

Since eachFiF\_\{i\}isLL\-smooth, every mixture objectiveΦλ\\Phi\_\{\\lambda\}isLL\-weakly convex\. Hence,Φλ\+\(L/2\)∥⋅∥2\\Phi\_\{\\lambda\}\+\(L/2\)\\\|\\cdot\\\|^\{2\}is convex\. Taking the pointwise supremum overλ∈Λ\\lambda\\in\\Lambdashows thatΨ\+\(L/2\)∥⋅∥2\\Psi\+\(L/2\)\\\|\\cdot\\\|^\{2\}is convex, and thereforeΨ\\PsiisLL\-weakly convex\. Moreover,∑i=1nFi​\(wi\)\\sum\_\{i=1\}^\{n\}F\_\{i\}\(w\_\{i\}\)isLL\-weakly convex, whileℛ𝒢\\mathcal\{R\}\_\{\\mathcal\{G\}\}is convex\. Thus,ℒ\\mathcal\{L\}is\(1\+ρ\)​L\(1\+\\rho\)L\-weakly convex\. ∎

###### Lemma 2\(Approximate Fairness\-Direction Inequality\)\.

Letλ^t\\widehat\{\\lambda\}^\{\\,t\}be the empirical fairness mixture selected by Algorithm[1](https://arxiv.org/html/2608.26493#alg1), and define the corresponding population\-gradient direction by

ξ¯it:=λ^it∇Fi\(wit\),𝝃¯t:=\(ξ¯1t,…,ξ¯nt\)\.\\bar\{\\xi\}\_\{i\}^\{t\}:=\\widehat\{\\lambda\}\_\{i\}^\{t\}\\nabla F\_\{i\}\(w\_\{i\}^\{t\}\),\\qquad\\bar\{\\bm\{\\xi\}\}^\{t\}:=\(\\bar\{\\xi\}\_\{1\}^\{t\},\\ldots,\\bar\{\\xi\}\_\{n\}^\{t\}\)\.\(32\)Then, for every𝐮∈ℝn​d\\mathbf\{u\}\\in\\mathbb\{R\}^\{nd\},

Ψ⁡\(𝐮\)≥Ψ⁡\(𝐰t\)\+⟨𝝃¯t,𝐮−𝐰t⟩−L2​‖𝐮−𝐰t‖2−εΨ,t,\\Psi\(\\mathbf\{u\}\)\\geq\\Psi\(\\mathbf\{w\}^\{t\}\)\+\\left\\langle\\bar\{\\bm\{\\xi\}\}^\{t\},\\mathbf\{u\}\-\\mathbf\{w\}^\{t\}\\right\\rangle\-\\frac\{L\}\{2\}\\\|\\mathbf\{u\}\-\\mathbf\{w\}^\{t\}\\\|^\{2\}\-\\varepsilon\_\{\\Psi,t\},\(33\)whereεΨ,t\\varepsilon\_\{\\Psi,t\}is defined in \([13](https://arxiv.org/html/2608.26493#S3.E13)\)\. Moreover, under Assumption[2](https://arxiv.org/html/2608.26493#Thmassmon2),

𝔼\[𝝃^t\|ℱt,λ^t\]=𝝃¯t\.\\mathbb\{E\}\\left\[\\widehat\{\\bm\{\\xi\}\}^\{t\}\\,\\middle\|\\,\\mathcal\{F\}\_\{t\},\\widehat\{\\lambda\}^\{\\,t\}\\right\]=\\bar\{\\bm\{\\xi\}\}^\{t\}\.\(34\)

###### Proof\.

By the definition ofΨ\\Psiand Lemma[1](https://arxiv.org/html/2608.26493#Thmlemma1),

Ψ⁡\(𝐮\)\\displaystyle\\Psi\(\\mathbf\{u\}\)≥Φλ^t​\(𝐮\)\\displaystyle\\geq\\Phi\_\{\\widehat\{\\lambda\}^\{\\,t\}\}\(\\mathbf\{u\}\)≥Φλ^t​\(𝐰t\)\+⟨∇Φλ^t​\(𝐰t\),𝐮−𝐰t⟩−L2​‖𝐮−𝐰t‖2\.\\displaystyle\\geq\\Phi\_\{\\widehat\{\\lambda\}^\{\\,t\}\}\(\\mathbf\{w\}^\{t\}\)\+\\left\\langle\\nabla\\Phi\_\{\\widehat\{\\lambda\}^\{\\,t\}\}\(\\mathbf\{w\}^\{t\}\),\\mathbf\{u\}\-\\mathbf\{w\}^\{t\}\\right\\rangle\-\\frac\{L\}\{2\}\\\|\\mathbf\{u\}\-\\mathbf\{w\}^\{t\}\\\|^\{2\}\.\(35\)UsingΦλ^t​\(𝐰t\)=Ψ⁡\(𝐰t\)−εΨ,t\\Phi\_\{\\widehat\{\\lambda\}^\{\\,t\}\}\(\\mathbf\{w\}^\{t\}\)=\\Psi\(\\mathbf\{w\}^\{t\}\)\-\\varepsilon\_\{\\Psi,t\}and∇Φλ^t​\(𝐰t\)=𝝃¯t\\nabla\\Phi\_\{\\widehat\{\\lambda\}^\{\\,t\}\}\(\\mathbf\{w\}^\{t\}\)=\\bar\{\\bm\{\\xi\}\}^\{t\}gives \([33](https://arxiv.org/html/2608.26493#S5.E33)\)\. Equation \([34](https://arxiv.org/html/2608.26493#S5.E34)\) follows from the conditional independence of the selection and gradient mini\-batches\. ∎

###### Lemma 3\(Moreau\-Envelope Properties\)\.

Letℒ\\mathcal\{L\}beκ\\kappa\-weakly convex and letμ∈\(0,1/κ\)\\mu\\in\(0,1/\\kappa\)\. Then the proximal mapping is single\-valued,ℒμ\\mathcal\{L\}\_\{\\mu\}is continuously differentiable, and

∇ℒμ​\(𝐰\)=1μ​\(𝐰−proxμ​ℒ⁡\(𝐰\)\)\.\\nabla\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}\)=\\frac\{1\}\{\\mu\}\\left\(\\mathbf\{w\}\-\\operatorname\{prox\}\_\{\\mu\\mathcal\{L\}\}\(\\mathbf\{w\}\)\\right\)\.\(36\)Moreover,1μ​\(𝐰−proxμ​ℒ⁡\(𝐰\)\)∈∂ℒ⁡\(proxμ​ℒ⁡\(𝐰\)\)\\frac\{1\}\{\\mu\}\\left\(\\mathbf\{w\}\-\\operatorname\{prox\}\_\{\\mu\\mathcal\{L\}\}\(\\mathbf\{w\}\)\\right\)\\in\\partial\\mathcal\{L\}\\left\(\\operatorname\{prox\}\_\{\\mu\\mathcal\{L\}\}\(\\mathbf\{w\}\)\\right\)\.

### V\-BTechnical Lemmas for the Convergence Analysis

We next present three technical lemmas that form the backbone of the convergence analysis\. Lemma[4](https://arxiv.org/html/2608.26493#Thmlemma4)establishes a one\-step descent inequality for the Moreau envelopeℒμ\\mathcal\{L\}\_\{\\mu\}\. Lemma[5](https://arxiv.org/html/2608.26493#Thmlemma5)controls the error caused by evaluating the graph\-coupling direction using stale and compressed neighbor\-model copies\. Lemma[6](https://arxiv.org/html/2608.26493#Thmlemma6)provides a cumulative bound on the resulting communication residual under the finite\-horizon stepsize and trigger schedule\. Together, these results separate the optimization, fairness\-selection, stochastic, and communication\-induced errors\.

Letℋt:=σ⁡\(ℱt,\{Bi,selt\}i=1n\)\\mathcal\{H\}\_\{t\}:=\\sigma\\left\(\\mathcal\{F\}\_\{t\},\\\{B\_\{i,\\mathrm\{sel\}\}^\{t\}\\\}\_\{i=1\}^\{n\}\\right\)denote the information available after the fairness\-selection mini\-batches have been observed\. Hence,λ^t\\widehat\{\\lambda\}^\{\\,t\}andεΨ,t\\varepsilon\_\{\\Psi,t\}areℋt\\mathcal\{H\}\_\{t\}\-measurable\.

Define the stochastic\-gradient noise by

qit:=\(1\+ρ​λ^it\)​\(git−∇Fi​\(wit\)\),𝐪t:=\(q1t,…,qnt\)\.q\_\{i\}^\{t\}:=\\left\(1\+\\rho\\widehat\{\\lambda\}\_\{i\}^\{t\}\\right\)\\left\(g\_\{i\}^\{t\}\-\\nabla F\_\{i\}\(w\_\{i\}^\{t\}\)\\right\),\\qquad\\mathbf\{q\}^\{t\}:=\(q\_\{1\}^\{t\},\\ldots,q\_\{n\}^\{t\}\)\.\(37\)By Assumption[2](https://arxiv.org/html/2608.26493#Thmassmon2),𝔼⁡\[𝐪t\|ℋt\]=𝟎,\\mathbb\{E\}\\left\[\\mathbf\{q\}^\{t\}\\,\\middle\|\\,\\mathcal\{H\}\_\{t\}\\right\]=\\mathbf\{0\},and

𝔼⁡\[‖𝐪t‖2\|ℋt\]≤\(1\+ρ\)2​∑i=1nσi2=n​\(1\+ρ\)2​σ¯2\.\\mathbb\{E\}\\left\[\\\|\\mathbf\{q\}^\{t\}\\\|^\{2\}\\,\\middle\|\\,\\mathcal\{H\}\_\{t\}\\right\]\\leq\(1\+\\rho\)^\{2\}\\sum\_\{i=1\}^\{n\}\\sigma\_\{i\}^\{2\}=n\(1\+\\rho\)^\{2\}\\bar\{\\sigma\}^\{2\}\.\(38\)Define the graph\-coupling approximation error by

dit:=rit−∇wiℛ𝒢​\(𝐰t\),𝐝t:=\(d1t,…,dnt\)\.d\_\{i\}^\{t\}:=r\_\{i\}^\{t\}\-\\nabla\_\{w\_\{i\}\}\\mathcal\{R\}\_\{\\mathcal\{G\}\}\(\\mathbf\{w\}^\{t\}\),\\qquad\\mathbf\{d\}^\{t\}:=\(d\_\{1\}^\{t\},\\ldots,d\_\{n\}^\{t\}\)\.\(39\)Using the definition ofritr\_\{i\}^\{t\}, this error can be written as

dit=α​∑j∈𝒩iai​j​\(wjt−w^j→it\)\.d\_\{i\}^\{t\}=\\alpha\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}a\_\{ij\}\\left\(w\_\{j\}^\{t\}\-\\widehat\{w\}\_\{j\\to i\}^\{t\}\\right\)\.\(40\)The local direction used by Algorithm[1](https://arxiv.org/html/2608.26493#alg1)therefore admits the decomposition

hit=Vit\+qit\+dit,𝐡t=𝐕t\+𝐪t\+𝐝t\.h\_\{i\}^\{t\}=V\_\{i\}^\{t\}\+q\_\{i\}^\{t\}\+d\_\{i\}^\{t\},\\qquad\\mathbf\{h\}^\{t\}=\\mathbf\{V\}^\{t\}\+\\mathbf\{q\}^\{t\}\+\\mathbf\{d\}^\{t\}\.\(41\)Define the average weighted communication residual as

Et:=1n​∑i=1n∑j∈𝒩iai​j​‖𝐰jt−𝐰^j→it‖2\.E^\{t\}:=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}a\_\{ij\}\\left\\\|\\mathbf\{w\}\_\{j\}^\{t\}\-\\widehat\{\\mathbf\{w\}\}\_\{j\\to i\}^\{t\}\\right\\\|^\{2\}\.\(42\)
###### Lemma 4\(One\-Step Moreau\-Envelope Descent\)\.

Let0<μ<1κ,κ=\(1\+ρ\)​L\.0<\\mu<\\frac\{1\}\{\\kappa\},\\kappa=\(1\+\\rho\)L\.Under Assumptions[1](https://arxiv.org/html/2608.26493#Thmassmon1),[2](https://arxiv.org/html/2608.26493#Thmassmon2), and[5](https://arxiv.org/html/2608.26493#Thmassmon5), the iterates generated by Algorithm[1](https://arxiv.org/html/2608.26493#alg1)satisfy

𝔼⁡\[ℒμ​\(𝐰t\+1\)\]\\displaystyle\\mathbb\{E\}\\left\[\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\+1\}\)\\right\]≤𝔼⁡\[ℒμ​\(𝐰t\)\]−1−κ​μ4​γt​𝔼​\[‖∇ℒμ​\(𝐰t\)‖2\]\\displaystyle\\leq\\mathbb\{E\}\\left\[\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\}\)\\right\]\-\\frac\{1\-\\kappa\\mu\}\{4\}\\gamma\_\{t\}\\mathbb\{E\}\\left\[\\left\\\|\\nabla\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\}\)\\right\\\|^\{2\}\\right\]\+ρ​γtμ​𝔼​\[εΨ,t\]\+\(γt1−κ​μ\+γt2μ\)​𝔼​\[‖𝐝t‖2\]\\displaystyle\\quad\+\\frac\{\\rho\\gamma\_\{t\}\}\{\\mu\}\\mathbb\{E\}\[\\varepsilon\_\{\\Psi,t\}\]\+\\left\(\\frac\{\\gamma\_\{t\}\}\{1\-\\kappa\\mu\}\+\\frac\{\\gamma\_\{t\}^\{2\}\}\{\\mu\}\\right\)\\mathbb\{E\}\\left\[\\\|\\mathbf\{d\}^\{t\}\\\|^\{2\}\\right\]\+γt2μ​H2\+n​\(1\+ρ\)2​γt22​μ​σ¯2\.\\displaystyle\\quad\+\\frac\{\\gamma\_\{t\}^\{2\}\}\{\\mu\}H^\{2\}\+\\frac\{n\(1\+\\rho\)^\{2\}\\gamma\_\{t\}^\{2\}\}\{2\\mu\}\\bar\{\\sigma\}^\{2\}\.\(43\)

###### Proof:

See Appendix[B\-A](https://arxiv.org/html/2608.26493#A2.SS1)\. ∎

###### Lemma 5\(Communication\-Residual Recursion\)\.

Under Assumption[4](https://arxiv.org/html/2608.26493#Thmassmon4)and the event\-trigger condition in Algorithm[1](https://arxiv.org/html/2608.26493#alg1), there exist constantsχ0,χ1,χ2\>0\\chi\_\{0\},\\chi\_\{1\},\\chi\_\{2\}\>0, independent ofttandTT, such thatχ0​ω∈\(0,1\]\\chi\_\{0\}\\omega\\in\(0,1\]and

𝔼⁡\[ℰt\+1\]\\displaystyle\\mathbb\{E\}\[\\mathcal\{E\}^\{t\+1\}\]≤\(1−χ0​ω\)​𝔼​\[ℰt\]\+χ1​γt2​1n​𝔼​‖𝐡t‖2\+χ2​ϑt2\.\\displaystyle\\leq\(1\-\\chi\_\{0\}\\omega\)\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]\+\\chi\_\{1\}\\gamma\_\{t\}^\{2\}\\frac\{1\}\{n\}\\mathbb\{E\}\\\|\\mathbf\{h\}^\{t\}\\\|^\{2\}\+\\chi\_\{2\}\\vartheta\_\{t\}^\{2\}\.\(44\)Moreover, the graph\-direction approximation error defined in \([39](https://arxiv.org/html/2608.26493#S5.E39)\) satisfies

‖𝐝t‖2≤n​K𝒢​ℰt,\\\|\\mathbf\{d\}^\{t\}\\\|^\{2\}\\leq nK\_\{\\mathcal\{G\}\}\\mathcal\{E\}^\{t\},\(45\)whereK𝒢\>0K\_\{\\mathcal\{G\}\}\>0depends only on the graph weights and the coupling parameterα\\alpha\.

###### Proof:

See Appendix[B\-B](https://arxiv.org/html/2608.26493#A2.SS2)\. ∎

Using the decomposition𝐡t=𝐕t\+𝐪t\+𝐝t\\mathbf\{h\}^\{t\}=\\mathbf\{V\}^\{t\}\+\\mathbf\{q\}^\{t\}\+\\mathbf\{d\}^\{t\}, the conditional zero\-mean property of𝐪t\\mathbf\{q\}^\{t\}, and Assumption[5](https://arxiv.org/html/2608.26493#Thmassmon5), we obtain

1n​𝔼​‖𝐡t‖2≤Bh\+2​K𝒢​𝔼​\[ℰt\],\\frac\{1\}\{n\}\\mathbb\{E\}\\\|\\mathbf\{h\}^\{t\}\\\|^\{2\}\\leq B\_\{h\}\+2K\_\{\\mathcal\{G\}\}\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\],\(46\)whereBh:=2​H2/n\+\(1\+ρ\)2​σ¯2B\_\{h\}:=2H^\{2\}/n\+\(1\+\\rho\)^\{2\}\\bar\{\\sigma\}^\{2\}\. Indeed,𝔼​‖𝐡t‖2=𝔼​‖𝐕t\+𝐝t‖2\+𝔼​‖𝐪t‖2≤2​H2\+2​𝔼​‖𝐝t‖2\+n​\(1\+ρ\)2​σ¯2\\mathbb\{E\}\\\|\\mathbf\{h\}^\{t\}\\\|^\{2\}=\\mathbb\{E\}\\\|\\mathbf\{V\}^\{t\}\+\\mathbf\{d\}^\{t\}\\\|^\{2\}\+\\mathbb\{E\}\\\|\\mathbf\{q\}^\{t\}\\\|^\{2\}\\leq 2H^\{2\}\+2\\mathbb\{E\}\\\|\\mathbf\{d\}^\{t\}\\\|^\{2\}\+n\(1\+\\rho\)^\{2\}\\bar\{\\sigma\}^\{2\}, and \([45](https://arxiv.org/html/2608.26493#S5.E45)\) completes the bound\. Substituting \([46](https://arxiv.org/html/2608.26493#S5.E46)\) into \([44](https://arxiv.org/html/2608.26493#S5.E44)\) gives

𝔼⁡\[ℰt\+1\]\\displaystyle\\mathbb\{E\}\[\\mathcal\{E\}^\{t\+1\}\]≤\(1−χ0​ω\+2​χ1​K𝒢​γt2\)​𝔼​\[ℰt\]\+χ1​Bh​γt2\\displaystyle\\leq\\left\(1\-\\chi\_\{0\}\\omega\+2\\chi\_\{1\}K\_\{\\mathcal\{G\}\}\\gamma\_\{t\}^\{2\}\\right\)\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]\+\\chi\_\{1\}B\_\{h\}\\gamma\_\{t\}^\{2\}\+χ2​ϑt2\.\\displaystyle\\quad\+\\chi\_\{2\}\\vartheta\_\{t\}^\{2\}\.\(47\)
###### Lemma 6\(Average Communication\-Residual Bound\)\.

Suppose the conditions of Lemma[5](https://arxiv.org/html/2608.26493#Thmlemma5)hold, and letγt=γ/T\\gamma\_\{t\}=\\gamma/\\sqrt\{T\}andϑt=ϑ0/t\+1\\vartheta\_\{t\}=\\vartheta\_\{0\}/\\sqrt\{t\+1\}\. Ifγ\>0\\gamma\>0is sufficiently small such that2​χ1​K𝒢​γ2≤χ0​ω2,2\\chi\_\{1\}K\_\{\\mathcal\{G\}\}\\gamma^\{2\}\\leq\\frac\{\\chi\_\{0\}\\omega\}\{2\},then the closed recursion in \([47](https://arxiv.org/html/2608.26493#S5.E47)\) satisfies

𝔼⁡\[ℰt\+1\]≤\(1−χ0​ω2\)​𝔼​\[ℰt\]\+χ1​Bh​γt2\+χ2​ϑt2\.\\mathbb\{E\}\[\\mathcal\{E\}^\{t\+1\}\]\\leq\\left\(1\-\\frac\{\\chi\_\{0\}\\omega\}\{2\}\\right\)\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]\+\\chi\_\{1\}B\_\{h\}\\gamma\_\{t\}^\{2\}\+\\chi\_\{2\}\\vartheta\_\{t\}^\{2\}\.\(48\)Consequently,

1T​∑t=0T−1𝔼⁡\[ℰt\]≤2χ0​ω​T​\[𝔼⁡\[ℰ0\]\+χ1​Bh​γ2\+χ2​ϑ02​\(1\+log⁡T\)\]\.\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]\\leq\\frac\{2\}\{\\chi\_\{0\}\\omega T\}\\left\[\\mathbb\{E\}\[\\mathcal\{E\}^\{0\}\]\+\\chi\_\{1\}B\_\{h\}\\gamma^\{2\}\+\\chi\_\{2\}\\vartheta\_\{0\}^\{2\}\(1\+\\log T\)\\right\]\.\(49\)

###### Proof:

See Appendix[B\-C](https://arxiv.org/html/2608.26493#A2.SS3)\. ∎

### V\-CConvergence Rate

Since the objective is defined over client\-specific models, stationarity is measured with respect to the stacked variable𝐰=\(w1,…,wn\)\\mathbf\{w\}=\(w\_\{1\},\\ldots,w\_\{n\}\)rather than a network\-average model\. For0<μ<1/κ0<\\mu<1/\\kappa, define the normalized Moreau\-envelope stationarity measure

𝒮μt=1n​‖∇ℒμ​\(𝐰t\)‖2\.\\mathcal\{S\}\_\{\\mu\}^\{t\}=\\frac\{1\}\{n\}\\left\\\|\\nabla\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\}\)\\right\\\|^\{2\}\.By Lemma[3](https://arxiv.org/html/2608.26493#Thmlemma3), a small value of𝒮μt\\mathcal\{S\}\_\{\\mu\}^\{t\}implies that𝐰t\\mathbf\{w\}^\{t\}is close to a point that is nearly stationary for the original nonsmooth and possibly nonconvex objective\. Thus, unlike consensus\-based decentralized optimization, the analysis characterizes approximate stationarity of the full personalized objective rather than of a consensus\-restricted objective\.

Remark on fairness selection\.ForΛ=Δn\\Lambda=\\Delta\_\{n\}, if the scalar max\-consensus step satisfiesKΨ≥diam⁡\(𝒢\)K\_\{\\Psi\}\\geq\\operatorname\{diam\}\(\\mathcal\{G\}\), all clients agree on the same empirical fairness maximizerλ^t\\widehat\{\\lambda\}^\{\\,t\}\. This agreement does not necessarily imply that the selected component maximizes the population fairness envelope\. The resulting discrepancy is quantified byεΨ,t\\varepsilon\_\{\\Psi,t\}and controlled through Assumption[6](https://arxiv.org/html/2608.26493#Thmassmon6)\. Approximate or delayed fairness consensus can similarly be incorporated into an enlarged fairness\-selection gap\.

###### Theorem 2\(Convergence of DMFL\-SQ\)\.

Suppose Assumptions[1](https://arxiv.org/html/2608.26493#Thmassmon1)–[7](https://arxiv.org/html/2608.26493#Thmassmon7)hold\. Assume also thatℒ⋆:=inf𝐰ℒ⁡\(𝐰\)\>−∞\\mathcal\{L\}^\{\\star\}:=\\inf\_\{\\mathbf\{w\}\}\\mathcal\{L\}\(\\mathbf\{w\}\)\>\-\\infty\. Letμ∈\(0,1/κ\)\\mu\\in\(0,1/\\kappa\), whereκ=\(1\+ρ\)​L\\kappa=\(1\+\\rho\)L, and run DMFL\-SQ forTTiterations withγt=γ/T\\gamma\_\{t\}=\\gamma/\\sqrt\{T\}andϑt=ϑ0/t\+1\\vartheta\_\{t\}=\\vartheta\_\{0\}/\\sqrt\{t\+1\}\.

Ifγ\>0\\gamma\>0is sufficiently small such that2​χ1​K𝒢​γ2≤χ0​ω/22\\chi\_\{1\}K\_\{\\mathcal\{G\}\}\\gamma^\{2\}\\leq\\chi\_\{0\}\\omega/2, then

1T​∑t=0T−1𝔼⁡\[𝒮μt\]\\displaystyle\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\mathcal\{S\}\_\{\\mu\}^\{t\}\]≤4​Δμ0n⁡\(1−κ​μ\)​γ​T\+4​ρ​BΨn​μ​\(1−κ​μ\)​T\\displaystyle\\leq\\frac\{4\\Delta\_\{\\mu\}^\{0\}\}\{n\(1\-\\kappa\\mu\)\\gamma\\sqrt\{T\}\}\+\\frac\{4\\rho B\_\{\\Psi\}\}\{n\\mu\(1\-\\kappa\\mu\)\\sqrt\{T\}\}\+4​γ​H2n​μ​\(1−κ​μ\)​T\+2​γ​\(1\+ρ\)2​σ¯2μ⁡\(1−κ​μ\)​T\\displaystyle\\quad\+\\frac\{4\\gamma H^\{2\}\}\{n\\mu\(1\-\\kappa\\mu\)\\sqrt\{T\}\}\+\\frac\{2\\gamma\(1\+\\rho\)^\{2\}\\bar\{\\sigma\}^\{2\}\}\{\\mu\(1\-\\kappa\\mu\)\\sqrt\{T\}\}\+8​K𝒢χ0​ω​T​\(1−κ​μ\)​\(11−κ​μ\+γμ​T\)\\displaystyle\\quad\+\\frac\{8K\_\{\\mathcal\{G\}\}\}\{\\chi\_\{0\}\\omega T\(1\-\\kappa\\mu\)\}\\left\(\\frac\{1\}\{1\-\\kappa\\mu\}\+\\frac\{\\gamma\}\{\\mu\\sqrt\{T\}\}\\right\)×\[𝔼⁡\[ℰ0\]\+χ1​Bh​γ2\+χ2​ϑ02​\(1\+log⁡T\)\],\\displaystyle\\qquad\\qquad\\times\\left\[\\mathbb\{E\}\[\\mathcal\{E\}^\{0\}\]\+\\chi\_\{1\}B\_\{h\}\\gamma^\{2\}\+\\chi\_\{2\}\\vartheta\_\{0\}^\{2\}\(1\+\\log T\)\\right\],\(50\)whereΔμ0:=ℒμ​\(𝐰0\)−ℒ⋆\\Delta\_\{\\mu\}^\{0\}:=\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{0\}\)\-\\mathcal\{L\}^\{\\star\}and𝒮μt=n−1​‖∇ℒμ​\(𝐰t\)‖2\\mathcal\{S\}\_\{\\mu\}^\{t\}=n^\{\-1\}\\\|\\nabla\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\}\)\\\|^\{2\}\. All constants in \([50](https://arxiv.org/html/2608.26493#S5.E50)\) are independent ofTT\.

Consequently,

1T∑t=0T−1𝔼\[𝒮μt\]=𝒪\(T−1/2\)\+𝒪\(log⁡TT\)\.\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\mathcal\{S\}\_\{\\mu\}^\{t\}\]=\\mathcal\{O\}\(T^\{\-1/2\}\)\+\\mathcal\{O\}\\left\(\\frac\{\\log T\}\{T\}\\right\)\.\(51\)
###### Proof:

See Appendix[B](https://arxiv.org/html/2608.26493#A2)\. ∎Theorem[2](https://arxiv.org/html/2608.26493#Thmtheorem2)establishes an𝒪\(T−1/2\)\\mathcal\{O\}\(T^\{\-1/2\}\)rate in expected squared Moreau\-envelope stationarity for the nonsmooth and nonconvex objective\. The communication terms decay as𝒪⁡\(\(1\+log⁡T\)/T\)\\mathcal\{O\}\(\(1\+\\log T\)/T\)and are therefore lower order\. The bound explicitly captures the effects of stochastic gradients, empirical fairness selection, compression, and event\-triggered neighbor\-model communication\.

## VISimulation Results

We evaluateDMFL\-SQin decentralized, heterogeneous, and communication\-constrained settings using CIFAR\-10 and the real MUSMET EEG dataset\. For CIFAR\-10, we use the standard 50k/10k train/test split and partition the data acrossn=20n=20clients using a Dirichlet distribution withαDir=0\.1\\alpha\_\{\\rm Dir\}=0\.1, inducing strong non\-iid heterogeneity\. Each client trains a CNN with two convolutional layers followed by two fully connected layers, and clients communicate over an undirected peer\-to\-peer graph; a ring topology is used by default unless otherwise specified\.

For MUSMET, we consider a 4\-class EEG\-based emotion recognition task corresponding to aggressive, happy, relax, and sad states\[[31](https://arxiv.org/html/2608.26493#bib.bib31)\]\. The dataset contains synchronized EEG/audio recordings from2020musicians\. Labels are obtained from the recording protocol, with each\.xdffile corresponding to one emotional condition\. EEG signals are bandpass\-filtered over 1–40 Hz, segmented into 2\-second non\-overlapping windows, and represented using Welch PSD features over standard frequency bands\. Each musician is treated as one client and trains a 3\-layer MLP with hidden dimensions\[256,128\]\[256,128\]\. For each musician and emotional\-condition recording, the non\-overlapping 2\-s windows are split chronologically, with the first 80% used for training and the remaining 20% for testing\. Feature normalization is computed from the training portion only and then applied to both sets\.

All methods are implemented in PyTorch\. DMFL\-SQ uses mini\-batch stochastic gradient descent with one local update per communication round\. For computational efficiency, the experiments use independent fixed\-size fairness\-selection and gradient mini\-batches, each of size3232\. This fixed\-batch implementation is used for finite\-horizon empirical evaluation and is not claimed to verify the asymptotic selection\-gap condition in Assumption[6](https://arxiv.org/html/2608.26493#Thmassmon6)\. The learning rate isγ=0\.1\\gamma=0\.1\. Unless otherwise stated, we useα=0\.4\\alpha=0\.4,ρ=0\.5\\rho=0\.5, a sparsity ratiok/P=0\.01k/P=0\.01, an88\-bit quantizer, and the event\-trigger scheduleϑt=0\.5/t\+1\\vartheta\_\{t\}=0\.5/\\sqrt\{t\+1\}\. The experimental compressor is𝒞=Q8∘Topk\\mathcal\{C\}=Q\_\{8\}\\circ\\operatorname\{Top\}\_\{k\}, whereQ8Q\_\{8\}is the support\-preserving uniform 8\-bit quantizer\. Since‖Q8​\(u\)−u‖2≤k2​\(28−1\)2​‖u‖2\\\|Q\_\{8\}\(u\)\-u\\\|^\{2\}\\leq\\frac\{k\}\{2\(2^\{8\}\-1\)^\{2\}\}\\\|u\\\|^\{2\}and‖Topk⁡\(v\)‖2≥\(k/P\)​‖v‖2\\\|\\operatorname\{Top\}\_\{k\}\(v\)\\\|^\{2\}\\geq\(k/P\)\\\|v\\\|^\{2\}, the composite compressor satisfies Assumption[4](https://arxiv.org/html/2608.26493#Thmassmon4)by takingω=kP​\(1−k2​\(28−1\)2\)\>0\\omega=\\frac\{k\}\{P\}\\left\(1\-\\frac\{k\}\{2\(2^\{8\}\-1\)^\{2\}\}\\right\)\>0; for CIFAR\-10, this givesω≈8\.37×10−3\\omega\\approx 8\.37\\times 10^\{\-3\}\.

All communication\-efficiency plots reportB¯client\\overline\{B\}\_\{\\mathrm\{client\}\}, which accounts for sparse\-quantized model\-copy messages, coordinate\-index or mask overhead, quantizer scales, message headers, graph\-degree effects, and scalar fairness\-consensus messages; we usebq=8b\_\{q\}=8,bscale=32b\_\{\\mathrm\{scale\}\}=32,bloss=32b\_\{\\mathrm\{loss\}\}=32,bhdr=32b\_\{\\mathrm\{hdr\}\}=32, andbtrig=0b\_\{\\mathrm\{trig\}\}=0bits, with silence indicating an inactive trigger\.

Fig\. 2:Test accuracy versus average communicated bits per client on the CIFAR\-10 dataset\.
Fig\. 3:Ablation on the event\-trigger thresholdϑ0\\vartheta\_\{0\}on the CIFAR\-10 dataset\.
Fig\. 4:Average training loss versus communication rounds on the CIFAR\-10 dataset\.

![Refer to caption](https://arxiv.org/html/2608.26493v1/fig4_cvar.png)

Fig\. 5:Effect of fairness regularization on CIFAR\-10 tail risk \(CVaR@0\.1\)\.
Fig\. 6:Client\-level test\-accuracy distribution on non\-iid CIFAR\-10\.
![Refer to caption](https://arxiv.org/html/2608.26493v1/fig6_topology.png)

Fig\. 7:Effect of network topology on DMFL\-SQ \(CIFAR\-10, non\-iid\)\.

Fig\. 8:Test accuracy versus average communicated bits per client on the MUSMET dataset\.
Fig\. 9:Ablation on the event\-trigger thresholdϑ0\\vartheta\_\{0\}on the MUSMET dataset\.
Fig\. 10:Average training loss versus communication rounds on the MUSMET dataset\.

Fig\. 11:Test accuracy versus training rounds on the MUSMET dataset\.
Fig\. 12:Sensitivity of DMFL\-SQ to the fairness weightρ\\rhoon the MUSMET dataset\.
Fig\. 13:Sensitivity of DMFL\-SQ to the graph\-coupling weightα\\alphaon the MUSMET dataset\.

The proposed DMFL\-SQ algorithm is compared with the following benchmark algorithms:

- •D\-PSGD performs local stochastic\-gradient updates followed by dense model mixing among neighboring clients\[[29](https://arxiv.org/html/2608.26493#bib.bib29)\]\.
- •CHOCO\-SGDis a communication\-efficient decentralized method that replaces dense neighbor exchanges with compressed gossip updates\[[13](https://arxiv.org/html/2608.26493#bib.bib13)\]\.
- •DSGTaugments decentralized optimization with an auxiliary gradient\-tracking variable to better estimate the network\-wide descent direction\[[32](https://arxiv.org/html/2608.26493#bib.bib32)\]\.
- •q\-FFL is server\-based and is included primarily as a fairness\-aware reference\[[9](https://arxiv.org/html/2608.26493#bib.bib9)\]\. Its communication cost includes both uplink and downlink: each client transmits and receives one full\-precisionPP\-dimensional vector per round, givingBqFFL,client=2​T​P​bfpB\_\{\\rm qFFL,client\}=2TPb\_\{\\rm fp\}withbfp=32b\_\{\\rm fp\}=32\.

For baseline parity, all methods use the same MUSMET client partitions, model architecture, mini\-batch size, and five random seeds, while all decentralized methods use the same communication graph\. The principal baseline settings are: D\-PSGD with learning rate0\.0350\.035; CHOCO\-SGD with learning rate0\.0180\.018, consensus step0\.250\.25, sparsity ratio0\.10\.1, 8\-bit quantization, and weight decay5×10−55\\times 10^\{\-5\}; q\-FFL with learning rate0\.180\.18,q=1q=1,L=2L=2, and gradient clipping at0\.50\.5; and DSGT with stepsize0\.045/t\+10\.045/\\sqrt\{t\+1\}and tracker clipping at1\.21\.2\. All methods use the same fixed training horizonT=500T=500and perform one local optimization update per training round, while DSGT requires one additional gradient evaluation per client for tracker initialization\. For each realized graph, we setKΨ=diam⁡\(𝒢\)K\_\{\\Psi\}=\\operatorname\{diam\}\(\\mathcal\{G\}\)and use the realized\|ℰ\|\|\\mathcal\{E\}\|in the communication accounting\. Withn=20n=20andbloss=32b\_\{\\rm loss\}=32, the fairness\-consensus cost isBΨ,t=74​KΨ​\|ℰ\|B\_\{\\Psi,t\}=74K\_\{\\Psi\}\|\\mathcal\{E\}\|bits per round\. For the ring, ER\(p=0\.5\)\(p=0\.5\), and RGG\(r=0\.4\)\(r=0\.4\)realizations,\(\|ℰ\|,KΨ\)\(\|\\mathcal\{E\}\|,K\_\{\\Psi\}\)is\(20,10\)\(20,10\),\(95,2\)\(95,2\), and\(60,4\)\(60,4\), respectively, corresponding to740740,703703, and888888bits/client/round\. The reported total communication uses the actual per\-round trigger indicators inBmodel,tB\_\{\\rm model,t\}\.

TABLE II:Accuracy\-fairness\-communication comparison on the MUSMET dataset\. Performance metrics are reported as mean±\\pm95% CI acrossR=5R=5seeds\.MethodAvg\. Acc\.↑\\uparrowWorst Acc\.↑\\uparrowBottom\-10% Acc\.↑\\uparrowAcc\. Std↓\\downarrowJain’s Index↑\\uparrowBits/Client↓\\downarrowDSGT0\.3132±0\.05410\.3132\\pm 0\.05410\.0501±0\.05860\.0501\\pm 0\.05860\.0610±0\.05920\.0610\\pm 0\.05920\.1981±0\.06660\.1981\\pm 0\.06660\.7103±0\.17190\.7103\\pm 0\.17191998\.31998\.3Mq\-FFL0\.4114±0\.01550\.4114\\pm 0\.01550\.0716±0\.11730\.0716\\pm 0\.11730\.1091±0\.08490\.1091\\pm 0\.08490\.2025±0\.08800\.2025\\pm 0\.08800\.7983±0\.13850\.7983\\pm 0\.13851618\.01618\.0MD\-PSGD0\.5716±0\.05560\.5716\\pm 0\.05560\.1957±0\.10040\.1957\\pm 0\.10040\.2236±0\.09320\.2236\\pm 0\.09320\.2156±0\.03040\.2156\\pm 0\.03040\.8713±0\.05200\.8713\\pm 0\.05201438\.61438\.6MCHOCO\-SGD0\.6928±0\.03420\.6928\\pm 0\.03420\.1935±0\.12050\.1935\\pm 0\.12050\.2320±0\.06650\.2320\\pm 0\.06650\.2163±0\.02560\.2163\\pm 0\.02560\.9074±0\.01280\.9074\\pm 0\.012897\.497\.4MDMFL\-SQ0\.8030±0\.00960\.8030\\pm 0\.00960\.4810±0\.02250\.4810\\pm 0\.02250\.5160±0\.02600\.5160\\pm 0\.02600\.1210±0\.00580\.1210\\pm 0\.00580\.9779±0\.00470\.9779\\pm 0\.00470\.60\.6M

TABLE III:Component ablation of DMFL\-SQ on MUSMET Dataset\. Performance metrics are reported as mean±\\pm95% CI acrossR=5R=5seeds\.VariantAvg\. Acc\.↑\\uparrowWorst Acc\.↑\\uparrowBottom\-10% Acc\.↑\\uparrowJain’s Index↑\\uparrowBits/Client↓\\downarrowNo fairness \(ρ=0\\rho=0\)0\.7869±0\.00240\.7869\\pm 0\.00240\.3743±0\.02410\.3743\\pm 0\.02410\.4065±0\.02410\.4065\\pm 0\.02410\.9521±0\.00410\.9521\\pm 0\.00410\.3 MNo graph coupling \(α=0\\alpha=0\)0\.7474±0\.01570\.7474\\pm 0\.01570\.3862±0\.02630\.3862\\pm 0\.02630\.4206±0\.02070\.4206\\pm 0\.02070\.9554±0\.00420\.9554\\pm 0\.00420\.2 MNo event\-triggering0\.8044±0\.02110\.8044\\pm 0\.02110\.4397±0\.01780\.4397\\pm 0\.01780\.4850±0\.02110\.4850\\pm 0\.02110\.9748±0\.00460\.9748\\pm 0\.00464\.0 MNo sparsification0\.8187±0\.01690\.8187\\pm 0\.01690\.4504±0\.02180\.4504\\pm 0\.02180\.4905±0\.02110\.4905\\pm 0\.02110\.9737±0\.00430\.9737\\pm 0\.00432\.2 MNo quantization0\.8074±0\.00670\.8074\\pm 0\.00670\.4405±0\.02360\.4405\\pm 0\.02360\.4820±0\.02100\.4820\\pm 0\.02100\.9746±0\.00450\.9746\\pm 0\.00455\.3 MDMFL\-SQ0\.8030±0\.00960\.8030\\pm 0\.00960\.4810±0\.02250\.4810\\pm 0\.02250\.5160±0\.02600\.5160\\pm 0\.02600\.9779±0\.00470\.9779\\pm 0\.00470\.60\.6MFig\.[4](https://arxiv.org/html/2608.26493#S6.F4)shows the test accuracy versus average communicated bits per client on non\-iid CIFAR\-10\. DMFL\-SQ reaches higher accuracy with substantially fewer communicated bits than the dense decentralized baselines D\-PSGD and DSGT\. Compared with CHOCO\-SGD, DMFL\-SQ achieves better final accuracy while remaining communication efficient, illustrating the benefit of combining personalization, fairness, compression, and event\-triggered communication\.

Remark\.The modest CIFAR\-10 accuracies reflect the challenging setting considered here:2020clients, a strongly non\-iid Dirichlet split withαDir=0\.1\\alpha\_\{\\mathrm\{Dir\}\}=0\.1, peer\-to\-peer communication only, and no central server or global aggregation\. Thus, the results should be interpreted as communication\-fairness\-accuracy trade\-offs under severe decentralized heterogeneity rather than as centralized CIFAR\-10 benchmarks\. We further complement this setting with MUSMET experiments on naturally heterogeneous EEG data\.

Fig\.[4](https://arxiv.org/html/2608.26493#S6.F4)evaluates the sensitivity of DMFL\-SQ to the initial event\-trigger thresholdϑ0\\vartheta\_\{0\}\. Smaller thresholds trigger more frequent communication and lead to faster early convergence, while larger thresholds reduce communication but slow the initial learning progress\. The results show that DMFL\-SQ remains stable across different threshold choices, with larger thresholds eventually catching up when more communication rounds are allowed\.

Fig\.[4](https://arxiv.org/html/2608.26493#S6.F4)shows the average training loss on non\-iid CIFAR\-10\. DMFL\-SQ achieves a faster decrease and a lower final loss than D\-PSGD, suggesting improved optimization behavior under heterogeneous client data\. The curve remains stable despite the use of sparse\-quantized event\-triggered communication\.

Fig\.[7](https://arxiv.org/html/2608.26493#S6.F7)shows the effect of fairness regularization on the CVaR@0\.1 test loss under non\-iid CIFAR\-10\. After an initial transient, both variants reduce tail risk as training progresses\. The fairness\-aware variant withρ\>0\\rho\>0remains slightly below theρ=0\\rho=0variant for most rounds, indicating that the agnostic mixture term helps reduce upper\-tail client loss\.

Fig\.[7](https://arxiv.org/html/2608.26493#S6.F7)reports the client\-level test\-accuracy distribution on non\-iid CIFAR\-10\. Each point corresponds to one client\. DMFL\-SQ shifts the lower tail of the distribution upward compared with the baselines\.

Remark:The PAC\-Bayes result is stated for bounded lossesℓi​\(wi,z\)∈\[0,1\]\\ell\_\{i\}\(w\_\{i\};z\)\\in\[0,1\], as is standard in PAC\-Bayes analysis\. In the experiments, worst\-client loss and CVaR@0\.1 are computed using raw cross\-entropy loss and are reported only as empirical diagnostic metrics; they are not used as the bounded loss in Theorem[1](https://arxiv.org/html/2608.26493#Thmtheorem1)\. To apply the theorem directly to cross\-entropy, one may use the clipped and normalized lossℓi,B​\(wi,z\)=B−1​min⁡\{ℓiCE​\(wi,z\),B\},\\ell\_\{i,B\}\(w\_\{i\};z\)=B^\{\-1\}\\min\\\{\\ell\_\{i\}^\{\\rm CE\}\(w\_\{i\};z\),B\\\},which lies in\[0,1\]\[0,1\]\.

Fig\.[7](https://arxiv.org/html/2608.26493#S6.F7)evaluates the sensitivity of DMFL\-SQ to the communication topology\. Ring, Erdős–Rényi \(ER\), and random geometric graph \(RGG\) topologies all show stable learning, indicating that DMFL\-SQ is not restricted to a specific graph structure\. The different transient behaviors reflect how graph connectivity affects neighbor\-copy propagation and graph\-regularized coupling under non\-iid data\.

Fig\.[10](https://arxiv.org/html/2608.26493#S6.F10)shows the communication\-accuracy trade\-off on the MUSMET dataset\. Similar to the CIFAR\-10 results in Fig\.[4](https://arxiv.org/html/2608.26493#S6.F4), DMFL\-SQ attains the highest accuracy among the compared methods while requiring fewer communicated bits than dense decentralized baselines, showing a favorable accuracy\-communication trade\-off on this naturally heterogeneous real\-world dataset\.

Fig\.[10](https://arxiv.org/html/2608.26493#S6.F10)shows the sensitivity of DMFL\-SQ to the initial event\-trigger thresholdϑ0\\vartheta\_\{0\}on MUSMET\. Across the tested thresholds, the method exhibits stable learning behavior, showing that the event\-triggered mechanism remains effective on naturally heterogeneous EEG data\.

Fig\.[10](https://arxiv.org/html/2608.26493#S6.F10)shows the average training loss on MUSMET\. DMFL\-SQ converges faster and reaches a much lower loss than D\-PSGD, demonstrating stable optimization under naturally heterogeneous EEG data despite sparse\-quantized event\-triggered communication\.

Fig\.[13](https://arxiv.org/html/2608.26493#S6.F13)shows that DMFL\-SQ consistently attains the highest test accuracy and converges stably compared with the considered baselines on the MUSMET dataset\.

Figs\.[13](https://arxiv.org/html/2608.26493#S6.F13)and[13](https://arxiv.org/html/2608.26493#S6.F13)further show that increasing the fairness weightρ\\rhomainly improves worst\-client accuracy while preserving average accuracy, whereas stronger graph couplingα\\alphaimproves both metrics, with diminishing gains beyond moderate values\. These results support the robustness of DMFL\-SQ to its key hyperparameters\.

Table[II](https://arxiv.org/html/2608.26493#S6.T2)shows that DMFL\-SQ achieves the highest average accuracy, worst\-client accuracy, bottom\-10% accuracy, and Jain’s index and lowest standard deviation of client accuracies on the heterogeneous MUSMET dataset\. Moreover, DMFL\-SQ substantially reduces the communication cost compared with both dense decentralized baselines and compressed CHOCO\-SGD, demonstrating a favorable accuracy\-fairness\-communication trade\-off\.

Table[III](https://arxiv.org/html/2608.26493#S6.T3)shows that each component contributes to the overall performance of DMFL\-SQ\. The DMFL\-SQ algorithm achieves the highest worst\-client and bottom\-10% accuracies, as well as the highest Jain’s index, while requiring only0\.60\.6Mbits per client\. Although removing sparsification or quantization slightly improves the average accuracy, it substantially increases the communication cost and reduces the tail\-client performance\. Similarly, removing the fairness or graph\-coupling terms degrades the worst\-client and bottom\-10% accuracies\. These results demonstrate that the DMFL\-SQ design provides the most favorable overall balance among average accuracy, client fairness, and communication efficiency\.

## VIIConclusion

We presented DMFL\-SQ, a decentralized multi\-task fair learning framework with sparsified, quantized, and event\-triggered communication\. The algorithm unifies personalization, fairness, and communication efficiency within a single decentralized optimization paradigm in the*non\-convex*regime\. Our theoretical results establish an𝒪\(T−1/2\)\\mathcal\{O\}\(T^\{\-1/2\}\)rate in expected squared Moreau\-envelope stationarity, while explicitly accounting for compression, network topology, and triggering effects\. Extensive experiments demonstrate that DMFL\-SQ achieves a favorable accuracy\-fairness\-communication trade\-off under the considered heterogeneous settings while reducing communication cost compared to existing baselines\.

## Appendix AProof of Theorem[1](https://arxiv.org/html/2608.26493#Thmtheorem1)& Corollary[1](https://arxiv.org/html/2608.26493#Thmcorollary1)

We first recall the population and empirical agnostic mixture risks:Ψ⁡\(𝐰\)=supλ∈Λ∑i=1nλi​Fi​\(wi\)\\Psi\(\\mathbf\{w\}\)=\\sup\_\{\\lambda\\in\\Lambda\}\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}F\_\{i\}\(w\_\{i\}\),Ψ^​\(𝐰\)=supλ∈Λ∑i=1nλi​F^i​\(wi\)\\widehat\{\\Psi\}\(\\mathbf\{w\}\)=\\sup\_\{\\lambda\\in\\Lambda\}\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}\\widehat\{F\}\_\{i\}\(w\_\{i\}\)\. For a fixed mixture vectorλ∈Λ\\lambda\\in\\Lambda, defineΦλ​\(𝐰\):=∑i=1nλi​Fi​\(wi\)\\Phi\_\{\\lambda\}\(\\mathbf\{w\}\):=\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}F\_\{i\}\(w\_\{i\}\),Φ^λ​\(𝐰\):=∑i=1nλi​F^i​\(wi\)\\widehat\{\\Phi\}\_\{\\lambda\}\(\\mathbf\{w\}\):=\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}\\widehat\{F\}\_\{i\}\(w\_\{i\}\)\.

#### Case 1:Λ=Δn\\Lambda=\\Delta\_\{n\}

Sinceλ↦Φλ​\(𝐰\)\\lambda\\mapsto\\Phi\_\{\\lambda\}\(\\mathbf\{w\}\)is linear over the simplexΔn\\Delta\_\{n\}, its supremum is attained at a vertex\. Hence,

Ψ⁡\(𝐰\)=max1≤i≤n⁡Fi​\(wi\),Ψ^​\(𝐰\)=max1≤i≤n⁡F^i​\(wi\)\.\\Psi\(\\mathbf\{w\}\)=\\max\_\{1\\leq i\\leq n\}F\_\{i\}\(w\_\{i\}\),\\qquad\\widehat\{\\Psi\}\(\\mathbf\{w\}\)=\\max\_\{1\\leq i\\leq n\}\\widehat\{F\}\_\{i\}\(w\_\{i\}\)\.For each𝐰\\mathbf\{w\}, leti⋆​\(𝐰\)∈arg⁡max1≤i≤n​Fi​\(wi\)i^\{\\star\}\(\\mathbf\{w\}\)\\in\\arg\\max\_\{1\\leq i\\leq n\}F\_\{i\}\(w\_\{i\}\)be a measurable selection of a worst\-case client\. Define the augmented posterior and prior over pairs\(𝐰,i\)\(\\mathbf\{w\},i\)byQ~​\(d​𝐰,d​i\)=Q⁡\(d​𝐰\)​δi⋆​\(𝐰\)​\(d​i\)\\widetilde\{Q\}\(d\\mathbf\{w\},di\)=Q\(d\\mathbf\{w\}\)\\delta\_\{i^\{\\star\}\(\\mathbf\{w\}\)\}\(di\),Π~​\(d​𝐰,d​i\)=Π⁡\(d​𝐰\)​π​\(d​i\)\\widetilde\{\\Pi\}\(d\\mathbf\{w\},di\)=\\Pi\(d\\mathbf\{w\}\)\\pi\(di\), whereπ⁡\(i\)=1/n\\pi\(i\)=1/nis the uniform distribution over clients\. Then

𝔼\(𝐰,i\)∼Q~​\[Fi​\(wi\)\]=𝔼𝐰∼Q​\[Ψ⁡\(𝐰\)\]\.\\mathbb\{E\}\_\{\(\\mathbf\{w\},i\)\\sim\\widetilde\{Q\}\}\\\!\\left\[F\_\{i\}\(w\_\{i\}\)\\right\]=\\mathbb\{E\}\_\{\\mathbf\{w\}\\sim Q\}\\\!\\left\[\\Psi\(\\mathbf\{w\}\)\\right\]\.Moreover, sinceF^i⋆​\(𝐰\)​\(wi⋆​\(𝐰\)\)≤max1≤i≤n⁡F^i​\(wi\)=Ψ^​\(𝐰\)\\widehat\{F\}\_\{i^\{\\star\}\(\\mathbf\{w\}\)\}\\\!\\left\(w\_\{i^\{\\star\}\(\\mathbf\{w\}\)\}\\right\)\\leq\\max\_\{1\\leq i\\leq n\}\\widehat\{F\}\_\{i\}\(w\_\{i\}\)=\\widehat\{\\Psi\}\(\\mathbf\{w\}\), we have

𝔼\(𝐰,i\)∼Q~​\[F^i​\(wi\)\]≤𝔼𝐰∼Q​\[Ψ^​\(𝐰\)\]\.\\mathbb\{E\}\_\{\(\\mathbf\{w\},i\)\\sim\\widetilde\{Q\}\}\\\!\\left\[\\widehat\{F\}\_\{i\}\(w\_\{i\}\)\\right\]\\leq\\mathbb\{E\}\_\{\\mathbf\{w\}\\sim Q\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}\)\\right\]\.The KL divergence of the augmented posterior satisfies

KL\(Q~∥Π~\)=KL\(Q∥Π\)\+lnn\.\\mathrm\{KL\}\(\\widetilde\{Q\}\\\|\\widetilde\{\\Pi\}\)=\\mathrm\{KL\}\(Q\\\|\\Pi\)\+\\ln n\.For every fixed pair\(𝐰,i\)\(\\mathbf\{w\},i\), Hoeffding’s lemma gives

𝔼𝒟​exp⁡\(τPB​\[Fi​\(wi\)−F^i​\(wi\)\]\)≤exp⁡\(τPB28​mi\)≤exp⁡\(τPB28​mmin\)\.\\mathbb\{E\}\_\{\\mathcal\{D\}\}\\exp\\left\(\\tau\_\{\\mathrm\{PB\}\}\\left\[F\_\{i\}\(w\_\{i\}\)\-\\widehat\{F\}\_\{i\}\(w\_\{i\}\)\\right\]\\right\)\\leq\\exp\\left\(\\frac\{\\tau\_\{\\mathrm\{PB\}\}^\{2\}\}\{8m\_\{i\}\}\\right\)\\leq\\exp\\left\(\\frac\{\\tau\_\{\\mathrm\{PB\}\}^\{2\}\}\{8m\_\{\\min\}\}\\right\)\.Integrating with respect to the data\-independent augmented priorΠ~\\widetilde\{\\Pi\}and applying the PAC–Bayes change\-of\-measure inequality together with Markov’s inequality yields, with probability at least1−δ1\-\\delta, simultaneously for all augmented posteriorsQ~\\widetilde\{Q\}, \]

𝔼\(𝐰,i\)∼Q~​\[Fi​\(wi\)\]\\displaystyle\\mathbb\{E\}\_\{\(\\mathbf\{w\},i\)\\sim\\widetilde\{Q\}\}\\\!\\left\[F\_\{i\}\(w\_\{i\}\)\\right\]≤𝔼\(𝐰,i\)∼Q~​\[F^i​\(wi\)\]\+KL\(Q~∥Π~\)\+ln\(1/δ\)τPB\\displaystyle\\leq\\mathbb\{E\}\_\{\(\\mathbf\{w\},i\)\\sim\\widetilde\{Q\}\}\\\!\\left\[\\widehat\{F\}\_\{i\}\(w\_\{i\}\)\\right\]\+\\frac\{\\mathrm\{KL\}\(\\widetilde\{Q\}\\\|\\widetilde\{\\Pi\}\)\+\\ln\(1/\\delta\)\}\{\\tau\_\{\\mathrm\{PB\}\}\}\+τPB8​mmin\.\\displaystyle\+\\frac\{\\tau\_\{\\mathrm\{PB\}\}\}\{8m\_\{\\min\}\}\.\(52\)This proves the first claim\.

#### Case 2: General compactΛ⊆Δn\\Lambda\\subseteq\\Delta\_\{n\}

Let𝒩ϵ\(Λ,∥⋅∥1\)⊆Λ\\mathcal\{N\}\_\{\\epsilon\}\(\\Lambda,\\\|\\cdot\\\|\_\{1\}\)\\subseteq\\Lambdabe a fixed deterministicϵ\\epsilon\-net ofΛ\\Lambdain theℓ1\\ell\_\{1\}norm, constructed independently of the observed datasets\. Sinceℓi​\(wi,z\)∈\[0,1\]\\ell\_\{i\}\(w\_\{i\};z\)\\in\[0,1\], we haveFi​\(wi\),F^i​\(wi\)∈\[0,1\]F\_\{i\}\(w\_\{i\}\),\\widehat\{F\}\_\{i\}\(w\_\{i\}\)\\in\[0,1\]\. Therefore, for any fixed𝐰\\mathbf\{w\}, both mapsλ↦Φλ​\(𝐰\)\\lambda\\mapsto\\Phi\_\{\\lambda\}\(\\mathbf\{w\}\),λ↦Φ^λ​\(𝐰\)\\lambda\\mapsto\\widehat\{\\Phi\}\_\{\\lambda\}\(\\mathbf\{w\}\)are11\-Lipschitz with respect to∥⋅∥1\\\|\\cdot\\\|\_\{1\}\. Indeed, for anyλ,λ′∈Λ\\lambda,\\lambda^\{\\prime\}\\in\\Lambda,

\|Φλ​\(𝐰\)−Φλ′​\(𝐰\)\|≤∑i=1n\|λi−λi′\|​Fi​\(wi\)≤‖λ−λ′‖1,\|\\Phi\_\{\\lambda\}\(\\mathbf\{w\}\)\-\\Phi\_\{\\lambda^\{\\prime\}\}\(\\mathbf\{w\}\)\|\\leq\\sum\_\{i=1\}^\{n\}\|\\lambda\_\{i\}\-\\lambda\_\{i\}^\{\\prime\}\|F\_\{i\}\(w\_\{i\}\)\\leq\\\|\\lambda\-\\lambda^\{\\prime\}\\\|\_\{1\},and similarly\|Φ^λ​\(𝐰\)−Φ^λ′​\(𝐰\)\|≤‖λ−λ′‖1\|\\widehat\{\\Phi\}\_\{\\lambda\}\(\\mathbf\{w\}\)\-\\widehat\{\\Phi\}\_\{\\lambda^\{\\prime\}\}\(\\mathbf\{w\}\)\|\\leq\\\|\\lambda\-\\lambda^\{\\prime\}\\\|\_\{1\}\.

For each𝐰\\mathbf\{w\}, letλ⋆​\(𝐰\)∈arg⁡maxλ∈Λ​Φλ​\(𝐰\)\\lambda^\{\\star\}\(\\mathbf\{w\}\)\\in\\arg\\max\_\{\\lambda\\in\\Lambda\}\\Phi\_\{\\lambda\}\(\\mathbf\{w\}\)be a measurable selection\. Using a fixed deterministic tie\-breaking rule, choosevϵ\(𝐰\)∈𝒩ϵ\(Λ,∥⋅∥1\)v\_\{\\epsilon\}\(\\mathbf\{w\}\)\\in\\mathcal\{N\}\_\{\\epsilon\}\(\\Lambda,\\\|\\cdot\\\|\_\{1\}\)such that‖λ⋆​\(𝐰\)−vϵ​\(𝐰\)‖1≤ϵ\\\|\\lambda^\{\\star\}\(\\mathbf\{w\}\)\-v\_\{\\epsilon\}\(\\mathbf\{w\}\)\\\|\_\{1\}\\leq\\epsilon\. By the Lipschitz property,

Ψ⁡\(𝐰\)=Φλ⋆​\(𝐰\)​\(𝐰\)≤Φvε​\(𝐰\)​\(𝐰\)\+ε\.\\Psi\(\\mathbf\{w\}\)=\\Phi\_\{\\lambda^\{\\star\}\(\\mathbf\{w\}\)\}\(\\mathbf\{w\}\)\\leq\\Phi\_\{v\_\{\\varepsilon\}\(\\mathbf\{w\}\)\}\(\\mathbf\{w\}\)\+\\varepsilon\.Also,Φ^vε​\(𝐰\)​\(𝐰\)≤Ψ^​\(𝐰\)\\widehat\{\\Phi\}\_\{v\_\{\\varepsilon\}\(\\mathbf\{w\}\)\}\(\\mathbf\{w\}\)\\leq\\widehat\{\\Psi\}\(\\mathbf\{w\}\)\. Define the augmented posterior and prior over pairs\(𝐰,v\)\(\\mathbf\{w\},v\)byQ~ε​\(d​𝐰,d​v\)=Q⁡\(d​𝐰\)​δvε​\(𝐰\)​\(d​v\)\\widetilde\{Q\}\_\{\\varepsilon\}\(d\\mathbf\{w\},dv\)=Q\(d\\mathbf\{w\}\)\\delta\_\{v\_\{\\varepsilon\}\(\\mathbf\{w\}\)\}\(dv\),Π~ε​\(d​𝐰,d​v\)=Π⁡\(d​𝐰\)​πε​\(d​v\)\\widetilde\{\\Pi\}\_\{\\varepsilon\}\(d\\mathbf\{w\},dv\)=\\Pi\(d\\mathbf\{w\}\)\\pi\_\{\\varepsilon\}\(dv\), whereπε\\pi\_\{\\varepsilon\}is the uniform distribution over𝒩ε\(Λ,∥⋅∥1\)\\mathcal\{N\}\_\{\\varepsilon\}\(\\Lambda,\\\|\\cdot\\\|\_\{1\}\)\. Then

KL\(Q~ε∥Π~ε\)=KL\(Q∥Π\)\+ln\|𝒩ε\(Λ,∥⋅∥1\)\|\.\\mathrm\{KL\}\(\\widetilde\{Q\}\_\{\\varepsilon\}\\\|\\widetilde\{\\Pi\}\_\{\\varepsilon\}\)=\\mathrm\{KL\}\(Q\\\|\\Pi\)\+\\ln\|\\mathcal\{N\}\_\{\\varepsilon\}\(\\Lambda,\\\|\\cdot\\\|\_\{1\}\)\|\.For eachv∈𝒩ε\(Λ,∥⋅∥1\)v\\in\\mathcal\{N\}\_\{\\varepsilon\}\(\\Lambda,\\\|\\cdot\\\|\_\{1\}\), definemv:=mλ\|λ=vm\_\{v\}:=m\_\{\\lambda\}\|\_\{\\lambda=v\}\. Since theε\\varepsilon\-net is contained inΛ\\Lambda, the definition ofmeffm\_\{\\mathrm\{eff\}\}givesmv≥meffm\_\{v\}\\geq m\_\{\\mathrm\{eff\}\}\.

For every fixed pair\(𝐰,v\)\(\\mathbf\{w\},v\), wherev∈𝒩ϵ\(Λ,∥⋅∥1\)v\\in\\mathcal\{N\}\_\{\\epsilon\}\(\\Lambda,\\\|\\cdot\\\|\_\{1\}\),

Φv​\(𝐰\)−Φ^v​\(𝐰\)=∑i=1n∑k=1mivimi​\[Fi​\(wi\)−ℓi​\(wi,zi,k\)\]\.\\displaystyle\\Phi\_\{v\}\(\\mathbf\{w\}\)\-\\widehat\{\\Phi\}\_\{v\}\(\\mathbf\{w\}\)=\\sum\_\{i=1\}^\{n\}\\sum\_\{k=1\}^\{m\_\{i\}\}\\frac\{v\_\{i\}\}\{m\_\{i\}\}\\left\[F\_\{i\}\(w\_\{i\}\)\-\\ell\_\{i\}\(w\_\{i\};z\_\{i,k\}\)\\right\]\.Since the losses lie in\[0,1\]\[0,1\]and the client samples are mutually independent, Hoeffding’s lemma gives

𝔼𝒟​exp⁡\(τPB​\[Φv​\(𝐰\)−Φ^v​\(𝐰\)\]\)\\displaystyle\\mathbb\{E\}\_\{\\mathcal\{D\}\}\\exp\\left\(\\tau\_\{\\mathrm\{PB\}\}\\left\[\\Phi\_\{v\}\(\\mathbf\{w\}\)\-\\widehat\{\\Phi\}\_\{v\}\(\\mathbf\{w\}\)\\right\]\\right\)≤exp⁡\(τPB28​∑i=1nvi2mi\)=exp⁡\(τPB28​mv\),\\displaystyle\\qquad\\leq\\exp\\left\(\\frac\{\\tau\_\{\\mathrm\{PB\}\}^\{2\}\}\{8\}\\sum\_\{i=1\}^\{n\}\\frac\{v\_\{i\}^\{2\}\}\{m\_\{i\}\}\\right\)=\\exp\\left\(\\frac\{\\tau\_\{\\mathrm\{PB\}\}^\{2\}\}\{8m\_\{v\}\}\\right\),\(53\)wheremv=\(∑i=1nvi2mi\)−1m\_\{v\}=\\left\(\\sum\_\{i=1\}^\{n\}\\frac\{v\_\{i\}^\{2\}\}\{m\_\{i\}\}\\right\)^\{\-1\}\. Sincemv≥meffm\_\{v\}\\geq m\_\{\\mathrm\{eff\}\}, integrating with respect toΠ~ϵ\\widetilde\{\\Pi\}\_\{\\epsilon\}and applying the PAC\-Bayes change\-of\-measure inequality and Markov’s inequality yields, with probability at least1−δ1\-\\delta, simultaneously for all augmented posteriorsQ~ϵ\\widetilde\{Q\}\_\{\\epsilon\},

𝔼\(𝐰,v\)∼Q~ϵ​\[Φv​\(𝐰\)\]≤\\displaystyle\\mathbb\{E\}\_\{\(\\mathbf\{w\},v\)\\sim\\widetilde\{Q\}\_\{\\epsilon\}\}\\\!\\left\[\\Phi\_\{v\}\(\\mathbf\{w\}\)\\right\]\\leq\{\}𝔼\(𝐰,v\)∼Q~ϵ​\[Φ^v​\(𝐰\)\]\+τPB8​meff\\displaystyle\\mathbb\{E\}\_\{\(\\mathbf\{w\},v\)\\sim\\widetilde\{Q\}\_\{\\epsilon\}\}\\\!\\left\[\\widehat\{\\Phi\}\_\{v\}\(\\mathbf\{w\}\)\\right\]\+\\frac\{\\tau\_\{\\mathrm\{PB\}\}\}\{8m\_\{\\mathrm\{eff\}\}\}\+KL\(Q~ϵ∥Π~ϵ\)\+ln\(1/δ\)τPB\.\\displaystyle\+\\frac\{\\mathrm\{KL\}\(\\widetilde\{Q\}\_\{\\epsilon\}\\\|\\widetilde\{\\Pi\}\_\{\\epsilon\}\)\+\\ln\(1/\\delta\)\}\{\\tau\_\{\\mathrm\{PB\}\}\}\.\(54\)
This proves the second claim\.

### A\-AProof of Corollary[1](https://arxiv.org/html/2608.26493#Thmcorollary1)

The PAC\-Bayes bound in Theorem[1](https://arxiv.org/html/2608.26493#Thmtheorem1)is algorithm\-agnostic\. We now quantify the one\-step perturbation of the empirical agnostic mixture risk caused by evaluating the graph regularizer using stale and compressed neighbor copies\.

First, we establish a Lipschitz property of the empirical fairness envelope\. For any two stacked model collections𝐰=\(w1,…,wn\)\\mathbf\{w\}=\(w\_\{1\},\\ldots,w\_\{n\}\)and𝐯=\(v1,…,vn\)\\mathbf\{v\}=\(v\_\{1\},\\ldots,v\_\{n\}\), we have

\|Ψ^​\(𝐰\)−Ψ^​\(𝐯\)\|\\displaystyle\\left\|\\widehat\{\\Psi\}\(\\mathbf\{w\}\)\-\\widehat\{\\Psi\}\(\\mathbf\{v\}\)\\right\|=\|supλ∈Λ∑i=1nλi​F^i​\(wi\)−supλ∈Λ∑i=1nλi​F^i​\(vi\)\|\\displaystyle=\\left\|\\sup\_\{\\lambda\\in\\Lambda\}\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}\\widehat\{F\}\_\{i\}\(w\_\{i\}\)\-\\sup\_\{\\lambda\\in\\Lambda\}\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}\\widehat\{F\}\_\{i\}\(v\_\{i\}\)\\right\|≤supλ∈Λ\|∑i=1nλi​\(F^i​\(wi\)−F^i​\(vi\)\)\|\\displaystyle\\leq\\sup\_\{\\lambda\\in\\Lambda\}\\left\|\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}\\left\(\\widehat\{F\}\_\{i\}\(w\_\{i\}\)\-\\widehat\{F\}\_\{i\}\(v\_\{i\}\)\\right\)\\right\|≤Glip​supλ∈Λ∑i=1nλi​‖wi−vi‖\.\\displaystyle\\leq G\_\{\\mathrm\{lip\}\}\\sup\_\{\\lambda\\in\\Lambda\}\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}\\\|w\_\{i\}\-v\_\{i\}\\\|\.\(55\)SinceΛ⊆Δn\\Lambda\\subseteq\\Delta\_\{n\},

supλ∈Λ∑i=1nλi​‖wi−vi‖\\displaystyle\\sup\_\{\\lambda\\in\\Lambda\}\\sum\_\{i=1\}^\{n\}\\lambda\_\{i\}\\\|w\_\{i\}\-v\_\{i\}\\\|≤maxi∈\[n\]⁡‖wi−vi‖≤‖𝐰−𝐯‖\.\\displaystyle\\leq\\max\_\{i\\in\[n\]\}\\\|w\_\{i\}\-v\_\{i\}\\\|\\leq\\\|\\mathbf\{w\}\-\\mathbf\{v\}\\\|\.\(56\)Consequently,

\|Ψ^​\(𝐰\)−Ψ^​\(𝐯\)\|≤Glip​‖𝐰−𝐯‖\.\\displaystyle\\left\|\\widehat\{\\Psi\}\(\\mathbf\{w\}\)\-\\widehat\{\\Psi\}\(\\mathbf\{v\}\)\\right\|\\leq G\_\{\\mathrm\{lip\}\}\\\|\\mathbf\{w\}\-\\mathbf\{v\}\\\|\.\(57\)
Recall that the implemented direction can be decomposed as𝐡t=𝐕t\+𝐪t\+𝐝t\\mathbf\{h\}^\{t\}=\\mathbf\{V\}^\{t\}\+\\mathbf\{q\}^\{t\}\+\\mathbf\{d\}^\{t\}, where𝐝t=𝐫t−∇ℛ𝒢​\(𝐰t\)\\mathbf\{d\}^\{t\}=\\mathbf\{r\}^\{t\}\-\\nabla\\mathcal\{R\}\_\{\\mathcal\{G\}\}\(\\mathbf\{w\}^\{t\}\)is the graph\-coupling approximation error\. Define the exact\-neighbor shadow update

𝐰ext\+1:=𝐰t−γt​\(𝐡t−𝐝t\)\.\\displaystyle\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{t\+1\}:=\\mathbf\{w\}^\{t\}\-\\gamma\_\{t\}\\left\(\\mathbf\{h\}^\{t\}\-\\mathbf\{d\}^\{t\}\\right\)\.\(58\)Thus,𝐰ext\+1\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{t\+1\}starts from the same current iterate𝐰t\\mathbf\{w\}^\{t\}and uses the same stochastic gradients and fairness selection as DMFL\-SQ, but evaluates the graph\-coupling direction using the exact current neighbor models\.

Since the DMFL\-SQ update is𝐰t\+1=𝐰t−γt​𝐡t\\mathbf\{w\}^\{t\+1\}=\\mathbf\{w\}^\{t\}\-\\gamma\_\{t\}\\mathbf\{h\}^\{t\}, we obtain

𝐰t\+1−𝐰ext\+1=−γt​𝐝t\.\\displaystyle\\mathbf\{w\}^\{t\+1\}\-\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{t\+1\}=\-\\gamma\_\{t\}\\mathbf\{d\}^\{t\}\.\(59\)Applying \([57](https://arxiv.org/html/2608.26493#A1.E57)\) gives

\|Ψ^​\(𝐰t\+1\)−Ψ^​\(𝐰ext\+1\)\|\\displaystyle\\left\|\\widehat\{\\Psi\}\(\\mathbf\{w\}^\{t\+1\}\)\-\\widehat\{\\Psi\}\(\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{t\+1\}\)\\right\|≤Glip​γt​‖𝐝t‖\.\\displaystyle\\leq G\_\{\\mathrm\{lip\}\}\\gamma\_\{t\}\\\|\\mathbf\{d\}^\{t\}\\\|\.\(60\)Taking expectations and using Jensen’s inequality yields

\|𝔼⁡\[Ψ^​\(𝐰t\+1\)\]−𝔼⁡\[Ψ^​\(𝐰ext\+1\)\]\|\\displaystyle\\left\|\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}^\{t\+1\}\)\\right\]\-\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{t\+1\}\)\\right\]\\right\|≤Glip​γt​𝔼​‖𝐝t‖\\displaystyle\\leq G\_\{\\mathrm\{lip\}\}\\gamma\_\{t\}\\mathbb\{E\}\\\|\\mathbf\{d\}^\{t\}\\\|≤Glip​γt​\(𝔼​‖𝐝t‖2\)1/2\.\\displaystyle\\leq G\_\{\\mathrm\{lip\}\}\\gamma\_\{t\}\\left\(\\mathbb\{E\}\\\|\\mathbf\{d\}^\{t\}\\\|^\{2\}\\right\)^\{1/2\}\.\(61\)
The graph\-direction approximation bound gives‖𝐝t‖2≤n​K𝒢​ℰt\\\|\\mathbf\{d\}^\{t\}\\\|^\{2\}\\leq nK\_\{\\mathcal\{G\}\}\\mathcal\{E\}^\{t\}, whereℰt=1n​∑i=1n∑j∈𝒩iai​j​‖wjt−w^j→it‖2\\mathcal\{E\}^\{t\}=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}a\_\{ij\}\\left\\\|w\_\{j\}^\{t\}\-\\widehat\{w\}\_\{j\\to i\}^\{t\}\\right\\\|^\{2\}\. Therefore,

\|𝔼⁡\[Ψ^​\(𝐰t\+1\)\]−𝔼⁡\[Ψ^​\(𝐰ext\+1\)\]\|≤Glip​γt​n​K𝒢​𝔼​\[ℰt\]\.\\displaystyle\\left\|\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}^\{t\+1\}\)\\right\]\-\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{t\+1\}\)\\right\]\\right\|\\leq G\_\{\\mathrm\{lip\}\}\\gamma\_\{t\}\\sqrt\{nK\_\{\\mathcal\{G\}\}\\,\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]\}\.\(62\)
Letτ∼Unif​\{0,…,T−1\}\\tau\\sim\\mathrm\{Unif\}\\\{0,\\ldots,T\-1\\\}, independently of the algorithmic randomness\. Averaging \([62](https://arxiv.org/html/2608.26493#A1.E62)\) overt=0,…,T−1t=0,\\ldots,T\-1gives

\|𝔼⁡\[Ψ^​\(𝐰τ\+1\)\]−𝔼⁡\[Ψ^​\(𝐰exτ\+1\)\]\|≤Glip​n​K𝒢T​∑t=0T−1γt​𝔼⁡\[ℰt\]\.\\displaystyle\\left\|\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}^\{\\tau\+1\}\)\\right\]\-\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{\\tau\+1\}\)\\right\]\\right\|\\leq\\frac\{G\_\{\\mathrm\{lip\}\}\\sqrt\{nK\_\{\\mathcal\{G\}\}\}\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\gamma\_\{t\}\\sqrt\{\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]\}\.\(63\)
Under Assumption[7](https://arxiv.org/html/2608.26493#Thmassmon7),γt=γ/T\\gamma\_\{t\}=\\gamma/\\sqrt\{T\}\. Hence, by the Cauchy\-Schwarz inequality,

\|𝔼⁡\[Ψ^​\(𝐰τ\+1\)\]−𝔼⁡\[Ψ^​\(𝐰exτ\+1\)\]\|≤Glip​γ​n​K𝒢T​\(1T​∑t=0T−1𝔼⁡\[ℰt\]\)1/2\.\\displaystyle\\left\|\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}^\{\\tau\+1\}\)\\right\]\-\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{\\tau\+1\}\)\\right\]\\right\|\\leq\\frac\{G\_\{\\mathrm\{lip\}\}\\gamma\\sqrt\{nK\_\{\\mathcal\{G\}\}\}\}\{\\sqrt\{T\}\}\\left\(\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]\\right\)^\{1/2\}\.\(64\)
By Lemma[6](https://arxiv.org/html/2608.26493#Thmlemma6),

1T​∑t=0T−1𝔼⁡\[ℰt\]≤2χ0​ω​T​\(𝔼⁡\[ℰ0\]\+χ1​Bh​γ2\+χ2​ϑ02​\(1\+log⁡T\)\)\.\\displaystyle\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]\\leq\\frac\{2\}\{\\chi\_\{0\}\\omega T\}\\left\(\\mathbb\{E\}\[\\mathcal\{E\}^\{0\}\]\+\\chi\_\{1\}B\_\{h\}\\gamma^\{2\}\+\\chi\_\{2\}\\vartheta\_\{0\}^\{2\}\(1\+\\log T\)\\right\)\.\(65\)Substituting \([65](https://arxiv.org/html/2608.26493#A1.E65)\) into \([64](https://arxiv.org/html/2608.26493#A1.E64)\) yields

\|𝔼⁡\[Ψ^​\(𝐰τ\+1\)\]−𝔼⁡\[Ψ^​\(𝐰exτ\+1\)\]\|\\displaystyle\\left\|\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}^\{\\tau\+1\}\)\\right\]\-\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{\\tau\+1\}\)\\right\]\\right\|≤Glip​γ​2​n​K𝒢χ0​ω​T​\(𝔼⁡\[ℰ0\]\+χ1​Bh​γ2\+χ2​ϑ02​\(1\+log⁡T\)\)1/2\.\\displaystyle\\qquad\\leq\\frac\{G\_\{\\mathrm\{lip\}\}\\gamma\\sqrt\{2nK\_\{\\mathcal\{G\}\}\}\}\{\\sqrt\{\\chi\_\{0\}\\omega\}\\,T\}\\left\(\\mathbb\{E\}\[\\mathcal\{E\}^\{0\}\]\+\\chi\_\{1\}B\_\{h\}\\gamma^\{2\}\+\\chi\_\{2\}\\vartheta\_\{0\}^\{2\}\(1\+\\log T\)\\right\)^\{1/2\}\.\(66\)Therefore, the one\-step empirical agnostic mixture\-risk perturbation caused by stale, sparse, quantized, and event\-triggered neighbor\-model exchange satisfies

\|𝔼⁡\[Ψ^​\(𝐰τ\+1\)\]−𝔼⁡\[Ψ^​\(𝐰exτ\+1\)\]\|=𝒪⁡\(log⁡TT\),\\left\|\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}^\{\\tau\+1\}\)\\right\]\-\\mathbb\{E\}\\\!\\left\[\\widehat\{\\Psi\}\(\\mathbf\{w\}\_\{\\mathrm\{ex\}\}^\{\\tau\+1\}\)\\right\]\\right\|=\\mathcal\{O\}\\\!\\left\(\\frac\{\\sqrt\{\\log T\}\}\{T\}\\right\),and hence vanishes asymptotically asT→∞T\\to\\infty\. This proves the corollary\.□\\square

## Appendix BProof of Theorem[2](https://arxiv.org/html/2608.26493#Thmtheorem2)\(Convergence Analysis\)

We prove Theorem[2](https://arxiv.org/html/2608.26493#Thmtheorem2)using the notation introduced in Section[V](https://arxiv.org/html/2608.26493#S5)\. For eachtt, define the proximal point and the associated Moreau\-envelope gradient as𝐰~t:=proxμ​ℒ⁡\(𝐰t\)\\widetilde\{\\mathbf\{w\}\}^\{t\}:=\\operatorname\{prox\}\_\{\\mu\\mathcal\{L\}\}\(\\mathbf\{w\}^\{t\}\),𝐬t:=1μ​\(𝐰t−𝐰~t\)=∇ℒμ​\(𝐰t\)\\mathbf\{s\}^\{t\}:=\\frac\{1\}\{\\mu\}\\left\(\\mathbf\{w\}^\{t\}\-\\widetilde\{\\mathbf\{w\}\}^\{t\}\\right\)=\\nabla\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\}\)\.

### B\-AProof of Lemma[4](https://arxiv.org/html/2608.26493#Thmlemma4)

Combining Lemma[2](https://arxiv.org/html/2608.26493#Thmlemma2)with theLL\-weak convexity of∑i=1nFi​\(wi\)\\sum\_\{i=1\}^\{n\}F\_\{i\}\(w\_\{i\}\)and the convexity ofℛ𝒢\\mathcal\{R\}\_\{\\mathcal\{G\}\}gives, for every𝐮∈ℝn​d\\mathbf\{u\}\\in\\mathbb\{R\}^\{nd\},

ℒ⁡\(𝐮\)\\displaystyle\\mathcal\{L\}\(\\mathbf\{u\}\)≥ℒ⁡\(𝐰t\)\+⟨𝐕t,𝐮−𝐰t⟩−κ2​‖𝐮−𝐰t‖2−ρ​εΨ,t,\\displaystyle\\geq\\mathcal\{L\}\(\\mathbf\{w\}^\{t\}\)\+\\left\\langle\\mathbf\{V\}^\{t\},\\mathbf\{u\}\-\\mathbf\{w\}^\{t\}\\right\\rangle\-\\frac\{\\kappa\}\{2\}\\\|\\mathbf\{u\}\-\\mathbf\{w\}^\{t\}\\\|^\{2\}\-\\rho\\varepsilon\_\{\\Psi,t\},\(67\)whereκ=\(1\+ρ\)​L\\kappa=\(1\+\\rho\)Land𝐕t\\mathbf\{V\}^\{t\}is defined in Assumption[5](https://arxiv.org/html/2608.26493#Thmassmon5)\. Applying \([67](https://arxiv.org/html/2608.26493#A2.E67)\) with𝐮=𝐰~t\\mathbf\{u\}=\\widetilde\{\\mathbf\{w\}\}^\{\\,t\}gives

ℒ⁡\(𝐰~t\)\\displaystyle\\mathcal\{L\}\(\\widetilde\{\\mathbf\{w\}\}^\{\\,t\}\)≥ℒ⁡\(𝐰t\)\+⟨𝐕t,𝐰~t−𝐰t⟩\\displaystyle\\geq\\mathcal\{L\}\(\\mathbf\{w\}^\{t\}\)\+\\left\\langle\\mathbf\{V\}^\{t\},\\widetilde\{\\mathbf\{w\}\}^\{\\,t\}\-\\mathbf\{w\}^\{t\}\\right\\rangle−κ2​‖𝐰~t−𝐰t‖2−ρ​εΨ,t\.\\displaystyle\\quad\-\\frac\{\\kappa\}\{2\}\\left\\\|\\widetilde\{\\mathbf\{w\}\}^\{\\,t\}\-\\mathbf\{w\}^\{t\}\\right\\\|^\{2\}\-\\rho\\varepsilon\_\{\\Psi,t\}\.\(68\)On the other hand, by the definition of the proximal point and by using𝐰t\\mathbf\{w\}^\{t\}as a feasible candidate,

ℒ⁡\(𝐰~t\)\+12​μ​‖𝐰~t−𝐰t‖2≤ℒ⁡\(𝐰t\)\.\\mathcal\{L\}\(\\widetilde\{\\mathbf\{w\}\}^\{\\,t\}\)\+\\frac\{1\}\{2\\mu\}\\left\\\|\\widetilde\{\\mathbf\{w\}\}^\{\\,t\}\-\\mathbf\{w\}^\{t\}\\right\\\|^\{2\}\\leq\\mathcal\{L\}\(\\mathbf\{w\}^\{t\}\)\.\(69\)Therefore,

ℒ⁡\(𝐰t\)−ℒ⁡\(𝐰~t\)≥12​μ​‖𝐰~t−𝐰t‖2\.\\mathcal\{L\}\(\\mathbf\{w\}^\{t\}\)\-\\mathcal\{L\}\(\\widetilde\{\\mathbf\{w\}\}^\{\\,t\}\)\\geq\\frac\{1\}\{2\\mu\}\\left\\\|\\widetilde\{\\mathbf\{w\}\}^\{\\,t\}\-\\mathbf\{w\}^\{t\}\\right\\\|^\{2\}\.\(70\)
Rearranging \([68](https://arxiv.org/html/2608.26493#A2.E68)\) and using \([70](https://arxiv.org/html/2608.26493#A2.E70)\), we obtain

⟨𝐕t,𝐰t−𝐰~t⟩\\displaystyle\\left\\langle\\mathbf\{V\}^\{t\},\\mathbf\{w\}^\{t\}\-\\widetilde\{\\mathbf\{w\}\}^\{\\,t\}\\right\\rangle≥\(12​μ−κ2\)​‖𝐰t−𝐰~t‖2−ρ​εΨ,t\.\\displaystyle\\geq\\left\(\\frac\{1\}\{2\\mu\}\-\\frac\{\\kappa\}\{2\}\\right\)\\left\\\|\\mathbf\{w\}^\{t\}\-\\widetilde\{\\mathbf\{w\}\}^\{\\,t\}\\right\\\|^\{2\}\-\\rho\\varepsilon\_\{\\Psi,t\}\.\(71\)Substituting𝐰t−𝐰~t=μ​𝐬t\\mathbf\{w\}^\{t\}\-\\widetilde\{\\mathbf\{w\}\}^\{\\,t\}=\\mu\\mathbf\{s\}^\{t\}yields

⟨𝐕t,𝐬t⟩≥1−κ​μ2​‖𝐬t‖2−ρμ​εΨ,t\.\\left\\langle\\mathbf\{V\}^\{t\},\\mathbf\{s\}^\{t\}\\right\\rangle\\geq\\frac\{1\-\\kappa\\mu\}\{2\}\\\|\\mathbf\{s\}^\{t\}\\\|^\{2\}\-\\frac\{\\rho\}\{\\mu\}\\varepsilon\_\{\\Psi,t\}\.\(72\)
The vector form of the algorithmic update is

𝐰t\+1=𝐰t−γt​𝐡t,𝐡t=𝐕t\+𝐪t\+𝐝t\.\\mathbf\{w\}^\{t\+1\}=\\mathbf\{w\}^\{t\}\-\\gamma\_\{t\}\\mathbf\{h\}^\{t\},\\qquad\\mathbf\{h\}^\{t\}=\\mathbf\{V\}^\{t\}\+\\mathbf\{q\}^\{t\}\+\\mathbf\{d\}^\{t\}\.\(73\)Using𝐰~t\\widetilde\{\\mathbf\{w\}\}^\{\\,t\}as a feasible candidate in the definition ofℒμ​\(𝐰t\+1\)\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\+1\}\), we have

ℒμ​\(𝐰t\+1\)\\displaystyle\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\+1\}\)≤ℒ⁡\(𝐰~t\)\+12​μ​‖𝐰~t−𝐰t\+1‖2\\displaystyle\\leq\\mathcal\{L\}\(\\widetilde\{\\mathbf\{w\}\}^\{\\,t\}\)\+\\frac\{1\}\{2\\mu\}\\left\\\|\\widetilde\{\\mathbf\{w\}\}^\{\\,t\}\-\\mathbf\{w\}^\{t\+1\}\\right\\\|^\{2\}=ℒ⁡\(𝐰~t\)\+12​μ​‖−μ​𝐬t\+γt​𝐡t‖2\\displaystyle=\\mathcal\{L\}\(\\widetilde\{\\mathbf\{w\}\}^\{\\,t\}\)\+\\frac\{1\}\{2\\mu\}\\left\\\|\-\\mu\\mathbf\{s\}^\{t\}\+\\gamma\_\{t\}\\mathbf\{h\}^\{t\}\\right\\\|^\{2\}=ℒμ​\(𝐰t\)−γt​⟨𝐬t,𝐡t⟩\+γt22​μ​‖𝐡t‖2\.\\displaystyle=\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\}\)\-\\gamma\_\{t\}\\left\\langle\\mathbf\{s\}^\{t\},\\mathbf\{h\}^\{t\}\\right\\rangle\+\\frac\{\\gamma\_\{t\}^\{2\}\}\{2\\mu\}\\\|\\mathbf\{h\}^\{t\}\\\|^\{2\}\.\(74\)
Recall thatℋt\\mathcal\{H\}\_\{t\}contains the historyℱt\\mathcal\{F\}\_\{t\}and the fairness\-selection mini\-batches at roundtt\. Conditional onℋt\\mathcal\{H\}\_\{t\}, the quantities𝐕t\\mathbf\{V\}^\{t\},𝐝t\\mathbf\{d\}^\{t\},𝐬t\\mathbf\{s\}^\{t\}, andεΨ,t\\varepsilon\_\{\\Psi,t\}are measurable, while𝔼⁡\[𝐪t\|ℋt\]=𝟎\.\\mathbb\{E\}\\left\[\\mathbf\{q\}^\{t\}\\,\\middle\|\\,\\mathcal\{H\}\_\{t\}\\right\]=\\mathbf\{0\}\.Consequently,𝔼⁡\[⟨𝐬t,𝐪t⟩\|ℋt\]=0\\mathbb\{E\}\\left\[\\left\\langle\\mathbf\{s\}^\{t\},\\mathbf\{q\}^\{t\}\\right\\rangle\\,\\middle\|\\,\\mathcal\{H\}\_\{t\}\\right\]=0\. Taking the conditional expectation of \([74](https://arxiv.org/html/2608.26493#A2.E74)\) and using \([72](https://arxiv.org/html/2608.26493#A2.E72)\) gives

𝔼⁡\[ℒμ​\(𝐰t\+1\)\|ℋt\]\\displaystyle\\mathbb\{E\}\\left\[\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\+1\}\)\\,\\middle\|\\,\\mathcal\{H\}\_\{t\}\\right\]≤ℒμ​\(𝐰t\)−1−κ​μ2​γt​‖𝐬t‖2\+ρ​γtμ​εΨ,t\\displaystyle\\leq\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\}\)\-\\frac\{1\-\\kappa\\mu\}\{2\}\\gamma\_\{t\}\\\|\\mathbf\{s\}^\{t\}\\\|^\{2\}\+\\frac\{\\rho\\gamma\_\{t\}\}\{\\mu\}\\varepsilon\_\{\\Psi,t\}−γt​⟨𝐬t,𝐝t⟩\+γt22​μ​𝔼​\[‖𝐡t‖2\|ℋt\]\.\\displaystyle\-\\gamma\_\{t\}\\left\\langle\\mathbf\{s\}^\{t\},\\mathbf\{d\}^\{t\}\\right\\rangle\+\\frac\{\\gamma\_\{t\}^\{2\}\}\{2\\mu\}\\mathbb\{E\}\\left\[\\\|\\mathbf\{h\}^\{t\}\\\|^\{2\}\\,\\middle\|\\,\\mathcal\{H\}\_\{t\}\\right\]\.\(75\)
By Young’s inequality,

−⟨𝐬t,𝐝t⟩≤1−κ​μ4​‖𝐬t‖2\+11−κ​μ​‖𝐝t‖2\.\-\\left\\langle\\mathbf\{s\}^\{t\},\\mathbf\{d\}^\{t\}\\right\\rangle\\leq\\frac\{1\-\\kappa\\mu\}\{4\}\\\|\\mathbf\{s\}^\{t\}\\\|^\{2\}\+\\frac\{1\}\{1\-\\kappa\\mu\}\\\|\\mathbf\{d\}^\{t\}\\\|^\{2\}\.\(76\)Combining this with \([75](https://arxiv.org/html/2608.26493#A2.E75)\) yields

𝔼⁡\[ℒμ​\(𝐰t\+1\)\|ℋt\]\\displaystyle\\mathbb\{E\}\\left\[\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\+1\}\)\\,\\middle\|\\,\\mathcal\{H\}\_\{t\}\\right\]≤ℒμ​\(𝐰t\)−1−κ​μ4​γt​‖𝐬t‖2\+ρ​γtμ​εΨ,t\\displaystyle\\leq\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\}\)\-\\frac\{1\-\\kappa\\mu\}\{4\}\\gamma\_\{t\}\\\|\\mathbf\{s\}^\{t\}\\\|^\{2\}\+\\frac\{\\rho\\gamma\_\{t\}\}\{\\mu\}\\varepsilon\_\{\\Psi,t\}\+γt1−κ​μ​‖𝐝t‖2\+γt22​μ​𝔼​\[‖𝐡t‖2\|ℋt\]\.\\displaystyle\\quad\+\\frac\{\\gamma\_\{t\}\}\{1\-\\kappa\\mu\}\\\|\\mathbf\{d\}^\{t\}\\\|^\{2\}\+\\frac\{\\gamma\_\{t\}^\{2\}\}\{2\\mu\}\\mathbb\{E\}\\left\[\\\|\\mathbf\{h\}^\{t\}\\\|^\{2\}\\,\\middle\|\\,\\mathcal\{H\}\_\{t\}\\right\]\.\(77\)
Using𝐡t=𝐕t\+𝐝t\+𝐪t\\mathbf\{h\}^\{t\}=\\mathbf\{V\}^\{t\}\+\\mathbf\{d\}^\{t\}\+\\mathbf\{q\}^\{t\}and the conditional zero\-mean property of𝐪t\\mathbf\{q\}^\{t\}, we obtain

𝔼⁡\[‖𝐡t‖2\|ℋt\]\\displaystyle\\mathbb\{E\}\\left\[\\\|\\mathbf\{h\}^\{t\}\\\|^\{2\}\\,\\middle\|\\,\\mathcal\{H\}\_\{t\}\\right\]=‖𝐕t\+𝐝t‖2\+𝔼⁡\[‖𝐪t‖2\|ℋt\]\\displaystyle=\\\|\\mathbf\{V\}^\{t\}\+\\mathbf\{d\}^\{t\}\\\|^\{2\}\+\\mathbb\{E\}\\left\[\\\|\\mathbf\{q\}^\{t\}\\\|^\{2\}\\,\\middle\|\\,\\mathcal\{H\}\_\{t\}\\right\]≤2​‖𝐕t‖2\+2​‖𝐝t‖2\+𝔼⁡\[‖𝐪t‖2\|ℋt\]\.\\displaystyle\\leq 2\\\|\\mathbf\{V\}^\{t\}\\\|^\{2\}\+2\\\|\\mathbf\{d\}^\{t\}\\\|^\{2\}\+\\mathbb\{E\}\\left\[\\\|\\mathbf\{q\}^\{t\}\\\|^\{2\}\\,\\middle\|\\,\\mathcal\{H\}\_\{t\}\\right\]\.\(78\)Furthermore, by Assumption[2](https://arxiv.org/html/2608.26493#Thmassmon2),

𝔼⁡\[‖𝐪t‖2\|ℋt\]\\displaystyle\\mathbb\{E\}\\left\[\\\|\\mathbf\{q\}^\{t\}\\\|^\{2\}\\,\\middle\|\\,\\mathcal\{H\}\_\{t\}\\right\]=∑i=1n\(1\+ρ​λ^it\)2​𝔼​\[‖git−∇Fi​\(wit\)‖2\|ℋt\]\\displaystyle=\\sum\_\{i=1\}^\{n\}\\left\(1\+\\rho\\widehat\{\\lambda\}\_\{i\}^\{t\}\\right\)^\{2\}\\mathbb\{E\}\\left\[\\left\\\|g\_\{i\}^\{t\}\-\\nabla F\_\{i\}\(w\_\{i\}^\{t\}\)\\right\\\|^\{2\}\\,\\middle\|\\,\\mathcal\{H\}\_\{t\}\\right\]≤\(1\+ρ\)2​∑i=1nσi2\\displaystyle\\leq\(1\+\\rho\)^\{2\}\\sum\_\{i=1\}^\{n\}\\sigma\_\{i\}^\{2\}=n​\(1\+ρ\)2​σ¯2\.\\displaystyle=n\(1\+\\rho\)^\{2\}\\bar\{\\sigma\}^\{2\}\.\(79\)
Substituting \([78](https://arxiv.org/html/2608.26493#A2.E78)\) and \([79](https://arxiv.org/html/2608.26493#A2.E79)\) into \([77](https://arxiv.org/html/2608.26493#A2.E77)\) gives

𝔼⁡\[ℒμ​\(𝐰t\+1\)\|ℋt\]\\displaystyle\\mathbb\{E\}\\left\[\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\+1\}\)\\,\\middle\|\\,\\mathcal\{H\}\_\{t\}\\right\]≤ℒμ​\(𝐰t\)−1−κ​μ4​γt​‖𝐬t‖2\+ρ​γtμ​εΨ,t\\displaystyle\\leq\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\}\)\-\\frac\{1\-\\kappa\\mu\}\{4\}\\gamma\_\{t\}\\\|\\mathbf\{s\}^\{t\}\\\|^\{2\}\+\\frac\{\\rho\\gamma\_\{t\}\}\{\\mu\}\\varepsilon\_\{\\Psi,t\}\+\(γt1−κ​μ\+γt2μ\)​‖𝐝t‖2\+γt2μ​‖𝐕t‖2\\displaystyle\\quad\+\\left\(\\frac\{\\gamma\_\{t\}\}\{1\-\\kappa\\mu\}\+\\frac\{\\gamma\_\{t\}^\{2\}\}\{\\mu\}\\right\)\\\|\\mathbf\{d\}^\{t\}\\\|^\{2\}\+\\frac\{\\gamma\_\{t\}^\{2\}\}\{\\mu\}\\\|\\mathbf\{V\}^\{t\}\\\|^\{2\}\+n​\(1\+ρ\)2​γt22​μ​σ¯2\.\\displaystyle\\quad\+\\frac\{n\(1\+\\rho\)^\{2\}\\gamma\_\{t\}^\{2\}\}\{2\\mu\}\\bar\{\\sigma\}^\{2\}\.\(80\)
Finally, taking the total expectation, using𝔼​‖𝐕t‖2≤H2\\mathbb\{E\}\\\|\\mathbf\{V\}^\{t\}\\\|^\{2\}\\leq H^\{2\}, and recalling that𝐬t=∇ℒμ​\(𝐰t\)\\mathbf\{s\}^\{t\}=\\nabla\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\}\)proves \([43](https://arxiv.org/html/2608.26493#S5.E43)\)\.

### B\-BProof of Lemma[5](https://arxiv.org/html/2608.26493#Thmlemma5)

Letd¯𝒢:=max⁡∑j∈𝒩ii⁡ai​j\\bar\{d\}\_\{\\mathcal\{G\}\}:=\\max\_\{i\}\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}a\_\{ij\}denote the maximum weighted degree\. Using the communication\-copy erroreit=wit−w^ite\_\{i\}^\{t\}=w\_\{i\}^\{t\}\-\\widehat\{w\}\_\{i\}^\{t\}, the common\-copy property of the broadcast communication mechanism gives

ℰt=1n∑i=1ndi𝒢∥eit∥2,di𝒢:=∑j:i∈𝒩jaj​i≤d¯𝒢\.\\mathcal\{E\}^\{t\}=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}d\_\{i\}^\{\\mathcal\{G\}\}\\\|e\_\{i\}^\{t\}\\\|^\{2\},\\qquad d\_\{i\}^\{\\mathcal\{G\}\}:=\\sum\_\{j:i\\in\\mathcal\{N\}\_\{j\}\}a\_\{ji\}\\leq\\bar\{d\}\_\{\\mathcal\{G\}\}\.\(81\)
After the local update, the innovation relative to the most recently reconstructed copy is

sit\+1=wit\+1−w^it=eit−γt​hit\.s\_\{i\}^\{t\+1\}=w\_\{i\}^\{t\+1\}\-\\widehat\{w\}\_\{i\}^\{t\}=e\_\{i\}^\{t\}\-\\gamma\_\{t\}h\_\{i\}^\{t\}\.\(82\)
Suppose first that the event\-trigger condition is satisfied\. Clientiitransmitscit=𝒞⁡\(sit\+1\),c\_\{i\}^\{t\}=\\mathcal\{C\}\(s\_\{i\}^\{t\+1\}\),and henceeit\+1=sit\+1−𝒞⁡\(sit\+1\)e\_\{i\}^\{t\+1\}=s\_\{i\}^\{t\+1\}\-\\mathcal\{C\}\(s\_\{i\}^\{t\+1\}\)\. Assumption[4](https://arxiv.org/html/2608.26493#Thmassmon4)therefore gives

𝔼⁡\[‖eit\+1‖2\|sit\+1\]≤\(1−ω\)​‖sit\+1‖2\.\\mathbb\{E\}\\left\[\\\|e\_\{i\}^\{t\+1\}\\\|^\{2\}\\,\\middle\|\\,s\_\{i\}^\{t\+1\}\\right\]\\leq\(1\-\\omega\)\\\|s\_\{i\}^\{t\+1\}\\\|^\{2\}\.\(83\)For0<ω<10<\\omega<1, applying Young’s inequality withη=ω/\[2​\(1−ω\)\]\\eta=\\omega/\[2\(1\-\\omega\)\]yields

\(1−ω\)​‖eit−γt​hit‖2\\displaystyle\(1\-\\omega\)\\\|e\_\{i\}^\{t\}\-\\gamma\_\{t\}h\_\{i\}^\{t\}\\\|^\{2\}≤\(1−ω2\)​‖eit‖2\+2​γt2ω​‖hit‖2\.\\displaystyle\\leq\\left\(1\-\\frac\{\\omega\}\{2\}\\right\)\\\|e\_\{i\}^\{t\}\\\|^\{2\}\+\\frac\{2\\gamma\_\{t\}^\{2\}\}\{\\omega\}\\\|h\_\{i\}^\{t\}\\\|^\{2\}\.\(84\)The same inequality holds trivially whenω=1\\omega=1\. Consequently, in the triggered case,

𝔼​‖eit\+1‖2\\displaystyle\\mathbb\{E\}\\\|e\_\{i\}^\{t\+1\}\\\|^\{2\}≤\(1−ω2\)​𝔼​‖eit‖2\+2​γt2ω​𝔼​‖hit‖2\.\\displaystyle\\leq\\left\(1\-\\frac\{\\omega\}\{2\}\\right\)\\mathbb\{E\}\\\|e\_\{i\}^\{t\}\\\|^\{2\}\+\\frac\{2\\gamma\_\{t\}^\{2\}\}\{\\omega\}\\mathbb\{E\}\\\|h\_\{i\}^\{t\}\\\|^\{2\}\.\(85\)
If the event\-trigger condition is not satisfied, no message is sent, so thateit\+1=sit\+1e\_\{i\}^\{t\+1\}=s\_\{i\}^\{t\+1\}\. The trigger rule implies‖eit\+1‖2=‖sit\+1‖2<ϑt2\\\|e\_\{i\}^\{t\+1\}\\\|^\{2\}=\\\|s\_\{i\}^\{t\+1\}\\\|^\{2\}<\\vartheta\_\{t\}^\{2\}\. Since the remaining terms below are nonnegative, this further gives

‖eit\+1‖2\\displaystyle\\\|e\_\{i\}^\{t\+1\}\\\|^\{2\}≤\(1−ω2\)​‖eit‖2\+2​γt2ω​‖hit‖2\+ϑt2\.\\displaystyle\\leq\\left\(1\-\\frac\{\\omega\}\{2\}\\right\)\\\|e\_\{i\}^\{t\}\\\|^\{2\}\+\\frac\{2\\gamma\_\{t\}^\{2\}\}\{\\omega\}\\\|h\_\{i\}^\{t\}\\\|^\{2\}\+\\vartheta\_\{t\}^\{2\}\.\(86\)
Combining the triggered and non\-triggered cases, multiplying bydi𝒢d\_\{i\}^\{\\mathcal\{G\}\}, summing over the clients, and usingdi𝒢≤d¯𝒢d\_\{i\}^\{\\mathcal\{G\}\}\\leq\\bar\{d\}\_\{\\mathcal\{G\}\}gives

𝔼⁡\[ℰt\+1\]\\displaystyle\\mathbb\{E\}\[\\mathcal\{E\}^\{t\+1\}\]≤\(1−ω2\)​𝔼​\[ℰt\]\+2​d¯𝒢ω​γt2​1n​𝔼​‖𝐡t‖2\+d¯𝒢​ϑt2\.\\displaystyle\\leq\\left\(1\-\\frac\{\\omega\}\{2\}\\right\)\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]\+\\frac\{2\\bar\{d\}\_\{\\mathcal\{G\}\}\}\{\\omega\}\\gamma\_\{t\}^\{2\}\\frac\{1\}\{n\}\\mathbb\{E\}\\\|\\mathbf\{h\}^\{t\}\\\|^\{2\}\+\\bar\{d\}\_\{\\mathcal\{G\}\}\\vartheta\_\{t\}^\{2\}\.\(87\)Thus, \([44](https://arxiv.org/html/2608.26493#S5.E44)\) holds withχ0=12,χ1=2​d¯𝒢ω,χ2=d¯𝒢\\chi\_\{0\}=\\frac\{1\}\{2\},\\quad\\chi\_\{1\}=\\frac\{2\\bar\{d\}\_\{\\mathcal\{G\}\}\}\{\\omega\},\\quad\\chi\_\{2\}=\\bar\{d\}\_\{\\mathcal\{G\}\}\.

It remains to relate the graph\-direction approximation error to the communication residual\. From \([40](https://arxiv.org/html/2608.26493#S5.E40)\) and the weighted Cauchy\-Schwarz inequality,

‖dit‖2\\displaystyle\\\|d\_\{i\}^\{t\}\\\|^\{2\}=α2​‖∑j∈𝒩iai​j​\(wjt−w^j→it\)‖2\\displaystyle=\\alpha^\{2\}\\left\\\|\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}a\_\{ij\}\\left\(w\_\{j\}^\{t\}\-\\widehat\{w\}\_\{j\\to i\}^\{t\}\\right\)\\right\\\|^\{2\}≤α2​\(∑j∈𝒩iai​j\)​∑j∈𝒩iai​j​‖wjt−w^j→it‖2\\displaystyle\\leq\\alpha^\{2\}\\left\(\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}a\_\{ij\}\\right\)\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}a\_\{ij\}\\left\\\|w\_\{j\}^\{t\}\-\\widehat\{w\}\_\{j\\to i\}^\{t\}\\right\\\|^\{2\}≤α2​d¯𝒢​∑j∈𝒩iai​j​‖wjt−w^j→it‖2\.\\displaystyle\\leq\\alpha^\{2\}\\bar\{d\}\_\{\\mathcal\{G\}\}\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}a\_\{ij\}\\left\\\|w\_\{j\}^\{t\}\-\\widehat\{w\}\_\{j\\to i\}^\{t\}\\right\\\|^\{2\}\.\(88\)Summing overiiyields‖𝐝t‖2≤n​α2​d¯𝒢​ℰt\\\|\\mathbf\{d\}^\{t\}\\\|^\{2\}\\leq n\\alpha^\{2\}\\bar\{d\}\_\{\\mathcal\{G\}\}\\mathcal\{E\}^\{t\}\. Therefore, \([45](https://arxiv.org/html/2608.26493#S5.E45)\) holds withK𝒢=α2​d¯𝒢K\_\{\\mathcal\{G\}\}=\\alpha^\{2\}\\bar\{d\}\_\{\\mathcal\{G\}\}\. This completes the proof\.□\\square

### B\-CProof of Lemma[6](https://arxiv.org/html/2608.26493#Thmlemma6)

From \([47](https://arxiv.org/html/2608.26493#S5.E47)\),

𝔼⁡\[ℰt\+1\]\\displaystyle\\mathbb\{E\}\[\\mathcal\{E\}^\{t\+1\}\]≤\(1−χ0​ω\+2​χ1​K𝒢​γt2\)​𝔼​\[ℰt\]\+χ1​Bh​γt2\+χ2​ϑt2\.\\displaystyle\\leq\\left\(1\-\\chi\_\{0\}\\omega\+2\\chi\_\{1\}K\_\{\\mathcal\{G\}\}\\gamma\_\{t\}^\{2\}\\right\)\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]\+\\chi\_\{1\}B\_\{h\}\\gamma\_\{t\}^\{2\}\+\\chi\_\{2\}\\vartheta\_\{t\}^\{2\}\.Sinceγt2=γ2/T≤γ2\\gamma\_\{t\}^\{2\}=\\gamma^\{2\}/T\\leq\\gamma^\{2\}and2​χ1​K𝒢​γ2≤χ0​ω/22\\chi\_\{1\}K\_\{\\mathcal\{G\}\}\\gamma^\{2\}\\leq\\chi\_\{0\}\\omega/2, we have

1−χ0​ω\+2​χ1​K𝒢​γt2≤1−χ0​ω2\.1\-\\chi\_\{0\}\\omega\+2\\chi\_\{1\}K\_\{\\mathcal\{G\}\}\\gamma\_\{t\}^\{2\}\\leq 1\-\\frac\{\\chi\_\{0\}\\omega\}\{2\}\.Therefore,

𝔼⁡\[ℰt\+1\]\\displaystyle\\mathbb\{E\}\[\\mathcal\{E\}^\{t\+1\}\]≤\(1−χ0​ω2\)​𝔼​\[ℰt\]\+χ1​Bh​γt2\+χ2​ϑt2\.\\displaystyle\\leq\\left\(1\-\\frac\{\\chi\_\{0\}\\omega\}\{2\}\\right\)\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]\+\\chi\_\{1\}B\_\{h\}\\gamma\_\{t\}^\{2\}\+\\chi\_\{2\}\\vartheta\_\{t\}^\{2\}\.\(89\)
Rearranging and summing overt=0,…,T−1t=0,\\ldots,T\-1gives

χ0​ω2​∑t=0T−1𝔼⁡\[ℰt\]\\displaystyle\\frac\{\\chi\_\{0\}\\omega\}\{2\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]≤𝔼⁡\[ℰ0\]−𝔼⁡\[ℰT\]\+χ1​Bh​∑t=0T−1γt2\+χ2​∑t=0T−1ϑt2\\displaystyle\\leq\\mathbb\{E\}\[\\mathcal\{E\}^\{0\}\]\-\\mathbb\{E\}\[\\mathcal\{E\}^\{T\}\]\+\\chi\_\{1\}B\_\{h\}\\sum\_\{t=0\}^\{T\-1\}\\gamma\_\{t\}^\{2\}\+\\chi\_\{2\}\\sum\_\{t=0\}^\{T\-1\}\\vartheta\_\{t\}^\{2\}≤𝔼⁡\[ℰ0\]\+χ1​Bh​γ2\+χ2​ϑ02​\(1\+log⁡T\),\\displaystyle\\leq\\mathbb\{E\}\[\\mathcal\{E\}^\{0\}\]\+\\chi\_\{1\}B\_\{h\}\\gamma^\{2\}\+\\chi\_\{2\}\\vartheta\_\{0\}^\{2\}\(1\+\\log T\),\(90\)where we used∑t=0T−1γt2=γ2\\sum\_\{t=0\}^\{T\-1\}\\gamma\_\{t\}^\{2\}=\\gamma^\{2\}and∑t=0T−1\(t\+1\)−1≤1\+log⁡T\\sum\_\{t=0\}^\{T\-1\}\(t\+1\)^\{\-1\}\\leq 1\+\\log T\. Dividing \([90](https://arxiv.org/html/2608.26493#A2.E90)\) byχ0​ω​T/2\\chi\_\{0\}\\omega T/2yields

1T​∑t=0T−1𝔼⁡\[ℰt\]\\displaystyle\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]≤2χ0​ω​T​\[𝔼⁡\[ℰ0\]\+χ1​Bh​γ2\+χ2​ϑ02​\(1\+log⁡T\)\],\\displaystyle\\leq\\frac\{2\}\{\\chi\_\{0\}\\omega T\}\\left\[\\mathbb\{E\}\[\\mathcal\{E\}^\{0\}\]\+\\chi\_\{1\}B\_\{h\}\\gamma^\{2\}\+\\chi\_\{2\}\\vartheta\_\{0\}^\{2\}\(1\+\\log T\)\\right\],\(91\)which proves \([49](https://arxiv.org/html/2608.26493#S5.E49)\)\.□\\square

### B\-DProof of Theorem[2](https://arxiv.org/html/2608.26493#Thmtheorem2)

Summing the one\-step inequality \([43](https://arxiv.org/html/2608.26493#S5.E43)\) overt=0,…,T−1t=0,\\ldots,T\-1and usingℒμ​\(𝐰T\)≥ℒ⋆\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{T\}\)\\geq\\mathcal\{L\}^\{\\star\}gives

1−κ​μ4​∑t=0T−1γt​𝔼​\[‖∇ℒμ​\(𝐰t\)‖2\]\\displaystyle\\frac\{1\-\\kappa\\mu\}\{4\}\\sum\_\{t=0\}^\{T\-1\}\\gamma\_\{t\}\\mathbb\{E\}\\left\[\\left\\\|\\nabla\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\}\)\\right\\\|^\{2\}\\right\]≤Δμ0\+ρμ​∑t=0T−1γt​𝔼​\[εΨ,t\]\\displaystyle\\leq\\Delta\_\{\\mu\}^\{0\}\+\\frac\{\\rho\}\{\\mu\}\\sum\_\{t=0\}^\{T\-1\}\\gamma\_\{t\}\\mathbb\{E\}\[\\varepsilon\_\{\\Psi,t\}\]\+∑t=0T−1\(γt1−κ​μ\+γt2μ\)𝔼∥𝐝t∥2\\displaystyle\\quad\+\\sum\_\{t=0\}^\{T\-1\}\\left\(\\frac\{\\gamma\_\{t\}\}\{1\-\\kappa\\mu\}\+\\frac\{\\gamma\_\{t\}^\{2\}\}\{\\mu\}\\right\)\\mathbb\{E\}\\\|\\mathbf\{d\}^\{t\}\\\|^\{2\}\+n​\(1\+ρ\)2​σ¯22​μ∑t=0T−1γt2\\displaystyle\\quad\+\\frac\{n\(1\+\\rho\)^\{2\}\\bar\{\\sigma\}^\{2\}\}\{2\\mu\}\\sum\_\{t=0\}^\{T\-1\}\\gamma\_\{t\}^\{2\}\+H2μ∑t=0T−1γt2,\\displaystyle\\quad\+\\frac\{H^\{2\}\}\{\\mu\}\\sum\_\{t=0\}^\{T\-1\}\\gamma\_\{t\}^\{2\},\(92\)whereΔμ0=ℒμ​\(𝐰0\)−ℒ⋆\\Delta\_\{\\mu\}^\{0\}=\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{0\}\)\-\\mathcal\{L\}^\{\\star\}\.

Sinceγt=γ/T\\gamma\_\{t\}=\\gamma/\\sqrt\{T\}, dividing \([92](https://arxiv.org/html/2608.26493#A2.E92)\) byn​T​γt​\(1−κ​μ\)/4nT\\gamma\_\{t\}\(1\-\\kappa\\mu\)/4and using𝒮μt=n−1​‖∇ℒμ​\(𝐰t\)‖2\\mathcal\{S\}\_\{\\mu\}^\{t\}=n^\{\-1\}\\\|\\nabla\\mathcal\{L\}\_\{\\mu\}\(\\mathbf\{w\}^\{t\}\)\\\|^\{2\}yields

1T​∑t=0T−1𝔼⁡\[𝒮μt\]\\displaystyle\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\mathcal\{S\}\_\{\\mu\}^\{t\}\]≤4​Δμ0n⁡\(1−κ​μ\)​γ​T\+4​ρn​μ​\(1−κ​μ\)​1T​∑t=0T−1𝔼⁡\[εΨ,t\]\\displaystyle\\leq\\frac\{4\\Delta\_\{\\mu\}^\{0\}\}\{n\(1\-\\kappa\\mu\)\\gamma\\sqrt\{T\}\}\+\\frac\{4\\rho\}\{n\\mu\(1\-\\kappa\\mu\)\}\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\varepsilon\_\{\\Psi,t\}\]\+4n⁡\(1−κ​μ\)\(11−κ​μ\+γμ​T\)1T∑t=0T−1𝔼∥𝐝t∥2\\displaystyle\\quad\+\\frac\{4\}\{n\(1\-\\kappa\\mu\)\}\\left\(\\frac\{1\}\{1\-\\kappa\\mu\}\+\\frac\{\\gamma\}\{\\mu\\sqrt\{T\}\}\\right\)\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\\\|\\mathbf\{d\}^\{t\}\\\|^\{2\}\+4​γ​H2n​μ​\(1−κ​μ\)​T\+2​γ​\(1\+ρ\)2​σ¯2μ⁡\(1−κ​μ\)​T\.\\displaystyle\\quad\+\\frac\{4\\gamma H^\{2\}\}\{n\\mu\(1\-\\kappa\\mu\)\\sqrt\{T\}\}\+\\frac\{2\\gamma\(1\+\\rho\)^\{2\}\\bar\{\\sigma\}^\{2\}\}\{\\mu\(1\-\\kappa\\mu\)\\sqrt\{T\}\}\.\(93\)
By the tower property and Assumption[6](https://arxiv.org/html/2608.26493#Thmassmon6),

1T​∑t=0T−1𝔼⁡\[εΨ,t\]≤BΨT\.\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\varepsilon\_\{\\Psi,t\}\]\\leq\\frac\{B\_\{\\Psi\}\}\{\\sqrt\{T\}\}\.\(94\)Moreover, Lemma[5](https://arxiv.org/html/2608.26493#Thmlemma5)gives‖𝐝t‖2≤n​K𝒢​ℰt\\\|\\mathbf\{d\}^\{t\}\\\|^\{2\}\\leq nK\_\{\\mathcal\{G\}\}\\mathcal\{E\}^\{t\}, and hence

1T​∑t=0T−1𝔼​‖𝐝t‖2≤n​K𝒢​1T​∑t=0T−1𝔼⁡\[ℰt\]\.\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\\\|\\mathbf\{d\}^\{t\}\\\|^\{2\}\\leq nK\_\{\\mathcal\{G\}\}\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\mathcal\{E\}^\{t\}\]\.\(95\)
Substituting \([94](https://arxiv.org/html/2608.26493#A2.E94)\) and \([95](https://arxiv.org/html/2608.26493#A2.E95)\) into \([93](https://arxiv.org/html/2608.26493#A2.E93)\), and then applying Lemma[6](https://arxiv.org/html/2608.26493#Thmlemma6), yields

1T​∑t=0T−1𝔼⁡\[𝒮μt\]\\displaystyle\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\mathcal\{S\}\_\{\\mu\}^\{t\}\]≤4​Δμ0n⁡\(1−κ​μ\)​γ​T\+4​ρ​BΨn​μ​\(1−κ​μ\)​T\\displaystyle\\leq\\frac\{4\\Delta\_\{\\mu\}^\{0\}\}\{n\(1\-\\kappa\\mu\)\\gamma\\sqrt\{T\}\}\+\\frac\{4\\rho B\_\{\\Psi\}\}\{n\\mu\(1\-\\kappa\\mu\)\\sqrt\{T\}\}\+4​γ​H2n​μ​\(1−κ​μ\)​T\+2​γ​\(1\+ρ\)2​σ¯2μ⁡\(1−κ​μ\)​T\\displaystyle\\quad\+\\frac\{4\\gamma H^\{2\}\}\{n\\mu\(1\-\\kappa\\mu\)\\sqrt\{T\}\}\+\\frac\{2\\gamma\(1\+\\rho\)^\{2\}\\bar\{\\sigma\}^\{2\}\}\{\\mu\(1\-\\kappa\\mu\)\\sqrt\{T\}\}\+8​K𝒢χ0​ω​T​\(1−κ​μ\)​\(11−κ​μ\+γμ​T\)\\displaystyle\\quad\+\\frac\{8K\_\{\\mathcal\{G\}\}\}\{\\chi\_\{0\}\\omega T\(1\-\\kappa\\mu\)\}\\left\(\\frac\{1\}\{1\-\\kappa\\mu\}\+\\frac\{\\gamma\}\{\\mu\\sqrt\{T\}\}\\right\)×\[𝔼⁡\[ℰ0\]\+χ1​Bh​γ2\+χ2​ϑ02​\(1\+log⁡T\)\]\.\\displaystyle\\qquad\\qquad\\times\\left\[\\mathbb\{E\}\[\\mathcal\{E\}^\{0\}\]\+\\chi\_\{1\}B\_\{h\}\\gamma^\{2\}\+\\chi\_\{2\}\\vartheta\_\{0\}^\{2\}\(1\+\\log T\)\\right\]\.This is precisely \([50](https://arxiv.org/html/2608.26493#S5.E50)\)\. Since all problem\-dependent constants are independent ofTT, the first four terms are𝒪\(T−1/2\)\\mathcal\{O\}\(T^\{\-1/2\}\), while the communication term is𝒪⁡\(\(1\+log⁡T\)/T\)\\mathcal\{O\}\(\(1\+\\log T\)/T\)\. Therefore,

1T∑t=0T−1𝔼\[𝒮μt\]=𝒪\(T−1/2\)\+𝒪\(log⁡TT\)\.\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[\\mathcal\{S\}\_\{\\mu\}^\{t\}\]=\\mathcal\{O\}\(T^\{\-1/2\}\)\+\\mathcal\{O\}\\left\(\\frac\{\\log T\}\{T\}\\right\)\.This completes the proof\.□\\square

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