WHERE to Generate Matters: Budget-Aware Synthetic Augmentation for Label Skewed Federated Learning

# WHERE to Generate Matters: Budget-Aware Synthetic Augmentation for Label Skewed Federated Learning
Source: [https://arxiv.org/html/2607.06616](https://arxiv.org/html/2607.06616)
11institutetext:Chung\-Ang University, Seoul, South Korea11email:\{dltkddn,tjdghks994,jaewoolee\}@cau\.ac\.kr###### Abstract

Label skew in federated learning \(FL\) causes client drift and degrades global accuracy\. Synthetic data augmentation can reduce this imbalance; however, full class balancing requires substantial computation cost\. We propose FedEAS, a policy that assigns each client an entropy\-adaptive per\-class generation budget computed from its local label distribution\. The budget jointly decides*how much*each client generates and*WHERE*the samples go\. Accordingly, the total generation budget follows from the per\-client budgets rather than being fixed in advance\. FedEAS recovers most of the accuracy gain of full class balancing while reducing the generation budget by 94\.1%\. At the same total generation budget, it outperforms Uniform allocation by up to 18\.82% across CIFAR\-10 and CIFAR\-100\.

## 1Introduction

Label skew is a fundamental challenge in federated learning \(FL\)\[kairouz,lifl\]\. Clients hold locally imbalanced class distributions, so their local updates drift apart and degrade the global model\[zhao,liconv,hsu,noniidsilos\]\. Optimization\- and model\-level methods mitigate this client drift with proximal regularization, control variates, contrastive objectives, and classifier calibration, leaving the local data distribution unchanged\[fedprox,scaffold,moon,classcal,nofear\]\.

Existing synthetic data augmentation methods correct the imbalance itself, generating the samples that each client’s local distribution lacks\[genfedsd,feddda\]\. The dominant policy is full class balancing, where every client generates until each of its classes reaches the size of its largest one\[genfedsd,feddda\]\.

However, full class balancing is an expensive policy\. As shown in Fig\.[1](https://arxiv.org/html/2607.06616#S1.F1)\(a\), reaching it on a 20\-client CIFAR\-10 partition under Dirichlet label skew withα=0\.1\\alpha=0\.1requires 226,783 synthetic samples\. The cost stems from treating the amount to generate as a balancing requirement rather than a decision\. Recent studies relax this requirement by treating generation budget and client resources as explicit constraints\[fedgc,fimi\]\. Their allocations are uniform or follow data size and device resources rather than label skew, and the total generation budget is fixed in advance\. As illustrated in Fig\.[1](https://arxiv.org/html/2607.06616#S1.F1)\(b\), the same total generation budget yields very different accuracy depending on where the samples go\. Therefore, we ask the central question of this paper\.*How much synthetic data does each client actually need, and WHERE should the samples go?*

![Refer to caption](https://arxiv.org/html/2607.06616v1/x1.png)\(a\)
![Refer to caption](https://arxiv.org/html/2607.06616v1/x2.png)\(b\)

Figure 1:Budget\-aware synthetic augmentation on a CIFAR\-10 Dirichlet partition \(K=20K=20,α=0\.1\\alpha=0\.1\)\.\(a\)Reaching full class balancing requires 226,783 synthetic samples, whereas the FedEAS\(β=12\\beta\\\!=\\\!12\) budgets require 13,437; hatched bars show per\-client full\-balancing demand, solid bars the FedEAS budgets\.\(b\)Allocation decides how much of the budget turns into accuracy\. FedAvg trains without synthetic data, while Uniform, Missing\-only, and FedEAS\(β=12\\beta\\\!=\\\!12\) share the same 13,437\-sample total and differ only in where the samples go \(last\-20\-round mean\)\.To answer this question, we propose FedEAS \(Federated Entropy\-Adaptive Synthesis\), which assigns each client a per\-class generation budget computed from its local label distribution\. The budget jointly decides*how much*each client generates and*WHERE*the samples go\. A more skewed client receives a larger budget, a near\-balanced client receives almost none, and in the IID \(Independent and Identically Distributed\) limit FedEAS reduces to standard FedAvg\[fedavg\]\. Within each client, generation fills only the classes that fall below the budget\. These are exactly the client’s absent and scarce classes\. Accordingly, the total generation budget follows from the per\-client budgets rather than being fixed in advance\.

On CIFAR\-10 and CIFAR\-100 under Dirichlet label skew, FedEAS recovers most of the accuracy gain of full class balancing while generating 94\.1% fewer synthetic samples \(13,437 vs\. 226,783 samples on CIFAR\-10; Section[4\.2](https://arxiv.org/html/2607.06616#S4.SS2)\)\. The saving translates into 3\.7×\\timeslower end\-to\-end runtime on a single GPU \(Section[4\.3](https://arxiv.org/html/2607.06616#S4.SS3)\)\. Accuracy also rises monotonically as the budget grows, so the operating point can be chosen by the available generation resources alone \(Section[4\.3](https://arxiv.org/html/2607.06616#S4.SS3)\)\.

This efficiency depends on*WHERE*the samples go\. We therefore compare allocation policies under a matched\-budget protocol, in which every policy shares the same generator and the same total generation budget, so that accuracy differences reflect allocation alone\. Under this protocol, FedEAS outperforms Uniform allocation by up to 18\.82% across CIFAR\-10 and CIFAR\-100, and reaches the 70% accuracy threshold on CIFAR\-10 up to2×2\\timesfaster than Missing\-only generation \(Sections[4\.2](https://arxiv.org/html/2607.06616#S4.SS2)and[4\.3](https://arxiv.org/html/2607.06616#S4.SS3)\)\.

In summary, this paper makes the following contributions\.

- •Entropy\-adaptive budget allocation policy\.We propose FedEAS, which decides*how much*and*WHERE*to generate for reducing imbalance in label skewed FL\. It gives skewed clients larger budgets and allocates samples only to each client’s scarce classes\. The total generation budget follows from the per\-client budgets \(Section[3\.1](https://arxiv.org/html/2607.06616#S3.SS1)\)\.
- •Generation\-efficient accuracy\.FedEAS recovers most of the accuracy gain of full class balancing with 94\.1% less generation budget and 3\.7×\\timeslower end\-to\-end runtime \(Sections[4\.2](https://arxiv.org/html/2607.06616#S4.SS2)and[4\.3](https://arxiv.org/html/2607.06616#S4.SS3)\)\.
- •Allocation dominance at matched budgets\.Under the same total generation budget, FedEAS outperforms Uniform allocation by up to 18\.82% and converges up to2×2\\timesfaster than Missing\-only generation \(Sections[4\.2](https://arxiv.org/html/2607.06616#S4.SS2)and[4\.3](https://arxiv.org/html/2607.06616#S4.SS3)\)\.

## 2Preliminaries and Motivation

### 2\.1Federated Setup and Client Drift

We consider a standard FL setup\[lifl\]withKKclients, where clientkkholds a local dataset𝒟k\\mathcal\{D\}\_\{k\}of sizeNkN\_\{k\}\. The global objective is

minθ∈ℝd⁡F​\(θ\):=∑k=1Kpk​Fk​\(θ\),pk=Nk∑jNj,\\min\_\{\\theta\\in\\mathbb\{R\}^\{d\}\}F\(\\theta\):=\\sum\_\{k=1\}^\{K\}p\_\{k\}F\_\{k\}\(\\theta\),\\quad p\_\{k\}=\\frac\{N\_\{k\}\}\{\\sum\_\{j\}N\_\{j\}\},\(1\)whereFk​\(θ\)=∑c=1Cqkc​ℓkc​\(θ\)F\_\{k\}\(\\theta\)=\\sum\_\{c=1\}^\{C\}q\_\{k\}^\{c\}\\ell\_\{k\}^\{c\}\(\\theta\)denotes clientkk’s local objective overCCclasses,qkc=nkc/Nkq\_\{k\}^\{c\}=n\_\{k\}^\{c\}/N\_\{k\}denotes its class distribution withnkcn\_\{k\}^\{c\}the number of class\-ccsamples, andℓkc\\ell\_\{k\}^\{c\}denotes the class\-ccloss\. Under non\-IID data, FedAvg’s convergence bound contains the cross\-client heterogeneity term

Γ=∑kpk​‖∇Fk−∇F‖2\.\\Gamma=\\sum\_\{k\}p\_\{k\}\\\|\\nabla F\_\{k\}\-\\nabla F\\\|^\{2\}\.\(2\)Γ\\Gammaquantifies this heterogeneity in FedAvg convergence analysis\[scaffold,fedopt,wangfedavg\]and is the starting point for our analysis \(Section[2\.2](https://arxiv.org/html/2607.06616#S2.SS2)\)\.

Label skew and entropy\.We induce label skew via Dirichlet partitioning\[hsu,noniidsilos\]with concentrationα\\alpha\(smallerα⇒\\alpha\\Rightarrowmore severe skew\)\. Clientkk’s skew is summarized by the normalized Shannon entropy

H~k=Hklog⁡C∈\[0,1\],Hk=−∑c=1CnkcNk​log⁡nkcNk,\\tilde\{H\}\_\{k\}=\\frac\{H\_\{k\}\}\{\\log C\}\\in\[0,1\],\\quad H\_\{k\}=\-\\sum\_\{c=1\}^\{C\}\\frac\{n\_\{k\}^\{c\}\}\{N\_\{k\}\}\\log\\frac\{n\_\{k\}^\{c\}\}\{N\_\{k\}\},\(3\)with0​log⁡0:=00\\log 0:=0, and the per\-class mean countn¯k=Nk/C\\bar\{n\}\_\{k\}=N\_\{k\}/C, used by the budget rule of Section[3](https://arxiv.org/html/2607.06616#S3)\. Throughout, bars denote means \(n¯k\\bar\{n\}\_\{k\},q¯\\bar\{q\}\) and tildes normalized or post\-augmentation quantities \(H~k\\tilde\{H\}\_\{k\},q~kc\\tilde\{q\}\_\{k\}^\{c\}\)\. The dominant policy is Full\-Balance\[genfedsd\], the full class balancing of Section[1](https://arxiv.org/html/2607.06616#S1), which augments untilq~kc≈1/C\\tilde\{q\}\_\{k\}^\{c\}\\approx 1/Cfor everycc\. As shown in Fig\.[1](https://arxiv.org/html/2607.06616#S1.F1)\(a\), it can require tens of thousands of samples for the most skewed clients, motivating a budget\-aware reformulation\.

Research focus\.We study how to allocate synthetic generation across clients and classes, not generator design\. All augmentation methods share one pre\-trained class\-conditional denoising diffusion probabilistic model \(DDPM\)\[ddpm,cfg,diffbeatgan\]as a fixed black\-box generator, so performance differences primarily reflect allocation\.

### 2\.2Motivation: How Much and WHERE to Generate

Section[1](https://arxiv.org/html/2607.06616#S1)asked*how much*synthetic data each client needs and*WHERE*the samples should go\. This subsection answers both parts before any training and arrives at three observations\.Observation 1\.Imbalance is concentrated in a few clients\.Observation 2\.Imbalance decreases only when samples go to scarce classes\.Observation 3\.Imbalance becomes more expensive to remove as generation proceeds\. Observations 1 and 3 answer the*how much*part, and Observation 2 answers the*WHERE*part\. To verify them, we first build a score for allocations and show that it tracks the client drift of Section[1](https://arxiv.org/html/2607.06616#S1)\.

Scoring an allocation\.We score an allocation by how strongly it pulls every client toward class balance\. For a total generation budgetBB, we define the imbalance score

ℐ​\(B\)=∑k=1Kpk​‖qk​\(B\)−u‖22,\\mathcal\{I\}\(B\)=\\sum\_\{k=1\}^\{K\}p\_\{k\}\\\|q\_\{k\}\(B\)\-u\\\|\_\{2\}^\{2\},\(4\)whereqk​\(B\)q\_\{k\}\(B\)denotes clientkk’s class distribution after addingBBsynthetic samples in total under a given allocation,u=\(1/C,…,1/C\)u=\(1/C,\\ldots,1/C\)denotes the uniform distribution, and the weightspkp\_\{k\}keep their original values from Eq\. \([1](https://arxiv.org/html/2607.06616#S2.E1)\)\.ℐ\\mathcal\{I\}needs only label counts, so allocations can be compared before any generation\.

Why the score tracks client drift\.ℐ\\mathcal\{I\}tracks the drift termΓ\\Gammaof Eq\. \([2](https://arxiv.org/html/2607.06616#S2.E2)\)\. To see the link, splitℐ\\mathcal\{I\}into cross\-client heterogeneity and global imbalance:

∑kpk​‖qk−u‖22=∑kpk​‖qk−q¯‖22\+‖q¯−u‖22,\\sum\_\{k\}p\_\{k\}\\\|q\_\{k\}\-u\\\|\_\{2\}^\{2\}=\\sum\_\{k\}p\_\{k\}\\\|q\_\{k\}\-\\bar\{q\}\\\|\_\{2\}^\{2\}\+\\\|\\bar\{q\}\-u\\\|\_\{2\}^\{2\},\(5\)because∑kpk​\(qk−q¯\)=0\\sum\_\{k\}p\_\{k\}\(q\_\{k\}\-\\bar\{q\}\)=0withq¯:=∑kpk​qk\\bar\{q\}:=\\sum\_\{k\}p\_\{k\}q\_\{k\}\. The first term is the distributional counterpart ofΓ\\Gamma, and the second measures global deviation from balance, which matters for generalization on a balanced test set\. We referenceuurather thanq¯\\bar\{q\}so that the reference stays invariant under generation andℐ=0\\mathcal\{I\}=0exactly when every client is balanced\. The link becomes explicit under three idealizing assumptions, namely \(i\) approximately orthogonal per\-class loss gradients\{∇ℓc\}c\\\{\\nabla\\ell^\{c\}\\\}\_\{c\}, \(ii\) a shared gradient norm‖g‖\\\|g\\\|, and \(iii\) per\-class losses shared across clients \(ℓkc≡ℓc\\ell\_\{k\}^\{c\}\\equiv\\ell^\{c\}\):

Γ≈‖g‖2​∑k=1Kpk​‖qk−q¯‖22,\\Gamma\\approx\\\|g\\\|^\{2\}\\sum\_\{k=1\}^\{K\}p\_\{k\}\\\|q\_\{k\}\-\\bar\{q\}\\\|\_\{2\}^\{2\},\(6\)in which the first term of Eq\. \([5](https://arxiv.org/html/2607.06616#S2.E5)\) appears exactly\. The derivation expands∇Fk−∇F=∑c\(qkc−q¯c\)​∇ℓc\\nabla F\_\{k\}\-\\nabla F=\\sum\_\{c\}\(q\_\{k\}^\{c\}\-\\bar\{q\}^\{c\}\)\\nabla\\ell^\{c\}, where cross\-terms vanish under \(i\) and diagonal terms share the scale‖g‖2\\\|g\\\|^\{2\}under \(ii\)\. These assumptions do not hold tightly in deep networks, so we claim no formal bound, and the empirical counterpartΓ^\\hat\{\\Gamma\}is measured directly in Section A of the supplementary material\. Reducing class imbalance therefore reduces client drift at its distributional source, andℐ\\mathcal\{I\}serves as its training\-free surrogate\.

![Refer to caption](https://arxiv.org/html/2607.06616v1/x3.png)\(a\)
![Refer to caption](https://arxiv.org/html/2607.06616v1/x4.png)\(b\)

Figure 2:Imbalance concentration and reduction on the partition of Fig\.[1](https://arxiv.org/html/2607.06616#S1.F1)\.\(a\)Cumulative fraction of the imbalance score and of the FedEAS generation budget \(β=12\\beta=12\), adding clients in decreasing order of imbalance; five clients hold 53\.6% of the imbalance score and receive 45\.1% of the budget\.\(b\)Normalized imbalance reduction1−ℐ​\(B\)/ℐ​\(0\)1\-\\mathcal\{I\}\(B\)/\\mathcal\{I\}\(0\)as the total generation budgetBBgrows\. Uniform allocation and the FedEAS budget path of Section[3](https://arxiv.org/html/2607.06616#S3)are evaluated at the same matched budgets \(markers\), the dashed curve shows the greedy reference of Eq\. \([7](https://arxiv.org/html/2607.06616#S2.E7)\), and the bracket marks the \+14\.6% gap at the matched 13,437\-sample budget\.Observation 1: Imbalance is concentrated in a few clients\.As shown in Fig\.[2](https://arxiv.org/html/2607.06616#S2.F2)\(a\), the top 25% of clients hold 53\.6% of the total imbalance score on this CIFAR\-10 partition \(K=20K\{=\}20,α=0\.1\\alpha\{=\}0\.1\)\. Accordingly, the generation required to fully balance a client varies by more than an order of magnitude, as illustrated in Fig\.[1](https://arxiv.org/html/2607.06616#S1.F1)\(a\)\. The normalized entropyH~k\\tilde\{H\}\_\{k\}captures this concentration from local counts alone\. A single balancing policy therefore conflates clients with very different needs\.

Observation 2: Imbalance decreases only when samples go to scarce classes\.Consider one synthetic sample of classccadded to clientkk\. To first order,ℐ\\mathcal\{I\}changes by−Δkc\-\\Delta\_\{k\}^\{c\}with

Δkc=2​pkNk​\(‖qk‖22−qkc\),\\Delta\_\{k\}^\{c\}=\\frac\{2p\_\{k\}\}\{N\_\{k\}\}\\left\(\\\|q\_\{k\}\\\|\_\{2\}^\{2\}\-q\_\{k\}^\{c\}\\right\),\(7\)which follows from the first\-order form of the updateqk→\(Nk​qk\+ec\)/\(Nk\+1\)q\_\{k\}\\rightarrow\(N\_\{k\}q\_\{k\}\+e\_\{c\}\)/\(N\_\{k\}\+1\), whereece\_\{c\}denotes the one\-hot vector of classcc\. The gain is positive only whileqkcq\_\{k\}^\{c\}lies below the client\-specific level‖qk‖22\\\|q\_\{k\}\\\|\_\{2\}^\{2\}, so a sample placed in an already\-large class increases imbalance\. Uniform allocation pays for this directly\. On this partition it places 1,679 of its 13,437 samples, 12\.5%, at pairs where the gain is negative\. Sincepk=Nk/∑jNjp\_\{k\}=N\_\{k\}/\\sum\_\{j\}N\_\{j\}, the leading factor equals2/∑jNj2/\\sum\_\{j\}N\_\{j\}for every client, and the gain grows with the client’s skew, confirming Observation 1 at the sample level\. Efficient generation therefore fills scarce classes toward a per\-client threshold instead of spreading over all classes\.

Observation 3: Imbalance becomes more expensive to remove as generation proceeds\.Fig\.[2](https://arxiv.org/html/2607.06616#S2.F2)\(b\) traces the normalized imbalance reduction1−ℐ​\(B\)/ℐ​\(0\)1\-\\mathcal\{I\}\(B\)/\\mathcal\{I\}\(0\)as the total generation budget grows, for uniform allocation and for a greedy reference that repeatedly places one sample at the pair maximizing Eq\. \([7](https://arxiv.org/html/2607.06616#S2.E7)\)\. Both curves are steepest at small budgets and flatten because the removable imbalance depletes\. A two\-client case in Section B of the supplementary material shows the same diminishing\-returns pattern in closed form\. The third curve in Fig\.[2](https://arxiv.org/html/2607.06616#S2.F2)\(b\) previews the policy of Section[3](https://arxiv.org/html/2607.06616#S3), which tracks the greedy reference using only local statistics and removes 51\.8% of the initial imbalance with 13,437 samples, 5\.9% of what Full\-Balance requires\.

Consequently, Observations 1–3 fix the requirements of an efficient policy\. It should give skewed clients larger budgets \(Observation 1\), allocate samples only to each client’s scarce classes \(Observation 2\), and stop well before full class balancing \(Observation 3\)\. Section[3](https://arxiv.org/html/2607.06616#S3)meets all three requirements with a single rule\.

![Refer to caption](https://arxiv.org/html/2607.06616v1/x5.png)Figure 3:Overview of FedEAS\. Each client summarizes its skew by the normalized entropyH~k\\tilde\{H\}\_\{k\}\(top left\), computes the per\-class generation budgetbkb\_\{k\}of Eq\. \([8](https://arxiv.org/html/2607.06616#S3.E8)\) \(top right\), and allocates samples only to the classes belowbkb\_\{k\}\(bottom\)\. Full\-Balance fills every class to its local maximum, while Uniform, Missing\-only, and FedEAS place the same total generation budget differently, as compared in Section[4\.2](https://arxiv.org/html/2607.06616#S4.SS2)\.

## 3FedEAS: Federated Entropy\-Adaptive Synthesis

FedEAS meets the three observations of Section[2\.2](https://arxiv.org/html/2607.06616#S2.SS2)with a single rule\. As illustrated in Fig\.[3](https://arxiv.org/html/2607.06616#S2.F3), each client computes a per\-class generation budgetbkb\_\{k\}from its local label distribution, allocates samples only to the classes belowbkb\_\{k\}, and trains as in standard FedAvg\.

### 3\.1Entropy\-Adaptive Budget Allocation Policy

FedEAS assigns clientkkthe per\-class generation budget

bk=⌊n¯k⋅\(1−H~k\)⋅β⌋,b\_\{k\}=\\left\\lfloor\\sqrt\{\\bar\{n\}\_\{k\}\}\\cdot\(1\-\\tilde\{H\}\_\{k\}\)\\cdot\\beta\\right\\rfloor,\(8\)wheren¯k\\bar\{n\}\_\{k\}andH~k\\tilde\{H\}\_\{k\}are the per\-class mean count and the normalized entropy of Section[2\.1](https://arxiv.org/html/2607.06616#S2.SS1), andβ\>0\\beta\>0is the scalar budget parameter shared by all clients\. Herebkb\_\{k\}is a per\-client, per\-class count, and the floor makes it integer\-valued\. Each factor of Eq\. \([8](https://arxiv.org/html/2607.06616#S3.E8)\) answers one observation of Section[2\.2](https://arxiv.org/html/2607.06616#S2.SS2)\.

Skewed clients receive larger budgets \(Observation 1\)\.The entropy factor\(1−H~k\)\(1\-\\tilde\{H\}\_\{k\}\)grows with skew, so a more skewed client receives a larger budget and a near\-balanced client receives almost none\. Then¯k\\sqrt\{\\bar\{n\}\_\{k\}\}factor scales the budget with client size sub\-linearly\. A linearn¯k\\bar\{n\}\_\{k\}rule would stretch the cross\-client budget ratio to38\.5×38\.5\\times, against7\.2×7\.2\\timesunder the sub\-linear rule \(CIFAR\-10,K=20K\{=\}20,α=0\.1\\alpha\{=\}0\.1\), over\-allocating to large clients while under\-serving small ones\. In the IID limit, everybkb\_\{k\}reaches zero and FedEAS trains exactly as standard FedAvg\.

Samples go only to classes below the budget \(Observation 2\)\.Givenbkb\_\{k\}, classccat clientkkreceives

nk,genc=max⁡\{0,bk−nkc\},n\_\{k,\\mathrm\{gen\}\}^\{c\}=\\max\\\{0,\\,b\_\{k\}\-n\_\{k\}^\{c\}\\\},\(9\)the fill\-to\-threshold form that Observation 2 derived\. Classes at or abovebkb\_\{k\}receive nothing, so the samples concentrate on the client’s absent and scarce classes\. Summing over clients and classes gives the realized total generation budgetBFE=∑k,cnk,gencB\_\{\\mathrm\{FE\}\}=\\sum\_\{k,c\}n\_\{k,\\mathrm\{gen\}\}^\{c\}, the value ofBBthat Section[2\.2](https://arxiv.org/html/2607.06616#S2.SS2)scores and that every baseline shares under the matched\-budget protocol of Section[4](https://arxiv.org/html/2607.06616#S4)\. If the pre\-floor value ofbkb\_\{k\}is below one,bk=0b\_\{k\}=0and clientkkgenerates nothing\.

Early stopping of generation \(Observation 3\)\.The scalarβ\\betasets the level of every budget, so generation ends in the steep region of Fig\.[2](https://arxiv.org/html/2607.06616#S2.F2)\(b\), while imbalance is still cheap to remove\. Raisingβ\\betascales everybkb\_\{k\}together, so a single parameter adapts the total generation budget to the available resources \(Section[4\.3](https://arxiv.org/html/2607.06616#S4.SS3)\)\.

### 3\.2FedEAS in the Training Loop

FedEAS changes only how each client builds its local dataset\. Algorithm[1](https://arxiv.org/html/2607.06616#alg1)shows the full procedure\. The server broadcastsβ\\betaonce at initialization\. On first participation, clientkkcomputes Eqs\. \([8](https://arxiv.org/html/2607.06616#S3.E8)\)–\([9](https://arxiv.org/html/2607.06616#S3.E9)\), generates the required images, and stores them in the cache𝒞k\\mathcal\{C\}\_\{k\}\. The client then trains on𝒟k∪𝒞k\\mathcal\{D\}\_\{k\}\\cup\\mathcal\{C\}\_\{k\}, and aggregation remains standard FedAvg weighted by the original dataset sizesNkN\_\{k\}\.

One\-time generation\.Generation runs once per client and the cache is reused in all later rounds\. The saving over Full\-Balance therefore comes from the smaller cache, and the communication protocol does not change\.

Generator\-agnostic\.Eqs\. \([8](https://arxiv.org/html/2607.06616#S3.E8)\)–\([9](https://arxiv.org/html/2607.06616#S3.E9)\) read only local label counts\. Any class\-conditional generator\[iddpm,scoresde,gan,stylegan2\]can be substituted without changing the policy \(Section[4](https://arxiv.org/html/2607.06616#S4.F4)\)\.

Algorithm 1FedEAS: entropy\-adaptive budget allocation\.0:Clients

k=1,…,Kk=1,\\ldots,K; classes

c=1,…,Cc=1,\\ldots,C; local counts

nkcn\_\{k\}^\{c\}and

Nk=∑cnkcN\_\{k\}=\\sum\_\{c\}n\_\{k\}^\{c\}; rounds

TT; local epochs

EE; budget parameter

β\\beta; pre\-trained class\-conditional generator

GϕG\_\{\\phi\}\.

0:Global model

θT\\theta^\{T\}\.

1:Server initializes

θ0\\theta^\{0\}and broadcasts

β\\beta; each client initializes synthetic cache

𝒞k←∅\\mathcal\{C\}\_\{k\}\\leftarrow\\emptyset
2:for

t=0,1,…,T−1t=0,1,\\ldots,T\-1do

3:Server broadcasts

θt\\theta^\{t\}to selected clients

StS\_\{t\}
4:foreach client

k∈Stk\\in S\_\{t\}in paralleldo

5:if

𝒞k=∅\\mathcal\{C\}\_\{k\}=\\emptyset\(one\-time generation\)then

6:

n¯k←Nk/C\\bar\{n\}\_\{k\}\\leftarrow N\_\{k\}/C,

H~k←Hk/log⁡C\\tilde\{H\}\_\{k\}\\leftarrow H\_\{k\}/\\log C
7:

bk←⌊n¯k⋅\(1−H~k\)⋅β⌋b\_\{k\}\\leftarrow\\lfloor\\sqrt\{\\bar\{n\}\_\{k\}\}\\cdot\(1\-\\tilde\{H\}\_\{k\}\)\\cdot\\beta\\rfloor
8:foreach class

ccwith

nkc<bkn\_\{k\}^\{c\}<b\_\{k\}do

9:

nk,genc←bk−nkcn\_\{k,\\mathrm\{gen\}\}^\{c\}\\leftarrow b\_\{k\}\-n\_\{k\}^\{c\}
10:Generate

nk,gencn\_\{k,\\mathrm\{gen\}\}^\{c\}images of class

ccvia

GϕG\_\{\\phi\}
11:endfor

12:

𝒞k←\\mathcal\{C\}\_\{k\}\\leftarrowall generated images

13:endif

14:Train on

𝒟k∪𝒞k\\mathcal\{D\}\_\{k\}\\cup\\mathcal\{C\}\_\{k\}for

EElocal epochs; send

θkt\+1\\theta^\{t\+1\}\_\{k\}
15:endfor

16:

wk←Nk/∑j∈StNjw\_\{k\}\\leftarrow N\_\{k\}/\\sum\_\{j\\in S\_\{t\}\}N\_\{j\}for each

k∈Stk\\in S\_\{t\}
17:

θt\+1←∑k∈Stwk​θkt\+1\\theta^\{t\+1\}\\leftarrow\\sum\_\{k\\in S\_\{t\}\}w\_\{k\}\\theta^\{t\+1\}\_\{k\}
18:endfor

## 4Experiments

In this section, we validate the entropy\-adaptive budget allocation policy on both parts of the central question of Section[1](https://arxiv.org/html/2607.06616#S1)\. Section[4\.2](https://arxiv.org/html/2607.06616#S4.SS2)measures*how much*generation budget FedEAS needs and isolates the effect of*WHERE*the samples go under the matched\-budget protocol\. Section[4\.3](https://arxiv.org/html/2607.06616#S4.SS3)sweeps the budget parameter, selects the default, and reports the end\-to\-end cost, and Section[4\.4](https://arxiv.org/html/2607.06616#S4.SS4)ablates the entropy andn¯k\\sqrt\{\\bar\{n\}\_\{k\}\}factors of the budget rule\.

### 4\.1Setup and Matched\-Budget Protocol

Datasets and partitioning\.We evaluate on CIFAR\-10 \(C=10C=10\) and CIFAR\-100 \(C=100C=100\)\[cifar\], inducing label skew via Dirichlet partitioning with concentrationα\\alpha\(primaryα=0\.1\\alpha=0\.1, with sweepsα∈\{0\.05,0\.1,0\.3\}\\alpha\\in\\\{0\.05,0\.1,0\.3\\\}on CIFAR\-10\)\. We reportK=20K=20\(50% participation\) andK=100K=100\(10% participation\), each forT=200T=200rounds\. Accuracy, in this section, is the last\-20\-round mean, the mean global test accuracy over rounds 181–200\.

Model and generator\.The local model is ResNet\-18\[resnet\], and each client trainsE=10E=10local epochs per round with SGD \(lr1×10−31\\\!\\times\\\!10^\{\-3\}\) and batch size 64\. For the primary experiments we train a separate class\-conditional DDPM\[ddpm,ddim\]on each dataset’s full training set, isolating allocation from generator design\. Uncurated samples from this DDPM appear in Section C of the supplementary material\. Section[4](https://arxiv.org/html/2607.06616#S4.F4)validates the allocation gap under a generator change with an off\-the\-shelf Stable Diffusion \(SD\-turbo\) backbone\[sdturbo\], and deployment limitations are discussed in Section[6](https://arxiv.org/html/2607.06616#S6)\. Remaining hyperparameters appear in Section D of the supplementary material\.

Baselines and budget matching\.We compare against six established FL baselines\[fedavg,fedprox,scaffold,moon,fedgen,fedmix\]and three augmentation policies under the same DDPM backbone and schedule\. Uniform and Missing\-only policies are compared under the matched\-budget protocol\. Each shares the same generator and the same total generation budgetBFEB\_\{\\mathrm\{FE\}\}realized by FedEAS \(Section[3\.1](https://arxiv.org/html/2607.06616#S3.SS1)\), so accuracy differences reflect allocation alone\.

*Uniform*assigns⌊BFE/\(K​C\)⌋\\lfloor B\_\{\\mathrm\{FE\}\}/\(KC\)\\rfloorsamples to every client–class pair and distributes the remainder one at a time, spending exactlyBFEB\_\{\\mathrm\{FE\}\}samples\. This is the equal\-allocation rule that prior budget\-constrained generation work recommends\[fedgc\]\.*Missing\-only*applies the same rule over absent pairs \(nkc=0n\_\{k\}^\{c\}=0\) only\.*Full\-Balance*generates until each class reachesmaxc⁡nkc\\max\_\{c\}n\_\{k\}^\{c\}, instantiating Gen\-FedSD’s policy\[genfedsd\]\. It exceeds the shared budget and serves as an upper\-bound reference rather than a matched competitor\. Unless stated otherwise, FedEAS usesβ=12\\beta=12, selected in Section[4\.3](https://arxiv.org/html/2607.06616#S4.SS3)\. Samples are cached before a client’s first update, and aggregation weights use the original local dataset sizes\.

### 4\.2Main Accuracy and Generation Budget

Table 1:Accuracy \(%, last\-20\-round mean\) across datasets, heterogeneity levels, and client scales\.Boldandunderlineindicate the best and second\-best results, respectively, excluding Full\-Balance\. The last two rows report total generation budgets\.FedEAS recovers most of the Full\-Balance accuracy gain at a small fraction of its generation budget\. Table[1](https://arxiv.org/html/2607.06616#S4.T1)reports the main comparison across datasets, heterogeneity levels, and client scales\. On CIFAR\-10 \(α=0\.1\\alpha=0\.1,K=20K=20\), FedEAS reaches 71\.65% against Full\-Balance’s 79\.91%, two thirds of the gain over FedAvg \(55\.52%\)\. It generates 13,437 samples against 226,783, a 94\.1% reduction in generation budget\. The recovered portion of the gain grows with scale\. AtK=100K=100, FedEAS reaches 72\.55% against 79\.33%, three quarters of the gain over FedAvg \(50\.18%\)\. On CIFAR\-100 it recovers half of the gain atK=20K=20and two thirds atK=100K=100\. Full\-Balance remains the accuracy upper bound\.

Table[1](https://arxiv.org/html/2607.06616#S4.T1)also shows under which skew levels synthetic data augmentation is required\. Under severe CIFAR\-10 skew \(α≤0\.1\\alpha\\leq 0\.1\), every reported augmentation policy, including Uniform, which ignores label skew, exceeds all six FL baselines\. Atα=0\.05\\alpha=0\.05,K=20K=20, Uniform reaches 54\.97% against 39\.16% for the strongest FL baseline\. On CIFAR\-100, Missing\-only and FedEAS exceed the strongest FL baseline \(SCAFFOLD, 45\.92% atK=20K=20\) by more than 5%, and Uniform matches it\. FedEAS itself improves over FedAvg by up to 29\.09% \(CIFAR\-10,α=0\.05\\alpha=0\.05,K=20K=20\) and stays competitive under mild skew \(α=0\.3\\alpha=0\.3\), where SCAFFOLD becomes strong\. Peak accuracies over all rounds appear in Section E of the supplementary material\.

Under the matched\-budget protocol, Uniform and FedEAS share the total generation budget, so their accuracy differences reflect allocation alone\. FedEAS outperforms Uniform at everyK=20K=20matched budget, by \+13\.28% atα=0\.05\\alpha=0\.05, \+7\.22% atα=0\.1\\alpha=0\.1, and \+1\.81% atα=0\.3\\alpha=0\.3on CIFAR\-10, and by \+5\.16% on CIFAR\-100\. AtK=100K=100every gap widens, reaching \+18\.82% atα=0\.05\\alpha=0\.05on CIFAR\-10 and \+8\.32% on CIFAR\-100\. The smaller gap atα=0\.3\\alpha=0\.3is expected\. Near\-balanced clients leave less room for allocation, and restoring absent classes already captures much of the gain\.

![Refer to caption](https://arxiv.org/html/2607.06616v1/x6.png)Figure 4:FedEAS converges faster than Missing\-only under matched budgets\. The dashed line marks the 70% global test accuracy threshold; dotted vertical lines mark the first round reaching it—30 vs\. 60 rounds atK=20K=20and 59 vs\. 100 atK=100K=100\.FedEAS converges faster than Missing\-only at a matched budget\.Missing\-only restores absent classes but leaves scarce classes deficient, whereas FedEAS fills both toward the budgetbkb\_\{k\}\. The difference appears in convergence speed\. As shown in Fig\.[4](https://arxiv.org/html/2607.06616#S4.F4), FedEAS reaches the 70% global test accuracy threshold up to2×2\\timesfaster than Missing\-only on CIFAR\-10 \(α=0\.1\\alpha=0\.1\), with last\-20\-round means of 71\.65 vs\. 70\.16 atK=20K=20and 72\.55 vs\. 70\.18 atK=100K=100\. The budget rule of Eq\. \([8](https://arxiv.org/html/2607.06616#S3.E8)\) therefore accelerates convergence beyond Missing\-only generation\.

The allocation gap holds under a generator change\.We repeat the matched\-budget comparison with an off\-the\-shelf SD\-turbo backbone\[sdturbo\]at the matched 13,437\-sample budget \(CIFAR\-10,K=20K=20,α=0\.1\\alpha=0\.1\)\. The same pattern appears in peak accuracy\. FedEAS outperforms Uniform by 2\.46% under SD\-turbo, against 6\.34% under the controlled DDPM, while absolute accuracy drops from generator and downsampling mismatch\.

![Refer to caption](https://arxiv.org/html/2607.06616v1/x7.png)\(a\)
![Refer to caption](https://arxiv.org/html/2607.06616v1/x8.png)\(b\)

Figure 5:Budget sensitivity on CIFAR\-10 \(α=0\.1\\alpha=0\.1,K=20K=20\) under matched FedEAS–Uniform budgets\.\(a\)Last\-20\-round accuracy as the total generation budget grows\.\(b\)Selection of the default budget parameter\. The budget path traces the imbalance reduction1−ℐ​\(B\)/ℐ​\(0\)1\-\\mathcal\{I\}\(B\)/\\mathcal\{I\}\(0\)against the synthetic fraction of the training data asβ\\betagrows; markers are the sweep values\. The dashed lines mark the two selection conditions\. Onlyβ=12\\beta=12falls in the shaded region that removes at least 50% of the initial imbalance while keeping synthetic samples below 25% of the training data\.
### 4\.3Budget Sensitivity and Selection

Accuracy rises monotonically as the budget grows\. We vary the budget parameter overβ∈\{4,8,12,16,20\}\\beta\\in\\\{4,8,12,16,20\\\}on CIFAR\-10 \(K=20K=20,α=0\.1\\alpha=0\.1\)\. As shown in Fig\.[5](https://arxiv.org/html/2607.06616#S4.F5)\(a\), the accuracy increases from 66\.11% to 74\.05%, exceeding Uniform by 5\.30–7\.59% at every matched budget, so the operating point can be chosen by the available generation resources alone\.

The default budget parameter follows from two conditions\.The realized budget should remove at least 50% of the initial imbalance scoreℐ​\(0\)\\mathcal\{I\}\(0\)of Section[2\.2](https://arxiv.org/html/2607.06616#S2.SS2), and synthetic samples should stay below 25% of the combined training data, so real samples outnumber synthetic ones at least three to one\. Both conditions read only label counts\. As shown in Fig\.[5](https://arxiv.org/html/2607.06616#S4.F5)\(b\), the budget path satisfies both conditions only betweenβ≈11\.4\\beta\\approx 11\.4andβ≈13\.4\\beta\\approx 13\.4on this partition, and the sweep grid meets them only atβ=12\\beta=12, which we adopt as the default\. At this operating point FedEAS removes 51\.8% of the initial imbalance while synthetic samples stay at 21\.2% of the combined training data\. A practitioner can re\-run the same selection under different resource limits, and Fig\.[5](https://arxiv.org/html/2607.06616#S4.F5)\(a\) shows the accuracy consequence of varyingβ\\beta\.

Efficiency falls as generation proceeds\.We define the efficiencyη=Δ​Acc×103/total generation budget\\eta=\\Delta\\mathrm\{Acc\}\\times 10^\{3\}/\\text\{total generation budget\}, whereΔ​Acc\\Delta\\mathrm\{Acc\}is the accuracy gain over FedAvg \(55\.52%\)\. Across this sweep,η\\etadecreases monotonically from 2\.575 to 0\.788 while remaining1\.7×1\.7\\times–2\.0×2\.0\\timesthe Uniform efficiency at each matched budget, detailed in Section F of the supplementary material\. The imbalance that remains at larger budgets is more expensive to remove\. Full\-Balance, the upper\-bound reference, uses 226,783 samples and hasη=0\.108\\eta=0\.108, so full class balancing raises accuracy but lowers efficiency\.

Wall\-clock cost\.Under the same implementation on a single NVIDIA RTX 4090 GPU, FedEAS’s runtime, including one\-time generation and FL training, is 3h7m against 11h36m for Full\-Balance\. The3\.7×3\.7\\timesreduction shows the budget saving translates into measured end\-to\-end cost\.

### 4\.4Budget\-Factor Ablation

The gains trace to allocation quality rather than to the budget level alone\. Table[2](https://arxiv.org/html/2607.06616#S4.T2)compares FedEAS atβ=8\\beta=8with two variants on CIFAR\-10 \(K=20K=20,α=0\.1\\alpha=0\.1\), one without the entropy factor and one with the linearn¯k\\bar\{n\}\_\{k\}rule\.

FedEAS with both factors reaches 70\.86%, exceeding the variant without the entropy factor \(70\.34%\) and the linear\-rule variant \(70\.33%\) despite generating fewer samples\. Removing the entropy factor lowers the last\-20\-round mean by 0\.52% while generating 1,664 additional samples\. The entropy factor therefore improves allocation rather than merely reducing the budget\.

Replacing then¯k\\sqrt\{\\bar\{n\}\_\{k\}\}factor with the linearn¯k\\bar\{n\}\_\{k\}rule lowers the last\-20\-round mean by 0\.53% while generating 1,525 additional samples\. The linear rule also widens the cross\-client budget ratio, as quantified in Section[3\.1](https://arxiv.org/html/2607.06616#S3.SS1)\. Together, these results support both factors of Eq\. \([8](https://arxiv.org/html/2607.06616#S3.E8)\)\.

Table 2:Budget\-factor ablation on CIFAR\-10 \(K=20K=20,α=0\.1\\alpha=0\.1\)\.

## 5Related Work

Optimization\- and model\-level FL under label skew\.Methods that do not alter local datasets intervene through the optimizer, representation, classifier, or loss\[feddf,fedrep,ditto,perfedavg\]\. FedProx\[fedprox\], SCAFFOLD\[scaffold\], and MOON\[moon\]use proximal regularization, control variates, and contrastive learning, respectively, while FedDyn\[feddyn\]and FedNova\[fednova\]correct client drift through dynamic regularization and normalized aggregation\. Label\-skew\-specific methods refine this via restricted softmax\[fedrs\], logits calibration\[fedlc\], classifier calibration\[classcal,ldam,decouple\], fixed classifiers\[nofear\], and label\-distribution\-aware concatenation\[diao\]\. FedVLS\[guo\]targets vacant classes via distillation and logit suppression at the model and loss level rather than by augmentation\. These are complementary to FedEAS, which intervenes at the data level before local training\.

Data\-level augmentation in FL\.Existing FL augmentation methods adapt classical augmentation\[augsurvey,cutmix,autoaugment,randaugment,cutout\]but generally do not formulate augmentation as a client–class budget\-allocation problem\. Gen\-FedSD\[genfedsd\]balances every client with text\-to\-image diffusion\. Our experiments evaluate this full class balancing as the upper\-bound reference\. FedMix\[fedmix\]shares averaged data to approximate cross\-client Mixup\[mixup\], and FedGen\[fedgen\]learns a data\-free server\-side generator over latent features\. FedFA\[fedfa\]augments in feature space and FedRDN\[fedrdn\]injects federation\-wide statistics at the input level, both targeting feature rather than label skew, while Hong et al\.\[hong\]learn a shared augmentation policy\. Budget\-constrained generation\[fedgc,fimi\]fixes the total generation budget in advance and allocates it uniformly or by data size and device resources rather than by label skew\. FedEAS instead derives both the budget and its allocation from each client’s label skew\. Generating only for absent pairs, our Missing\-only baseline, is the closest policy but leaves scarce classes deficient \(Section[4\.2](https://arxiv.org/html/2607.06616#S4.SS2)\)\.

Centralized synthetic augmentation\.Centralized studies show both the promise and the setup dependence of synthetic augmentation\[sariyildiz,stablerep\]\. He et al\.\[he\]examine its benefits and shortcomings for recognition, Azizi et al\.\[azizi\]show ImageNet gains from diffusion samples, and Trabucco et al\.\[trabucco\]use diffusion edits for few\-shot augmentation\. These studies motivate the budget\-aware premise but not allocation across federated clients\.

## 6Discussion

Limitations\.Fixing a separate class\-conditional DDPM\[ddpm\]per dataset isolates allocation from generator design, but each DDPM is trained on its dataset’s full training set, which limits deployment realism\. The SD\-turbo comparison of Section[4](https://arxiv.org/html/2607.06616#S4.F4)partially decouples the allocation gap from the DDPM, yet privacy\-preserving generator training remains open\. Related FL augmentation methods share this limitation\[genfedsd,fedgen,fedmix\]\. The policy reads only local label counts, so a generator trained with privacy\-preserving methods\[dpdiffusion,dpsgd,dpfl\]can replace the DDPM\. DDPM sampling is also stochastic with no per\-sample fidelity guarantee, which adds a generator\-quality variance orthogonal to allocation\. Quality control such as FID filtering can be combined with the policy\.

Future work\.Three directions extend this framework\. Federated\-trained or stronger generators would address the deployment limitation above\. A tunable exponentn¯kγ\\bar\{n\}\_\{k\}^\{\\gamma\}would generalize then¯k\\sqrt\{\\bar\{n\}\_\{k\}\}factor\. Budget allocation can also be paired with optimization\-level corrections, and a preliminary FedEAS\+SCAFFOLD run on CIFAR\-10 \(α=0\.3\\alpha=0\.3\) reaches 78\.18%, exceeding either alone\.

## 7Conclusion

We recast synthetic augmentation for label skewed FL as a budget\-allocation problem\. FedEAS addresses it with a single entropy\-adaptive rule computed from local label counts\. Skewed clients receive larger budgets, samples go only to each client’s scarce classes, and generation stops well before full class balancing\. The rule changes only the synthetic cache of each client, and the total generation budget follows from the per\-client budgets\.

FedEAS reduces the generation budget by 94\.1% and the end\-to\-end runtime by3\.7×3\.7\\timeswhile recovering most of the accuracy gain of full class balancing\. With the total generation budget held equal, FedEAS outperforms Uniform allocation by up to 18\.82% on CIFAR\-10 and CIFAR\-100\. Allocation therefore decides how much of a fixed budget turns into accuracy, and*WHERE*to generate is a primary design problem for synthetic augmentation in label skewed FL\.

## References