FedCW: Client Selection with Adaptive Weight in Heterogeneous Federated Learning

1  Introduction

Federated Learning (FL) [1] has emerged as a promising paradigm to enable distributed model training across multiple devices without the need to centralize data [2]. This characteristic is especially appealing in vehicular networks, where data privacy and communication costs are significant concerns [3,4]. With an increasing number of connected vehicles, FL can provide an efficient solution to collaboratively train models, such as those needed for traffic prediction, autonomous driving, and vehicular safety, while preserving individual data privacy [5]. The decentralized nature of FL reduces the risk of data breaches and ensures that sensitive information remains on local devices, which is particularly important for vehicular networks, where data privacy regulations are stringent, and the costs of data transmission are high [6]. However, the inherent heterogeneity in vehicular data and the variance in computational capabilities across different clients pose significant challenges to conventional FL algorithms [7–9].

A fundamental issue faced by traditional FL algorithms in vehicular networks is their inability to effectively address the heterogeneity of data and client capabilities. The variability in data distribution across clients, which is frequently non-independent and non-identically distributed (Non-IID), can significantly impede convergence and lead to biased global models [10–12]. In vehicular networks, data collected by individual vehicles may differ markedly due to factors such as geographic location, driving behavior, and environmental conditions, further complicating the learning process. Additionally, the random selection of clients without accounting for their potential contributions, coupled with static weight allocation that disregards clients’ relevance to the global model, exacerbates these challenges, particularly in resource-constrained vehicular contexts [13,14]. Consequently, the global model may converge slowly or fail to generalize effectively across all clients, resulting in suboptimal performance in real-world deployments. These challenges underscore a significant research gap in the domain of FL for vehicular networks. In Federated Learning, two central problems are client selection and the appropriate distribution of aggregation weights, particularly in heterogeneous environments where devices have diverse computational capacities and data distributions. The challenge of client selection lies in determining which edge devices should participate in each training round, as involving all clients can be computationally expensive and inefficient. Similarly, the aggregation of model updates must reflect the varying importance of contributions from different clients, considering that clients with more representative data or superior computational resources should have a greater influence on the global model. Fundamentally, both issues revolve around the optimal allocation of weights: client selection can be interpreted as assigning binary weights (0 or 1) to clients to determine their participation, while weight aggregation entails determining the proportional contribution of each client’s update. Studies have shown that unbalanced client participation and inappropriate weighting can detrimentally impact model convergence and accuracy. However, existing methods often treat client selection and weight aggregation independently, resulting in suboptimal outcomes that fail to capitalize on the interconnected nature of these processes. A holistic approach that jointly optimizes both client selection and weight aggregation could yield more efficient and accurate federated learning in heterogeneous edge environments.

In this paper, we aim to unify these two subproblems by presenting a new algorithm, FedCW (Federated Learning with Client Selection and Adaptive Weighting). This algorithm enhances Federated Learning in two key stages: client selection and weight allocation. we calculate the Euclidean distance between each client’s local model and the global model, prioritizing clients with greater divergence for selection. By dynamically adjusting the number of clients involved in each round and allocating weights based on both data volume and model divergence, the algorithm ensures that clients contributing more significant updates have a larger impact on the global model. This integrated approach simultaneously addresses both challenges, optimizing the learning process and improving global model convergence in heterogeneous federated learning environments. To validate our approach, we conduct experiments on various datasets, including MNIST, CIFAR-10, ImageNet-100, and CelebA. We evaluate the performance of FedCW against state-of-the-art algorithms. The experimental results demonstrate that FedCW significantly improves global model accuracy and convergence speed, especially in non-IID data scenarios. In the following, we provide a summary of contributions:

•   We integrate client selection and weight allocation into a unified optimization framework for Federated Learning. This approach effectively addresses the interdependent challenges of heterogeneous client environments, improving both the efficiency and accuracy of model training.

•   We introduce a novel client selection mechanism that prioritizes clients with diverse updates based on their Euclidean distance from the global model. Simultaneously, FedCW dynamically adjusts aggregation weights according to both data volume and model divergence, enhancing model convergence and generalization in non-IID environments.

•   We demonstrate that FedCW significantly outperforms existing algorithms through extensive experiments. Our method achieves faster convergence, higher model accuracy, and reduced communication overhead, particularly in non-IID federated learning scenarios.

2  Related Works

Client selection and weight allocation are considered open optimization problems in federated learning (FL), driven by the constraints of limited bandwidth and the necessity for selected participation in training. The problem of client selection is regarded as an ongoing optimization challenge within the FL process. Initial discussions on client selection, such as those by [15], introduced the concept of selecting clients based on their resource efficiency to enhance training convergence and model accuracy. This client selection paradigm has been foundational, sparking widespread research into optimizing FL with heterogeneous resources. However, choosing clients based merely on performance metrics such as speed or computational resources might lead to biases, potentially compromising data diversity. This challenge was addressed by [16], who provided a convergence analysis for client selection, advocating for a balanced approach that considers both computational and communication efficiencies. Further developments by [17] introduced a hybrid FL mechanism that aims to increase the number of participating clients and thus improve the accuracy of the aggregated model by enhancing the diversity of client data used during training. On the other hand, reference [18] explored dynamic data profiling to optimize client selection further, suggesting that understanding data characteristics dynamically could lead to more informed client choices in FL environments. Additionally, recent studies have proposed various frameworks to refine client selection further. For instance, reference [19]’s VFedCS framework focuses on optimizing client selection to manage the volatility and dynamic nature of client availability and resource allocation in federated settings. At the same time, recent vehicular/ITS surveys observe that most systems still decouple the decision of who participates from how much their updates should count during aggregation, which can entrench selection bias under non-IID and volatile participation [20,21].

Several algorithms have been developed to improve the aggregation phase in federated learning. Reference [22] proposed FedProx, which introduces a proximal term to the objective function, mitigating the impact of local updates diverging from the global model. This modification leads to more robust convergence in heterogeneous environments compared to FedAvg, particularly when devices perform a variable amount of local work. To tackle the client-drift problem caused by non-IID data, reference [23] introduced SCAFFOLD, which incorporates control variates to correct local updates. By reducing the variance of updates, SCAFFOLD improves the accuracy of the global model while requiring fewer communication rounds. In the domain of adaptive optimization, reference [24] presented FedOpt, which applies server-side optimization techniques such as Adam and Yogi during the aggregation process, leading to better performance in heterogeneous client environments. Additionally, reference [11] proposed FedNova, which normalizes local updates to eliminate objective inconsistency, ensuring that the global model converges to a solution of the correct objective function despite differences in local workloads. Reference [25] introduced FedMA, a novel aggregation technique that constructs a global model by matching and averaging neurons across layers of client models, effectively aligning the structure of local neural networks. This method enhances performance and reduces communication costs, particularly for complex neural network architectures such as CNNs and LSTMs. Nevertheless, these aggregation methods typically assume uniform or data-size-proportional weights and random/greedy participation, which implicitly presumes representativeness of the selected clients; vehicular studies indicate this assumption often breaks under mobility, skewed labels, and intermittent connectivity [20,21].

Security-oriented ITS works adopt FL for intrusion detection in in-/inter-vehicle networks, confirming feasibility but also revealing robustness and privacy tensions when participation is static and weights are near-uniform [26,27]. Resource-latency studies in vehicular edge computing (VEC) emphasize that client scheduling/offloading co-determines both spectral efficiency and learning dynamics; misaligned aggregation can offset scheduling benefits [28,29]. Communication-efficient FL with gradient quantization and DRL-based compression reduces bandwidth, yet injects additional noise/bias that calls for adaptive weighting and selection to prevent oscillation under harsh non-IID [30]. Application-driven ITS systems—traffic management and mobility/location prediction—report gains with FL but still rely on static participation and weighting, limiting resilience to churn and distribution drift [31,32].

Across client selection, aggregation, and vehicular deployments, three gaps emerge: (i) selection-aggregation decoupling—who participates is decided without calibrating how strongly their updates should influence the global model; (ii) sensitivity to non-IID and volatility—static policies amplify bias when participation is skewed or data distributions drift; (iii) engineering blind spots—scheduling/offloading and compression alter update statistics, yet server-side aggregation remains oblivious to these shifts.

Although the aforementioned works focus on client selection or weight allocation, they treat these as separate issues rather than addressing them as a unified problem. In this paper, we integrate both subproblems into a global optimization framework and propose the FedCW algorithm, which simultaneously tackles both challenges. Through extensive experiments, we compare our method with several of the aforementioned algorithms and demonstrate that FedCW effectively improves global accuracy and accelerates convergence.

3  Preliminaries

Training Process

FL involves multiple clients that train on locally generated data and periodically communicate with a global server to aggregate models. Specifically, the training process in each round comprises the following steps:

•   Initialize (Round 0): The global model is initialized on the server side and then distributed to all participating clients.

•   Client Selection and Model Distribution: The server selects the client according to the provided client model and information, and sends the aggregated global model to the selected client.

•   Local Training: Each selected client trains the received global model on their local data. This step involves running several epochs of training to adjust the model parameters based on the client’s unique data distribution.

•   Model Transmission: each client sends their updated local model back to the server after local training.

•   Model Aggregation: The server receives the successfully returned model and aggregates the updates to the new global model based on this algorithm.

•   Repeat Process: The new round of training begins with the client selection step, repeating steps 2 to 5 until the desired convergence or accuracy is achieved.

To graphically understand FL’s training process, we refer the reader to the Fig, which demonstrates each step of the training process through examples.

4  Problem Formulation

Global Objective

In the standard FL framework, our objective is to minimize the global loss function F(w), which for a fixed set of clients over T rounds. It can be expressed as:

minw∑k=1KpkFk(w)(1)

where Fk(w) is the local loss function of the k-th client, pk = (|D(k)|]/D) which is the weight of the k-th client’s data relative to the entire dataset, and w represents the parameters of the global model.

In our proposed method, the weights pk are no longer fixed or uniformly distributed among the clients. Instead, they are dynamically adjusted based on the Euclidean distance dk between the local model parameters wk of the k-th client and the global model parameters w. The modified weight pk is defined as:

pk=exp⁡(−λdk)∑j=1Kexp⁡(−λdj)(2)

where λ is a hyperparameter controlling the rate of exponential decay, and dk is the Euclidean distance between the local model of the k-th client and the global model. This formulation ensures that clients with models that are further from the global model have a larger influence on the model update, potentially accelerating convergence by incorporating more diverse updates into the global model. The updated global objective is then:

minwF(w)=∑k=1K(exp⁡(−λdk)∑j=1Kexp⁡(−λdj))Fk(w)(3)

where:

•   Fk(w) is the local loss function for the k-th client.

•   w represents the parameters of the global model.

•   dk is the Euclidean distance between the local model parameters wk of the k-th client and the global model parameters w.

•   λ is a hyperparameter that controls the rate of exponential decay, influencing how the distances affect the weights.

5  Client Selection with Adaptive Weight in FL

In this section, we introduce an enhanced FL algorithm. It can be conceptually divided into two stages: client selection and weight allocation. Algorithm 1 illustrates the operational procedure of our method. In the following, we will elucidate how Euclidean distances influence client selection and weight Allocation; we will also discuss how these elements integrate to form a cohesive whole.

5.1 Euclidean Distance Calculation

The server typically possesses significantly greater computational power and better resource allocation than individual clients. This enables efficient execution of complex mathematical computations, especially when involving multiple clients. Furthermore, performing these calculations on the server side also reduces the communication overhead by eliminating the need to transmit the computed distances. Therefore, starting from Round 1, server will compute the Euclidean distance from each client model to the current global model once the server receives and aggregates the models from all participating clients. The calculation is formulated as follows:

di=∑k(wk−wi,k)2(4)

The equation defined above calculates the Euclidean distance di between the global model and the model parameters of the i-th client. Here is a breakdown of the components within the equation:

•   wk represents the parameter vector of the global model. These parameters are aggregated values derived from the previous round of updates received from all or a subset of clients.

•   wi,k denotes the parameter vector of the i-th client’s model following local training on its dataset.

After the calculation is complete, the server will sort the di from largest to smallest. We will explain why later.

5.2 Client Selection

In this approach, we select clients based on a descending order of their Euclidean distance from the global model. Intuitively, selecting clients with the greatest divergence from the global model (i.e., the largest Euclidean distance) can introduce more information and data diversity. This is beneficial for the global model as it aids in learning and adapting to a broader data distribution, thereby reducing the risk of over-fitting. By prioritizing clients whose models significantly differ from the global model, each iteration can incorporate more substantial updates, potentially allowing the global model to adapt more quickly to the entire data distribution. We introduce a decay function to dynamically adjust the number of clients selected in each round. The decay function allows the algorithm to adjust the number of participating clients dynamically based on the progress of training or the performance of clients. This method can specifically increase or decrease the number of clients involved in training at different stages to meet the needs of the model during the training process. Given an initial selection fraction α and a decay rate λ, the client selection fraction in round t is:

αt=α×exp⁡(−λt)(5)

Adjusting the number of participating clients can more effectively utilize limited computational and communication resources. Initially, more clients may need to be involved to enhance the diversity of the model, while later in training, the number might be reduced to concentrate resources on optimizing a model that is nearing convergence. This method, compared to randomly selecting a fixed number of clients, improves the algorithm’s adaptability to different training stages and network conditions, particularly when facing non-IID data distributions. Let N be the total number of clients. The number of clients selected in round t is:

nt=max(⌈N×αt⌉,min_clients)(6)

where min_clients is a hyperparameter representing the minimum number of clients participating in each round.

To ensure the robustness and efficacy of the algorithm, especially when the total number of clients N is large and the proportion αt might be set relatively small. This parameter ensures sufficient model update diversity: it prevents inadequate model updates or excessive bias due to too few participating clients. It also ensures that enough clients participate to maintain the continuity and stability of model training, even if some clients may be unavailable due to technical issues.

5.3 Weight Allocation

In the FedAvg algorithm, the aggregation formula for updating the global model is traditionally given by:

wt+1=∑k=1Knknwkt(7)

This formula ensures that each client’s contribution to the global model is weighted by the proportion of data they hold relative to the total dataset. However, the sheer volume of data does not always reflect a client’s potential impact on the model. Therefore, in our algorithm, the weight allocation formula is modified as follows:

wk=nk⋅exp⁡(βdk)∑i=1Kni⋅exp⁡(βdi)(8)

•   nk is the size of the local dataset of client k.

•   dk is the Euclidean distance between the local model of client k and the global model.

•   β is an adjustment parameter that controls the influence of the Euclidean distance.

•   K is the number of participating clients.

Then the global model parameters are updated as follows:

wt+1=∑k=1Kwkwk(t)(9)

This weight allocation formula not only considers the volume of data each client contributes but also the divergence of each client’s model from the global model. It introduces a corrective mechanism that allows models with greater Euclidean distances (i.e., more divergent client models) to have a more significant voice in the aggregation process. By employing an exponential function regulated by β, we can flexibly adjust the impact of the Euclidean distance. This provides a method to balance the respect between data-rich clients and those clients whose model parameters significantly differ. The denominator ∑i=1Kni⋅exp⁡(βdi) acts as a normalization factor, ensuring that the sum of all client weights equals 1, thus maintaining stability in the model update steps. By combining data volume and model divergence, the algorithm can balance the influence of different clients on the global model, utilizing the information provided by data-rich clients while preventing the model from deviating from the optimization path due to the excessive influence of outlier clients.

When β is set to a positive value, it implies that as the Euclidean distance dk between a client’s local model and the global model increases, so does their weight in the model update. In this setting, as dk increases (i.e., the greater the divergence of the client model from the global model), the greater their influence on the global model. This approach prioritizes clients with substantial differences from the global model, theoretically assisting the global model to quickly adapt and learn from the unique data patterns represented by a minority of clients. Applicable scenarios: This is particularly suitable in situations where the global model needs to learn from significant differences in client models, especially in cases of highly non-independent and identically distributed (Non-IID) data, helping to enhance the model’s diversity and generalization capabilities.

When β is set to a negative value, the weights decrease as dk increases. This means that clients with smaller differences from the global model gain greater weight, aiding the global model to learn more robustly towards the central tendency of the data, reducing fluctuations caused by extreme or deviating clients.

This method of weight allocation can enhance the robustness and generalization ability of the global model in federated learning. It ensures that all relevant characteristics, such as data volume and model uniqueness, are appropriately considered in weighting each client’s contribution. This method is particularly effective in heterogeneous environments, as client data distributions and system characteristics often vary greatly in reality. We will demonstrate this in the experiments.

5.4 Synergy of Client Selection and Weight Allocation

In this algorithm, client selection and weight allocation are two interdependent components that together determine the efficiency and quality of the global model updates. They are less effective when existing alone while these components can be understood independently. And they complement each other when integrated, achieving optimal learning outcomes.

Limitations of Client Selection: Client selection alone, without appropriate weight allocation, may lead to an over-reliance on certain clients’ data. Particularly in scenarios where client data is highly imbalanced, selecting clients with larger data volumes can result in a model bias towards these clients’ characteristics, overlooking other important but smaller data volume clients. Purely basing client selection on criteria such as data volume or update frequency may neglect the differences between client models and the global model, which is detrimental to capturing the diversity of data across the entire network.

Limitations of Weight Allocation: If there is only weight allocation without an effective client selection mechanism, then weights might be allocated to clients that contribute minimally to model updates. For instance, clients with minimal divergence from the global model might receive higher weights, but these minor updates may not be sufficient to significantly enhance the model. Relying solely on weight allocation to handle all clients can lead to inefficient computations, especially when the number of clients is very large, resulting in significant computational and communication overhead that is largely unnecessary.

Client selection can reduce the number of clients that need to be processed each round, thereby reducing computational and communication burdens. On the other hand, weight allocation ensures that the maximum learning benefit is extracted from these selected clients, particularly by weighting updates from those clients who differ significantly from the global model, thereby accelerating model convergence. This combined strategy effectively learns from the data across the entire network, avoiding model over-fitting to specific client data and thus improving the model’s performance on unseen data.

5.5 Communication Complexity

FedCW requires each selected client to send its local model parameters to the server as in standard FL. The server computes the Euclidean distance dk for all participating clients, with complexity O(Nd) for N clients and d-dimensional parameters, followed by sorting in O(Nlog⁡N). These costs are negligible compared to the communication of model parameters. Furthermore, since FedCW dynamically reduces the number of participating clients via αt, the overall communication overhead is significantly reduced compared to FedAvg, as also validated in our experiments.

5.6 Security and Privacy Considerations

While FedCW demonstrates clear advantages in convergence and communication efficiency, its deployment in real-world vehicular networks also raises important concerns related to security, privacy, and system-level robustness. In particular, the Euclidean distance ranking strategy, although effective for selecting diverse clients, may inadvertently expose distributional characteristics of local model updates, thereby creating potential side-channel privacy risks. To mitigate this, future implementations of FedCW could be combined with secure aggregation protocols or differential privacy mechanisms that protect individual updates while still allowing the server to compute distance-based rankings. Another issue is the vulnerability of distance-based selection and adaptive weighting to malicious or Byzantine clients, who may inject manipulated updates to distort the aggregation process. Addressing this challenge requires the integration of anomaly detection techniques or Byzantine-resilient aggregation rules that can identify and suppress adversarial contributions without significantly increasing communication overhead. Beyond adversarial risks, large-scale vehicular deployments often encounter system bottlenecks such as channel congestion, synchronization delays, and frequent client churn, which can compromise the effectiveness of synchronous distance-based selection. To address these challenges, FedCW can be extended with asynchronous update strategies or hierarchical aggregation frameworks, where edge servers first coordinate subsets of vehicles before forwarding updates to the global server, thereby reducing latency and improving scalability.

5.7 Hyperparameter Discussion

The hyperparameters λ and β play central roles in balancing convergence and stability. The parameter λ controls the decay rate of client participation: larger λ leads to fewer clients in later rounds, reducing communication but potentially harming diversity, while smaller λ retains more clients, improving robustness but increasing overhead. The parameter β regulates the impact of Euclidean distance in weight allocation: positive β emphasizes clients with divergent models, accelerating adaptation to non-IID data but risking instability when extreme; negative β favors alignment with the global model, improving stability but slowing adaptation. Hence, λ and β jointly determine the trade-off between convergence speed, stability, and communication cost, and must be tuned according to heterogeneity levels and system constraints.

5.8 Theorem Analysis

Notation and update rule.

Let 𝒦t be the set of selected clients at round t. Denote server iterate by wt and client k’s local iterate after its local training by wkt. Let akt≥0 be the server aggregation weights with ∑k∈𝒦takt=1 (in FedCW, akt∝nkexp⁡(βdk)). Define the aggregated direction

gt = ∑k∈𝒦takt∇fk(wkt).(10)

The server performs the update

wt+1 = wt − ηtgt,(11)

where ηt>0 is the stepsize.

Assumption 1 (Smoothness): Each fk is L-smooth, i.e., for all w1,w2,

‖∇fk(w1)−∇fk(w2)‖ ≤ L‖w1−w2‖.(12)

Consequently, the global objective F(w)=1N∑k=1Nfk(w) is also L-smooth.

Assumption 2 (Bounded gradients and variance): There exists G>0 such that for all k,w,

‖∇fk(w)‖ ≤ G.(13)

Let ζt:=gt−E[gt|wt] be the zero-mean noise conditioned on wt. There exists σ2≥0 such that

E[‖ζt‖2|wt] ≤ σ2.(14)

Assumption 3 (Selection/weighting and local-drift bias): Define the conditional-mean decomposition at the server point wt:

E[gt|wt] = ∇F(wt) + εt,(15)

where εt collects (i) selection/weighting mismatch (𝒦t⊂[N], non-uniform akt) and (ii) local drift because ∇fk(wkt) is evaluated at wkt≠wt. Assume the average bias energy is bounded:

1T∑t=0T−1E‖εt‖2 ≤ ε2¯.(16)

Step 1: Descent lemma (exact algebra, no convexity).

By L-smoothness of F and the update (11),

F(wt+1)≤F(wt)+⟨∇F(wt),wt+1−wt⟩+L2‖wt+1−wt‖2=F(wt)−ηt⟨∇F(wt),gt⟩+Lηt22‖gt‖2.(17)

Step 2: Decompose gt and expand the inner product.

From (15), write

gt = ∇F(wt) + εt + ζt,E[ζt|wt]=0.(18)

Then

⟨∇F(wt),gt⟩=⟨∇F(wt),∇F(wt)⟩+⟨∇F(wt),εt⟩+⟨∇F(wt),ζt⟩=‖∇F(wt)‖2+⟨∇F(wt),εt⟩+⟨∇F(wt),ζt⟩.(19)

Taking conditional expectation given wt, the noise term vanishes:

E[⟨∇F(wt),ζt⟩|wt]=0.(20)

Using Young’s inequality ⟨a,b⟩≥−12‖a‖2−12‖b‖2 with a=∇F(wt) and b=εt gives

−⟨∇F(wt),εt⟩ ≤ 12‖∇F(wt)‖2+12‖εt‖2.(21)

Multiplying (19) by −ηt and taking expectation yields

E[−ηt⟨∇F(wt),gt⟩] ≤ −ηt2E‖∇F(wt)‖2 + ηt2E‖εt‖2.(22)

Step 3: Bound the quadratic term ‖gt‖2.

From (18) and (x+y+z)2≤3(x2+y2+z2),

‖gt‖2 ≤ 3‖∇F(wt)‖2 + 3‖εt‖2 + 3‖ζt‖2.(23)

Taking expectation and using (14),

E‖gt‖2 ≤ 3E‖∇F(wt)‖2 + 3E‖εt‖2 + 3σ2.(24)

Step 4: One-step expected descent inequality.

Taking expectation of (17) and substituting (22) and (24),

EF(wt+1)≤EF(wt)−ηt2E‖∇F(wt)‖2+ηt2E‖εt‖2+Lηt22(3E‖∇F(wt)‖2+3E‖εt‖2+3σ2).(25)

Group coefficients of each term:

EF(wt+1) ≤ EF(wt)−(ηt2−3Lηt22)E‖∇F(wt)‖2+(ηt2+3Lηt22)E‖εt‖2+3Lηt22σ2.(26)

Step 5: Stepsize choice and telescoping.

Choose a constant stepsize satisfying

ηt≡η ≤ 16L⟹η2−3Lη22 ≥ η4,η2+3Lη22 ≤ 3η4.(27)

Then (26) becomes

EF(wt+1) ≤ EF(wt)−η4E‖∇F(wt)‖2+3η4E‖εt‖2+3Lη22σ2.(28)

Summing (28) over t=0,…,T−1 and telescoping gives

η4∑t=0T−1E‖∇F(wt)‖2 ≤ EF(w0)−EF(wT)+3η4∑t=0T−1E‖εt‖2+3Lη22Tσ2.(29)

Using F(wT)≥F⋆ and dividing by T yields the ergodic stationarity bound

1T∑t=0T−1E‖∇F(wt)‖2 ≤ 4(F(w0)−F⋆)ηT + 31T∑t=0T−1E‖εt‖2 + 6Lησ2.(30)

Theorem 1 (Stationarity with selection/weighting bias): Under (12)–(16) and η≤1/(6L),

1T∑t=0T−1E‖∇F(wt)‖2 ≤ 4(F(w0)−F⋆)ηT + 3ε2¯ + 6Lησ2.(31)

Remark 1 (Interpretation and roles of λ,β): In FedCW, the bias εt is controlled by the coverage of selected clients and the skewness of weights akt. A larger decay rate λ (fewer clients in later rounds) or a larger |β| (heavier emphasis on dk) typically increases ‖εt‖, enlarging the 3ε2¯ term and slowing convergence (or enlarging the neighborhood). Conversely, ensuring a minimal selection size and tempering |β| reduce ε2¯.

Corollary 1 (Convex case): If, additionally, F is convex and ηt is diminishing with ∑tηt=∞, ∑tηt2<∞, and 1T∑t=0T−1E‖εt‖2→0, then every limit point of {wt} is stationary; for convex F, stationary points are globally optimal.

Additional nonconvex result: iteration complexity and neighborhood size

Goal.

We turn (30) into an explicit iteration complexity statement under nonconvexity by optimizing the stepsize. Let A:=4(F(w0)−F⋆) and B:=3ε2¯. Then for constant η≤1/(6L),

1T∑t=0T−1E‖∇F(wt)‖2 ≤ AηT + B + 6Lησ2.(32)

Optimizing the constant stepsize.

For fixed T, the right-hand side of (32) as a function of η is

Φ(η) = AηT + 6Lησ2 + B,η∈(0,16L].(33)

Ignoring the box constraint momentarily, the unconstrained minimizer solves −Aη2T+6Lσ2=0, i.e.,

η⋆ = A6Lσ2T.(34)

If η⋆≤16L, plugging (34) back into (32) yields

1T∑t=0T−1E‖∇F(wt)‖2 ≤ 26LAσ2T + B = CT⏟stochastic term + B⏟bias floor,(35)

where C:=26LAσ2. Thus the ergodic stationarity measure decays as O(1/T) until it hits the bias floor B=3ε2¯.

Reaching an ϵ-stationary point (ergodic).

Given a target ϵ>0, to ensure

1T∑t=0T−1E‖∇F(wt)‖2 ≤ ϵ2,(36)

we need ϵ2>B and

CT ≤ ϵ2−B⟺T ≥ C2(ϵ2−B)2 = 24 L(F(w0)−F⋆)σ2(ϵ2−B)2.(37)

Hence, under nonconvexity and stochasticity, FedCW attains an ϵ-stationarity level in T=O(1/(ϵ2−B)2) rounds, provided the bias floor B=3ε2¯ is below ϵ2. Reducing B (e.g., by ensuring a minimum selection size and tempering |β|) directly improves the attainable accuracy and the iteration complexity.

When η⋆>1/(6L).

If (34) violates the box constraint, set η=1/(6L) in (32) to obtain

1T∑t=0T−1E‖∇F(wt)‖2 ≤ 24L(F(w0)−F⋆)T + B + σ2L,(38)

which shows O(1/T) decay of the optimization term plus a fixed variance floor σ2/L and the bias floor B.

Diminishing stepsizes.

Alternatively, choosing ηt=η0/t+1 (with η0 small enough) yields the classical nonconvex rate

1T∑t=0T−1E‖∇F(wt)‖2 ≤ O(1T) + B,(39)

again saturating at the bias floor B.

Remark 2 (Practical knobs): The bias floor B=3ε2¯ shrinks when (i) the per-round selection size does not become too small (controlled by λ), and (ii) the weight skewness is moderated (controlled by |β|). These knobs trade off communication/computation with convergence speed and final accuracy. Moreover, λ directly controls the number of participating clients per round: a larger λ lowers communication overhead but increases the risk of slower convergence due to reduced diversity, while a smaller λ improves convergence at the cost of higher communication. Similarly, |β| determines the emphasis on divergent updates: larger values may accelerate convergence under moderate non-IID but risk instability under extreme heterogeneity, which could increase the total number of communication rounds required. Therefore, the choice of λ and β balances communication cost against convergence efficiency in non-IID environments.

6  Experiment

In this section, we will introduce the experiment setup and the results.

6.1 Experiment Set

The experiments were conducted on a machine equipped with an AMD Ryzen 5600X processor, 32 GB DDR4 RAM, a 1 TB SSD, and an NVIDIA GeForce RTX 4070ti super GPU. The experimental environment was set up using Python 3.8, with machine learning frameworks TensorFlow 2.5 and PyTorch 1.9 employed for implementing and training neural networks. We also used CUDA 11.8 with cuDNN support, and fixed random seeds (set to 42) for NumPy, PyTorch, and TensorFlow to ensure reproducibility. These tools and hardware configurations provided an efficient environment to conduct the experiments and analyze the results.

We utilized the LEAF framework [33], which is a benchmarking framework for learning in federated settings. LEAF provides tools and datasets for various applications, including federated learning, multi-task learning, meta-learning. This framework allowed us to simulate realistic federated environments and evaluate the performance of FedCW and baseline algorithms under different data distributions and client behaviors.

We evaluated FedCW on four image classification tasks using the MNIST, CIFAR-10, ImageNet-100, and CelebA datasets. To simulate different degrees of Non-IID across clients, we used a Dirichlet distribution to partition the data. Specifically, the training data for each dataset was divided among clients by sampling from a Dirichlet distribution with varying concentration parameters (α). For reproducibility, we set α=0.1,0.5,1.0,10 in experiments to represent strong non-IID, moderate non-IID, near-IID, and IID-like distributions, respectively. Lower values of α correspond to higher degrees of Non-IID, meaning that clients receive data concentrated on fewer classes. Conversely, higher values of α lead to a more IID-like distribution, where each client receives a more balanced share of the data across all classes. This method allows us to systematically control the degree of data heterogeneity among clients and assess the performance of FedCW under different Non-IID settings.

In our experiments, we focused on full client participation for each communication round. We compared FedCW with several popular baselines, including FedAvg, FedProx, SCAFFOLD, FedOpt, FedNova, and FedMA, across all datasets. Each algorithm was implemented using the same neural network architectures to ensure fair and consistent comparisons. The hyperparameters for each method, such as learning rate, batch size, and number of communication rounds, were tuned empirically based on prior experience. For FedCW specifically, we report the exact values used for λ and β in the experiment tables to ensure reproducibility.

6.2 Experiment Results

6.2.1 Accuracy and Communication Efficiency Comparison

Table 1 shows the performance of FedCW across all four datasets (MNIST, CIFAR-10, ImageNet-100, and CelebA) in terms of communication overhead, computation time, FLOPs, and accuracy. On the MNIST dataset, FedCW achieves a communication overhead of 198.6 MB, a 35% reduction compared to FedAvg’s 305.3 MB. Similar reductions are seen on CIFAR-10 (401.8 MB for FedCW vs. 603.2 MB for FedAvg, a 33% decrease). This substantial reduction is due to FedCW’s selective client participation strategy based on Euclidean distance, which prioritizes clients contributing the most valuable updates, thus minimizing unnecessary data exchange.

In terms of computation time, FedCW completes training faster; on MNIST, it takes 4173.25 s, a 17% reduction from FedAvg’s 5025.6 s. This speed-up comes from focusing only on clients whose updates are most informative, enhancing convergence and reducing the need for prolonged training. Furthermore, FedCW demonstrates computational efficiency with a 9.4% reduction in FLOPs on MNIST compared to FedAvg, owing to its ability to learn quickly from divergent updates and avoid redundant computations.

The higher accuracy achieved by FedCW—92.27% on MNIST, which is 1.84% higher than FedAvg—is attributed to its dynamic client selection process. By selecting clients that provide the most significant contributions to the model, FedCW maintains robustness, especially in non-IID data scenarios. In contrast, other algorithms like FedAvg involve random client selection, which may include clients with less impactful updates, resulting in slower convergence and lower accuracy. Thus, FedCW’s strategic client selection and efficient aggregation contribute to its superior performance.

6.2.2 Ablation Study on FedCW Components

The plot in Fig. 1 illustrates the results of an ablation study comparing different versions of the FedCW algorithm over 100 communication rounds. It includes the complete FedCW algorithm and versions where client selection or weighted averaging is removed, as well as a baseline random client selection. From the plot, the complete FedCW algorithm achieves the highest accuracy, starting at around 67% and steadily rising to the capped maximum of 95%. In comparison, the version of FedCW without weighted averaging starts lower, around 65%, and improves more slowly, ultimately plateauing below 90%. Similarly, the version without client selection performs better than random selection but still underperforms compared to the complete FedCW algorithm, showing the importance of both components. Random client selection achieves the lowest accuracy, indicating that without any strategic selection mechanism, the model struggles to achieve high accuracy, with slower improvements over the rounds.

Figure 1: Impact of different components on FedCW performance. Different lines shows the performance of FedCW when key components such as client selection and weighted averaging are removed

The superior performance of FedCW can be attributed to two key components: client selection and weighted averaging. The client selection strategy, which prioritizes clients with the most informative updates, ensures that each round of communication focuses on maximizing the global model’s progress. By selecting clients with more divergent local data, the algorithm captures a wider range of variations in the dataset, leading to faster convergence. The weighted averaging mechanism dynamically adjusts the importance of client contributions based on their relevance, ensuring that significant updates have a larger impact on the global model. This helps maintain balance and avoids overfitting to any particular subset of clients, which is crucial in non-IID federated learning environments where client data distributions are heterogeneous. In contrast, removing these components diminishes the model’s ability to effectively utilize client data, resulting in slower convergence and lower final accuracy, as shown by the other lines in the plot. Overall, FedCW’s combination of these strategies allows it to leverage diverse and non-IID data efficiently, leading to faster convergence and higher overall performance compared to ablated versions of the algorithm.

6.2.3 Impact of Non-IID Data Distribution on Algorithm Performance

The experimental results illustrated in Fig. 2 demonstrate the robustness and adaptability of the FedCW algorithm across multiple datasets. The goal of this experiment was to assess the performance of FedCW under different levels of Non-IID data distributions. As observed, FedCW consistently outperforms the baseline algorithms such as FedAvg, FedProx, SCAFFOLD, FedOpt, FedNova, and FedMA across all degrees of Non-IID. For example, in the MNIST dataset, FedCW reaches an accuracy of 97% under IID conditions, while FedAvg only achieves 91.5%. A similar trend can be observed across other datasets like CIFAR-10, where FedCW outperforms FedAvg significantly, achieving 75% accuracy compared to 60% for FedAvg under IID conditions. These results also highlight the intrinsic relationship between non-IID distributions, convergence, and communication overhead. As the degree of non-IID increases (lower α), all algorithms require more communication rounds to reach the same accuracy, reflecting slower convergence. However, FedCW mitigates this cost by selecting only the most informative clients and adaptively weighting their updates, which reduces redundant transmissions and accelerates progress. In practice, this means that FedCW can achieve a target accuracy with fewer communication rounds than baselines under strong non-IID, thereby lowering the overall communication overhead while maintaining robust convergence behavior.

Figure 2: Accuracy of federated learning algorithms under varying degrees of Non-IID Data. The degrees of Non-IID are controlled by four different values of the dirichlet distribution’s concentration parameter (α). In this experiment, α values of 0.1, 0.5, 1.0, and 10 were used to simulate increasing levels of data heterogeneity among clients. A lower α (e.g., 0.1) represents a highly Non-IID distribution, while a higher α (e.g., 10) leads to a more IID-like distribution

6.2.4 Convergence Speed Comparison across Algorithms

In this experiment, we compare the convergence speed of several federated learning algorithms, including FedCW, FedAvg, FedProx, SCAFFOLD, FedOpt, FedNova, and FedMA, by tracking their training loss over 100 communication rounds on the MNIST dataset. As illustrated in Fig. 3, FedCW demonstrates the fastest convergence, rapidly reducing its training loss to around 0.5 within the first 10 rounds and maintaining a stable performance throughout the remaining rounds on the MNIST dataset. This behavior is attributable to its client selection strategy and dynamic weighted averaging, which ensures that the most informative clients are selected, and their updates are efficiently aggregated. FedProx and FedMA follow in terms of convergence speed, with FedProx benefiting from the addition of a proximal term that stabilizes local updates in non-IID environments, leading to faster convergence compared to FedAvg. FedMA, with its layer-wise neuron matching, also performs well by preserving more structural information during aggregation, thereby reducing the loss more effectively than the other methods. SCAFFOLD exhibits a slightly slower convergence than FedCW and FedProx, although it manages to address client drift by utilizing control variates. FedOpt and FedNova show moderate improvements over FedAvg but are slower than FedCW due to their simpler aggregation mechanisms. FedAvg, as the baseline, has the slowest convergence, as it relies on random client selection and equal-weight averaging, which are less effective in non-IID settings. In several other data sets, the experimental results show similar characteristics. The results highlight the importance of sophisticated client selection and weighted aggregation strategies, as seen in FedCW, in achieving faster convergence, especially in heterogeneous federated learning environments.

Figure 3: Convergence speed of algorithms (Training Loss vs. Communication Rounds)

6.2.5 Effect of Client Participation on Accuracy

As previously discussed, client selection and weight allocation are fundamentally the same problem. Therefore, the FedCW algorithm, which addresses both issues simultaneously, demonstrates a significant advantage over other algorithms. Fig. 4 clearly reveal this. The results from the experiment testing the impact of varying client numbers on accuracy reveal that FedCW consistently achieves superior performance compared to other algorithms across different datasets, including MNIST, CelebA, CIFAR-10, and ImageNet-100. As the number of clients increases, FedCW is better able to aggregate diverse and representative data, allowing the model to generalize more effectively and maintain high accuracy. In contrast, algorithms like FedAvg, which use simple averaging across all clients, struggle to maintain high accuracy as the number of clients increases, particularly in non-IID settings. FedProx, SCAFFOLD, FedOpt, FedNova, and FedMA also benefit from their respective aggregation and optimization strategies, but none match FedCW’s ability to handle a large number of clients while maintaining robust performance. FedCW’s ability to adaptively adjust weight allocation during aggregation ensures that the most valuable updates are emphasized, further enhancing the model’s convergence speed and overall accuracy. This makes FedCW particularly advantageous in federated learning environments with a varying number of participating clients.

Figure 4: Accuracy vs. number of clients across multiple datasets. Each bar represents the performance of different algorithms, including FedCW, FedAvg, FedProx, SCAFFOLD, FedOpt, FedNova, and FedMA

7  Conclusion

In this paper, we proposed FedCW. It is an algorithm designed to simultaneously address the challenges of client selection and weight allocation in heterogeneous Federated Learning (FL) environments. Our approach leverages digital twins to assist in real-time computation offloading, while selecting clients based on their Euclidean distance from the global model and dynamically adjusting aggregation weights to balance data volume and model divergence. Through extensive experiments, we demonstrated that FedCW significantly improves model accuracy and reduces convergence time compared to existing methods such as FedAvg, FedProx, and SCAFFOLD, particularly in non-IID settings. In the future, further work can focus on enhancing the adaptability of FedCW to even more dynamic edge environments and exploring more advanced techniques for optimizing resource allocation in large-scale FL systems.

Acknowledgement: Not applicable.

Funding Statement: The authors received no specific funding for this study.

Author Contributions: The authors confirm contribution to the paper as follows: Conceptualization, Haotian Wu and Jiaming Pei; methodology, Haotian Wu; software, Haotian Wu; validation, Haotian Wu; formal analysis, Haotian Wu; investigation, Haotian Wu; resources, Jinhai Li; data curation, Jinhai Li; writing—original draft preparation, Haotian Wu and Jiaming Pei; writing—review and editing, Haotian Wu and Jiaming Pei; visualization, Haotian Wu; supervision, Jiaming Pei and Jinhai Li; project administration, Jinhai Li; funding acquisition, Jinhai Li. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: The data that support the findings of this study are available from the corresponding author upon reasonable request.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest to report regarding the present study.

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