arXiv CSopen access

FedDRAW: Federated Dual Reputation Annealing Weighting for Heterogeneous Multi-Institutional Chest Radiograph Classification

Highlights FedDRAW: Federated Dual Reputation Annealing Weighting for Heterogeneous MultiInstitutional Chest Radiograph Classification Maryam Moradpour, Anne-Christin Hauschild • FedDRAW: federated learning that schedules aggregation influence across rounds instead of fixing it by hospital size

arXiv:2609.05223v1 [cs.LG] 4 Sep 2026

• Client weights shift from data size to model similarity during training • A deferred schedule returns all client weights to equal at convergence • Small hospitals are not permanently outweighed by larger ones • Best average rank compared to eight state-of-the-art methods on twelve chest X-ray client partitions

FedDRAW: Federated Dual Reputation Annealing Weighting for Heterogeneous Multi-Institutional Chest Radiograph Classification Maryam Moradpoura , Anne-Christin Hauschilda,b a Institute for Predictive Deep Learning in Medicine and Healthcare, Justus-Liebig University, Gießen, Germany, b Institute for Medical Informatics, University Medical Center Göttingen, Germany,

ARTICLE INFO

ABSTRACT

Keywords: Federated learning Aggregation weighting Simulated annealing Non-IID data Chest radiograph classification Medical image analysis

Artificial intelligence models are promising for medical diagnosis, but they require large numbers of unbiased data, which in medicine are distributed across hospitals and cannot be centralized to protect patient privacy. Federated Learning (FL) addresses this, since hospitals train one shared diagnostic model while patient data remain local. Training proceeds in communication rounds, in which each hospital trains the shared model locally and returns it to the server for merging by weighted average. This aggregation weight determines whose institutional knowledge shapes the result. Federated averaging (FedAvg) sets it in proportion to local sample count, so a small but informative hospital is permanently assigned a small influence, and largest clients could dominate the global model even when they are less informative. We propose Federated Dual Reputation Annealing Weighting (FedDRAW), a server-side aggregation method that combines a data-size prior with the cosine similarity between client and global parameters under two coupled annealing schedules. An inner schedule shifts client reputation from the size prior towards similarity. An outer, deferred annealing schedule on the softmax inverse temperature keeps the weighting selective in the early and middle rounds and relaxes it to uniformity at convergence. We evaluate FedDRAW on 12 simulated client-partition scenarios of two chest radiograph datasets (CheXpert and ChestMNIST), against seven federated baselines under identical local training settings. FedDRAW achieved the highest average rank among all eight methods under both AUC and the geometric mean (GM) of sensitivity and specificity, which a Friedman test with Nemenyi post-hoc analysis confirmed to be a statistically significant difference between the methods. Scheduling two signals, rather than fixing the weights by sample count alone, could enable less biased diagnostic models.

1. Introduction Machine learning, and deep learning in particular, has shown promising results for clinical tasks that include diagnosis, risk prediction, and treatment planning (Rajpurkar et al., 2022), and medical image analysis is one of the areas in which deep neural networks have been applied most extensively (Litjens et al., 2017). Such models generally require large numbers of annotated cases, and the training data should cover the variation that occurs across the patient populations on which the model is later used. A single institution may not satisfy either requirement, because its cohort could reflect one catchment population, one set of scanners and acquisition protocols, and one annotation practice; hence, a model trained at one location might carry the bias of that location and perform worse elsewhere (Tayebi Arasteh et al., 2023; Li et al., 2021c). Conventional centralized training assumes that the data required to build such models can be assembled in one location. In healthcare, this assumption rarely holds. Clinical data remain distributed across hospitals and medical centers and cannot readily be pooled because of patient privacy, legal and regulatory constraints, institutional data-governance policies, data ownership, and security considerations (Kairouz et al., 2021; Guan et al., 2024). Privacy, however, is not the only obstacle. Clinical data are naturally distributed among institutions; an individual site often holds a dataset that is small, imbalanced, or unrepresentative of the wider patient population, and a locally trained model therefore generalizes poorly (Guan et al., 2024). Transferring large clinical archives to a single center may also be impractical. In multi-institutional chest-radiograph analysis, for example, collaborative training has been shown to improve diagnostic performance on an unseen institution relative to single-site training, with the largest on-domain benefit observed for institutions holding smaller local datasets (Tayebi Arasteh et al., 2023). Federated learning (FL) (McMahan et al., 2017) was introduced as a collaborative learning paradigm in which participating institutions train a shared model while their raw data remain local (Hauschild et al., 2022; Park et al., ORCID (s): 0009-0001-0906-5842 (M. Moradpour); 0000-0002-7499-4373 (A. Hauschild)

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2024; Moradpour et al., 2026). FL has accordingly been taken up across multi-institutional healthcare applications, with medical imaging among the most active (Guan et al., 2024). Therefore, in this study we use public medical imaging datasets as examples. The data held by participating institutions, however, are not statistically identical. Heterogeneity in FL is not a single phenomenon but comprises feature-distribution skew, label-distribution skew, concept drift, concept shift, and quantity skew (Kairouz et al., 2021). In clinical imaging, these forms occur together, because institutions differ in the number of patients they serve, in disease prevalence and class proportions, in scanners, acquisition protocols and contrast settings, in patient populations, and in annotation protocols and the interpretation of ambiguous findings (Li et al., 2021c; Liu et al., 2026; Tayebi Arasteh et al., 2023). FL settings are commonly distinguished by how these data are partitioned across participants, and three categories are conventionally identified, namely horizontal FL, vertical FL, and federated transfer learning (Yang et al., 2019; Kairouz et al., 2021). In horizontal FL, the participants hold different samples that are described by the same features and address the same learning task. In vertical FL, they hold complementary feature subsets for a largely overlapping set of samples, and each participant therefore optimizes a different part of the model. In federated transfer learning, the participants differ in both their samples and their feature spaces, and knowledge is transferred between the resulting heterogeneous domains rather than merged into one shared model (Yang et al., 2019). This distinction determines what the server is able to aggregate, and horizontal FL is the relevant category for the present work. Because horizontally partitioned clients share a feature space and a prediction task, they can train an identical network architecture whose parameters are dimensionally and semantically comparable, and local models and their differences can therefore be compared and merged directly at the server (Shi et al., 2025; Rehman et al., 2023). This study adopts a horizontal cross-silo setting, in which healthcare institutions hold different patient cohorts and collaboratively train a compatible global model. A communication round then proceeds in four steps. The server distributes the current global model to the participating clients, each client trains that model on its local data, the clients return their locally updated parameters, and the server aggregates the received models to form the next global model (McMahan et al., 2017). Because heterogeneous clients produce different local models, the server must decide how strongly each client should affect the result. We refer to this quantity as the aggregation weight, that is, the relative influence of a client’s local update on the next global model, and it is the component of the federated pipeline that this work modifies. Federated Averaging (FedAvg) (McMahan et al., 2017) is the standard reference for this step. The server averages the returned local models with weights proportional to the number of local training samples; hence the aggregation criterion is the client sample count and nothing else. The rule is inexpensive, requires no information beyond quantities the server already knows, and reflects the reasonable expectation that larger local datasets yield statistically more stable local estimates. Its criterion, however, remains a function of sample quantity throughout training, and several studies have argued that this quantity is an incomplete indicator of a client’s value. Dataset size does not reveal how representative a client’s class distribution is of the intended global objective (Ye et al., 2023); larger local datasets do not necessarily produce more reliable updates, and size-based weighting can amplify poorly trained or distributionally biased client models (Kundroo et al., 2026). Because weight is tied to volume, methods of this type judge a client by the amount of data it holds rather than by how informative that data is, so a small institution may retain persistently limited influence even when its updates are valuable (Zeng et al., 2021). This concern is not only theoretical, as increasing the sample counts of clients whose local models perform poorly has been shown to destabilize size-weighted averaging while leaving distance-based weighting stable (Yeganeh et al., 2020). Subsequent FedAvg-based methods addressed related but distinct difficulties. FedProx (Li et al., 2020a) constrains excessive local deviation through a proximal penalty on the local objective and admits partial work from slower devices, targeting statistical and systems heterogeneity together, whereas SCAFFOLD (Karimireddy et al., 2020) corrects client drift by adjusting each local gradient with server and client control variates, at the cost of persistent per-client state. Meanwhile, FedNova (Wang et al., 2020) addresses the objective inconsistency caused by unequal local computation, normalizing each client’s accumulated update by its local progress, hence clients performing more local steps do not acquire disproportionate influence. These methods improve local optimization stability, drift correction, and update normalization, yet none of them reconsiders the criterion by which client importance is assigned. FedNova is instructive in this respect, because it restores the intended sample-size objective rather than asking whether that objective distributes influence appropriately. Conventional FedAvg-based approaches therefore improve federated optimization under heterogeneous conditions while leaving unresolved the persistent dominance of data-size-based influence and the limited contribution assigned to small but potentially informative clients.

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A natural response is to replace or supplement the sample count with a signal derived from the client models themselves, and model similarity has become the most widely used such signal because it relates local client knowledge to the evolving global model. An early and influential step in this direction is FedAdp (Wu and Wang, 2021), which measures the angle between each inferred local gradient and a sample-size-weighted estimate of the global gradient and converts a smoothed version of this angle into contribution scores through a decreasing nonlinear mapping and a softmax. The same work, however, also exposed a weakness of the signal itself. The angle is highly random at the beginning of federated training, and FedAdp restrains this instability by averaging the angle over its history, a remedy that still lets similarity act on aggregation from the very first round, when the smoothed value coincides with the instantaneous one. Later methods refined where and how similarity is measured. L-DAWA (Rehman et al., 2023), developed for federated self-supervised representation learning, moves the comparison to the level of individual layers and scales each layer’s contribution by its angular alignment with the previous global model, while SimProx (ElNiss et al., 2024) combines cosine and Gaussian similarity after finding a single metric incomplete, at the price of a pairwise matrix whose cost grows quadratically with the number of clients. FedSim (Palihawadana et al., 2022) instead turns similarity into a clustering criterion, grouping clients by their gradients in every round partly to prevent a large group of similar clients from dominating the global update, although the benefit of this clustering depends on the heterogeneity regime. A further question concerns what similarity actually rewards. FedSaC (Yan et al., 2024) shows that collaboration among the most similar clients may add no information, whereas moderate differences between client feature distributions can supply complementary knowledge, and personalized federated methods have been designed explicitly to avoid the negative transfer that collaboration with unrelated clients can induce (Xu et al., 2023). A client that diverges from the current global model may hold rare presentations, underrepresented populations, or unusual acquisition conditions; hence, low similarity should not by itself reduce its contribution to a negligible level. Taken together, these studies establish similarity as an informative signal, yet none of them adjusts the trust placed in that signal according to the stage of training, even though the signal is measured against a global model that is itself still forming in the early rounds. Similarity is not the only information that has been incorporated into the weights. FedDisco (Ye et al., 2023) augments the size prior with a measure of how far each client’s label distribution lies from a target distribution, although this discrepancy is computed once before training and the resulting weights are reused across all rounds. FedLBW (Kundroo et al., 2026) goes further and discards the sample count altogether, weighting clients by the inverse validation loss of their models on a class-balanced dataset held at the server. However, the need for such a labeled evaluation set at the server becomes a concern, because the weights then depend on how representative that set is. FedLAW (Li et al., 2023) shares this dependence, learning the client weights and a global shrinking factor on a serverside proxy dataset. Other approaches let the weights emerge from the models themselves. FedAWA (Shi et al., 2025) optimizes the weights of each round in favor of updates aligned with the merged global direction, and FedLaw (Lu et al., 2025) scores each client by the agreement of its update with those of the other clients, smoothed over the client’s contribution history. Two observations follow. First, although the numerical weights of these methods differ between rounds, they differ because the client state has changed and not because the weighting rule itself is scheduled. The scoring or optimization criterion is applied identically in every round, and none of these formulations contains a coefficient that varies deliberately with training progress, and the relative trust placed in each signal therefore remains fixed even though the information those signals carry is not equally reliable early and late (Ye et al., 2023; Shi et al., 2025; Li et al., 2023; Lu et al., 2025). Second, adaptation does not by itself distribute influence. A criterion that rewards agreement with the majority direction can repeatedly favor the same clients, because alignment-based weighting reduces the influence of updates that differ from the merged direction (Shi et al., 2025), and contribution defined through similarity to other clients treats conformity with the collective update pattern as the measure of value (Lu et al., 2025). Three limitations therefore remain. Data-size weighting is stable and immediately available, yet it systematically restricts the contribution of small but informative clients. Model similarity carries information that sample count cannot express, yet the one method that examines its early behavior reports that its alignment signal is full of randomness at the beginning of training and restrains this by averaging over history rather than by adapting the influence of the signal to its maturity (Wu and Wang, 2021). We further hypothesize that one contributing factor is the immaturity of the evolving global reference against which the signal is measured. Unconstrained dynamic weighting, finally, may repeatedly favor large, highly similar, or majority-aligned clients, leaving rare or complementary institutional knowledge with little opportunity to shape the global model. Existing methods thus do not provide a stage-aware aggregation mechanism that exploits the stability of data quantity early, progressively incorporates model similarity as the global model matures, and M. Moradpour et al.: Preprint

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at the same time controls how strongly aggregation influence may concentrate. This motivates the question addressed here, namely how federated aggregation can coordinate data quantity and model similarity according to their changing reliability while preserving a meaningful contribution opportunity for heterogeneous clients. Annealing offers a principled way to introduce a component gradually rather than at full strength from the first iteration. The idea originates in simulated annealing, where the system is allowed to move to a high-risk state with high probability during an initial high-temperature stage in order to escape local optima, and this exploration is progressively reduced as the temperature falls (Kirkpatrick et al., 1983; Nguyen et al., 2021). In federated learning, the mechanism has so far served either as a search procedure over training configurations such as participant subsets, hyperparameters, and quantization schemes (Cunha Neto et al., 2022; Li et al., 2024; Peleg et al., 2025), or as a client-side schedule that gradually shifts each client’s reliance from its local model toward the global model as that model matures (Nguyen et al., 2021). In all of these, the server-side aggregation rule itself remains unchanged. Notably, the client-side transition rests on the observation that the averaged global model is not necessarily better than local models in the early stage of training (Nguyen et al., 2021), an observation that concerns the very model against which client-to-global similarity is measured, and which, together with the stability of the data-size prior from the first round, suggests that the relative importance of the two criteria should follow the training stage rather than remain fixed. We accordingly propose Federated Dual Reputation Annealing Weighting (FedDRAW), a server-side aggregation method that combines client data size with client-to-global model similarity and anneals their roles across communication rounds without modifying client-side training. The term dual denotes two coupled schedules. An inner schedule governs the mixture of the two criteria, anchoring the early rounds in the data-size prior and smoothly increasing the contribution of similarity as the global reference matures. An outer schedule governs the temperature at which the resulting client scores are converted into aggregation weights, and it is annealed in a deferred manner, preserving a discriminative weight distribution in the early and middle rounds and relaxing it toward a uniform distribution in the final rounds. The two schedules address the identified limitations jointly. The inner schedule matches each criterion to the stage of training in which it is reliable, while the outer schedule controls concentration, because a mechanism that only increased the influence of similarity would risk transferring dominance from the largest clients to the most globally aligned ones. Fairness carries different meanings across federated learning studies, ranging from a more uniform distribution of predictive performance across clients to parity between demographic groups and to rewards proportional to measured contribution, and we review these notions in Section 2.3. In this work, fairness refers to aggregation-level equal treatment, meaning that every client participating in a communication round receives the same server-side aggregation coefficient in the later rounds, irrespective of its local dataset size or the alignment of its update. This is a statement about server-assigned coefficients alone, not a claim of equal predictive performance across clients, of demographic parity, of contribution-proportional reward, or of equal selection frequency, and because local updates may differ in magnitude, equality of coefficients is not equality of numerical influence. Relaxing the weights toward uniformity also raises the question of whether a poorly performing client could then harm the global model. In our setting, every client begins each round from the shared global model and clients are assumed to be honest rather than adversarial; hence, their updates are refinements of the current global model rather than arbitrary or corrupted contributions, and equal late-round coefficients do not admit harmful updates by construction; robustness to adversarial participants is outside the scope of this work. We evaluate FedDRAW on multi-label chest-radiograph classification using CheXpert and on ChestMNIST, under heterogeneous client partitions, and compare it with representative conventional, adaptive, and similarity-based aggregation methods. The contributions of this work are as follows. • Dual-criterion server-side aggregation. We propose FedDRAW, a server-side aggregation method that jointly considers client data quantity and client-to-global model similarity when determining client influence, requiring no change to client-side training and no proxy dataset at the server. • Stage-aware inner annealing. We introduce a schedule that progressively reduces reliance on the data-size prior and increases the contribution of model similarity as federated training advances, introducing the signal gradually rather than at full strength from the first round, in contrast to a fixed combination of the two criteria. • Outer annealing toward uniform weighting. We introduce a second, outer schedule on the aggregation temperature that relaxes the weight distribution received from the inner annealing toward uniformity in the later

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rounds, and we define the resulting property as aggregation-level equal-treatment fairness, distinguishing it from performance, group, and contribution fairness. • Empirical validation on medical imaging. We evaluate FedDRAW on twelve heterogeneous federated partitions of CheXpert and ChestMNIST against seven conventional, adaptive, and similarity-based aggregation baselines under identical local hyperparameters. • Statistical evaluation. We compare the eight methods across all partition scenarios with per-metric Friedman tests followed by post-hoc analysis, and we additionally examine the behavior of the weighting mechanism itself, including the measured heterogeneity of the client partitions and the distribution of aggregation weights across clients and rounds. The remainder of this paper is organized as follows. Section 2 reviews related work on federated optimization, aggregation weighting, fairness, and federated learning in medical imaging. Section 3 presents the datasets and the FedDRAW method, including the two annealing schedules and their configuration. Section 4 describes the experimental setup, the client partitions, the baselines, and the evaluation protocol, and reports the experimental results together with the statistical analysis. Section 5 concludes the paper and outlines directions for future work.

2. Related Work This section reviews prior work according to the component of the federated pipeline that each method modifies, since that is the axis along which the present contribution is positioned. Section 2.1 covers methods that act on local optimization, which improve how client updates are produced while leaving the rule that combines them intact. Section 2.2 turns to methods that modify the aggregation rule itself by deriving client influence from information other than the local sample count. Section 2.3 surveys the incompatible senses in which fairness has been used in federated learning and states which of them the property claimed for FedDRAW does and does not correspond to. Section 2.4 situates these developments in multi-institutional medical imaging, the setting in which FedDRAW is evaluated, and examines the evaluation metrics adopted in that literature. Throughout, each subsection closes by identifying what the reviewed methods leave unresolved with respect to the assignment of client influence.

2.1. Federated Optimization under Heterogeneous Client Data Federated learning must account for statistical and systems heterogeneity, because clients may differ in data distributions, dataset sizes, computational resources, local update counts, and participation patterns (Kairouz et al., 2021; Li et al., 2020a). Within this setting, FedAvg (McMahan et al., 2017) established the standard iterative formulation in which participating clients perform local optimization and the server aggregates their models using sample-sizeproportional weights. Under heterogeneous client objectives, however, local optimization may produce client drift or objective inconsistency, which can slow convergence and degrade global-model performance (Karimireddy et al., 2020; Wang et al., 2020). Several established methods address heterogeneity by modifying local optimization. FedProx (Li et al., 2020a) introduces proximal regularization to constrain local models toward the current global model and tolerates partial work from slower devices, whereas SCAFFOLD (Karimireddy et al., 2020) uses server and client control variates to correct client-specific update directions at the cost of persistent per-client state. Similarly, FedDyn (Acar et al., 2021) dynamically regularizes the local objectives so that local stationary points are better aligned with the stationary point of the global objective, while MOON (Li et al., 2021a) applies model-level contrastive learning, using the similarity between the representations of the current local model, the global model, and the previous local model to correct local training under heterogeneous client data. The central intervention of these methods is the generation of local updates rather than the construction of a round-dependent rule for client influence. It is worth noting that MOON also uses representation similarity, although as a local regularizer rather than as a server-side weighting signal. FedDRAW leaves the local training objective unchanged and acts only on the weights assigned to the resulting client models, so the two families of methods are compatible rather than competing. By contrast, FedNova (Wang et al., 2020) occupies an intermediate position, since it normalizes each client’s accumulated update according to its effective local progress, so that clients performing more local optimization steps do not acquire disproportionate influence, thereby restoring the sample-size objective that unequal local computation would otherwise distort. This correction protects the intended weighting rather than reconsidering whether that rule assigns influence appropriately, which is the question FedDRAW addresses. M. Moradpour et al.: Preprint

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2.2. Aggregation Weighting beyond Sample Count A second line of work modifies the aggregation rule itself by using client information beyond the local sample count. For instance, FedDisco (Ye et al., 2023) combines relative dataset size with a scalar discrepancy between each client’s category distribution and a target distribution, so that larger datasets and lower discrepancies receive higher weight. It is conceptually close to FedDRAW in that both challenge pure sample-size weighting, although the discrepancy depends on the fixed local label distribution and is therefore determined before training and reused across communication rounds. FedDRAW instead uses a signal that is recomputed from the current client and global models and, more importantly, changes the relative influence of the two signals as training progresses. Some methods derive client influence from model geometry or from measured performance. IDA (Yeganeh et al., 2020) weights each client by the inverse distance between its parameters and the mean model of the participating cohort, and reports that increasing the sample counts of poorly performing clients destabilizes size-weighted averaging while leaving distance-based weighting stable. Similarly, FedAdp (Wu and Wang, 2021) measures the angle between each inferred local gradient and a sample-size-weighted estimate of the global gradient, maps a smoothed angle through a decreasing nonlinear function, and normalizes the resulting contribution scores with a softmax, and it is the most directly related adaptive-weighting baseline evaluated in this work. When the participating clients differ in dataset size, the FedAdp weight retains a multiplicative sample-size factor alongside the contribution score, so the method combines quantity with alignment rather than replacing the former by the latter (Wu and Wang, 2021). FedDRAW also combines these two signals, and differs in that the relative influence of each is scheduled across communication rounds rather than fixed by a single expression. It is also the only method reviewed here that reports an early-training difficulty with its own signal, since the angle between the local and estimated global gradient is highly random at the beginning of training and is therefore replaced by a cumulative smoothed angle. Although this restrains the instability, it does so through historical averaging rather than through a stage-dependent activation of the signal, since the smoothed value coincides with the instantaneous angle at the first round and the signal influences aggregation from the outset. On the other hand, FedLAW (Li et al., 2023) learns a global shrinking factor together with the relative client weights by optimizing a loss on a server-side proxy dataset, and FedLBW (Kundroo et al., 2026) replaces sample-count weighting entirely with normalized inverse validation losses obtained by evaluating each uploaded model on a labeled serverside proxy set. Both therefore make the representativeness of the proxy set a determinant of the weights, whereas FedDRAW requires no server-side data. For instance, FedAWA (Shi et al., 2025) forms a client vector as the difference between each returned local model and the incoming global model and optimizes the weights of the round so that vectors aligned with the merged global direction receive greater influence, while FedDRW (Li et al., 2025) retains a sample-size component but modulates it with a softmax over the similarity between each client’s feature representation and its label-distribution cluster centre, for long-tailed federated classification. Server-side methods may also exploit client diversity through model integration rather than through scalar reweighting, as in FedEL (Wu et al., 2024), which first obtains a conventional federated model and subsequently constructs a server-side ensemble from client-specific classifier modules. This changes the structure of the deployed model, whereas FedDRAW retains a single shared architecture and modifies only the aggregation coefficients. Across the methods surveyed in this subsection, the numerical weights may differ between rounds, although they differ because the client state has changed rather than because the weighting rule itself is scheduled. The scoring or optimization criterion is applied identically in every round, and none of these formulations contains a coefficient that varies deliberately with training progress (Ye et al., 2023; Shi et al., 2025; Li et al., 2023; Kundroo et al., 2026). A second common characteristic is that the criteria these methods adopt tend, by construction, to reduce the influence of clients whose updates diverge from the prevailing consensus, since weight is granted for proximity to a cohort mean (Yeganeh et al., 2020), for alignment with an estimated global direction (Wu and Wang, 2021; Shi et al., 2025), or for agreement with a target distribution (Ye et al., 2023). A small institution holding rare presentations or an underrepresented population is therefore liable to receive persistently low influence precisely because its data are distinctive, and the reviewed methods do not treat the preservation of such contributions as an explicit design objective. Although these methods adapt client influence according to the current state or measured behavior of the clients, FedDRAW differs in two respects. It schedules the composition of the client score across communication rounds rather than applying a stationary criterion, and it separately schedules the concentration of the resulting weight distribution so that no client retains disproportionate influence over the final model.

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2.3. Fairness Notions in Federated Learning Fairness has been used in federated learning for several incompatible objectives, which differ in which quantity is equalized and for whom. Client-level performance fairness balances predictive performance across participating clients, whether by up weighting clients with higher losses (Li et al., 2020b), by measuring the dispersion of client accuracy (Li et al., 2021b), by optimizing against the worst-case mixture of client distributions (Mohri et al., 2019), by seeking an update direction that increases no participant’s loss (Hu et al., 2022), or by penalizing the spread of losses across clients (Yue et al., 2023). Group fairness instead equalizes prediction-rate disparities between demographic groups defined by a sensitive attribute, adjusting the aggregation weights from client-reported fairness statistics (Ezzeldin et al., 2023). Collaborative fairness allocates model benefit in proportion to measured contribution and therefore regards identical treatment of unequal contributors as unfair (Lyu et al., 2020; Lu et al., 2025), while a fourth line equalizes how often clients are selected rather than how their updates are weighted once selected (Peleg et al., 2025). Equal aggregation weights are consequently not what most of this literature means by fairness, since several of these methods make weights deliberately unequal in order to equalize outcomes (Li et al., 2020b; Wang et al., 2024). The property claimed for FedDRAW is distinct and procedural. We use aggregation-level equal treatment to denote the condition in which every client participating in a communication round receives the same server-side aggregation coefficient in the later rounds, irrespective of its dataset size or the alignment of its update, which prevents any client from retaining disproportionate influence over the final model. The closest precedent is FedEBA+ (Wang et al., 2024), which likewise limits the concentration of the weight vector, although it ties the weight itself to client underperformance and applies its entropy term as a fixed regularizer, whereas FedDRAW derives the weight from data size and client-toglobal similarity and schedules the permitted concentration across communication rounds. The claim concerns serverassigned coefficients alone. It is not a claim of equal predictive performance across clients, of demographic parity, of contribution-proportional reward, or of equal selection frequency, and because local updates may differ in magnitude, equality of coefficients is not equality of numerical influence.

2.4. Federated Learning in Medical Imaging Medical imaging is an important application of federated learning, because clinically relevant data are often distributed across institutions and difficult to pool for reasons of privacy, governance, and data-sharing constraints (Guan et al., 2024). Institutional datasets may differ in scanner hardware, acquisition protocols, preprocessing procedures, patient populations, disease prevalence, and annotation practices, producing feature, label, and domain shifts (Guan et al., 2024; Tayebi Arasteh et al., 2023; Li et al., 2021c). Moreover, in some federations the annotated label sets themselves do not fully overlap (Kulkarni et al., 2024). A multi-institutional chest-radiograph study reports that on-domain performance was influenced primarily by the amount of training data, whereas off-domain performance benefited more strongly from greater training-data diversity in collaborative learning (Tayebi Arasteh et al., 2023). This distinction between volume and diversity is directly relevant to aggregation design, since a weighting rule tied to volume alone rewards the former and not the latter. FedBN (Li et al., 2021c) addresses a related feature-shift problem by retaining batch-normalization parameters locally while aggregating the remaining model parameters. These methods adapt what is shared or how local training proceeds, rather than redefining the relative aggregation coefficient assigned to each participating client, which is the question FedDRAW addresses. Work on chest radiography has converged on a small number of benchmarks. CheXpert provides 224,316 radiographs from 65,240 patients labeled for fourteen observations (Irvin et al., 2019), whereas ChestMNIST provides a standardized lower-resolution multi-label chest-radiograph task within the MedMNIST collection (Yang et al., 2023). Both are framed as multi-label thoracic-pathology classification, and both are substantially imbalanced at the label level, since several findings are present in only a small fraction of studies. Evaluation on these benchmarks is dominated by the area under the receiver operating characteristic curve. The MedMNIST benchmark protocol reports the area under the curve together with accuracy (Yang et al., 2023), cross-institutional federated evaluation on chest radiographs is reported in terms of the area under the curve (Tayebi Arasteh et al., 2023; Kulkarni et al., 2024), and comparisons between supervised and label-efficient methods on the CheXpert test set are likewise reported as a mean area under the curve averaged over the competition pathologies (Tiu et al., 2022). Both metrics respond weakly to the class imbalance that characterizes these datasets, and this limitation has been examined directly in the medical-informatics literature. Optimizing and reporting a global area under the curve implicitly assumes that all decision thresholds are equally relevant, whereas clinicians operate within ranges of M. Moradpour et al.: Preprint

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acceptable thresholds that reflect the differing consequences of false positives and false negatives (Cabitza et al., 2026). Accuracy, for its part, can be artificially inflated under class imbalance, and threshold-dependent metrics are frequently computed at an arbitrary cut-off that rarely corresponds to an actual decision trade-off (Cabitza et al., 2026). The consequence is that a favorable value under threshold-free evaluation does not establish that an operating point exists at which an infrequent finding is detected at a clinically acceptable rate, so reported performance may be optimistic relative to what a deployed system would achieve once a threshold is fixed (Cabitza et al., 2026). The point is practical rather than formal, since a system supporting a clinical decision necessarily operates at a chosen threshold, and the findings whose detection motivates such a system are frequently the infrequent ones. For this reason we complement the macro-averaged area under the curve with the geometric mean of sensitivity and specificity, computed at per-class thresholds selected on a validation split and then held fixed for test evaluation. Selecting condition-specific thresholds on a validation set that does not overlap the test set is established practice on CheXpert (Tiu et al., 2022). The geometric mean is small whenever either sensitivity or specificity is small, so it cannot be inflated by strong performance on the majority class alone, and reporting it alongside the area under the curve characterizes performance both across all thresholds and at the single operating point that a deployed system would use.

3. Method In this section, we present the proposed method in full detail. We first introduce the two chest radiograph datasets on which our study is built, since their characteristics motivate several design decisions of the method. We then formalize the federated learning setting and the aggregation problem, and introduce FedDRAW, a server-side aggregation framework built on two coupled annealing mechanisms, namely an inner annealing that governs the informational basis of client reputation, and an outer annealing that governs how reputation is translated into aggregation weights. Each component of the framework is described in a dedicated subsection, followed by the complete algorithm.

3.1. Datasets We evaluate the proposed FedDRAW framework on two publicly available multi-label chest radiograph datasets, CheXpert (Irvin et al., 2019) and ChestMNIST (Yang et al., 2023). The two datasets differ in size, image resolution, patient population, and, importantly, in the automated labeling pipelines from which their annotations were derived; consistent behavior across both therefore indicates that the evaluated aggregation strategies are not tied to the characteristics of a single data source or label-extraction process. For consistency across experiments, we consider the same five thoracic findings in both datasets, namely Atelectasis, Edema, Pleural Effusion, Cardiomegaly, and Consolidation. Figure 1 shows representative test-set examples from both datasets together with their binary labels for the five findings. Table 1. Label distribution of the training data used in this study for (a) CheXpert and (b) ChestMNIST. Each count gives the number of images for which the corresponding finding is positive. Since the task is multi-label, an image may be positive for several findings simultaneously, so the counts are not mutually exclusive and do not sum to the total number of images. The final row of each subtable gives the number of images that are negative for all five findings. (a) CheXpert (191,027 frontal radiographs used, after uncertainty mapping).

Class Atelectasis Edema Pleural Effusion Cardiomegaly Consolidation All five labels = 0

Positive images 59,583 61,493 76,899 23,385 12,983 51,052

(b) ChestMNIST (official training split of 78,468 images, used in its entirety).

Class Atelectasis Edema Pleural Effusion Cardiomegaly Consolidation All five labels = 0

Positive images 7,996 1,690 9,261 1,950 3,263 59,438

CheXpert. CheXpert is a large-scale chest radiograph dataset containing 224,316 chest X-ray images from 65,240

patients (Irvin et al., 2019). Its labels were extracted automatically from radiology reports by a rule-based labeler and take one of three values per finding, namely positive, negative, or uncertain, the last of which reflects ambiguous report language. The dataset provides annotations for fourteen observations, which its authors selected on the basis of their prevalence in the reports and their clinical relevance (Irvin et al., 2019). Among these fourteen, the CheXpert authors designated five observations, namely Atelectasis, Cardiomegaly, Consolidation, Edema, and Pleural Effusion, as the competition tasks of the associated benchmark, selected according to their clinical importance and prevalence (Irvin M. Moradpour et al.: Preprint

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et al., 2019). We adopt these same five findings as our prediction targets. For label preprocessing, the five target observations were converted to binary labels. Following the uncertainty-handling strategy of Pillai (Pillai, 2022), uncertain labels (−1) for Atelectasis and Edema were mapped to the positive class (U-Ones), whereas uncertain labels for Cardiomegaly, Consolidation, and Pleural Effusion were mapped to the negative class (U-Zeros); missing label entries were likewise assigned to the negative class, yielding binary labels for all five findings. In this study, we work with the 191,027 frontal chest radiographs of the dataset. The selected five pathologies exhibit substantial differences in prevalence, providing a naturally imbalanced multi-label setting, since Pleural Effusion and Edema are among the more prevalent findings, whereas Consolidation is considerably less frequent. Images for which none of the five selected findings is positive are retained as negative samples for the considered label set. Table 1(a) reports the resulting label distribution of the CheXpert training data used in this work.

Atelectasis: 1 Edema: 0 Pleural Effusion: 1 Cardiomegaly: 1 Consolidation: 1

Atelectasis: 0 Edema: 1 Pleural Effusion: 0 Cardiomegaly: 0 Consolidation: 0

Atelectasis: 0 Edema: 1 Pleural Effusion: 0 Cardiomegaly: 1 Consolidation: 0

Atelectasis: 0 Edema: 0 Pleural Effusion: 0 Cardiomegaly: 0 Consolidation: 0

Atelectasis: 0 Edema: 0 Pleural Effusion: 0 Cardiomegaly: 1 Consolidation: 0

Atelectasis: 0 Edema: 0 Pleural Effusion: 1 Cardiomegaly: 1 Consolidation: 0

Atelectasis: 0 Edema: 0 Pleural Effusion: 0 Cardiomegaly: 0 Consolidation: 0

(a) CheXpert

Atelectasis: 1 Edema: 0 Pleural Effusion: 0 Cardiomegaly: 0 Consolidation: 1

Atelectasis: 1 Edema: 0 Pleural Effusion: 1 Cardiomegaly: 0 Consolidation: 1

Atelectasis: 0 Edema: 0 Pleural Effusion: 1 Cardiomegaly: 0 Consolidation: 0

(b) ChestMNIST Figure 1. Representative test-set examples from CheXpert (a) and ChestMNIST (b) with their binary labels for the five considered findings. For the CheXpert test set, the reference labels were obtained from radiologist annotations, with the majority vote of five board-certified radiologists serving as the ground truth (Irvin et al., 2019). The ChestMNIST labels originate from the automated text mining of radiology reports in the source ChestX-ray14 dataset (Yang et al., 2023; Wang et al., 2017), and its Effusion category is reported here under the harmonized name Pleural Effusion for consistency with CheXpert.

ChestMNIST. ChestMNIST is the chest X-ray subset of the MedMNIST benchmark and is derived from the NIH ChestX-ray14 dataset (Yang et al., 2023; Wang et al., 2017), whose labels were obtained through automated text mining of radiology reports. The source dataset provides annotations for fourteen thoracic findings; from these, we retain the same five pathologies used for CheXpert, providing a consistent prediction task across the two datasets. The label nomenclature of the two datasets differs slightly for one of the selected findings, since the ChestX-ray14 label set, which ChestMNIST inherits, denotes pleural fluid accumulation as Effusion, whereas CheXpert uses the term Pleural Effusion. We treat these two categories as corresponding to the same finding and refer to it throughout the paper as Pleural Effusion. In contrast to CheXpert, the labels distributed with ChestMNIST are already binary, and each finding is annotated as either present (1) or absent (0), with no uncertain category, since the task is formulated by the benchmark providers as multi-label binary-class classification (Yang et al., 2023). No additional uncertainty handling is therefore required on our side, which also means that any label noise present in ChestMNIST stems entirely from the automated M. Moradpour et al.: Preprint

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report-mining pipeline of its source dataset rather than from our preprocessing choices. We use the official training split of the benchmark in its entirety, without any further image selection. Like CheXpert, ChestMNIST exhibits substantial class imbalance, with positive findings representing only a fraction of the available images for several pathologies; this imbalance, combined with the smaller image resolution, makes ChestMNIST a complementary and more challenging counterpart to CheXpert for evaluating aggregation behavior. Table 1(b) reports the label distribution of the ChestMNIST training data used in this work. The construction of federated clients from these centralized datasets is described in Section 4.3.

3.2. Preliminaries and Problem Formulation Federated learning (FL) is a distributed learning paradigm in which multiple clients collaboratively train a shared model under the coordination of a central server, without ever exchanging their raw data (McMahan et al., 2017; Kairouz et al., 2021). Its central goal is to obtain a global model that benefits from the collective data of all participants while respecting data locality, a property of particular importance in medical imaging, where patient data cannot leave the acquiring institution. In each communication round, clients train the current model on their private data and return only the updated parameters; the server then combines these local models into a new global model through weighted aggregation. The aggregation weights determine how strongly each client influences the global model, and this influence is decisive for both convergence and final performance, and assigning weights that reflect each client’s actual contribution has been shown to accelerate convergence under heterogeneous data (Wu and Wang, 2021). The de facto standard aggregation rule, FedAvg (McMahan et al., 2017), weights each client by its dataset size. As we argue below, this choice embeds two assumptions that become problematic under realistic data heterogeneity. Addressing these two limitations at the aggregation level is the starting point of the proposed method. We consider a federated learning system of 𝐾 clients indexed by 𝑖 ∈ {1, … , 𝐾}, each holding a private local ∑ dataset 𝐷𝑖 of size 𝑛𝑖 = |𝐷𝑖 |, with total data volume 𝑁 = 𝐾 𝑖=1 𝑛𝑖 . All clients share an identical neural network architecture; in this work a DenseNet backbone (Huang et al., 2017), so that the parameter vectors 𝜃𝑖 ∈ ℝ𝑑 of all clients are dimensionally and structurally aligned; here 𝜃𝑖 denotes the parameters of client 𝑖’s network. Before the first round, the server distributes this common architecture together with its pretrained initialization 𝜃 0 to all clients, so that training starts from a shared pretrained backbone rather than from a random initialization. Training proceeds over 𝑇 synchronous communication rounds. At the beginning of round 𝑡, the server broadcasts the current global model 𝜃 𝑡 to all clients; each client 𝑖 initializes its local network from 𝜃 𝑡 and performs 𝐸 epochs of local training on 𝐷𝑖 , producing updated parameters 𝜃𝑖𝑡 . All clients participate in every round; no client selection or filtering is applied. The server then forms the next global model by weighted aggregation: 𝜃 𝑡+1 ←

𝐾 ∑

𝜔𝑡𝑖 𝜃𝑖𝑡 ,

s.t.

𝑖=1

𝐾 ∑

𝜔𝑡𝑖 = 1, 𝜔𝑡𝑖 ≥ 0,

(1)

𝑖=1

where 𝜔𝑡𝑖 is the aggregation weight assigned to client 𝑖 at round 𝑡. In FedAvg (McMahan et al., 2017), the weights are fixed to 𝜔𝑖 = 𝑛𝑖 ∕𝑁 for all rounds. Under non-IID client data, this static, purely size-proportional rule is known to degrade the quality of the global model, and the resulting accuracy loss has been attributed to the divergence of client weights, which grows with the statistical distance between each client’s label distribution and the population distribution (Zhao et al., 2018). We attribute this limitation to two implicit assumptions embedded in the rule, namely (a) that dataset size is a sufficient proxy for contribution quality, and (b) that the relative importance of clients is constant throughout training. Regarding (a), weighting client updates purely by data volume treats the merit of a client as the amount of data it holds, which has been argued to be inappropriate in realistic deployments where clients differ in the informativeness and not only in the size of their data (Zeng et al., 2021). A client with a large but homogeneous dataset may produce updates that are largely redundant with respect to the global model, while a small client may hold underrepresented patterns, for instance uncommon pathologies in a medical imaging federation, whose contribution would be systematically undervalued by size-based weighting. Regarding (b), client utility is not a fixed property, since recent analysis of weighted aggregation shows that the relative importance of a client to the global model changes over the course of training, with a critical point in the learning dynamics before which more coherent clients play the most essential role in generalization (Li et al., 2023). In our setting, clients with larger datasets tend to produce more stable, statistically representative updates in early rounds, whereas as the global model matures, every client, including those with small datasets, benefits from the stronger initialization it receives, and a client that was initially unreliable may subsequently M. Moradpour et al.: Preprint

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produce updates that are well aligned with the learning objective. Fixed weights are blind to this evolution and permanently marginalize clients whose initial conditions were unfavorable; adapting the weights to the per-round contribution of each client has accordingly been shown to accelerate convergence under non-IID data (Wu and Wang, 2021). Methods such as FedProx (Li et al., 2020a) and SCAFFOLD (Karimireddy et al., 2020) address heterogeneity from a different angle. FedProx modifies the local objective with a proximal term, and SCAFFOLD corrects client drift using control variates. Neither is designed to adapt the aggregation weights according to the round-specific contribution of each client, and both therefore retain the static weighting of FedAvg at the aggregation step. A natural alternative is to replace dataset size with a measure of how well each client’s updated model aligns with the global model. L-DAWA, for instance, aggregates client models according to the angular divergence between each client’s parameters and the global model (Rehman et al., 2023), and FedSim guides aggregation using similarity inferred from client gradients (Palihawadana et al., 2022). However, similarity-based weighting introduces a limitation of its own, which we refer to as a cold-start effect. In the earliest rounds, the global model, even when initialized from a pretrained backbone, is still task-agnostic and has not yet incorporated meaningful knowledge from any client. At this stage, high similarity between a client’s model and the global model does not necessarily represent a useful update, since it may merely reflect that the client’s data induced minimal parameter change, leaving its model close to the uninformative initialization. Conversely, low similarity may signal that the client’s data contains novel patterns the global model has not yet learned, which are precisely the updates most needed for generalization. Trusting similarity prematurely may therefore reward passivity and penalize informativeness. This interpretation is our own design rationale rather than an established result; it is, however, consistent with two independent observations in the literature. First, using model similarity directly as an aggregation weight has been reported to lack sufficient adaptability and to introduce instability in the early stages of training (Shi et al., 2025). Second, similarity-based federated methods commonly precede the similarity computation by a number of standard warm-up rounds, computing client similarities only after the shared model has been trained collaboratively (Bdair et al., 2021), an implicit acknowledgment that similarity becomes meaningful only once the global model has matured. These complementary limitations, the staleness of fixed size-based weights and the unreliability of premature similarity, motivate an aggregation mechanism that (i) uses dataset size as a stabilizing prior in early rounds when similarity is unreliable, (ii) progressively transitions to similarity-based evaluation as the global model becomes a credible reference, and (iii) ensures that no client is permanently excluded from meaningful participation. We realize all three objectives through two coupled annealing mechanisms operating simultaneously, an inner annealing governing the informational basis of client reputation, and an outer annealing governing the distributional shape of the resulting weights. We call their joint operation Federated Dual Reputation Annealing Weighting (FedDRAW), which we describe in the following subsections.

3.3. Overview Figure 2 gives an overview of the proposed framework. We use the term dual annealing for the joint operation of two schedules that are annealed at the same time over the communication rounds: (1) an inner annealing, which governs what information defines the reputation of a client, and (2) an outer annealing, which governs how strongly differences in reputation are turned into differences in aggregation weight. At each round 𝑡, FedDRAW computes the aggregation weights 𝜔𝑡𝑖 through two operations performed on the server: 1. Inner annealing (reputation scoring). A reputation score 𝜉𝑖𝑡 is computed for every client as a combination of its static data-size prior and its dynamic similarity to the global model, with a schedule that progressively shifts the reputation from the data-size prior toward the similarity signal as training advances. 2. Outer annealing (weight shaping). The reputation scores are converted into normalized aggregation weights through a temperature-controlled softmax whose schedule progressively reduces weight concentration, promoting an increasingly balanced and ultimately equal influence of all participating clients on the global model, which realizes the fairness objective of the method. Both schedules are governed by exponential functions of the round index, in analogy with the cooling process of simulated annealing (Kirkpatrick et al., 1983), where a decreasing temperature drives the transition from global exploration to local refinement. In our setting, cooling captures the system’s progression from cautious, size-stabilized aggregation toward confident, similarity-informed, and ultimately fair aggregation. The two stages are not sequential phases, since they operate simultaneously at every round, and their coupling is essential to the method (Section 3.7). In the following subsections, we explain the inner and the outer annealing in detail. M. Moradpour et al.: Preprint

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FedDRAW FedDRAW Server

𝐂𝐞𝐧𝐭𝐞𝐫 𝐈

Outer Annealing

Local Update 𝜃1𝑡

Inner Annealing 𝐂𝐞𝐧𝐭𝐞𝐫 𝐈𝐈 Local Update 𝜃2𝑡

.

.

. .

. .

Input 1: Data Size Prior 𝑛𝑖 𝑝𝑖 = 𝐾 σ𝑗=1 𝑛𝑗

Input2: Similarity Score 𝑠𝑖𝑡 = 𝑆𝑖𝑚(𝜃𝑖𝑡 ℓ , 𝜽𝑡 ℓ ) (Last layer ℓ only)

Reputation Score 𝜉𝑖𝑡 = 𝛾𝑡 𝑝𝑖 + (1 − 𝛾𝑡 )𝑠𝑖𝑡

Control Transition 𝛾𝑡 = 𝑒 −𝜂𝑡 , 0 ≤ 𝛾𝑡 ≤ 1

𝐂𝐞𝐧𝐭𝐞𝐫 𝑲 Local Update 𝜃𝑘𝑡

Aggregation (Server)

Uniformity/concentration Control

ξ𝑖𝑡

Temperature-controlled Softmax 𝑒𝑥𝑝(𝛽𝑡 ξ𝑖𝑡 )  𝑡𝑖 = 𝑘 σ𝑗 𝑒𝑥𝑝(𝛽𝑡 𝜉𝑗𝑡 )

 𝑡𝑖

Weighted Aggregation 𝐾

𝜽𝑡+1 = ෍  𝑖𝑡 𝜃𝑖𝑡 𝑖=1

Control Transition 𝛽𝑡 = 𝛽max 1 − 𝑒 − 𝑇−𝑡 , 𝛽𝑡 ,  > 0

Global Model 𝜽𝑡+1

Evaluate & Distributed

Figure 2. Overview of the FedDRAW framework. Clients perform unmodified local training and upload their parameters 𝜃𝑖𝑡 . On the server, the inner annealing combines the static data-size prior 𝑝𝑖 with the last-layer similarity 𝑠𝑡𝑖 into a reputation score 𝜉𝑖𝑡 , and the outer annealing converts reputations into aggregation weights 𝜔𝑡𝑖 through a temperature-controlled softmax. The global model 𝜃 𝑡+1 is then formed by weighted aggregation and broadcast to all clients.

3.4. Algorithm Algorithm 1 summarizes the complete FedDRAW procedure; every step follows directly from the equations of Sections 3.5–3.7. Before walking through it, we summarize the implementation characteristics of the method. FedDRAW introduces no additional server-side model and no client-to-server communication beyond the standard parameter upload of any federated protocol, since all reputation and weight computations (Eqs. (4)–(7)) are carried out on the server from the parameters it receives in any case. Similarity is computed on the parameters of the final classification layer only (Section 3.8), since this layer carries the task-specific representations most sensitive to each client’s local label distribution; restricting the computation to it also makes the weighting step computationally efficient, costing (𝐾 ⋅𝑑𝓁 ), where 𝑑𝓁 is the dimensionality of the final layer, which is negligible relative to local training. Because only the server-side weight computation is modified and the client training procedure is left unchanged, FedDRAW is fully compatible with client-side drift-mitigation methods (e.g., FedProx (Li et al., 2020a)) and serves as a drop-in replacement for the FedAvg (McMahan et al., 2017) aggregation step in any standard federated pipeline. Here ClientUpdate(𝐷𝑖 , 𝜃 𝑡 , 𝐸) denotes 𝐸 epochs of local training on 𝐷𝑖 starting from the broadcast global model 𝑡 𝜃 . All 𝐾 clients execute this update in parallel at every round; the size priors 𝑝𝑖 are computed once from 𝐷𝑖 before training and never recomputed.

3.5. Inner Annealing: Reputation Scoring The central idea of the inner annealing stage is a gradual shift of the weighting criterion. Aggregation initially relies on client data size, exactly as in FedAvg (McMahan et al., 2017), where the weight of client 𝑖 is proportional to its sample count 𝑛𝑖 . It then progressively transitions toward relying on the similarity between each client’s updated model and the global model. The stage thus answers the question of what information should determine client importance at each point in training. Its design principle is that the reputation score should rest on the dataset size while the global model is still immature, and should incorporate the client-to-global similarity only once that reference has stabilized. We state this as a design principle of FedDRAW rather than as an established result; the practice of postponing similarity-based reasoning until after a warm-up phase (Bdair et al., 2021) and the finding that the relative importance of clients changes over the course of training (Li et al., 2023) both point in the same direction. To see why this direction of transition is appropriate, consider the alternatives. Using only dataset size throughout reduces to FedAvg (McMahan et al., 2017), with all of its limitations, namely static weights that cannot adapt to evolving client quality, and permanent marginalization of small but informative clients. Using only similarity throughout lets the cold-start effect dominate the early rounds, because a client whose data induces small parameter changes appears highly similar to the immature global model not because it is well aligned with the objective, but because it barely moved from the starting point, while a client with diverse or complex data produces large updates that lower its similarity despite being exactly what the global model needs. The proposed schedule therefore begins M. Moradpour et al.: Preprint

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Algorithm 1 FedDRAW: Federated Dual Reputation Annealing Weighting Require: local datasets {𝐷𝑖 }𝐾 ; rounds 𝑇 ; local epochs 𝐸; inner annealing rate 𝜂 > 0; outer annealing rate 𝜆 > 0; 𝑖=1 maximum inverse temperature 𝛽max > 0 Ensure: final global model 𝜃 𝑇 1: 𝜃 0 ← pretrained parameters of the DenseNet121 backbone, shared by all clients /∑ 2: 𝑝𝑖 ← 𝑛𝑖 ⊳ Data Size Prior, Eq. (2) 𝑗 𝑛𝑗 for each client 𝑖 3: for 𝑡 = 0, 1, … , 𝑇 − 1 do // Local training (in parallel on clients) 4: for each client 𝑖 = 1, …(, 𝐾 in parallel do ) 5: 𝜃𝑖𝑡 ← ClientUpdate 𝐷𝑖 , 𝜃 𝑡 , 𝐸 6: send 𝜃𝑖𝑡 to the server 7: end for // Inner annealing (on server) 8: 𝛾𝑡 ← 𝑒−𝜂𝑡 ⊳ Inner Annealing, Eq. (5) 9: for each client( 𝑖 = 1, … , 𝐾) do 10: 𝑠𝑡𝑖 ← Sim 𝜃𝑖𝑡 [𝓁], 𝜃 𝑡 [𝓁] ⊳ Similarity Score, Eq. (3) 11: 𝜉𝑖𝑡 ← 𝛾𝑡 𝑝𝑖 + (1 − 𝛾𝑡 ) 𝑠𝑡𝑖 ⊳ Reputation Score, Eq. (4) 12: end for // Outer annealing (on server) ) ( ⊳ Outer Annealing, Eq. (7) 13: 𝛽𝑡 ← 𝛽max 1 − 𝑒−𝜆 (𝑇 −𝑡) 𝐾 ) ( ( )/∑ ⊳ emperature-Controlled Softmax, Eq. (6) exp 𝛽𝑡 𝜉𝑗𝑡 for each client 𝑖 14: 𝜔𝑡𝑖 ← exp 𝛽𝑡 𝜉𝑖𝑡 𝑗=1

15:

// Weighted aggregation (on server) 𝐾 ∑ 𝜃 𝑡+1 ← 𝜔𝑡𝑖 𝜃𝑖𝑡

⊳ Weighted aggregation(global update), Eq. (8)

𝑖=1

16: broadcast 𝜃 𝑡+1 to all clients 17: end for 18: return 𝜃 𝑇

with the size prior for stability and shifts, in an annealing-inspired manner, toward similarity once the global model provides a stable reference for the comparison.

3.5.1. Data Size Prior For each client 𝑖, we define the normalized size prior 𝑛 𝑝 𝑖 = ∑𝐾 𝑖

,

𝑗=1 𝑛𝑗

(2)

computed once before training and held constant. It encodes the assumption that, absent any other information, a client with more data is expected to produce a more representative update.

3.5.2. Similarity Score At each round 𝑡, after receiving the updated parameters 𝜃𝑖𝑡 , the server computes ( ) 𝑠𝑡𝑖 = Sim 𝜃𝑖𝑡 , 𝜃 𝑡 ,

(3)

where Sim(⋅, ⋅) denotes cosine similarity applied to model parameters. The restriction of this computation to the parameters of the final layer is an implementation choice orthogonal to the annealing framework; we motivate it in Section 3.8. Unlike the size prior, 𝑠𝑡𝑖 is fundamentally dynamic, since it changes at every round because both 𝜃𝑖𝑡 and 𝜃 𝑡 evolve, reflecting the current state of learning rather than a static property of the data. M. Moradpour et al.: Preprint

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3.5.3. Reputation Score We define the reputation score of client 𝑖 at round 𝑡 as the convex linear combination 𝜉𝑖𝑡 = 𝛾𝑡 𝑝𝑖 + (1 − 𝛾𝑡 ) 𝑠𝑡𝑖 ,

(4)

where 𝛾𝑡 ∈ [0, 1] is the inner annealing parameter, which controls the transition within the inner annealing, in that it determines how much the combination relies on the data-size prior versus the similarity signal at each round. At 𝛾𝑡 = 1 the reputation is the pure size prior; at 𝛾𝑡 = 0 it is the pure similarity score; intermediate values blend the two. Our design requires 𝛾𝑡 to decrease from 1 toward 0 over the course of training, encoding the progressive shift of trust from the static prior to the dynamic similarity signal.

3.5.4. Control Transition: Exponential Decay Schedule for 𝛾𝑡 We require a schedule with three properties: (i) 𝛾0 = 1, ensuring pure size-based reputation at the first round; (ii) 𝛾𝑡 → 0 as 𝑡 → ∞, ensuring eventual reliance on similarity; (iii) smooth decay without abrupt transitions that could destabilize training. Inspired by the exponential cooling schedule of simulated annealing (Kirkpatrick et al., 1983; van Laarhoven and Aarts, 1987), we adopt 𝛾𝑡 = exp(−𝜂 𝑡),

(5)

where 𝜂 > 0 is the inner annealing rate. The exponential form of Eq. (5) shapes the transition in a specific way, since the decrease of 𝛾𝑡 is fastest in the first rounds and becomes progressively slighter thereafter, so that the bulk of the shift away from the size prior happens early, followed by a long, gentle approach toward pure similarity-based reputation. The analogy to simulated annealing is more than superficial. At high temperature, that is, for large 𝛾𝑡 in the early rounds, the aggregation draws broadly from all clients weighted by data size, a form of exploration that does not prejudge any client based on similarity to an immature model, and as 𝛾𝑡 cools, the aggregation exploits the similarity signal to focus on clients aligned with the now-mature global model.

Role of 𝜂. The rate 𝜂 governs the speed of the transition and should be calibrated to the convergence dynamics of the learning task. A large 𝜂 yields rapid decay, appropriate when the global model converges quickly, for instance when training starts from a strong pretrained backbone, and a stable reference for the similarity computation is therefore reached early. A small 𝜂 maintains the stabilizing size prior for many rounds, appropriate for highly heterogeneous settings where the global model needs longer to become a meaningful reference. More generally, the suitable range of 𝜂 depends on the total number of communication rounds of the federation and on the characteristics and quality of the client data, and should be chosen with both in mind. We used a single intermediate value throughout all experiments without scenario-specific tuning; the exact value is reported in the experimental setup.

3.6. Outer Annealing: From Selective to Fair Aggregation The inner stage determines the informational content of client reputation; a separate concern is how reputation scores translate into weights, that is, the distributional shape of the weight vector. Even perfectly accurate reputations, if converted into a distribution that assigns negligible weight to most clients, raise a fundamental fairness concern. Client-level fairness has been formalized in federated learning in several ways, for instance through minimax objectives over mixtures of client distributions (Mohri et al., 2019) or through the uniformity of model performance across devices (Li et al., 2021b). Our concern here is a distinct, aggregation-level notion, which we operationalize as the requirement that no client should be permanently excluded from meaningful participation in the global model, that is, that no client should be assigned a persistently negligible aggregation weight. Ethically, federated collaborations are formed under the expectation that participation benefits all parties; an aggregation scheme that systematically ignores certain clients undermines that cooperative foundation. Technically, permanent exclusion creates a self-reinforcing negative cycle, in which a client receiving near-zero weight exerts no influence on 𝜃 𝑡+1 , so the next global model is poorly aligned with its local data, producing larger local deviations, lower similarity scores, and still lower weights. This compounding marginalization is especially damaging because marginalized clients often hold precisely the rare or underrepresented patterns (e.g., uncommon diagnoses) that the global model needs to generalize. The outer stage controls how strongly reputation differences translate into weight differences, and ensures this strength decreases over time. Early in training, a peaked distribution favoring high-reputation clients is desirable, because the inner stage is supplying size-based reputations, and amplifying size differences helps the global model M. Moradpour et al.: Preprint

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stabilize by attending to the largest, most representative datasets. Late in training, a near-uniform distribution is desirable, because all clients have by then benefited from many rounds of global updates and improved their local models, and the initially weak or dissimilar among them should no longer be marginalized. The transition between the two regimes should be gradual, avoiding abrupt weight redistribution that could destabilize the global model.

3.6.1. Temperature-Controlled Softmax We convert the reputation scores obtained from the inner annealing (Eq. (4)) into normalized weights through a softmax with time-dependent inverse temperature 𝛽𝑡 > 0, that is, 𝛽𝑡 = 1∕𝑇𝑡 in the temperature notation of Hinton et al. (Hinton et al., 2015): ( ) exp 𝛽𝑡 𝜉𝑖𝑡 𝑡 (6) 𝜔𝑖 = ∑ ( ). 𝐾 𝑡 exp 𝛽 𝜉 𝑡 𝑗 𝑗=1 Here, 𝜔𝑡𝑖 is the final aggregation weight of client 𝑖. In the early and middle rounds, it is driven by the reputation produced by the inner annealing, that is, the blend of data size and similarity, whereas in the late rounds the outer schedule takes over and drives all weights toward equality. The parameter 𝛽𝑡 is thus responsible for the shift from reputation-driven to equal weighting, which is precisely the notion of fairness realized in our method. Large 𝛽𝑡 amplifies even small reputation differences; in the limit 𝛽𝑡 → ∞, the entire weight concentrates on the single highest-reputation client. Small 𝛽𝑡 compresses differences; in the limit 𝛽𝑡 → 0, the weights become uniform, 𝜔𝑡𝑖 → 1∕𝐾. This makes 𝛽𝑡 the natural control parameter for the peaked-to-uniform transition we seek.

3.6.2. Control Transition: Deferred Annealing Schedule for 𝛽𝑡 The outer temperature 𝛽𝑡 controls how sharply the softmax converts reputation scores into aggregation weights. The desired behavior is that the weights remain selective through the early and middle rounds and equalize only toward the end of training, since premature equalization risks discarding the contribution of small but informative clients before it has been absorbed into the global model. We therefore employ the schedule ) ( 𝑡 = 1, … , 𝑇 , (7) 𝛽𝑡 = 𝛽max 1 − 𝑒−𝜆 (𝑇 −𝑡) , where 𝛽max > 0 denotes the maximum inverse temperature, 𝜆 > 0 the outer annealing rate, and 𝑇 the total number of communication rounds. The schedule is parameterized by the number of remaining rounds 𝑇 − 𝑡 rather than by elapsed time. Consequently, 𝛽𝑡 remains close to 𝛽max throughout the early and middle phases of training, preserving selective aggregation, and decreases only as 𝑡 approaches 𝑇 . At the final round, the exponent of the softmax vanishes, so that 𝛽𝑇 = 0 and the aggregation weights become exactly uniform, 𝜔𝑇𝑖 = 1∕𝐾. Differentiated weighting in the early and middle rounds and equal weighting at convergence are thus obtained by construction, independently of the values of 𝛽max and 𝜆. Because the flattening of the weight distribution is postponed to the final rounds rather than beginning immediately, we refer to Eq. (7) as a deferred annealing schedule.

Role of 𝛽𝑡 . At each round, 𝛽𝑡 acts as the inverse temperature of the softmax and thus determines how strongly differences in reputation are translated into differences in aggregation weight, since for large 𝛽𝑡 the weights concentrate on high-reputation clients, whereas 𝛽𝑡 → 0 yields the uniform distribution 1∕𝐾 regardless of the reputation scores. The magnitude of this selectivity is governed by 𝛽max , since a larger value concentrates aggregation weight more heavily on high-reputation clients, whereas a smaller value keeps the distribution closer to uniform throughout. The rate 𝜆 sets how gradually the flattening unfolds over the final rounds, since a larger 𝜆 keeps 𝛽𝑡 near 𝛽max until immediately before the last round, whereas a smaller 𝜆 begins the descent earlier. Both parameters affect only the magnitude and timing of the deviation from uniform weighting, not the terminal behavior; the values used in our experiments are reported in the experimental setup. Interaction with the inner schedule. The two annealing schedules operate on complementary time scales. The inner

schedule 𝛾𝑡 = 𝑒−𝜂𝑡 shifts the reputation scores from the data-size prior toward the similarity signal during the early rounds, whereas the outer schedule in Eq. (7) flattens the weight distribution only in the final rounds. The requirement that the transition from the size prior to the similarity signal completes before the weights equalize therefore holds structurally for any 𝜂 > 0 and 𝜆 > 0, and no coupling constraint of the form 𝜆 < 𝜂 needs to be imposed between the two schedules. M. Moradpour et al.: Preprint

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3.7. Weighted Aggregation The inner and outer stages operate simultaneously at every round, forming a coupled dual annealing process. At round 𝑡: (1) the inner stage computes 𝛾𝑡 and the reputation vector {𝜉𝑖𝑡 }𝐾 via Eq. (4); (2) the outer stage computes 𝛽𝑡 and 𝑖=1 𝑡 𝐾 the weight vector {𝜔𝑖 }𝑖=1 via Eq. (6). The two parameters decay on their own schedules but govern complementary aspects of aggregation, in that 𝛾𝑡 controls the content of client evaluation, that is, what defines reputation, while 𝛽𝑡 controls its consequence, that is, how sharply reputation differences become weight differences. An analogy clarifies the interaction. The inner annealing acts as a scorer evaluating each client’s contribution quality, while the outer annealing acts as a supervisor deciding how strongly to act on those scores. Early on, the scorer relies on data size, and the supervisor amplifies the resulting differences, yielding stable, size-driven, selective aggregation. As training advances, the scorer shifts toward similarity while the supervisor simultaneously relaxes toward equal treatment of all clients. The combined trajectory moves from informed and selective aggregation to informed and fair aggregation, realizing a stabilize-first, equalize-later principle. The coupling is essential. Outer annealing without the inner stage would still equalize weights over time, but early-round reputation would rest on a single static signal, missing the opportunity to leverage the right information at the right time. Inner annealing without the outer stage would transition to similarity-based reputation but could permanently suppress low-similarity clients even at late rounds, violating fairness. Only the joint operation achieves all three design goals, namely early stability, a gradual mid-training change of criterion, and late-training fairness. After computing the weights {𝜔𝑡𝑖 } through dual annealing, the server forms 𝜃 𝑡+1 ←

𝐾 ∑

𝜔𝑡𝑖 𝜃𝑖𝑡 .

(8)

𝑖=1

This aggregation step is identical in form to standard federated averaging; the entirety of FedDRAW’s novelty lies in the dynamic, round-dependent computation of 𝜔𝑡𝑖 .

3.8. Last-Layer Similarity We compute cosine similarity using only the parameters of the final (classification) layer rather than the full parameter vector. This choice is grounded in what different layers of a deep network actually represent, and is particularly consequential in federated settings. Representations learned by deep networks progress from generic to task-specific with depth, since early layers encode features such as edges and colors that are applicable to many datasets and tasks, and the representation must transition from general to specific by the final layers of the network (Yosinski et al., 2014). When all clients start each round from the same global model 𝜃 𝑡 and perform only a small number of local epochs 𝐸, the generic earlylayer parameters therefore remain both close to their round-start values and highly similar across clients, since they encode features that are largely independent of the particular local dataset. When similarity is computed over the full parameter vector 𝜃𝑖𝑡 ∈ ℝ𝑑 , this large mass of nearly identical generic parameters dominates the score and drives it toward artificially high values, so that the discriminative signal of client-specific learning is drowned out. Consistent with this, using whole-model similarity directly has been reported to insufficiently capture the relationship between clients (Shi et al., 2025). The final layer, in contrast, maps learned feature representations to output predictions and is the component most directly tied to each client’s local label distribution, and it has been shown to be the part of the network most strongly biased by label-distribution skew across clients (Luo et al., 2021). This has also been exploited directly in federated settings, where the weights of the last layer of a local update are more sensitive to the local data distribution than those of other layers, which is why the cosine similarity of the last layer between the global model and each client update carries a usable signal about that client’s data (Yaldiz et al., 2023). Computing similarity on this layer therefore yields a sharper, more discriminative measure of client-specific learning dynamics, and does so at negligible cost. We additionally verified this behavior directly. Two models pre-trained on ImageNet (Deng et al., 2009) and fine-tuned on disjoint 5-class datasets (animals versus cars) yield a full-network cosine similarity close to 1.0, misleadingly high and dominated by the shared pre-trained parameters, whereas restricting the computation to the last layer drops the score to approximately 0.2, correctly reflecting the substantial divergence in what the two models learned during fine-tuning.

M. Moradpour et al.: Preprint

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4. Experimental setup and evaluation In this section, we evaluate the proposed method and, in particular, the interaction between its inner and outer annealing schedules, which jointly control how the aggregation weights evolve from size-prior-dominated to similarityinformed and finally to near-uniform values over the course of federated training. FedDRAW is compared against seven well-established federated learning algorithms, introduced in Section 2: FedAvg (McMahan et al., 2017), FedNova (Wang et al., 2020), FedAdp (Wu and Wang, 2021), FedProx (Li et al., 2020a), SCAFFOLD (Karimireddy et al., 2020), MOON (Li et al., 2021a), and FedDyn (Acar et al., 2021). The evaluation is carried out on two public multi-label chest radiograph datasets, CheXpert (Irvin et al., 2019) and ChestMNIST (Yang et al., 2023), under twelve client-partition scenarios of varying difficulty. Because all methods were trained with identical local hyperparameters (Section 4.1), the comparison isolates the effect of the respective heterogeneity-handling strategy. We first describe the experimental setup, the evaluation metrics, and the design of the partition scenarios, and then present the results on each dataset, a statistical comparison across all scenarios, and an analysis of the FedDRAW aggregation dynamics.

4.1. Experimental setup All experiments were conducted with a DenseNet121 backbone initialized with weights pretrained on ImageNet, in which the final classification layer was adapted to the five target pathologies (Atelectasis, Edema, Pleural Effusion, Cardiomegaly, and Consolidation). We replaced the softmax cross-entropy loss of DenseNet121 with a multi-label binary cross-entropy loss with sigmoid activation and trained the resulting model on the clients. Uncertain and missing CheXpert labels were converted to binary values using the U-Ones/U-Zeros mapping described in Section 3.1. Label smoothing with a factor of 0.05 and gradient-norm clipping at 5.0 were applied to stabilize local training, and a weight decay of 5 × 10−3 was used. All images were converted to three-channel grayscale, resized to the network input resolution, and enhanced with contrast-limited adaptive histogram equalization (CLAHE; kernel size 28, clip limit 0.02) before being normalized with the channel mean and standard deviation of the training data (0.5146 and 0.2548, respectively). This preprocessing is deterministic and was applied identically to the training, validation, and test images. During local training, the images were additionally augmented geometrically with a random rotation of up to ±8◦ and a random affine transformation with translations of up to 9% of the image size and isotropic scaling in [0.95, 1.05], applied stochastically. Validation and test images received no augmentation. The same pipeline was used at every client, and no artificial domain shift was introduced between clients, so that the heterogeneity in each scenario arose exclusively from the partitioning of the samples and their label distributions. For CheXpert, the 191,027 frontal training radiographs were partitioned among the federated clients, while the frontal subsets of the validation and test partitions provided with the dataset (202 of 234 and 518 of 668 images, respectively) were used for threshold selection and final evaluation (Irvin et al., 2019). For ChestMNIST, we used the official MedMNIST split of 78,468 training, 11,219 validation, and 22,433 test images at a resolution of 224 × 224 pixels from the large-size release (MedMNIST+) (Yang et al., 2023); only the training partition was distributed among the clients. In each communication round, every client performed 𝐸 = 2 local epochs of training with the Adam optimizer, a learning rate of 1 × 10−5 , and a batch size of 64, and federated training was run for 𝑇 = 15 communication rounds. Identical training hyperparameters were used for FedDRAW and for all competing methods, so that the reported differences are attributable to the aggregation strategies rather than to differences in local optimization. Each experiment was repeated with five independent runs, and we report the mean and standard deviation across runs. The FedDRAW-specific hyperparameters were kept fixed across all scenarios and both datasets, CheXpert and ChestMNIST (𝜂 = 0.4, 𝜆 = 0.2, 𝛽max = 3).

4.2. Evaluation metrics Model performance was evaluated using two complementary metrics, namely AUC and GM, that is, the square root of the product of sensitivity and specificity, both macro-averaged over the five target pathologies. AUC is the standard metric in chest radiograph classification, and prior studies applying federated learning to chest X-ray datasets such as CheXpert report their evaluation primarily in terms of AUC (Ziegler et al., 2022; Zulqarnain, 2026; Tayebi Arasteh et al., 2023). AUC alone, however, does not reveal the consequences of severe class imbalance, and the same holds for accuracy, which is dominated by the prevalent negative class, so that a classifier which predicts every finding as absent still attains a high value. AUC conceals the imbalance for a different and less obvious reason. The AUC is M. Moradpour et al.: Preprint

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constructed from the true-positive and false-positive rates, each of which is normalized by the size of its own class, so the curve, and hence the area beneath it, is invariant to the ratio of positive to negative samples. A model can therefore rank the positive cases well overall, and be credited with a high AUC, while at every threshold that could actually be deployed it detects only a small fraction of the positive cases. Because AUC additionally aggregates over all possible thresholds, it also does not describe the behavior of the classifier at the single threshold at which it is operated. This limitation is particularly important for chest X-ray datasets, whose labels are both highly imbalanced and affected by label noise arising from automated report labeling; under such conditions, evaluations restricted to prevalence-invariant and threshold-independent metrics can give an incomplete or misleading picture of clinical usefulness (Cabitza et al., 2026). We therefore additionally report GM (Kubat and Matwin, 1997), defined for a pathology 𝑐 as GM𝑐 =

√

Se𝑐 ⋅ Sp𝑐 ,

Se𝑐 =

TP𝑐 , TP𝑐 + FN𝑐

Sp𝑐 =

TN𝑐 , TN𝑐 + FP𝑐

(9)

where Se𝑐 and Sp𝑐 denote the sensitivity and specificity of pathology 𝑐 at its decision threshold, and the reported value is the macro-average of GM𝑐 over the five target pathologies. Because the GM collapses toward zero whenever either sensitivity or specificity degenerates, GM cannot be inflated by biasing predictions toward the majority class, which makes it a necessary complement to AUC on strongly imbalanced data. Importantly, a model can attain a high AUC in a given round while simultaneously exhibiting a low GM, because AUC summarizes the ranking of predicted probabilities over all possible thresholds, whereas GM evaluates the single threshold at which the classifier is actually operated. Evaluating both therefore reveals operating-point behavior that AUC alone cannot capture. Any threshold-dependent metric requires an explicit decision rule, and the default threshold of 0.5 is not appropriate for chest radiograph classification. That value implicitly assumes balanced classes and calibrated probabilities, neither of which holds here. With positive prevalences well below 50% for several of the target pathologies, a fixed threshold of 0.5 systematically biases predictions toward the negative class, suppressing sensitivity while leaving specificity near-saturated, so that the resulting GM reflects the prevalence of the data rather than the discriminative quality of the model. The threshold must therefore be determined from data, and it must be determined per pathology, since the five findings differ substantially in prevalence and in the sharpness of their probability distributions. Inspired by the threshold-selection protocol used for chest radiograph classification by Tiu et al. (Tiu et al., 2022), we determine the decision thresholds on the validation set and evaluate on the test set. For every algorithm and every run, the per-pathology thresholds are selected at the final communication round subject to maximizing GM𝑐 on the validation set, and are then applied once, without modification, to the held-out test set.

4.3. Client partition scenarios The two datasets alone do not cover the range of client configurations under which the objective of FedDRAW becomes testable, since neither provides a predefined partition into institutions with differing sizes or differing pathology distributions. It is therefore necessary to construct federated scenarios artificially, so that the behavior of FedDRAW and of the competing methods can be compared under controlled and reproducible degrees of heterogeneity. The scenarios were designed to test the central hypothesis of this work, namely that a client with few local samples may nevertheless be highly informative, either because its data quality is high relative to its size or because it holds samples of a pathology that is scarce or absent at the larger clients, and that such a client should not be marginalized by samplesize-proportional aggregation. In a centralized reference evaluation, models trained on the data of the designated small clients achieved higher performance than models trained on the data of the individual larger clients. This gap was built into the partition design and describes the data rather than the behavior of the aggregation method. For each dataset, we constructed a dedicated set of scenarios. Tables 2 and 3 give the client-level sample counts. We denote the CheXpert scenarios CHX-1 to CHX-7 and the ChestMNIST scenarios CHM-1 to CHM-5. The CheXpert scenarios, listed in Table 2, follow a deliberate progression. CHX-1 to CHX-4 vary the ratio of large to small clients in order to test whether FedDRAW recognizes the valuable information of the smaller clients, rather than letting the larger clients dominate the aggregation, across increasingly asymmetric federations. CHX-4 is the most severe of these configurations, as the small client contributes only 2.7% of the total training data. CHX-5 and CHX-6 simulate the clinically relevant situation in which a small client holds informative samples of a specific pathology that the larger clients largely lack. The small client of CHX-5 was enriched with 3,500 Atelectasis-positive samples and the small client of CHX-6 with 3,500 Edema-positive samples, in both cases alongside samples of the remaining four pathologies. In these two scenarios, the small client holds 20% of the training data but approximately 65% of all positive samples of the enriched pathology, whereas each large client contains only 950 of them. Finally, in CHX-7, M. Moradpour et al.: Preprint

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the CheXpert training set was split into four clients uniformly at random; this homogeneous reference scenario verifies that FedDRAW does not degrade performance when no meaningful heterogeneity is present and can compete closely with the other methods. Table 2. Client sample sizes in the CheXpert scenarios. C𝑖 denotes the 𝑖-th federated client. Each scenario represents a different heterogeneity setting: CHX-1 to CHX-4 vary the ratio of large to small clients, CHX-5 and CHX-6 enrich the small client with informative samples of a specific pathology (Atelectasis and Edema, respectively), and CHX-7 is a homogeneous random split. In all heterogeneous scenarios, the small clients carry a disproportionate share of informative samples relative to their size.

Scenario

Composition

CHX-1 CHX-2 CHX-3 CHX-4 CHX-5 CHX-6 CHX-7

1 large + 3 small C1: 20,000; C2–C4: 5,000 each 35,000 2 large + 2 small C1–C2: 20,000 each; C3–C4: 5,000 each 50,000 3 large + 1 small C1–C3: 10,000 each; C4: 2,500 32,500 9 large + 1 small C1–C9: 10,000 each; C10: 2,500 92,500 Atelectasis-enriched small client C1: 5,000; C2–C3: 10,000 each 25,000 Edema-enriched small client C1: 5,000; C2–C3: 10,000 each 25,000 4 equal (random split) C1: 47,756; C2–C4: 47,757 each 191,027

Samples per client

Total

Table 3. Client sample sizes in the ChestMNIST scenarios. C𝑖 denotes the 𝑖-th federated client. CHM-1 to CHM-3 create increasingly severe quantity imbalance; CHM-4 enriches the small client with informative Atelectasis samples, and CHM-5 is a homogeneous reference split. In all heterogeneous scenarios, the small clients carry a disproportionate share of informative samples relative to their size.

Scenario

Composition

CHM-1 CHM-2 CHM-3 CHM-4 CHM-5

1 large + 1 small C1: 18,000; C2: 60,468 78,468 1 large + 3 small C1–C3: 9,500 each; C4: 49,968 78,468 5 large + 1 small C1–C5: 14,000 each; C6: 8,468 78,468 Atelectasis-enriched small client C1: 7,000; C2–C4: 15,000 each 52,000 4 equal C1–C4: 19,617 each 78,468

Samples per client

Total

The ChestMNIST scenarios, listed in Table 3, mirror this design on a smaller and noisier dataset. CHM-1 and CHM-2 create pronounced quantity imbalance with one and three small clients, respectively, and CHM-3 extends the imbalance to a six-client federation in which the small client holds only 10.8% of the data, making it the most severe ChestMNIST configuration. CHM-4 is the class-enriched counterpart of CHX-5, with the small client enriched for Atelectasis (4,000 positive samples, i.e., more than half of its local data and the majority of all Atelectasis-positive samples in the federation), and CHM-5 is the homogeneous four-client reference split.

4.4. Results on CheXpert Table 4 summarizes the performance of FedDRAW and the competing methods on CheXpert across the seven scenarios. FedDRAW attained the highest GM in every evaluated configuration and the highest AUC in five of the seven; in the remaining two scenarios its AUC was within 0.19 and 0.16 percentage points of the best value. The magnitude of the advantage varied systematically with the degree and type of heterogeneity, and in the quantityskewed scenarios the improvements in GM consistently exceeded those in AUC. This pattern is directly aligned with the central design objective of FedDRAW. Under sample-size-proportional aggregation, the updates of a small client are strongly diluted regardless of their quality, whereas FedDRAW can temporarily amplify such updates when their similarity to the current global model is high and subsequently anneals the weights back toward uniformity. Since the reallocated influence primarily changes where the model operates rather than how it ranks, the benefit concentrates in the threshold-dependent metric. CHX-4 is the most severe quantity-skewed configuration, in which the small client holds only 2.7% of the training data and faces nine large clients that could easily dominate the aggregation. Even under this extreme asymmetry, FedDRAW remains reliable. It attained the highest GM of 77.35 ± 0.13%, improving over FedAvg (75.61 ± 0.27%) by 1.74 percentage points, while its AUC of 85.57 ± 0.60% lay within 0.19 percentage points of the best value in this scenario, a difference well inside one standard deviation. The two metrics therefore do not agree on the ordering of the methods, and the disagreement is informative rather than incidental. Because AUC is invariant to prevalence and averages over thresholds that are never applied, it cannot register the effect that the reallocated influence has on the model, since the small client shifts how positive and negative cases are separated at the selected operating points, which is exactly what GM measures and what AUC is blind to. Under the metric that reflects the deployed decision rule, FedDRAW is the leading method even in the most asymmetric federation considered here. The largest overall gains arose in CHX-1, where a single large client faces three small ones. FedDRAW obtained 86.67 ± 0.25% AUC and 78.76 ± 0.18% GM, against 84.98 ± 0.68% and 76.01 ± 1.31% for FedAvg, corresponding to improvements of 1.69 and 2.75 percentage points. The size of this improvement is consistent with the structure of the M. Moradpour et al.: Preprint

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FedDRAW Table 4. Performance (%) on CheXpert across the seven client-partition scenarios (mean ± standard deviation over five runs). Best values per column are shown in bold.

CHX-1

CHX-2

CHX-3

CHX-4

Algorithm

AUC

GM

AUC

GM

AUC

GM

AUC

GM

FedDRAW FedAvg FedNova FedProx MOON FedDyn SCAFFOLD FedAdp

86.67 ± 0.25 84.98 ± 0.68 85.33 ± 0.75 85.40 ± 0.33 83.86 ± 0.83 86.08 ± 0.13 67.77 ± 3.07 85.01 ± 0.79

78.76 ± 0.18 76.01 ± 1.31 76.68 ± 0.35 76.26 ± 1.46 76.36 ± 0.64 72.11 ± 0.52 60.27 ± 1.61 76.73 ± 1.59

87.46 ± 0.30 85.77 ± 0.33 86.16 ± 0.24 86.00 ± 0.98 84.93 ± 0.50 84.45 ± 0.36 74.80 ± 0.77 85.85 ± 0.33

77.83 ± 0.81 75.41 ± 1.32 76.40 ± 1.00 76.52 ± 0.41 76.61 ± 0.42 69.88 ± 2.19 63.46 ± 1.22 75.53 ± 1.01

86.64 ± 0.18 85.71 ± 0.43 85.63 ± 0.78 85.89 ± 0.61 83.66 ± 0.19 85.70 ± 0.56 73.21 ± 3.63 83.89 ± 3.65

78.03 ± 0.91 76.71 ± 1.80 77.03 ± 0.39 77.14 ± 1.01 75.17 ± 0.66 69.98 ± 1.10 65.92 ± 1.42 76.09 ± 1.23

85.57 ± 0.60 85.76 ± 0.68 82.88 ± 3.15 85.10 ± 0.32 82.95 ± 0.33 81.01 ± 0.05 73.45 ± 3.22 84.85 ± 0.66

77.35 ± 0.13 75.61 ± 0.27 76.20 ± 0.27 75.53 ± 0.38 72.58 ± 1.50 65.12 ± 1.65 66.40 ± 1.71 76.28 ± 0.06

Algorithm

AUC

GM

AUC

GM

AUC

GM

FedDRAW FedAvg FedNova FedProx MOON FedDyn SCAFFOLD FedAdp

80.59 ± 1.95 79.04 ± 1.29 78.95 ± 1.03 79.17 ± 1.10 79.50 ± 0.28 77.91 ± 1.05 69.12 ± 1.55 78.55 ± 1.47

73.50 ± 0.89 71.11 ± 2.59 70.18 ± 0.27 70.22 ± 1.93 71.59 ± 1.72 66.07 ± 6.53 63.17 ± 3.70 70.47 ± 1.68

83.91 ± 0.57 81.53 ± 1.91 81.78 ± 1.82 81.71 ± 0.46 79.21 ± 1.65 78.37 ± 0.64 72.74 ± 1.17 81.16 ± 2.13

73.56 ± 2.18 71.86 ± 1.41 71.23 ± 1.63 69.98 ± 0.91 66.17 ± 3.05 67.95 ± 3.63 65.52 ± 2.13 71.89 ± 2.62

89.64 ± 0.37 89.80 ± 0.23 89.16 ± 1.01 89.33 ± 1.08 85.70 ± 1.05 85.48 ± 0.44 61.03 ± 3.56 89.71 ± 0.23

79.76 ± 0.13 79.59 ± 1.07 79.02 ± 0.63 78.29 ± 0.98 77.15 ± 0.69 75.34 ± 1.51 58.85 ± 3.88 79.49 ± 1.16

CHX-5

CHX-6

CHX-7

partition, since three of the four clients are small, so a majority of the potentially informative updates are down-weighted under proportional aggregation, and correcting this under-representation affects a correspondingly large share of the federation. In addition, the standard deviations of FedDRAW were markedly smaller than those of FedAvg, indicating that redistributing influence away from a single dominant client also reduced run-to-run variability. The intermediate settings CHX-2 and CHX-3 followed the same pattern. In CHX-2, FedDRAW reached 87.46 ± 0.30% AUC and 77.83 ± 0.81% GM, improving over FedAvg (85.77 ± 0.33% and 75.41 ± 1.32%) by 1.69 and 2.42 percentage points. In CHX-3, FedDRAW achieved 86.64 ± 0.18% AUC and 78.03 ± 0.91% GM, compared with 85.71 ± 0.43% and 76.71 ± 1.80% for FedAvg, corresponding to gains of 0.93 and 1.32 percentage points. Taken together, CHX-1 to CHX-4 show that the advantage of FedDRAW grows with the fraction of the federation that is under-represented by proportional weighting, and that the method remains effective even in the most asymmetric configuration. The class-enriched scenarios CHX-5 and CHX-6 provide a more targeted assessment under distribution heterogeneity. FedDRAW assigns a single aggregation weight to each client and performs no class-wise aggregation; the purpose of these scenarios is therefore not to determine whether FedDRAW identifies an individually dominant pathology, but whether the overall model update of a small yet informative client can contribute effectively to global optimization despite its limited local dataset. In CHX-5, FedDRAW achieved 80.59±1.95% AUC and 73.50±0.89% GM, improving over FedAvg (79.04 ± 1.29% and 71.11 ± 2.59%) by 1.55 and 2.39 percentage points; in CHX-6, FedDRAW reached 83.91±0.57% AUC and 73.56±2.18% GM, compared with 81.53±1.91% and 71.86±1.41% for FedAvg, corresponding to gains of 2.38 and 1.70 percentage points. In both scenarios, no competing method exceeded FedDRAW in either metric. These improvements can be attributed to the fact that the enriched client contributes gradient information for a pathology that is comparatively scarce at the large clients; under proportional weighting this information is diluted by a factor determined solely by sample count, whereas FedDRAW allows it to enter the global model with greater influence during the rounds in which it is most useful. Notably, the enriched pathology constitutes only one component of the small client’s multi-label data, and its local update is shaped by the complete joint label distribution, which also includes the remaining four target pathologies. The results therefore support the client-level design of FedDRAW rather than a class-specific interpretation of its weighting mechanism. Finally, the homogeneous reference scenario CHX-7 confirms that the adaptive mechanism is safe when no meaningful heterogeneity is present. FedDRAW obtained the highest GM of 79.76 ± 0.13%, while its AUC of 89.64 ± 0.37% lay within 0.16 percentage points of the best value in this scenario. Here again the two metrics do not agree on the ordering of the methods, although the clients are of equal size and no method has a structural advantage. The proportional weights of FedAvg are already close to uniform in this setting, so there is little imbalance for the adaptive mechanism to compensate; the relevant observation is that FedDRAW retains the leading value under the M. Moradpour et al.: Preprint

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FedDRAW Table 5. Performance (%) on ChestMNIST across the five client-partition scenarios (mean ± standard deviation over five runs). Best values per column are shown in bold. CHM-1

CHM-2

CHM-3

CHM-4

CHM-5

Algorithm

AUC

GM

AUC

GM

AUC

GM

AUC

GM

AUC

GM

FedDRAW FedAvg FedNova FedProx MOON FedDyn SCAFFOLD FedAdp

81.78 ± 0.53 80.13 ± 0.17 78.36 ± 1.87 80.33 ± 0.67 80.37 ± 0.41 72.99 ± 0.80 63.36 ± 0.92 80.25 ± 0.48

74.84 ± 0.29 73.06 ± 0.16 71.77 ± 1.49 73.16 ± 0.28 73.04 ± 0.50 67.53 ± 0.41 59.83 ± 0.73 72.95 ± 0.36

82.91 ± 0.13 80.14 ± 0.60 80.69 ± 0.12 80.96 ± 0.55 78.60 ± 0.51 71.79 ± 0.35 61.62 ± 1.39 80.20 ± 0.43

75.86 ± 0.40 73.22 ± 0.45 73.26 ± 0.11 73.58 ± 0.52 71.40 ± 0.34 66.68 ± 0.11 58.31 ± 1.52 73.85 ± 0.45

84.18 ± 0.14 84.10 ± 0.27 84.25 ± 0.21 84.13 ± 0.25 80.03 ± 0.60 74.63 ± 1.37 68.14 ± 1.03 83.97 ± 0.24

76.61 ± 0.34 76.01 ± 0.28 75.62 ± 0.06 75.72 ± 0.25 72.95 ± 0.59 68.57 ± 1.09 63.36 ± 0.81 76.15 ± 0.37

82.67 ± 0.77 81.69 ± 0.21 81.37 ± 0.34 81.22 ± 0.68 79.51 ± 0.43 75.74 ± 0.54 65.23 ± 1.37 81.26 ± 0.41

75.48 ± 0.42 73.67 ± 0.41 74.18 ± 0.09 72.12 ± 1.04 72.50 ± 0.37 72.39 ± 1.88 62.70 ± 0.63 73.64 ± 2.01

83.86 ± 0.69 83.97 ± 0.10 83.77 ± 0.13 84.22 ± 0.04 82.27 ± 0.16 74.86 ± 0.85 60.56 ± 3.87 83.83 ± 0.10

76.53 ± 0.37 76.30 ± 0.14 76.24 ± 0.15 76.61 ± 0.06 74.99 ± 0.23 68.89 ± 0.65 57.45 ± 2.99 76.66 ± 0.14

metric that reflects the deployed operating points, and that the adaptive weighting incurs no performance penalty when its principal advantage is not required.

4.5. Results on ChestMNIST Table 5 reports the ChestMNIST results under the five scenarios. Because ChestMNIST is the noisier of the two datasets, it permits client partitions with more pronounced heterogeneity and therefore allows FedDRAW to be examined under more severe conditions than CheXpert. FedDRAW attained the highest GM in the four heterogeneous configurations, while the AUC differences between the leading methods were small; this pattern is examined in detail below and is itself informative about the relative sensitivity of the two metrics. In CHM-1, which represents a pronounced two-client quantity imbalance, FedDRAW achieved 81.78 ± 0.53% AUC and 74.84 ± 0.29% GM, compared with 80.13 ± 0.17% and 73.06 ± 0.16% for FedAvg, corresponding to improvements of 1.65 and 1.78 percentage points. The advantage widened further in CHM-2, in which FedDRAW obtained 82.91 ± 0.13% AUC and 75.86 ± 0.40% GM, against 80.14 ± 0.60% and 73.22 ± 0.45% for FedAvg, corresponding to improvements of 2.77 and 2.64 percentage points; in both scenarios, no competing method reached a higher value of either metric. As in the corresponding CheXpert experiments, the gains were largest when small clients collectively held a substantial fraction of the informative updates, supporting the interpretation that the benefit of FedDRAW stems from counteracting the systematic dilution of small-client contributions rather than from a datasetspecific effect. FedDRAW also achieved the best performance in the class-enriched scenario CHM-4, in which the smaller client was intentionally enriched for Atelectasis while retaining examples of the other target pathologies. FedDRAW reached an AUC of 82.67 ± 0.77% and a GM of 75.48 ± 0.42%, compared with 81.69 ± 0.21% and 73.67 ± 0.41% for FedAvg, corresponding to improvements of 0.98 and 1.81 percentage points. Together with the corresponding CheXpert experiments, this finding indicates that the benefit of adaptive client weighting is not restricted to a single dataset and is consistent with the hypothesis that a client’s contribution to federated optimization cannot always be adequately represented by its local sample count alone. CHM-3 is the most severe configuration, in which the small client holds only 10.8% of the data in a six-client federation and the five large clients jointly dominate the sample count. In this scenario, FedDRAW attained the highest GM of 76.61±0.34% among all methods, while its AUC of 84.18±0.14% lay 0.07 percentage points below the strongest AUC (84.25 ± 0.21%), a difference well within one standard deviation. This is precisely the situation anticipated in Section 4.2. Under strong class imbalance combined with client heterogeneity, AUC is not sufficient to characterize a classifier, because it is invariant to prevalence and aggregates over thresholds that are never used in practice. Methods that appear indistinguishable, or even marginally preferable, under AUC are separated once the operating points are taken into account, and it is at those operating points that FedDRAW realizes the benefit of reallocating influence toward the informative small client. The result therefore confirms both the necessity of reporting GM in this setting and the effectiveness of FedDRAW under the most severe heterogeneity considered here. The homogeneous reference scenario CHM-5 confirms the same conclusion from the opposite direction. Here the ordering of the methods under AUC again differs from the ordering under GM, even though all clients hold equal amounts of data and no method has a structural advantage. Taken together with CHM-3, this shows that AUC alone cannot decide between aggregation strategies on imbalanced multi-label chest radiographs, since the metric is invariant to prevalence and averages over thresholds that are never used, so it neither exposes the disadvantage of proportional weighting under heterogeneity nor credits the correction that FedDRAW applies. GM, which evaluates the classifiers at the thresholds under which they would be deployed, is therefore the metric that reflects the objective of this work.

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FedDRAW Table 6. Per-scenario ranks of every method used in the Friedman test (AUC / GM; rank 1 = best, tied methods receive the average of the tied ranks). The last two rows report the average AUC and GM ranks across the twelve scenarios. Scenario

FedDRAW FedAvg FedNova FedProx MOON FedDyn SCAFFOLD FedAdp

CHX-1 CHX-2 CHX-3 CHX-4 CHX-5 CHX-6 CHX-7 CHM-1 CHM-2 CHM-3 CHM-4 CHM-5

1/1 1/1 1/1 2/1 1/1 1/1 3/1 1/1 1/1 2/1 1/1 3/3

6/6 5/6 3/4 1/4 4/3 4/3 1/2 5/3 5/5 4/3 2/3 2/4

4/3 2/4 5/3 6/3 5/6 2/4 5/4 6/6 3/4 1/5 3/2 5/5

3/5 3/3 2/2 3/5 3/5 3/5 4/5 3/2 2/3 3/4 5/7 1/2

7/4 6/2 7/6 5/6 2/2 6/7 6/6 2/4 6/6 6/6 6/5 6/6

2/7 7/7 4/7 7/8 7/7 7/6 7/7 7/7 7/7 7/7 7/6 7/7

8/8 8/8 8/8 8/7 8/8 8/8 8/8 8/8 8/8 8/8 8/8 8/8

5/2 4/5 6/5 4/2 6/4 5/2 2/3 4/5 4/2 5/2 4/4 4/1

Average rank (AUC) Average rank (GM)

1.50 1.17

3.50 3.83

3.92 4.08

2.92 4.00

5.42 5.00

6.33 6.92

8.00 7.92

4.42 3.08

4.6. Statistical analysis To assess whether the observed differences between the algorithms are systematic across the full experimental design, we used the Friedman test (Demšar, 2006), a non-parametric test that compares multiple algorithms over multiple datasets without assuming normality or homogeneity of variances. Because AUC and GM capture two different properties of the trained models, we analyzed the two metrics separately. For each metric, the eight methods were ranked within every scenario, with rank 1 denoting the best method and tied methods receiving the average of the tied ranks, yielding 𝑁 = 12 ranking blocks over 𝑘 = 8 algorithms per metric. Table 6 reports the AUC and GM ranks of every method in every scenario together with the per-metric average ranks. The Friedman test rejected the null hypothesis of equal performance for both metrics (𝜒𝐹2 (7) = 58.61 for AUC and 𝜒𝐹2 (7) = 63.50 for GM, both 𝑝 < 0.001), confirming that the choice of aggregation strategy has a significant effect on performance. FedDRAW obtained the best average rank for both metrics, 1.50 for AUC and 1.17 for GM, ranking first in eight of the twelve scenarios for AUC and in eleven of the twelve for GM; the next-best average ranks were 2.92 for AUC and 3.08 for GM. After establishing significant differences with the Friedman test, we performed the Nemenyi post-hoc test (Demšar, 2006). If the difference in average ranks between two algorithms exceeds a critical difference (CD), the difference is significant. Given the moderate number of scenarios (𝑁 = 12) relative to the number of compared algorithms (𝑘 = 8), we used a significance level of 𝛼 = 0.10, for which the critical value 𝑞𝛼 for 𝑘 = 8 algorithms is 2.780 (Demšar, 2006). Eq. 10 is the result of our calculation of the critical difference. √ 𝑘(𝑘 + 1) CD = 𝑞𝛼 = 2.78, (10) 6𝑁 This indicates that any pairwise difference in average ranks larger than 2.78 is statistically significant. On AUC, FedDRAW performed significantly better than FedAdp (Wu and Wang, 2021), MOON (Li et al., 2021a), FedDyn (Acar et al., 2021), and SCAFFOLD (Karimireddy et al., 2020). On GM, FedDRAW performed significantly better than FedProx (Li et al., 2020a), FedNova (Wang et al., 2020), MOON (Li et al., 2021a), FedDyn (Acar et al., 2021), and SCAFFOLD (Karimireddy et al., 2020). The stronger separation obtained under GM is consistent with the design of FedDRAW, whose reallocation of aggregation weight primarily improves the sensitivity–specificity balance at the operating thresholds. Figure 3 visualizes both analyses as critical-difference diagrams.

Figure 3. Critical-difference diagrams of the Nemenyi post-hoc test over the twelve scenarios for AUC (left) and GM (right) (𝛼 = 0.10, CD = 2.78). Algorithms connected by a horizontal bar are not significantly different.

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FedDRAW

4.7. Analysis of FedDRAW aggregation dynamics The central design objective of FedDRAW is that a small client whose updates are informative should not be permanently constrained to the contribution implied by its sample count. Figure 4 examines this behavior in the two CheXpert scenarios with the most pronounced size asymmetry. In CHX-4, Clients 1–9 each hold 10,000 samples and Client 10 holds 2,500, so that Client 10 represents approximately 2.7% of the training data against approximately 10.8% for each larger client. Under sample-sizeproportional aggregation this ratio would be fixed for the entire run, and FedDRAW indeed begins from exactly this allocation, assigning Client 10 an initial weight of 0.0270 because the round-0 weights are given by the data-size prior alone. Within four rounds, the weight of Client 10 rises above the uniform level 1∕𝐾 = 0.10 and reaches a maximum of 0.1037 in round 6, at which point the small client receives a larger share of the aggregate than any of the four times larger clients (Figure 4a). The weight then declines and reaches 0.1000 by round 14. The reallocation is therefore not a transient artifact of initialization but a sustained reweighting over seven consecutive communication rounds, driven by the inner schedule as reliance shifts from the data-size prior toward last-layer similarity, and subsequently released by the outer schedule. CHX-1 exhibits the same mechanism from the opposite direction. A single client of 20,000 samples faces three clients of 5,000 samples each and would hold a fixed weight of 0.5714 under proportional aggregation. FedDRAW starts from this same weight and reduces it rapidly; the three small clients overtake it in round 5, and the large client reaches its minimum of 0.2334 in round 6 before returning to 0.2500 as the outer schedule drives the weights toward uniformity (Figure 4c). The initial dominance is thus dismantled within five rounds, consistent with the pronounced improvement observed for this configuration (Section 4.4). t)

1e 4

FedDRAW weight ( it)

0.08 0.06 Client 1 Client 2 Client 3 Client 4 Client 5

0.04 0.02

Client 6 Client 7 Client 8 Client 9 Client 10

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Deviation from mean weight ( it

4

0.10

2 0 2 4 6 8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Communication round (t)

Communication round (t)

Client 1 Client 2 Client 3 Client 4

0.5 0.4 0.3 0.2 0.1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Communication round (t) (c)

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Deviation from mean weight ( it

FedDRAW weight ( it)

0.6

(b)

t)

(a)

1.5 1e 4 1.0 0.5 0.0 0.5 1.0 1.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Communication round (t) (d)

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FedDRAW Figure 4. Evolution of FedDRAW aggregation weights 𝜔𝑡𝑖 across communication rounds on CheXpert. (a) CHX-4, Clients 1–9 with 10,000 samples each and Client 10 with 2,500 samples; the weight of the small client rises above the uniform level and exceeds that of every larger client between rounds 4 and 10. (b) Deviation of Clients 1–9 from their mean weight in CHX-4. (c) CHX-1, Client 1 with 20,000 samples and Clients 2–4 with 5,000 samples each; the initial dominance of the large client is dismantled within five rounds. (d) Deviation of Clients 2–4 from their mean weight in CHX-1. Panels (b) and (d) show that equally sized clients are differentiated during training on a scale that is not resolvable in panels (a) and (c). All panels share the client colors and line styles given in the legends of panels (a) and (c).

On the scale of Figures 4a and 4c the curves of the equally sized clients appear to overlap, but they are not identical; panels (b) and (d) therefore plot their deviation from the group mean, which resolves differences that the common scale cannot display. The curves are distinct rather than degenerate, since FedDRAW separates same-size clients on the order of 10−4 , and in CHX-4 the separation is systematic, with Clients 7–9 remaining below the mean in every round. The spread reaches 9.98 × 10−4 in round 6 in CHX-4 and 3.60 × 10−4 in round 4 in CHX-1, and vanishes exactly in the final round of both scenarios, where 𝛽𝑇 −1 = 0 enforces uniform weights. Equally sized clients are therefore distinguished while the similarity signal is informative and equalized once the outer schedule has annealed. These dynamics should be read together with the predictive results. In CHX-4, FedDRAW achieved 85.57 ± 0.60% AUC and 77.35 ± 0.13% GM against 85.76 ± 0.68% and 75.61 ± 0.27% for FedAvg, and the reallocation of weight toward the small client improved the sensitivity–specificity balance by 1.74 percentage points of GM at essentially unchanged discrimination. Overall, the experiments demonstrate that the advantage of FedDRAW is most evident under heterogeneous client configurations, particularly when comparatively small clients contain useful information that would receive limited influence under conventional sample-size-based aggregation.

5. Conclusion In this work, we addressed a structural limitation of sample-size-proportional aggregation in federated learning. Weighting clients by their number of local samples means that influence is granted to the largest participants by construction, although a smaller client may hold informative samples, including presentations of a pathology that are scarce or entirely absent at the larger clients. Moreover, because all clients continue to train throughout the federation, a weight that is fixed in advance cannot reflect how informative a client’s updates actually are at a given stage of training. We proposed FedDRAW, a server-side aggregation strategy that blends the cosine similarity between client and global classification parameters with a data-size prior under two coupled exponential annealing schedules. The resulting weights follow a deliberate trajectory over training, in which the size prior anchors aggregation in the early rounds, model similarity gains influence as training advances, and a decaying temperature drives all weights toward uniformity as training converges. FedDRAW leaves client-side training unchanged, introduces no additional communication, and was applied with a single fixed hyperparameter configuration across all experiments. Across twelve client-partition scenarios on CheXpert and ChestMNIST, FedDRAW achieved the highest GM in all seven CheXpert configurations and the highest AUC in five of them, as well as the highest values of both metrics in the principal heterogeneous ChestMNIST configurations, with the largest gains arising precisely where proportional weighting is most restrictive, that is, when a substantial fraction of the federation consists of small or class-enriched clients. In homogeneous settings, where little imbalance exists to compensate, FedDRAW remained competitive with the strongest baselines, indicating that the adaptive mechanism incurs no penalty when its principal advantage is not required. The analysis of the aggregation dynamics confirmed that the mechanism operates as designed, since a client holding only 2.7% of the training data transiently received an above-proportional share of influence before the weights converged toward uniformity. Our evaluation further showed that the relative ranking of federated methods can differ between threshold-independent discrimination and threshold-dependent operating performance, supporting the practice of reporting GM alongside AUC for imbalanced, noisily labeled chest radiograph data. As future work, we plan to extend FedDRAW from client-level to class-level adaptivity. In its present form, the method assigns a single scalar weight to each client, so a client that is informative for one pathology but not for others still receives one aggregate weight. A class-wise variant, in which the aggregation weight is computed separately for each pathology, would allow every client to contribute selectively to the findings for which its data are most informative, and is a natural direction for extending the reputation-based weighting proposed here.

Acknowledgment This work was supported by BMFTR (grants 01D2208A and 01KD2414A/FAIrPaCT), the hessian.AI Innovation Lab (Hessian Ministry for Digital Strategy and Innovation, grant S-DIW04/0013/003), and NHR-Nord@Göttingen (Emmy/Grete) as part of the NHR infrastructure, partly funded by the Deutsche Forschungsgemeinschaft (DFG, grant 405797229). M. Moradpour et al.: Preprint

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