FedMaJ: Ordinal Meta-Aggregation for Federated Learning Using Majority-Judgment Voting

FedMaJ: Ordinal Meta-Aggregation for Federated Learning Using Majority-Judgment Voting Driss Choukri•◦ , Essaid Sabir• , Halima Elbiaze⋄ and Abdelkrim Haqiq◦ • Department of Science and Technology, TÉLUQ, University of Quebec, Montreal, H2S 3L4, Canada ◦ Computer Networks, Mobility and Modeling Laboratory, FST, Hassan I University of Settat, Morocco ⋄ Department of Computer Science, University of Quebec at Montreal, Montreal, Canada

arXiv:2610.39985v1 [cs.NI] 29 Oct 2026

Emails: driss.choukri.doc@uhp.ac.ma, essaid.sabir@teluq.ca, elbiaze.halima@uqam.ca, abdelkrim.haqiq@uhp.ac.ma

Abstract—Federated learning traditionally relies on a fixed aggregation rule despite shifting client state and wireless dynamics. We present FedMaJ, a framework that dynamically selects aggregation rules via Majority Judgment voting. During voting rounds, clients evaluate candidate models across performance, resource utilization, channel condition, and fairness metrics. The server adopts the majority-preferred rule and retains it until the next vote, scheduled dynamically based on accuracy trends. Evaluated in over-the-air federated learning with explicit energy and resource cost modeling, synthetic and Fashion-MNIST experiments show that FedMaJ achieves competitive learning performance and superior aggregation-influence fairness. Under extreme-evaluation perturbations, Majority Judgment yields significantly more stable rule selections than mean-based scoring, establishing ordinal voting as a robust control layer for adaptive aggregation. Index Terms—Federated learning, aggregation rule selection, Majority Judgment, dynamic aggregation, communication-aware learning, client heterogeneity.

I. I NTRODUCTION Federated learning (FL) enables distributed clients to train a shared model without transmitting raw data. Its performance depends on how client updates are aggregated. Existing methods account for dataset size [1], data quality [2], resource availability [3], wireless conditions [4], or participation history [5]. However, the preferred criterion may change as learning progresses and client resources, channels, and participation patterns evolve. In over-the-air FL (OTA-FL), this choice also interacts with transmit scaling, receiver denoising, and aggregation distortion [6], [7]. Adaptive aggregation addresses part of this challenge. For example, FedMcon learns an aggregator through a meta-controller trained on proxy data [8]. Here, we investigate a complementary approach: selecting among a finite library of aggregation rules using client evaluations. Averaging numerical evaluations can be sensitive to extreme scores and requires comparable scoring scales. This motivates an ordinal decision mechanism that reflects broad client support. We propose FedMaJ, a client-informed framework for dynamic aggregation-rule selection based on Majority Judgment [9]. At voting rounds, selected clients digitally upload their updates, and the parameter server (PS) constructs and

broadcasts one candidate model per rule. Clients evaluate these candidates locally and assign ordinal grades using learning, communication, resource, and fairness indicators. The PS selects the rule with the highest majority grade, resolves ties through iterative median comparison, and retains the rule until the next vote. An accuracy-dependent voting interval adjusts the frequency of candidate evaluation and selection. FedMaJ separates the choice of aggregation policy from its execution. It supports conventional digital FL and is instantiated here in OTA-FL, where the selected logical weights interact with wireless transmission and aggregation distortion. By involving clients in policy selection, FedMaJ makes their evolving preferences part of the aggregation decision rather than relying solely on server-defined criteria. The main contributions are: • Client-informed ordinal selection: We formulate aggregation-rule selection as a repeated decision in which clients grade candidate models and Majority Judgment identifies the winning rule. • OTA-FL integration and overhead modeling: We integrate rule selection with wireless-aware indicators and adaptive voting, and characterize its communication, computation, and energy costs. • Experimental evaluation: Synthetic and FashionMNIST experiments compare FedMaJ with fixed rules, random selection, and average-score selection, assessing learning performance, influence fairness, system costs, client satisfaction, and sensitivity to extreme evaluations. II. M AJORITY-J UDGMENT-BASED F EDERATED L EARNING A. General FL Setting and Candidate Rules Consider a PS and clients K = {1, 2, . . . , K}. At communication round t, a subset St ⊆ K participates. Client k owns dataset Dk with size nk and computes a local update ∆t,k from the current global model. FedMaJ assumes a finite candidate set A = {A1 , A2 , . . . , AM }, and M = |A|. Each (m) rule Am produces logical aggregation weights pt,k satisfying P (m) (m) pt,k ≥ 0 and k∈St pt,k = 1. If rule Am is selected, the P (m) (m) desired aggregate update is ∆t = k∈St pt,k ∆t,k followed

TABLE I: Candidate rules and unnormalized scores. (m)

Rule Principle

Score ωt,k

A1 A2 A3

Dataset size [1] Uniform influence Loss awareness [2]

nk 1 ℓt,k + ϵℓ

A4

Resources [3], [10]

nk Bt,k bt,k +ϵE E

A5 A6

Link reliability [4] nk ct,k OTA distortion [7], [11] d nk+ϵ

A7

Participation [5], [12]

t,k

d

nk Nt,k +ϵN

(m)

by the global update wt+1 = wt + ∆t or an equivalent model-averaging form. Table I summarizes the candidate rules used by FedMaJ and provides representative references for their underlying design principles. The cited works support these principles but do not necessarily use the exact proxy formulations adopted here. Each rule defines a nonnegative (m) (m) (m) P (m) score ωt,k , normalized as pt,k = ωt,k / j∈St ωt,j . bt,k Here, ℓt,k denotes local loss, Bt,k residual resources, E estimated client cost, ct,k link reliability, dt,k a distortion or communication-cost indicator, and Nt,k past participation count. The ϵ terms are positive stabilizers. All quantities are bt,k does not available before rule selection; in particular, E depend on the realized cost of the rule selected in the same round. Algorithm 1 summarizes the procedure. B. Majority-Judgment Rule Selection At a voting round t, selected clients digitally upload their local updates through orthogonal links, allowing the PS to reuse each update under all M candidate weight vectors. Dataset sizes, battery states, current channel indicators, and previous participation or influence are available before grading; postround battery and influence updates are excluded. FedMaJ supports any finite ordered grade set Tn = {T1 , T2 , . . . , Tn }, with T1 ≺ T2 ≺ · · · ≺ Tn , where T1 and Tn are the lowest and highest grades. Each selected client assigns one grade to each candidate rule Am , and Majority Judgment selects the winner using the grade distributions and their medians rather than arithmetic averages. At a voting round, after receiving the individual updates, for any candidate m ∈ {1, 2, . . . , M }, the PS constructs X (m) (m) (m) (m) ∆t = pt,j ∆wt,j , wt+1 = wt + ∆t (1) j∈St

The PS broadcasts all M full candidate models to every selected client. Client k, not the PS, then  evaluates  (m) (m) each model on its private dataset ℓt,k = Fk wt+1 =   P (m) 1 ξ∈Dk f wt+1 ; ξ . Each candidate is evaluated locally nk on private data; raw datasets are never transmitted. For scalars {xi }i∈I , let xmin = minj∈I xj and xmax = maxj∈I xj . Define NI (xi ) = (xi − xmin )/(xmax − xmin ) when xmax > xmin , and 0.5 otherwise; reverse normalization is NI− (xi ) = 1 − NI (xi ). For each candidate, use

(m) (m) (m) (m) pet,k = NSt (pt,k ) and ℓet,k = NSt (ℓt,k ). For x ∈ {c, B, I}, − write x e− t,k = NSt (xt,k ), where It,k is cumulative aggregation influence before round t; participation count Nt,k may serve as a proxy when influence history is unavailable. The learning (m) (m) (m) indicator Qt,k = 1 − |e pt,k − ℓet,k | matches influence to learning need, with larger loss indicating greater need. Com(m) (m) − munication and resource indicators, Ct,k = 1 − pet,k e ct,k and (m) (m) e − Rt,k = 1 − pet,k Bt,k , penalize high influence on unreliable links and resource-depleted clients. The fairness indicator (m) (m) − Ft,k = 1 − |e pt,k − Iet,k | favors previously underrepresented clients. The combined grading score is (m)

(m)

(m)

(m)

(m)

Ut,k = αQ Qt,k + αC Ct,k + αR Rt,k + αF Ft,k ,

(2)

where αQ , αC , αR , αF ≥ 0 and αQ + αC + αR + αF = 1. (m) Before grading, Ut,k is clipped to [0, 1]. For a grade set with n grades, define the threshold vector 0 = θ0 < θ1 < · · · < θn−1 < θn = 1. The grading map is Γn (U ) = Ti when θi−1 ≤ U < θi , with Γn (1) = Tn . Thus, each  scalar  score is (m) (m) mapped to an ordinal grade by gt,k = Γn Ut,k . The PS (m)

collects Gt

(m)

= {gt,k : k ∈ St } and selects (m)
A⋆t = arg max MJ Gt . Am ∈A

(3)

Here, MJ ranks candidates by their median grade. Ties are resolved by repeatedly removing one occurrence of the common median from each tied candidate and comparing the remaining medians. C. Client Satisfaction with the Selected Rule Let ι(Ti ) = i. For M ≥ 2, client k’s satisfaction with any candidate A is M h i X 1 (m) (A) I ι(gt,k ) > ι(gt,k ) . (4) St,k (A) = 1 − M − 1 m=1 It lies in [0, 1] and equals one when A receives the client’s highest grade, including ties. This diagnostic does not affect selection. At voting rounds, write St,k = St,k (A⋆t ) and Zv = PT t=1 vt |St |. Mean satisfaction and top-choice rate are S client = Zv−1

T X

vt

t=1

Ptop = Zv−1

T X t=1

X

St,k ,

(5)

I[St,k = 1].

(6)

k∈St

vt

X k∈St

active et ) for the For all-round comparisons, set St,k = St,k (A et , and average as active rule A PT P active t=1 k∈St St,k S active = . (7) PT t=1 |St |

These all-round grades are reconstructed offline for fixed and adaptive methods, adding no operational communication, energy, latency, or voting overhead.

D. Dynamic and Resilient Voting Mechanism To reduce decision-layer overhead, FedMaJ adapts the voting interval to the accuracy margin:    [Acct−1 − Accmin ]+ vote , (8) τt = min τmax , 1 + ∆acc where [x]+ = max{x, 0}, Accmin is the required accuracy, ∆acc > 0 is the accuracy-margin step, and τmax is the maximum voting interval. Here, Acct−1 is the accuracy of the current global model wt ; for t = 1, the PS evaluates the initial model to obtain Acc0 . This accuracy is supplied by the ordinary global-model monitoring mechanism and is distinct from the candidate losses used in client grading. When Acct−1 ≤ Accmin , voting occurs every round. Above this threshold, each additional ∆acc increases the interval by one, up to τmax , thereby reducing candidate-evaluation, reporting, broadcasting, and decision overhead. Let tlast be the most recent voting round. With v1 = 1, the voting indicator for t ≥ 2 is ( 1, t − tlast ≥ τtvote , vt = (9) 0, otherwise, and tlast ← t whenever vt = 1. et = A⋆t At a voting round, Majority Judgment selects A e e according to (3); otherwise, the PS retains At = At−1 . The resulting logical aggregate is X (Ae ) ∆t = pt,kt ∆t,k . (10) k∈St

By relying on median-supported grades rather than average scores, Majority Judgment limits the effect of extreme evaluations and favors broadly supported rules. Clients are assumed to follow the prescribed grading protocol; strategic manipulation and incentive-compatible voting are left for future work. E. Conventional, Hybrid, and OTA-FL Interpretations In conventional digital FL, the PS directly applies the e ) (A selected weights pt,kt to the received updates. In wireless or hybrid FL, the grades and indicators ct,k and dt,k can also capture link reliability, delay, packet loss, quantization, and retransmissions. In OTA-FL, the selected logical weights are realized through physical-layer variables. For a single-antenna system, X e ) (A b t = at ∆ ht,k bt,k pt,kt ∆t,k + at zt , (11) k∈St

where ht,k , bt,k , at , and zt denote the OTA channel, transmit scaling, receiver denoising, and noise, respectively. Digital dl grade reporting and control broadcasting use hul t,k and ht,k . H For multiple antennas, ht,k is replaced by mt ht,k . Here, ct,k represents link reliability, while dt,k measures the mismatch between desired and realized aggregation coefficients. Thus, OTA-FL candidates jointly account for logical weights, wireless controls, energy, and aggregation distortion, covering representative OTA-FL designs [6], [7], [11]. The remainder focuses on this OTA-FL setting.

Algorithm 1 FedMaJ with Adaptive Voting Require: A, T , w1 , Accmin , ∆acc , τmax e0 , tlast ← 1, Nvote ← 0; evaluate Acc0 1: Initialize A 2: for t = 1, . . . , T do 3: Select St , broadcast wt , and compute local updates 4: Set v1 = 1; for t > 1, use (8)–(9) 5: if vt = 1 then 6: Clients digitally upload individual updates 7: PS constructs and broadcasts all M candidate models using (1) 8: Clients evaluate candidates and report ordinal grades et = A⋆t using (3) PS selects A tlast ← t; Nvote ← Nvote + 1 else et = A et−1 12: Retain A 13: end if et through (11); update wt+1 and evaluate Acct 14: Apply A 15: end for 16: return wT +1 , rule trace, Nvote 9: 10: 11:

III. P ROPOSED F ED M A J A LGORITHM A. FedMaJ Procedure Algorithm 1 summarizes the accuracy-adaptive FedMaJ procedure. Local training, OTA update transmission, and aggregation occur at every communication round, whereas candidate evaluation is performed only at voting rounds. Between voting rounds, the PS reuses the most recently selected rule. At each voting round, a run uses either ordinal grades with Majority Judgment or numerical scores with average-score selection (FedMaJ-AvgScore); the two reporting schemes are never combined. The PS also broadcasts compact control information required for normalization, such as selected-set minima, maxima, and denominators. Each client combines this (m) (m) information with its local statistics to compute pet,k , ℓet,k , e ct,k , et,k , and Ie− . Therefore, clients do not receive the raw local B t,k statistics of other participants. This limits direct information disclosure but does not provide a formal privacy guarantee. B. Communication Overhead At each vote, a selected client reports M evaluations: grade score Bt,k = M ⌈log2 n⌉ bits for MJ, or Bt,k = M bscore bits for AvgScore. For S selected clients per round, total x reporting is Bxul = Nvote SBt,k , x ∈ {grade, score}; with P variable participation, replace Nvote S by t vt |St |. This reporting payload excludes model updates and routine metadata (dataset sizes, participation, influence, resources, and channel estimates), assumed available through ordinary operation. The downlink carries one model at non-voting rounds and M candidates plus Lctrl control bits at voting rounds: t dl Btotal = T B model +

T X t=1

  vt (M − 1)B model + Lctrl . (12) t

For constant control size, the additional downlink payload is Nvote [(M − 1)B model + Lctrl ]. C. Complexity Analysis For S selected clients, M rules, n grades, and model dimension dw , local training is unchanged and ordinary PT aggregation costs O(T Sdw ). The voting count Nvote = t=1 vt satisfies ⌈T /τmax ⌉ ≤ Nvote ≤ T , with voting rate ρvote = Nvote /T . Candidate construction costs O(Nvote M Sdw ) in total. At each vote, every selected client performs M model evaluations and O(M ) grading operations; evaluation cost depends on the model and local dataset. Sorting-based Majority Judgment costs O(Nvote M S log S), while histogram-based processing costs O(Nvote M (S + n)), linear in client–candidate pairs for fixed n. Voting-schedule updates cost O(T ) overall. Globalmodel monitoring is common to all methods and excluded from the additional decision overhead. D. Energy Model We sum selected-client and PS energy over T rounds and (x) = E (x) /Eref , using one fixed Eref > 0 for report E all components and configurations. Physical energies are in joules; normalized values are dimensionless model estimates, not hardware measurements. The system total is " # T 1 X X comp dl Et,k + EPS,t + EPS,t . (13) E sys = Eref t=1 k∈St

Client and PS computation costs are decomposed as
loc OTA eval x Et,k = Et,k + Et,k + vt Et,k + Et,k , comp agg dec EPS,t = EPS,t + vt EPS,t ,

(14) (15)

where x = grade for FedMaJ-MJ and x = score for FedMaJAvgScore. Local training and OTA transmission occur every round; candidate evaluation, reporting, and rule selection occur only at voting rounds. o For any computing node i and operation o, we use Ei,t = o 2 o κi Ci,t fi,t , where κi , Ci,t , and fi,t are effective switched capacitance, CPU cycles, and frequency [13]. This covers client training/evaluation and PS aggregation/selection. For training, cpu loc Ck,t = νt,k nproc t,k χk : local iterations times processed samples per iteration times cycles per sample. Evaluation cycles OTA OTA OTA τt . cover all M candidates. OTA energy is Et,k = Pt,k For q ∈ {ul, dl}, digital link rates are ! P q |hqt,k |2 q q rt,k = W log2 1 + , (16) N0 W q where W q , P q , and hqt,k denote bandwidth, transmit power, dl and channel coefficient, with P dl = PPS and noise spectral density N0 . Using the reporting payloads defined above, x x ul Et,k = P ul Bt,k /rt,k . The PS broadcasts at the weakest selected-client rate:
dl PPS [1 + vt (M − 1)]B model + vt Lctrl t dl EPS,t = . (17) dl mink∈St rt,k

Thus, fixed rules avoid voting-specific costs, while less frequent voting reduces candidate-evaluation, reporting, control, and decision overhead. Normalization preserves additivity and is used only for reporting, not rule selection. IV. E XPERIMENTAL VALIDATION We evaluate FedMaJ using both synthetic data and the Fashion-MNIST dataset under the same OTA-FL experimental configuration; only the underlying data are replaced between the two experiments. The synthetic experiment captures controlled data, resource, and wireless heterogeneity, with proxy accuracy measured from the distance between the global and target parameter vectors. To validate FedMaJ on real-world data, Fig. 1 reports the test accuracy, aggregation-influence fairness, and test loss obtained using Fashion-MNIST. The remaining results are obtained from the synthetic experiment. Results are averaged over ten random seeds, with shaded regions representing one standard deviation. A. Experimental Setup 1) Synthetic OTA-FL Environment: We consider K = 40 clients, select S = 7 clients uniformly without replacement per round, and use a dw = 60 dimensional model over T = 120 rounds. Dataset sizes are sampled between 60 and 219, while local targets wk⋆ are generated around a common center with standard deviation 3.0 to induce strong non-IID heterogeneity. The global target w⋆ is their dataset-size-weighted average. At round t, client k computes ∆t,k = ηt (wk⋆ − wt ) + εt,k , where εt,k is zero-mean Gaussian√ noise with standard deviation 0.04, and ηt = max{0.10, 0.22/ 1 + 0.015t}. The OTA channel gains follow a Gamma distribution with shape 1.0 and scale 0.30, clipped to [0.02, 3]. We set bt,k ≤ 1.4, at = 1, and the OTA noise standard deviation to 0.02. The communication parameters are W ul = 1 MHz, W dl = 2 MHz, dl = 0.5 W, and N0 = 10−12 W/Hz. Each P ul = 0.1 W, PPS model model contains B = 1920 bits. The main parameters are summarized in Table II. 2) Accuracy-Aware Grading Instantiation: The gradingweight vector αt = (αQ , αC , αR , αF ) is adapted to the current learning state as ( (0.55, 0.15, 0.10, 0.20), Acct−1 < Accmin , αt = (18) (0.30, 0.20, 0.20, 0.30), otherwise. This prioritizes learning-quality recovery below the accuracy threshold and restores a balanced learning–communication– resource–fairness tradeoff afterward. To prevent a candidate with insufficient learning quality from winning only because of favorable resource or fairness indicators, Majority Judgment is applied to the learning-safe set   (m) safe safe At = Am ∈ A : median Qt,k ≥ Qt,k,m = 0.55 . (19) k∈St

If this set is empty, the candidate with the smallest mean client-local candidate loss is retained as an emergency fallback. The six ordinal grades are Very Poor, Poor, Fair,

TABLE II: Main parameter values. Parameter

Value

Parameter

Value

Independent seeds Non-IID target standard deviation Learning-step decay coefficient Maximum transmit scaling Accuracy-margin step ∆acc Score quantization bscore

10 3.0 0.015 1.4 0.01 32 bits

Model dimension dw Initial-model error standard deviation Local-update noise standard deviation Channel Gamma shape/scale Maximum voting interval τmax Uplink/downlink bandwidth

60 1.4 0.04 1.0/0.30 10 1/2 MHz

Parameter

Value

Client dataset size Initial/minimum local step OTA-noise standard deviation Required accuracy Accmin Number of candidates/grades (M, n) Client/PS transmit power

60–219 0.22/0.10 0.02 0.80 (7, 6) 0.1/0.5 W

TABLE III: Synthetic OTA-FL proxy results. Values are mean ± standard deviation over ten seeds. Method

Final accuracy

Normalized energy

OTA distortion

Fairness Jain’s index

Vacc

Nvote

FedMaJ-MJ FedMaJ-AvgScore Random adaptive FedAvg Uniform aggregation Loss-aware aggregation Resource-aware aggregation Communication-aware aggregation OTA-distortion-aware aggregation Participation-aware aggregation

0.9095 ± 0.0259 0.9131 ± 0.0246 0.9044 ± 0.0391 0.9294 ± 0.0257 0.9113 ± 0.0224 0.9107 ± 0.0215 0.9258 ± 0.0270 0.8621 ± 0.0453 0.7592 ± 0.0437 0.9286 ± 0.0223

0.4061 ± 0.0072 0.4043 ± 0.0056 0.4523 ± 0.0291 0.4077 ± 0.0034 0.3809 ± 0.0028 0.3880 ± 0.0027 0.3967 ± 0.0034 0.5202 ± 0.0128 0.9420 ± 0.0422 0.4350 ± 0.0109

0.0080 ± 0.0007 0.0079 ± 0.0006 0.0070 ± 0.0008 0.0081 ± 0.0007 0.0074 ± 0.0005 0.0080 ± 0.0006 0.0074 ± 0.0006 0.0024 ± 0.0003 0.0013 ± 0.0003 0.0091 ± 0.0011

0.9535 ± 0.0106 0.9524 ± 0.0139 0.9293 ± 0.0163 0.8871 ± 0.0158 0.9615 ± 0.0066 0.9459 ± 0.0130 0.8932 ± 0.0153 0.8690 ± 0.0143 0.8369 ± 0.0232 0.9064 ± 0.0112

0.0775 ± 0.0219 0.0733 ± 0.0156 0.1025 ± 0.0518 0.0700 ± 0.0090 0.0733 ± 0.0207 0.0700 ± 0.0125 0.0717 ± 0.0098 0.2450 ± 0.0565 0.9225 ± 0.0379 0.0658 ± 0.0114

22.9 ± 4.0 22.8 ± 2.9 25.7 ± 5.3 0 0 0 0 0 0 0

Good, Very Good, and Excellent, with score thresholds {0, 0.40, 0.55, 0.65, 0.75, 0.85, 1}. 3) Compared Methods: We compare FedMaJ-MJ with FedMaJ-AvgScore, which uses the same candidate library, indicators, safety filter, and adaptive voting schedule but averages the numerical client scores, and Random adaptive, which randomly selects a rule at each adaptive voting round. We also consider fixed rules A1 –A7 , each applied over the entire training horizon. All methods use identical selectedclient sequences, channels, initial models, update noise, and OTA noise for each seed. For readability, the figures show representative fixed rules, while Table III reports all fixed-rule results. B. Evaluation Metrics The synthetic test loss and proxy accuracy are 1 1 2 Ltest = . ∥wt+1 − w⋆ ∥2 , Acct = t dw 1 + Ltest t

(20)

b 2 The OTA distortion is Dt = d−1 w ∥∆t − ∆t ∥2 . Given the cumulative client influence It,k , Jain fairness writes P 2 K It,k k=1 Jtinf = . (21) PK 2 K k=1 It,k The accuracy-threshold violation rate is estimated by T

Vacc =

1X I[Acct < Accmin ] . T t=1

(22)

At voting rounds, we report client satisfaction (4), its client– vote mean (5), and the top-choice rate (6). These diagnostics apply to FedMaJ-MJ, FedMaJ-AvgScore, and random adaptive selection because they evaluate all candidates. For fixed rules, candidate grades are reconstructed offline to compute (7), without adding modeled overhead or energy. Energy is normalized using the common reference defined in the energy model, chosen as the maximum final physical

system energy across all methods, seeds, and fixed-interval configurations. C. Results and Discussion 1) Synthetic Learning and Fairness: Most learningoriented methods reach the accuracy threshold within 10– 20 rounds (Fig. 2(a)). FedMaJ-MJ achieves 0.9095 ± 0.0259, close to FedMaJ-AvgScore (0.9131 ± 0.0246), exceeding communication-aware and OTA-distortion-aware aggregation by 4.74 and 15.03 percentage points. Its influence fairness reaches 0.9535 ± 0.0106, close to uniform aggregation and 7.48% above FedAvg, versus 0.9293 for random selection (Fig. 2(b)). Violation rates are 7.75%, 7.33%, and 10.25% for FedMaJ-MJ, FedMaJ-AvgScore, and random selection, versus 24.50% and 92.25% for the two wireless-oriented rules (Fig. 3(b)). FedAvg and participation-aware aggregation achieve slightly fewer violations, but lower influence fairness than FedMaJ-MJ. 2) Client Preferences and Robustness: Loss-aware aggregation receives the most Very Good and Excellent grades, while OTA-distortion-aware aggregation receives the most Very Poor and Poor grades (Fig. 3(a)). These distributions pool rounds and seeds; voting uses round-specific profiles and MJ tiebreaking. FedMaJ-MJ, FedMaJ-AvgScore, and fixed A3 show similar offline satisfaction, consistent with A3 dominating selections (Fig. 3(d)). We test sensitivity by assigning maximum scores to one non-winning candidate for a fraction of clients at saved votes. At 10% perturbation, MJ reduces the rule-change rate from 0.1723 to 0.0753 (56.3%); at 30%, rates fall from 0.3964 to 0.2188 (Fig. 3(c)), indicating lower sensitivity to the tested extreme evaluations. 3) Fashion-MNIST Validation: Fig. 1 shows that FedMaJMJ and FedMaJ-AvgScore converge to accuracy and loss levels comparable to learning-oriented fixed rules while maintaining high influence fairness. FedMaJ-MJ satisfies the accuracy threshold and distributes cumulative influence more evenly than several fixed rules. Random adaptive selection

1.0

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Fig. 1: Fashion-MNIST evaluation of FedMaJ and the considered baselines in terms of test accuracy, Jain aggregation-influence fairness, and test loss over communication rounds. 1.0

1.0 Jain influence fairness

Proxy accuracy

0.9 0.8 0.7 0.6 0.5

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0.4 0.3 0

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Fig. 2: Synthetic OTA-FL proxy accuracy and Jain influence fairness over ten seeds (mean ± standard deviation).

A7

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J e g ore are J-M ware ware dAv ptiv Ma -aw gSc /Fe n-a n-a ada ion Fed J-Av size atio atio Ma tort dom icip setunic Fed -dis Ran Part Data Comm OTA

(a) Ordinal-grade distributions

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Score Random

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A7

(d) Client satisfaction

Fig. 3: Evaluation of FedMaJ in terms of ordinal-grade distributions, accuracy-threshold violations, robustness to extreme client evaluations, and active-rule client satisfaction. is more variable, while wireless-oriented rules retain higher losses. Together, these results support adaptive rule selection as a way to balance learning and influence fairness, with MJ additionally reducing sensitivity to extreme client evaluations. V. C ONCLUSION This paper introduced FedMaJ, which dynamically selects aggregation rules via client grading and Majority Judgment, with an accuracy-adaptive voting schedule. Synthetic OTA-FL evaluations demonstrate competitive proxy accuracy and superior influence fairness compared to FedAvg. Fashion-MNIST

experiments further confirm stable learning performance and high fairness. Under extreme-evaluation perturbations, Majority Judgment reduces the rule-change rate by 56.3% over average-score selection, establishing ordinal voting as a robust control layer. These findings highlight ordinal consensus as an effective mechanism for balancing client preferences and system objectives. Future work will explore larger datasets, complex models, time-varying channels, and strategic grading. R EFERENCES [1] B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas, “Communication-efficient learning of deep networks from decentralized data,” in Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, ser. Proceedings of Machine Learning Research, vol. 54. PMLR, 2017, pp. 1273–1282. [2] H. Zhang, X. Yu, and T. Westerlund, “Dual-criterion model aggregation in federated learning: Balancing data quantity and quality,” in IEEE 11th World Forum on Internet of Things (WF-IoT), 2025, pp. 1–3. [3] T. Nishio and R. Yonetani, “Client selection for federated learning with heterogeneous resources in mobile edge,” in IEEE International Conference on Communications, 2019. [4] M. M. Amiri and D. Gündüz, “Federated learning over wireless fading channels,” IEEE Transactions on Wireless Communications, vol. 19, no. 5, pp. 3546–3557, 2020. [5] L. Tan, X. Zhang, Y. Zhou, X. Che, M. Hu, X. Chen, and D. Wu, “Adafed: Optimizing participation-aware federated learning with adaptive aggregation weights,” IEEE Transactions on Network Science and Engineering, vol. 9, no. 4, pp. 2708–2720, 2022. [6] K. Yang, T. Jiang, Y. Shi, and Z. Ding, “Federated learning via overthe-air computation,” IEEE Transactions on Wireless Communications, vol. 19, no. 3, p. 2022–2035, Mar. 2020. [7] X. Cao, G. Zhu, J. Xu, and S. Cui, “Transmission power control for overthe-air federated averaging at network edge,” IEEE Journal on Selected Areas in Communications, vol. 40, no. 5, pp. 1571–1586, 2022. [8] T. Shen, Z. Li, Z. Zhao, D. Zhu, Z. Lv, K. Kuang, S. Zhang, C. Wu, and F. Wu, “Fedmcon: an adaptive aggregation method for federated learning via meta controller,” Frontiers of Information Technology &amp; Electronic Engineering, vol. 26, no. 8, p. 1378–1393, Aug. 2025. [9] M. Balinski and R. Laraki, Majority Judgment: Measuring, Ranking, and Electing. The MIT Press, 01 2011. [10] Y. Sun, S. Zhou, Z. Niu, and D. Gündüz, “Dynamic scheduling for overthe-air federated edge learning with energy constraints,” IEEE Journal on Selected Areas in Communications, vol. 40, no. 1, pp. 227–242, 2022. [11] S. M. Azimi-Abarghouyi and L. Tassiulas, “Over-the-air federated learning via weighted aggregation,” IEEE Transactions on Wireless Communications, vol. 23, no. 12, pp. 18 240–18 253, 2024. [12] S. Mohajer Hamidi, A. Bereyhi, S. Asaad, and H. Vincent Poor, “Overthe-air fair federated learning via multi-objective optimization,” IEEE Communications Letters, vol. 29, no. 7, pp. 1549–1553, 2025. [13] Z. Yang, M. Chen, W. Saad, C. S. Hong, and M. Shikh-Bahaei, “Energy efficient federated learning over wireless communication networks,” IEEE Transactions on Wireless Communications, vol. 20, no. 3, pp. 1935–1949, 2021.

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