Air-FEEL: Over-the-Air Federated Edge Learning

• Over-the-Air Federated Edge Learning (Air-FEEL) is a distributed training framework that aggregates model updates in one shot via analog over-the-air computation while preserving data privacy.
• It leverages both coherent and non-coherent aggregation schemes, using optimized power control and channel inversion to manage fading and alignment issues.
• Air-FEEL improves communication efficiency and reduces latency in multi-device environments, serving as a key enabler for advanced 5G/6G wireless AI applications.

Over-the-Air Federated Edge Learning (Air-FEEL) combines the privacy-preserving, distributed learning paradigm of federated edge learning (FEEL) with over-the-air computation (AirComp) to achieve “one-shot” analog aggregation of model updates. In Air-FEEL, distributed edge devices use their local data to collaboratively train AI models while preserving data privacy, in which the over-the-air model/gradient aggregation is exploited for enhancing the learning efficiency. The topic spans coherent analog aggregation, non-coherent majority-vote schemes, digital over-the-air computation, and convergence-aware resource management for beyond 5G and 6G networks (Cao et al., 2022).

1. Canonical architecture and training procedure

Air-FEEL consists of one Parameter Server (PS) and $K$ edge devices, each holding its own local data. Training proceeds in global rounds $t=0,1,2,\dots$ until convergence. The PS holds the current global model $w^{(t)}\in\mathbb{R}^N$ (or $\mathbb{C}^N$) and broadcasts it to all devices. Each device $k$ initializes its local model as $w_k^{(t,0)}\leftarrow w^{(t)}$, runs $\tau$ local SGD steps on its private dataset $D_k$,

$$w_k^{(t,i+1)} = w_k^{(t,i)} - \eta \nabla F_k(w_k^{(t,i)}), \quad i=0,\dots,\tau-1,$$

and computes its update $\Delta_k^{(t)}$, either as a gradient estimate for Air-FedSGD with $t=0,1,2,\dots$0, or as a model difference for Air-FedAvg with $t=0,1,2,\dots$1 (Cao et al., 2022).

Without orthogonal multiple-access, all $t=0,1,2,\dots$2 devices simultaneously transmit their analog-modulated updates over a multiple-access channel. By exploiting the linear superposition property, the PS receives a noisy, weighted sum of the local updates in one “shot,” with no per-device decoding. The PS then applies a scalar post-processing or denoising factor $t=0,1,2,\dots$3 and computes

$t=0,1,2,\dots$4

followed by the global update

$t=0,1,2,\dots$5

The process repeats until fluctuations of the test accuracy or the loss fall below a preset threshold (Cao et al., 2022).

This workflow already identifies the central distinction between Air-FEEL and conventional FEEL. The communication phase is not organized around recovering each device’s message. It is organized around recovering the aggregate update itself. A plausible implication is that the learning algorithm, the wireless alignment rule, and the resource allocation policy must be designed jointly, because the object delivered by the channel is an estimate of an average rather than a collection of individual packets.

2. Wireless aggregation model and error structure

A representative narrowband complex-baseband Air-FEEL model writes the channel from device $t=0,1,2,\dots$6 to the PS as $t=0,1,2,\dots$7, with additive noise $t=0,1,2,\dots$8. Each device scales its $t=0,1,2,\dots$9-dimensional update $w^{(t)}\in\mathbb{R}^N$0 by a complex scalar $w^{(t)}\in\mathbb{R}^N$1,

$w^{(t)}\in\mathbb{R}^N$2

where $w^{(t)}\in\mathbb{R}^N$3 is an amplitude-control factor and $w^{(t)}\in\mathbb{R}^N$4. The PS receives

$w^{(t)}\in\mathbb{R}^N$5

and forms

$w^{(t)}\in\mathbb{R}^N$6

where $w^{(t)}\in\mathbb{R}^N$7 captures both mis-alignment bias and noise (Cao et al., 2022).

The key alignment condition is explicit: perfect unbiased aggregation requires $w^{(t)}\in\mathbb{R}^N$8 for some constant $w^{(t)}\in\mathbb{R}^N$9. This inverts all $\mathbb{C}^N$0 but amplifies noise from the weakest device. In practice, $\mathbb{C}^N$1 trades off bias and noise. That trade-off is the core wireless bottleneck in coherent Air-FEEL, and it explains why channel inversion can be effective in moderate fading yet highly suboptimal in deep fading scenarios (Cao et al., 2022).

The same bias–variance structure reappears at the learning level. If the aggregation error at round $\mathbb{C}^N$2 is written as $\mathbb{C}^N$3, then both deterministic bias and mean-squared error matter. This makes Air-FEEL fundamentally different from a pure PHY-layer function-computation problem: an estimator with small instantaneous MSE can still be undesirable if it creates persistent bias that degrades convergence.

3. Convergence behavior and performance metrics

For Air-FedSGD under $\mathbb{C}^N$4-smoothness and the Polyak–Łojasiewicz inequality, a representative optimality-gap bound is

$\mathbb{C}^N$5

where $\mathbb{C}^N$6 depends on $\mathbb{C}^N$7 and $\mathbb{C}^N$8, $\mathbb{C}^N$9, and $k$0. Two regimes emerge: biased aggregation implies an error-floor $k$1, whereas unbiased aggregation implies the error-floor tends to $k$2 as $k$3 but may slow early-stage convergence (Cao et al., 2022).

This learning-theoretic split is sharpened in convergence-aware power-control studies. When aggregation estimates are unbiased, the training algorithm would converge exactly to the optimal point with mild conditions; when they are biased, the algorithm would converge with an error floor determined by the accumulated estimate bias over communication rounds. Structured solutions then appear in a form of regularized channel inversion under average and maximum power constraints (Cao et al., 2021). Earlier optimized-power-control formulations reached the same conclusion from a successive convex approximation and trust-region route, emphasizing that convergence-optimal transmission generally differs from per-round MSE minimization (Cao et al., 2020).

The performance metrics used in Air-FEEL are likewise joint communication–learning metrics. Learning accuracy is measured on an external test set and is inversely related to the training loss $k$4. Training latency is

$k$5

where $k$6 is local-compute time of the slowest device and $k$7 is one-shot aggregation time, independent of $k$8 in Air-FEEL. Communication efficiency is the resource-block usage per aggregated update, $k$9 in Air-FEEL versus $w_k^{(t,0)}\leftarrow w^{(t)}$0 in conventional FEEL. Energy consumption includes both compute energy and communication energy (Cao et al., 2022).

Several non-coherent and quantized variants retain explicit non-convex convergence guarantees. The circularly-shifted-chirp majority-vote scheme preserves the $w_k^{(t,0)}\leftarrow w^{(t)}$1 convergence rate of signSGD for non-convex losses under standard smoothness and bounded-noise assumptions (Hoque et al., 2022). The balanced-number-system digital scheme gives $w_k^{(t,0)}\leftarrow w^{(t)}$2, up to the additive “noise-ball” determined by quantization and channel MSE, and introduces Adaptive Absolute Maximum (AAM) to reduce the additive MSE term (Sahin, 2022).

4. Waveform, coding, and receiver designs

A major branch of Air-FEEL uses non-coherent majority-vote aggregation rather than coherent analog summation. In the FSK-based majority-vote scheme, each edge device maps the sign of each local stochastic gradient coordinate to one of two orthogonal OFDM subcarriers, and the edge server computes the global vote by comparing received energies. The scheme eliminates the need for channel state information at the edge devices and edge server, and residual timing misalignment does not bias the energy sums. Its convergence theorem gives an averaged gradient-norm rate of order $w_k^{(t,0)}\leftarrow w^{(t)}$3 under non-convex smoothness assumptions (Sahin et al., 2021). A closely related DFT-spread OFDM construction uses pulse-position modulation symbols, energy detection, and guard bins to absorb delay spread and synchronization error, while also targeting lower peak-to-mean envelope power ratio (Sahin et al., 2021).

Long-range operation motivated a chirp-based variant. The circularly-shifted chirp majority-vote scheme maps votes to linear CSCs constructed with a DFT-s-OFDM transmitter and computes the majority vote with an energy detector. For an ACLR specification of e.g. $w_k^{(t,0)}\leftarrow w^{(t)}$4 dB, the minimum OBO is approximately $w_k^{(t,0)}\leftarrow w^{(t)}$5 dB for OBDA and $w_k^{(t,0)}\leftarrow w^{(t)}$6 dB for CSC-MV for $w_k^{(t,0)}\leftarrow w^{(t)}$7, i.e. a $w_k^{(t,0)}\leftarrow w^{(t)}$8–$w_k^{(t,0)}\leftarrow w^{(t)}$9 dB OBO saving. Numerically, CSC-MV extends the $\tau$0 dB SNR coverage from approximately $\tau$1 m to approximately $\tau$2 m, roughly doubling the coverage area, and in heterogeneous data settings it recovers full accuracy, for example approximately $\tau$3 at $\tau$4 dB, without CSI at either the edge devices or the edge server (Hoque et al., 2022). The same work states that the scheme is fully compatible with 4G/5G DFT-s-OFDM transceivers and only needs a special FDSS filter $\tau$5.

Another branch replaces one-bit sign aggregation with richer digital encodings. The balanced-number-system scheme encodes real gradients into balanced $\tau$6-ary digits, carries each digit by activating one of $\tau$7 subcarriers per digit-slot, and aggregates by energy detection across fading channels with a non-coherent receiver. With AAM, the test accuracy can reach up to $\tau$8 for heterogeneous data distribution; on MNIST with 25 edge devices and 18 MHz OFDM, Air-FEEL without AAM reaches about $\tau$9 in 100 rounds, while with AAM it reaches about $D_k$0, and in heterogeneous labels signSGD-MV stalls at less than $D_k$1 while Air-FEEL with AAM still reaches about $D_k$2 (Sahin, 2022).

Digital over-the-air computation has also moved toward coded random access and lattice methods. UMA-based generalized digital OAC lets all devices share the same non-orthogonal UMA codebook, uses approximate message passing for joint detection and aggregation, and can match OBDA’s uplink resource usage by setting $D_k$3, so that the total number of channel uses equals $D_k$4. In a CIFAR-10 task with the same uplink communication resources, OBDA converges to approximately $D_k$5 test accuracy in 200 rounds, whereas GD-OAC with $D_k$6 bits and $D_k$7 converges to approximately $D_k$8 (Qiao et al., 2023). A different digital line uses balanced-numeral lattice codes, whose error-probability bound does not depend on the number of clients $D_k$9, in contrast to nested-lattice codes whose rate and reliability degrade as $$w_k^{(t,i+1)} = w_k^{(t,i)} - \eta \nabla F_k(w_k^{(t,i)}), \quad i=0,\dots,\tau-1,$$0 grows (Egger et al., 2024). More recently, a learned digital OTA framework combined an unrolled decoder with a jointly learnt unsourced random access codebook, reporting an extension of reliable operation by more than $$w_k^{(t,i+1)} = w_k^{(t,i)} - \eta \nabla F_k(w_k^{(t,i)}), \quad i=0,\dots,\tau-1,$$1 dB with the same uplink overhead as state-of-the-art (Tarizzo et al., 20 Sep 2025).

These families make clear that Air-FEEL is not a single physical-layer recipe. It includes coherent analog AirComp, non-coherent sign-based schemes, non-coherent digit-based OFDM activation, lattice-coded digital summation, and learned codebooks. The choice among them depends on whether the limiting factor is CSI acquisition, power-amplifier nonlinearity, low-SNR reliability, or quantization distortion.

5. Resource management and system-level extensions

Resource management is central because Air-FEEL couples channel heterogeneity, data heterogeneity, and device heterogeneity. The overview literature lists joint power control, beamforming and RIS, device scheduling, and analog compression via sparsification as primary levers. Closed-form power-control solutions exhibit threshold-based channel inversion: devices with $$w_k^{(t,i+1)} = w_k^{(t,i)} - \eta \nabla F_k(w_k^{(t,i)}), \quad i=0,\dots,\tau-1,$$2 use $$w_k^{(t,i+1)} = w_k^{(t,i)} - \eta \nabla F_k(w_k^{(t,i)}), \quad i=0,\dots,\tau-1,$$3; otherwise $$w_k^{(t,i+1)} = w_k^{(t,i)} - \eta \nabla F_k(w_k^{(t,i)}), \quad i=0,\dots,\tau-1,$$4. In multi-antenna settings, transmit beamformers and aggregation vectors are optimized to minimize MSE, and RIS phase-shift matrices can be jointly optimized with beamforming (Cao et al., 2022).

RIS-empowered Air-FEEL develops this idea at the system level. With direct uplink channel $$w_k^{(t,i+1)} = w_k^{(t,i)} - \eta \nabla F_k(w_k^{(t,i)}), \quad i=0,\dots,\tau-1,$$5, device-RIS channel $$w_k^{(t,i+1)} = w_k^{(t,i)} - \eta \nabla F_k(w_k^{(t,i)}), \quad i=0,\dots,\tau-1,$$6, RIS-PS channel $$w_k^{(t,i+1)} = w_k^{(t,i)} - \eta \nabla F_k(w_k^{(t,i)}), \quad i=0,\dots,\tau-1,$$7, and RIS phase-shift matrix $$w_k^{(t,i+1)} = w_k^{(t,i)} - \eta \nabla F_k(w_k^{(t,i)}), \quad i=0,\dots,\tau-1,$$8, the received signal is

$$w_k^{(t,i+1)} = w_k^{(t,i)} - \eta \nabla F_k(w_k^{(t,i)}), \quad i=0,\dots,\tau-1,$$9

Joint power and RIS phase design minimizes the aggregation MSE by alternating optimization. In the reported example with $\Delta_k^{(t)}$0, no RIS requires 1000 training rounds for $\Delta_k^{(t)}$1 accuracy, whereas RIS-assisted channel alignment reduces MSE by $\Delta_k^{(t)}$2 dB and only about $\Delta_k^{(t)}$3 rounds are needed, a $\Delta_k^{(t)}$4 reduction. In MNIST experiments with FedAvg and 200 rounds, final accuracy is $\Delta_k^{(t)}$5 for digital FEEL, $\Delta_k^{(t)}$6 for analog without RIS, and $\Delta_k^{(t)}$7 for analog with RIS; RIS cuts the required bandwidth by approximately $\Delta_k^{(t)}$8 to reach $\Delta_k^{(t)}$9 accuracy (Liu et al., 2021).

Scheduling addresses another persistent bottleneck: not all devices should participate in every round, and discarded updates need not be lost. An energy-aware dynamic scheduling algorithm that accounts for both communication energy and computation energy increases the accuracy by $t=0,1,2,\dots$00 on CIFAR-10 compared with the myopic benchmark under a highly unbalanced local data distribution, while satisfying the energy constraints (Sun et al., 2021). A gradient- and channel-aware scheduling mechanism measures data importance by the gradient of local model parameter, channel condition, and energy consumption jointly; it also retains and accumulates the local updates of unselected devices for future potential transmission. In the reported results, the residual-feedback mechanism yields $t=0,1,2,\dots$01–$t=0,1,2,\dots$02 accuracy gain over no-residual in mid and late training, and in non-i.i.d. settings the joint DSI+CSI metric outperforms CSI-only or random by up to $t=0,1,2,\dots$03 test accuracy (Du et al., 2022). Semi-asynchronous AirComp further addresses stragglers: PAOTA achieves convergence performance close to that of the ideal Local SGD, and for a target $t=0,1,2,\dots$04 accuracy requires $t=0,1,2,\dots$05 rounds / $t=0,1,2,\dots$06 s versus $t=0,1,2,\dots$07 rounds / $t=0,1,2,\dots$08 s for Local SGD and $t=0,1,2,\dots$09 rounds / $t=0,1,2,\dots$10 s for synchronous COTAF (Kou et al., 2023).

System scope has also expanded beyond a single synchronous cell. A multi-cell non-coherent framework performs over-the-air computation in both uplink and downlink, with multiple edge servers functioning as aggregation nodes in the uplink and inter-cell interference exploited for the OAC. In MNIST experiments, single-cell homogeneous training leaves cell-edge devices at approximately $t=0,1,2,\dots$11–$t=0,1,2,\dots$12 accuracy, whereas in $t=0,1,2,\dots$13 cells nearly all devices converge to approximately $t=0,1,2,\dots$14 global test accuracy in about 200 rounds; in heterogeneous data, multi-cell personalized test accuracy reaches more than $t=0,1,2,\dots$15 (Adeli et al., 2022). Energy-harvesting Air-FEEL, by contrast, shows that weighted averaging with respect to latest energy arrivals and data cardinalities mitigates device bias but does not eliminate channel-induced degradation: numerical experiments report a $t=0,1,2,\dots$16–$t=0,1,2,\dots$17 performance loss in energy harvesting OTA FL (Aygün et al., 2022).

6. Limitations, misconceptions, and active directions

Several common simplifications are inaccurate. Air-FEEL is not synonymous with uncoded analog averaging: the literature explicitly includes one-bit digital aggregation, balanced-number-system non-coherent aggregation, UMA-based generalized digital OAC, lattice-based sum decoding, and learned digital codes (Qiao et al., 2023). Air-FEEL is also not inherently CSI-free. Coherent analog designs depend on phase inversion, amplitude control, and denoising factors, while CSI-free operation is obtained by changing the aggregation primitive itself, for example by majority vote or energy detection (Cao et al., 2022). Nor does communication efficiency automatically resolve learning pathologies: under heterogeneous labels, OBDA can fail at about $t=0,1,2,\dots$18 accuracy, signSGD-MV can stall at less than $t=0,1,2,\dots$19, and energy-harvesting OTA FL still exhibits a $t=0,1,2,\dots$20–$t=0,1,2,\dots$21 performance loss (Hoque et al., 2022).

Current research directions follow directly from these limitations. The overview literature points to secure aggregation under jamming and Byzantine attacks, large-scale and hierarchical Air-FEEL, green Air-FEEL with energy harvesting and wireless power transfer, and integrated sensing and communications (Cao et al., 2022). Later work makes these directions concrete. Integrated sensing, communication, and computation for Air-FEEL uses an FMCW radar waveform, models sensing noise and AirComp distortion jointly in the convergence bound, and derives an $t=0,1,2,\dots$22 convergence result together with an alternating algorithm for batch-size control and resource allocation on a human-motion recognition task (Wen et al., 21 Aug 2025). Blind asynchronous Air-FEEL removes prior information about the time misalignments and recovers the global model by solving a convex semi-definite program; its reported performance is close to the ideal synchronized scenario by $t=0,1,2,\dots$23 and $t=0,1,2,\dots$24 better than the case where no recovering method is used (Razavikia et al., 2022). Temporal-structure-assisted gradient aggregation models the aggregation series with a Markovian temporal prior, develops a message-passing algorithm with state evolution analysis, and achieves learning performance comparable to the error-free benchmark in both convergence rate and final test accuracy (Fan et al., 2021).

These developments suggest that Air-FEEL is best understood not as a single aggregation protocol but as a design space. Its defining feature is the direct coupling of wireless superposition and distributed optimization; its unresolved questions concern how to preserve that coupling under fading, power-amplifier nonlinearity, heterogeneity, asynchronous arrivals, adversarial interference, and increasingly integrated sensing–communication–computation workloads.