AirGC-CD: Gaussian-Circulant Precoding for Exactly Debiasable PAPR Reduction in Over-the-Air Federated Learning
Authors: Jonggyu Jang, Hyeonsu Lyu, Hyun Jong Yang
Organizations: Department of Electronics Engineering, Chung-nam National University, Daejeon 34134, Republic of Korea · Institute of New Media and Communications, Seoul 08826, Republic of Korea · Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, Republic of Korea
Over-the-air federated learning lets edge devices transmit their local updates simultaneously, reducing the communication overhead. The resulting waveform, however, has a peak-to-average power ratio (PAPR) that grows with the model dimension, and keeping the amplifier in its linear range leaves two remedies: clipping the peaks or backing off the transmit power. Neither remedy is without cost: i) the clipping distortion appears at the receiver as a bias that cannot be removed, and ii) back-off keeps the signal intact but degrades the average signal-to-noise ratio (SNR). Independent of this trade-off, the transmission remains uncompressed, spending one channel use per model parameter, which keeps large-model training out of reach. To address these challenges, we propose AirGC-CD, an over-the-air scheme that precodes each local update with a partial Gaussian circulant matrix before clipping. In AirGC-CD, the precoder's output is exactly Gaussian regardless of the update's sparsity, so the clipping function is designed for a known distribution instead of inheriting it from the data. This enables the clipping to be inverted on average by a single scalar Bussgang gain in closed form, and we prove that the resulting aggregate is exactly unbiased, with clipping adding only variance. The clipping ratio is then the only free parameter left, trading the variance of the clipping against the SNR loss from back-off, and we derive its near-optimum in closed form. Since the precoder is linear, it also acts as a compressor, reducing the transmission from the model dimension d to the sketch dimension m at a cost of only O(dlog d) via two fast Fourier transforms, whereas a Gaussian sketch costs O(md). Experiments on five image datasets show that AirGC-CD outperforms baseline over-the-air FL schemes in most settings, particularly at low SNR, while using fewer channel uses per round.
Figures & tables
Scheme
Compression
Superposition
PAPR reduction
Unbiased aggregation
Encoding complexity
Year
AirComp-FL [ 5 , 11 ]
✗
✓
✗
✓
O(d)
2020
Sparsification + CS [ 7 ]
✓
✓
✗
✗
O(md)
2019
One-bit aggregation [ 12 ]
✓
✓
✓
✗
O(d)
2021
Power control / scheduling [ 13 ]
✗
✓
✗
✓
O(d)
2022
1 -bit CS AirComp [ 14 ]
✓
✓
✗
✗
O(md)
2023
Sparse one-bit + error feedback [ 15 ]
✓
✓
✗
✗
O(d)
2024
TABLE I: Positioning of AirGC-CD against representative AirComp-FL and PAPR-reduction schemes.
Fig. 1: Schematic diagram of AirGC-CD . At each ED, the local model weight wk,t,i is trained on the local dataset via SGD. The EDs then encode their local update Δwk,t via Gaussian-circulant sketching. Next, to adjust the MSE tradeoff between the channel noise and clipping loss, we apply clipping, where αγ denotes the Bussgang gain. In the AirComp uplink phase, the EDs simultaneously transmit sk,t over the shared channel. Then, the PS broadcasts the aggregated sketch to EDs via a reliable digital link. Finally, the EDs desketch and recover the average update.
Fig. 2: Test accuracy of the proposed method and baselines for various SNRs.
Fig. 3: Relative MSE ( E[∥ΔwG,t−K1∑kΔwk,t∥2/∥K1∑kΔwk,t∥2] ) of update aggregation using AirGC-CD for various clipping ratios γ and SNRs. The dotted vertical lines denote the clipping ratio γ⋆ in ( 39 ).
Fig. 4: PAPR of the transmitted updates.
Fig. 5: Test accuracy evaluation with the uplink channel uses per round of AirGC-CD and baselines. The SNR is assumed to be 0 dB and 20 dB for (a) and (b), respectively. For this evaluation, AirGC-CD and AirFL-Sketch are evaluated for m∈{512,1024,2048,4096,16384,65536} .
Fig. 6: Per-round system overhead: local training, sketching, desketching, and communication.
Fig. 7: The convergence of AirGC-CD and baselines for SNR ∈{0,10} dB. Subfigures (a) and (b) depict the test accuracy versus communication rounds, and subfigures (c) and (d) depict the test accuracy versus total communication overhead. AirFL-LoRA is omitted from (a) and (c) because it diverges at 0 dB.
Fig. 8: The test accuracy of the trained model for various learning rates η and sketch dimensions m . The best test accuracy for each m is highlighted.
Fig. 9: Test accuracy of AirGC-CD and baselines for various datasets.
This paper investigates over-the-air federated learning (AirFL) in wireless systems where the access point is equipped with a multi-waveguide pinching antenna system (PASS). We adopt the widely studied learning-oriented AirFL formulation, which seeks to maximize the number of selected devices while keeping the aggregation distortion below a prescribed threshold. The resulting joint optimization of device selection, receive beamforming, and pinching-antenna placement is highly nonconvex due to the intricate coupling among these system variables. To address this challenge, we develop AirPASS, an alternating optimization framework with two main components: a homotopy-Riemannian margin-consolidation method for device selection and receive beamforming under a fixed PASS configuration, and a homotopy-assisted geometry optimization method for updating the pinching-antenna positions under fixed selected devices and beamformer.
Seyed Mohammad Azimi-Abarghouyi, Christopher G. Brinton
Department of Electrical Engineering, Chalmers University of Technology, Gothenburg, Sweden · School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN USA
Over-the-air computation has emerged as a scalable and efficient solution for deploying federated learning algorithms in wireless networks by exploiting waveform superposition for simultaneous model aggregation. Most existing work struggles with heterogeneous fading channels. These approaches either enforce unbiased updates from all devices or allow partial device contributions, requiring careful tuning of the convergence bound to mitigate bias under specific fading models. However, the former significantly amplifies receiver noise due to the weakest channel, whereas the latter is sensitive to fading model mismatch and converges only to a biased objective. To tackle these challenges, we propose FedOAG, which employs algorithmic components to automatically satisfy energy constraints via gradient normalization and evenly mix devices' updates through implicit gossiping. Importantly, FedOAG does not require transmission from all devices, nor does it rely on a specific fading model or knowledge of time-varying statistical channel distributions. We show that FedOAG converges to a stationary point of an unbiased non-convex objective at the best possible rate O(1/T) for any stochastic first-order method. We corroborate our analysis with numerical experiments over dynamic wireless conditions on real-world datasets.
Ming Xiang, Nicolò Michelusi, Yonina C. Eldar +1
Northeastern University, Boston, MA, USA · Arizona State University, Tempe, AZ, USA · Weizmann Institute of Science, Rehovot, Israel
To mitigate the scalability bottleneck in the radio access network (RAN) in federated edge learning (FEEL), over-the-air federated learning (AirFL) exploits waveform superposition over multiple-access channels (MACs) for analog model aggregation. However, coherent AirFL typically relies on stringent PHY-layer conditions such as accurate channel state information (CSI), tight time/frequency synchronization, and frequent transceiver calibration for signal alignment. However, these requirements, if not impossible to be met, incur substantial communication and computation overhead. In this paper, we propose a non-coherent AirFL (NCAirFL) protocol over a broadband single-antenna MAC, leveraging binary dithering, unbiased non-coherent detection, and long-term error feedback to waive the need for instantaneous CSI. For NCAirFL with general smooth non-convex objectives and a constant learning rate, we establish a convergence bound achieving the convergence rate in the same order of O(1/T) as communication-ideal FedAvg, where T is the total number of communication rounds. To further improve communication efficiency under data and wireless resource heterogeneity, we also derive a lower bound on the expected single-round objective decrease in the global loss conditioned on device scheduling, building upon which a surrogate objective function is obtained for jointly optimal device selection and power control. Experimental results on MNIST and CIFAR-10 corroborate that NCAirFL achieves learning performance close to FedAvg in practical settings, with the proposed device scheduling policy substantially accelerating convergence.
Haifeng Wen, Nicolò Michelusi, Osvaldo Simeone +2
IoT Thrust, The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, 511453, China · School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85281 USA · Institute for Intelligent Networked Systems, Northeastern University London, London, E1 8PH, UK +2