cs.LGSep 19, 2026

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

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Abstract

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

SchemeCompressionSuperpositionPAPR reductionUnbiased aggregationEncoding complexityYear
AirComp-FL [ 5 , 11 ]✗✓✗✓O(d)\mathcal{O}(d)2020
Sparsification + CS [ 7 ]✓✓✗✗O(md)\mathcal{O}(md)2019
One-bit aggregation [ 12 ]✓✓✓✗O(d)\mathcal{O}(d)2021
Power control / scheduling [ 13 ]✗✓✗✓O(d)\mathcal{O}(d)2022
11 -bit CS AirComp [ 14 ]✓✓✗✗O(md)\mathcal{O}(md)2023
Sparse one-bit + error feedback [ 15 ]✓✓✗✗O(d)\mathcal{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{\mathbf{w}}_{k,t,i} is trained on the local dataset via SGD. The EDs then encode their local update Δwk,t\Delta{\mathbf{w}}_{k,t} via Gaussian-circulant sketching. Next, to adjust the MSE tradeoff between the channel noise and clipping loss, we apply clipping, where αγ\alpha_{\gamma} denotes the Bussgang gain. In the AirComp uplink phase, the EDs simultaneously transmit sk,t{\mathbf{s}}_{k,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[∥Δw^G,t−1K∑kΔwk,t∥2/∥1K∑kΔwk,t∥2]\mathbb{E}\big[\|\Delta\widehat{{\mathbf{w}}}_{G,t}-\tfrac{1}{K}\textstyle\sum_{k}\Delta{\mathbf{w}}_{k,t}\|^{2}\big/\|\tfrac{1}{K}\textstyle\sum_{k}\Delta{\mathbf{w}}_{k,t}\|^{2}\big] ) of update aggregation using AirGC-CD for various clipping ratios γ\gamma and SNRs. The dotted vertical lines denote the clipping ratio γ⋆\gamma^{\star} 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}m\in\{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}\in\{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 00 dB.
Fig. 8: The test accuracy of the trained model for various learning rates η\eta and sketch dimensions mm . The best test accuracy for each mm is highlighted.
Fig. 9: Test accuracy of AirGC-CD and baselines for various datasets.
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