# AirGC-CD: Gaussian-Circulant Precoding for Exactly Debiasable PAPR Reduction in Over-the-Air Federated Learning

arXiv: https://arxiv.org/abs/2609.23084 · HTML: https://mlsanity.com/paper/2609.23084/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 45 references that are in the map.
- First posted on arXiv 19 Sep 2026.

## Builds on (most cited first, 22 of 22)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Federated Learning via Over-the-Air Computation](https://mlsanity.com/paper/1812.11750/) (2018)
- [Broadband Analog Aggregation for Low-Latency Federated Edge Learning (Extended Version)](https://mlsanity.com/paper/1812.11494/) (2018)
- [One-Bit Over-the-Air Aggregation for Communication-Efficient Federated Edge Learning: Design and Convergence Analysis](https://mlsanity.com/paper/2001.05713/) (2020)
- [On the Convergence of SGD with Biased Gradients](https://mlsanity.com/paper/2008.00051/) (2020)
- [On Analog Gradient Descent Learning over Multiple Access Fading Channels](https://mlsanity.com/paper/1908.07463/) (2019)
- [Federated Full-Parameter Tuning of Billion-Sized Language Models with Communication Cost under 18 Kilobytes](https://mlsanity.com/paper/2312.06353/) (2023)
- [Optimized Power Control Design for Over-the-Air Federated Edge Learning](https://mlsanity.com/paper/2106.09316/) (2021)
- [Federated LoRA with Sparse Communication](https://mlsanity.com/paper/2406.05233/) (2024)
- [Massive Digital Over-the-Air Computation for Communication-Efficient Federated Edge Learning](https://mlsanity.com/paper/2405.15969/) (2024)
- [1-Bit Compressive Sensing for Efficient Federated Learning Over the Air](https://mlsanity.com/paper/2103.16055/) (2021)
- [Over-the-Air Federated Learning via Second-Order Optimization](https://mlsanity.com/paper/2203.15488/) (2022)
- [Key Focus Areas and Enabling Technologies for 6G](https://mlsanity.com/paper/2412.07029/) (2024)
- [Over-the-Air Federated Learning via Weighted Aggregation](https://mlsanity.com/paper/2409.07822/) (2024)
- [Joint Design of Coding and Modulation for Digital Over-the-Air Computation](https://mlsanity.com/paper/2311.06829/) (2023)
- [Rendering Wireless Environments Useful for Gradient Estimators: A Zero-Order Stochastic Federated Learning Method](https://mlsanity.com/paper/2401.17460/) (2024)
- [Communication-Efficient Byzantine-Resilient Federated Zero-Order Optimization](https://mlsanity.com/paper/2406.14362/) (2024)
- [Challenges and Opportunities for Beyond-5G Wireless Security](https://mlsanity.com/paper/2303.00727/) (2023)
- [Over-the-Air Federated Learning Over MIMO Channels: A Sparse-Coded Multiplexing Approach](https://mlsanity.com/paper/2304.04402/) (2023)
- [Fed-ZOE: Communication-Efficient Over-the-Air Federated Learning via Zeroth-Order Estimation](https://mlsanity.com/paper/2412.16779/) (2024)
- [On Signal Peak Power Constraint of Over-the-Air Federated Learning](https://mlsanity.com/paper/2512.23381/) (2025)

## Abstract (arXiv)

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.