Bayesian Over-The-Air Computation

Abstract

Analog over-the-air computation (OAC) is an efficient solution to a class of uplink data aggregation tasks over a multiple-access channel (MAC), wherein the receiver, dubbed the fusion center, aims to reconstruct a function of the data distributed at edge devices rather than the individual data themselves. Existing OAC relies exclusively on the maximum likelihood (ML) estimation at the fusion center to recover the arithmetic sum of the transmitted signals from different devices. ML estimation, however, is much susceptible to noise. In particular, in the misaligned OAC where there are channel misalignments among transmitted signals, ML estimation suffers from severe error propagation and noise enhancement. To address these challenges, this paper puts forth a Bayesian approach for OAC by letting each edge device transmit two pieces of prior information to the fusion center. Three OAC systems are studied: the aligned OAC with perfectly-aligned signals; the synchronous OAC with misaligned channel gains among the received signals; and the asynchronous OAC with both channelgain and time misalignments. Using the prior information, we devise linear minimum mean squared error (LMMSE) estimators and a sum-product maximum a posteriori (SP-MAP) estimator for the three OAC systems. Numerical results verify that, 1) For the aligned and synchronous OAC, our LMMSE estimator significantly outperforms the ML estimator. In the low signalto-noise ratio (SNR) regime, the LMMSE estimator reduces the mean squared error (MSE) by at least 6 dB; in the high SNR regime, the LMMSE estimator lowers the error floor on the MSE by 86.4%; 2) For the asynchronous OAC, our LMMSE and SP-MAP estimators are on an equal footing in terms of the MSE performance, and are significantly better than the ML estimator. On the other hand, in terms of the computational complexity, the SP-MAP estimator is much more efficient than the LMMSE estimator.

CHARLES: Channel-Quality-Adaptive Over-the-Air Federated Learning over Wireless Networks

“…Most of these existing works assume perfect CSI at the transmitter side, which allows precoding designs to mitigate channel fading. Research on OTA-FL with imperfect CSI has also been ongoing [7]- [13], [19], [20]). In [9], the performance of the proposed compressed analog DSGD algorithm under imperfect CSI was considered in experiments only, which shows that imperfect CSI could lead to signal misalignment at the server-side.…”

“…In [7], a whitened matched filtering and a sampling scheme is proposed to deal with imperfect CSI on the receiver side. A Bayesian approach was further proposed in the follow-up work of the same authors [19]. In [10], a receive beamforming pattern is proposed to compensate the server-side imperfect CSI by equipping the server with a sufficiently large number of antennas.…”

Broadband Digital Over-the-Air Computation for Asynchronous Federated Edge Learning

“…Goldenbaum et al, [17], [18] design and implement an analog-modulated AirComp system that only needs coarse symbol-level synchronization, and leverages direct spread spectrum sequences with random phases for modulation. Shao et al, [19], [20] leverages oversampling and sum-product ML estimators to estimate the arithmetic sum under symbol misalignment and phase misalignment. However, their designs only work for conventional timedomain narrowband systems, and cannot directly apply to OFDM-based frequency-domain broadband systems.…”

Blind Asynchronous Over-the-Air Federated Edge Learning

“…These challenges are particularly important because, in practice, signal precoding is imperfect due to inaccurate channel estimation and non-ideal hardware, and the synchronization across different devices is costly because it requires all devices to time their transmissions accurately based on the propagation time. Therefore, in a realistic scenario, there appears some channel-gain mismatches, time asynchronies, or both [9].…”

“…The work in [12] utilizes digital modulation and channel coding to combat any channel or time misalignments at the receiver, but it still requires decoding the transmitted signals, thus not fully leveraging the communication benefits of analog signals with OAC. The most related work up to date is [9], which tackles the problem of misalignment of analog OAC by constructing matched filters at the receiver, but it assumes that the sampling process knows the exact time misalignments from the different users.…”