In this study, we propose a digital over-the-air computation (OAC) scheme for achieving continuous-valued (analog) aggregation for federated edge learning (FEEL). We show that the average of a set of real-valued parameters can be calculated approximately by using the average of the corresponding numerals, where the numerals are obtained based on a balanced number system. By exploiting this key property, the proposed scheme encodes the local stochastic gradients into a set of numerals. Next, it determines the positions of the activated orthogonal frequency division multiplexing (OFDM) subcarriers by using the values of the numerals. To eliminate the need for precise sample-level time synchronization, channel estimation overhead, and channel inversion, the proposed scheme also uses a non-coherent receiver at the edge server (ES) and does not utilize a pre-equalization at the edge devices (EDs). We theoretically analyze the MSE performance of the proposed scheme and the convergence rate for a non-convex loss function. To improve the test accuracy of FEEL with the proposed scheme, we introduce the concept of adaptive absolute maximum (AAM). Our numerical results show that when the proposed scheme is used with AAM for FEEL, the test accuracy can reach up to 98% for heterogeneous data distribution.

翻译：本研究提出了一种数字空中计算（OAC）方案，用于实现联邦边缘学习（FEEL）中的连续值（模拟）聚合。我们证明，一组实值参数的平均值可以通过对应数字的平均值近似计算，其中这些数字基于平衡数系获得。通过利用这一关键特性，所提方案将本地随机梯度编码为一组数字，接着根据数字值确定激活的正交频分复用（OFDM）子载波位置。为了消除对精确样本级时间同步、信道估计开销和信道反转的需求，该方案还在边缘服务器（ES）端采用非相干接收机，且不利用边缘设备（ED）端的预均衡。我们从理论上分析了所提方案的均方误差性能以及非凸损失函数的收敛速率。为提升采用该方案的FEEL测试精度，我们引入自适应绝对最大值（AAM）概念。数值结果表明，当该方案与AAM结合用于FEEL时，对于异构数据分布，测试精度可达98%。