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：本研究提出一种数字式空中计算方案，用于实现联邦边缘学习中的连续值（模拟）聚合。我们证明，通过利用平衡数系统获取的数码，一组实值参数的平均值可近似由其对应数码的平均值计算得出。基于这一关键特性，所提方案将局部随机梯度编码为数码集，进而利用数码值确定激活的正交频分复用子载波位置。为消除精确样本级时间同步、信道估计开销及信道反转需求，该方案在边缘服务器采用非相干接收机，且边缘设备不进行预均衡。我们在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。为提升基于该方案的联邦边缘学习测试精度，我们引入自适应绝对最大值概念。数值结果表明，当所提方案结合自适应绝对最大值用于联邦边缘学习时，在异构数据分布下测试精度可达98%。