Over-the-air federated learning (OTA-FL) is an emerging technique to reduce the computation and communication overload at the PS caused by the orthogonal transmissions of the model updates in conventional federated learning (FL). This reduction is achieved at the expense of introducing aggregation error that can be efficiently suppressed by means of receive beamforming via large array-antennas. This paper studies OTA-FL in massive multiple-input multiple-output (MIMO) systems by considering a realistic scenario in which the edge server, despite its large antenna array, is restricted in the number of radio frequency (RF)-chains. For this setting, the beamforming for over-the-air model aggregation needs to be addressed jointly with antenna selection. This leads to an NP-hard problem due to the combinatorial nature of the optimization. We tackle this problem via two different approaches. In the first approach, we use the penalty dual decomposition (PDD) technique to develop a two-tier algorithm for joint antenna selection and beamforming. The second approach interprets the antenna selection task as a sparse recovery problem and develops two iterative joint algorithms based on the Lasso and fast iterative soft-thresholding methods. Convergence and complexity analysis is presented for all the schemes. The numerical investigations depict that the algorithms based on the sparse recovery techniques outperform the PDD-based algorithm, when the number of RF-chains at the edge server is much smaller than its array size. However, as the number of RF-chains increases, the PDD approach starts to be superior. Our simulations further depict that learning performance with all the antennas being active at the PS can be closely tracked by selecting less than 20% of the antennas at the PS.

翻译：空中联邦学习（OTA-FL）是一种新兴技术，旨在降低传统联邦学习（FL）中由模型更新的正交传输造成的参数服务器（PS）计算与通信开销。这种降低以引入聚合误差为代价，而该误差可通过大型阵列天线的接收波束成形有效抑制。本文研究了大规模多输入多输出（MIMO）系统中的OTA-FL，考虑了一个现实场景：边缘服务器尽管配备大型天线阵列，但其射频（RF）链数量受限。在此设定下，空中模型聚合的波束成形需与天线选择联合处理。由于优化问题的组合特性，这导致了一个NP-hard问题。我们通过两种不同方法解决该问题。第一种方法采用罚对偶分解（PDD）技术，开发了一种用于联合天线选择与波束成形的双层算法。第二种方法将天线选择任务解释为稀疏恢复问题，并基于Lasso和快速迭代软阈值方法开发了两种迭代联合算法。对所有方案进行了收敛性与复杂度分析。数值研究表明，当边缘服务器RF链数量远小于阵列大小时，基于稀疏恢复技术的算法优于基于PDD的算法。然而，随着RF链数量增加，PDD方法开始显现优势。我们的仿真进一步表明，当PS处仅选择少于20%的天线时，学习性能可紧密跟踪所有天线均激活时的表现。