Joint Antenna Selection and Beamforming for Parallel Over-The-Air Computing With Limited Rf Chains

This paper addresses the challenge of training multiple models in parallel via federated learning with over-the-air computation (AirComp) in multiuser multi-stream multiple-input multiple-output (MIMO) systems. With the rapid development of artificial intelligence (AI), many Internet-of-Things (IoT) and industrial applications require training and updating several specialized models. As state-of-the-art sensors usually equip with multiple antennas, it is possible to transmit multiple data streams to carry the parameters of multiple models and to be aggregated in parallel via AirComp at the access point (AP). However, existing AirComp techniques primarily focus on single-model training, and it remains a significant challenge for aggregating signals dedicated for training multiple models separately and simultaneously. In addition, due to hardware complexity in practice, a small number of radio frequency (RF) chains need to be shared among the array of antennas at the AP. As such, this paper proposes a joint antenna selection (AS) and beamforming design for AirComp that leverages antenna selection to minimize model-by-model signal aggregation error. Specifically, a mixed integer programming is formulated. Based on the trace inverse property and an integration of semidefinite relaxation (SDR), Schur decomposition, and Lagrangian relaxation (LR), a new optimization strategy is developed to transform and relax the problem into a series of easy-to-solve convex subproblems, leading to an efficient algorithm. Numerical results confirm that with AS, when doubling the antennas from the RF chains, more than 3dB gain can be achieved.