This paper considers uplink communication design for training multiple independent machine learning models via wireless federated learning. To reduce communication delay, we utilize multi-antenna beamforming and propose a multi-model over-the-air (M-OTA) uplink transmission and aggregation method for simultaneous training of multiple models. Devices are formed into groups for local model training based on the model assignments per communication round and simultaneously perform OTA aggregation over a multi-access channel. Based on the M-OTA transmission and reception process, we formulate a joint optimization of transmit and receive beamforming and device grouping over $T$ communication rounds to maximize the training convergence rates of all models. We analyze the per-model optimality gap, which captures the impact of noisy aggregation and inter-model interference on the learning performance. We further show that the $T$-round joint optimization problem can be decomposed into per-round online optimization to devise solutions. Following this, we propose a device grouping strategy that explores channel correlation among devices to maximize OTA aggregation performance. Furthermore, we present a low-complexity iterative algorithm for joint transmit-receive beamforming with closed-form updates. Simulation shows that our proposed M-OTA approach substantially outperforms the conventional single-model sequential OTA approach and other commonly used beamforming schemes.

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