  We study joint downlink-uplink beamforming design for wireless federated
learning (FL) with a multi-antenna base station. Considering analog
transmission over noisy channels and uplink over-the-air aggregation, we derive
the global model update expression over communication rounds. We then obtain an
upper bound on the expected global loss function, capturing the downlink and
uplink beamforming and receiver noise effect. We propose a low-complexity joint
beamforming algorithm to minimize this upper bound, which employs alternating
optimization to breakdown the problem into three subproblems, each solved via
closed-form gradient updates. Simulation under practical wireless system setup
shows that our proposed joint beamforming design solution substantially
outperforms the conventional separate-link design approach and nearly attains
the performance of ideal FL with error-free communication links.
