Federated Learning With Controlled Descent Under Fading: Convergence and Energy Implications

In over-the-air computation-assisted federated learning (OTA-FL), devices transmit their local models to a parameter server over a shared time-frequency resource. Model aggregation occurs automatically due to the superposition property of the wireless channel. We derive a novel upper bound on the convergence of the optimality gap of OTA-FL that applies to any choice of device transmit powers and receiver scaling. The bound is based on less restrictive assumptions compared to the literature. It leads to the insightful concept of an effective learning rate that captures the dependence of the convergence of OTA-FL on the gains of the channels between the devices and the parameter server. We jointly optimize the transmit powers and the receiver scaling to minimize the error floor implied by the bound while controlling the effective learning rate. This leads to a novel controlled descent algorithm (CDA) and a new variant that adapts the effective learning rate. CDA can be implemented using a low overhead protocol. We benchmark CDA against several transmit power, receiver scaling, and device selection schemes. For both linear regression and multi-class logistic regression, CDA requires fewer iterations and a lower sum energy to achieve a target optimality gap or testing accuracy.