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.

We propose a novel energy-efficient feedback-enhanced successively reordered transmissions scheme (FE-SRTS) that combines distributed multiple access-based ordered channel access, feedback from the fusion node (FN), and quantized payloads. In FE-SRTS, the sensor nodes sequentially transmit their log-likelihood ratios (LLRs) to the FN until the latter decides. The order in which the nodes transmit is updated based on feedback from the FN and is implemented in a distributed manner using the timer scheme. We derive novel decision rules that enable FE-SRTS to achieve the detection error probability of the optimal rule in which the FN knows the LLRs of all nodes, but with a substantially lower average number of transmissions than conventional ordered and unordered schemes. We also account for retransmissions and power control due to fading, quantized payloads, and feedback. We analyze the average number of sensor transmissions and the total energy consumed by FE-SRTS. The analysis leads to insightful asymptotic results that establish the efficacy of FE-SRTS for Gaussian statistics. Our simulations, based on the Zigbee standard, show that the total energy consumed by FE-SRTS is markedly lower than by conventional schemes.

We propose a novel energy-efficient controlled descent algorithm (EECDA) for over-the-air computation-assisted federated learning. In EECDA, the computing devices transmit their local parameters to the parameter server using amplitude modulation over a common time-frequency resource. As a result, a computation that involves adding the data of multiple users occurs automatically over the wireless channel since the signals superimpose. EECDA adapts the transmit powers of the devices and the amplification at the receiver to minimize the error floor on the optimality gap, which measures the performance of the federated learning algorithm. We derive the transmit powers and receiver amplification in closed-form. This is based on a novel recursive upper bound on the optimality gap that characterizes how wireless channel fades, device transmit powers, receiver amplification, noise variance, and batch selection variance determine the effective learning rate and error floor. For a small total energy, EECDA achieves a markedly lower optimality gap than the conventional minimum mean square error scheme.

For the binary hypothesis testing problem, we propose a novel feedback-enhanced successively reordered transmissions scheme (FE-SRTS), in which the nodes change the order in which they transmit based on the feedback from the fusion node (FN) in each step. It can be implemented in a distributed manner using the timer scheme without any node knowing the measurement of any other node. We derive novel decision rules for it that enable the FN to decide on a hypothesis after receiving only a subset of measurements. For the Bayesian detection framework, FE-SRTS achieves the same optimal error probability as the unordered transmissions scheme (UTS), in which all the nodes transmit their log-likelihood ratios to the FN. However, it requires far fewer nodes to transmit, on average, than UTS, which leads to a much higher energy-efficiency. As the signal-to-noise ratio increases, the average number of transmissions of FE-SRTS decreases to two. This is much lower than the average number of transmissions of the conventional ordered transmissions scheme, which does not employ feedback and does not update the order in which the nodes transmit.