  In this paper, we study the performance of federated learning over wireless
networks, where devices with a limited energy budget train a machine learning
model. The federated learning performance depends on the selection of the
clients participating in the learning at each round. Most existing studies
suggest deterministic approaches for the client selection, resulting in
challenging optimization problems that are usually solved using heuristics, and
therefore without guarantees on the quality of the final solution. We formulate
a new probabilistic approach to jointly select clients and allocate power
optimally so that the expected number of participating clients is maximized. To
solve the problem, a new alternating algorithm is proposed, where at each step,
the closed-form solutions for user selection probabilities and power
allocations are obtained. Our numerical results show that the proposed approach
achieves a significant performance in terms of energy consumption, completion
time and accuracy as compared to the studied benchmarks.
