Federated learning (FL) is a useful tool that enables the training of machine learning models over distributed data without having to collect data centrally. When deploying FL in constrained wireless environments, however, intermittent connectivity of devices, heterogeneous connection quality, and non-i.i.d. data can severely slow convergence. In this paper, we consider FL with arbitrary device participation probabilities for each round and show that by weighing each device's update by the reciprocal of their per-round participation probability, we can guarantee convergence to a stationary point. Our bound applies to non-convex loss functions and non-i.i.d. datasets and recovers state-of-the-art convergence rates for both full and uniform partial participation, including linear speedup, with only a single-sided learning rate. Then, using the derived convergence bound, we develop a new online client selection and power allocation algorithm that utilizes the Lyapunov drift-plus-penalty framework to opportunistically minimize a function of the convergence bound and the average communication time under a transmit power constraint. We use optimization over manifold techniques to obtain a solution to the minimization problem. Thanks to the Lyapunov framework, one key feature of the algorithm is that knowledge of the channel distribution is not required and only the instantaneous channel state information needs to be known. Using the CIFAR-10 dataset with varying levels of data heterogeneity, we show through simulations that the communication time can be significantly decreased using our algorithm compared to uniformly random participation, especially for heterogeneous channel conditions.

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