Federated learning (FL) is a new distributed machine learning framework known for its benefits on data privacy and communication efficiency. Since full client participation in many cases is infeasible due to constrained resources, partial participation FL algorithms have been investigated that proactively select/sample a subset of clients, aiming to achieve learning performance close to the full participation case. This paper studies a passive partial client participation scenario that is much less well understood, where partial participation is a result of external events, namely client dropout, rather than a decision of the FL algorithm. We cast FL with client dropout as a special case of a larger class of FL problems where clients can submit substitute (possibly inaccurate) local model updates. Based on our convergence analysis, we develop a new algorithm FL-FDMS that discovers friends of clients (i.e., clients whose data distributions are similar) on-the-fly and uses friends' local updates as substitutes for the dropout clients, thereby reducing the substitution error and improving the convergence performance. A complexity reduction mechanism is also incorporated into FL-FDMS, making it both theoretically sound and practically useful. Experiments on MNIST and CIFAR-10 confirmed the superior performance of FL-FDMS in handling client dropout in FL.

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