  In many federated learning (FL) models, a common strategy employed to ensure
the progress in the training process, is to wait for at least $M$ clients out
of the total $N$ clients to send back their local gradients based on a
reporting deadline $T$, once the parameter server (PS) has broadcasted the
global model. If enough clients do not report back within the deadline, the
particular round is considered to be a failed round and the training round is
restarted from scratch. If enough clients have responded back, the round is
deemed successful and the local gradients of all the clients that responded
back are used to update the global model. In either case, the clients that
failed to report back an update within the deadline would have wasted their
computational resources. Having a tighter deadline (small $T$) and waiting for
a larger number of participating clients (large $M$) leads to a large number of
failed rounds and therefore greater communication cost and computation resource
wastage. However, having a larger $T$ leads to longer round durations whereas
smaller $M$ may lead to noisy gradients. Therefore, there is a need to optimize
the parameters $M$ and $T$ such that communication cost and the resource
wastage is minimized while having an acceptable convergence rate. In this
regard, we show that the average age of a client at the PS appears explicitly
in the theoretical convergence bound, and therefore, can be used as a metric to
quantify the convergence of the global model. We provide an analytical scheme
to select the parameters $M$ and $T$ in this setting.
