  We consider the open federated learning (FL) systems, where clients may join
and/or leave the system during the FL process. Given the variability of the
number of present clients, convergence to a fixed model cannot be guaranteed in
open systems. Instead, we resort to a new performance metric that we term the
stability of open FL systems, which quantifies the magnitude of the learned
model in open systems. Under the assumption that local clients' functions are
strongly convex and smooth, we theoretically quantify the radius of stability
for two FL algorithms, namely local SGD and local Adam. We observe that this
radius relies on several key parameters, including the function condition
number as well as the variance of the stochastic gradient. Our theoretical
results are further verified by numerical simulations on both synthetic and
real-world benchmark data-sets.
