  We propose a novel framework to study asynchronous federated learning
optimization with delays in gradient updates. Our theoretical framework extends
the standard FedAvg aggregation scheme by introducing stochastic aggregation
weights to represent the variability of the clients update time, due for
example to heterogeneous hardware capabilities. Our formalism applies to the
general federated setting where clients have heterogeneous datasets and perform
at least one step of stochastic gradient descent (SGD). We demonstrate
convergence for such a scheme and provide sufficient conditions for the related
minimum to be the optimum of the federated problem. We show that our general
framework applies to existing optimization schemes including centralized
learning, FedAvg, asynchronous FedAvg, and FedBuff. The theory here provided
allows drawing meaningful guidelines for designing a federated learning
experiment in heterogeneous conditions. In particular, we develop in this work
FedFix, a novel extension of FedAvg enabling efficient asynchronous federated
training while preserving the convergence stability of synchronous aggregation.
We empirically demonstrate our theory on a series of experiments showing that
asynchronous FedAvg leads to fast convergence at the expense of stability, and
we finally demonstrate the improvements of FedFix over synchronous and
asynchronous FedAvg.
