  Federated learning enables a large amount of edge computing devices to learn
a model without data sharing jointly. As a leading algorithm in this setting,
Federated Average FedAvg, which runs Stochastic Gradient Descent (SGD) in
parallel on local devices and averages the sequences only once in a while, have
been widely used due to their simplicity and low communication cost. However,
despite recent research efforts, it lacks theoretical analysis under
assumptions beyond smoothness. In this paper, we analyze the convergence of
FedAvg. Different from the existing work, we relax the assumption of strong
smoothness. More specifically, we assume the semi-smoothness and semi-Lipschitz
properties for the loss function, which have an additional first-order term in
assumption definitions. In addition, we also assume bound on the gradient,
which is weaker than the commonly used bounded gradient assumption in the
convergence analysis scheme. As a solution, this paper provides a theoretical
convergence study on Federated Learning.
