  Federated Learning is a popular distributed learning paradigm in machine
learning. Meanwhile, composition optimization is an effective hierarchical
learning model, which appears in many machine learning applications such as
meta learning and robust learning. More recently, although a few federated
composition optimization algorithms have been proposed, they still suffer from
high sample and communication complexities. In the paper, thus, we propose a
class of faster federated compositional optimization algorithms (i.e., MFCGD
and AdaMFCGD) to solve the nonconvex distributed composition problems, which
builds on the momentum-based variance reduced and local-SGD techniques. In
particular, our adaptive algorithm (i.e., AdaMFCGD) uses a unified adaptive
matrix to flexibly incorporate various adaptive learning rates. Moreover, we
provide a solid theoretical analysis for our algorithms under non-i.i.d.
setting, and prove our algorithms obtain a lower sample and communication
complexities simultaneously than the existing federated compositional
algorithms. Specifically, our algorithms obtain lower sample complexity of
$\tilde{O}(\epsilon^{-3})$ with lower communication complexity of
$\tilde{O}(\epsilon^{-2})$ in finding an $\epsilon$-stationary solution. We
conduct the numerical experiments on robust federated learning and distributed
meta learning tasks to demonstrate the efficiency of our algorithms.
