  Conditional stochastic optimization has found applications in a wide range of
machine learning tasks, such as invariant learning, AUPRC maximization, and
meta-learning. As the demand for training models with large-scale distributed
data grows in these applications, there is an increasing need for
communication-efficient distributed optimization algorithms, such as federated
learning algorithms. This paper considers the nonconvex conditional stochastic
optimization in federated learning and proposes the first federated conditional
stochastic optimization algorithm (FCSG) with a conditional stochastic gradient
estimator and a momentum-based algorithm (FCSG-M). To match the lower bound
complexity in the single-machine setting, we design an accelerated algorithm
(Acc-FCSG-M) via the variance reduction to achieve the best sample and
communication complexity. Compared with the existing optimization analysis for
MAML in FL, federated conditional stochastic optimization considers the sample
of tasks. Extensive experimental results on various tasks validate the
efficiency of these algorithms.
