  In the paper, we propose an effective and efficient Compositional Federated
Learning (ComFedL) algorithm for solving a new compositional Federated Learning
(FL) framework, which frequently appears in many data mining and machine
learning problems with a hierarchical structure such as distributionally robust
FL and model-agnostic meta learning (MAML). Moreover, we study the convergence
analysis of our ComFedL algorithm under some mild conditions, and prove that it
achieves a convergence rate of $O(\frac{1}{\sqrt{T}})$, where $T$ denotes the
number of iteration. To the best of our knowledge, our new Compositional FL
framework is the first work to bridge federated learning with composition
stochastic optimization. In particular, we first transform the distributionally
robust FL (i.e., a minimax optimization problem) into a simple composition
optimization problem by using KL divergence regularization. At the same time,
we also first transform the distribution-agnostic MAML problem (i.e., a minimax
optimization problem) into a simple yet effective composition optimization
problem. Finally, we apply two popular machine learning tasks, i.e.,
distributionally robust FL and MAML to demonstrate the effectiveness of our
algorithm.
