  Federated learning (FL) aims to minimize the communication complexity of
training a model over heterogeneous data distributed across many clients. A
common approach is local methods, where clients take multiple optimization
steps over local data before communicating with the server (e.g., FedAvg).
Local methods can exploit similarity between clients' data. However, in
existing analyses, this comes at the cost of slow convergence in terms of the
dependence on the number of communication rounds R. On the other hand, global
methods, where clients simply return a gradient vector in each round (e.g.,
SGD), converge faster in terms of R but fail to exploit the similarity between
clients even when clients are homogeneous. We propose FedChain, an algorithmic
framework that combines the strengths of local methods and global methods to
achieve fast convergence in terms of R while leveraging the similarity between
clients. Using FedChain, we instantiate algorithms that improve upon previously
known rates in the general convex and PL settings, and are near-optimal (via an
algorithm-independent lower bound that we show) for problems that satisfy
strong convexity. Empirical results support this theoretical gain over existing
methods.
