  Federated Learning (FL) is a distributed machine learning approach that
enables model training in communication efficient and privacy-preserving
manner. The standard optimization method in FL is Federated Averaging (FedAvg),
which performs multiple local SGD steps between communication rounds. FedAvg
has been considered to lack algorithm adaptivity compared to modern first-order
adaptive optimizations. In this paper, we propose new communication-efficient
FL algortithms based on two adaptive frameworks: local adaptivity (PreFed) and
server-side adaptivity (PreFedOp). Proposed methods adopt adaptivity by using a
novel covariance matrix preconditioner. Theoretically, we provide convergence
guarantees for our algorithms. The empirical experiments show our methods
achieve state-of-the-art performances on both i.i.d. and non-i.i.d. settings.
