  In this paper, we study communication efficient distributed algorithms for
distributionally robust federated learning via periodic averaging with adaptive
sampling. In contrast to standard empirical risk minimization, due to the
minimax structure of the underlying optimization problem, a key difficulty
arises from the fact that the global parameter that controls the mixture of
local losses can only be updated infrequently on the global stage. To
compensate for this, we propose a Distributionally Robust Federated Averaging
(DRFA) algorithm that employs a novel snapshotting scheme to approximate the
accumulation of history gradients of the mixing parameter. We analyze the
convergence rate of DRFA in both convex-linear and nonconvex-linear settings.
We also generalize the proposed idea to objectives with regularization on the
mixture parameter and propose a proximal variant, dubbed as DRFA-Prox, with
provable convergence rates. We also analyze an alternative optimization method
for regularized cases in strongly-convex-strongly-concave and non-convex (under
PL condition)-strongly-concave settings. To the best of our knowledge, this
paper is the first to solve distributionally robust federated learning with
reduced communication, and to analyze the efficiency of local descent methods
on distributed minimax problems. We give corroborating experimental evidence
for our theoretical results in federated learning settings.
