  We study a family of algorithms, which we refer to as local update methods,
that generalize many federated learning and meta-learning algorithms. We prove
that for quadratic objectives, local update methods perform stochastic gradient
descent on a surrogate loss function which we exactly characterize. We show
that the choice of client learning rate controls the condition number of that
surrogate loss, as well as the distance between the minimizers of the surrogate
and true loss functions. We use this theory to derive novel convergence rates
for federated averaging that showcase this trade-off between the condition
number of the surrogate loss and its alignment with the true loss function. We
validate our results empirically, showing that in communication-limited
settings, proper learning rate tuning is often sufficient to reach near-optimal
behavior. We also present a practical method for automatic learning rate decay
in local update methods that helps reduce the need for learning rate tuning,
and highlight its empirical performance on a variety of tasks and datasets.
