  We study a family of algorithms, which we refer to as local update methods,
generalizing many federated and meta-learning algorithms. We prove that for
quadratic models, local update methods are equivalent to first-order
optimization on a surrogate loss we exactly characterize. Moreover, fundamental
algorithmic choices (such as learning rates) explicitly govern a trade-off
between the condition number of the surrogate loss and its alignment with the
true loss. We derive novel convergence rates showcasing these trade-offs and
highlight their importance in communication-limited settings. Using these
insights, we are able to compare local update methods based on their
convergence/accuracy trade-off, not just their convergence to critical points
of the empirical loss. Our results shed new light on a broad range of
phenomena, including the efficacy of server momentum in federated learning and
the impact of proximal client updates.
