  We consider the problem of federated learning in a one-shot setting in which
there are $m$ machines, each observing $n$ sample functions from an unknown
distribution on non-convex loss functions. Let $F:[-1,1]^d\to\mathbb{R}$ be the
expected loss function with respect to this unknown distribution. The goal is
to find an estimate of the minimizer of $F$. Based on its observations, each
machine generates a signal of bounded length $B$ and sends it to a server. The
server collects signals of all machines and outputs an estimate of the
minimizer of $F$. We show that the expected loss of any algorithm is lower
bounded by $\max\big(1/(\sqrt{n}(mB)^{1/d}), 1/\sqrt{mn}\big)$, up to a
logarithmic factor. We then prove that this lower bound is order optimal in $m$
and $n$ by presenting a distributed learning algorithm, called Multi-Resolution
Estimator for Non-Convex loss function (MRE-NC), whose expected loss matches
the lower bound for large $mn$ up to polylogarithmic factors.
