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

Order Optimal Bounds for One-Shot Federated Learning over non-Convex Loss Functions