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

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→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(1/(√(n)(mB)^1/d), 1/√(mn)), 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.