Title: Federated Learning's Blessing: FedAvg has Linear Speedup. Abstract: Federated learning (FL) learns a model jointly from a set of participating devices without sharing each other's privately held data. The characteristics of non-\textit{i.i.d.} data across the network, low device participation, high communication costs, and the mandate that data remain private bring challenges in understanding the convergence of FL algorithms, particularly in regards to how convergence scales with the number of participating devices. In this paper, we focus on Federated Averaging (FedAvg)--arguably the most popular and effective FL algorithm class in use today--and provide a unified and comprehensive study of its convergence rate. Although FedAvg has recently been studied by an emerging line of literature, it remains open as to how FedAvg's convergence scales with the number of participating devices in the fully heterogeneous FL setting--a crucial question whose answer would shed light on the performance of FedAvg in large FL systems. We fill this gap by providing a unified analysis that establishes convergence guarantees for FedAvg under three classes of problems: strongly convex smooth, convex smooth, and overparameterized strongly convex smooth problems. We show that FedAvg enjoys linear speedup in each case, although with different convergence rates and communication efficiencies. While there have been linear speedup results from distributed optimization that assumes full participation, ours are the first to establish linear speedup for FedAvg under both statistical and system heterogeneity.  For strongly convex and convex problems, we also characterize the corresponding convergence rates for the Nesterov accelerated FedAvg algorithm, which are the first linear speedup guarantees for momentum variants of FedAvg in the convex setting. To provably accelerate FedAvg, we design a new momentum-based FL algorithm that further improves the convergence rate in overparameterized linear regression problems. Empirical studies of the algorithms in various settings have supported our theoretical results.