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We consider a standard federated learning (FL) architecture where a group ofclients periodically coordinate with a central server to train a statisticalmodel. We develop a general algorithmic framework called FedLin to tackle someof the key challenges intrinsic to FL, namely objective heterogeneity, systemsheterogeneity, and infrequent and imprecise communication. Our framework ismotivated by the observation that under these challenges, various existing FLalgorithms suffer from a fundamental speed-accuracy conflict: they eitherguarantee linear convergence but to an incorrect point, or convergence to theglobal minimum but at a sub-linear rate, i.e., fast convergence comes at theexpense of accuracy. In contrast, when the clients' local loss functions aresmooth and strongly convex, we show that FedLin guarantees linear convergenceto the global minimum, despite arbitrary objective and systems heterogeneity.We then establish matching upper and lower bounds on the convergence rate ofFedLin that highlight the effects of intermittent communication. Finally, weshow that FedLin preserves linear convergence rates under aggressive gradientsparsification, and quantify the effect of the compression level on theconvergence rate. Our work is the first to provide tight linear convergencerate guarantees, and constitutes the first comprehensive analysis of gradientsparsification in FL.

Linear Convergence in Federated Learning: Tackling Client Heterogeneity and Sparse Gradients