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As a prevalent distributed learning paradigm, Federated Learning (FL) trainsa global model on a massive amount of devices with infrequent communication.This paper investigates a class of composite optimization and statisticalrecovery problems in the FL setting, whose loss function consists of adata-dependent smooth loss and a non-smooth regularizer. Examples includesparse linear regression using Lasso, low-rank matrix recovery using nuclearnorm regularization, etc. In the existing literature, federated compositeoptimization algorithms are designed only from an optimization perspectivewithout any statistical guarantees. In addition, they do not consider commonlyused (restricted) strong convexity in statistical recovery problems. We advancethe frontiers of this problem from both optimization and statisticalperspectives. From optimization upfront, we propose a new algorithm named\textit{Fast Federated Dual Averaging} for strongly convex and smooth loss andestablish state-of-the-art iteration and communication complexity in thecomposite setting. In particular, we prove that it enjoys a fast rate, linearspeedup, and reduced communication rounds. From statistical upfront, forrestricted strongly convex and smooth loss, we design another algorithm, namely\textit{Multi-stage Federated Dual Averaging}, and prove a high probabilitycomplexity bound with linear speedup up to optimal statistical precision.Experiments in both synthetic and real data demonstrate that our methodsperform better than other baselines. To the best of our knowledge, this is thefirst work providing fast optimization algorithms and statistical recoveryguarantees for composite problems in FL.

Fast Composite Optimization and Statistical Recovery in Federated Learning