Improved Convergence Rates for Non-Convex Federated Learning with Compression

Federated learning is a new distributed learning paradigm that enables efficient training of emerging large-scale machine learning models. In this paper, we consider federated learning on non-convex objectives with compressed communication from the clients to the central server. We propose a novel first-order algorithm (\texttt{FedSTEPH2}) that employs compressed communication and achieves the optimal iteration complexity of $\mathcal{O}(1/\epsilon^{1.5})$ to reach an $\epsilon$-stationary point (i.e. $\mathbb{E}[\|\nabla f(\bm{x})\|^2] \leq \epsilon$) on smooth non-convex objectives. The proposed scheme is the first algorithm that attains the aforementioned optimal complexity {with compressed communication}. The key idea of \texttt{FedSTEPH2} that enables attaining this optimal complexity is applying judicious momentum terms both in the local client updates and the global server update. As a prequel to \texttt{FedSTEPH2}, we propose \texttt{FedSTEPH} which involves a momentum term only in the local client updates. We establish that \texttt{FedSTEPH} enjoys improved convergence rates under various non-convex settings (such as the Polyak-\L{}ojasiewicz condition) and with fewer assumptions than prior work.