Inspired by a recent breakthrough of Mishchenko et al. [2022], who for the first time showed that local gradient steps can lead to provable communication acceleration, we propose an alternative algorithm which obtains the same communication acceleration as their method (ProxSkip). Our approach is very different, however: it is based on the celebrated method of Chambolle and Pock [2011], with several nontrivial modifications: i) we allow for an inexact computation of the prox operator of a certain smooth strongly convex function via a suitable gradient-based method (e.g., GD or Fast GD), ii) we perform a careful modification of the dual update step in order to retain linear convergence. Our general results offer the new state-of-the-art rates for the class of strongly convex-concave saddle-point problems with bilinear coupling characterized by the absence of smoothness in the dual function. When applied to federated learning, we obtain a theoretically better alternative to ProxSkip: our method requires fewer local steps ($\mathcal{O}(\kappa^{1/3})$ or $\mathcal{O}(\kappa^{1/4})$, compared to $\mathcal{O}(\kappa^{1/2})$ of ProxSkip), and performs a deterministic number of local steps instead. Like ProxSkip, our method can be applied to optimization over a connected network, and we obtain theoretical improvements here as well.

Inspired by a recent breakthrough of Mishchenko et al. [2022], who for the first time showed that local gradient steps can lead to provable communication acceleration, we propose an alternative algorithm which obtains the same communication acceleration as their method (ProxSkip). Our approach is very different, however: it is based on the celebrated method of Chambolle and Pock [2011], with several nontrivial modifications: i) we allow for an inexact computation of the prox operator of a certain smooth strongly convex function via a suitable gradient-based method (e.g., GD or Fast GD), ii) we perform a careful modification of the dual update step in order to retain linear convergence. Our general results offer the new state-of-the-art rates for the class of strongly convex-concave saddle-point problems with bilinear coupling characterized by the absence of smoothness in the dual function. When applied to federated learning, we obtain a theoretically better alternative to ProxSkip: our method requires fewer local steps ( \mathcal{O}(\kappa {1/3}) or \mathcal{O}(\kappa {1/4}), compared to \mathcal{O}(\kappa {1/2}) of ProxSkip), and performs a deterministic number of local steps instead. Like ProxSkip, our method can be applied to optimization over a connected network, and we obtain theoretical improvements here as well.

We propose a new training algorithm, named DualFL (Dualized Federated Learning), for solving distributed optimization problems in federated learning. DualFL achieves communication acceleration for very general convex cost functions, thereby providing a solution to an open theoretical problem in federated learning concerning cost functions that may not be smooth nor strongly convex. We provide a detailed analysis for the local iteration complexity of DualFL to ensure the overall computational efficiency of DualFL. Furthermore, we introduce a completely new approach for the convergence analysis of federated learning based on a dual formulation. This new technique enables concise and elegant analysis, which contrasts the complex calculations used in existing literature on convergence of federated learning algorithms.

Distributed optimization methods with probabilistic local updates have recently gained attention for their provable ability to communication acceleration. Nevertheless, this capability is effective only when the loss function is smooth and the network is sufficiently well-connected. In this paper, we propose the first linear convergent method MG-Skip with probabilistic local updates for nonsmooth distributed optimization. Without any extra condition for the network connectivity, MG-Skip allows for the multiple-round gossip communication to be skipped in most iterations, while its iteration complexity is $\mathcal{O}\left(\kappa \log \frac{1}{\epsilon}\right)$ and communication complexity is only $\mathcal{O}\left(\sqrt{\frac{\kappa}{(1-\rho)}} \log \frac{1}{\epsilon}\right)$, where $\kappa$ is the condition number of the loss function and $\rho$ reflects the connectivity of the network topology. To the best of our knowledge, MG-Skip achieves the best communication complexity when the loss function has the smooth (strongly convex)+nonsmooth (convex) composite form.