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Inspired by a recent breakthrough of Mishchenko et al (2022), who for thefirst time showed that local gradient steps can lead to provable communicationacceleration, we propose an alternative algorithm which obtains the samecommunication acceleration as their method (ProxSkip). Our approach is verydifferent, however: it is based on the celebrated method of Chambolle and Pock(2011), with several nontrivial modifications: i) we allow for an inexactcomputation of the prox operator of a certain smooth strongly convex functionvia a suitable gradient-based method (e.g., GD, Fast GD or FSFOM), ii) weperform a careful modification of the dual update step in order to retainlinear convergence. Our general results offer the new state-of-the-art ratesfor the class of strongly convex-concave saddle-point problems with bilinearcoupling characterized by the absence of smoothness in the dual function. Whenapplied to federated learning, we obtain a theoretically better alternative toProxSkip: our method requires fewer local steps ($O(\kappa^{1/3})$ or$O(\kappa^{1/4})$, compared to $O(\kappa^{1/2})$ of ProxSkip), and performs adeterministic number of local steps instead. Like ProxSkip, our method can beapplied to optimization over a connected network, and we obtain theoreticalimprovements here as well.

Communication Acceleration of Local Gradient Methods via an Accelerated Primal-Dual Algorithm with Inexact Prox