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The primal-dual method of multipliers (PDMM) was originally designed forsolving a decomposable optimisation problem over a general network. In thispaper, we revisit PDMM for optimisation over a centralized network. We firstnote that the recently proposed method FedSplit [1] implements PDMM for acentralized network. In [1], Inexact FedSplit (i.e., gradient based FedSplit)was also studied both empirically and theoretically. We identify the cause forthe poor reported performance of Inexact FedSplit, which is due to the improperinitialisation in the gradient operations at the client side. To fix the issueof Inexact FedSplit, we propose two versions of Inexact PDMM, which arereferred to as gradient-based PDMM (GPDMM) and accelerated GPDMM (AGPDMM),respectively. AGPDMM accelerates GPDMM at the cost of transmitting two timesthe number of parameters from the server to each client per iteration comparedto GPDMM. We provide a new convergence bound for GPDMM for a class of convexoptimisation problems. Our new bounds are tighter than those derived forInexact FedSplit. We also investigate the update expressions of AGPDMM andSCAFFOLD to find their similarities. It is found that when the number K ofgradient steps at the client side per iteration is K=1, both AGPDMM andSCAFFOLD reduce to vanilla gradient descent with proper parameter setup.Experimental results indicate that AGPDMM converges faster than SCAFFOLD whenK>1 while GPDMM converges slightly worse than SCAFFOLD.

Revisiting the Primal-Dual Method of Multipliers for Optimisation over Centralised Networks