The present work introduces the hybrid consensus alternating direction method of multipliers (H-CADMM), a novel framework for optimization over networks which unifies existing distributed optimization approaches, including the centralized and the decentralized consensus ADMM. H-CADMM provides a flexible tool that leverages the underlying graph topology in order to achieve a desirable sweet-spot between node-to-node communication overhead and rate of convergence -- thereby alleviating known limitations of both C-CADMM and D-CADMM. A rigorous analysis of the novel method establishes linear convergence rate, and also guides the choice of parameters to optimize this rate. The novel hybrid update rules of H-CADMM lend themselves to in-network acceleration that is shown to effect considerable -- and essentially free-of-charge -- performance boost over the fully decentralized ADMM. Comprehensive numerical tests validate the analysis and showcase the potential of the method in tackling efficiently, widely useful learning tasks.
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