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We study the decentralized consensus and stochastic optimization problemswith compressed communications over static directed graphs. We propose aniterative gradient-based algorithm that compresses messages according to adesired compression ratio. The proposed method provably reduces thecommunication overhead on the network at every communication round. Contrary toexisting literature, we allow for arbitrary compression ratios in thecommunicated messages. We show a linear convergence rate for the proposedmethod on the consensus problem. Moreover, we provide explicit convergencerates for decentralized stochastic optimization problems on smooth functionsthat are either (i) strongly convex, (ii) convex, or (iii) non-convex. Finally,we provide numerical experiments to illustrate convergence under arbitrarycompression ratios and the communication efficiency of our algorithm.

On Arbitrary Compression for Decentralized Consensus and Stochastic Optimization over Directed Networks