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In decentralized optimization, multiple nodes in a network collaborate tominimize the sum of their local loss functions. The information exchangebetween nodes required for this task, is often limited by network connectivity.We consider a setting in which communication between nodes is hindered by both(i) a finite rate-constraint on the signal transmitted by any node, and (ii)additive noise corrupting the signal received by any node. We propose a novelalgorithm for this scenario: Decentralized Lazy Mirror Descent withDifferential Exchanges (DLMD-DiffEx), which guarantees convergence of the localestimates to the optimal solution under the given communication constraints. Asalient feature of DLMD-DiffEx is the introduction of additional proxyvariables that are maintained by the nodes to account for the disagreement intheir estimates due to channel noise and rate-constraints. Convergence to theoptimal solution is attained by having nodes iteratively exchange thesedisagreement terms until consensus is achieved. In order to prevent noiseaccumulation during this exchange, DLMD-DiffEx relies on two sequences; onecontrolling the power of the transmitted signal, and the other determining theconsensus rate. We provide clear insights on the design of these two sequenceswhich highlights the interplay between consensus rate and noise amplification.We investigate the performance of DLMD-DiffEx both from a theoreticalperspective as well as through numerical evaluations.

Decentralized optimization over noisy, rate-constrained networks: Achieving consensus by communicating differences