  In decentralized optimization, multiple nodes in a network collaborate to
minimize the sum of their local loss functions. The information exchange
between 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 novel
algorithm for this scenario: Decentralized Lazy Mirror Descent with
Differential Exchanges (DLMD-DiffEx), which guarantees convergence of the local
estimates to the optimal solution under the given communication constraints. A
salient feature of DLMD-DiffEx is the introduction of additional proxy
variables that are maintained by the nodes to account for the disagreement in
their estimates due to channel noise and rate-constraints. Convergence to the
optimal solution is attained by having nodes iteratively exchange these
disagreement terms until consensus is achieved. In order to prevent noise
accumulation during this exchange, DLMD-DiffEx relies on two sequences; one
controlling the power of the transmitted signal, and the other determining the
consensus rate. We provide clear insights on the design of these two sequences
which highlights the interplay between consensus rate and noise amplification.
We investigate the performance of DLMD-DiffEx both from a theoretical
perspective as well as through numerical evaluations.
