  We consider network-based decentralized optimization problems, where each
node in the network possesses a local function and the objective is to
collectively attain a consensus solution that minimizes the sum of all the
local functions. A major challenge in decentralized optimization is the
reliance on communication which remains a considerable bottleneck in many
applications. To address this challenge, we propose an adaptive randomized
communication-efficient algorithmic framework that reduces the volume of
communication by periodically tracking the disagreement error and judiciously
selecting the most influential and effective edges at each node for
communication. Within this framework, we present two algorithms: Adaptive
Consensus (AC) to solve the consensus problem and Adaptive Consensus based
Gradient Tracking (AC-GT) to solve smooth strongly convex decentralized
optimization problems. We establish strong theoretical convergence guarantees
for the proposed algorithms and quantify their performance in terms of various
algorithmic parameters under standard assumptions. Finally, numerical
experiments showcase the effectiveness of the framework in significantly
reducing the information exchange required to achieve a consensus solution.
