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.

翻译：我们考虑基于网络的分散式优化问题，其中网络中的每个节点拥有一个局部函数，目标是通过集体协作达成共识解，使所有局部函数之和最小化。分散式优化的主要挑战在于对通信的依赖，这在许多应用中仍是一个显著瓶颈。为解决此问题，我们提出一种自适应随机化高效通信算法框架，通过周期性跟踪不一致误差并审慎选择每个节点上最具影响力和最有效的边进行通信，从而降低通信量。在此框架下，我们提出两种算法：用于求解共识问题的适应性共识（AC），以及用于求解光滑强凸分散式优化问题的基于适应性共识的梯度追踪算法（AC-GT）。我们为所提算法建立了坚实的理论收敛性保证，并在标准假设下通过多种算法参数量化其性能。最后，数值实验展示了该框架在显著减少达成共识所需信息交换方面的有效性。