In decentralized optimization, nodes of a communication network privately possess a local objective function, and communicate using gossip-based methods in order to minimize the average of these per-node objectives. While synchronous algorithms can be heavily slowed down by a few nodes and edges in the graph (the straggler problem), their asynchronous counterparts lack from a sharp analysis taking into account heterogeneous delays in the communication network. In this paper, we propose a novel continuous-time framework to analyze asynchronous algorithms, which does not require to define a global ordering of the events, and allows to finely characterize the time complexity in the presence of (heterogeneous) delays. Using this framework, we describe a fully asynchronous decentralized algorithm to minimize the sum of smooth and strongly convex functions. Our algorithm (DCDM, Delayed Coordinate Dual Method), based on delayed randomized gossip communications and local computational updates, achieves an asynchronous speed-up: the rate of convergence is tightly characterized in terms of the eigengap of the graph weighted by local delays only, instead of the global worst-case delays as in previous analyses.

翻译：在优化方面,通信网络的分权节点私自拥有一个本地客观功能,并使用八卦为基础的方法进行交流,以最大限度地减少这些每个节点目标的平均值。同步算法可以因图形中的几个节点和边缘(分流问题)而大大减速,但考虑到通信网络中各种延误,对对应方的无序同步分析缺乏精确的分析。在本文件中,我们提出了一个新的连续时间框架,用于分析无同步算法,它不需要确定事件的全球顺序,并允许在出现(偏差的)延迟的情况下对时间复杂性进行细化描述。我们使用这个框架描述一种完全不同步的分散算法,以最大限度地减少顺畅和强烈的convex功能的总和。我们的算法(DCDM,延迟协调的双重方法)基于延迟随机流言通信和本地计算更新,实现了一个不同步的加速率:趋同率在仅按本地延迟加权的图表的微插图上是紧密的,而不是在以往分析中最坏的拖延。