We consider the problem of distributed online optimization, with a group of learners connected via a dynamic communication graph. The goal of the learners is to track the global minimizer of a sum of time-varying loss functions in a distributed manner. We propose a novel algorithm, termed Distributed Online Mirror Descent with Multiple Averaging Decision and Gradient Consensus (DOMD-MADGC), which is based on mirror descent but incorporates multiple consensus averaging iterations over local gradients as well as local decisions. The key idea is to allow the local learners to collect a sufficient amount of global information, which enables them to more accurately approximation the time-varying global loss, so that they can closely track the dynamic global minimizer over time. We show that the dynamic regret of DOMD-MADGC is upper bounded by the path length, which is defined as the cumulative distance between successive minimizers. The resulting bound improves upon the bounds of existing distributed online algorithms and removes the explicit dependence on $T$.

翻译：我们考虑了分布式在线优化的问题,通过动态通信图将一组学习者连接在一起。学习者的目标是以分布式方式跟踪一个时间变化损失函数总和的全球最小化因素。我们提议了一个新颖的算法,叫做“分布式在线镜源,具有多种挥发性决定和渐进共识(DOMD-MADGC)”,它以镜底为根据,但包含对本地梯度和本地决定平均迭代的多重共识。关键的想法是让本地学习者收集足够数量的全球信息,使他们能够更准确地近近近时间变化的全球损失,从而能够密切跟踪动态全球最小化器。我们表明,DOMD-MADGC的动态遗憾被路径长度(以相继最小化器之间的累积距离为定义)所覆盖的上限。由此形成的界限改善了现有分布式在线算法的界限,并消除了对$T的明显依赖性。