  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 $T$.
