  Stochastic optimization methods such as mirror descent have wide applications
due to low computational cost. Those methods have been well studied under
assumption of the independent and identical distribution, and usually achieve
sublinear rate of convergence. However, this assumption may be too strong and
unpractical in real application scenarios. Recent researches investigate
stochastic gradient descent when instances are sampled from a Markov chain.
Unfortunately, few results are known for stochastic mirror descent. In the
paper, we propose a new version of stochastic mirror descent termed by MarchOn
in the scenario of the federated learning. Given a distributed network, the
model iteratively travels from a node to one of its neighbours randomly.
Furthermore, we propose a new framework to analyze MarchOn, which yields best
rates of convergence for convex, strongly convex, and non-convex loss. Finally,
we conduct empirical studies to evaluate the convergence of MarchOn, and
validate theoretical results.
