  We consider a decentralized online convex optimization problem in a network
of agents, where each agent controls only a coordinate (or a part) of the
global decision vector. For such a problem, we propose two decentralized
variants (ODA-C and ODA-PS) of Nesterov's primal-dual algorithm with dual
averaging. In ODA-C, to mitigate the disagreements on the primal-vector
updates, the agents implement a generalization of the local
information-exchange dynamics recently proposed by Li and Marden over a static
undirected graph. In ODA-PS, the agents implement the broadcast-based push-sum
dynamics over a time-varying sequence of uniformly connected digraphs. We show
that the regret bounds in both cases have sublinear growth of $O(\sqrt{T})$,
with the time horizon $T$, when the stepsize is of the form $1/\sqrt{t}$ and
the objective functions are Lipschitz-continuous convex functions with
Lipschitz gradients. We also implement the proposed algorithms on a sensor
network to complement our theoretical analysis.
