  In this paper, we consider a distributed online convex optimization problem
over a time-varying multi-agent network. The goal of this network is to
minimize a global loss function through local computation and communication
with neighbors. To effectively handle the optimization problem with a
high-dimensional and structural constraint set, we develop a distributed online
multiple Frank-Wolfe algorithm to avoid the expensive computational cost of
projection operation. The dynamic regret bounds are established as
$\mathcal{O}(T^{1-\gamma}+H_T)$ with the linear oracle number $\mathcal{O}
(T^{1+\gamma})$, which depends on the horizon (total iteration number) $T$, the
function variation $H_T$, and the tuning parameter $0<\gamma<1$. In particular,
when the prior knowledge of $H_T$ and $T$ is available, the bound can be
enhanced to $\mathcal{O} (1+H_T)$. Moreover, we illustrate the significant
advantages of the multiple iteration technique and reveal a trade-off between
dynamic regret bound, computational cost, and communication cost. Finally, the
performance of our algorithm is verified and compared through the distributed
online ridge regression problems with two constraint sets.
