  We consider distributed online convex optimization problems, where the
distributed system consists of various computing units connected through a
time-varying communication graph. In each time step, each computing unit
selects a constrained vector, experiences a loss equal to an arbitrary convex
function evaluated at this vector, and may communicate to its neighbors in the
graph. The objective is to minimize the system-wide loss accumulated over time.
We propose a decentralized algorithm with regret and cumulative constraint
violation in $\mathcal{O}(T^{\max\{c,1-c\} })$ and $\mathcal{O}(T^{1-c/2})$,
respectively, for any $c\in (0,1)$, where $T$ is the time horizon. When the
loss functions are strongly convex, we establish improved regret and constraint
violation upper bounds in $\mathcal{O}(\log(T))$ and
$\mathcal{O}(\sqrt{T\log(T)})$. These regret scalings match those obtained by
state-of-the-art algorithms and fundamental limits in the corresponding
centralized online optimization problem (for both convex and strongly convex
loss functions). In the case of bandit feedback, the proposed algorithms
achieve a regret and constraint violation in $\mathcal{O}(T^{\max\{c,1-c/3 \}
})$ and $\mathcal{O}(T^{1-c/2})$ for any $c\in (0,1)$. We numerically
illustrate the performance of our algorithms for the particular case of
distributed online regularized linear regression problems.
