  In this paper, we consider an online distributed composite optimization
problem over a time-varying multi-agent network that consists of multiple
interacting nodes, where the objective function of each node consists of two
parts: a loss function that changes over time and a regularization function.
This problem naturally arises in many real-world applications ranging from
wireless sensor networks to signal processing. We propose a class of online
distributed optimization algorithms that are based on approximate mirror
descent, which utilize the Bregman divergence as distance-measuring function
that includes the Euclidean distances as a special case. We consider two
standard information feedback models when designing the algorithms, that is,
full-information feedback and bandit feedback. For the full-information
feedback model, the first algorithm attains an average regularized regret of
order $\mathcal{O}(1/\sqrt{T})$ with the total number of rounds $T$. The second
algorithm, which only requires the information of the values of the loss
function at two predicted points instead of the gradient information, achieves
the same average regularized regret as that of the first algorithm. Simulation
results of a distributed online regularized linear regression problem are
provided to illustrate the performance of the proposed algorithms.
