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 O(T max{c,1−c} ) and O(T 1−c/2 ), respectively, for any c ∈ (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 O(log(T )) and O( 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 O(T max{c,1−c/3} ) and O(T 1−c/2 ) for any c ∈ (0, 1). We numerically illustrate the performance of our algorithms for the particular case of distributed online regularized linear regression problems.
Summary: 
The paper considers a distributed variant of online convex optimization problem over multiple players, where, at each trial t, convex loss_l_t.i is revealed to player i and but evaluated by sum of loss functions sum_i=1^n l_t,i. The players can communicate with their neighborhood and share their decisions. Under the problems setting and some assumption on the neighborhood graph structure, the authors prove regret bounds for convex/strongly-convex losses and full-info/bandit settings. Specifically, the paper allows the algorithm to violate domain constraints but the sum of violation has to be sublinear in the number of trials. They also show the violation bounds simultaneously. 
Comments:
The key assumption is Assumption 4, that the players share a doubly-stochastic matrix which is used to mix neighbors’ decisions. This assumption allows to mix all players’ decisions in a long run and the derived regret bounds make sense. The theoretical results are non-trivial. 
As a summary, I feel the results are beyond standard previous work and has certain values. Note that I have not evaluated correctness of the results, but the results are likely under Assumption 4.