We study decentralized online optimization of upper-linearizable payoffs over an action set under efficient separation access, with applications to online continuous diminishing-return (DR) submodular maximization. We propose Decentralized Barrier Follow-the-Regularized-Leader (Dec-BFTRL), and evaluate each agent's played action against the average of all local objectives.

Each agent maps an internal iterate to a feasible action through an approximate gauge projection, communicates only a cumulative surrogate-gradient dual state, and invokes the local HybridNewton procedure to approximately minimize its post-communication BFTRL potential. For every agent, we achieve expected network-aggregate regret of $\widetilde O(\sqrt{T})$.

Over $T$ rounds, each agent uses $T$ neighbor-mixing steps and $\widetilde O(T)$ separation-oracle calls. We give four wrapper instantiations covering three DR-submodular maximization problems.