This paper describes a novel approach to decentralized online optimization in a large network of agents. At each stage, the agents face a new objective function that reflects the effects of a changing environment, and each agent can share information pertaining to past decisions and cost functions only with his neighbors. These operating conditions arise in many practical applications, but introduce challenging questions related to the performance of distributed strategies relative to impractical centralized approaches. The proposed algorithm yields small regret (i.e., the difference between the total cost incurred using causally available information and the total cost that would have been incurred in hindsight had all the relevant information been available all at once) and is robust to evolving network topologies. It combines a subgradient-based sequential convex optimization scheme with decentralized decision-making via approximate dynamic programming.