Regret has been widely adopted as the metric of choice for evaluating the performance of online optimization algorithms for distributed, multi-agent systems. However, variations in data or model associated with each agent can significantly impact decisions, and requires consensus among agents. Moreover, most existing works have focused on developing approaches for (strongly) convex losses, and very few results have been obtained in terms of regret bounds in distributed online optimization for general non-convex losses. To address these two issues, we propose a novel composite regret with a new network-based metric to evaluate distributed online optimization algorithms. We concretely define static and dynamic forms of the composite regret. By leveraging the dynamic form of our composite regret, we develop a consensus-based online normalized gradient (CONGD) approach for pseudo-convex losses and then show a sublinear behavior for CONGD relating to a regularity term for the path variation of the optimizer. For general non-convex losses, we first explore the regrets defined based on the recent advances such that no deterministic algorithm can achieve the sublinear regret. We then develop the distributed online non-convex optimization with composite regret (DINOCO) without access to the gradients, depending on an offline optimization oracle. We show that DINOCO can achieve sublinear regret; to our knowledge, this is the first regret bound for general distributed online non-convex learning.