Imitation learning (IL) is a general learning paradigm for sequential decision-making problems. Interactive imitation learning, where learners can interactively query for expert annotations, has been shown to achieve provably superior sample efficiency guarantees compared with its offline counterpart or reinforcement learning. In this work, we study classification-based online imitation learning (abbrev. COIL) and the fundamental feasibility to design oracle-efficient regret-minimization algorithms in this setting, with a focus on the general non-realizable case. We make the following contributions: (1) we show that in the COIL problem, any proper online learning algorithm cannot guarantee a sublinear regret in general; (2) we propose Logger, an improper online learning algorithmic framework, that reduces COIL to online linear optimization, by utilizing a new definition of mixed policy class; (3) we design two oracle-efficient algorithms within the Logger framework that enjoy different sample and interaction round complexity tradeoffs, and show their improvements over behavior cloning; (4) we show that under standard complexity-theoretic assumptions, efficient dynamic regret minimization is infeasible in the Logger framework.