We initiate a study of computable online (c-online) learning, which we analyze under varying requirements for "optimality" in terms of the mistake bound. Our main contribution is to give a necessary and sufficient condition for optimal c-online learning and show that the Littlestone dimension no longer characterizes the optimal mistake bound of c-online learning. Furthermore, we introduce anytime optimal (a-optimal) online learning, a more natural conceptualization of "optimality" and a generalization of Littlestone's Standard Optimal Algorithm. We show the existence of a computational separation between a-optimal and optimal online learning, proving that a-optimal online learning is computationally more difficult. Finally, we consider online learning with no requirements for optimality, and show, under a weaker notion of computability, that the finiteness of the Littlestone dimension no longer characterizes whether a class is c-online learnable with finite mistake bound. A potential avenue for strengthening this result is suggested by exploring the relationship between c-online and CPAC learning, where we show that c-online learning is as difficult as improper CPAC learning.

翻译：我们启动了对可计算在线（c-online）学习的研究，并在关于“最优性”的不同要求下（以错误界为衡量标准）对其进行了分析。我们的主要贡献是给出了最优c-online学习的必要充分条件，并证明了利特斯通维数不再表征c-online学习的最优错误界。此外，我们引入了任意时间最优（a-optimal）在线学习，这是一种更自然的“最优性”概念，也是利特斯通标准最优算法的推广。我们证明了a-最优与最优在线学习之间存在计算分离，证实a-最优在线学习在计算上更为困难。最后，我们考虑了无最优性要求的在线学习，并在较弱的可计算概念下证明：利特斯通维数的有限性不再决定一个类别是否具有有限错误界的c-online可学习性。通过探索c-online学习与CPAC学习之间的关系，我们为加强这一结果指明了潜在路径，其中我们证明c-online学习与不当CPAC学习同样困难。