We consider online learning in the model where a learning algorithm can access the class only via the consistency oracle -- an oracle, that, at any moment, can give a function from the class that agrees with all examples seen so far. This model was recently considered by Assos et al. (COLT'23). It is motivated by the fact that standard methods of online learning rely on computing the Littlestone dimension of subclasses, a problem that is computationally intractable. Assos et al. gave an online learning algorithm in this model that makes at most $C^d$ mistakes on classes of Littlestone dimension $d$, for some absolute unspecified constant $C > 0$. We give a novel algorithm that makes at most $O(256^d)$ mistakes. Our proof is significantly simpler and uses only very basic properties of the Littlestone dimension. We also observe that there exists no algorithm in this model that makes at most $2^{d+1}-2$ mistakes. We also observe that our algorithm (as well as the algorithm of Assos et al.) solves an open problem by Hasrati and Ben-David (ALT'23). Namely, it demonstrates that every class of finite Littlestone dimension with recursively enumerable representation admits a computable online learner (that may be undefined on unrealizable samples).

翻译：摘要： 我们考虑在一类学习算法仅能通过一致性预言机（即在任意时刻均可返回与当前所有已见样本一致的类内函数的预言机）访问类别这一模型下的在线学习问题。该模型由 Assos 等人（COLT'23）近期提出，其动机源于标准在线学习方法依赖于计算子类的Littlestone维数——这是一个计算上不可解的问题。Assos等人给出了该模型下的一种在线学习算法，对于Littlestone维数为d的类别，该算法最多犯C^d次错误（其中C>0为某个绝对未指定常数）。我们提出了一种新算法，其错误上界至多为O(256^d)。我们的证明显著简化，且仅需使用Littlestone维数最基本的性质。我们还观察到，在该模型下不存在任何算法能将错误数控制在2^{d+1}-2次以内。此外，我们的算法（以及Assos等人的算法）解决了Hasrati和Ben-David（ALT'23）提出的一个开放问题：即证明每个具有递归可枚举表示且Littlestone维数有限的类别均存在可计算的在线学习器（该学习器在不可实现样本上可能无定义）。