We consider online learning in the model where a learning algorithm can access the class only via the \emph{consistent 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 computationally intractable problem. 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 show that there exists no algorithm in this model that makes less than $3^d$ mistakes.

翻译：我们考虑一种在线学习模型，其中学习算法仅能通过*一致性预言机*访问类别——该预言机可在任意时刻给出一个与当前所有观测样本一致的类别函数。此模型由Assos等人（COLT'23）近期提出，其动机源于标准在线学习方法依赖计算子类别的Littlestone维度，而这在计算上是一个棘手问题。Assos等人为该模型设计了一种在线学习算法，对于Littlestone维度为$d$的类别，该算法最多犯$C^d$次错误，其中$C>0$为某个未明确指定的绝对常数。我们提出了一种新算法，其错误次数上限为$O(256^d)$。我们的证明显著简化，仅需使用Littlestone维度的基本性质。此外，我们证明在该模型中，不存在错误次数少于$3^d$的算法。