Effective Littlestone Dimension
This work addresses the problem of characterizing computable online learning for theoretical computer scientists, but it is incremental as it builds on prior effective VC dimension research.
The paper introduces an effective version of the Littlestone dimension and shows that finite effective Littlestone dimension is necessary but not sufficient for computable online learning, with sufficiency proven only in special cases such as classes of Littlestone dimension 1 or when learners have an upper bound on numbers to guess.
Delle Rose et al.~(COLT'23) introduced an effective version of the Vapnik-Chervonenkis dimension, and showed that it characterizes improper PAC learning with total computable learners. In this paper, we introduce and study a similar effectivization of the notion of Littlestone dimension. Finite effective Littlestone dimension is a necessary condition for computable online learning but is not a sufficient one -- which we already establish for classes of the effective Littlestone dimension 2. However, the effective Littlestone dimension equals the optimal mistake bound for computable learners in two special cases: a) for classes of Littlestone dimension 1 and b) when the learner receives as additional information an upper bound on the numbers to be guessed. Interestingly, finite effective Littlestone dimension also guarantees that the class consists only of computable functions.