We study the classical apple-tasting setting, a partial feedback model in which the learner observes the true label only when it predicts
1. A combinatorial characterization for the rates of this problem in the online adversarial setting has been studied by Raman et al. (2024). We study the self-directed setting, in which the learner can adaptively choose the order of the points it predicts on. In the realizable setting, we establish a tetrachotomy on the expected minimax number of mistakes made by any learner. Depending on the VC dimension, the
∅-centered star number, and the self-directed dimension, we show that the rates could possibly be
Θ(1),
O(logn)∩ω(1),
Θ~(n), or
Θ(n). In the agnostic setting, the minimax regret is
Θ~(n) or
Θ(n), depending on the VC dimension.