Apple Tasting: Combinatorial Dimensions and Minimax Rates
Vinod Raman, Unique Subedi, Ananth Raman, Ambuj Tewari
TL;DR
This work analyzes online binary classification under apple tasting feedback, where the label is revealed only when the predictor outputs 1. It develops a geometric, combinatorial theory around Littlestone dimension and a new Effective width to characterize learnability: agnostic learning has minimax regret $\Theta\big(\sqrt{L(\mathcal{H})T}\big)$ up to polylog factors, while realizable learning is governed by the Effective width via a sharp trichotomy of $\Theta(1)$, $\Theta(\sqrt{T})$, or $\Theta(T)$ depending on $W(\mathcal{H})$. The paper introduces AL trees and the EXP4.AT algorithm to achieve the stated bounds, and it provides both upper and lower bounds that are tight up to constants or polylog factors. These results advance understanding of partial feedback in online learning and connect classic dimensions to minimax performance with asymmetric feedback. The work also outlines open questions about determinism, log-factor removal, and broader partial-feedback settings.
Abstract
In online binary classification under \emph{apple tasting} feedback, the learner only observes the true label if it predicts ``1". First studied by \cite{helmbold2000apple}, we revisit this classical partial-feedback setting and study online learnability from a combinatorial perspective. We show that the Littlestone dimension continues to provide a tight quantitative characterization of apple tasting in the agnostic setting, closing an open question posed by \cite{helmbold2000apple}. In addition, we give a new combinatorial parameter, called the Effective width, that tightly quantifies the minimax expected mistakes in the realizable setting. As a corollary, we use the Effective width to establish a \emph{trichotomy} of the minimax expected number of mistakes in the realizable setting. In particular, we show that in the realizable setting, the expected number of mistakes of any learner, under apple tasting feedback, can be $Θ(1), Θ(\sqrt{T})$, or $Θ(T)$. This is in contrast to the full-information realizable setting where only $Θ(1)$ and $Θ(T)$ are possible.