The Complete Phase Diagram of Deterministic Apple Tasting with Expert Advice
Year: 2026
Type: preprint
Abstract: In prediction with expert advice under apple tasting feedback, the learner observes the true label only in rounds where it predicts $1$. For deterministic learners and $n$ experts over $T$ rounds, Chase and Mehalel (COLT 2025) determined the optimal realizable mistake bound $\Theta(\sqrt{T\log n})$ for $T\le n\le 2^T$, and stated as an open problem the regime of "slightly more than $\sqrt T$ many experts", namely $n=T^{1/2+\epsilon}$ with $\epsilon=o(1)$. We close this regime: for all $\sqrt T\le n\le T$ the optimal deterministic mistake bound is $\Theta(\sqrt{T\log_+(n/\sqrt T)})$, where $\log_+ x=\max{1,\ln x}$. For example, with $n=\lceil\sqrt T\ln T\rceil$ experts the answer is $\Theta(\sqrt{T\ln\ln T})$. The lower bound is a single self-contained "lowest-load committee" argument which also re-derives, with explicit constants, the known $\Omega(\sqrt{T\log n})$ bound for $n\ge T$. We then extend the picture to the $k$-realizable (agnostic-budget) setting and prove a lower bound of $\Omega(\sqrt{T(k+\log_+(n/\sqrt T))})$ throughout $\sqrt T\le n\le 2^T$, which is tight in the critical band $\sqrt T\le n\le T$ by a tuning of a lemma of Chase and Mehalel; for $n\ge T$ this recovers their known $\Theta(\sqrt{T(k+\log n)})$ and contains no new content. Along the way we isolate an expert-budget threshold phenomenon: with two experts $M^(2,k,T)=\Theta(k+1)$ regardless of $T$, and forcing the $\sqrt{Tk}$ term requires $n\asymp\sqrt{T/(k+1)}$ experts. Combining all regimes yields, for every $n\ge 2$, $T\ge 2$ and $0\le k\le T$: $M^(n,k,T)=\Theta(\min{T,\ (n-1)(k+1)+k,\ \sqrt{T(k+\log_+(n/\sqrt T))}})$.
Author: Zhang, Guangjian
Institution: UNSW Sydney
Language: English
Field: Decision Sciences
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7204949749
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