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Life sciences · Preprint
arXiv · September 9, 2026
Posted before peer review. The findings may change or fail to hold.
This is a theoretical mathematics preprint that proves the gap-entropy conjecture for a specific class of best-arm identification problems with Gaussian arms. The work establishes tight bounds on the expected sample complexity in terms of a quantity involving the arm gaps and an entropy term, but has not yet been peer reviewed and carries no clinical, biological, or empirical evidence.
Preprint.
Proved that optimal expected sample complexity is within absolute constant factors of H(log(1/δ)+Ent(I)) across all permutations of arm labels Constructed an algorithm whose expected sample count is bounded by a constant multiple of H(log(1/δ)+Ent(I)) plus g^{−2}log log(e^e/g)
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This is a theoretical mathematics preprint proving a conjecture in optimal stopping and information theory; it has not undergone peer review and does not involve clinical, biological, or empirical data.
Quoted from the source exactly as published.
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We prove the gap-entropy conjecture for fixed-confidence best-arm identification with independent unit-variance Gaussian arms, means in $[0,1]$, and a unique optimal arm. For each suboptimal arm $i$, let $Δ_i=μ_*-μ_i$ be its gap from the optimal mean, and write $H=\sum_{i\ne *}Δ_i^{-2}$. Let $p_r$ be the fraction of $H$ contributed by arms with $2^{-(r+1)}<Δ_i\le2^{-r}$, and let $\mathrm{Ent}(I)=\sum_{r:p_r>0} p_r\log(1/p_r)$. Among all algorithms that identify the optimal arm with probability at least $1-δ$ on every Gaussian instance, the optimal expected number of samples on a given instance, averaged over all permutations of the arm labels, is within absolute constant factors of $H(\log(1/δ)+\mathrm{Ent}(I))$. Moreover, there is an algorithm, independent of the instance, whose expected number of samples is bounded by a constant multiple of this quantity plus $g^{-2}\log\log(e^e/g)$, where $g=\min_{i\ne *}Δ_i$ is the gap to the closest competitor.
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