gap-entropy 猜想的正面解决

Computer Science > Machine Learning arXiv:2609.10529v1 (cs) [Submitted on 9 Sep 2026] Title:A positive resolution of the gap-entropy conjecture Authors:P. M. Aronow, Nathan Kallus, Patrick Lopatto View a PDF of the paper titled A positive resolution of the gap-entropy conjecture, by P. M. Aronow and 2 other authors View PDF HTML (experimental) Abstract: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 $\Delta_i=\mu_*-\mu_i$ be its gap from the optimal mean, and write $H=\sum_{i\ne *}\Delta_i^{-2}$. Let $p_r$ be the fraction of $H$ contributed by arms with $2^{-(r+1)}0} p_r\log(1/p_r)$. Among all algorithms that identify the optimal arm with probability at least $1-\delta$ 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 $H(\log(1/\delta)+\mathrm{Ent}(I))$. 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 $g=\min_{i\ne *}\Delta_i$ is the gap to the closest competitor. Subjects: Machine Learning (cs.LG); Machine Learning (stat.ML) Cite as: arXiv:2609.10529 [cs.LG] (or arXiv:2609.10529v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2609.10529 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Nathan Kallus [view email] [v1] Wed, 9 Sep 2026 17:57:11 UTC (39 KB) Full-text links: Access Paper: View a PDF of the paper titled A positive resolution of the gap-entropy conjecture, by P. M. Aronow and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG new | recent | 2026-09 Change to browse by: cs stat stat.ML References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) IArxiv recommender toggle IArxiv Recommender (What is IArxiv?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)