AI サービスが一時的に利用できないため、復旧後に翻訳を補完します。
[Submitted on 12 Sep 2026] Title:Prior-Free Competitive Ratios for Improving Bandits: Scale, Curvature and Horizon Are Free, but Not Jointly Under Noise View a PDF of the paper titled Prior-Free Competitive Ratios for Improving Bandits: Scale, Curvature and Horizon Are Free, but Not Jointly Under Noise, by Xuan Li View PDF HTML (experimental) Abstract:In the improving multi-armed bandits problem, each of $k$ arms has an unknown nondecreasing, discretely concave reward curve $f_i$, and pulling arm $i$ for the $t$-th time yields $f_i(t)$. For sufficiently long horizons, Blum and Ravichandran (ALT 2025) proved that randomized algorithms achieve an $O(\sqrt k)$ approximation to the best single arm when the scale $m=f^*(T)$ of the optimal arm is known ($T\ge2k$), and $O(\sqrt k\log k)$ when it is not ($T>4k$), against an $\Omega(\sqrt k)$ lower bound. The logarithmic factor is unnecessary: a one-page \emph{probe-and-commit} algorithm achieves competitive ratio $4\sqrt3\,\sqrt k$ for $T\ge2\lfloor\sqrt k\rfloor$, without any knowledge of the scale, and we determine the optimal ratio for every horizon, $\Theta(\sqrt k+k/T)$, also for unknown horizons. Without noise, \emph{no prior is needed at all}: a random-marginal probing algorithm reading neither the scale $m$, nor the concavity-envelope exponent $\beta$ of Blum, Garicano, Ravichandran and Sharma (UAI 2026), nor the horizon $T$, achieves the optimal $\Theta(k^{\beta/(1+\beta)}+k/T)$ simultaneously for every $\beta$ and every horizon. Under the multiplicative noise model of Blum and Ravichandran, probe-and-commit keeps the same all-horizon order $\Theta(\sqrt k+k/T)$ without knowing the noise level (and $\Theta(\sqrt k)$ on the same range), but the price of priors jumps: for any fixed noise level $\varepsilon\in(0,1/2]$, the uniform price of adaptation $\phi_\varepsilon(k)$ --- the worst case over horizons $T\ge16k$ of the loss relative to $k^{\beta/(1+\beta)}$ for algorithms knowing neither $m$ nor $\beta$ --- is $\Theta_\varepsilon(\sqrt{\log k/\log\log k})$, the lower bound asymptotic in $k$ at fixed positive $\varepsilon$ and matched by a nested random-permutation probing algorithm, whereas knowing either $m$ or $\beta$ alone restores a constant price. Subjects: Machine Learning (cs.LG) Cite as: arXiv:2609.17595 [cs.LG] (or arXiv:2609.17595v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2609.17595 arXiv-issued DOI via DataCite Submission history From: Xuan Li [view email] [v1] Sat, 12 Sep 2026 11:38:57 UTC (39 KB) Full-text links: Access Paper: View a PDF of the paper titled Prior-Free Competitive Ratios for Improving Bandits: Scale, Curvature and Horizon Are Free, but Not Jointly Under Noise, by Xuan Li View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG new | recent | 2026-09 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar Loading... Data provided by: 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?)