{"paper":{"title":"Manifold-Aware Information Gain and Lower Bounds for Gaussian-Process Bandits on Riemannian Quotient Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A regret lower bound for Gaussian-process bandits on Riemannian manifolds includes an explicit factor of the manifold volume raised to the power ν/(2ν+d).","cross_cats":[],"primary_cat":"eess.SP","authors_text":"Changsheng Chen, Ning Xie, Yuriy Dorn","submitted_at":"2026-05-13T13:38:29Z","abstract_excerpt":"We prove a regret lower bound for Gaussian-process bandits on a smooth compact Riemannian manifold $\\M$ of dimension $d$ with intrinsic Mat\\'ern-$\\nu$ kernel ($\\nu>d/2$) that exposes how the geometry of the arm space enters the constant. For any algorithm and time horizon $T$ exceeding an explicit threshold, the worst-case expected regret over the RKHS-ball $\\|f\\|_{\\Hil_{k_\\nu}}\\!\\le\\!B$ satisfies \\begin{multline*} \\E[R_T(f)]\\;\\ge\\;c_*(d,\\nu)\\,B^{d/(2\\nu+d)}\\,\\sigma_n^{2\\nu/(2\\nu+d)} \\\\ \\cdot\\,\\vol_g(\\M)^{\\nu/(2\\nu+d)}\\,T^{(\\nu+d)/(2\\nu+d)}(\\log T)^{\\nu/(2\\nu+d)}. \\end{multline*} The exponent "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For any algorithm and time horizon T exceeding an explicit threshold, the worst-case expected regret over the RKHS-ball ||f||_{H_{k_ν}} ≤ B satisfies E[R_T(f)] ≥ c_*(d,ν) B^{d/(2ν+d)} σ_n^{2ν/(2ν+d)} · vol_g(M)^{ν/(2ν+d)} T^{(ν+d)/(2ν+d)} (log T)^{ν/(2ν+d)}.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The unknown function f lies inside the RKHS ball of the intrinsic Matérn-ν kernel with ν > d/2 on a smooth compact Riemannian manifold; if the kernel does not faithfully encode the manifold geometry, the explicit volume factor may not hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Derives an explicit volume-dependent lower bound on regret for GP bandits on Riemannian manifolds that matches the exponent of known upper bounds and includes a new geometric constant.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A regret lower bound for Gaussian-process bandits on Riemannian manifolds includes an explicit factor of the manifold volume raised to the power ν/(2ν+d).","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"965ccf9eab722b7b214b45fa9c5e0bc127d67b4774bda56c845147ead1f3d7f3"},"source":{"id":"2605.13524","kind":"arxiv","version":1},"verdict":{"id":"b723c03b-5de7-4158-a7ca-74e0be36d54e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:07:02.440379Z","strongest_claim":"For any algorithm and time horizon T exceeding an explicit threshold, the worst-case expected regret over the RKHS-ball ||f||_{H_{k_ν}} ≤ B satisfies E[R_T(f)] ≥ c_*(d,ν) B^{d/(2ν+d)} σ_n^{2ν/(2ν+d)} · vol_g(M)^{ν/(2ν+d)} T^{(ν+d)/(2ν+d)} (log T)^{ν/(2ν+d)}.","one_line_summary":"Derives an explicit volume-dependent lower bound on regret for GP bandits on Riemannian manifolds that matches the exponent of known upper bounds and includes a new geometric constant.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The unknown function f lies inside the RKHS ball of the intrinsic Matérn-ν kernel with ν > d/2 on a smooth compact Riemannian manifold; if the kernel does not faithfully encode the manifold geometry, the explicit volume factor may not hold.","pith_extraction_headline":"A regret lower bound for Gaussian-process bandits on Riemannian manifolds includes an explicit factor of the manifold volume raised to the power ν/(2ν+d)."},"references":{"count":33,"sample":[{"doi":"","year":2021,"title":"On information gain and regret bounds in Gaussian process bandits,","work_id":"cc1df3cc-dff2-47ec-866b-9190118a6e50","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2014,"title":"Stochastic multi-armed-bandit prob- lem with non-stationary rewards,","work_id":"c5c63e94-d677-4a99-b6a3-1924a594b699","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"A domain-shrinking based Bayesian optimization algorithm with order-optimal regret performance,","work_id":"2239b39f-8a71-4c7e-bbaf-29559e7be180","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"Gaussian process optimization in the bandit setting: no regret and experimental design,","work_id":"001b24c9-34e0-4fcf-a957-0b5d1c17a7d7","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"Mat´ern Gaussian processes on Riemannian manifolds,","work_id":"b1bc2654-9ea1-4a50-aebc-144a6cae5f9c","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":33,"snapshot_sha256":"950fe74c2ebfaa6a071472f5b0d16588fdab9010ca8ffb76f0aa1ddb8df69aa3","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}