{"paper":{"title":"Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","stat.ML"],"primary_cat":"cs.DS","authors_text":"Santosh S. Vempala, Zongchen Chen","submitted_at":"2019-05-07T01:20:59Z","abstract_excerpt":"We study Hamiltonian Monte Carlo (HMC) for sampling from a strongly logconcave density proportional to $e^{-f}$ where $f:\\mathbb{R}^d \\to \\mathbb{R}$ is $\\mu$-strongly convex and $L$-smooth (the condition number is $\\kappa = L/\\mu$). We show that the relaxation time (inverse of the spectral gap) of ideal HMC is $O(\\kappa)$, improving on the previous best bound of $O(\\kappa^{1.5})$; we complement this with an example where the relaxation time is $\\Omega(\\kappa)$. When implemented using a nearly optimal ODE solver, HMC returns an $\\varepsilon$-approximate point in $2$-Wasserstein distance using "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.02313","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"}