{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/does-hamiltonian-monte-carlo-mix-faster-than","title":"Does Hamiltonian Monte Carlo mix faster than a random walk on multimodal densities?","arxiv_id":"1808.03230","date":"2018-08-09","proceeding":null,"authors":["Oren Mangoubi","Natesh S. Pillai","Aaron Smith"],"abstract":"Hamiltonian Monte Carlo (HMC) is a very popular and generic collection of\nMarkov chain Monte Carlo (MCMC) algorithms. One explanation for the popularity\nof HMC algorithms is their excellent performance as the dimension $d$ of the\ntarget becomes large: under conditions that are satisfied for many common\nstatistical models, optimally-tuned HMC algorithms have a running time that\nscales like $d^{0.25}$. In stark contrast, the running time of the usual\nRandom-Walk Metropolis (RWM) algorithm, optimally tuned, scales like $d$. This\nsuperior scaling of the HMC algorithm with dimension is attributed to the fact\nthat it, unlike RWM, incorporates the gradient information in the proposal\ndistribution. In this paper, we investigate a different scaling question: does\nHMC beat RWM for highly $\\textit{multimodal}$ targets? We find that the answer\nis often $\\textit{no}$. We compute the spectral gaps for both the algorithms\nfor a specific class of multimodal target densities, and show that they are\nidentical. The key reason is that, within one mode, the gradient is effectively\nignorant about other modes, thus negating the advantage the HMC algorithm\nenjoys in unimodal targets. We also give heuristic arguments suggesting that\nthe above observation may hold quite generally. Our main tool for answering\nthis question is a novel simple formula for the conductance of HMC using\nLiouville's theorem. This result allows us to compute the spectral gap of HMC\nalgorithms, for both the classical HMC with isotropic momentum and the recent\nRiemannian HMC, for multimodal targets.","url_abs":"http://arxiv.org/abs/1808.03230v2","url_pdf":"http://arxiv.org/pdf/1808.03230v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"does-hamiltonian-monte-carlo-mix-faster-than","repo_url":"https://github.com/sir-deenicus/EvolutionaryBayes","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}