{"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/non-asymptotic-convergence-bounds-for-1","title":"Non-asymptotic convergence bounds for modified tamed unadjusted Langevin algorithm in non-convex setting","arxiv_id":"2207.02600","date":"2022-07-06","proceeding":null,"authors":["Ariel Neufeld","Matthew Ng Cheng En","Ying Zhang"],"abstract":"We consider the problem of sampling from a high-dimensional target distribution $\\pi_\\beta$ on $\\mathbb{R}^d$ with density proportional to $\\theta\\mapsto e^{-\\beta U(\\theta)}$ using explicit numerical schemes based on discretising the Langevin stochastic differential equation (SDE). In recent literature, taming has been proposed and studied as a method for ensuring stability of Langevin-based numerical schemes in the case of super-linearly growing drift coefficients for the Langevin SDE. In particular, the Tamed Unadjusted Langevin Algorithm (TULA) was proposed in [Bro+19] to sample from such target distributions with the gradient of the potential $U$ being super-linearly growing. However, theoretical guarantees in Wasserstein distances for Langevin-based algorithms have traditionally been derived assuming strong convexity of the potential $U$. In this paper, we propose a novel taming factor and derive, under a setting with possibly non-convex potential $U$ and super-linearly growing gradient of $U$, non-asymptotic theoretical bounds in Wasserstein-1 and Wasserstein-2 distances between the law of our algorithm, which we name the modified Tamed Unadjusted Langevin Algorithm (mTULA), and the target distribution $\\pi_\\beta$. We obtain respective rates of convergence $\\mathcal{O}(\\lambda)$ and $\\mathcal{O}(\\lambda^{1/2})$ in Wasserstein-1 and Wasserstein-2 distances for the discretisation error of mTULA in step size $\\lambda$. High-dimensional numerical simulations which support our theoretical findings are presented to showcase the applicability of our algorithm.","url_abs":"https://arxiv.org/abs/2207.02600v2","url_pdf":"https://arxiv.org/pdf/2207.02600v2.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"non-asymptotic-convergence-bounds-for-1","repo_url":"https://github.com/ng-cheng-en-matthew/mtula-simulations","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2207.02600","atlas_url":"https://app.syntology.ai/?focus=2207.02600","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}