{"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/convergence-of-langevin-simulated-annealing","title":"Convergence of Langevin-Simulated Annealing algorithms with multiplicative noise","arxiv_id":"2109.11669","date":"2021-09-23","proceeding":null,"authors":["Pierre Bras","Gilles Pagès"],"abstract":"We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for $V : \\mathbb{R}^d \\to \\mathbb{R}$ a potential function to minimize, we consider the stochastic equation $dY_t = - \\sigma \\sigma^\\top \\nabla V(Y_t) dt + a(t)\\sigma(Y_t)dW_t + a(t)^2\\Upsilon(Y_t)dt$, where $(W_t)$ is a Brownian motion, where $\\sigma : \\mathbb{R}^d \\to \\mathcal{M}_d(\\mathbb{R})$ is an adaptive (multiplicative) noise, where $a : \\mathbb{R}^+ \\to \\mathbb{R}^+$ is a function decreasing to $0$ and where $\\Upsilon$ is a correction term. This setting can be applied to optimization problems arising in Machine Learning. The case where $\\sigma$ is a constant matrix has been extensively studied however little attention has been paid to the general case. We prove the convergence for the $L^1$-Wasserstein distance of $Y_t$ and of the associated Euler-scheme $\\bar{Y}_t$ to some measure $\\nu^\\star$ which is supported by $\\text{argmin}(V)$ and give rates of convergence to the instantaneous Gibbs measure $\\nu_{a(t)}$ of density $\\propto \\exp(-2V(x)/a(t)^2)$. To do so, we first consider the case where $a$ is a piecewise constant function. We find again the classical schedule $a(t) = A\\log^{-1/2}(t)$. We then prove the convergence for the general case by giving bounds for the Wasserstein distance to the stepwise constant case using ergodicity properties.","url_abs":"https://arxiv.org/abs/2109.11669v2","url_pdf":"https://arxiv.org/pdf/2109.11669v2.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":"convergence-of-langevin-simulated-annealing","repo_url":"https://github.com/bras-p/langevin-simulated-annealing","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"tf","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}