{"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/computing-the-bias-of-constant-step","title":"Computing the Bias of Constant-step Stochastic Approximation with Markovian Noise","arxiv_id":"2405.14285","date":"2024-05-23","proceeding":null,"authors":["Sebastian Allmeier","Nicolas Gast"],"abstract":"We study stochastic approximation algorithms with Markovian noise and constant step-size $\\alpha$. We develop a method based on infinitesimal generator comparisons to study the bias of the algorithm, which is the expected difference between $\\theta_n$ -- the value at iteration $n$ -- and $\\theta^*$ -- the unique equilibrium of the corresponding ODE. We show that, under some smoothness conditions, this bias is of order $O(\\alpha)$. Furthermore, we show that the time-averaged bias is equal to $\\alpha V + O(\\alpha^2)$, where $V$ is a constant characterized by a Lyapunov equation, showing that $\\mathbb{E}[\\bar{\\theta}_n] \\approx \\theta^*+V\\alpha + O(\\alpha^2)$, where $\\bar{\\theta}_n=(1/n)\\sum_{k=1}^n\\theta_k$ is the Polyak-Ruppert average. We also show that $\\bar{\\theta}_n$ converges with high probability around $\\theta^*+\\alpha V$. We illustrate how to combine this with Richardson-Romberg extrapolation to derive an iterative scheme with a bias of order $O(\\alpha^2)$.","url_abs":"https://arxiv.org/abs/2405.14285v2","url_pdf":"https://arxiv.org/pdf/2405.14285v2.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":"computing-the-bias-of-constant-step","repo_url":"https://github.com/ngast/paper_bias_stochastic_approximation2024","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2405.14285","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}