{"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-analysis-of-the-dynamics-of-a","title":"Convergence Analysis of the Dynamics of a Special Kind of Two-Layered Neural Networks with $\\ell_1$ and $\\ell_2$ Regularization","arxiv_id":"1711.07005","date":"2017-11-19","proceeding":null,"authors":["Zhifeng Kong"],"abstract":"In this paper, we made an extension to the convergence analysis of the\ndynamics of two-layered bias-free networks with one $ReLU$ output. We took into\nconsideration two popular regularization terms: the $\\ell_1$ and $\\ell_2$ norm\nof the parameter vector $w$, and added it to the square loss function with\ncoefficient $\\lambda/2$. We proved that when $\\lambda$ is small, the weight\nvector $w$ converges to the optimal solution $\\hat{w}$ (with respect to the new\nloss function) with probability $\\geq (1-\\varepsilon)(1-A_d)/2$ under random\ninitiations in a sphere centered at the origin, where $\\varepsilon$ is a small\nvalue and $A_d$ is a constant. Numerical experiments including phase diagrams\nand repeated simulations verified our theory.","url_abs":"http://arxiv.org/abs/1711.07005v1","url_pdf":"http://arxiv.org/pdf/1711.07005v1.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":"convergence-analysis-of-the-dynamics-of-a","repo_url":"https://github.com/FengNiMa/ReLU_Convergence","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","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}