{"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/implicit-regularization-of-discrete-gradient","title":"Implicit Regularization of Discrete Gradient Dynamics in Linear Neural Networks","arxiv_id":"1904.13262","date":"2019-04-30","proceeding":"NeurIPS 2019 12","authors":["Gauthier Gidel","Francis Bach","Simon Lacoste-Julien"],"abstract":"When optimizing over-parameterized models, such as deep neural networks, a large set of parameters can achieve zero training error. In such cases, the choice of the optimization algorithm and its respective hyper-parameters introduces biases that will lead to convergence to specific minimizers of the objective. Consequently, this choice can be considered as an implicit regularization for the training of over-parametrized models. In this work, we push this idea further by studying the discrete gradient dynamics of the training of a two-layer linear network with the least-squares loss. Using a time rescaling, we show that, with a vanishing initialization and a small enough step size, this dynamics sequentially learns the solutions of a reduced-rank regression with a gradually increasing rank.","url_abs":"https://arxiv.org/abs/1904.13262v2","url_pdf":"https://arxiv.org/pdf/1904.13262v2.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":"implicit-regularization-of-discrete-gradient","repo_url":"https://github.com/GauthierGidel/Implicit-Regularization-of-Discrete-Gradient-Dynamics-in-Linear-Neural-Networks","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1904.13262","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}