{"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/gradient-flow-dynamics-of-shallow-relu","title":"Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs","arxiv_id":"2206.00939","date":"2022-06-02","proceeding":null,"authors":["Etienne Boursier","Loucas Pillaud-Vivien","Nicolas Flammarion"],"abstract":"The training of neural networks by gradient descent methods is a cornerstone of the deep learning revolution. Yet, despite some recent progress, a complete theory explaining its success is still missing. This article presents, for orthogonal input vectors, a precise description of the gradient flow dynamics of training one-hidden layer ReLU neural networks for the mean squared error at small initialisation. In this setting, despite non-convexity, we show that the gradient flow converges to zero loss and characterise its implicit bias towards minimum variation norm. Furthermore, some interesting phenomena are highlighted: a quantitative description of the initial alignment phenomenon and a proof that the process follows a specific saddle to saddle dynamics.","url_abs":"https://arxiv.org/abs/2206.00939v2","url_pdf":"https://arxiv.org/pdf/2206.00939v2.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":"gradient-flow-dynamics-of-shallow-relu","repo_url":"https://github.com/eboursier/gfdynamics","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/2206.00939","atlas_url":"https://app.syntology.ai/?focus=2206.00939","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}