Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs

[2206.00939] Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Statistics > Machine Learning arXiv:2206.00939 (stat) [Submitted on 2 Jun 2022 ( v1 ), last revised 14 Apr 2026 (this version, v3)] Title: Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs Authors: Etienne Boursier , Loucas Pillaud-Vivien , Nicolas Flammarion View a PDF of the paper titled Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs, by Etienne Boursier and Loucas Pillaud-Vivien and Nicolas Flammarion View PDF HTML (experimental) 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. Comments: corrected Proposition 1 Subjects: Machine Learning (stat.ML) ; Machine Learning (cs.LG) Cite as: arXiv:2206.00939 [stat.ML] (or arXiv:2206.00939v3 [stat.ML] for this version) https://doi.org/10.48550/arXiv.2206.00939 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Etienne Boursier [ view email ] [v1] Thu, 2 Jun 2022 09:01:25 UTC (1,871 KB) [v2] Mon, 31 Oct 2022 16:42:53 UTC (2,754 KB) [v3] Tue, 14 Apr 2026 08:05:07 UTC (2,756 KB) Full-text links: Access Paper: View a PDF of the paper titled Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs, by Etienne Boursier and Loucas Pillaud-Vivien and Nicolas Flammarion View PDF HTML (experimental) TeX Source view license Current browse context: stat.ML < prev | next > new | recent | 2022-06 Change to browse by: cs cs.LG stat References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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