{"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/a-principle-of-least-action-for-the-training","title":"A Principle of Least Action for the Training of Neural Networks","arxiv_id":"2009.08372","date":"2020-09-17","proceeding":null,"authors":["Skander Karkar","Ibrahim Ayed","Emmanuel de Bézenac","Patrick Gallinari"],"abstract":"Neural networks have been achieving high generalization performance on many tasks despite being highly over-parameterized. Since classical statistical learning theory struggles to explain this behavior, much effort has recently been focused on uncovering the mechanisms behind it, in the hope of developing a more adequate theoretical framework and having a better control over the trained models. In this work, we adopt an alternate perspective, viewing the neural network as a dynamical system displacing input particles over time. We conduct a series of experiments and, by analyzing the network's behavior through its displacements, we show the presence of a low kinetic energy displacement bias in the transport map of the network, and link this bias with generalization performance. From this observation, we reformulate the learning problem as follows: finding neural networks which solve the task while transporting the data as efficiently as possible. This offers a novel formulation of the learning problem which allows us to provide regularity results for the solution network, based on Optimal Transport theory. From a practical viewpoint, this allows us to propose a new learning algorithm, which automatically adapts to the complexity of the given task, and leads to networks with a high generalization ability even in low data regimes.","url_abs":"https://arxiv.org/abs/2009.08372v4","url_pdf":"https://arxiv.org/pdf/2009.08372v4.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":"a-principle-of-least-action-for-the-training","repo_url":"https://github.com/skander-karkar/LAP","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"pytorch","reach":null}],"tasks":[{"task_slug":"learning-theory","task_name":"Learning Theory"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2009.08372","atlas_url":"https://app.syntology.ai/?focus=2009.08372","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2009.08372"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-25T09:33:49+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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