{"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/geometry-of-the-loss-landscape-in","title":"Geometry of the Loss Landscape in Overparameterized Neural Networks: Symmetries and Invariances","arxiv_id":"2105.12221","date":"2021-05-25","proceeding":null,"authors":["Berfin Şimşek","François Ged","Arthur Jacot","Francesco Spadaro","Clément Hongler","Wulfram Gerstner","Johanni Brea"],"abstract":"We study how permutation symmetries in overparameterized multi-layer neural networks generate `symmetry-induced' critical points. Assuming a network with $ L $ layers of minimal widths $ r_1^*, \\ldots, r_{L-1}^* $ reaches a zero-loss minimum at $ r_1^*! \\cdots r_{L-1}^*! $ isolated points that are permutations of one another, we show that adding one extra neuron to each layer is sufficient to connect all these previously discrete minima into a single manifold. For a two-layer overparameterized network of width $ r^*+ h =: m $ we explicitly describe the manifold of global minima: it consists of $ T(r^*, m) $ affine subspaces of dimension at least $ h $ that are connected to one another. For a network of width $m$, we identify the number $G(r,m)$ of affine subspaces containing only symmetry-induced critical points that are related to the critical points of a smaller network of width $r<r^*$. Via a combinatorial analysis, we derive closed-form formulas for $ T $ and $ G $ and show that the number of symmetry-induced critical subspaces dominates the number of affine subspaces forming the global minima manifold in the mildly overparameterized regime (small $ h $) and vice versa in the vastly overparameterized regime ($h \\gg r^*$). Our results provide new insights into the minimization of the non-convex loss function of overparameterized neural networks.","url_abs":"https://arxiv.org/abs/2105.12221v2","url_pdf":"https://arxiv.org/pdf/2105.12221v2.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":"geometry-of-the-loss-landscape-in","repo_url":"https://github.com/jbrea/symmetrysaddles.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2105.12221","atlas_url":"https://app.syntology.ai/?focus=2105.12221","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}