{"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/an-analytical-formula-of-population-gradient","title":"An Analytical Formula of Population Gradient for two-layered ReLU network and its Applications in Convergence and Critical Point Analysis","arxiv_id":"1703.00560","date":"2017-03-02","proceeding":"ICML 2017 8","authors":["Yuandong Tian"],"abstract":"In this paper, we explore theoretical properties of training a two-layered\nReLU network $g(\\mathbf{x}; \\mathbf{w}) = \\sum_{j=1}^K\n\\sigma(\\mathbf{w}_j^T\\mathbf{x})$ with centered $d$-dimensional spherical\nGaussian input $\\mathbf{x}$ ($\\sigma$=ReLU). We train our network with gradient\ndescent on $\\mathbf{w}$ to mimic the output of a teacher network with the same\narchitecture and fixed parameters $\\mathbf{w}^*$. We show that its population\ngradient has an analytical formula, leading to interesting theoretical analysis\nof critical points and convergence behaviors. First, we prove that critical\npoints outside the hyperplane spanned by the teacher parameters\n(\"out-of-plane\") are not isolated and form manifolds, and characterize in-plane\ncritical-point-free regions for two ReLU case. On the other hand, convergence\nto $\\mathbf{w}^*$ for one ReLU node is guaranteed with at least\n$(1-\\epsilon)/2$ probability, if weights are initialized randomly with standard\ndeviation upper-bounded by $O(\\epsilon/\\sqrt{d})$, consistent with empirical\npractice. For network with many ReLU nodes, we prove that an infinitesimal\nperturbation of weight initialization results in convergence towards\n$\\mathbf{w}^*$ (or its permutation), a phenomenon known as spontaneous\nsymmetric-breaking (SSB) in physics. We assume no independence of ReLU\nactivations. Simulation verifies our findings.","url_abs":"http://arxiv.org/abs/1703.00560v2","url_pdf":"http://arxiv.org/pdf/1703.00560v2.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":"an-analytical-formula-of-population-gradient","repo_url":"https://github.com/yuandong-tian/ICML17_ReLU","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}},{"paper_slug":"an-analytical-formula-of-population-gradient","repo_url":"https://github.com/MindSpore-scientific/code-14/tree/main/Two-Layer-ReLU-Network-Analytically","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"mindspore","reach":null}],"tasks":[],"methods":[{"method_slug":"relu","method_name":"ReLU"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1703.00560","atlas_url":"https://app.syntology.ai/?focus=1703.00560","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}