{"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/who-is-afraid-of-big-bad-minima-analysis-of-1","title":"Who is Afraid of Big Bad Minima? Analysis of gradient-flow in spiked matrix-tensor models","arxiv_id":null,"date":"2019-12-01","proceeding":"NeurIPS 2019 12","authors":["Stefano Sarao Mannelli","Giulio Biroli","Chiara Cammarota","Florent Krzakala","Lenka Zdeborová"],"abstract":"Gradient-based algorithms are effective for many machine learning tasks, but despite ample recent effort and some progress, it often remains unclear why they work in practice in optimising high-dimensional non-convex functions and why they find good minima instead of being trapped in spurious ones.Here we present a quantitative theory explaining this behaviour in a spiked matrix-tensor model.Our framework is based on the Kac-Rice analysis of stationary points and a closed-form analysis of  gradient-flow originating from statistical physics. We show that there is a well defined region of parameters where the gradient-flow algorithm finds a good global minimum despite the presence of exponentially many spurious local minima.\nWe show that this is achieved by surfing on saddles that have strong negative direction towards the global minima, a phenomenon that is connected to a BBP-type threshold in the Hessian describing the critical points of the landscapes.","url_abs":"http://papers.nips.cc/paper/9073-who-is-afraid-of-big-bad-minima-analysis-of-gradient-flow-in-spiked-matrix-tensor-models","url_pdf":"http://papers.nips.cc/paper/9073-who-is-afraid-of-big-bad-minima-analysis-of-gradient-flow-in-spiked-matrix-tensor-models.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":"who-is-afraid-of-big-bad-minima-analysis-of-1","repo_url":"https://github.com/sphinxteam/spiked_matrix-tensor_T0","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}