{"paper":{"title":"A fast and simple modification of Newton's method helping to avoid saddle points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA","math.DS","math.NA","stat.ML"],"primary_cat":"math.OC","authors_text":"Hoang Phuong Nguyen, Maged Helmy, Tat Dat To, Thu Hang Nguyen, Tuan Hang Nguyen, Tuyen Trung Truong","submitted_at":"2020-06-02T10:38:00Z","abstract_excerpt":"We propose in this paper New Q-Newton's method. The update rule is very simple conceptually, for example $x_{n+1}=x_n-w_n$ where $w_n=pr_{A_n,+}(v_n)-pr_{A_n,-}(v_n)$, with $A_n=\\nabla ^2f(x_n)+\\delta _n||\\nabla f(x_n)||^2.Id$ and $v_n=A_n^{-1}.\\nabla f(x_n)$. Here $\\delta _n$ is an appropriate real number so that $A_n$ is invertible, and $pr_{A_n,\\pm}$ are projections to the vector subspaces generated by eigenvectors of positive (correspondingly negative) eigenvalues of $A_n$.\n  The main result of this paper roughly says that if $f$ is $C^3$ (can be unbounded from below) and a sequence $\\{x_n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.01512","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2006.01512/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}