{"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-modification-of-quasi-newton-s-methods","title":"A fast and simple modification of Newton's method helping to avoid saddle points","arxiv_id":"2006.01512","date":"2020-06-02","proceeding":null,"authors":["Tuyen Trung Truong","Tat Dat To","Tuan Hang Nguyen","Thu Hang Nguyen","Hoang Phuong Nguyen","Maged Helmy"],"abstract":"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 $A_n$. The main result of this paper roughly says that if $f$ is $C^3$ (can be unbounded from below) and a sequence $\\{x_n\\}$, constructed by the New Q-Newton's method from a random initial point $x_0$, {\\bf converges}, then the limit point is a critical point and is not a saddle point, and the convergence rate is the same as that of Newton's method. The first author has recently been successful incorporating Backtracking line search to New Q-Newton's method, thus resolving the convergence guarantee issue observed for some (non-smooth) cost functions. An application to quickly finding zeros of a univariate meromorphic function will be discussed. Various experiments are performed, against well known algorithms such as BFGS and Adaptive Cubic Regularization are presented.","url_abs":"https://arxiv.org/abs/2006.01512v4","url_pdf":"https://arxiv.org/pdf/2006.01512v4.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-modification-of-quasi-newton-s-methods","repo_url":"https://github.com/hphuongdhsp/Q-Newton-method","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"protein-folding","task_name":"Protein Folding"},{"task_slug":"stochastic-optimization","task_name":"Stochastic Optimization"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}