A fast and simple modification of Newton's method helping to avoid saddle points

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 $w_n=p{r}_{A_n,+}(v_n)-p{r}_{A_n,-}(v_n)$, with $A_n={\nabla }^{2}f(x_n)+{\delta }_n||\nabla f(x_n)|{|}^2.Id$ 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 $p{r}_{A_n,\pm }$ 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.

Main part: 43 pages, Appendix: 21 pages. Some misprints corrected. Add comparison to a relevant circle of ideas by Goldfarb, Gould et al, and others. Add applications to finding roots of a univariate meromorphic function

References in corpus (8)

• A Literature Survey of Benchmark Functions For Global Optimization Problems
• Identifying and attacking the saddle point problem in high-dimensional non-convex optimization
• Convergence to minima for the continuous version of Backtracking Gradient Descent
• Some convergent results for Backtracking Gradient Descent method on Banach spaces
• Backtracking Gradient Descent allowing unbounded learning rates
• Asymptotic behaviour of learning rates in Armijo's condition
• Unconstrained optimisation on Riemannian manifolds
• Coordinate-wise Armijo's condition

Cited by in corpus (2)

• New Q-Newton's method meets Backtracking line search: good convergence guarantee, saddle points avoidance, quadratic rate of convergence, and easy implementation
• Generalisations and improvements of New Q-Newton's method Backtracking