Generalisations and improvements of New Q-Newton's method Backtracking

TL;DR

This paper introduces a flexible and generalized framework for the New Q-Newton's method Backtracking algorithm, enhancing its adaptability and theoretical guarantees for optimization and solving polynomial systems.

Contribution

It extends the original method by allowing more general parameter choices and basis selections, broadening its applicability and theoretical robustness.

Findings

The generalized method maintains convergence properties.

Application demonstrated in solving polynomial systems.

Enhanced flexibility improves practical performance.

Abstract

In this paper, we propose a general framework for the algorithm New Q-Newton's method Backtracking, developed in the author's previous work. For a symmetric, square real matrix $A$, we define $minsp(A):=\min _{||e||=1} ||Ae||$. Given a $C^2$ cost function $f:\mathbb{R}^m\rightarrow \mathbb{R}$ and a real number $0<\tau $, as well as $m+1$ fixed real numbers $\delta _0,\ldots ,\delta _m$, we define for each $x\in \mathbb{R}^m$ with $\nabla f(x)\not= 0$ the following quantities: $\kappa :=\min _{i\not= j}|\delta _i-\delta _j|$; $A(x):=\nabla ^2f(x)+\delta ||\nabla f(x)||^{\tau}Id$, where $\delta$ is the first element in the sequence $\{\delta _0,\ldots ,\delta _m\}$ for which $minsp(A(x))\geq \kappa ||\nabla f(x)||^{\tau}$; $e_1(x),\ldots ,e_m(x)$ are an orthonormal basis of $\mathbb{R}^m$, chosen appropriately; $w(x)=$ the step direction, given by the formula: $$w(x)=\sum _{i=1}^m\frac{<\nabla f(x),e_i(x)>}{||A(x)e_i(x)||}e_i(x);$$ (we can also normalise by $w(x)/\max \{1,||w(x)||\}$ when needed) $\gamma (x)>0$ learning rate chosen by Backtracking line search so that Armijo's condition is satisfied: $$f(x-\gamma (x)w(x))-f(x)\leq -\frac{1}{3}\gamma (x)<\nabla f(x),w(x)>.$$ The update rule for our algorithm is $x\mapsto H(x)=x-\gamma (x)w(x)$. In New Q-Newton's method Backtracking, the choices are $\tau =1+\alpha >1$ and $e_1(x),\ldots ,e_m(x)$'s are eigenvectors of $\nabla ^2f(x)$. In this paper, we allow more flexibility and generality, for example $\tau$ can be chosen to be $<1$ or $e_1(x),\ldots ,e_m(x)$'s are not necessarily eigenvectors of $\nabla ^2f(x)$. New Q-Newton's method Backtracking (as well as Backtracking gradient descent) is a special case, and some versions have flavours of quasi-Newton's methods. Several versions allow good theoretical guarantees. An application to solving systems of polynomial equations is given.