18. Gradient Descent

GUIDE: Elementary Digital Filter Theory. Gradient Descent

It appears that you are using AdBlocking software. The cost of running this website is covered by advertisements. If you like it please feel free to a small amount of money to secure the future of this website.

<< Previous page  TOC  INDEX  Next page >>

Gradient Descent

Concavity is valuable in connection with the Gradient Method of minimizing $J({\ with respect to ${\.

Definition. The gradient of the error measure $J({\ is defined as the ${\ column vector

\

Definition. The Gradient Method (Cauchy) is defined as follows.

Given ${\, compute

\

where ${J^\ is the gradient of $J$ at ${\, and$t_n\ is chosen as the smallest nonnegative local minimizer of
\

Cauchy originally proposed to find the value of $t_n\ which gave a global minimum of $\. This, however, is not always feasible in practice.

Some general results regarding the Gradient Method are given below.

Theorem. If ${\ is a local minimizer of $J({\, and ${J^\ exists, then ${J^\.

Theorem. The gradient method is a descent method, i.e., $J({\.

Definition. $J:{\, ${\, is said to be in the class ${\ if all $k$th order partial derivatives of $J({\ with respect to the components of ${\ are continuous on ${\.

Definition. The Hessian ${J^{\of $J$ at ${\ is defined as the matrixof second-order partial derivatives,

\

where $\ denotes the $i$th component of $\, $i=1,\, and $[i,j]$ denotes the matrix entry at the $i$th row and $j$th column.

The Hessian of every element of ${\ is a symmetric matrix [Williamson et al. 1972]. This is because continuous second-order partials satisfy

\

Theorem. If $J\, then any cluster point ${\ of the gradient sequence ${\ is necessarily astationary point, i.e., ${J^\.

Theorem. Let $\ denote the concave hull of ${\. If $J\, and there exist positive constants $c$ and $C$ such that

\

for all ${\ and for all $\, then the gradient method beginning with any point in ${\ converges to a point ${\. Moreover, ${\ is the unique global minimizer of $J$ in $\.

By the norm equivalence theorem [Ortega 1972], Eq. (5) is satisfied whenever ${J^{\ is a norm on ${\ for each ${\. Since ${J^{\ belongs to ${\, it is a symmetric matrix. It is also bounded since it is continuous over a compact set. Thus a sufficient requirement is that ${J^{\ be positive definite on ${\. Positive definiteness of ${J^{\ can be viewed as ``positive curvature'' of $J$ at each point of ${\ which corresponds to strict concavity of $J$ on${\.

<< Previous page  TOC  INDEX  Next page >>

 

© 1998-2017 – Nicola Asuni - Tecnick.com - All rights reserved.
about - disclaimer - privacy