Description of The Method
As in Newton's method, one uses a second order approximation to find the minimum of a function . The Taylor series of around an iterate is:
where is the gradient and an approximation to the Hessian matrix. The gradient of this approximation (with respect to ) is
and setting this gradient to zero provides the Newton step:
The Hessian approximation is chosen to satisfy
which is called the secant equation (the Taylor series of the gradient itself). In more than one dimension is under determined. In one dimension, solving for and applying the Newton's step with the updated value is equivalent to the secant method. The various quasi-Newton methods differ in their choice of the solution to the secant equation (in one dimension, all the variants are equivalent). Most methods (but with exceptions, such as Broyden's method) seek a symmetric solution ; furthermore, the variants listed below can be motivated by finding an update that is as close as possible to in some norm; that is, where is some positive definite matrix matrix that defines the norm. An approximate initial value of is often sufficient to achieve rapid convergence. The unknown is updated applying the Newton's step calculated using the current approximate Hessian matrix
- , with chosen to satisfy the Wolfe conditions;
- ;
- The gradient computed at the new point, and
is used to update the approximate Hessian, or directly its inverse using the Sherman-Morrison formula.
- A key property of the BFGS and DFP updates is that if is positive definite and is chosen to satisfy the Wolfe conditions then is also positive definite.
The most popular update formulas are:
| Method | ||
|---|---|---|
| DFP | ||
| BFGS |  |
|
| Broyden | ||
| Broyden family | ||
| SR1 |
Read more about this topic: Quasi-Newton Method
Famous quotes containing the words description of the, description of, description and/or method:
“The next Augustan age will dawn on the other side of the Atlantic. There will, perhaps, be a Thucydides at Boston, a Xenophon at New York, and, in time, a Virgil at Mexico, and a Newton at Peru. At last, some curious traveller from Lima will visit England and give a description of the ruins of St. Paul’s, like the editions of Balbec and Palmyra.”
—Horace Walpole (1717–1797)
“God damnit, why must all those journalists be such sticklers for detail? Why, they’d hold you to an accurate description of the first time you ever made love, expecting you to remember the color of the room and the shape of the windows.”
—Lyndon Baines Johnson (1908–1973)
“Why does philosophy use concepts and why does faith use symbols if both try to express the same ultimate? The answer, of course, is that the relation to the ultimate is not the same in each case. The philosophical relation is in principle a detached description of the basic structure in which the ultimate manifests itself. The relation of faith is in principle an involved expression of concern about the meaning of the ultimate for the faithful.”
—Paul Tillich (1886–1965)
“In child rearing it would unquestionably be easier if a child were to do something because we say so. The authoritarian method does expedite things, but it does not produce independent functioning. If a child has not mastered the underlying principles of human interactions and merely conforms out of coercion or conditioning, he has no tools to use, no resources to apply in the next situation that confronts him.”
—Elaine Heffner (20th century)