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Newton's method uses curvature information (i.e. the second derivative) to take a more direct route. In calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding the roots of a differentiable function, which are solutions to the equation =.
The line-search method first finds a descent direction along which the objective function will be reduced, and then computes a step size that determines how far should move along that direction. The descent direction can be computed by various methods, such as gradient descent or quasi-Newton method. The step size can be determined either ...
The properties of gradient descent depend on the properties of the objective function and the variant of gradient descent used (for example, if a line search step is used). The assumptions made affect the convergence rate, and other properties, that can be proven for gradient descent. [33]
L-BFGS shares many features with other quasi-Newton algorithms, but is very different in how the matrix-vector multiplication = is carried out, where is the approximate Newton's direction, is the current gradient, and is the inverse of the Hessian matrix. There are multiple published approaches using a history of updates to form this direction ...
In optimization, a gradient method is an algorithm to solve problems of the form with the search directions defined by the gradient of the function at the current point. Examples of gradient methods are the gradient descent and the conjugate gradient.
The learning rate and its adjustments may also differ per parameter, in which case it is a diagonal matrix that can be interpreted as an approximation to the inverse of the Hessian matrix in Newton's method. [5] The learning rate is related to the step length determined by inexact line search in quasi-Newton methods and related optimization ...
It is easy to find situations for which Newton's method oscillates endlessly between two distinct values. For example, for Newton's method as applied to a function f to oscillate between 0 and 1, it is only necessary that the tangent line to f at 0 intersects the x-axis at 1 and that the tangent line to f at 1 intersects the x-axis at 0. [19]
Numerous methods exist to compute descent directions, all with differing merits, such as gradient descent or the conjugate gradient method. More generally, if P {\displaystyle P} is a positive definite matrix, then p k = − P ∇ f ( x k ) {\displaystyle p_{k}=-P\nabla f(x_{k})} is a descent direction at x k {\displaystyle x_{k}} . [ 1 ]