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The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a trajectory that maximizes that function; the procedure is then known as gradient ascent .
A comparison of gradient descent (green) and Newton's method (red) for minimizing a function (with small step sizes). Newton's method uses curvature information (i.e. the second derivative) to take a more direct route.
The Barzilai-Borwein method [1] is an iterative gradient descent method for unconstrained optimization using either of two step sizes derived from the linear trend of the most recent two iterates. This method, and modifications, are globally convergent under mild conditions, [ 2 ] [ 3 ] and perform competitively with conjugate gradient methods ...
For the case of a function with at most countably many critical points (such as a Morse function) and compact sublevels, as well as with Lipschitz continuous gradient where one uses standard GD with learning rate <1/L (see the section "Stochastic gradient descent"), then convergence is guaranteed, see for example Chapter 12 in Lange (2013 ...
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 ...
where is an approximation to the Hessian matrix at , which is updated iteratively at each stage, and () is the gradient of the function evaluated at x k. A line search in the direction p k is then used to find the next point x k +1 by minimizing f ( x k + γ p k ) {\displaystyle f(\mathbf {x} _{k}+\gamma \mathbf {p} _{k})} over the scalar γ ...
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 ]
In optimization, a gradient method is an algorithm to solve problems of the form min x ∈ R n f ( x ) {\displaystyle \min _{x\in \mathbb {R} ^{n}}\;f(x)} with the search directions defined by the gradient of the function at the current point.