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Gradient descent is a method for unconstrained mathematical optimization. It is a first-order iterative algorithm for minimizing a differentiable multivariate function . 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 ...
The method of steepest descent is a method to approximate a complex integral of the form = () for large , where () and () are analytic functions of . Because the integrand is analytic, the contour C {\displaystyle C} can be deformed into a new contour C ′ {\displaystyle C'} without changing the integral.
Computing by an iterative method, such as line search defines a descent direction at the th iterate to be any such that , <, where , denotes the inner product. The motivation for such an approach is that small steps along p k {\displaystyle \mathbf {p} _{k}} guarantee that f {\displaystyle \displaystyle f} is reduced, by Taylor's theorem .
Coordinate descent is an optimization algorithm that successively minimizes along coordinate directions to find the minimum of a function.At each iteration, the algorithm determines a coordinate or coordinate block via a coordinate selection rule, then exactly or inexactly minimizes over the corresponding coordinate hyperplane while fixing all other coordinates or coordinate blocks.
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction [1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. [2]
Vs. the locally optimal steepest descent method [ edit ] In both the original and the preconditioned conjugate gradient methods one only needs to set β k := 0 {\displaystyle \beta _{k}:=0} in order to make them locally optimal, using the line search , steepest descent methods.
Stochastic gradient descent competes with the L-BFGS algorithm, [citation needed] which is also widely used. Stochastic gradient descent has been used since at least 1960 for training linear regression models, originally under the name ADALINE. [25] Another stochastic gradient descent algorithm is the least mean squares (LMS) adaptive filter.
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.