A line search is a crucial technique used in Optimierungsalgorithmen, particularly in iterative methods for minimizing or maximizing a function. The primary goal of a line search is to determine the best step size along a given direction in the search space, which leads to a more efficient convergence towards an optimale Lösung.
Mathematisch betrachtet, wenn man eine Funktion optimiert f(x), where x is a vector of parameters, the line search focuses on a specific direction d (which is typically derived from the gradient of the function). Starting from a current point xk, the line search seeks to minimize the function along the line defined by xk + αd, where α represents the step size. The optimal step size α is the value that minimizes f(xk + αd).
Es gibt verschiedene Methoden, um eine Linien-Suche durchzuführen, darunter:
- Exakte Linien-Suche: This method finds the step size α that exactly minimizes the function along the line. It can be computationally expensive, especially in high-dimensional spaces.
- Unexakte Linien-Suche: Instead of finding the exact minimum, this approach looks for a step size that sufficiently reduces the function value, often using criteria such as the Wolfe conditions or Armijo rule.
Die Linien-Suche wird häufig in Optimierungsalgorithmen wie Gradientenabstieg, Newton’s Method, and Conjugate Gradient. By effectively finding the right step size, line search helps improve the speed and efficiency of convergence towards an optimal solution, making it an essential tool in machine learning, künstliche Intelligenz, and various fields requiring optimization.