GradientVerfahren

GradientVerfahren, also known as the gradient descent method, is a fundamental iterative optimization algorithm used to find a local minimum of a differentiable function. The core idea is to repeatedly take steps in the direction of the steepest descent of the function. This direction is determined by the negative of the gradient of the function at the current point. The gradient is a vector that points in the direction of the greatest rate of increase of the function. Therefore, moving in the opposite direction of the gradient leads to a decrease in the function's value.

The algorithm starts with an initial guess for the minimum. In each iteration, it calculates the gradient of the function at the current point and updates the point by subtracting a fraction of the gradient. This fraction is known as the learning rate, and it controls the size of the steps taken. A small learning rate can lead to slow convergence, while a large learning rate might cause the algorithm to overshoot the minimum or even diverge. The process continues until a stopping criterion is met, such as when the magnitude of the gradient is sufficiently small or when the change in the function value between iterations is negligible.

GradientVerfahren is widely applied in various fields, including machine learning for training models, signal processing, and economics. Its simplicity and effectiveness make it a cornerstone of many optimization problems, although it can get stuck in local minima for non-convex functions. Variations like stochastic gradient descent and mini-batch gradient descent have been developed to address challenges related to large datasets and computational efficiency.