gradienttienlaskusmenetelmä

Gradienttienlaskusmenetelmä, known in English as the gradient descent method, is an iterative optimization algorithm used to find a local minimum of a differentiable function. It is a fundamental technique in machine learning and many other fields that deal with optimization problems. The core idea is to repeatedly move in the direction of the steepest descent, which is determined by the negative of the gradient of the function at the current point.

The process begins with an initial guess for the parameters of the function. In each iteration, the gradient of the function is calculated with respect to these parameters. The gradient indicates the direction of the greatest increase of the function. To find a minimum, we move in the opposite direction of the gradient, i.e., the direction of the steepest decrease. The size of the step taken in this direction is controlled by a learning rate, a hyperparameter that needs to be chosen carefully. A small learning rate may lead to slow convergence, while a large learning rate might cause the algorithm to overshoot the minimum or even diverge.

The algorithm continues until a convergence criterion is met, such as when the change in the function's value or the magnitude of the gradient becomes sufficiently small. While gradient descent guarantees convergence to a local minimum for convex functions, for non-convex functions, it may converge to a local minimum that is not the global minimum. Various extensions and variants of gradient descent, such as stochastic gradient descent and mini-batch gradient descent, have been developed to address computational efficiency and to improve convergence properties, especially for large datasets.