Gradiëntafdaling

Gradiëntafdaling is a technique used in various fields, including mathematics, physics, and optimization algorithms, to find the minimum or maximum of a function. The term "gradient" refers to the vector of partial derivatives of a function, which points in the direction of the steepest ascent. By moving in the opposite direction of the gradient, one can descend towards a local minimum.

In mathematical terms, for a function f(x), the gradient is denoted as ∇f(x). The gradient descent algorithm iteratively updates the variable x by moving it in the direction of the negative gradient, scaled by a learning rate α. The update rule is given by:

x = x - α * ∇f(x)

The learning rate controls the size of the steps taken towards the minimum. If it is too large, the algorithm may overshoot the minimum. If it is too small, the algorithm may converge very slowly.

Gradient descent is widely used in machine learning for training models, such as linear regression and neural networks. It is also employed in physics for simulating the behavior of particles in a potential field.

One of the main advantages of gradient descent is its simplicity and efficiency. However, it can get stuck in local minima, especially in non-convex functions. To mitigate this, variations of gradient descent, such as stochastic gradient descent and mini-batch gradient descent, have been developed. These methods introduce randomness into the update process, allowing the algorithm to escape local minima and potentially find a global minimum.