Numerical Analysis Questions Medium
The gradient descent method is a numerical optimization algorithm used to find the minimum of a function. It is commonly used in machine learning and data analysis to optimize models and find the best set of parameters.
The method starts with an initial guess for the minimum and iteratively updates the guess by taking steps proportional to the negative gradient of the function at that point. The gradient represents the direction of steepest ascent, so by moving in the opposite direction, the algorithm aims to reach the minimum.
At each iteration, the algorithm calculates the gradient of the function at the current guess and multiplies it by a learning rate, which determines the size of the step taken. The learning rate is a hyperparameter that needs to be carefully chosen, as a small value may result in slow convergence, while a large value may cause the algorithm to overshoot the minimum.
The process continues until a stopping criterion is met, such as reaching a maximum number of iterations or when the change in the function value between iterations becomes sufficiently small. The final guess obtained is considered an approximation of the minimum of the function.
The gradient descent method is an iterative process that can be computationally expensive for large datasets or complex functions. However, it is widely used due to its simplicity and effectiveness in finding local minima. Various extensions and modifications, such as stochastic gradient descent and mini-batch gradient descent, have been developed to improve its efficiency and performance in different scenarios.