Gradient descent is an optimization algorithm commonly used in machine learning to minimize a cost function. The main purpose is to find the optimal weights (or parameters) for a specific model that will produce the least amount of error in the predictions made by the model.
The basic idea of gradient descent is to iteratively adjust the weights of the model in the direction of the steepest descent (i.e., the opposite direction of the gradient) of the cost function. This process continues until the optimal set of weights are found or some other stopping criterion is met.
More formally, given a cost function J(θ), where θ represents the weights of the model, the goal is to find the value of θ that minimizes J(θ). The gradient of the cost function is the vector of partial derivatives of J(θ) with respect to each weight:
$$\nabla J(\theta) = \left( \frac{\partial J(\theta)}{\partial \theta_1}, \frac{\partial J(\theta)}{\partial \theta_2},..., \frac{\partial J(\theta)}{\partial \theta_n}\right)$$
where n is the number of weights in the model.
The gradient descent algorithm starts by initializing the weights to some random values. Then, in each iteration, it updates the weights by subtracting a small step size, known as the learning rate α, multiplied by the gradient of the cost function, as follows:
θi + 1 = θi − α∇J(θi)
where i is the current iteration number.
The learning rate determines the step size of the update and can significantly affect the convergence of the algorithm. If the learning rate is too small, the algorithm can take a long time to converge to the optimal weights. On the other hand, if it is too large, the algorithm can overshoot the minimum and may not converge at all.
Below is an example of a cost function with two parameters, and the gradient descent algorithm steps in finding the optimal weights:
In summary, gradient descent is an iterative optimization algorithm that seeks to minimize a cost function by updating the weights of a model in the direction of the steepest descent of the cost function. It is an essential technique used in many machine learning algorithms, such as linear regression, logistic regression, and artificial neural networks.