Gradient descent : an optimization algorithm

Gradient descent is an optimization algorithm used in machine learning and mathematical optimization to minimize a function iteratively. The basic idea is to adjust the parameters of a model in the opposite direction of the gradient (slope) of a cost function. This process continues until the algorithm converges to a minimum, ideally reaching the global minimum if the function is convex.

Here's a simplified explanation:

• Initialization:

• Start with initial values for the parameters of your model.

• Compute the Gradient:

• Calculate the gradient of the cost function with respect to each parameter. The gradient points in the direction of the steepest increase of the function.

• Update Parameters:

• Adjust the parameters in the opposite direction of the gradient to move towards the minimum. This adjustment is done using a learning rate, which determines the size of the steps taken.

• Repeat:

• Repeat steps 2 and 3 until the algorithm converges, reaching a minimum.

The learning rate is a crucial hyperparameter. If it's too small, the algorithm may take a long time to converge, and if it's too large, it might overshoot the minimum. Different variations of gradient descent, such as stochastic gradient descent (SGD) and mini-batch gradient descent, adapt the way updates are made to enhance efficiency.

In summary, gradient descent is a fundamental optimization technique employed in training machine learning models to find the optimal parameters that minimize a given cost or loss function.

Let's consider a simple linear regression problem where we want to find the best-fitting line to a set of data points. The goal is to minimize the mean squared error (MSE) as our cost function. The MSE for a linear regression problem is given by:

Now, the parameter update in the gradient descent algorithm is performed as follows:

The convex function property is crucial for the success of gradient descent. A function is considered convex if, when you draw a straight line between any two points on the function, the line lies entirely below the function. In the context of gradient descent, convexity ensures that there is only one global minimum, and the algorithm will converge to that minimum regardless of the initial parameters.

In the case of linear regression and MSE, the convexity property ensures that gradient descent will converge to the best-fitting parameters for the linear model.